Realizing Third Order Sliding Mode Control for a Hydraulic ...projekter.aau.dk/projekter/files/77410482/MCE_1025_2013_Msc.pdf · Realizing Third Order Sliding Mode Control for a Hydraulic

Realizing Third Order Sliding Mode Control for a Hydraulic Multibody

Servo System

1st of February 2013 - 11th of June 2013Master Thesis

Mathias Friis Junge Kasper Bitsch LundStudyboard of Energy Technology

Title: Realizing Third Order Sliding Mode Controlfor a Hydraulic Multibody Servo System

Semester: 10th semester 2013Project period: 01.02.2013 to 11.06.2013ECTS: 30Supervisors: Torben Ole Andersen & Lasse SchmidtProject group: MCE4-1025

Mathias Friis Junge

Kasper Bitsch Lund

SYNOPSIS:

The objective of this thesis is to investigate thecontrol performance of a Third Order Sliding ControlAlgorithm in comparison to an advanced industry-like linear controller for a hydralic servo application.A CASE 580 backhoe loader is utilized as testcase, where an assymtric unmatched valve-cylinderconfiguration along with coupled dynamics of themultibody backhoe configuration give rise to highlynon-linear system characteristics. To accomplisha proper designed linear reference controller, anon-linear model of the mechanics, hydralics andpower unit is put forth. The model also providebasis for simulation of controller performance. Thecontrollers are evaluated by emulating industry-like work conditions. It is found, that the linearreference controller performes slightly better in termsof tracking performance. Suggestions of methods ofhow to increase the performance of the 3SMC arestated in the end of the thesis

Copies: 5Pages, total: 171Appendices: 3Supplements: 1 CD

By signing this document, both members of the group confirms to have par-ticipated in the project work and are thereby collectively liable for the contentof the report.

iii

Preface

This thesis is composed during a four month time period in the spring of 2013, and doc-uments the research conducted in connection with the final project of the MechatronicMaster Programme at the Department of Energy, Aalborg University. The thesis dealswith performance of a non-linear sliding mode control algorithm on an electro-hydraulicactuator in comparison to industry-like linear controller performance. This is effectively re-alized by utilizing a CASE 580 backhoe loader in connection with a Hydraulic Power Unit(HPU) provided by Bosch Rexroth©. The Authors would like to thank Bosch Rexroth©

for providing the test equipment.

Much of the experimental work at the test facility, as well as key concepts of systemmodelling was carried out in close collaboration with other project groups at AalborgUniversity. The Authors would like to thank Christian Jeppesen, Claus Vad and NielsHaldrup for providing the basis of knowledge exchange during the project period.

June, 2013

Mathias F. Junge & Kasper B. Lund

v

Summary

This thesis investigates how a third order sliding algorithm featuring easy controlparameter tuning and disturbance robustness compares to an advanced industry-likelinear controller in terms of tracking performance. A CASE 580 backhoe loader is utilizedapplication for the comparison. The report is divided into three parts regarding SystemModelling, Controller Design and Controller Performance.In part 1, the system modelling is further divided into three submodels each with thepurpose of describing the dynamic behavior of the mechanics, hydraulics and power packunit. The mechanical model originates in the Euler-Langrange equations applied to themultibody backhoe. Kinematic relations in terms of the Denavit-Hartenberg conventionare used for establishing the relation between the mechanical- and hydraulic part of thebackhoe system. The hydraulic submodel contains 4 separate straight forward valve-cylinder drive models, whereas the model of the hydraulic power unit is based on asimple control of a variable displacement piston pump. The submodels are combined ina MathWorks Simulink environment to provide a basis for system simulation. Part 1 iscompleted with a linear system analysis which serves as tool when designing the linearreference controller.In part 2, linear- and the sliding controllers are designed based on three trajectories whichaims to emulate real industry-like applications. The trajectories are scaled to match thepower limitation of the system based on a QP-analysis. A linear PI-controller utilizing ahigh-pass pressure feedback filter and velocity feed-forward are designed to represent amore advanced industrial reference controller. The third order sliding control algorithm isafterwards presented. To the knowledge of the authors, the control algorithm has neverbefore been implemented on any physical system, why the chapter of the sliding controlalgorithm also contains experience of practical issues regarding discrete time systems offinite resolution.In part 3, the performance of the linear reference controller and the third order slidingcontrol algorithm are compared, along with a simple proportional controller representingthe most primitive control topology. The comparison is based on the trajectories designedin part 2, featuring large acceleration, progressive load and abrupt disturbance.It is found, that the linear controller is slightly better in terms of tracking performance.It is also shown that cumbersome calculations are needed to derive the linear controllerwhich is not the case for the sliding mode controller. The sliding mode controller proved tobe nearly as good as the advanced linear controller and superior compared to the simpleproportional controller. Due to practical limitation of the system by virtue of both discretetime and state measurement, the tracking performance of the sliding mode controller wasdecreased.

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Reader Guidance

This report is conducted in accordance with a convention of naming, typesetting andreferences which is found suitable and intuitive by the authors. For reader convenience theconvention is explained in the following along with a nomenclature list of symbols usedthroughout this report.

Naming

1. Variables are assigned in accordance with common practice within the subject of thecontext at which the variable interacts. As this thesis deals with multiple fields ofresearch, variables may be multiple defined but should be read into context, e.g. Pis a measure of potential energy in mechanics and a measure of pressure in the fieldof hydraulics.

2. The commonly used cosine- and sine function are sometimes abbreviated with c ands respectively, e.g cos(θ) is abbreviated cθ. Further, cos(θ1 + θ2) is abbreviated withcθ12.

3. The yth figure of the xth chapter are named Figure x.y. Tables and equation arenamed in a similar way.

4. The piston side of any cylinder is referred to as the A-side, where B-side denotes therod side of the cylinder.

5. The concept of simulation refers to a numerical analysis in the MathWorksSimulink® environment.

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Typesetting

1. Variables are written in italic.

2. Subscripts of a single character are written in Italic, whereas subscripts of multiplecharacters are written in Roman, e.g. xp for piston position and meq for mass equiv-alent.

3. Vector quantities are underlined italic-type, where matrices are written in italic-bolde.g. a multi state linear SISO system is written as x = Ax+ Bu

4. The x-, y-, and z-axis of coordinate systems are in Figures illustrated as red, greenand blue respectively.

5. Arguments to variables and function are often omitted if not important to thecontext, e.g. f(t) = f .

Reference

Sources will be referred to in accordance with the Harvard Method, e.g. [Spong, 2006]. Allsources are listed in the Bibliography on page 109 with appropriate source information.

x

Nomenclatureα Piston Area Ration [−]α Sliding Alogrithm Parameter [−]αi Twist of link i [rad]αs Relative Swash Plate Angle [−]ai Length of link i [m]AA Piston Area Side A [m2]AB Piston Area Side B [m2]βF Fluid Bulk Modulus [Pa]

B Damping Coeficient [N ·sm ]

CLe Leakage Coefficient [ m3

Pa·s ]CMi Center of mass for link i [m]di Offset of link i [m]

Dp Displacement Coefficient [m3

rad ]ε Air Content ratio [−]γ Sliding Alogrithm Parameter [−]Ii Moment of inertia, rotational axis i [kg ·m2]K Kinetic Energy [J ]

KB A-side Valve Coefficient [ m3

s√Pa%

]

KB B-side Valve Coefficient [ m3

s√Pa%

]

L Langrangian [J ]λ Sliding Alogrithm Parameter [−]meq Equivalent Mass [kg]Mi Mass of link i [kg]ωm Shaft Speed Pump [rad/s]ωp Pump Related Eigenfrequency [rad/s]P Potential Energy [J ]PA A-side Pressure [Pa]PB B-side Pressure [Pa]Ps Supply Pressure [Pa]

QA A-side Flow [m3

s ]

QB B-side Flow [m3

s ]

QLe Leakage Flow [m3

s ]

QP Pump Flow [m3

s ]

QT Tank Flow [m3

s ]σ Flow Gain Coefficient [−]θi Joint angle of link i [rad]τd Filter Time Constant [−]VA A-side Volume [m3]VB B-side Volume [m3]xp Cylinder Piston Position [m]xv Valve position [m]ζp Pump Related Damping Ratio [−]

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Table of contents

1. Introduction 11.1. System Configuration and Limitations . . . . . . . . . . . . . . . . . . . . . 2

I. Modelling of an Electro-Hydraulic Multi-Body Manipulator 5

2. Non-linear Model of the Backhoe 72.1. Mechanical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1. Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2. Motion Dynamics of the Backhoe . . . . . . . . . . . . . . . . . . . . 14

2.2. Effects of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3. Modelling of Hydraulic Servo Valve . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1. Fluid Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4. Simple Model of the HPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3. Parameter Estimation and Verification of Non-linear Model 293.1. Verification of Mechanical Model . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1. Bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2. Extender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.3. Dipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.4. Boom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2. Verification of the Hydraulic Model . . . . . . . . . . . . . . . . . . . . . . . 333.2.1. 4WRKE and Boom Cylinder . . . . . . . . . . . . . . . . . . . . . . 343.2.2. 4WRTE and Dipper Cylinder . . . . . . . . . . . . . . . . . . . . . . 363.2.3. 4WREE10 and Extender Cylinder . . . . . . . . . . . . . . . . . . . 373.2.4. 4WREE6 and Bucket Cylinder . . . . . . . . . . . . . . . . . . . . . 38

3.3. Verification of the HPU Model . . . . . . . . . . . . . . . . . . . . . . . . . 40

4. Linearised Model 434.1. Simplified Hydraulic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.1. Positive Spool Displacement . . . . . . . . . . . . . . . . . . . . . . . 444.1.2. Negative Spool Displacement . . . . . . . . . . . . . . . . . . . . . . 45

4.2. Linearised Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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II. Controller Design 51

5. Trajectory Planning 535.1. Trajectory Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.1. Large Acceleration of Heavy Duty . . . . . . . . . . . . . . . . . . . 545.1.2. Progressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.3. Abrupt Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2. Trajectory Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3. QP-analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1. Large Acceleration Of Heavy Duty . . . . . . . . . . . . . . . . . . . 595.3.2. Progressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.3. Abrupt Disturbance in Inertia Load . . . . . . . . . . . . . . . . . . 62

6. Linear Controller Design 636.1. Determining the Worst Case Operating Point . . . . . . . . . . . . . . . . . 636.2. Controller Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3. Discrete Implementation of the Linear Controllers . . . . . . . . . . . . . . 71

6.3.1. Pressure-feedback Controller . . . . . . . . . . . . . . . . . . . . . . 716.3.2. PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7. Sliding Mode Control Design 737.1. Proof of Convergence for Third Order Sliding Mode . . . . . . . . . . . . . 73

7.1.1. Definition and Problem Statement . . . . . . . . . . . . . . . . . . . 747.1.2. Proof of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2. Realizing Third Order Sliding Mode . . . . . . . . . . . . . . . . . . . . . . 827.3. Practical Sliding Mode Implementation . . . . . . . . . . . . . . . . . . . . 85

7.3.1. Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 857.3.2. Calculating Velocity- and Acceleration Error in Discrete Time . . . . 867.3.3. Practial Concerns and Delimitations . . . . . . . . . . . . . . . . . . 86

III. Controller Performance 91

8. Controller Performance 938.1. Controller Performance at Large Acceleration of Heavy Duty . . . . . . . . 958.2. Controller Performance at Progressive Load . . . . . . . . . . . . . . . . . . 988.3. Controller Performance at Abrupt Disturbance . . . . . . . . . . . . . . . . 1018.4. Controller Comparrison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9. Conclusion 107

A. Mechanical Properties of the 580 Backhoe Loader 111A.1. Dimension of the Backhoe Loader . . . . . . . . . . . . . . . . . . . . . . . . 112A.2. Dimension of the Hydraulic Cylinders . . . . . . . . . . . . . . . . . . . . . 113A.3. Centre of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.4. Cylinder Extension and Joint Angles . . . . . . . . . . . . . . . . . . . . . . 121A.5. Cylinder Force and Joint Torques . . . . . . . . . . . . . . . . . . . . . . . . 128A.6. Jacobians of the Centre of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 131

xiv

B. Valve Parameters 135B.1. General Valve Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.1.1. Parameters for the 4WRTE Valve . . . . . . . . . . . . . . . . . . . 136B.1.2. 4WRKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.1.3. 4WREE6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.1.4. 4WREE10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

C. Optimization and Verification of the Model Parameters 145C.1. Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

C.1.1. Strategy for the Optimization Scheme . . . . . . . . . . . . . . . . . 148C.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

C.2.1. Bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.2.2. Dipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152C.2.3. Boom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.2.4. Extender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xv

Chapter 1Introduction

Hydraulics find its way into numerous applications as the high power-to-size ratio makesit suitable for mobile- and industrial applications of heavy assignments [Merrit, 1967].Parallel to the development of hydraulic actuated machines has been the need for hydrauliccontrol. For this purpose electro-hydraulic servo drives have been the preferred choice, asthey allow for greater precision and faster operation[Rydbjerg, 2008]. However, the designof proper linear feedback control in a servo mechanism requires detailed knowledge of thesystem and cumbersome modelling effort[Rydbjerg, 2008]. Further, in classical controldesign the distinct non-linearities of hydraulic systems are approximated with linearrelations. All together, this may cause expensive- and inefficient control design. Utilizinga non-linear sliding control algorithm might reduce the extend of system modelling, andbe more robust towards non-linearities[Schmidt, 2013]. Based on these assumptions, theaim of this thesis is to investigate whether the following hypothesis holds true.

Hypothesis

- It is possible to achieve similar or better tracking performance in a hydraulic servosystem utilizing a sliding mode control algorithm compared to model-based linearcontroller design

To effectively evaluate the hypothesis, this thesis centers around a hydraulic showcaseconsisting of a CASE 580 Prestige Backhoe Loader depicted in Figure 1.1, with thehydraulics provided by Bosch Rexroth. The multibody configuration of the backhoe addsanother degree of non-linearity compared to constant inertia applications. In order todesign a linear controller as reference- and for simulation evaluation, the backhoe withappurtenant hydraulics is modeled in Part 1 of the report. This provides basis for thelinear controller design in Part 2 where the sliding algorithm is derived as well. Thecomparison of the controllers is carried out in Part 3.

1

1.1. System Configuration and Limitations

The bucket of the backhoe is attached at the end of an extension arm, where it rotatesaround a horizontal axis with the force of the bucket cylinder. This extension arm is capableof sliding along the dipper arm with the force of an extension cylinder. The dipper arm isattached at the end of the boom, where it rotates around a horizontal axis with force inputfrom the dipper cylinder. The boom is attached to the backhoe where it rotates around ahorizontal axis with force input from the boom cylinder. The whole arm configuration iscapable of swinging around a vertical axis with force input from the swing cylinder, whichis not shown in Figure1.1.

Boom

Dipper

Extender

Bucket

Swing

Boom cylinder

Dipper cylinder

Bucket cylinder

Figure 1.1.: The backhoe consist of 4 differents links - a boom, dipper, extender anda bucket. Each link is actuated by a cylinder of correpsonding name. Theextender cylinder is not depicted as it is placed inside the dipper/extenderbeam.

From Figure 1.1 it can be seen that the backhoe arm is capable of operating in a threedimensional space without the backhoe having to move. Rather than utilizing the hydraulicpump fitted to the engine of the Backhoe Loader, the cylinders of the backhoe are suppliedby a Bosch Rexroth© Hydraulic Power Unit (HPU) with appurtenant data acquisitionequipment. The HPU is placed next to the Backhoe Loader, where the hydraulic powerlines and position sensors are fed to the backhoe arm. A picture of the system setup canbe seen in Figure 1.2

2

Figure 1.2.: The backhoe arm is not powered by the hydrualic pump on the backhoeloader, but a Bosch Rexroth HPU placed next to the Backhoe Loader.

The HPU is connected to the 4 cylinders of the backhoe arm, but not the swing mechanismshown in Figure 1.1. This rules out operation in three dimension, if the Loader are toremain stagnant during operation. Hence, the operation is limited to the vertical plane ofFigure 1.2. A more precise definition of the workspace of the backhoe will be derived inthis next part, along with a model of the dynamics of the backhoe arm - also referred toas backhoe manipulator or simply backhoe. Further, the driving hydraulic system and theHPU are modelled to make control design possible.

Part I.

Modelling of an Electro-HydraulicMulti-Body Manipulator

5

Chapter 2Non-linear Model of the BackhoeContents

2.1. Mechanical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1. Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2. Motion Dynamics of the Backhoe . . . . . . . . . . . . . . . . . . 14

2.2. Effects of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3. Modelling of Hydraulic Servo Valve . . . . . . . . . . . . . . . . 22

2.3.1. Fluid Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4. Simple Model of the HPU . . . . . . . . . . . . . . . . . . . . . . 26

The non-linear model of the Backhoe is divided into three sub-models, as shown inFigure 2.1. The model of the HPU includes the transients in the supply pressure atalternating flow requirements. The hydraulic model includes the four proportional valvesutilized to control the flow to the cylinders of the backhoe. The mechanical model includesthe relation between the motion of the cylinder and the backhoe together with thedynamical response to cylinder force inputs. The states connecting the three sub-modelscan be seen in Figure 2.1

Hydraulics MechanicsPower Pack

Power Pack Valve Cylinder LoadP s QA

QBF

xP

xP

xPP A P B

Figure 2.1.: Illustration of the three submodels and the interconnecting states

7

2.1. Mechanical model

The mechanical model serves to establish first the relation between the motion of thecylinder pistons and the motion of the backhoe, and then to effectively determine theresponse of the backhoe arm to a cylinder force input. The first includes an establishmentof a systematic way to determine the position of each joint in the vertical plane of operation.This is referred to as forward kinematics of the backhoe. Due to the nature of a roboticarm, this can be coupled to the extension of all four cylinder. The second concerns derivinga dynamical model of the backhoe coupling the action forces of the cylinders to the reactionforces of the backhoe arm. Before deriving any of these, it is appropriate to express somegeneral definitions utilized throughout the rest of the report. Key points have been assignedto the backhoe arm as can be seen in Figure 2.2. The ends of each cylinder have beenassigned with point A through F, with the exception of the extender cylinder. The lengthof the fully retracted boom cylinder is denotes ABmin, and the stroke capability ABstroke.Similar nominations are made for the dipper and bucket cylinder. As no points havebeen assigned to the extender cylinder, |O2O3|min and |O2O3|stroke are utilized. Point O0

through O4 are special points of interest, and will be further introduced in section 2.1.1.The distance between these point are i.e denoted |O0O1|. The points G, H and I arereference points utilized during trigonometric relations of the bucket mechanism.

O0

O1O2 O3

O4

A

B

C

D

EF

G H

I

Figure 2.2.: To establish the actuator to joint relation, key reference points have beenassigned to the backhoe for use in latter calculations

8

The hydraulic force that each cylinder exerts on the backhoe load are general denoted FLand should be read into the context of the appearance. The mechanical dynamic resistanceforce of the backhoe is denoted Fmech and all friction related force are collected under theterm Ff . A close up of the dipper joint shown in Figure 2.3 illustrate the convention.As shown in Figure 2.3, positive revolution in each joint is defined as anti-clockwise. Allmechanical properties of the backhoe employed in the following are found in Appendix Aon page 111, along with derivations of trigonometric- and force-torque relations.

FL

Fmech Ff w2

Figure 2.3.: Illustration of the force convention applied throughout the report

2.1.1. Forward Kinematics

This section aims to describe the orientation and position of each joint in relationto the base of the backhoe. This description will be carried out through a kinematicanalysis based on the Denavit-Hartenberg-Convention, or simply DH-Convention[Spong,2006]. According to the convention it is possible to make a transformation between anytwo coordinate systems in terms of only four parameters. Generally, transformations inthree dimensions requires six parameters, three for position transformations and anotherthree for orientation transformations. In order to utilize only four parameters, the DH-Convention states two constraints for the relation between the coordinate systems. [Spong,2006]. That is

• The axis x1 is perpendicular to the axis z0• The axis x1 intersects the axis z0

Two coordinate systems that satisfy these constraints are shown in Figure 2.4. It can beseen, that the transformation can be described in terms of α, a, d and θ which is denotedthe link twist, link length, link offset and joint angle, respectively.

9

Figure 2.4.: The coordinates systems depicted satisfies the two constraints of the DH-Convention [Spong, 2006].

In order to utilize the DH-convention to describe the kinematics of the backhoe, coordi-nate systems have been assigned to the backhoe arm in accordance with the conventionconstraints. Figure 2.5 on the facing page shows the backhoe arm with the DH-coordinateframes superimposed. The base-, or the world reference frame, - is chosen to coincide withthe zero coordinate system [x0, y0, z0], where the x0-axis and y0-axis span the operationplan of the backhoe. The position of reference system [x1, y1, z1] is determined by only1 variable, in this case θ1, and is referred to as joint variable 1. For revolute joints, θwill be the joint variable where d denotes the joint variable for the prismatic joint. TheDH-parameters for the backhoe are given in Table 2.1, and the dimensions of the backhoeare listed in Appendix A on page 111.

Link(i) αi ai di θi1 0 |O0O1| 0 θ1

2 π2 0 0 θ2

3 −π2 0 d3 0

4 0 −|O3O4| 0 θ4

Table 2.1.: DH-parameters for the Case 580 Prestige according to Figure 2.5 on the facingpage.

10

x0

y0

y1

x1,x2

z2

x4

y4

x3

y3

x0

y0

θ1

x2

z2

x1y1

y3

x3

x4

y3

θ4

θ2

d3

Figure 2.5.: Sketch of the Case 580 Prestige backhoe with the DH-coordinate framessuperimposed in the position of Figure 2.2 on page 8 and in a zero-configuration where all joint variables are zero.

The neutral posture of the backhoe is obtained when all joint variables are set to zero.However, d3 cannot assume the value zero, but is set at its minimum value of 2.00 m. Thezero-posture is shown in Figure 2.5.

Describing the origin of the ith reference frame in the ith-1 coordinate system can be doneby using the homogenous transformation matrix. This matrix is in general described as,

Ti−1i =

[Ri−1i oi−1i

0 1

](2.1)

where, Ri−1i describes the orientation of the ith coordinate frame with repsect to coordinate

frame i−1, and oi−1i is a distance vector between the origins with respect to the coordinatesystem om i− 1. The DH-Convention is a meassure of two rotations and two translationin the order,

Ti−1i = Rot, zθiTrans, zdiTrans, xaiRot, xαi (2.2)

where the rotation and the translation matrix describing the above are given as,

Rot, zθi =

cθi −sθi 0 0sθi cθi 0 00 0 1 00 0 0 1

Rot, xαi =

1 0 0 00 cθi −sθi 00 sθi cθi 00 0 0 1

11

Trans, zdi =

0 0 0 00 1 0 00 0 1 di0 0 0 1

Trans, xai =

1 0 0 ai0 1 0 00 0 1 00 0 0 1

This yields a homogenous transformation matrix T i−1i in terms of DH-paramters as,

Ti−1i =

cθi −sθi · cαi sθi · sαi aicθisθi cθi · cαi −cθi · sαi ai · sθi0 sαi cαi di0 0 0 1

(2.3)

Using the matrix given in ((2.3)) and the Table 2.1 on page 10, each coordinate frameis described in relation to the previous frame. The homogenous transformation matricesT 01 , T

12 , T

23 , T

34 are listed below

T01 =

cθ1 −sθ1 0 |O0O1| · cθ1sθ1 cθ1 0 |O0O1| · sθ10 0 1 00 0 0 1

T12 =

cθ2 0 sθ2 0sθ2 0 −cθ2 00 1 0 00 0 0 1

T23 =

1 0 0 00 0 1 00 −1 0 d30 0 0 1

T34 =

cθ4 −sθ4 0 −|O3O4| · cθ4sθ4 cθ4 0 −|O3O4| · sθ40 0 1 00 0 0 1

(2.4)

When planning a trajectory it is desired to describe each coordinate frame in the basereference frame. Each coordinate frame is described in reference to the base coordinateframe as:

T01 as shown in Equation ((2.4))

T02 = T0

1T12

=

c(θ1 + θ2) 0 s(θ1 + θ2) |O0O1|cθ1s(θ1 + θ2) 0 −c(θ1 + θ1) |O0O1|sθ1

0 1 0 00 0 0 1

(2.5)

12

T03 = T0

2T23

=

c(θ1 + θ2) −s(θ1 + θ2) 0 |O0O1|cθ1 + s(θ1 + θ2)d3s(θ1 + θ2) c(θ1 + θ1) 0 |O0O1|sθ1 − c(θ1 + θ2)d3

0 0 1 00 0 0 1

(2.6)

T04 = T0

3T34

=

cθ124 −sθ1 0 cθ1|O0O1|+ sθ12d3 − cθ124|O3O4|sθ124 cθ124 0 sθ1|O0O1| − cθ12d3 − sθ124|O3O4|

0 0 1 00 0 0 1

(2.7)

As seen in Equation ( (2.4) on the facing page) position of the tool point depends onall four joint variables. Constraining the robot to work within its physical limits yields aboundary tool point workspace as shown in Figure 2.6. The workspace has been plottedby using a Monte Carlo inspired method where a large but finite amount of manipulatorconfigurations are arbitrarily selected in order to calculate the position of the tool pointin Equation (2.4) on the facing page. Only the boundary points are shown in Figure 2.6.This plot is valuable when designing a trajectory for the backhoe as it is only possible torealize trajectories within the bounds of the plot in Figure 2.6

−4 −2 0 2 4 6

−6

−4

−2

0

2

4

6Workspace for the 580 Case Prestige backhoe loader

Horizontal position of toolpoint [m]

Vertica

lpositionofto

olp

oint[m

]

Figure 2.6.: End effector workspace for the backhoe with three configuration examplessuperimposed.

13

2.1.2. Motion Dynamics of the Backhoe

In this section the dynamics of motion for the backhoe will be derived based on the Euler-Langrange equations. Similar to Newton’s second law of motion, the differential equationsof the Euler-Langrange method describes the evolution of a mechanical system in time,but is more suitable to describe the motion of the manipulator links containing both linearand angular velocities. Defining the Langrangian as,

L = K − P (2.8)

where K and P are the kinetic and potential energy of the system, respectively. Then, Ifthe kinetic and potential energy can be described by a set of generalized coordinates theEuler-Langragian equation states that

d

dt

δLδq− δLδq

= τ (2.9)

where q is a vector of generalized coordinates and τ is a vector of external force which canneither be described in terms of kinetic nor potential energy.

For the backhoe manipulator of four links it is possible to determine the kinetic energyin terms of the generalized coordinate vector, since the position of the backhoe can bedescribed by the joint variable vector q∗ = [θ1 θ2 d3 θ4]

T . As the backhoe consist ofprismatic- and angular joints, both the linear- and angular velocity of each link contributesto the kinetic energy. If s0i is a vector of 6 entries that denotes the linear and angularvelocity of the centre of mass of joint i with respect to the base reference frame, the thecorrelation between s0i and q can be stated as,

s0i =

[v0iω0i

]=

[Jvi(q)

Jωi(q)

]q = Ji(q)q where v0i =

v0ixv0iyv0iz

, ω0i =

ω0ix

ω0iy

ω0i

(2.10)

Ji(q) is a [6× 4] Jacobian matrix of the manipulator for link i for the backhoe. The Jvi(q)Jacobian relates the joint velocities to the linear velocity of the center of mass for link i.If k denotes the links from 1 through i , then the linear velocity Jacobian for link k canbe written as,

Jvi =[jvk

04−i

](2.11)

where 04−i is a [6× (4− i)] matrix with all entries equal to 0. Depending on whether jointk is revolute or prismatic, the vector components of Jvi can be calculated as

jvk

=

{zk−1 × (oi − ok−1) if joint k is revolute

zk−1 if joint k is prismatic, k = 1, ..., i (2.12)

14

where zk−1 is the unit vector of reference frame k−1. From Figure 2.2 on page 8 the linearJacobians for the center of mass of link 1 and 2 can be written as,

Jv1 =

−|CM1| · sθ1 0 0 0|CM1| · cθ1 0 0 0

0 0 0 0

(2.13)

Jv2 =

|CM2| · cθ12 − |O0O1| · sθ1 |CM2| · cθ12 0 0|CM2| · sθ12 + |O0O1| · cθ1 |CM2| · sθ12 0 0

0 0 0 0

(2.14)

where CM1 and CM2 are the center of mass of link 1 and 2, respectively as found inAppendix A on page 111. The sizes of the linear Jacobian of link 3 and force makes themunsuitable for printing here, but can be found in Appendix A on page 111

In a corresponding manner, the Jωi(q) Jacobian can be found as,

Jω =[jωk

0n−i

]where the vector components of Jω can be found as,

jωi

=

{zk−1 if joint k is revolute

0 if joint k is prismatic(2.15)

Using these Jacobians of linear and angular velocities, the kinetic energy of the backhoecan be described as

K =1

2qT

D(q)︷︸︸︷[4∑i=1

{miJvi(q)

TJvi(q) + Jωi(q)TRi(q)IiRi(q)

TJωl(q)}]

q (2.16)

=1

2qTD(q)q =

1

2

∑i,j

di,j(q)qiqj (2.17)

where mi is the mass of joint i, Ii is the inertia matrix of link i and di,j is the entries in thesymmetric and positive definite D(q) matrix. If the position of the center of mass for linki with respect to the base frame is denoted rci(q) and the gravitation vector with respectto the base frame is g, then the potential energy of the backhoe can be calculated as,

P =

4∑i=1

migT rci(q) (2.18)

Using the obtained expression for the kinetics and knowing that the potential energy ofthe backhoe is a function of q, the Langrangian of (2.8) on the preceding page can bewritten as,

15

L = K − P =1

2

∑i,j

di,j(q)qiqj − P (2.19)

If qk and qk denotes the velocity and position of joint k ∈ [1 : 4], respectively, thenequation (2.9) on page 14 can be written as

d

dt

δ

δqk

1

2

∑i,j

di,j(q)qiqj − P

− δ

δqk

1

2

∑i,j

di,j(q)qiqj − P

= τk (2.20)

d

dt

∑j

dk,j(q)qj

−1

2

∑i,j

δdi,j(q)

δqkqiqj −

δP

δqk

= τk (2.21)

∑j

dk,j(q)qj +∑j

d

dtdk,j(q)qj −

1

2

∑i,j

δdi,j(q)

δqkqiqj +

δP

δqk= τk (2.22)

Utilizing the chain rule of differentiation on the second term of the expression above leadsto

∑j

dk,j(q)qj +∑i,j

δdk,j(q)

δqiqiqj −

1

2

∑i,j

δdi,j(q)

δqkqiqj +

δP (q)

δqk= τk (2.23)

Due to symmetry of the D(q) matrix and that the summation runs over all i and j for{i, j} ∈ [1; 4], i and j can be interchanged in the second term of (2.23) to obtain

∑j

dk,j(q)qj +1

2

∑i,j

(δdk,j(q)

δqi+δdk,i(q)

δqj

)qiqj −

1

2

∑i,j

δdi,j(q)

δqkqiqj +

δP (q)

δqk= τk

∑j

dk,j(q)qj +∑i,j

1

2

(δdk,j(q)

δqi+δdk,i(q)

δqj−δdi,j(q)

δqk

)qiqj +

δP (q)

δqk= τk (2.24)

k = 1, ..., 4

Equation (2.24) describes the dynamic response of link k of the backhoe in terms of thegeneralized joint variable if given a torque input of τk at joint k. It can be seen, thatthe response is coupled to the position, velocity and acceleration of the other links of thebackhoe. The above expression can be abbreviated and put into matrix form by utilizingequation (2.17) and by defining,

C(q, q) =

c1,1 c1,2 c1,3 c1,4c2,1 c2,2 c2,3 c2,4c3,1 c3,2 c3,3 c3,4c4,1 c4,2 c4,3 c4,4

(2.25)

16

where

ck,j =∑i

1

2

(δdk,j(q)

δqi+δdk,i(q)

δqj−δdi,j(q)

δqk

)qi

If we further define

G(q) =[δP (q)

δq1

δP (q)

δq2

δP (q)

δq3

δP (q)

δq4

]T (2.26)

then Equation (2.24) on the preceding page can be written as

D(q) q + C(q, q) q + G(q) = τmech (2.27)

The D matrix relates to the joint accelerations and can be perceived as an inertia matrix.The C matrix relates to the product of joint velocities i and j. Since the diagonal ofthe C matrix describes quadratic terms of the joint velocities, that is i = j, these arereferred to as centrifugal terms, where the off-diagonal denotes velocities of i 6= j andare referred to as Coriolis terms. The G vector is the torque contribution associated withthe gravitational forces acting on the backhoe link, and τ is a vector of external inducedjoint torques. For the backhoe, the external joint torques are induced by the forces of thehydraulic actuators. The correlations between the cylinder forces and the joint torques arederived in Appendix A.5 on page 128 and it is shown that,

τ = M · F (2.28)

where M is a position dependent, diagonal torque multiplier matrix. The correlationbetween the generalized joint-variable positions q, and the cylinder positions xp, are inAppendix A.5 on page 128 determined as,

q = P (d), d =

xp1 + |AB|minxp2 + |CD|minxp3 + |O2O3|minxp3 + |EF |min

(2.29)

where P (d) is a position transformation matrix. The corresponding velocity andacceleration transformation are given as,

q = V(d)xp (2.30)

q = V(d, xp)xp + V(d)xp (2.31)

Utilizing Equation (2.29) the third term of Equation (2.27) describing the gravitationcontribution can be determined with respect to cylinder position as,

G(q) = G(P (d)) (2.32)

In a corresponding manner, Equation (2.29) and (2.30) can be utilized to make the secondterm of Equation (2.27) a function of cylinder position and velocity

17

C(q, q)q = C(V(d)xp, P (d)) V(d)xp (2.33)

Lastly, the first term of Equation (2.27) can in cylinder space be expressed by utilizingEquation (2.29), (2.30) and (2.31)

D(q) q = D(P (d)) V(d, xp)xp + D(P (d))V(d)xp (2.34)

Now, as all terms of Equation (2.27) can be expressed in cylinder space, an equivalent forceequilibrium of the equation can be found by combining Equation (2.32), (2.33), (2.34) and(2.28)

Dxp + Cxp + G = Fmech (2.35)

where

D = M−1D(P (d))V(d)

C = M−1[C(V(d)d, P (d)) V(d) + D(P (d)) V(d, xp)

]G = M−1G(P (d))

This equation describes the dynamics of the backhoe in the hydraulic actuator space,linking the linear movement of the cylinder to the force of the cylinder, rather thanangular movement to the cylinder produced joint torques. The magnitude of each term inthe torque equation of (2.35) are all position dependent. However, the D- and C matricesdepend on coupled acceleration- and velocity, respectively. This makes it difficult to makea direct comparison of the magnitude among the three terms of equation (2.35). In anattempt to evaluate the influence of the three force terms, the magnitudes are calculatedfor simultaneous sinusoidal movement of all cylinders. The magnitude of the sine wavesare set to cover the stroke capability of the corresponding cylinder. That is,

xp1 = 12 · xp1,max · sin(ω1t)


xp3 = 12 · xp3,max · sin(ω3t) (2.36)


With steady-state assumption, the maximum sum of velocities for ω1, ω2, ω3 and ω4

depend on the maximum hydraulic flow into cylinders, and the piston- and rod areas.The hydraulic flow is either limited by the pump displacement or the valve capabilities atoperating pressure. In section 5.3 on page 58 it will be shown that the flow is primarilyrestricted by the maximum pump flow of 2.42 · 10−3 [m3/s]. The magnitude of forces byterms of Equation (2.35) are in the following illustrated for maximum linear velocities of0.1ms of the boom and dipper cylinder, and a maximum linear velocities of 0.3ms for theextender- and bucket cylinder. This yields a maximum flow demand that exceeds the pump

18

capability of almost a factor 2, and is on this basis assumed representable for a motion withhigher dynamical forces. The frequency of the trajectory described by Equation (2.36) canthen be determined as,

0.1ms = xp1,max = 12 · ω1xp1,max


0.3ms = xp3,max = 12 · ω3xp3,max (2.37)


The derivatives of the trajectories describe in (2.36) on the preceding page are easilyobtained since all derivative are defined for a sine function. This allows to plot the forceterms of equation (2.35) on the facing page, as illustrated in Figure 2.7

G d ( )

C d ( )

D d ( )

MUX

xp

xp

xp

Figure 2.7.: Illustration of how the reaction forces of equation (2.35) on the facing page.

0 10 20 30 40 50 60−10

−8

−6

−4

−2

0

2

4x 10

4

Time [s]

C,D

,G [N

]

Acceleration term

Centrifugal and Coriolis term

Gravitation term

0 10 20 30 40 50 60−6

−4

−2

0

2

4

6x 10

4

Time [s]

C,D

,G [N

]

Acceleration term


Gravitation term

Figure 2.8.: Comparison of force terms of Equation (2.35) for the Boom cylinder (left)and Dipper cylinder (right) with the backhoe trajectory shown in Equation(2.36).

0 10 20 30 40 50 60−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

C,D

,G [N

]

Acceleration term


Gravitation term

0 10 20 30 40 50 60

−2000

−1000

0

1000

2000

3000

4000

Time [s]

C,D

,G [N

]

Acceleration term


Gravitation term

Figure 2.9.: Comparison of force terms of Equation (2.35) for the boom Extender (left)and Bucket cylinder (right) with the backhoe trajectory shown in Equation(2.36).

19

From Figure 2.8 and 2.9 it can be seen, that gravitation is the dominating force for all fourcylinders. This is consistent with the fact, that the gravitation force is neither velocity noracceleration dependent. With the relatively slow cylinder velocities and accelerations fromthe trajectory of Equation (2.35), the outcome of the centrifugal, Coriolis and accelerationterms will be non-dominant, which can also be seen in Figure 2.8 and 2.9. The gravitationforce acting on the bucket cylinder as a function of the position of both the boom- anddipper cylinder can be seen in Figure 2.10. Here the extender- and bucket cylinder arefixed at zero stroke.

0.050.1

0.150.2

0.250.3

0.350.4

00.1

0.20.3

0.40.5

0.6

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

x 104

xP1

[m]xP2

[m]

G1 [N

]

Figure 2.10.: Gravitation force acting on the boom cylinder as a function of xP1 andxP2.

The negative value of the force correspond with the conventions of forces described inFigure 2.3 on page 9. From the boom cylinder perspective, the backhoe arm appearsheaviest when the boom cylinder is fully retracted and the dipper cylinder extensionmakes the dipper configuration horizontal. Correspondingly, the gravitation force actingon the dipper cylinder can be plotted as a function of the position of the boom- and dippercylinder and can be seen in Figure 2.11

0.050.1

0.150.2

0.250.3

0.350.4

00.1

0.20.3

0.40.5

0.6

−4

−3

−2

−1

0

1

x 104

XP1

[m]XP2

[m]

Figure 2.11.: Gravitation force acting on the dipper cylinder as a function of xP1 andxP2.

20

2.2. Effects of Friction

Friction is often an important model property when designing a controller for a mechanicalsystem. Friction is highly non-linear and may result in steady state errors, limit cycles andpoor performance.[Olsson, 1997] It is often advantageous to reduce friction through goodhardware design or by lubricating bearings and other moveable parts. The latter has beendone for the backhoe, but despite the effort friction proved to be predominant. A classicalway of describing friction is by means of sticktion, coloumb friction, viscous damping andthe Stribeck effect. For the backhoe this is modeled as,

Ff = B · xp + sgn(xp)

(Fc + Fs · e

xpc1

)(2.38)

Ff Is the net force due to the effects of friction [N ]

B Is the viscous damping coefficient [kgs ]Fs Is the static friction force, predominant at zero velocity [N ]c1 Is the Stribeck coefficient. The Stribeck effect is predominant at small velocities [·]Fc Is the coulomb friction acting as a bias for the net friction force [kg]

The friction effect as a function of velocity is shown in figure 2.12

Ff

xp

Figure 2.12.: Friction curve described via sticktion, coloumb friction and viscousdamping

21

In simulation problems may occur when modeling friction as a sign function. Instead ahyperbolic tangent function is used to realize Equation (2.38) on the previous page

2.3. Modelling of Hydraulic Servo Valve

Hydraulic systems are usually described through two main equations, namely the orificeequation and the continuity equation. The orifice equation is shown in general form below.

Q = Cd · ω · xv ·√

2

ρ

√∆P

Collecting the terms Cd · ω√

2ρ yields a simpler equation for the flow.

Q = K · xv ·√

∆P (2.39)

In the backhoe system, the direction of flow is controlled via a directional valve. The valveconnects a pump to either of two cylinder chambers and the remaining side to tank. Ahydraulic sketch of any given valve and cylinder combination for the backhoe is shown inFigure 2.13

Figure 2.13.: Hydraulic sketch of any given valve-cylinder configuration of the backhoe

From the sketch it is seen that the A chamber is connected to the pump side for xv > 0and to tank for xv < 0. The B chamber is connected to the pump side for xv < 0 and totank for xv > 0. Utilizing this information and equation (2.39) yields the following.

22

QA = sg(xv) ·KA · xv ·√PS − PA − sg(−xv) ·KA · xv ·

√PA

QB = sg(−xv) ·KA · σ · xv ·√PB − sg(xv) ·KA · σ · xv ·

√PS − PB

where:

sg(x) =

{1 for x > 0

0 for x ≤ 0

QA Is the flow through valve port A [m3

s ]

QB Is the flow through valve port B [m3

s ]

KA Is the flow gain coefficient for orifice A [ m3

s√Pa%

]

σ Is the ratio of the flow gain coefficient for orifice A and B, KBKA

·xv Is the spool position relative to the maximum spool stroke [%]PS Is the supply pressure (relative to tank pressure) [Pa]PA Is the pressure in chamber A (relative to tank pressure) [Pa]PB Is the pressure in chamber B (relative to tank pressure) [Pa]

While the orifice equation states the flow through the valve based on the pressuredifferential across the valve, the continuity equation states the pressure gradient in thechambers based on the flow. This may be written as:

QA +QLe = xp ·AA +VA0 +Ap · xp

βPA

QB −QLe = −α · xp ·AA +VB0 − α ·AA · xp

βPB

rewriting this in terms of PA and PB

PA =β

VA0 +Ap · xp(QA − xpAA +QLe) (2.40)

PB =β

VB0 − α ·AA · xp(QB + α · xpAA −QLe) (2.41)

The parameters for the above expressions are either given through data sheet figures orcan be estimated through measurements. The bulk modulus for the system is however afunction of pressure. The following section will describe how a model for the bulk modulusβ is established.

23

2.3.1. Fluid Stiffness

The stiffness of the fluid utilized in the backhoe system is both temperature- and pressuredependent [Hansen and Andersen, 2007]. For simplicity, the fluid temperature is assumedconstant during operation. Further, the fluid density is assumed constant, and accordingto the definition of fluid compressibility,

κF =1

βF=

1

ρ· δρδP

(2.42)

this will yield a constant fluid stiffness. However, as the hydraulic fluid in the systemcontains entrapped air, the effective fluid stiffness, or bulk modulus, is also a measure ofthe amount of entrapped air in the fluid. Assuming the compressibility of the air to bemuch larger than that of fluid, the effective fluid stiffness can be described as, [Hansenand Andersen, 2007].

βF,eff =1

1βF

+ εAβA

(2.43)

where εA is the pressure dependent volumetric ratio of entrapped air in the fluid, and βAis the stiffness of the air. If the density of the fluid is assumed pressure independent, εA0is the volumetric ratio of entrapped air at atmospheric pressure and with the assumptionof constant operating temperature, εA is only a function of operating pressure and can bedetermined by [Hansen and Andersen, 2007]

εA =1

1−ε0ε0·(P0P

)−1cad + 1

(2.44)

where P0 is atmospheric pressure, and cad is an adiabatic constant for air of approximately1.4. With the assumption of temperature- and pressure independent fluid, the effective fluidstiffness is only a measure of the amount of air entrapped in the fluid. The stiffness of theair during adiabatic compression can be calculated as,

βA = cad · P (2.45)

and if the assumed constant fluid stiffness is 1 GPa, the pressure profile of the effectivebulk modulus can be seen in Figure

24

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 107

0

1

2

3

4

5

6

7

8

9

10x 10

8

Bulk

Modulus[P

a]

Pressure [Pa]

εA0

= 0.005

εA0

= 0.01

εA0

= 0.02

εA0

= 0.05

εA0

= 0.1

Figure 2.14.: Effective bulk modulus as a function of operating pressure,with βF = 1 · 109 Pa.

Even though the fluid supply to all cylinders originates from the same power pack, whichshould give the same fluid property for all cylinders, the effective bulk modulus mightvary between the cylinders due to model simplification and power hose dilation, which isnot modeled. The air content of the fluid then becomes a tuning parameter which shouldreflect the fluid property experienced in measurements rather than the actual air contentratio of the fluid. The volumetric air ratio will be tuned based on experimental data insection 3.2 on page 33

25

2.4. Simple Model of the HPU

The hydraulic pump utilized in the HPU is a axial piston variable pump powered by anAC machine. By alternating the swash plate angle at which the pistons are attached, thesupply flow, and consequently the supply pressure can be controlled. Consider a controlvolume between the hydraulic pump and the valves, as shown in Figure 2.15. The outputflow of the pump can be established by means of the pump specific displacement coefficient,the relative swash plate angle and the motor speed. Assuming a control volume of constantsize, the pressure gradient in the control volume can then be establish based on the flowdifference.

QP = DPωmαs, PS =βeVC

√(QP −QV −QL) (2.46)

VC

PS

awm

S

Q

QV

P

QL

Figure 2.15.: Schematic of the hydraulic pump utilized in the HPU [Schmidt, 2013]

The pump specific displacement coefficient and the constant motor speed is set to

DP = 0.1 [L/rev] ωm = 1450 [rpm] (2.47)

yielding a maximum pump flow of

QP,max = 2.42 · 10−3 [m3/s] for αs = 1 (2.48)

The external pressure compensation of the pump is modeled as a simple proportionalcontroller of the form [Centinkunt, 2007]

Uα = KP · (Pcmd − PS) (2.49)

The relative swash plate angle is altered by small control cylinders as seen in Figure 2.16.Rather than modeling the internal hydraulics of the pump, the dynamics of the swashplate angle is represented by a second order response of the form

αs + 2ζpωpαs + ω2pαs = ω2

pUα (2.50)

26

Figure 2.16.: The hydraulic pump utilized in the power pack allows for external pressurecontrol, as shown at port X in the diagram[A10VSO]

Determining the parameters and verifying the simple HPU model is done along with thehydraulic and the mechanical model in the following chapter.

27

Chapter 3Parameter Estimation and Verification ofNon-linear Model

Contents3.1. Verification of Mechanical Model . . . . . . . . . . . . . . . . . . 29

3.1.1. Bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2. Extender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.3. Dipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.4. Boom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2. Verification of the Hydraulic Model . . . . . . . . . . . . . . . . 333.2.1. 4WRKE and Boom Cylinder . . . . . . . . . . . . . . . . . . . . 343.2.2. 4WRTE and Dipper Cylinder . . . . . . . . . . . . . . . . . . . . 363.2.3. 4WREE10 and Extender Cylinder . . . . . . . . . . . . . . . . . 373.2.4. 4WREE6 and Bucket Cylinder . . . . . . . . . . . . . . . . . . . 38

3.3. Verification of the HPU Model . . . . . . . . . . . . . . . . . . . 40

The non-linear model described in the previous chapter is throughout this chapter verified,and the model parameters which makes the best coherence between the model andmeasured data are stated. The model is divided into a mechanical, hydraulic and HPUsubmodel as in the previous chapter.

3.1. Verification of Mechanical Model

The model of the mechanics of the backhoe is verified by applying four individual stepsto each of the hydraulic valves while measuring the pressure in the cylinder chambers.The pressure measurements for the chambers are then used as an input to the mechanicalmodel. The cylinder positions are further measured and compared with the output of themechanical model. Figure 3.1 shows the mechanical verification setup.

29

Uref

HydraulicsFL Mechanics

xp

xp

Mechanical model

xp,model

xp,model

xn

Figure 3.1.: Mechanical blockdiagram of the system considered while verifying themechanical model

Some mechanical parameters i.e. friction parameters and mass distributions are furtherobtained through an evolutionary optimization scheme. The scheme seeks to minimize thedifference between xp and xp,model seen in figure 3.1. This is done by tuning the unknownmodel parameters xn. The differential evolution optimization algorithm is a global minimaoptimization algorithm and it proved to work well for the problem in hand. Using evenmore steps than four to perform the optimization scheme might have given a more accuratemechanical model, but since every evaluation of the model took several minutes and thealgorithm needed in the range of hundreds to thousands of function calls to converge, foursteps was chosen as a compromise for accuracy versus speed. The optimization scheme isexplained in further details in Appendix C on page 145.

One dataset for each of the four links is shown below. All datasets used for the optimizationalgorithm are shown in Appendix C on page 145.

3.1.1. Bucket

One response for the bucket verification test is shown in Figure 3.2

0 0.5 1 1.5 2 2.5 3 3.50.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

EFstroke[m

]

Time [s]

MeasuredSimulated

Figure 3.2.: Simulated response (green) for the bucket link vs actual response for thebucket link (blue).

The maximum error between model and experimental data is 1cm for this dataset. Forother datasets the maximum error exceeded 5 cm. This is however considered to be

30

adequate to simulate the response of the bucket link.

3.1.2. Extender

One response for the extender verification test is shown in Figure 3.3

0 0.5 1 1.5 2 2.5 3 3.5

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

O2O

3,stroke[m

]

Time [s]

MeasuredSimulated

Figure 3.3.: Simulated response (green) for the extender link vs actual response for theextender link (blue).

The maximum error between model and experimental data is 0.5 cm for this dataset Forother datasets the maximum error exceeded 1 cm. This is considered to be adequate tosimulate the response of the extender link.

3.1.3. Dipper

One response for the dipper verification test is shown in Figure 3.4

0 0.5 1 1.5 2 2.5 3 3.5 4

0.15

0.2

0.25

0.3

0.35

CD

stro

ke[m

]

Time [s]

MeasuredSimulated

Figure 3.4.: Simulated response (green) for the dipper link vs actual response for thedipper link (blue).

31

The maximum error between model and experimental data is in the range of millimeters.For other datasets the maximum error exceeded 1 cm. This is considered to be adequateto simulate the response of the dipper later on and to design a model based controller forthe dipper cylinder.

3.1.4. Boom

One response for the boom verification test is shown in figure 3.5

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

AB

stro

ke[m

]

Time [s]

MeasuredSimulated

Figure 3.5.: Simulated response (green) for the boom link vs actual response for theboom link (blue).

The maximum error between the the cylinder stroke in this plot is approximately 2cm.For some other tests performed the maximum error exceeds 8 cm and no adequate modelfor sticktion was obtainable.

Modeling the boom based on the step responses proved difficult especially since the stepsexerted the eigenfrequency of the backhoe. This resulted in a rocking motion on the wheels.Adding some counterweight to the rear of the backhoe reduced the oscillations. Decreasingthe magnitude of the steps helped further. Using another test signal than the step wasalso tried out, but to keep consistency in the verification setup a small step is preferredover another type of input signal.

32

3.2. Verification of the Hydraulic Model

The model of the hydraulics of the backhoe are verified by four individual step test on eachvalve. The pressure in both cylinder chambers are measured and compared to simulatedpressure responses, where initial volumes, bulk moduli, leakage coefficients and flow gaincoefficients are varied to make the best correlation. To aviod the inaccuries associatedwith the mechanical model influence the hydraulic model, the measured positions of thecylinder pistons are used for verification. As both the position and velocity are neededin the verification, but only the position of the cylinder can be measured, the velocitymust be estimated based on the position measurements. As an attempt to differentiatethe position numerically to obtain the velocity, the measured position is fed through a firstorder differentiator of the form

xp =s

τd · s+ 1xp (3.1)

Ideally, the differentiator should lead with 90 degrees, so chosing the time constant τd suchthat it deviates no more than 1 degree from that at 10Hz yields a value of τd = 1.4 · 10−3.The bode plot of the first order differentiator can be seen in Figure 3.6 where it is comparedto an ideal differentiator. However, the filter effect for the first order differentiator is notsufficient to provide a smooth and continous position derivative. To acquire this, a 12thorder Butterworth filter is utilized with the Matlab command filtfilt which does notdistort the phase and yet provides an efficient filter effect, as can be seen in the filtercharacteristic in Figure 3.6 [MathWorks, 2013]. The cutoff frequency is chosen to be 10Hz, as it is later shown that the mechanical eigenfrequency of the system does not exceedthis value. The analysis of the mechanical eigenfrequency is made in Chapter 4 on page 43.

−10

0

10

20

Mag

nitu

de (

dB)

100

101

89

90

Pha

se (

deg)

Frequency (rad/s)10

010

1−25

−20

−15

−10

−5

0

5

Frequency (Hz)

Mag

nitu

de (

dB)

Figure 3.6.: Bode plot of a first order differentiator (blue) compared to an idealdiffenrentiator (green) is shown to the left. The filter effect of the first orderdifferentiator is not sufficient to provide a smooth continous velocity. AButterworth filter, with the characteristic shown to the right, is used toobtain the required filter effect to provide the piston velocities from positionmeasurements.

To verify the differentation method used to obtain the velocity, the differentiated andfiltered value of the velocity is integrated and compared to the original measured value. Theresult can be seen in Figure 3.7. The maximum deviation between the two is approximately

33

1 mm. and is evenly spread around zero, indicating that no phase shift occurs during thedifferentiation

1 1.5 2 2.5 3 3.5 4 4.5 5−0.1

0

0.1

0.2

0.3

0.4

Time [s]

Str

oke

[m

]

1 1.5 2 2.5 3 3.5 4 4.5 5−2

−1

0

1

2x 10

−3

Time [s]

Err

or

[m]

FilteredMeasured

Figure 3.7.: The filtered derivative of the position measurement is integrated andcompared to the orinal measured value.

Using the differentation method, the hydraulics model is verified based on step responsetest of each valve-cylinder configuration.

3.2.1. 4WRKE and Boom Cylinder

The 4WRKE valve is connected to the boom cylinder, and a step test in the control inputto the valve is carried out. The resulting pressure build up in the A- and B chamber areshown in Figure 3.8. Due to the limited work space of the boom, no more than a 20 %step input could be made in the control signal. During the verifcation process it was foundthat a timedelay of 35 ms to the control signal made a better match with the measureddata. The course of the delay might be put down to dynamics in pressure sensor andpressure distribution, but as the delay is fairly large compared to the common dynamicsof these, it is more likely to ascribe the delay to inaccuracies in the valve dynamics foundin the datasheet, which is described in Appendix B on page 135. It is out of scope toadress the direct course of the delay, but the delay time is utilized when comparingsimulated and measured data for the 4WRKE valve and boom cylinder configuration.Rather than assuming constant supply pressure, the measured supply pressure is utilizedin the simulation. It will later be shown, that the supply pressure alters significantly duringtransient.

34

3 4 5 6 7 8 90

2

4

6

8

10

12

14

x 106

Pre

ssure

[Pa]

Time [s]

3 4 5 6 7 8 9−40

−30

−20

−10

0

10

20

30

40

Uref[%

]

MeasuredSimulatedControl signal

3 4 5 6 7 8 90

0.5

1

1.5

2x 10

7

Pre

ssure

[Pa]

Time [s]

3 4 5 6 7 8 9−40

−20

0

20

40

Uref[%

]


Figure 3.8.: Simulated and measured response of the pressure in cylinder chamber A(upper) and chamber B (lower) for a 20% step in the command signal tothe boom valve (4WRKE).

The simulation data is based on the parameters, listed in table 3.1. The initials volumesare measures of both hydraulic hoses and initial volumes within the cylinders. The valvecoefficient gains are set as tuning parameter that allows to alter the flow coefficient foundin the valve datasheet, as seen in Appendix B on page 135. From table 3.1, the flowcoefficient for the B-side of the 4WRKE-valve is not gained for modelling purpose, hencethe valve coefficient gain assumes the value 1. An A-side flow gain of 90 % of the datasheetstated value made a better correlation between simulated and measured data. The oil aircontent and leakage coefficient are tuned to give the best match between the measuredand simulated response.

Parameter Symbol Value UnitInitial volume A-side VA0 0.98 · 10−3 m3

Initial volume B-side VA0 8.3 · 10−3 m3

Air content ratio εO 0.05 −Leakage coefficient cLE,1 3 · 10−14 m3

Pa·sA-side Valve coefficient gain KA,RKE 0.9 [-]B-side Valve coefficient gain KB,RKE 1 [-]

Table 3.1.: Table of hydraulic parameters used to obtain the simulated response ofFigure 3.8.

The simulated step response shown in Figure 3.8 shows good correlation to the measured,and on this basis the hydraulic model of the 4WRKE valve and boom cylinderconfiguration, with the values of Table 3.1, are assumed adequate for latter control design.

35

3.2.2. 4WRTE and Dipper Cylinder

The oil flow to the dipper cylinder is controlled by the 4WRTE valve. Like the verificationof the previous cylinder-valve configuration, the verification of the 4WRTE and dippercylinder model is based on a pressure response to a step input in the control signal to thevalve. The backhoe is set in a configuration where all cylinders are retracted as much aspossible. In this position the boom is lifted to a miximum angle, making the work spacefor the dipper link as large as possible. However, test showed that a step input of no morethan a 40 % could be made, if the backhoe loader were to remain stable during the steptest and not corrupting the measurements. Maintaining the same tuning paramters as forthe previous valve-cylinder verfification, the measured and simulated responses for a 40 %step test in the 4WRTE and dipper cylinder are seen in Figure 3.9. A time delay of 34 mshas been added to the control signal.

4 4.5 5 5.5 6 6.5 7 7.50

5

10x 10

6

Pre

ssure

[Pa]

Time [s]

4 4.5 5 5.5 6 6.5 7 7.5

4

6

8

10

12

14

16x 10

6

Pre

ssure

[Pa]

Time [s]

4 4.5 5 5.5 6 6.5 7 7.5−50

0

50

Uref[%

]


4 4.5 5 5.5 6 6.5 7 7.5−60

−40

−20

0

20

40

60

Uref[%

]


Figure 3.9.: Simulated and measured response of the pressure in cylinder chamber A(upper) and chamber B (lower) for a 40% step in the command signal tothe dipper valve (4WRTE).

The simulation data are based on the values of table 3.2. The initial volumes of the cylinderare based on measurements of hose lengths and cylinder data. The aircontent of the oilare allowed to deviate from that of the previous step test as for the reasons stated insection 2.3.1 on page 24, and is along with the leakage coefficient treated as a tuningparameter. It was found, that the datasheet obtained flow coefficient of the valve showedgood correlation with the measured response. The valve coefficient gains are on this basiskept at a value of 1.

36

Parameter Symbol Value UnitInitial volume A-side VA0,Dipper 1.2 · 10−3 m3

Initial volume B-side VA0,Dipper 6.5 · 10−3 m3


Pa·sA-side Valve coefficient gain KA,RKE 1 [-]B-side Valve coefficient gain KB,RKE 1 [-]


The simulation of the pressure response in the B-side of the cylinder show good correlationwith the measured. The pressure response in the A-side deviates more terms of steady statelevels, but shows similar oscillating behavior during transients and is on the basis assumedadequate for control design.

3.2.3. 4WREE10 and Extender Cylinder

The step test, at which the model for the 4WREE10 and Extender cylinder is verified,is conducted with the extender facing a horizontal action direction. In this configurationit was possible to achieve a step input of 75% control signal while still keeping in theextension working range. The measured and simulated step responses are shown in Figure3.10. A time dalay of 40ms has been added.

1 1.5 2 2.5 3 3.5 4 4.5 50

5

10x 10

6

Pre

ssure

[Pa]

Time [s]

1 1.5 2 2.5 3 3.5 4 4.5 5−100

0

100

Uref[%

]MeasuredSimulatedControl signal

1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

x 106

Pre

ssure

[Pa]

Time [s]

1 1.5 2 2.5 3 3.5 4 4.5 5−100

−50

0

50

100

Uref[%

]


Figure 3.10.: Simulated and measured response of the pressure in cylinder chamber A(upper) and chamber B (lower) for a 75% step in the command signal tothe extender valve (4WREE10).

The simulation is based on the values of table 3.3. The initial volumes are based onmeasurements, whereas the air content of the oil and the leakage coefficient are tuning

37

parameters. It was found that by increasing the flow coefficient of the A-side valve by 5 %made a better match between measured and simulated pressure response.

Parameter Symbol Value UnitInitial volume A-side VA0,Extender 1.2 · 10−3 m3

Initial volume B-side VA0,Extender 4.3 · 10−3 m3




The measured and simulated pressure responses to a 75 % step input shows similarcharacteristics, and are assumed adequate for latter control design.

3.2.4. 4WREE6 and Bucket Cylinder

The last cylinder-valve configuration to verify is the 4WREE6 and Bucket cylinderconfiguration. The step response is made where all other cylinder are fully retracted.It was possible to conduct a 50% step input in the control signal to 4WREE6 valve, whilekeeping the unloaded bucket within working range. The measured and simulated pressureresponses are shown in Figure 3.11. A time delay of 37 ms has been added.

4 4.5 5 5.5 6 6.5 7

11.5

22.5

33.5

44.5

55.5

6

x 106

Pre

ssure

[Pa]

Time [s]

4 4.5 5 5.5 6 6.5 7

2

4

6

8

10

12

14

x 106

Pre

ssure

[Pa]

Time [s]

4 4.5 5 5.5 6 6.5 7−50

−40

−30

−20

−10

0

10

20

30

40

50

Uref[%

]


4 4.5 5 5.5 6 6.5 7−60

−40

−20

0

20

40

60

Uref[%

]


Figure 3.11.: Simulated and meassured response of the pressure in cylinder chamber Bfor a 50% step in the command signal to the boom valve (4WREE6).

The flow coefficient of the A-side has been increased by 5% to make a better match betweenthe measured- and simulated response. The values utilized for simulation of a step in thecontrol signal to the 4WREE6 valve are shown in table 3.4

38

Parameter Symbol Value UnitInitial volume A-side VA0,Bucket 1.4 · 10−3 m3

Initial volume B-side VA0,Bucket 4.1 · 10−3 m3



Table 3.4.: Table of hydraulic parameters used to obtain the simulated response ofFigure 3.11

The measured and simulated pressure response show similar characteristics and thehydraulic model is assumed adequate for control design. During verification of all theabove datasets the measured supply pressure has been utilized in the simulation. In thefollowing, the simplified HPU model from section 2.4 on page 26 is verified to match thetransient of the supply pressure for all of the above datasets.

39

3.3. Verification of the HPU Model

Based on the model derived in section 2.4 on page 26, the HPU model is in the followingverified by utilizing the tuned parameters of table 3.5. The simulations are based on theverified valve models of the previous section. An illustration of the simulation is shown inFigure

U

PA

S

PBPS

QVDATA HPUQV PSValve

Figure 3.12.: The HPU model is verified using the same data as for the verification ofthe hydraulic model

Parameter Symbol Value UnitControl Volume VC 3 · 10−3 m3

Leakage Coefficient CLE 10 · 10−10 m3

Pa·sPressure regulator gain KP 1 · 10−6 Pa−1

Swash Plate Eigenfrequency ωp 1.8 Hz

Swash Plate Damping Ratio ζp 10 −

Table 3.5.: Tables of parameters utilized in the simulation of the supply pressure for stepinput to the boom, dipper, extender and bucket valve

Utilizing the datasets as for the verification of the hydraulics, the measured and simulatedsupply pressure is shown in Figure 3.13. Designing af linear reference controller calls fora linear model. The following next chapter will describe how the non-linear model of thebackhoe is transformed to a simpler and linear model needed for control purposes.

40

3 4 5 6 7 8 9

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 107

Supply

Pre

ssure

[Pa]

Time [s]

3 4 5 6 7 8 9

−50

0

50

Uref[%

]


4 4.5 5 5.5 6 6.5 7 7.5

1

2x 10

7

Supply

Pre

ssure

[Pa]

Time [s]

4 4.5 5 5.5 6 6.5 7 7.5−50

0

50

Uref[%

]


1 1.5 2 2.5 3 3.5 4 4.5 5

1

2x 10

7

Supply

Pre

ssure

[Pa]

Time [s]

1 1.5 2 2.5 3 3.5 4 4.5 5−100

0

100

Uref[%

]


4 4.5 5 5.5 6 6.5 7

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

7

Supply

Pre

ssure

[Pa]

Time [s]

4 4.5 5 5.5 6 6.5 7−100

−80

−60

−40

−20

0

20

40

60

80

100

Uref[%

]


Figure 3.13.: Supply pressure response to step test of boom, dipper, extender and bucketvalve, respectively.

41

Chapter 4Linearised Model

To design a linear controller using classical control theory, a linearized model is required.Based on the non-linear relations established in Chapter 2 on page 7, a linear model of theentire system will be derived throughout this section. The model will be divided into fourlinear systems describing the correlation between input voltage Uref and the position, xp,of each cylinder. The derivation is carried out in Actuator Space, which requires inertialmasses to be translated into linear masses, as demonstrated in equation (2.35) on page 18.Appendix A on page 111 describe the transformation in further details, along with anestablishment of the torque/force relations. A Matlab script for the numeric calculation isfound on the enclosed CD. Neglecting sticktion, coulomb friction, and the Coriolis term,the cylinder equation of motion is described as.

FL = meq(xp)xp + geq(xp) +Bxp

where:

xp,0 Is the operating point of the cylinder position [kg]FL Is the load force [N ]meq Is the mass equivalent, in the operating point xp,0 [kg]geq Is the graviational contribution in the operating point xp,0 [N ]

B Is the viscous friction coefficient [N ·sm ]

An equivalent diagram illustrating the relations of (4.1) is shown in Figure 4.1

Figure 4.1.: Principle drawing illusrating the linear equivalent of the load resistanceacting on any of the backhoe cylinders

43

The non-linear relations of the (4.1) is linearizing by means of a first order Taylor expansionwith respect to the variables xp, geq and FL to obtain,

meq∆xp + geq∆xp +B∆xp = ∆FL

(4.1)

which in the Laplace domain is equivalent to

meq∆xp · s2 + geq∆xp +B∆xp · s = ∆FL (4.2)m

∆xp =1

meq · s2 +B · s+ geq∆FL

=1

meq · s2 +B · s+ geq∆PL ·AA (4.3)

where:

PL Is the net pressure on the A side of the cylinder [kg]AA Is the Area of the A side of cylinder [N ]

For simplicity, the ∆ notation for change variable is omitted when the linear mechanicalrelation is reintroduced later in this chapter. The following sections will describe how todetermine the load force FL exerted by the cylinder on the mechanical system.

4.1. Simplified Hydraulic Model

The following derivation is based on the note by Torben Ole Andersen as seen on theenclosed cd [Andersen]

4.1.1. Positive Spool Displacement

Neglecting the compression flow in the continuity equation yields the following flowequations

QA = AAxp = KAxv√Ps − PA (4.4)

QB = −αAAxp = −σKAxv√PB (4.5)

where:

α Is the ratio between the A-side area and the B-side area, ABAA

[·]σ Is the ratio between the valve flow coefficient for the A- and B side, KB

KA[·]

Introducing the load pressure:

PL = PA − αPB ⇔ PA = PL + αPB, PB =PA − PL

α(4.6)

44

Rewriting equations (4.4) on the preceding page and (4.5) on the facing page to expressPB through PA and Ps.√

Ps − PA =σ

α

√PB ⇔ PB = (PS − PA)

(α2

σ2

)(4.7)

Expressing PA and PB as a function of the supply pressure and load pressure yields,

PA − PLα

=α2

σ2(Ps − Pa)

m

PA − PL =α3

σ2Ps −

α3

σ2PA

m

PA(1 +α3

σ2) =

α3

σ2Ps + PL

m

PA =α3Ps + σ2PLα3 + σ2

(4.8)

PB = (PS − PL − αPB)α2

σ2m

(1 +α3

σ2)PB =

α2

σ2(PS − PL)

m

PB =α2PS − α2PLα3 + σ2

(4.9)

Equation (4.8) and (4.9) restates the pressure difference across the valve to be a functionof the supply pressure and load pressure only. This enables a simplified transfer functionas may be seen later on. This is valid for xv > 0.

4.1.2. Negative Spool Displacement

A similar derivation for the negative spool displacement is carried out

QA = AA · xp = KA · xv√PA (4.10)

QB = −αAA · xp = −σKA · xv√Ps − PB (4.11)

Rewriting equations (4.10) and (4.11) to express PB through PA and Ps.√PA =

σ

α

√PB − Ps ⇔ PB = Ps +

α2

σ2PA (4.12)

(PA − PL)

α= Ps −

α2

σ2PA

m

PA − PL = αPs −α3

σ2PA

m (1 +

α3

σ2

)PA = αPs + PL

m

PA =σ2(αPs + PL)

α3 + σ2(4.13)

PB = Ps +α2

σ2(−PL − αPB)

m

(1 +α3

σ2)PB = Ps −

α2

σ2PL

m

PB =σ2Ps − α2PLα3 + σ2

(4.14)

45

4.2. Linearised Model

Rewriting the flow equations through a Taylor series expansion of first order yields thefollowing linear correlation.

∆QA = Kq,A ·∆xv +KQP,A∆PA (4.15)∆QB = Kq,B ·∆xv +KQP,B∆PB (4.16)

where:

Kq,A =

{KA

√Ps0 − PA0 xv > 0

KA

√PA0 xv < 0

KqP,A =

{ −KAxv02√Ps0−PA0

xv > 0KAxv02√PA0

xv < 0

Kq,B =

{−σKA

√PB0 xv > 0

−σKA

√Ps0 − PB0 xv < 0

KqP,B =

{−σKAxv02√PB0

xv > 0σKAxv0

2√Ps0−PB0

xv < 0

∆PA =1

1 + α3

σ2

∆PL (4.17) ∆PB =−α2

σ2

1 + α3

σ2

∆PL (4.18)

Differentiating (4.6) on page 44 and inserting (2.40) and (2.41) on page 23 leads to,

∆PL = PA − αPB =β

Va(∆QA −AA∆xp +Qle)− α ·

β

VB(∆QB + αAAxp −Qle)

=β

VA(Kq,A ·∆xv +KqP,A ·∆PA −AA∆xp + Cle(∆PB −∆PA))

− α · βVB

(Kq,B ·∆xv +KqP,B ·∆PB + αAA∆xp − Cle(∆PB −∆PA)) (4.19)

Eliminating ∆PA and ∆PB by substituting with ∆PL yields

∆PL =β

VA

(Kq,A ·∆xv +

KqP,A

1 + α3

σ2

∆PL −AA∆xp − Cle

(1 + α2

σ2

1 + α3

σ2

)∆PL

)

− α · βVB

(Kq,B ·∆xv −

KqP,Bα2

σ2

1 + α3

σ2

·∆PL + αAA∆xp + Cle

(1 + α2

σ2

1 + α3

σ2

)∆PL

)(4.20)

Substituting (4.17) and (4.18) into (4.20) while collecting the variables related to ∆xv,∆xp and ∆PL respectively yields

46

∆PL =

(β

VA·Kq,A − α ·

β

VB·Kq,B

)∆xv −

(β

VAAA +

β

VBα2AA

)∆xp

+

(βVaKqP,A

1 + α3

σ2

+

βVBKqP,B

α3

σ2

1 + α3

σ2

+ Cle

(1 + α2

σ2

1 + α3

σ2

)(− β

VA− α β

VB

))∆PL

m∆PL = KQ∆xv −KQx∆xp +KQP∆PL

where:

KQ =β

VA·Kq,A − α ·

β

VB·Kq,B (4.21)

KQx =β

VAAA +

β

VBα2AA (4.22)

KQP =

βVa

(KqP,A − Cle

(1 + α2

σ2

))+ β

VB

(KqP,B

α3

σ2 − αCle

(1 + α2

σ2

))1 + α3

σ2

(4.23)

Equivalently this can be stated in laplace.

∆PL · s = KQ∆xv +KQP∆PL −KQx∆xp · sm

∆PL =KQ∆xv −KQx∆xp · s

s−KQP(4.24)

Inserting this in (4.3) on page 44 yields

∆xp =1

meq · s2 +B · s+ geq·KQ∆xv −KQx∆xp · s

s−KQP·AA

=AAKQ∆xv −AAKQx∆xp · s

meq · s3 +B · s2 + geq · s−meqKQP · s2 −KQPB · s− geqKQP

=AAKQ∆xv

meq · s3 +B · s2 + geq · s−meqKQP · s2 −KQPB · s− glKQP +AAKQx · s

=AAKQ∆xv

meq · s3 + (B −meqKQP) · s2 + (geq −KQPB − geqKQP +AAKQx) · s− geqKQP(4.25)

It may be shown that geq does not affect the bode characteristic significantly. Figure 4.2shows a bode plot for the dipper cylinder with and without geq included. As it may beseen the two plots are virtually coinciding.

47

−110

−100

−90

−80

−70

−60

−50

Magnitude (

dB

)

101

102

−315

−270

−225

−180

−135

−90

−45

Phase (

deg)

Frequency (rad/s)

Without geq

With geq

Figure 4.2.: Bode plot for the dipper cylinder with and without the term geq

Simplifying the above system by excluding geq enables a transfer function from spoolposition xv to cylinder velocity xp · s as follows.

∆xp · s∆xv

=AAKQ

meq · s2 + (B −meqKQP) · s+ (−KQPB +AAKQx)

=K · ω2

n

s2 + 2ζωn · s+ ω2n

(4.26)

where

K =AA ·KQ

−KQP ·B +AA ·KQx(4.27)

ωn =

√−KQPB +AAKQx

meq(4.28)

ζ =B −KQP ·meq

2 ·√−KQP ·B ·meq +AA ·KQx ·meq

(4.29)

To visualize the system described in equation ((4.26)) the equation is described via a blockdiagram. This diagram is shown in figure 4.3

48

KQ Sxv 1M s+B

1s

PL AA

KQx

-

+ 1 s-KQP

Ghyd Gmech

xp

Figure 4.3.: Block diagram representation of the linearised hydraulic system.

Part II.

Controller Design

51

Chapter 5Trajectory Planning

Contents5.1. Trajectory Visualization . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.1. Large Acceleration of Heavy Duty . . . . . . . . . . . . . . . . . 545.1.2. Progressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.3. Abrupt Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2. Trajectory Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3. QP-analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1. Large Acceleration Of Heavy Duty . . . . . . . . . . . . . . . . . 595.3.2. Progressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.3. Abrupt Disturbance in Inertia Load . . . . . . . . . . . . . . . . 62

The model of the CASE 580 Prestige derived in the previous part makes it possible toemulate work condition for many different electro-hydraulic servo applications. In thefollowing, three different trajectories are put forth for the purpose of emulating industrylike work condition for a servo application. The trajectories will later provide basis forevaluating controller performance. To analyse whether the system is capable of realizing agiven trajectory in terms of required output power, a QP-analysis for the system is derivedin the following chapter. The trajectories are designed not to exceed the workspace plotseen in Figure 2.6 on page 13

5.1. Trajectory Visualization

The objective for the trajectory planning of the backhoe is to design a trajectory thatsimulates three industry-like applications. The applications in consideration have beenorally presented by PhD. fellow Lasse Schmidt [Schmidt, 2013] and is in short presentedhere.

• Large Acceleration of Heavy Duty: Isolation material moves in stacks alonga conveyor belt, when a large blade swings across to cut the material. This actionrequires fast performance to ensure an even sized division of material.

53

• Progressive Load: A scoop sweeps a conveyor belt, removing gravel and dirt. Theload becomes increasingly larger as the scoop catches more gravel and dirt along theconveyor belt.

• Abrupt Disturbance: A hydraulic actuated piston is used in a press application.The load resistance suddenly change as the press meets the item to be formed.

5.1.1. Large Acceleration of Heavy Duty

To emulate a large acceleration situation, the dipper cylinder is actuated with great speedfrom complete extension to complete retraction and vice versa. The inertia from the dipperlink and out acts as load during the operation.

5.1.2. Progressive Load

To emulate the Linear progressive load situation a heavy chain is hung from the bucketlink. As the backhoe retracts, the chain is lifted from the ground increasing the loadprogressively.

5.1.3. Abrupt Disturbance

To emulate an abrupt disturbance situation, a large pallet packed with rocks of 250kg isattached to a leash of negligible mass. As the dipper cylinder retracts and the bucket issituated approximately 2 meters above ground level, the pallet and rocks are lifted at onceinducing a sudden increase of inertia.

5.2. Trajectory Design

All three trajectories have been designed to follow the same path from an extendedposition of 0.5m to a retracted position at 0.05m. The difference lies in the velocity andacceleration profile. Figure 5.1 shows the designed trajectory with the backhoe workspaceplot superimposed.

54

−5 −3 −1 1 3 5 7 9 10−4

−2

0

2

4

6

Horizontal position of toolpoint [m]

Verticalpositionoftoolpoint[m

]

Workspace for the backhoe

Trajectory 1

Figure 5.1.: End effector workspace for the backhoe loader with three backhoe positionconfigurations along the trajectory superimposed.

During the trajectory it is needed to control the desired position, velocity and accelerationall together. On this basis, the trajectories are design from a quintic polynomial of theform[Craig, 200].

xp(t) = a0 + a1 · t+ a2 · t2 + a3 · t3 + a4 · t4 + a5 · t5 (5.1)

The trajectory is divided into 8 different stages. In the first stage, an acceleration is desiredto achieve a certain velocity within a specified time. In the following second stage, it isdesired to keep a constant cruising speed. The third stage is similar to the first, where adeceleration is desired to obtain zero velocity and constant position. The desired positionis kept constant as a pause, before 4 similar stages takes the desired position back to theorigin. Common to all stages are the need of specifying an initial position, velocity andacceleration, and a final position, velocity and acceleration. Here the quintic polynomialcomes in handy as the six parameters of the polynomial allows to construct six equation ofsix unknowns, i.e the quintic polynomial that satisfy the six constraints of the first stagecan be written as

xp(t = 0) = a0 (5.2)

xp(t = tf ) = a0 + a1 · tf + a2 · t2f + a3 · t3f + a4 · t4f + a5 · t5f (5.3)

xp(t = 0) = a1 (5.4)

xp(t = tf ) = a1 + 2a2 · tf + 3a3 · t2f + 4a4 · t3f + 5a5 · t4f (5.5)

xp(t = 0) = 2a2 (5.6)

xp(t = tf ) = 2a2 + 6a3 · tf + 12a4 · t2f + 20a5 · t3f (5.7)

The trajectory utilized for emulating the acceleration of heavy duty task is shown in Figure5.2, whereas Figure 5.3 shows the trajectory utilized for the progressive load and abrupt

55

disturbance scenario. The MatLab script utilized to plot both trajectories can be foundon the enclosed CD. The trajectory has been designed in an iterative process where atrajectory is chosen first, then plotted in a QP plot and then redesigned if needed. TheQP-plot will be described in details in the next section.

0 2 4 6 8 10 120

0.2

0.4

XP[m

]

Time [s]

0 2 4 6 8 10 12−0.2

−0.1

0

0.1

0.2

XP[m

/s]

Time [s]

0 2 4 6 8 10 12−0.4

−0.2

0

0.2

0.4

XP[m

/s2]

Time [s]

Figure 5.2.: Position, velocity and acceleration during the trajectory of large accelrationof heavy duty

56

0 5 10 15 20 250

0.2

0.4

XP[m

]

Time [s]

0 5 10 15 20 25−0.1

0

0.1

0.2

XP[m

/s]

Time [s]

0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

XP[m

/s2]

Time [s]

Figure 5.3.: Position, velocity and acceleration during the trajectory of progressive loadand abrupt disturbance

57

5.3. QP-analysis

When designing a trajectory it is important to analyse whether the system is capable ofdelivering the required output power necessary to overcome the load conditions inducedduring the trajectory. One way of doing so, is to construct a so called QP-plot forthe trajectory and compare it to the limitations of the system. However, there is nostandardized procedure of constructing a QP-plot for an asymmetric valve- and cylinderconfiguration. The following is a steady-state analysis utilizing the linearized equation ofChapter 4 on page 43, where compression flows are neglected and no cavitation situationscan occur. The output power of the cylinder, Wout - can be expressed as,

Wout = FL · xp (5.8)

With the steady state assumption, the velocity of the piston can be described in terms ofthe pump flow, which in the following is defined as,

Qp =

{QA = AA · xp for xv ≥ 0

−QB = α ·AA · xp for xv < 0(5.9)

Note that Qp is defined as negative for xv < 0. This is done to ease visualization of theQP-plot which will be evident later on. The output force can be determined in terms ofthe load pressure, that is

FL = PL ·AA (5.10)

Inserting equation (5.9) and (5.10) into equation (5.8), the output power from the cylindercan be determined in terms of pump flow and load pressure as,

Wout =

{PL ·AAQA

AA= PL ·QA for xv ≥ 0

PL ·AA −QBα·AA

= −QB ·PLα for xv < 0

(5.11)

Normalizing the load pressure with respect to the supply pressure, the maximum andminimum load pressure for xv ≥ 0 can be calculated by inserting PA = Ps and PA = 0into the expression of the load pressure in equation (4.8) on page 45, respectively.

PA(1 +α3

σ2) =

α3

σ2Ps + PL

m PLPs∈[−α3σ−2 ; 1

]for xv > 0 (5.12)

The corresponding maximum and minimum load pressure for xv < 0 can be determinedby inserting PB = 0 and PB = Ps into equation (4.14) on page 45, respectively

PB =1

1 + α3

σ2

(Ps −

α2

σ2PL

)

m PLPs∈[−α ; σ2α−2

]for xv < 0 (5.13)

58

Now, depending on the load conditions, the maximum flow capability is limited byeither the pump or the valve. The flow limitation of the pump is in section 2.4 onpage 26 determined as 2.42 · 10−3 [m3/s], and is in steady state considered load pressureindependent. However, the flow limitation of the valve relates to the load pressure andis in Figure 5.4 illustrated for xv = 100% and xv = −100% for the 4WRTE valve anddipper cylinder configuration. Limiting the load pressure not to exceed 2

3PL,max is oftengood practice as this ensures a surplus of available output power for control purposes.

−1.5 −1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8x 10−3

Qp[m

3 s]

PL

Ps[Pa]

xv = 100%xv = −100%

Figure 5.4.: QP-plot of the 4WRTE valve and dipper cylinder limitation, along withpump limitation(dashed lines). The vertical bars indicates the load pressurewhere the maximum power can be excerted by the dipper cylinder if thepump limitations are disregarded

This plot serves as guideline when designing a given trajectory. Limiting the pump flowand load pressure within the bounds of the dashed lines ensures a realizable trajectory.

5.3.1. Large Acceleration Of Heavy Duty

The trajectory designed for the large acceleration of heavy duty situation is set to followthe trajectory path shown in Figure 5.5. Applying this trajectory to the simulation modelyields the following QP-characteristic.

59

−1.5 −1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8x 10−3

Qp[m

3 s]

PL

Ps[Pa]

xv > 0xv < 0Trajectory

Figure 5.5.: QP-plot for trajectory 1

The backhoe is operating near the lower pressure limit when retracting the cylinder, andnear the maximum pump flow limit when extending the cylinder. It can be seen, thatthe trajectory surpasses the 2PL

3Pslimit, but only a flow requirements near zero. Hence,

the system is capable of realizing the required power. The trajectory may be altered torun slightly faster and still remain within limits, but to keep a surplus of available fluidpower to compensate for disturbances and model inaccuracies the designed trajectory isconsidered to be adequate.

60

5.3.2. Progressive Load

−1.5 −1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8x 10−3

Qp[m

3 s]

PL

Ps[Pa]



The backhoe is operating near the pressure limits when retracting the cylinder and nearthe maximum pump flow limit when extending the cylinder. As before, the trajectory maybe altered to run slightly faster, but the designed trajectory is considered adequate to keepa surplus of available fluid power.

61

5.3.3. Abrupt Disturbance in Inertia Load

−1.5 −1 −0.5 0 0.5 1−6

−4

−2

0

2

4

6

8x 10−3

Qp[m

3 s]

PL

Ps[Pa]



This QP-plot for this trajectory resembles the previous QP-plot and is thus considered tobe adequate. The design of the trajectories in this chapter provides basis for the controllerdesign in the following two chapters.

62

Chapter 6Linear Controller Design

Contents6.1. Determining the Worst Case Operating Point . . . . . . . . . . 636.2. Controller Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 666.3. Discrete Implementation of the Linear Controllers . . . . . . . 71

6.3.1. Pressure-feedback Controller . . . . . . . . . . . . . . . . . . . . 716.3.2. PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Generally, hydraulic plants exhibit significant non-linearities and have time-varyingparamets, which makes them difficult to control using linear controller, as these typicallyhave to be designed rather converatively in order to ensure stability. Still, linear controllersare often applied to hydraulic systems.

- [Torben O. Andersen]

The purpose of this chapter is to show what can be obtained when applying classicalcontrol theory on the backhoe. Linear controllers are designed based on a given operatingpoint. Determining this operating point is described in section 6.1. Based on the operatingpoint a controller is designed.

6.1. Determining the Worst Case Operating Point

The following section is partially based on an article written by Torben Ole Andersen,Michael R. Hansen, Henrik C. Pedersen and Finn Conrad. This note describes somegeneral considerations when choosing a suitable controller topology. [Torben O. Andersen]

In general linear controllers are designed for worst case situations , namely the situationwith the highest system gain, lowest eigenfrequency and lowest damping.

The lowest eigenfrequency for the backhoe is determined based on Equation (4.28) onpage 48 and is for clarity restated below.

63

ωn =

√−KQPB +AAKQx

meq

(6.1)

The coefficient meq is a function of the dipper piston position xp2,0, and it may be shownthat the term AAKQx is much larger that than the term −KQP · B. Hence, ωn may beapproximated as,

ωn ≈

√AAKQx

meq

(6.2)

KQx is a function of β2,0, which again is a function of the pressure in each of the chambersand the supply pressure Ps. The pressure in each of the chambers may be calculated as afunction of the load pressure PL and the supply pressure Ps as explained in section 4.1 onpage 44. Choosing the operating point for the pressure PL2,0 based on geq yields

PL2,0 =geq,0(xp2,0)

AA(6.3)

Using the load pressure to determine the pressure in each of the two chambers, β2,0 iscalculated based on the bulk modulus model presented in 2.3.1 on page 24.

Doing the above restates the eigenfrequency of the system ωn to be a function of thedipper cylinder stroke xp2,0 and the sign of xv.

ωn(xp,2,0) ≈

√AAKQx(xp2,0, sgn(xv2,0))

meq(xp2,0)(6.4)

Figure 6.1 shows ωn as a function of xp2,0. The eigenfrequency is lowest for the largestpossible dipper cylinder stroke. As the trajectory designed is bounded at xp2,0 =[0.05m, 0.5m], xp2,0 = 0.5m is chosen as the operating point for the dipper cylinder.As the eigenfrequency is generally lower for negative spool displacements (due to a lowervalue of bulk modulus), xv2,0 < 0 is considered to be the worst case situation for thebackhoe.

64

0 0.1 0.2 0.3 0.4 0.530

35

40

45

50

55

60

65

70

xp2,0

[m]

ωn [r

ad/s

]

xv<0

xv>0

Figure 6.1.: ωn as a function of the operating point for the dipper cylinder stroke xp2,0

The lowest damping coefficient may be calculated using equation (4.29). The equation isrestated below for clarity

ζ =B −KQP ·meq

2 ·√−KQP ·B ·meq +AA ·KQx ·meq

(6.5)

≈B −KQP ·meq

2 ·√AA ·KQx ·meq

(6.6)

ζ is a function of both the cylinder stroke xp2,0, β2,0 and the spool stroke xv2,0. Keepingthe obtained value for β2,0 and xp2,0 fixed, the damping ratio becomes a function of thespool position xv2,0 only. Figure 6.2 shows ζ as a function xv2,0. The damping is lowestfor the lowest value of xv2,0. A value of xv2,0 = ±1% is chosen as the operating point forxv since this yields the lowest combination of the damping and the eigenfrequency.

−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

xv2,0

[m]

ζ

Figure 6.2.: ζ as a function of the operating point for the spool displacement xv2,0

The gain K is chosen by equation (4.27) and is restated below for clarity

65

K =AA ·KQ

−KQP ·B +AA ·KQx(6.7)

≈KQ

KQx(6.8)

Locking xp2,0, xv2,0 and β2,0, K may be calculated. This leads to a worst case operatingpoint as follows.

Operating Point =

xp2,0β2,0PL2,0xv2,0

VA2,0(xv2,0)VB2,0(xv2,0)

=

0.5m7.62 · 108Pa−18 · 105Pa−1%

0.0075m3

0.0017m3

(6.9)

The above is made for a situation with no load attached to the backhoe. As two of the threetrajectories are performed with an attached load a new operating point must be calculated.Following the same procedure as above two new operating points were found, yieldingalmost no change in the values of the operating point. On this basis, above operatingpoint is used for all three trajectories. The bode plot for this system is shown in Figure6.3

101

102

−270

−225

−180

−135

−90

Ph

as

e (

de

g)

F requency (rad /s)

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

Ma

gn

itu

de

(d

B)

Figure 6.3.: Bode plot for the system at the worst case configuration

6.2. Controller Topologies

As it may be seen in Figure 6.2, the backhoe system suffers by a low damping ratio. Inorder to increase the damping ratio, a high pass pressure feedback filter is implemented.

66

The filter serves to reduce the flow during transient pressure periods and will, if designedcorrectly, increases the damping ratio of the system. An increased damping of the systemallows for higher proportional gain in a PI controller, which will be described later on.The filter has been implemented as shown in Figure 6.4.

KQ Sxv 1

s1

M s+BPL AA

KQx

-

+ xp

s-KQP

1-

t

ss+

Khp

1hp

Figure 6.4.: Block diagram with a pressure feedback filter implemented

The pressure loop is closed to include the high pass filter as follows,

Gpres =1

s−KQP·Khp · s

τhp · s+ 1=

1s−KQP

1 + 1s−KQP

· Khpτ ·s+1

(6.10)

Closing the entire system yields the following transfer function from xv to xp,

Gsys =xp(s)

xv(s)= KQ ·

Gpress ·AA · 1M ·s+B

1 +KQx ·Gpress ·AA · 1M ·s+B

(6.11)

The value for τhp is 1110ωn≈ 0.25. Khp is adjusted to obtain a sufficient phase margin and

bandwidth. A plot for τhp = 0.25 with various values for Khp is shown in Figure 6.5

67

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

20

Ma

gn

itu

de

(d

B)

10−3

10−2

10−1

100

101

102

103

−270

−225

−180

−135

−90

Ph

as

e (

de

g)

G m = 83 .2 dB (a t 50 .1 rad /s) , P m = 90 deg (a t 0 .00402 rad /s)

Frequency (rad /s)

Khp

= 5

Khp

= 15

Khp

= 25

Figure 6.5.: Bode plot for various khp gains coefficients tested for the pressure feedbackfilter

A value ofKhp = 14.75 is chosen as a good compromise between damping and phase. Whenimplementing the filter in practice the filter is added to Uref rather than the summationpoint shown in figure 6.4. The implemented filter value for Khp is calculated as,

Khp =1

KQ· 14.75 = 8.12 · 10−7 (6.12)

A PI controller is implemented to convert the system to a type 2 system hence removingsteady state errors for ramp inputs. [Charles L. Phillip, 2000]. The controller is designedto meet a design criterion of minimum 6dB gain margin and between 30◦ and 60◦ phasemargin. The PI controller CPI is stated below,

CPI =KP · s+KI

s(6.13)

Tuning the controller parameters results in a PI controller with KP = 3200 and KI =16000. The bode plot for the PI controller with the high-pass pressure filter included isshown in Figure 6.6

68

−100

−50

0

50

100

150

Ma

gn

itu

de

(d

B)

10−2

10−1

100

101

102

103

−270

−225

−180

−135

−90

Ph

as

e (

de

g)

G m = 6 .88 dB (a t 48 .4 rad /s) , P m = 55 .4 deg (a t 21 .7 rad /s)

Frequency (rad /s)

Figure 6.6.: Bode plot for the PI with high pass filter included.

A velocity feedforward compensator is added to the above system as an estimation ofthe reference voltage Uref needed to obtain the velocity specified by the given trajectory.This is done by utilizing model-based information. If this estimate is accurate, the taskof the controller is reduced to compensate for disturbances and unmodelled dynamics.Assuming the valve dynamics much faster than changes in the required velocity leadsto the simplification, Uref = xv,ref. The estimated reference voltage for a positive spooldisplacement may be calculated as,

QA = xv ·

√Ps −

α3Ps + σ2PLα3 + σ2

= xpAA , for xv > 0

m

xp ·AA√

Ps −α3Ps + σ2PLα3 + σ2

= xp ·KFFP = xv

m

KFFP =AA√


(6.14)

69

for a negative spool displacement

QA = xv ·

√Ps −

σ2(αPs + PL)

α3 + σ2= xpAA , for xv < 0

m

xp ·AA√

Ps −σ2(αPs + PL)

α3 + σ2

= xp ·KFFP = xv

m

KFFN =AA√


(6.15)

The above feedforward controller relies on information regarding the load pressure, thesupply pressure and the sign of the spool position. This information may be obtaineddirectly through measurements, which are all readily available. Using the load pressuremeasurements to calculate the feedforward will however yield a large reduction of thealready low damping ratio. In practice a slightly better result was obtained whencalculating the feedforward term for the operating point PL0. Using the same operatingpoint as found in the previous section yields the following feedforward terms.

KFFN = 323 (6.16)KFFP = 248 (6.17)

The controller structure for the linear reference controller is shown in Figure 6.7. The linearreference controller will be referenced as the VFF-PI-HP controller (Velocity feedforward- PI - High pass) from now on.

KQ Sxp,ref 1

s1

M s+BPL AA

KQx

-

+ xp

s-KQP

1

t

ss+

Khp

1hp

S PI

xp,ref

S

KFFP{KFFN

, x >0v

, x <0v

+ -+

Figure 6.7.: Block diagram for the linear reference controller

A simple P controller has further been implemented based on the bode plot shown inFIgure 6.3 on page 66. A KP = 600 was chosen to alter the gain margin of the system toapproximately 10dB.

70

6.3. Discrete Implementation of the Linear Controllers

To implement the designed linear controllers in the PLC a discrete time equivalent foreach controller type is designed. This is done by first transforming the controller from theLaplace-domain to the Z-domain and lastly by rewriting the equations in the z-domain todifference equations. The discrete time equivalent pressure-feedback controller and the PIcontroller are shown below.

6.3.1. Pressure-feedback Controller

The pressure feedback controller is stated in the Laplace-domain as,

C1(s) =Uref(s)

PL(s)=

Khp · sKQ(τhp · s+ 1)

Transforming this to the Z-domain yields,

C1(z) =Uref(z)

PL(z)=

Khp · 2T ·1−z−1

1+z−1

KQ(τhp · 2T ·1−z−1

1+z−1 + 1)(6.18)

=Khp · (1− z−1)

KQ(τhp · (1− z−1) + T2 · (1 + z−1))

m

Uref(z)·KQ(τhp · (1− z−1) +T

2(1 + z−1)) = PL(z) ·Khp · (1− z−1)

m

Uref(z)·KQ · (τhp +T

2)− Uref(z) · z−1 ·KQ · (τhp · −

T

2) = PL(z) ·Khp · (1− z−1)

This may be written in terms of difference equations as,

Uref(n) =Khp · (PL(n)− PL(n− 1)) + U(n− 1) ·KQ · (τhp − T

2 )

KQ · (τhp + T2 )

(6.19)

6.3.2. PI Controller

Th PI-controller is stated in the Laplace-domain as,

C2(s) =Uref(s)

e(s)=

Kp(s)s+Ki(s)

s

Transforming this to the Z-domain leads to,

C2(z) =Uref(z)

e(z)= Kp +Ki ·

T

2· 1 + z−1

1− z−1m

Uref(z)− Uref(z) · z−1 = e(z) ·Kp − e(z) ·Kp · z−1 +Ki ·T

2· e(z) +Ki ·

T

2· z−1 · e(z)

(6.20)

71

This may be written in terms of difference equations as,

Uref(n) = U(n− 1) +Kp · (e(n)− e(n− 1))+T

2·Ki · (e(n) + e(n− 1)) (6.21)

The linear controller designed throughout this chapter is now on a form applicable to thePLC system connected to the backhoe setup. The VFF-PI-HP controller design is believedto represent a higher industry controller standard and will be used as reference controllerto evaluated the performance of the sliding control algorithm derived in the followingchapter.

72

Chapter 7Sliding Mode Control Design

Contents7.1. Proof of Convergence for Third Order Sliding Mode . . . . . . 73

7.1.1. Definition and Problem Statement . . . . . . . . . . . . . . . . . 747.1.2. Proof of Convergence . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2. Realizing Third Order Sliding Mode . . . . . . . . . . . . . . . . 827.3. Practical Sliding Mode Implementation . . . . . . . . . . . . . . 85

7.3.1. Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . 857.3.2. Calculating Velocity- and Acceleration Error in Discrete Time . . 867.3.3. Practial Concerns and Delimitations . . . . . . . . . . . . . . . . 86

The concept of control is in principle the idea of manipulating the differential equationsgoverning a system of interest to acquire a certain behavior, i.e forcing a system stateto a acquire a desired value. In linear control design this is achieved by analysing alinear approximation of the differential relations, as presented in Chapter 4 on page 43.However, as the backhoe system along with other hydraulic systems show highly non-linear characteristics, as shown in Part 1, implementing a controller that assures stabilitythroughout the entire workspace often leads to inefficient controller performance. Anothercontrol approach is to utilize a sliding mode control scheme, where the control action caninclude the non-linearities of the governing differential equations. In the following, a thirdorder sliding control algorithm is proposed for comparison with the linear control trackingperformance [Schmidt and Andersen, 2013]

7.1. Proof of Convergence for Third Order Sliding Mode

In this section the convergence of the third order sliding control algorithm (3SMC) ultilizedfor control design is derived. The proof is founded on the research article Realizing Thirdorder Sliding Mode in dynamical Systems Using Strictly Switching Control by [Schmidtand Andersen, 2013], and should as such be ascribed to the authors of the research paper.

73

7.1.1. Definition and Problem Statement

Adopting the definition of arbitrary sliding mode order presented in the article Slidingorder and Sliding accuracy in Sliding Mode Control by [Levant, 1993] and interpreted by[Schmidt and Andersen, 2013], third order sliding mode is defined as,

Definition Consider a smooth dynamics system x = γ(t, x) with smooth outputfunction σ = σ(t, x), closed by some discontinuous control. Provided that σ, σ, σare continuous function of the closed system state variables, and the third ordersliding point set σ = σ = σ = 0 is non-empty and consists locally of Filippov sensetrajectories, then the motion on σ = σ = σ = 0 is called a third order sliding mode.

[Schmidt and Andersen, 2013]

To investigate the conditions for third order sliding mode to occur, a control system ofthird order stated in (7.1) is taken into consideration.

x(3)p = a(t, x) + b(t, x)u, x, u ∈ R (7.1)

where a(t, x) and b(t, x) are smooth function, and σ(t, x) is the output function governingthe control objective of obtaining σ(t, x) = 0 utilizing discontinuous feedback. The outputfunction is given as

σ(3) = c(t, x) + g(t, x)u, σ = σ(t, x) (7.2)

With a(t, x) and b(t, x) being smooth functions, then so are c(t, x) and g(t, x) which furtherare assumed bounded by,

C ≥ |c(t, x)| Km ≤ g(t, x) ≤ KM (7.3)

Defining the three dimensional state space of the output function as Σ = {(σ, σ, σ) ∈ R},the 8 subspaces of Σ are defined by

Σ+++ |σ>0,σ>0,σ>0 Σ−++ |σ<0,σ>0,σ>0

Σ++− |σ>0,σ>0,σ<0 Σ−+− |σ<0,σ>0,σ<0

Σ+−+ |σ>0,σ<0,σ>0 Σ−−+ |σ<0,σ<0,σ>0 (7.4)Σ+−− |σ>0,σ<0,σ<0 Σ−−− |σ<0,σ<0,σ<0

With these initial system considerations it is now left to prove the convergence of a controllaw candidate in finite time. Based on the structure of the twisting algorithm, a controllaw candidate is proposed as

u = −k1sgn(σ)− k2sgn(σ)− k3sgn(σ), k{1,2,3} ∈ R+ (7.5)

74

where

sgn(σ{1,2,3}) =

1 for σ{1,2,3} > 0

−1 for σ{1,2,3} < 0

Based on a geometrically approach it will in the following be proved that closing thesystem of (7.1) on the facing page by the control law in (7.5) will converge to a third ordersliding mode if the parameters k1, k2 and k3 are properly chosen.

7.1.2. Proof of Convergence

To simplify matters and reduce the complexity of the following analysis, the control lawof (7.5) is rewritten in terms of the coefficient λ = k1

k2,α = k2 and γ = k3k2 as,

u = −α{λ sgn(σ) + sgn(σ) + γ sgn(σ)} (7.6)

Utilizing the control law of (7.6) to close the system of (7.2) satisfying (7.3) yields thedifferential inclusion

σ(3) ∈ [−C,C]− α[Km,KM ]{λ sgn(σ) + sgn(σ) + γ sgn(σ} (7.7)

and provided that

α >C

Km

1

inf{|λ sgn(σ) + sgn(σ) + γ sgn(σ)|}(7.8)

the sign of σ(3) may be uniquely determined by the control law u. This leads to the mainTheorem to be proven.

Theorem 7.1.1

Let the system (7.2) be closed by the control (7.6). Then (7.9) provides for convergenceto a third order sliding mode σ = σ = σ = 0 in finite time.

α >C

Km(1 + λ− γ), ∧ 0 ≤ λ ≤ γ ≤ 1 + λ, ∧ 1 < λ+ γ (7.9)


If (7.8) is satisfied, then according to [Schmidt and Andersen, 2013] the system may beconsidered invariant with respect to variations of system parameters, and by normalizingwith respect to α with the simplified system σ(3) = u, the inclusion of ((7.7)) may bereduced to,

σ(3) ∈ −λ sgn(σ)− sgn(σ)− γ sgn(σ) (7.10)

75

Recalling the subspaces of the state vector σ = [σ, σ, σ] defined in (7.4), the inclusionof (7.10) states that σ(3) is constant and uniquely determined by σ in each of the eightsubspaces.

To avoid one state to dominate the sign of σ(3), the following restrictions of λ and γ areapplied.

λ < 1 + γ, 1 < λ+ γ, γ < λ+ 1 (7.11)

From [Schmidt and Andersen, 2013] and by inspection of (7.10), σ is diagonally equal inthe subspaces of Σ, i.e. σ ∈ Σ+−+ = −σ ∈ Σ−+−. This allows to only consider half of thestate space σ(σ+, σ, σ), and applying the restrictions of (7.11), σ(3) can be defined as,

σ(3) = −λ− 1− γ, for σ ∈ Σ+++

σ(3) = −λ− 1 + γ, for σ ∈ Σ++−

σ(3) = −λ+ 1− γ, for σ ∈ Σ+−+ (7.12)

σ(3) = −λ+ 1 + γ, for σ ∈ Σ+−−

Based on the expression of (7.10) and satisfying the restrictions of (7.11), it is possibleto construct the phase diagram illustrating the possible direction of σ in each of the 8subspaces, as illustrated in Figure 7.1

S

S

S

S

S

S

S

S

+_+

+_ _

_ __

_+

_+_+

+++

+_ _

+_+

s

s

Figure 7.1.: Possible direction of σ in the eight subspaces. The σ-axis is normal to thepaper

From (7.12) it is found that σ(3) ∈ Σ++− is strictly negative, which causes a transitionof σ ∈ Σ+++ into σ ∈ Σ++−. With the diagonal equivalent property of the σ subspaces,this is correspondent to σ ∈ Σ−−− → σ ∈ Σ−−+. Further inspection of (7.12) showsσ ∈ Σ++− → σ ∈ Σ+−− (with the diagonally equivalent σ ∈ Σ−−+ → σ ∈ Σ−++).However, with the strictly positive definition of σ(3) ∈ Σ+−− the direction of σ is no

76

longer unique, but can assume the transitions σ ∈ Σ+−− → σ ∈ Σ−−− ∨ σ ∈ Σ+−+

(σ ∈ Σ−++ → σ ∈ Σ+++ ∨ σ ∈ Σ−+−). The strictly negative σ(3) ∈ Σ+−+ again yieldsmultiple transition options found as σ ∈ Σ+−+ → σ ∈ Σ+++ ∨ σ ∈ Σ+−− ∨ σ ∈ Σ−−+

(σ ∈ Σ−+− → σ ∈ Σ−−− ∨ σ ∈ Σ−++ ∨ σ ∈ Σ++−)

It is now left to show, that the state trajectories of system (7.10) when satisfying (7.11)is unique and encircles the origin, with the origin being a stable focus of the state space,and converges to the origin within finite time. This task is divided into three separateTheorems.

Theorem 7.1.2

The flow map of σ is unique and continuously encircles the origin if the state spaceafter some initial stage, provided that 0 < λ < γ < 1 + λ and 1 < λ+ γ are satisfied.


With the diagonally property of the state vector in mind the flow map of σ is uniquelydefined in Σ+++ and Σ++− (hence, also in Σ−−− and Σ−−+ ), but is left to be evaluatedfor Σ+−− and Σ+−+, (Σ−++ and Σ−+−). If σ sets off in Σ++− and enters Σ+−−, thenwhether σ exits via σ = 0 or σ = 0 ( σ ∈ Σ+−− → σ ∈ Σ−−− ∨ σ ∈ Σ+−+) depends onthe initial conditions of |σ| and |σ|, and the value of λ and γ when σ ∈ Σ++− ∩ Σ+−−.Based on the definition by [Fillipov, 1988], a function x = f(x, t) that is discontinuous ona smooth surface, which divides the x-space into two region, here denoted G− and G+, canbe placed by the inclusion x ∈ F (x, t), with F (x, t) being bounded and convex. Further,the limit points for a continuity point contained in either G− or G+ satisfy [Schmidt andAndersen, 2013]

limx∗∈G−,x∗→x

f−(x, t) ∈ f(x, t) limx∗∈G+,x∗→x

f+(x, t) ∈ f(x, t) (7.13)

where x∗ denotes the continuity point. Applying the definition of (7.13), σ enters thesubspace Σ+−− with least possible magnitude of σ and σ if the initial condition isσ0 = [σ+ σ0 σ−] ∈ Σ++−, σ > 0. Consider the time instant σ = 0, with the correspondingstate σ1 = σ|σ=0 and σ1 = σ|σ=0, and let T1 denote the timespan from the initial set offat t = 0 to the time instant t1 = t|σ=0. Recalling ((7.12)), the state vector at t1 may befound as

77

σ1 = −∫∫

T1

(λ+ 1− γ)dt = −12(λ+ 1− γ) · t12 + σ0 = 0

m

t1 =

√2σ0

λ+ 1− γ(7.14)

σ1 = (λ+ 1− γ)

√−2σ1

λ+ 1 + γ= −

√2(λ+ 1− γ) · σ0 (7.15)

σ1 = −1

6(λ+ 1− γ)

(2σ0

λ+ 1− γ

)32 + σ0

(2σ0

λ+ 1− γ

)12 =

2

3

√2

λ+ 1− γ· σ03/2

(7.16)

To analyze whether σ exits via σ = 0 or σ = 0, let T2 define the relative time span fromt1 to t2, where t2 is the critical condition σ|t=t2 ∈ Σ+−− ∩ Σ+−+ ∩ Σ−−− or equivalentlyt2 = t||σ|,|σ|=0 where σ2 = σ||σ|,|σ|=0. By integration and recalling (7.12)

σ2 =

∫T2

(−λ+ 1 + γ)dt = (−λ+ 1 + γ)t2 − σ1 = 0 (7.17)

m

t2 =

√2(λ+ 1− γ)

−λ+ 1 + γ(7.18)

σ2 =

∫∫∫T2

(−λ+ 1 + γ)dt =

16(−λ+ 1 + γ)t2

3 − 12

√2(λ+ 1− γ)σ0 · t22 + 2

3

√2

(λ+ 1− γ)σ3/20 = 0 (7.19)

It can be shown, that inserting (7.18) into (7.19) and reducing leads to the criteria [Schmidtand Andersen, 2013]

λ = γ (7.20)

Hence, the critical condition of σ|t=t2 ∈ Σ+−−∩Σ+−+∩Σ−−− occurs if the initial conditionsσ0 = [σ+ σ0 σ−] is satisfied and λ = γ. The effect of choosing λ > γ and λ < γ is illustratedin Figure 7.2 on the next page.

78

0 0.5 1.0 1.5

0

Time [s]

σ,σ

σforλ < γσforλ < γσforλ > γσforλ > γ

Figure 7.2.: Illustration of how the relation between λ and σ influence the states σ andσ

Let λ < γ and recall from (7.12) that the sign of σ(3) switches as Σ+−− → Σ+−+ then σis instantaneously forced back toward Σ+−− creating a sliding mode along σ = 0 underthe assumption of infinite switching frequency. On this basis σ < 0 is a constant whichthen satisfy σσ < 0 driving σ → 0 by virtue of the sliding condition of [Slotine and Li,1991]. At the intersection σ−, the state vector σ can enter either Σ−−+ or Σ−−−, howeveras |σ|σ− = 0 then instantaneously σ ∈ Σ−−− → Σ−−+, yielding a unique path of σ giventhe initial conditions σ0 = [σ+ σ0 σ−] ∈ Σ++−, |σ| > 0 and λ < γ. The flow map of σ isfor these conditions illustrated in Figure 7.3.

S

S

S

S

S

S

+_

+

+_ _

_+

_+

_+

+_ _

+_

+

s

s

Slid

ing

Mod

e

Sliding M

ode

Figure 7.3.: Direction of σ with initial conditions σ0 = [σ+ σ0 σ−] ∈ Σ++−, |σ| > 0 andλ < γ

Consider the possible direction of σ depicted in Figure 7.1 on page 76 and let the initialconditions be σ ∈ Σ∗ where Σ∗ is any subspace where the directions of σ is not unique.Applying the property of diagonally identity, Σ∗ ∈ Σ+−−∨Σ+−−, and it is found that this

79

constitutes 6 different flow chains which all end in a subspace of the flow map in Figure7.3, with unique directions of σ. That is,

σ ∈ Σ+−− → Σ−−−

σ ∈ Σ+−− → Σ+−+

σ ∈ Σ+−+ → Σ+−− → Σ+−+

σ ∈ Σ+−+ → Σ−−+ → Σ−++ → Σ−+− (7.21)σ ∈ Σ+−+ → Σ+−− → Σ−−−

σ ∈ Σ+−+ → Σ−−+ → Σ−++ → Σ+++

No matter the initial condition, the direction af the state vector σ converges to the flowmap of Figure 7.3 after some initial stage, where it encircles the origin if λ < γ is satisfied.This concludes Theorem 7.1.2 and leads to the task of proving the origin being a stablefocus.

Theorem 7.1.3

The origin of the state space is a stable focus, provided that λ < γ, λ < 1+γ, γ < λ+1and 1 < λ+ γ are satisfied. [Schmidt and Andersen, 2013]

Recall the time instant t2 defined in Theorem 7.1.2 and stated in (7.18). Then σ2 may becalculated as

σ2 =

∫∫T2

(−λ+ 1 + γ)dt

= 12(−λ+ 1 + γ)t2

2 + σ1t2

= 12(−λ+ 1 + γ)

(√2(λ+ 1− γ)

−λ+ 1 + γ

)−√

2(1 + λ− γ)σ0

(√2(λ+ 1− γ)

−λ+ 1 + γ

)

= −(λ+ 1− γ−λ+ 1 + γ

)σ0 (7.22)

Provided that λ < γ it can be seen from (7.22) that

|σ2| < |σ0| (7.23)

which indicates that σ converges towards 0. For the time exceeding the instant t2 it wasshown in Theorem (7.1.2) that σ slides along σ = 0 driving σ → σ− at which the statevector assumes the values σ = [σ− σ0 σ+] before entering Σ−−+. Referring to the diagonallyproperty of the state space, this correspond to the initial critical condition of (7.15) forσ ∈ Σ++−. On this basis, σ ∈ Σ−−+ → Σ−++. If T3 is the time span from time instant t2to t3 = t|σ=0, and noting from (7.12) that σ(3) ∈ Σ++− = −σ(3) ∈ Σ−++, then σ3 and σ3may be found analogous to that σ1 and σ1

σ3 =√

2(λ+ 1− γ) · σ2 (7.24)

σ1 =2

3

√2

λ+ 1− γ· σ23/2 (7.25)

80

For the state vector to converge to the origin, it is required that |σ3| < |σ1| and |σ3| < |σ1|.Comparing (7.24) and (7.25) to (7.15) and (7.16)

√2(λ+ 1− γ) · σ2 <

√2(λ+ 1− γ) · σ0 (7.26)

2

3

√2

λ+ 1− γ· σ23/2 <

2

3

√2

λ+ 1− γ· σ03/2 (7.27)

yields σ2 < σ0, which is satisfied by (7.23). This concludes the proof of convergence towardthe origin, which only leaves the task of showing convergence in finte time remaining.

Theorem 7.1.4

The states σ,σ and σ converge to the origin of the state space in finite time, providedthat λ < γ, λ < 1 + γ, γ < λ+ 1 and 1 < λ+ γ are satisfied.


The time span T1 from initial set off to time was in (7.14) found as

T1 = t1 − 0 =

√2σ0

λ+ 1− γ<∞ (7.28)

Due to similarity properties of Σ++− and Σ−−+, the time span T3 may be calculatedanalogous as,

T3 = t3 − t2 =

√2σ2

λ+ 1− γ<∞ (7.29)

Since both T1 and T3 has finite magnitude, the states will converge to zero in finitetime. Since σ2 < σ0 holds true then T3 < T1 also is true, indicating that convergencespeed increases as |σ| → 0. Based on the above convergence analysis, a simplified controlalgorithm of the form in (7.30) is proposed for application to the backhoe system,

u = −α{λsgn(σ) + sgn(σ) + sgn(σ)}, 0 < λ < 1, α >C

Kmλ(7.30)

which satisfies Theorem D1, where γ = 1. The characteristic phase trajectories of thecontrol law can be seen in Figure 7.4.

81

s

s s

s

Figure 7.4.: Phase portrait of the characteristic trajectory of the control algorithm

7.2. Realizing Third Order Sliding Mode

To effectively apply the control law of equation (7.30) to the valve-cylinder configurationof the dipper, it is left to show that the model of Chapter 2 on page 7 can be put on theform of equation (7.1) on page 74. From section 4.1 on page 44 the hydraulic pressure inthe cylinder chambers can be expressed in terms of the load pressure, and is for readerconvenience repeated here

PA =

α3Ps + σ2PLα3 + σ2

for xv > 0

σ2(αPs + PL)

α3 + σ2for xv < 0

(7.31)

PB =

α2PS − α2PLα3 + σ2

for xv > 0

σ2Ps − α2PLα3 + σ2

for xv < 0

(7.32)

It should be noticed that equation (7.31) and (7.32) are only valid when fluid compressionand leakage flow are neglected. In such a system, the flow into each chamber can be writtenas

QA = −α−1QB =

σKAxv ·

√Ps−PLσ2+α3 for xv > 0

σKAxv

√αPs+PLσ2+α3 for xv < 0

(7.33)

82

Utilizing the definition of the load pressure as stated in equation (4.6) on page 44, theload pressure derivation may be found as

PL = PA − αPB (7.34)

With the assumption of negligible leakage- and compression flow, the pressure change inthe A- and B- chamber may be found as,

PA =β

VA(QA −AAxp), PB =

β

VB(QB +ABxp) (7.35)

Inserting equation (7.35) in (7.34), and introducing a chamber volume ratio as ν = VBVA

leads to

PL =β

VA(QA −AAxp)− α

β

VB(QB +ABxp)

=β

VA

[(QA −AAxp)−

α

ν(−QAα+AAαxp)

]=

β

VA

ν + α2

ν[QA −AAxp]

= Λ[QA −AAxp], Λ =β

VA

ν + α2

ν(7.36)

The equations of the above is a generalized representation of each valve and cylinderconfiguration. In the trajectory designed in Chapter 5 on page 53 only actuation ofthe dipper cylinder is utilized, while the 3 others are assumed in a fixed position. Inthe following, the system states and parameters relates to the dipper valve-cylinderconfiguration, i.e. xp = xp2. Rather than integrating the derivative of the load pressure toobtain the output force of the cylinder, the mechanical- and frictional load is differentiatedto obtain,

FL = PLAA = Fmech + Ff (7.37)

where

Fmech = meqxp + C(2,2)xp + G (7.38)

Ff = Bxp + tanh

(xpc2

){Fc + exp

(−|xp|c1

)Fs

}(7.39)

Disregarding the contribution of the C-term in Fmech, that is assuming no centrifugal forcepresent, and collecting terms with respect to system states, equation (7.37) can be writtenas,

PLAA = meqx(3)p + (meq +B + Fad)xp + ˙Gxp (7.40)

with Fad being the partial derivative of the stiction and coulomb friction with respect toxp. Combining equation (7.36) and (7.40) yields,

83

Λ[QA −AAxp]AA = meqx(3)p + (meq +B + Fad)xp + G(xp)xp

m xp =ΛQAAA

(meq +B + Fad)−meqx

(3)p +

{G(xp)− ΛA2

A

}xp

(meq +B + Fad)(7.41)

Disregarding the spool dynamics of the system by, and stating the model in SISO formwith piston acceleration as output to the valve input signal, a reduced order model can beobtained as

xp = F +Guv (7.42)

where

G =ΛAAσKA

(meq +B + Fad)√σ2 + α3

√Ps − PL for xv > 0√αPs − PL for xv < 0

(7.43)

F =−meqx

(3)p − G(xp)xp + ΛA2

Axp

(meq +B + Fad)(7.44)

It should be noticed, that G is a reduced order system parameter, whereas G is thegravitational force exerted on the dipper cylinder. In order to achieve system formdescribed in equation (7.1), the equation of (7.42) is differentiated to obtain

x(3)p = F − Guv (7.45)

By choosing the control signal as u = uv, then the reduced order system of (7.45) isindeed of the form of (7.1) on page 74. The procedure of expressing the system dynamicsin terms of the derivative of the control signal is utilized, since the control law of (7.30)is a discontinuous switching control by virtue of the sign functions. Hence, letting thederivative of the control signal, rather than the actual control signal, be discontinuousreduces the problem of chattering in the control signal, due to the integral filter effect. Ife(i) denotes the error x(i)p − x(i)d , the output function may the be expressed as

e(3) = x(3)p − x(3)d = F − x(3)d − Gu (7.46)

If F , x(3)d and G are smooth and bounded function, then the control law

u = −α(λsgn(e) + sgn(e) + sgn(e)) (7.47)

will converge to zero in finite time if,

α >C

Kmλ, |F |+ |x(3)d | ≤ C ∧ Km ≤ G (7.48)

The desired piston jerk is bounded and can be found by taking the derivative of the desiredacceleration, yielding a maximum value of 0.9481m

s3. However, even though F and G is

84

modelled by continuous functions, making both F and G bounded, both functions relateto the friction within the system, as can be seen in (7.39) and (7.40). Here the stictionphenomenon along with the coulomb friction makes the friction force on the very vergeof continuity or even discontinuous, depending on the modelling approach. On this basis,near zero velocity it may require an infinite value of α to guarantee convergence, makingit desirable to choose the largest value of α possible to minimize the region of this effect.Under the assumption of infinite switching frequency, α may be chosen arbitrarily large,but in systems of finite switching frequency (as in any physical system), α has a upperpractical bound, as will be describe in the following section, along with other practicalissues regarding implementation of sliding mode in real systems

7.3. Practical Sliding Mode Implementation

The 3SMC algorithm is a mathematical concept for which the proof of convergence is madeunder the assumption of a rather ideal system. Some of the requirements for the proof ofconvergence are not applicable for a physical system i.e. infinite sampling- and switchingfrequency. Some of the requirements may result in a decreased tracking performance, whileother effects may result in an unstable system. This section will describe how the slidingmode control algorithm has been implemented on the non-ideal backhoe system. The PLCcode is found on the enclosed CD.

7.3.1. Controller Parameters

For a system with infinite switching- and sampling frequency the sliding mode algorithmensures convergence as long as α is set sufficiently large to dominate the sign of the errorjerk.

α may be calculated to ensure this by utilizing Equation (7.48) on the preceding page. Itmay be seen that the calculation requires information about the derivative of the frictionparameters. These parameters have been modeled as a sign function and later implementedin the simulation model using a hyperbolic tangent function with a rather steep trend nearzero velocity. While the sign function is not differentiable, the derivative of the hyperbolictangent function is near infinite. It is mathematical possible to do calculation using thehyperbolic tangent function but it serves no practical purpose as the hyperbolic tangentfunction is only used to realize the sign function in simulation. Other parameters may bedifficult to obtain as well, i.e the triple derivative of the mass equivalent meq. Small modeluncertainties will affect the calculation of α to a high degree and it is therefore consideredto be of no purpose to do the calculation. In principle, α could be chosen infinitely large,but in a real system that is not realizable as the spool stroke is limited. Choosing an α toolarge in the system of finite switching frequency results in a maximum spool displacementin each switching cycle. It may further be noted that the proof is based on the assumptionthat the system can realize an infinite fast switching. Due to spool dynamics, oil stiffnessand the sampling limitations this is not realizable on the backhoe. Instead alpha is set asa tuning parameter in the laboratory.

85

7.3.2. Calculating Velocity- and Acceleration Error in Discrete Time

The 3SMC algorithm requires knowledge about the sign of the acceleration- and velocityerror. These quantities are not sensed on the backhoe system, but may instead becalculated based on the position error. Using a discrete differentiator and referring toFigure 7.5 these are calculated as shown in (7.49) and (7.50)

e =ek − ek-1

t1(7.49)

e =

ek−ek-1t1

− ek-1−ek-2t2−t1

t2(7.50)

T

x [s]p2

k

ek-1

ek-2

ek

t1

t2

Figure 7.5.: Figure showing an arbitrary distribution of samples.

Only the sign of the acceleration- and velocity error is needed, which simplifies (7.49) and(7.50) to,

sgn(e) = sgn(ek − ek-1) (7.51)

sgn(e) = sgn

(ek − ek-1

t1− ek-1 − ek-2

t2 − t1

)(7.52)

If the time between each sample is homogeneous, this may be further reduced as,

sgn(e) = sgn(ek − ek-1) (7.53)sgn(e) = sgn(ek − 2 · ek-1 + ek-2) (7.54)

7.3.3. Practial Concerns and Delimitations

While implementing the sliding mode controller on the backhoe a number of problemsoccurred. The problems may be explained in brief as sensor- and switching problems.

86

The position sensors attached to the backhoe all have a finite resolution of 0.1mm. Thesign of the velocity error and the sign of the acceleration error is calculated based on thismeasurement. Consider a case with a low velocity, where the position measurement is notupdated between each sample. Due to sensor quantification, the velocity is in the majorityof samples instants calculated as 0. Further, the error acceleration changes sign betweeneach quantification jump in the sensor feedback signal. A sketch illustrating this effect isshowed in Figure 7.6.

0.1 mm

0.2 mm

0.3 mm

0.4 mm

T

x [s]p2

k

T k

Actual position

Measured position

sign(e)

sign(e)

sign(e)

1

-1

et

Figure 7.6.: Figure showing position meassurement using a discrete sensor. When usingthe sensor to calculate the sign- of velocity and acceleration the resolutionof the sensor must be accounted for.

Even though 0.1mm may be thought off as a decent sensor resolution it is not well suitedto be used in a system with a high sampling frequency. At the time being, the PLC samplesat 1kHz. The minimum velocity required to ensure the position to be updated betweeneach sample can thus be calculated as

vmin =Tsxres

=1000Hz

0.1mm−1= 0.1

m

s(7.55)

Reducing the sample time Ts will reduce this effect, but as the proof of convergence forthe 3SMC relies on an infinite sampling time, this is not desirable.

87

A number of approaches have been tried to properly compensate for this problem withoutreplacing the position sensor in hand. In general all the solutions involves filteringtechniques. Two proposed solutions are listed below.

Variable Sampling Time

A buffer of size 3 with position- and time measurements is utilized. The buffer is updated ifa change in position is measured. The velocity and acceleration is calculated in accordancewith (7.51) and (7.52) on page 86 respectively. This alters the sampling frequency fromfixed to variable. An example using the buffer method is illustrated in Figure 7.7

1 2 3

0.20.1 0.3

e(k)

Error [mm]

0.1 0.4 0.8Time [s]

sgn(e) = 1

sgn(e) = -1

sgn(e) = 0.1-0.2

0.4-0.1sgn( 0.2-0.3

0.8-0.4

(

= -1

Figure 7.7.: Figure showing the calculated values of e and e for a sinusoidal input and asampling time of 1kHz

In simulations this method proved to cause some problems for the calculation of sgn(e)since this requires the velocity error to be calculated accurately. As the position sensor mayupdate its output between samples, the time measurements in the buffer is calculated witha small uncertainty, εt. This effect is shown in Figure 7.5 on page 86. This leads to a slighterror in the calculation of e and thereby an error in the calculation of sgn(e). This errormay be reduced by increasing the sampling frequency of the PLC, which is advantageousfrom a sliding mode point of view. This approach has been tested in simulation whenincluding a discrete state sensor and a discrete time controller. Figure 7.8 and Figure7.9 shows the sign of both the acceleration- and the velocity error for a situation with asampling frequency of 1kHz and 1MHz respectively.

0 0.2 0 .4 0 .6 0 .8 1 1.2 1 .4 1 .6 1 .8 2−0.01

−0.005

0

0.005

0.01

0.015

T im e [s ]

err

or

[m]

sgn(e) (scaled)sgn(e) (scaled)

Figure 7.8.: Figure showing the calculated values of e and e for a sinusoidal input and asampling time of 1kHz, when using a buffer solution

88

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.01

−0.005

0

0.005

0.01

0.015

Time [s]

Err

or

[m]

esgn(e) (scaled)sgn(e) (scaled)

Figure 7.9.: Figure showing the calculated values of e and e for a sinusoidal input and asampling time of 1MHz, when using a buffer solution

This method proved to work well in simulation as long as the sampling frequency is high,but unfortunately the PLC in hand is hardcoded to operate with a relatively low frequencyof 1kHz. In both simulation and in practice this proved to give a relatively large error inthe calculation of e and for this reason the solution is disregarded.

Reduced Sampling Average Filter

A reduced sampling average filter similar has been implemented. This filter utilizesinformation regarding the last n samples as a moving average filter, but instead of updatingthe control signal every sample this filter updates the control signal if the sample instancehas no remainder after division by n. Setting n sufficiently large ensures this calculationto be accurate, but does on the other hand introduce phase. The filter is illustrated inFigure 7.10.

1 2 3 4 5

0.10.1 0.2 0.2 0.2

Sample instance

Error [mm]

0.16

Store and latch for n samples

e(k)

Figure 7.10.: Figure showing the implemented filter for a situation of n = 5.

The filter has been tested in simulation for a sinusoidal input with a sampling frequencyof 1kHz and with a filter size of n = 5 and n = 20. Figure 7.12 and Figure 7.12 shows thesign of both the acceleration- and velocity error for n = 5 and n = 20 respectively.

89

0 0.2 0 .4 0 .6 0 .8 1 1.2 1 .4 1 .6 1 .8 2−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

T im e [s ]

Err

or

[m]


Figure 7.11.: Figure showing the calculated values of e and e for a sinusoidal input anda sampling time of 1kHz for filter solution with n=5

0 0.2 0 .4 0 .6 0 .8 1 1.2 1 .4 1 .6 1 .8 2−0.01

−0.005

0

0.005

0.01

0.015

T im e [s ]

err

or

[m]


Figure 7.12.: Figure showing the calculated values of e and e for a sinusoidal input anda sampling time of 1MHz for a filter solution with n=20

This filter was implemented in the PLC program, and empirical test showed that a valuen = 8 proved to work well. The filter solution is considered to be the worst of the twoproposed solutions, but as the PLC is limited at 1kHz the filter solution is implementedrather than the promising buffer solution. With the 3SMC algorithm implemented in thePLC, the controller is compared to the linear reference controller in the following part.

Part III.

Controller Performance

91

Chapter 8Controller PerformanceContents

8.1. Controller Performance at Large Acceleration of Heavy Duty 958.2. Controller Performance at Progressive Load . . . . . . . . . . . 988.3. Controller Performance at Abrupt Disturbance . . . . . . . . . 1018.4. Controller Comparrison . . . . . . . . . . . . . . . . . . . . . . . 104

Throughout this chapter, the linear controllers designed in Chapter 6 on page 63 andthe 3. Order sliding controller derived in Chapter 7 on page 73 are evaluated in terms oftracking performance when operating the trajectories designed in Chapter 5 on page 53.As the linear controllers are designed based on worst case considerations and a linearmodel that does not incorporate non-linearities, it was investigated whether a small degreeof parameter variation in the neighborhood of the calculated control parameters wouldprovide better tracking performance. It was found, that increasing both Ki and Kp with20% during the trajectory of large acceleration provided better tracking performance.During both the trajectory of progressive load and abrupt disturbance the calculatedvalues of Ki and Kp provided the best tracking performance of the tested controllers.Implementing the 3SMC controller in the system with a constant value of α showed anhighly oscillating behavior, which is believed to arise from the problem of determining thesign of the error acceleration, as was described in section 7.3 on page 85. On this basis,the control structure was altered to

u = −α|e|k(λsgn(e) + sgn(e) + sgn(e)), 0 < k ≤ 1 (8.1)

based on suggestion from [Schmidt, 2013]. The new α gain of the 3SMC controller wastuned and a value of α = 20000 provided the best tracking performance for all trajectories.The value of λ = 0.5 remained the same for all trajectories. The idea of adding an exponentk < 0 to make the controller more aggressive at low error values did not improve trackingperformance, hence k = 1 for all trajectories.

A simulation model for the entire system with either of the two controllers implementedhas been established in MATLAB® Simulink environment. The basic structure of thissimulation model is seen in figure 8.1 on the next page

93

Figure 8.1.: Basic structure of the MATLAB® Simulink model used for control purposes

The following 3 sections contains the tracking performance of the 3 controllers whenoperating the trajectory of large acceleration of heavy duty, the trajectory of progressiveload and the trajectory of abrupt disturbance, respectively. The performance will beevaluated for both the linear and the sliding mode controller. Simulation results aresuperimposed on controller for each trajectory. The section is followed by a comparison ofthe controllers with further elaboration of the individual performance. All tested controllersare found on the enclosed CD plotted in the same style as on the following pages

94

8.1. Controller Performance at Large Acceleration of HeavyDuty

Proportional Controller

0 2 4 6 8 10 120

0.2

0 .4

0 .6

0 .8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 2 4 6 8 10 12−0.1

−0.05

0

0.05

0.1

Err

or

[m]

R M S = 4 .900835e−02, P eak = 9 .298000e−02

0 2 4 6 8 10 12−60

−40

−20

0

20

40

Va

lve

ou

tpu

t [%

]

0 2 4 6 8 10 120

50

100

150

200

Pre

ss

ure

[B

ar]

T im e [s ]

P ressure A

P ressure B

R eference

E xperim enta l

S im ulated

E xperim enta l

S im ulated

Figure 8.2.: Tracking performance of proportional controller, when operating thetrajectory of large acceleration of heavy duty

95

VFF-PI-HP Controller

0 2 4 6 8 10 120

0.2

0 .4

0 .6

0 .8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 2 4 6 8 10 12

−2

0

2

4

6

x 10−3

Err

or

[m]

R M S = 5 .888572e−04, P eak = 2 .960000e−03

0 2 4 6 8 10 12−100

−50

0

50

Va

lve

ou

tpu

t [%

]

0 2 4 6 8 10 120

50

100

150

200

Pre

ss

ure

[B

ar]

T im e [s ]

P ressure A

P ressure B

R eference

E xperim enta l

S im ulated

E xperim enta l

S im ulated

Figure 8.3.: Tracking performance of VFF-PI_HP controller when operating thetrajectory of large acceleration of heavy duty

96

3SMC Controller

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Tra

ckin

g p

erf

orm

an

ce [

m]

0 2 4 6 8 10 12

−2

0

2

4

6

x 10−3

Err

or

[m]

RMS = 9.699960e−04, Peak = 2.980000e−03

0 2 4 6 8 10 12−60

−40

−20

0

20

40

Valv

e o

utp

ut

[%]

0 2 4 6 8 10 120

50

100

150

200

Pre

ssu

re [

Bar]

Time [s]

Pressure A

Pressure B

Reference

Experimental

Simulated

Experimental

Simulated

Figure 8.4.: Tracking performance of 3SMC controller when operating the trajectory oflarge acceleration of heavy duty

97

8.2. Controller Performance at Progressive Load


0 2 4 6 8 10 12 14 16 18 200

0.2

0 .4

0 .6

0 .8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05

0.1

Err

or

[m]

R M S = 2 .645889e−02, P eak = 5 .574000e−02

0 2 4 6 8 10 12 14 16 18 20−20

0

20

40

Va

lve

ou

tpu

t [%

]

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

Pre

ss

ure

[B

ar]

T im e [s ]

P ressure A

P ressure B

R eference

E xperim enta l

S im ulated

E xperim enta l

S im ulated

Figure 8.5.: Tracking performance of proportional controller when operating thetrajectory of progressive load

98


0 5 10 15 200

0.2

0 .4

0 .6

0 .8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 5 10 15 20−4

−2

0

2

4

6x 10

−3

Err

or

[m]

R M S = 4 .017875e−04, P eak = 4 .150000e−03

0 5 10 15 20−40

−20

0

20

40

Va

lve

ou

tpu

t [%

]

0 5 10 15 200

50

100

150

200

Pre

ss

ure

[B

ar]

T im e [s ]

P ressure A

P ressure B

R eference

E xperim enta l

S im ulated

E xperim enta l

S im ulated

Figure 8.6.: Tracking performance of VFF-PI_HP controller when operating thetrajectory of progressive load

99

3SMC Controller

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

Tra

ckin

g p

erf

orm

an

ce [

m]

0 2 4 6 8 10 12 14 16 18 20−5

0

5

10x 10

−3

Err

or

[m]

RMS = 9.515461e−04, Peak = 7.460000e−03

0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

40

60

Valv

e o

utp

ut

[%]

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

Pre

ssu

re [

Bar]

Time [s]

Pressure A

Pressure B

Reference

Experimental

Simulated

Experimental

Simulated

Figure 8.7.: Tracking performance of 3SMC controller when operating the trajectory ofprogressive load

100

8.3. Controller Performance at Abrupt Disturbance


0 5 10 15 200

0.2

0 .4

0 .6

0 .8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 5 10 15 20−0.1

−0.05

0

0.05

0.1

Err

or

[m]

R M S = 2 .617849e−02, P eak = 5 .629000e−02

0 5 10 15 20−20

0

20

40

Va

lve

ou

tpu

t [%

]

0 5 10 15 200

50

100

150

200

Pre

ss

ure

[B

ar]

T im e [s ]

P ressure A

P ressure B

R eference

E xperim enta l

S im ulated

E xperim enta l

S im ulated

Figure 8.8.: Tracking performance of proportional controller when operating thetrajectory of abrupt disturbance

101


0 2 4 6 8 10 12 14 16 18 200

0.2

0 .4

0 .6

0 .8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 2 4 6 8 10 12 14 16 18 20−4

−2

0

2

4

6x 10

−3

Err

or

[m]

R M S = 4 .604693e−04, P eak = 4 .480000e−03

0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

40

60

Va

lve

ou

tpu

t [%

]

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

Pre

ss

ure

[Ba

r]

T im e [s ]

P ressure A

P ressure B

R eference

E xperim enta l

S im ulated

E xperim enta l

S im ulated

Figure 8.9.: Tracking performance of VFF-PI-HP controller when operating thetrajectory of abrupt disturbance

102

3SMC Controller

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

Tra

ck

ing

pe

rfo

rma

nc

e [

m]

0 2 4 6 8 10 12 14 16 18 20−0.01

−0.005

0

0.005

0.01

Err

or

[m]

RMS = 1.077228e−03, Peak = 7.640000e−03

0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

40

60

Va

lve

ou

tpu

t [%

]

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

Pre

ss

ure

[B

ar]

Time [s]

Pressure A

Pressure B

Reference

Experimental

Simulated

Experimental

Simulated

Figure 8.10.: Tracking performance of 3SMC controller when operating the trajectoryof abrupt disturbance

103

8.4. Controller Comparrison

As expected the controller with the worst tracking performance for all the trajectories wasthe proportional controller. However, it should be noticed that no velocity feedforwardwas added to the proportional control signal, making it difficult to compare to neitherthe VFF-PI-HP controller nor the 3SMC controller. The proportional controller acts todemonstrate what can be obtained with the simplest control design and with no insight tothe system dynamics at all. Direct comparisons between the controller for all trajectoriesare illustrated in Figure 8.11, where the RMS- and peak errors are stated in table 8.1 forthe three trajectories.

0 2 4 6 8 10 12−0.1

−0.05

0

0.05

0.1

Trajectory 1

Err

or

[m]

3SMC

P

VFF−PI−HP

0 5 10 15 20−0.06

−0.04

−0.02

0

0.02

0.04

Err

or

[m]

Trajectory 2

3SMC

P

VFF−PI−HP

0 2 4 6 8 10 12 14 16 18 20−0.06

−0.04

−0.02

0

0.02

0.04

Err

or

[m]

Trajectory 3

3SMC

P

VFF−PI−HP

Figure 8.11.: Comparison of tracking error among the controller for all trajectories

104

Large Acceleration of Heavy Dutyepeak erms

P Controller 49E−3 92.9E−3

VFF-PI-HP Controller 0.59E−3 2.96E−3

3SMC Controller 0.97E−3 2.98E−3

Progressive Loadepeak erms

P Controller 26.5E−3 55.7E−3

VFF-PI-HP Controller 0.4.E−3 4.15E−3


Abrupt Disturbanceepeak erms

P Controller 26.2E−3 56.2E−3

VFF-PI-HP Controller 0.46E−3 4.48E−3


Table 8.1.: Performance of the controllers in terms of peak- and RMS error

From both Figure 8.11 and table 8.1 it can be seen, that the linear VFF-PI-HP controlleris superior for for all trajectories in both peak- and RMS-error. The 3SMC controller showalmost as good tracking performance for the trajectory of large acceleration, but suffersat the two other trajectories. In the latter two trajectories, the VFF-PI-HP- and 3SMCcontroller show similar tracking performance during the lifting part of the trajectories, withpeak error in the range of 1mm. As the dipper sets of to return to the original position,both the VFF-PI-HP- and 3SMC controller show oscillations in the tracking error, with themagnitude of the VFF-PI-HP controller being the smallest. The large acceleration underan increased gravitational force is difficult for both controllers to handle. The reason for the3SMC controller to induced higher magnitude of error oscillations may be put down to thedifficulties in determining the sign of the error acceleration from only position feedback,as was described in section 7.3 on page 85.

105

Chapter 9Conclusion

This thesis originated in pursuit of investigating whether a sliding mode control algorithmcould compete with, or even outmatch, a properly designed linear controller in termsof tracking performance. To accomplish this, a mathematical non-linear model of themechanics, hydraulics and HPU of the CASE 580 backhoe loader was derived. To makethe best coherence between the model and the observations from the setup, the modelparameters have been tuned based on experimental data both manually and by means ofa evolutionary optimization algorithm.

The obtained model includes all four bodies of the backhoe with appurtenant hydraulicsand is believed to be accurate for simulations of dynamics responses throughout theentire setup. This provided basis for designing a linear controller that is assumed tobe representative for the most advanced controller found in industry-like applications.Further, the model derived was used as a powerful tool for evaluating controllerperformance and predicting practical issues before implementation them on the system.A third order sliding mode-structure, founded on the research article of [Schmidt andAndersen, 2013], was proposed as a candidate for sliding mode control. The proof ofconvergence for this controller was derived based on ideal system considerations. However,practical issues during implementation of the control structure called for a filter solution inthe feedback signal. It was found, that utilizing a constant value of α, did not provide anyfeasible controller performance, and the structure was changed by varying α linearly as afunction of the position error. To illustrated the control performance that can be obtainedwhen using the simplest controller topology and with no insight to system dynamic, aproportional reference controller was evaluated in parallel. For evaluation of the controllerperformance, three different trajectories was designed to emulate work conditions ofindustry-like applications. Based on a QP-analysis, the trajectories were evaluated tobe obtainable within the limitation of the backhoe system. For all trajectories it wasshown, that the linear reference controller performed best in terms of tracking performanceillustrated by peak- and RMS error. Based on the practical issues during implementationof the third order sliding mode-structure. Simulations showed how increasing the samplingfrequency could solve some of the practical issues for the sliding mode controller, as forinstance by using a higher sampling frequency or using a sensor with a finer resolution. It isby the authors believed that the performance of the 3SMC can be enhanced in practice byutilizing a faster valve, increasing the sampling frequency and utilizing a sensor of greater

107

resolution. As the difference between the tracking performance of the VFF-PI-HP and3SMC controller was within the same range, which is particularly seen in the trajectory oflarge acceleration, the benefit of using the third order sliding mode featuring easy tuningrather than the cumbersome modeled based parameter calculations of the linear controllermay justify the utilization of the third order sliding mode control algorithm.

108

Bibliography

4WREE. Bosch Rexroth 4WREE. 4WREE-Datasheet. RE 29061/10.05, datasheet.Bosch Rexroth.

4WRKE. Bosch Rexroth 4WRKE. 4WRKE-Datasheet. RE 29075/08.04, datasheet.Bosch Rexroth.

4WRTE. Bosch Rexroth 4WRTE. 4WRTE-Datasheet. RE 29083/09.06, datasheet.Bosch Rexroth.

A10VSO. Bosch Rexroth A10VSO. A10VSO-Datasheet. RE 92711/01.12, datasheet.Bosch Rexroth.

Andersen. Torben Ole Andersen. Symmetrisk servoventil med asymmetrisk cylinder.handwritten note.

Centinkunt, 2007. Sabri Centinkunt. Mechatronics. ISBN: 978-0-471-47987-1. JohnWiley & Song, Inc, 2007.

Charles L. Phillip, 2000. Royce D. Harbor Charles L. Phillip. Feedback ControlSystems. ISBN: 0-13-949090-6, 4. Tom Robbins, 2000.

Craig, 200. John J. Craig. Introduction to Robotics. ISBN: 0-13-123629-6. PearsonEducation Inc., 200.

Fillipov, 1988. Aleksel Fedorovich Fillipov. Differential Equations with DiscontinuousRighthand Sides. ISBN: 90-277-2699-X, 1. Kluwer Academic Publishers, 1988.

Hansen and Andersen, 2007. Michael R. Hansen and Torben Ole Andersen. FluidPower Systems. Published Note, 2007.

Josefsen, 2004. Sigurdur Magni Benediktsson & Ole Søndergaard Josefsen. Softwaremodelling and tool centre control of hydraulic backhoe loader, 2004.

Levant, 1993. Ariel Levant. Sliding order and Sliding accuracy in Sliding ModeControl. http://www.tau.ac.il/ levant/slorder93.pdf, 1993.

MathWorks, 2013. MathWorks. Documentation: filtfilt.http://www.mathworks.se/help/signal/ref/filtfilt.html;jsessionid=222c4e2325aecfdae91659af2687,2013.

109

Merrit, 1967. Herbert E. Merrit. Hydraulic Control Systems. LCCCN: 66-28759. JohnWiley & Sons, Inc, 1967.

Olsson, 1997. H. Olsson. Friction Models and Friction Compensation, 1997.

Rasmus K. Ursem, 2003. Rasmus K. Ursem. Models for Evolutionary Algorithms andTheir Applications in System Identification and Control Optimization, 2003.

Rydbjerg, 2008. Karl Erik Rydbjerg. Hydraulic Servo Systems. LinköpingsUniversitet, 2008.

Schmidt, 2013. Lasse Schmidt. Oral Statement. PhD fellow, Bosch Rexroth, 2013.

Schmidt and Andersen, 2013. Lasse Schmidt and Torben Ole Andersen. RealizingThird Orden Sliding Modes in Dynamical Systems Using Stritly Switching Controls.Research Article, 2013.

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Torben O. Andersen, MIchael R. Hansen. Henrik C. Pedersen Finn ConradTorben O. Andersen, MIchael R. Hansen. Comparison of Linear Controllers for aHydraulic Servo System. 10.02, 6.

110

Appendix AMechanical Properties of the 580Backhoe Loader

ContentsA.1. Dimension of the Backhoe Loader . . . . . . . . . . . . . . . . . 112A.2. Dimension of the Hydraulic Cylinders . . . . . . . . . . . . . . . 113A.3. Centre of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.4. Cylinder Extension and Joint Angles . . . . . . . . . . . . . . . 121A.5. Cylinder Force and Joint Torques . . . . . . . . . . . . . . . . . 128A.6. Jacobians of the Centre of Mass . . . . . . . . . . . . . . . . . . 131

The purpose of the appendix is to establish the mechanical properties of link 1 through4 of the backhoe. At first, the mass, center of mass and inertia matrix of each link aredetermined based on dimensional measurements. This is followed by an establishmentof the relations between the angular (-and prismatic) position of the manipulator jointand the linear position of the hydraulic actuators. This appendix illustrates how thejoint position and actuator position relates, and entailing also the relation between thejoint-actuator speed and -acceleration. Lastly, the hydraulic force of the actuator andresulting torque (or force if prismatic) at the manipulator joint are also related throughtrigonometric which are derived.

111

A.1. Dimension of the Backhoe Loader

425

1170

1220

1491

185

487

350

229

474

380

375

9021133

Figure A.1.: Dimension of fixed lengths in mm.

112

A.2. Dimension of the Hydraulic Cylinders

Length Stroke

Rod Piston

Figure A.2.: Convention for cylinder dimensions

Length DiameterCylinder Length [mm] Stroke[mm] Piston[mm] Rod[mm]Boom 1170 842 127 57Dipper 930 583 127 63Extender 1470 1068 76 44Bucket 1185 872 89 63

Table A.1.: Cylinder dimensions

113

A.3. Centre of Mass

As the backhoe is an arm of 4 degrees of freedom, the location of the centre of massfor all 4 links are needed. However, an exact determination is difficult to obtain dueto complex shapes of the arms. In the following estimation of the centre of masses, aSolidWorks®CAD model is utilized. To verify the CAD model and to demonstrated analternative way of estimating the centre of mass, the shape of the boom is approximatedwith simple geometries in order to find its centre of mass. This result is then comparedto that of the CAD model. Further, the centre of masses are calculated for 3 differentdipper cylinder positions, namely at the minimum- halfway- and maximum cylinder strokecapability, to see variation of the position for the centre of mass. The boom cylinder isneglected in the centre of mass consideration as it is supported by the foundation of thebackhoe loader, as can be seen in Figure 1.1 on page 2. The center of mass of link i isdenoted CMi and is expressed in the D-H reference frame Oi.

Center of Mass of the Boom - Geometric Approximation

The center of mass for the boom is found with respect to O1 of the D-H reference frame.The x-, y- and z-axis are illustrated as red, green and blue, respectively. The boom consistof two parallel beams with a profile as seen in Figure A.3 - Side view, and 3 steel bracesjoining the beams, as can be seen in Figure A.3 - Upper view. The shape of the beam andbraces are approximated with 11 bodies (2 · 4 + 3) of simple geometries and center of massat CBi, where i ∈ [1 : 11] ∗.

Side View

Upper V

iew

CB1

CB2

CB4

CB3

CB5

CB6

CB7

O1

O1

O1 O1

Figure A.3.: The boom is divided into geometries of which the centre of masses areknown in order to estimate the centre of mass for the entire boom

The thickness of the beams varies within [1.1 − 2.5] cm, but is for simplicity assumeduniform at a value of 1.8 cm. The thickness of brace B5, B6 and B7 are 3.5cm, 6.5 cmand 6.5 cm, respectively. Further, the mass distribution is assumed homogenous alongeach body. Due to symmetry, the center of mass for the left-side beam has the same x-

∗Body B8 through B11 are not shown in Figure A.3 due to symmetry with body B1 through B4

114

and y-coordinates as the right-side beam, but with z-coordinates of same amplitude andopposite sign.

CB1 =

−1.2000

0.131

m CB8 = CB1 · (−k) =

−1.2000

−0.131

m

CB2 =

−0.7600.1980.131

m CB9 = CB2 · (−k) =

−0.7600.198−0.131

m

CB3 =

−1.307−0.1520.131

m CB10 = CB3 · (−k) =

−1.307−0.152−0.131

m (A.1)

CB4 =

−0.453−0.1520.131

m CB11 = CB4 · (−k) =

−0.453−0.152−0.131

mIn a similar way, the center of mass of the braces are found as,

CB5 =

−2.100

m CB6 =

−1.445−0.083

0

m CB7 =

−0.535−0.083

0

m (A.2)

Besides the stationary parts of the boom structure, the dipper cylinder influence theposition of the center of mass for the boom. The dipper cylinder is regarded only as a 2body element containing a chamber and a piston, where the chamber can be approximatedbe a circular cylindrical shell, and the rod as a circular cylinder. With this assumption,the mass of the hinge and the piston disc are neglected. Knowing that the total lengthof the dipper cylinder lies in the range of CD ∈ [0.930; 1.513] m, the cylinder angle withrespect to the stroke of the cylinder can be found as,

Half

MinMax

D

fmax

fmin

fhalf

C O1

Figure A.4.: Center of gravity for the boom at minimum-, half way- and maximum dippercylinder extension

115

ϕ(|CD|) = cos−1(|CO1|2 + |CD|2 − |DO|2

2 · |CO1| · |CD|

)

ϕmin = 10.6◦ ϕhalf = 16.5◦ ϕmax = 8.1◦ (A.3)

Due to symmetry, the distance between the hinge C and the center of mass of the cylinderchamber denoted Kcm is ∣∣KC

cm

∣∣ =1

2Lh +

1

2Lc (A.4)

As can be seen in Figure A.2 on page 113, the distance between C and the center of massof the rod denoted Rcm is ∣∣RCcm∣∣ =

1

2Lh +

1

2Lp + x1 + d2 (A.5)

wherex1 = x2 =

Lc − stroke2

(A.6)

and d2 is the cylinder extension. The centre of gravity for the cylinder chamber and therod can be found with respect to the hinge C as

KCcm =

cos[ϕ(|CD|)]sin[ϕ(|CD|)]

0

|CKcm| RCcm =

cos[ϕ(|CD|)]sin[ϕ(|CD|)]

0

|CRcm| (A.7)

On this basis, the center of mass of both the cylinder chamber and the rod with respectto reference frame O1 can be found as

Kcm = KCcm − i · |CO1| Rcm = RCcm − i · |CO1| (A.8)

where i is the unit vector of the x-axis in O1 and CO1 = 1.22m is the distance betweenhinge C and O1.

The united center of mass for all the bodies can be found as,

˜CM1 =1

Mtotal·∑i

(Vi · ρi · Pi) (A.9)

where Mtotal is the total mass of the boom including the cylinder, Vi is the volume of theith body, ρi the density of the ith body and Pi is the position of the centre of mass forthe ith body with respect to O1. The material thickness of the cylinder chamber is fromFigure A.2 on page 113 assumed uniform at 1

2(Do − Di) = 1.15cm. The density of boththe boom- and cylinder material are assumed equal to that of A1020 steel, or ρi = 7900 kg

m3

The calculated centre of mass of the boom including the cylinder is shown in the leftcolumn for a minimum, halfway and maximum cylinder extension, respectively. The CADestimation of (A.11) on the next page is for reader convenience shown in the right column.

116

˜CM1,min =

−1.030.020

m CM1,min =

−1.080.000

m˜CM1,half =

−1.020.040

m CM1,half =

−1.070.010

m (A.10)

˜CM1,max =

−1.010.010

m CM1,max =

−1.060.000

mCentre of Mass of The Boom - CAD

The SolidWorks® CAD model of the boom including the dipper cylinder can be seen inFigure A.5. The extension of the dipper cylinder is shown at half its stroke capability, butthe center of mass is also found for the minimum- and maximum stroke

Figure A.5.: CAD representation of the boom

CM1,min =

−1.080.000

m CM1,half =

−1.070.010

m CM1,max =

−1.060.000

m (A.11)

As the deviation between the CAD based- and the geometric based center of mass is within5 cm, only the CAD based center of mass is utilized throughout the report. The affectionof the cylinder extension alters the center of mass with 2 cm, but is neglected throughoutthe report. Instead, the centre of mass is assumed constant at a position corresponding tothat of a halfway extended position of the cylinder. Further, the use of the CAD modeleases the calculation of the inertia tensor of each link. As the rotation axes are all parallel,only the inertia of the rotation axis are printed and is denoted Ii. In the following, thelink mass, link centre of mass and the z-axis moment of inertia are printed for all 4 links

Link 1

M1 = 474kg CM1 =

−1.070.010

m I1 = 129.96kg ·m2 (A.12)

117

Link 2

Link 2 contains the dipper beam and the extender cylinder, as can be seen in Figure A.6.The reference frame O2 is shown in the Figure, where the red, green and blue denotes thex-axis, y-axis and z-axis, respectively.

Figure A.6.: Center of mass estimation of link 2 including extender cylinder

M2 = 278kg CM2 =

00

0.68

m I2 = 74.44kg ·m2 (A.13)

Link 3

Link 3 contains the extenders beam and the bucket cylinder, as can be seen in Figure A.6.The reference frame O3 is shown in the Figure, where the red, green and blue denotes thex-axis, y-axis and z-axis, respectively.

Figure A.7.: Center of mass estimation of link 3 including bucket cylinder

M3 = 172kg CM3 =

0.140.91

0

m I3 = 67.80kg ·m2 (A.14)

118

Link 4

The center of mass for the previous 3 links are considered stationary with respect to thelink itself, because they are treated like rigid object. But, as link 4 contains the bucket,the bucket arm and the 2 hinge arms, as can be seen in Figure A.6 on the preceding page,link 4 cannot be treated like a rigid body. On this basis, the center of mass is measurein SolidWorks with respect to the reference frame O4, as shown in Figure A.8, where thered, green and blue denotes the x-axis, y-axis and z-axis, respectively.

Figure A.8.: Center of mass estimation of link 4 including hinge arm

Due to symmetry, the z-component for the center of mass is 0 at all times. The variation ofthe x- and y-component for the center of mass are plotted in Figure A.9 and Figure A.10.Utilizing a least squares approximation, a second order polynomial shows good correlationwith the CAD obtained data for both the x- and y- component.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.7

0.75

0.8

0.85

0.9

xP4

[m]

CM

4 x [m

]

Polynomial fit

Points obtained from CAD

Figure A.9.: x-position of Centre of mass for the bucket with respect to cylinderextension

119

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.26

−0.24

−0.22

−0.2

−0.18

−0.16

−0.14

−0.12

−0.1

xp4

[m]

CM

4 y [m]

Polynomial fit

Points from CAD

Figure A.10.: y-position of Centre of mass for the bucket with respect to cylinderextension

The moment of inertia for the bucket alters from 86kg ·m2 at zero stroke to 53kg ·m2 atmaximum stroke of the bucket cylinder. However, the moment of inertia is a measure ofrotation resistance at the centre of mass and is fairly small compared to the linear inertia,and it only makes a contribution to the D- and C matrix of Equation (2.35) on page 18.As the amplitude of the D- and C- matrix was shown small compared to the G, it isassumed that no significant error are induced by assuming the moment of inertia as staticat a value of 70kg ·m2

M4 = 216kg CM4 =

−0.223 · xP42 − 0.065 · xP4 + 0.855

−0.308 · xP42 + 0.3917 · xP4 − 0.257

0

m I4 = 70kg ·m2

(A.15)

120

A.4. Cylinder Extension and Joint Angles

The Case 580 Prestige backhoe loader does not comply with angular actuation inputs,but is instead actuated through linear cylinders. The relation between the joint anglesor displacements, and the position of the linear actuators will be derived throughout thissection. Keypoints used to derive the relations between the joint angles and the cylinderpositions are illustrated in Figure A.11, where Oi for i ∈ [0; 4] denotes the origin ofreference frames according to the D-H-Convention shown in Figure 2.5 on page 11.

O0

O1O2 O3

O4

A

B

C

D

EF

G H

I

Figure A.11.: To establish the actuator to joint relation, key reference points have beenassigned to the backhoe for use in latter calculations

According to the D-H-Convention, the joint varible qi relates to a rotational or prismaticmotion with respect to reference frame Oi−1. The extensions and contractions of the4 cylinders of the backhoe loader are directly linked to the joint variables through thegeometrics of the backhoe arm.

Link 1

Link 1 is divided into 4 points of interest as shown in Figure A.12 on the next page, where|AB| denotes the total length of the boom cylinder, O0 is the origin of reference frame 0and C is a point on the line drawn from the origin of reference frame 0 to 1.

121

B

q1f01

f02

f03

A

O0

C

*

fB

Figure A.12.: The boom segment of Figure A.11. The total length of the boom cylinderis spanned by |AB|, and ϕ02 and θ∗1 are functions of the |AB| length.

Based on Figure A.12 the relation between the total cylinder length and first joint variablecan be found as,

|AO0| = 0.43m |BO0| = 1.58m |CO0| = 1.17m |BC| = 0.49m

|AB|min = 1.17m ϕ01 = 1.38rad ϕ03 = 0.19rad

ϕ02 = cos−1(|AO0|2 + |BO0|2 − |AB|2

2 · |AO0| · |BO0|

)= cos−1

(k1 − |AB|2

c1

)(A.16)

θ∗1 = π − (ϕ01 + ϕ02 + ϕ03) = Ψ1 − cos−1(k1 − |AB|2

c1

)By noting that the length |AB| is a meassure of the minimum cylinder length |AB|minand the cylinder stroke xp1, the final relation between joint angle 1 and the boom cylinderstroke is

θ∗1 = π − (ϕ01 + ϕ02 + ϕ03) = Ψ1 − cos−1(k1 − d12

c1

)(A.17)

where

d1 = |AB|min + xp1, Ψ1 = 1.57, k1 = 2.67, c1 = 1.32 (A.18)

Link 2

In a similar way as for link 1, the relation between the dipper cylinder stroke and jointangle 2 can be found based on trigonometrics. The triangle of interest for link 2 is shown

122

in Figure A.13, where O1 is the origin of reference frame 1. The base of the dipper cylinderis attached at point C, which coincides with the straight line between reference frame 0and 1. The cylinder rod is attached to the dipper link at point D.

C

D

f11

q2

O1

Figure A.13.: The dipper segment of Figure A.11. The total length of the dipper cylinderis spanned by |CD|, and ϕ11 and θ∗2 are functions of the |CD| length.

|CO1| = 1.22m |DO1| = 0.35m |CD|min = 0.93m

ϕ11 = cos−1(|CO1|2 + |DO1|2 − |CD|2

2 · |CO1| · |DO1|

)θ∗2 =

pi

2− ϕ11 (A.19)

Still, the length |CD| is a measure of the minimum cylinder length and the stroke, andthe relation between joint 2 and the dipper cylinder stroke can be expressed as

θ∗2 =pi

2− cos−1

(|CO1|2 + |DO1|2 − |d2|2

2 · |CO1| · |DO1|

)= Ψ2 − cos−1

(k2 − d22

c2

)(A.20)

where

d2 = |CD|min + xp2, Ψ2 =π

2, k2 = 1.61, c2 = 0.85 (A.21)

Link 3

Contrary to the other links of the backhoe, link 3 is prismatic rather than revolute, whichmakes it straight forward to establish the relation between the extender cylinder strokeand joint position 3.

O2

O2O3| |min xp3

O3 O3

Figure A.14.: The distance between origin 2 and 3 is linear proportional with thedisplacement of the extension cylinder

123

|O2O3|min = 2.00m (A.22)

otal Clearly, if the displacement of the extension cylinder is denoted d3, then the distancebetween the origin of reference frame 2 and 3 is given as,

d3 = |O2O3|min + xp3 (A.23)

Link 4

The bucket cylinder stroke and joint variable 4 are related through a more complexgeometry. In total, 7 points of interest are taken into account when determining tis relation,as can be seen in Figure A.15. The distance |EF | is spanned by the the total length ofthe bucket cylinder. Whereas point E,G,H,O3 are fixed regardless of the bucket cylinderstroke, the position of point F, I,O4 are a funktion of the bucket cylinder stroke.

O4

O3

f32

f33f31

H

fH3

fH2

fH1

G

E

F

I

q4*

Figure A.15.: The backhoe segment needed in order to determine joint variable 4 asa function of the bucket cylinder extension contains both point on theextender and the bucket. Fixed lengths are marked with andvariable length with ∼.

The relation betweenn the bucket cylinder stroke and the

124

|EG| = 0.23m |EH| = 1.63m |FH| = 0.48m |FI| = 0.38m

|GH| = 1.49m |HO3| = 0.18m |IO3| = 0.36m |IO4| = 1.13m

|O3O4| = 0.90m |EF |min = 1.19m ϕH1 = 0.12rad ϕ33 = 1.85rad

ϕH2 = cos−1(|EH|2 + |FH|2 − |EF |2

2 · |EH| · |FH|

)ϕH3 = π − (ϕH1 − ϕH2)

|FO3| =√|FH|2 + |HO3|2 − 2 · |FH| · |HO3| · cos(ϕH3)

ϕ31 = cos−1(|HO3|2 + |FO3|2 − |FH|2

2 · |HO3| · |FO3|

)

ϕ32 = cos−1(|FO3|2 + |IO3|2 − |FI|2

2 · |FO3 · |IO3|

)

ϕ33 = cos−1(|IO3|2 + |O3O4|2 − |IO4|2

2 · |IO3| · |O3O4|

)θ4 =

3π

2− (ϕ31 + ϕ32 + ϕ33) (A.24)

The cumbersome trigonometric relation between the cylinder stroke and joint variable oflink 4 makes it computational heavy to utilize in latter development of speed relationsbetween actuator and joint variables. On this basis, the relation of link 4 is approximatedwith a least squares algorithm providing a high order polynomial relation. The bucketcylinder stroke and joint variable relation is depicted in Figure A.16 on the next page. Afifth order approximation of

F (d4) = −10.9(d4)5 + 85.4(d4)

4 − 270.0(d4)3 + 428.1(d4)

2 − 342.1(d4) + 112.6 (A.25)where

d4 = |EFmin|+ xp4

shows good correlation with the trigometric calculation. The least squares approximatedpolynomial will be used thorughout the report.

125

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2−0.5

0

0.5

1

1.5

2

2.5

|EF| [m]

θ 4 [rad

]

Trigonometric

5th degree approximation

Figure A.16.: Least squares approximation of the trigonometric relations of link 4 areutilized to avoid cumbersome calculations

All of the above derived joint-actuator relation can be collected in a single expression as

q = P(d(xp)

)(A.26)

q =

θ1θ2d3θ4

, d(xp) =

d1d2d3d4

=

xp1 + |AB|minxp2 + |CD|minxp3 + |O2O3|minxp3 + |EF |min

(A.27)

P =

Ψ1 − cos−1

(k1−d12c1

)Ψ2 − cos−1

(k2−d22c2

)d3

F (d4)

(A.28)

The position transformation vector P is utilized to obtain an expression for the relationbetween the joint -and actuator velocity.

126

q =d

dtP(d(xp)

)= V(d)d = V(d)xp (A.29)

where

q =

θ1θ2d3θ4

, d = xp =

xp1xp2xp3xp4

, V(d) =

2c1d1√

1−(

k1−d12

c1

)2 0 0 0

02c2d2√

1−(

k2−d12

c2

)2 0 0

0 0 1 00 0 0 F ′(d4)

(A.30)

The V(d) is a velocity Jacobian matrix for the backhoe manipulator and will be utilizedto relate the velocities of the cylinder actuators to the joint velocities of the backhoe. Thetransformation of angular acceleration to linear acceleration can be found as

q =d

dt

(V(d)xp

)= V(d, xp)xp + V(d)xp (A.31)

The vast size of the V makes it unsuitable for printing here, but can be found inthe dynamics.m Matlab file located on the enclosed CD. Since the backhoe systemcontains both linear force excertments of the cylinder and revolute resistant force of themanipulator, the torque force relation of the system needs to be established for dynamicmodelling.

127

A.5. Cylinder Force and Joint Torques

The cylinders of the backhoe loader are operated by hydraulic pressure which exert a forceon the manipulator producing a torque in the appurtenant joint. The magnitude of thetorque is not only a measure of the magnitude of the force as the cylinder lever arm changesubstantially with the joint movement. If τ denotes a vector of link torque for revolute-and link force for prismatic joint, that is,

τ =

τ1τ2F3

τ4

(A.32)

it is desired to find the relation to the hydralic exerted force such that,

τ = MF (A.33)(A.34)

where the torque multiplier matrix M is diagonal 4 × 4 matrix. F is the force vectorrepresenting the magnitude of the force for each of the 4 cylinders of the backhoe, that is

F =

|~FAB||~FCD||~FO2O3 ||~FEF |

(A.35)

The entries of M is based on mechanics of each link.

Link 1

Figure A.17 on the facing page shows how the force of the boom cylinder acts on themanipulator at reference point B, and the cylinder induced torque component can becalculated as,

τ1 = M1,1 · |~FAB| = −sin(ϕB)|BO0| · |~FAB| (A.36)where

ϕB = cos−1(|BO0|2 + |AB|2 − |AO0|2

2 · |BO0| · |AB|

)

128

B

fB

fB

O3t1

A

FAB

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

M1,1[-]

xp1 [m]

Figure A.17.: Trigonometric illustration of how the cylinder force and joint torquerelation are established for Link 1

Link 2

In a corresponding way to that of Link 1, the cylinder induced torque component of thelink 2 can be calculated based on a geometric observation. Figure A.18 shows the dippercylinder force acting at reference point D and resulting torque component can be calculatedas,

τ2 = M1,1 · |~FCD| = −sin(ϕD2)|DO1| · ~FCD (A.37)

where

ϕD1 = π − ϕD1, ϕD1 = cos−1(|DO1|2 + |CD|2 − |CO1|2

2 · |DO1| · |CD|

)

DO1

fD1

fD2

FCD

t2

0 0.1 0.2 0.3 0.4 0.5−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

M2,2[-]

xp2 [m]


129

Link 3

The force of the extender cylinder is acting to produce a linear motion. Hence, the actuator-and link force is of same magnitude, that is

F3 = FO2O3 , M3,3 = 1 (A.38)

Link 4

The force of the bucket cylinder does not act on the bucket itself directly, but througha turning mechanism as can be seen in Figure A.11 on page 121. An illustration of thebucket mechanism is redrawn in Figure A.19.

FFEF

t

H

I

O3

fF1

FIF

FFI

fF2

fF3

fI1

H

t40 0.2 0.4 0.6 0.8

−0.5

−0.4

−0.3

−0.2

−0.1

M4,4[-]

xp4 [m]


The relation between the cylinder force and joint torque is based on a steady-stateconsideration as the force associated with acceleration of mass of the bucket mechanism isconsidered negligible. Based on this assumption a torque equilibrium around point H canbe made to find the force acting on the bucket.

∑MH = 0

−|FH| · ~FEF · sin(ϕF1) + |FH| · ~FIF · sin(ϕF2 + ϕF3) = 0 (A.39)

~FIF =~FEF · sin(ϕF1)

sin(ϕF2 + ϕF3)

130

where

ϕF1 = cos−1(|EF |2 + |FH|2 − |EH|2

2 · |EF | · |FH|

)

ϕF2 = cos−1(|FO3|2 + |FH|2 − |HO3|2

2 · |FO3| · |FH|

)

ϕF3 = cos−1(|FO3|2 + |FI|2 − |IO3|2

2 · |FO3| · |FI|

)

From Figure A.19 on the facing page it is clear that ~FIF = −~FFI and that the torque oflink 4 can be found as

τ4 = ~FFI · sin(ϕI1)m

τ4 = M4,4 · ~FEF = −sin(ϕI1) · sin(ϕF1)

sin(ϕF2 + ϕF3)· ~FEF (A.40)

where

ϕF3 = cos−1(|FI|2 + |IO3|2 − |FO3|2

2 · |FI| · |IO3|

)

With the final force/torque relation established, the torque multiplier matrix M can bewritten as

τ = MF

M = I4×4

−sin(ϕB)|BO0|−sin(ϕD2)|DO1|

1

− sin(ϕI1)·sin(ϕF1)sin(ϕF2+ϕF3)

(A.41)

The torque and force relation concludes this appendix, which provides basis for thedynamic analysis of the system modelling in section 2.1.2 on page 14

A.6. Jacobians of the Centre of Mass

As described in Section 2.1.2 on page 14 the Jacobian of the manipulator can be dividedinto the linear top half and the angular bottom half. If the linear top half in general isdescribed by

Jvi =

j1,1 j1,2 j1,3 j1,4j2,1 j2,2 j2,3 j2,4j3,1 j3,2 j3,3 j3,2

(A.42)

131

Then the entries of linear Jacobian for link 1 are given as

j1,1 = −|O0CM1| · sθ1j1,2 = |O0CM1| · cθ1j1,3 = 0

j1,4 = 0

j2,1 = 0

j2,2 = 0

j2,3 = 0

j2,4 = 0

j3,1 = 0

j3,2 = 0

j3,3 = 0

j3,4 = 0

(A.43)

The entries of linear Jacobian for link 2 are given as

j1,1 = |O1CM2| · cθ12 − |O0O1| · sθ1j1,2 = |O1CM2| · cθ12j1,3 = 0

j1,4 = |O1CM2| · sθ12 + |O0O1| · cθ1j2,1 = |O1CM2| · sθ12j2,2 = 0

j2,3 = 0

j2,4 = 0

j3,1 = 0

j3,2 = 0

j3,3 = 0

j3,4 = 0

(A.44)

The entries of linear Jacobian for link 3 are given as

132

j1,1 = cθ12 · |O2O3min + xp3| − |O3CM3y| · cθ12 − |O3CM3x| · sθ12 − |O0O1| · sθ1j1,2 = cθ12 · |O2O3min + xp3| − |O3CM3y| · cθ12 − |O3CM3x| · sθ12j1,3 = sθ12

j1,4 = 0

j2,1 = sθ12 · |O2O3min + xp3|+ |O3CM3x| · cθ12 − |O3CM3y| · sθ12 + |O0O1| · cθ1j2,2 = sθ12 · |O2O3min + xp3|+ |O3CM3x| · cθ12 − |O3CM3y| · sθ12j2,3 = −cθ12

j2,4 = 0

j3,1 = 0

j3,2 = 0

j3,3 = 0

j3,4 = 0

And lastly the entries of linear Jacobian for link 4 are given as

j1,1 = |O2O3,min| · cθ12 − sθ124 · |O4CM4x| − cθ124 · |O4CM4y|+ xp3cθ12 + |O3O4| · sθ124−O0O1 · sin θ1j1,2 = |O2O3,min| − sθ124 · |O4CM4x| − cθ124 · |O4CM4y + xp3cθ12 + |O3O4| · sθ124j1,3 = sθ12

j1,4 = |O2O3,min| · cθ12 − sθ124 · |O4CM4x| − cθ124 · |O4CM4y| − cθ12|O2O3,min + xp3|...+ xp3 · cθ12 + |O3O4| · sθ124

j2,1 = cθ124 · |O4CM4x| − sθ124 · |O4CM4y|+ |O2O3min| · sθ12 + |O0O1| · cθ1...+ xp3 · sθ12 − |O3O4|cθ124 · |O4CM4x|

j2,2 = cθ124 · |O4CM4x| − sθ124 · |O4CM4y|+ |O2O3,min| · sθ12 + xp3 · sθ12 − |O3O4|cθ124

j2,3 = −cθ12

j2,4 = cθ124 · |O4CM4x| − sθ124 · |O4CM4y| − sθ12 · |O2O3min + xp3|+ sθ12 · |O2O3min|+ xp3 · sθ12 − |O3O4| · cθ124

j3,1 = 0

j3,2 = 0

j3,3 = 0

j3,4 = 0

The angular Jacobian for the back is much easier calculated as the rotation axis for allrevolute joint are parallel. The lower half angular Jacobian for link 1 through 4 are givenas,

Jω1 =

0 0 0 00 0 0 01 0 0 0

(A.45)

133

Jω2 = Jω3 =

0 0 0 00 0 0 01 1 0 0

(A.46)

Jω4 =

0 0 0 00 0 0 01 1 0 1

(A.47)

The total Jacobian, e.g

J1 =

[Jv1Jω1

](A.48)

are used throughout the calculation of the dynamics of the back manipulator inChapter 2.1.2 on page 14. Based on the Jacobians of the manipulator it is possible tocalculate the D, C and G, which can be found on the enclosed CD

134

Appendix BValve Parameters

ContentsB.1. General Valve Equations . . . . . . . . . . . . . . . . . . . . . . . 135

B.1.1. Parameters for the 4WRTE Valve . . . . . . . . . . . . . . . . . 136B.1.2. 4WRKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.1.3. 4WREE6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.1.4. 4WREE10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

This appendix describes how the valve parameters for each of the four valves used forbackhoe are modeled.

The flow-regulating valves used for the backhoe loader are modeled in accordance withequation (B.1)and (B.2) [Andersen]

B.1. General Valve Equations

QA = KA · xv · sign(∆P ) ·√

∆P (B.1)

QB = KB · xv · sign(∆P ) ·√

∆P = KA · σ · xv · sign(∆P ) ·√

∆P (B.2)

where:

QA Is the flow through valve port A [m3

s ]

QB Is the flow through valve port B [m3

s ]

KA Is the flow gain coefficient for orifice A [ m3

s√Pa%

]

KB Is the flow gain coefficient for orifice B [ m3

s√Pa%

]

σ Is the ratio of the flow gain coefficient for orifice A and B, KBKA

·xv Is the spool position relative to the maximum spool stroke [%]∆P Is the pressure differential across the orifice in consideration [Pa]

135

The spool position xv can be described as

xv = Uref ·G(s) (B.3)(B.4)

where:

Uref Is a control input [%]G(s) Is a transfer function from input voltage to spool position [·]

For all four valves the transfer function G(s) has a unity gain resulting in xv = Uref insteady state. The flow gain coefficient KA is thus described as

KA =Q√

∆P · Uref(B.5)

The data sheet for each of the four valves specifies the frequency characteristics from inputvoltage to spool position when the A orifice is opened to the pump-side. This relationshipis rather non-linear and is thus plotted for three different inputs in the data sheet. Sincethe valves will be used in a closed loop context, only the 10% and 25% input curves willbe used to approximate a transfer function for the spool dynamics.

The facing pages provides the approximate values for G(s) and cv, respectively, for all fourvalves.

B.1.1. Parameters for the 4WRTE Valve

The 4WRTE 10 V1-100L-4X/6EG24EK31/F1M valve (from now on 4WRTE-valve) isa pilot operated 2-stage high response directional control valve with electrical positionfeedback of the main spool and integrated electronics. [4WRTE] The spool symbol isshown in Figure B.1

a 0 b a 0 b

A B

P T

2/24 Bosch Rexroth AG Hydraulics 4WRTE RE 29083/09.06

Ordering details

Electrically actuated 2-stage high response valve of 4-way design with integra-ted electronics (OBE)

Nominal size �0 = 10 Nominal size �6 = 16 Nominal size 25 = 25 Nominal size 27 = 27 Nominal size 32 = 32 Nominal size 35 = 35

Spool symbols

= E = E1-

= W6- = W8-

= V = V1-

= Q2-

With spool symbols E�-, W8-, V�-:

P → A :

P → B:

qVmax

qV/2

B → T:

A → T:

qV/2

qVmax

Note:With spools symbols W6-, W8- there is, in the neutral position, a connection from A to T and B to T with approx. 2 % of the relevant nominal cross-section.

Further details in clear text

M = 5) NBR-seals4) Electronic interfaces

A1 = Com./act. value ±�0 V

F1 = Com./act. value 4 to 20 mA

Electrical connectionsK31 = Without plug-in connector

with component plut to DIN EN �7520�-804

Plug-in connector – separate order, see page 7

Pilot oil supply and drain No code = External pilot oil supply,

external pilot oil drain E = Internal pilot oil supply,

external pilot oil drainT = External pilot oil supply,

internal pilot oil drainET = Internal pilot oil supply,

internal pilot oil drain

Supply voltageG24 = +24 V DC

6E = Pilot control valve size 6, Proportional solenoid with removable coil

4X = Component series 40 to 49 (40 to 49: unchanged installation and connection dimensions)

Characteristic curve form L = LinearP = Linear with fine control range

Ordering details: Nominal flow – see pages 11 to 15

25 = 1) or 50 = 2) or 100 = For nominal size �0

125 = 3) or 200 = For nominal size �6

220 = or 350 = For nominal size 25

500 = For nominal size 27



4WRTE 4X 6E G24 K31 M *

�) E, W6-, V, Q2- only available with characteristic curve form L (linear)2) E�-, W8-, V�- only available with characteristic curve form L (linear)3) V�-�25 only available with characteristic curve form L (linear)4) When replacing the component series 3X with component series 4X the

electronic interface is to be defined with A5 (enable signal at Pin C).5) Suitable for mineral oil (HL, HLP) to DIN 5�524

Figure B.1.: Hydraulic symbol for the 4WRTE-valve

The pilot-valve and the main-spool both accounts for some dynamics of the position of themain spool. The data sheet for the 4WRTE-valve states the combined frequency responsefrom electrical input Uref to the desired main spool position xv. The combined frequencyresponse superimposed on an approximated bode plot for the 4WRTE-valve is shown infigure B.2 on the next page. To get a satisfactory match for both the phase and themagnitude response, a first order system followed by a second order system is utilized.

136

+5

0

–5

–10

–15

–20

–25

–301 2 5 10 20 50 100 200

–270

–180

–90

0

Frequency in Hz →

Am

plitu

de rel

atio

nshi

p in

dB

→

Signal ± 0 %Signal ±25 %

Frequency response characteristic curves

Pha

se a

ngle

in ° →

Measured at: – Pilot control valve

Port „X“= 00 bar

– Main valvePort „P“= 0 bar

Figure B.2.: Frequency response for the 4WRTE valve with an approximated bode-plotsuperimposed

The transfer function for the spool dynamics is approximated as:

G(s) = G1(s) ·G2(s) (B.6)

where:

G1(s) = 11

110 · 2 · πs+1

G2(s) =(75 · 2 · π)2

s2 + 2 · 0.6 · 75 · 2 · π · s+ (75 · 2 · π)2

The data sheet specifies the flow as a function of input voltage to be linear. Thecharacteristics for the valve is shown in figure B.3. The plot is shown for a 5 bar pressuredifferential across land A.

100

80

60

40

20

10

0 25 50 75 100

Command value in % →

Flow

in %

→

Figure B.3.: Flow through the 4WRTE-valve as a function of input voltage in %. Theplot is valid in steady state only

137

KA is calculated by equation (B.14) on page 144

KA =Q√

∆P · U%=

100 · 10−3

60√5 · 105

= 2.36 · 10−8 (B.7)

where:

Qnom Is the nominal flow at a 10 bar pressure differential [m3

s ]Q% Is the percentile value of the nominal flow at a 10 bar pressure differential [%]

KB is stated to be half the value of KA [4WRTE]

B.1.2. 4WRKE

The 4WRKE 10 E1-100L-3X/6EG24EK31/F1D3M (from now on 4WRKE-valve) isa pilot operated 2-stage high response directional control valve with electrical positionfeedback of the main spool and integrated electronics. [4WRKE]. The spool symbol isshown in Figure B.4

2/22 Bosch Rexroth AG Industrial Hydraulics 4WRKE RE 29075/08.04

Ordering details

Electrically operated 2-stage proprtional directional valve of 4-way design with integrated electronics

Nominal size 10 = 10Nominal size 16 = 16Nominal size 25 = 25Nominal size 27 = 27Nominal size 32 = 32Nominal size 35 = 35

Symbols

P T

a 0 a 0

A B

P T

0 b 0 b

A B

a 0 b

P T

a 0 b

A B

= E= E1-

= E3-

= W6-= W8-

= R

= R3-

= EA 1)

= W6A

= EB 1)

= W6B

With symbols E1-, W8-:P → A :P → B:

qVmaxqV/2

B → T:A → T:

qV/2qVmax

With symbols R; R3:P → A :P → B :

qVmaxqVmax/2

B → P:A → T:

qV/2qVmax

Note:

With the spools W6, W8 and R3 there is a connection from A to T and B to T in the zero position with approx. 2 % of the applicable nominal cross-section.

1) Examples: Spool in switched position „a“ (P → B) ordering detail ..EA.. or W6A

Spool in switched position „b“ (P → A) ordering detail ..EB.. or W6B2) E and W6 only available with characteristic curve form L (linear) 3) E1 and W8 only available with characteristi curve form L (linear) 4) When replacing the component series 2X with component series 3X the

electrical interface is to be defined with A5 (enable signal at Pin C)5) For compatible pressure fluids see page 6

4WRKE 3X 6E G24 K31 D3 *Further details

in clear text

M = 5) NBR sealsV = 5) FKM seals

D3 = With pressure reducing valve

ZDR 6 DP0-4X/40YM-W80(fixed settting)

InterfacesC1 = Com./act. value ± 10 mAA1 4) = Com./act. value ± 10 VF1 = Com./act. value 4 to 20 mA

Electrical connections K31 = Without plug-in connector

with component plug to DIN EN 175201-804

Plug-in connector – separate order,see page 7

Pilot oil supply and drain No code = External pilot oil supply,

external pilot oil drain E = Internal pilot oil supply,

exernal pilot oil drainET = Internal pilot soil supply,

internal pilot oil drainT = External pilot oil supply,

internal pilot oil drain

Supply voltageG24 = + 24 V DC

6E = Proportional solenoid with removable coil

3X = Component series 30 to 39(30 to 39: unchanged installation and connection dimensions)

Characteristic curve form L = Linear P = Linear with fine control range

Ordering details: Nominal flow – see pages 10 to 14

25 = 2) or 50 = 3) or 100 = For nominal size 10

125 = 3) or 200 = For nominal size 16

220 = 3) or 350 = For nominal size 25




Figure B.4.: Hydraulic symbol for the 4WRKE-valve

The pilot-valve and the main-spool both accounts for some dynamics in regards to theposition of the main spool. No frequency response is available in the data sheet for the4WRKET but a step response is instead supplied. The data sheet for the 4WRKE-valvestates the combined step response from electrical input Uref to the desired main spoolposition xv. The combined step response superimposed on a simulated step responseshowed in Figure B.5. The valve dynamics is modeled as a second order system witha delay.

138

100

75

50

25

0 10 20 30

0 – 25

0 – 50

0 – 75

0 – 100

Signal change in %

Time in ms →

Strok

e in

% →

Transient function with a step form of electrical input signal

Measured at pS = 100 bar

Figure B.5.: Step response for the 4WRKE valve with a simulated response superim-posed [4WRKE]


G(s) = G1(s) ·G2(s) (B.8)

where:

G1(s) = exp−0.008∗s G2(s) =(75 · 2 · π)2

s2 + 2 · 0.6 · 75 · 2 · π · s+ (75 · 2 · π)2

The data sheet specifies the flow as a function of input voltage to be linear for input signalsgreater than or less to 15%. The region [-15%, 15%] . The characteristics for the valve isshown in figure B.3 on page 137. The plot is shown for a 5 bar pressure differential acrossland A.

139

100

80

60

40

20

10

15 36,25 57,5 78,75 100


Flow

in %

→

Figure B.6.: Flow through the4WRKE-valve as a function of input voltage in %. Theplot is valid in steady state only

Figure B.6 shows that a dead-band of ±15% of the maximum spool travel exists withinthe vale. This will be compensated for by adding a signal of ±15% to the control signalso:

U∗ref = Uref +Uref

|Uref|· 15 (B.9)

where:

U∗ref Is the control signal transmitted to the valve electronics [%]

By doing this the valve can be modeled as linear. The flow coefficient is modeled inaccordance with equation (B.14) on page 144

KA =Q√

∆P · Uref=

Qnom ·Q%

sign(∆P ) ·√

∆P · Uref

=d100·10−3

60

100 ·√

10 · 105· Q%

Uref=Q%

Uref· 2.36 · 10−8 (B.10)

where:Q%

Uref=

100

100− 0.15(B.11)

⇓KA =

20

17· 2.36 · 10−8 = 2.7730 · 10−8 (B.12)

140

KB is stated to be half the value of KA [4WRKE]

B.1.3. 4WREE6

The 4WREE 6 V32-2X/G24K31/F1V (from now on 4WREE6) is a direct operatedproportional directional valve with electrical position feedback and integrated electronics[4WREE]. The spool symbol is shown in Figure B.7.

P T

A B

a 0 b

A B

P T

a 0

2/20 Bosch Rexroth AG Hydraulics 4WRE; 4WREE RE 29061/10.05

Ordering details


Seal materialV = FKM seals,

suitable for mineral oil (HL, HLP) to DIN 51524

Electronic interfaces A1 or F1 For 4WREE

A1 = Command value input ± 10 VDCF1 = Command value input 4 to 20 mANo code = For 4WRE

Electrical connectionsFor 4WRE:

K4 = Without plug-in connector, with component plug

to DIN EN 175301-803Plug-in connector (solenoid, position

transducer). separate order, see page 7

For 4WREE:K31 = Without plug-in connector, with

component plug to DIN EN 175201-804Plug-in connector – separate order,

see page 8

G24 = Power supply voltage 24 VDC

2X = Component series 20 to 29(20 to 29: unchanged installation and

connection dimensions)

Nominal flow at a valve pressure differential ∆p = 10 barNS6

08 = 8 l/min16 = 16 l/min32 = 32 l/min

NS1025 = 25 l/min50 = 50 l/min75 = 75 l/min

4WRE 2X G24 V *

Without integratedelectronics (OBE) = No codeWith integratedelectronics (OBE) = E

Nominal size 6 = 6Nominal size 10 = 10

Spool symbols

= E E1–

= V

= W W1–

= EA

= WA

With symbols E1and W1:P → A: qV max B → T: qV/2P → B: qV/2 A → T: qV max

Note:For spools W and WA there is, in the neutral position, a connection between A to T and B to T with approx. 3 % of the relevant nominal cross-section.

Figure B.7.: Hydraulic symbol for the 4WREE6-valve

An approximate bode plot for the 4WREE6-valve is shown in Figure B.8.

–30

–25

–20

–15

–10

–5

0

5

1 10 100 20020 30 50

–3

–315

–270

–225

–180

–135

–90

–45

0

Frequency response characteristic curves for type 4WREE (measured with HLP46, ϑoil = 40 °C ± 5 °C, ps = 10 bar) NS6

Pha

se a

ngle

in ° →

Signal ± 10 %

Signal ± 25 %

Am

plitu

de rel

atio

nshi

p in

dB

→

Frequency in Hz →

4/3 valve version

Spool symbol „V“

Figure B.8.: Frequency response for the 4WREE6-valve with an approximate bode-plotsuperimposed (red is magnitude, blue is phase)

141

The flow Q through the valve as a function of input signal U% is shown for five differentpressure differentials in Figure B.9.. The curve is superimposed with read data points inorder to restore the data from the data sheet

30

20

10

0 10 20 30 40 50 60 70 80 90 100

40

50

60

4

3

2

1

5

70

80

90

100

110

120

15 20 30 40 50 60 70 80 90 100

Flow

in l/

min

→

32 l/min nominal flow at a 10 bar valve pressure differential

P → A / B → T

or

P → B / A → T

V spool

E and W spools


1 ∆p = 10 bar constant





Figure B.9.: Flow characteristic for the 4WREE10-valve in steady state. Read datapoints have been superimposed in order to restore the data from thedatasheet [4WREE]

In practice this is implemented in the non-linear model by a lookup-table, with pressureand reference voltage as input and flow as output.

B.1.4. 4WREE10

The 4WREE 10 V50-2X/G24K31/F1V valve (from now on 4WREE10-valve) is a directoperated proportional directional valve with electrical position feedback and integratedelectronics. [4WREE]. The spool symbol is shown in Figure B.10

P T

A B

a 0 b

A B

P T

a 0

2/20 Bosch Rexroth AG Hydraulics 4WRE; 4WREE RE 29061/10.05

Ordering details


Seal materialV = FKM seals,

suitable for mineral oil (HL, HLP) to DIN 51524

Electronic interfaces A1 or F1 For 4WREE

A1 = Command value input ± 10 VDCF1 = Command value input 4 to 20 mANo code = For 4WRE

Electrical connectionsFor 4WRE:

K4 = Without plug-in connector, with component plug

to DIN EN 175301-803Plug-in connector (solenoid, position

transducer). separate order, see page 7

For 4WREE:K31 = Without plug-in connector, with

component plug to DIN EN 175201-804Plug-in connector – separate order,

see page 8

G24 = Power supply voltage 24 VDC

2X = Component series 20 to 29(20 to 29: unchanged installation and

connection dimensions)

Nominal flow at a valve pressure differential ∆p = 10 barNS6

08 = 8 l/min16 = 16 l/min32 = 32 l/min

NS1025 = 25 l/min50 = 50 l/min75 = 75 l/min

4WRE 2X G24 V *

Without integratedelectronics (OBE) = No codeWith integratedelectronics (OBE) = E

Nominal size 6 = 6Nominal size 10 = 10

Spool symbols

= E E1–

= V

= W W1–

= EA

= WA

With symbols E1and W1:P → A: qV max B → T: qV/2P → B: qV/2 A → T: qV max

Note:For spools W and WA there is, in the neutral position, a connection between A to T and B to T with approx. 3 % of the relevant nominal cross-section.

Figure B.10.: Hydraulic symbol for the 4WREE10-valve

The data sheet states the frequency response from electrical input U% to the desired mainspool position xv%. The frequency response superimposed with an approximated bodeplot for is shown in Figure B.11 on the next page. To get a satisfactory match of both thephase and the magnitude response, a first order system followed by a second order systemis utilized.

142

–30

–25

–20

–15

–10

–5

0

5

1 10 100

–315

–270

–225

–180

–135

0

20020 30 50

–3

–45

–90

4/3 valve version

Spool symbol „V“

Pha

se a

ngle

in ° →

Signal ± 10 %

Signal ± 25 %

Am

plitu

de rel

atio

nshi

p in

dB

→

Frequency in Hz →

Frequency response characteristic curves for type 4WREE (measured with HLP46, ϑoil = 40 °C ± 5 °C, ps = 10 bar) NS10

Figure B.11.: Frequency response for the 4WREE10-valve with an approximate bode-plot superimposed (red is magnitude, blue is phase


G(s) = G1(s) ·G2(s) (B.13)

where:

G1(s) = 11

30·2·pi s+1, G2(s) = (69·2·π)2

s2+2·0.6·69 g2·π·s+(69·2·π)2

The flow Q through the valve as a function of input signal U% is shown for five differentpressure differentials in Figure B.12 on the following page. The curve is superimposed withread data points in order to restore the data from the data sheet

143

75

50

25

0 10 20 30 40 50 60 70 80 90 100

100

125

150

4

3

2

1

5

10 20 30 40 50 60 70 80 90 100

Flow

in l/

min

→

50 l/min nominal flow at a 10 bar valve pressure differential

P → A / B → T

or

P → B / A → T

V spool

E and W spools







Figure B.12.: Flow characteristic for the 4WREE6-valve in steady state. Read datapoints have been superimposed in order to restore the data from thedatasheet [4WREE]

By equation (B.5) on page 136 the flow gain coefficient KA is calculated as

cv =Q√

∆P · Uref(B.14)

In practice this is implemented in the non-linear model by a lookup-table, with pressureand reference voltage as input and flow as output.

144

Appendix COptimization and Verification of theModel Parameters

ContentsC.1. Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 145

C.1.1. Strategy for the Optimization Scheme . . . . . . . . . . . . . . . 148C.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

C.2.1. Bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.2.2. Dipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152C.2.3. Boom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.2.4. Extender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

During this appendix a scheme for determining the unknown model parameters ispresented. This scheme is based on an optimization algorithm, described in further detailsin section C.1. The final results of the parameter estimation scheme is presented insection C.2 on page 151

C.1. Parameter Estimation

The parameters for the mathematical model derived throughout this thesis, are mainlyobtained through one of the following methods.

• Physical measurements (e.g lenghts of each link)

• Datasheet figures (e.g valve coefficients)

• CAD drawings (e.g inertias of each link)

Determining friction parameters requires another method than one of the three statedabove.

One approach for determining the friction parameters involves the use of an optimizationalgorithm. One way of doing this is by comparing experimental data with model data,

145

utilizing the optimization algorithm to assign the missing parameters (x1, x2...xn) whileminimizing the difference between the model data (fmodel) and the experimental data(fdata). This difference may be squared to ensure a positive difference. Stating this in amore mathematical sense.

Minimize: f(x1, x2, ...xn) =∑(

fmodel(x1, x2, ...xn)− fdata

)2 (C.1)

The task, for determining the model parameters, hence reduces to choosing the rightalgorithm for determining the model parameters x1, x2, ...xn that satisfies ((C.1)).

Numerous of classic optimization algorithms exist, e.g. line search or trust region meth-ods. The classical methods are well suited for solving the problem in ((C.1)) for some localminima, but in general they fail to solve the problem for a global minima if several localminima’s exists. Some variations of the classical methods such as simulated annealing ora probabilistically restarted line search method may be suited for some problems, butoften they require the objective function to be continuous and/or differentiable. Since theobjective function is based on laboratory data this cannot be ensured. For this particularproblem using an evolutionary algorithm (EA) proved to converge towards a global min-ima (or at least a good solution), much faster and no knowledge regarding the derivativeof the objective function was required. [Rasmus K. Ursem, 2003]. A Ph.D dissertation byRasmus K. Ursem at Department of Computer Science, Aarhus University states such analgorithm as follows.

In short, evolutionary algorithms are iterative and stochastic optimization techniquesinspired by concepts from Darwinian evolution theory. An EA simulates an evolutionaryprocess on a population of individuals with the purpose of evolving the best possibleapproximate solution to the optimization problem at hand. In the simulation cycle, threeoperations are typically in play; recombination, mutation, and selection. Recombinationand mutation create new candidate solutions, whereas selection weeds out the candidateswith low fitness, which is evaluated by the objective function. Figure C.1 illustrates theinitialization and the iterative cycle in EAs.

- [Rasmus K. Ursem, 2003]

2 Chapter 1. Introduction

techniques (for a comprehensive survey, see [100]). In this context, the so-calledevolutionary algorithms (EAs) are a particularly promising approach, because thistechnique has shown good and robust performance on a broad range of real-worldproblems, e.g., [74; 104; 133; 52; 29; 113; 105; 78].

1.1 Evolutionary Algorithms

In short, evolutionary algorithms are iterative and stochastic optimization tech-niques inspired by concepts from Darwinian evolution theory. An EA simulates anevolutionary process on a population of individuals with the purpose of evolvingthe best possible approximate solution to the optimization problem at hand. Inthe simulation cycle, three operations are typically in play; recombination, muta-tion, and selection. Recombination and mutation create new candidate solutions,whereas selection weeds out the candidates with low fitness, which is evaluated bythe objective function1. Figure 1.1 illustrates the initialization and the iterativecycle in EAs. Chapter 2 gives an elaborate introduction to EAs.

69

46

2

3

PopulationFitness

Reproduction

SelectionEvaluation

Mutation

Initialization and evaluation

6

4

9

6

2 3

Figure 1.1: Initialization and the iterative cycle in evolutionary algorithms.

Historically, EAs were first suggested in the 1940’ties [51]. However, the found-ing fathers of modern EAs are considered to be Lawrence Fogel (Evolutionary Pro-gramming [53]), Ingo Rechenberg and Hans-Paul Schwefel (Evolution Strategies[113]), and by John Holland (Genetic Algorithms [68]). Several years later, Evo-lutionary Algorithms (EAs) and Evolutionary Computation (EC) were introducedas unifying terms for the forest of optimization techniques inspired by biologicalevolution. For a comprehensive overview, see [10].

1In EAs, the objective function is often referred to as the fitness function.

Figure C.1.: Evolutionary optimization algorithm in general.[Rasmus K. Ursem, 2003]

146

As with the classical approaches, a variety of evolutionary optimization algorithms exist.An algorithm called differential evolution presented by Kenneth Price and Rainer Stornat University of California, Berkeley has been awarded and recognized for its versatilityrobustness for a large number of problems. The algorithm is easy parallizable, so severalcpu kernels on one computer and even several computers can contribute in solving theproblem. This is seen as a huge advantage for the sake of this problem, since ((C.1)) isevaluated in the range of seconds to minutes and the algorithm may require thousandsof function calls to converge to a feasible solution. The algorithm has been proposedby professor at Aalborg University Erik Lund The differential evolution optimizationalgorithm will be used to estimate the friction parameters in the non-linear model.

During this report a number of parameters have been estimated through crude CADdrawings, while other parameters have been obtained through experiments performed inprevious work. Some assumptions are made for the CAD drawings e.g material thickness,the type of material etc. Some of the experiments carried out on the backhoe were made forthe sake of a master thesis made in 2004 [Josefsen, 2004] at Aalborg University. Althoughthe overall objective was similar to the objective of this report some differences applies.For instance.

• The group did not utilize the extender and therefore, the assembly of the extenderand the dipper links were weighed together

• From the report of 2004 it is not always clear how many parts were considered whenspeaking about the link of interest, e.g; Was the cylinder rod connected to the bucketlink contributed to the mass of the bucket or the mass of the extender assembly?

The algorithm is a constrained evolutionary algorithm so a confidence interval for eachparameter of interest must be supplied. Varying the parameters manually indicates a crudeconfidence interval.

Some critical parameters, xn, obtained through the CAD drawings, measurements or fromthe report from 2004 will be included in the optimization algorithm as an element ofuncertainty in the following way

xn = xn ±xn100· αn (C.2)

where:

xn Is the assumed value of the parameter of uncertaintyαn Is the range of deviation in %

The value of an will be assigned a high value if the level of uncertainty is high and a lowvalue if the opposite is the case.

The properties of the bucket and the extender cylinder are considered to contain the highestdegree of uncertainty, mainly due to the unclear mass properties proposed in the reportfrom 2004 and the combined weighing of the dipper- and extender link. The optimizationscheme is utilized to both obtain the friction parameters of the system and to estimatethe mass distribution of the backhoe. The total mass of the backhoe described in thereport from 2004 will be maintained, but the distribution will be varied. The cad drawings

147

and measurements made in this report will serve as a starting point for the optimizationscheme.

C.1.1. Strategy for the Optimization Scheme

Some parameters may be verified and estimated in a simpler system than the entirebackhoe system. Doing so reduces the complexity of the optimization scheme and tendsto converge towards a feasible solution in a faster manner.

The bucket-link will be treated as a separate system and a set of four steps on the bucketwill be carried out to determine the unknown friction parameters and the mechanicalparameters with uncertainty involved.

The parameters associated with the bucket-link, namely the mass m4, the center of mass|O4CM4y| and |O4CM4y| will be included in the optimization algorithm along with thefriction parameters for the bucket link. The parameters used in the optimization schemefor the bucket system is shown in table. C.1

Assumed parameter value xn Unit Variation in % αn

m4 216 kg 20

|O4CM4y| [-0.3083, 0.3917, -0.2569] m [15, 15, 15]

|O4CM4x| [-0.2228, 0.0648, 0.8553] m [15, 15, 15]

Bbucket 25000 kgs 50

FC,bucket 100 N 50

FS,bucket 3000 N 50

c1,bucket 0.01 · 95

Table C.1.: Parameters and variations used for the optimization scheme for the bucketlink

While applying steps to the bucket cylinder the other links were positioned as shown infigure C.2

Figure C.2.: Initial position of the backhoe during the bucket step-test

The dipper and the boom will be treated as another system and a set of four steps onthe dipper and three steps on the boom is carried out. Combining these systems provedpractical as the mass distribution of the links connected to the boom affects the responsefor the boom quite significantly. The parameters associated with this combined system is

148

the mass m2 and the center of mass for each link |O1CM1| and |O2CM2|. As the mass ofthe dipper and extender was previously weighed to 540kg and the bucket was weighed to135kg, the total mass of the dipper, bucket and extender is considered to be fixed. Themass of the extender is thus calculated as

m3 = 540kg −m2 −m4 + 135kg (C.3)

The parameters used in the Differential Evolution algorithm while testing the dipper andboom system is shown in table C.2

Assumed parameter value xn Unit Variation in % αn

m1 474 kg 10

m2 278 kg 15

CM3,y 0.91 m 10

CM2 0.68 m 10

CM1 -1.07 m 10

Bboom 2000 kgs 50

FC,boom 1500 N 50

F S,boom 20000 N 50

c1,boom 0.01 · 95

Bdipper 80000 kgs 50

FC,dipper 141 N 50

F S,dipper 14000 N 50

c1,dipper 0.01 · 95

Table C.2.: Parameters and variations used for the optimization scheme for the dipperlink

While applying steps to the dipper and boom cylinders the other links were positioned asshown in Figure C.3 and C.4

149

Figure C.3.: Initial position of the back-hoe during the boom step-test

Figure C.4.: Initial position of the back-hoe during the dipper step-test

Lastly, the friction parameters for the extender are estimated through four steps onthe extender cylinder. The parameters obtained during the previous two optimizationschemes are used in when applying steps to the extender link The parameters used in theoptimization scheme for the extender is shown in table C.3

Assumed parameter value xn Variation in % αn

Bextender 20000 50

FC,extender 4000 50

F S,extender 1500 50

c1,extender 0.05 95

Table C.3.: Final parameters for the extender link

While applying steps to the extender cylinder the other links were positioned as shown infigure C.5

Figure C.5.: Initial position of the backhoe during the extender step-test

150

C.2. Results

C.2.1. Bucket

After estimating the parameters for the bucket through the differential evolutionoptimization algorithm the parameters settled at.

Parameter Value Unitm4 213 kg

CM4,y [−0.2497, 0.2794, −0.2588] m

CM4,x [−0.2137, 0.0815, −0.6089] m

Bbucket 34133 kgs

FC,bucket 0 N

FS,bucket 4166 N

c1,bucket 0.05 ·

Table C.4.: Final parameters for the bucket link

0 0.5 1 1.5 2 2.5 3 3.5

0

0.05

0.1

0.15

0.2

0.25

0.3

EFstroke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

EFstroke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.50.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

EFstroke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

EFstroke[m

]

Time [s]

MeasuredSimulated

Figure C.6.: Comparison of force terms of Equation ((2.35)) for the boom Extender (left)and Bucket cylinder (right) with the backhoe trajectory shown in Equation((2.36))

151

C.2.2. Dipper

After estimating the parameters for the dipper through the differential evolutionoptimization algorithm the parameters settled at.

Parameter Value Unitm2 312.5 kg

Bdipper 95889 kgs

FC,dipper 0 N

FS,dipper 17700 N

c1,dipper 0.0051 ·

Table C.5.: Final parameter values for the dipper-link

0 0.5 1 1.5 2 2.5 3 3.5 4

0.15

0.2

0.25

0.3

0.35

CD

stro

ke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5 4

0.2

0.25

0.3

0.35

0.4

CD

stro

ke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5 4

0.25

0.3

0.35

0.4

0.45

CD

stro

ke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5 4

0.3

0.35

0.4

0.45

0.5

CD

stro

ke[m

]

Time [s]

MeasuredSimulated


152

C.2.3. Boom

After estimating the parameters for the bucket through the differential evolutionoptimization algorithm the parameters settled at.


Bboom 19722 kgs

FC,boom 2733 N

FS,boom 33332 N

c1,boom 0.0011 ·

Table C.6.: Final parameters for the boom link

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

AB

stro

ke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5 4

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

AB

stro

ke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5 4

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

AB

stro

ke[m

]

Time [s]

MeasuredSimulated


C.2.4. Extender

After estimating the parameters for the extender through the differential evolutionoptimization algorithm the parameters settled at.

153


Bextender 21740 kgs

FC,extender 5600 N

FS,extender 1750 N

c1,extender 0.0101 ·

Table C.7.: Parameters and variations used for the optimization scheme for the bucketlink

0 0.5 1 1.5 2 2.5 3 3.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

O2O

3,stroke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

O2O

3,stroke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

O2O

3,stroke[m

]

Time [s]

MeasuredSimulated

0 0.5 1 1.5 2 2.5 3 3.5

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

O2O

3,stroke[m

]

Time [s]

MeasuredSimulated


154

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Realizing Third Order Sliding Mode Control for a Hydraulic ...projekter.aau.dk/projekter/files/77410482/MCE_1025_2013_Msc.pdf · Realizing Third Order Sliding Mode Control for a Hydraulic