Mechanism and Machine Theory - profdoc.um.ac.irprofdoc.um.ac.ir/articles/a/1069771.pdf · Using this method and concept of WCMJ matrix, the compliance matrix of each ﬂexible body

Mechanism and Machine Theory 126 (2018) 108–140

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Mechanism and Machine Theory

journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Influence of joints flexibility on overall stiffness of a 3- P RUP

compliant parallel manipulator

Amir Rezaei, Alireza Akbarzadeh

∗

Ferdowsi University of Mashhad, Mechanical Engineering Department, Mashhad, Iran

a r t i c l e i n f o

Article history:

Received 23 August 2017

Revised 7 February 2018

Accepted 25 March 2018

Keywords:

Parallel manipulator

Distributed stiffness model

Joints stiffness

Energy method

Stiffness evaluation

Compliance errors

a b s t r a c t

The main aim of this paper is the influence assessment of the one of the most significant

non-geometrical errors called compliance errors on accuracy of a 3- P RUP parallel robot.

In this paper, a detailed and comprehensive study is performed to evaluate the effects of

the joints and bodies flexibility on position accuracy of the robot. For this purpose, a con-

tinuous modeling method based on the energy method and Castigliano’s 2nd theorem is

presented. The capabilities of this method significantly reduce the limiting assumptions to

obtain a highly accurate analytical stiffness model. Using the capabilities of this method,

the compliance errors modeling of flexible bodies with complex geometry will be possible

as well as the bending and torsional moments, shear forces and weight of all robot com-

pliant modules can be considered as continuous loads. Using the concept of Wrench Com-

pliant Module Jacobian, WCMJ, matrix and physical/structural properties matrix of each

module, the compliance matrix of each flexible module is independently obtained. Using a

computational algorithm, a comprehensive study is performed to obtain the contribution

of the joints flexibility on the robot accuracy throughout the workspace. Finally, to evaluate

the compliance errors boundaries throughout the workspace, the concept of General Com-

pliance Error Range, GCER, is presented. The GCER represents a range between maximum

and minimum accuracy of robot relative to the compliance errors.

© 2018 Published by Elsevier Ltd.

1. Introduction

Generally, the accuracy, repeatability and resolution of a robotic manipulator are of three important criteria to evaluate

the manipulators precision. Identification of the effective sources on the robot positioning errors is the first step to improve

the robot’s accuracy indices. Typically, the dimensional errors, assembling errors and joints clearance are significant types of

the geometrical errors as well as the actuators control errors, errors caused by flexibility of robot components, high or low

temperatures, external and internal noise in robot control system, measurement errors, errors caused by abrasion, wearing

and friction are significant types of the non-geometrical errors [1–4] .

In view of the importance and effectiveness of error sources, in robotic machine tools with high rigidity, the compliance

errors may be neglected. However, to achieve higher speed and acceleration, weight to stiffness ratio of the robots should

be reduced. Hence, in recent years, the applications of parallel robots as CNC machine tools are developed. Although, the

stiffness of parallel robots are inherently higher than serial ones, yet, the use of parallel robots as high-speed CNC machine

tools require to improve weight to stiffness ratio of these types of robots. When a parallel robot is utilized as high-speed

∗ Corresponding author.

E-mail addresses: [email protected] (A. Rezaei), [email protected] (A. Akbarzadeh).

https://doi.org/10.1016/j.mechmachtheory.2018.03.011

0094-114X/© 2018 Published by Elsevier Ltd.


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A. Rezaei, A. Akbarzadeh / Mechanism and Machine Theory 126 (2018) 108–140 109

Nomenclature

λi , ϕi , θ i rotation angles of the i th passive R- and U-joints about their z-, y- and x-axes

in local coordinate frames {T 1i }, {T 2i } and {T 3i }.

θ , ϕ, λ Euler angles about the x-, y- and z-axes of Moving Star, MS. i j R rotation matrix to transfer a vector defined in { j } to { i }.

U total strain energy of the manipulator.

U bodies , U joints strain energy of all compliant bodies and joints of the robot.

U MS , U LR , U BS , U AJ , U PJ strain energy of MS, LRs, ball screws, active and passive joints.

w applied external wrench, w = { f T ext m

T ext } T .

f ext , m ext external force and moment vectors applied to the end-effector.

K, C overall stiffness matrix and overall compliance matrix of the robot.

δs compliance errors vector of the end-effector due to flexibility of robot’s mod-

ules.

δχ, δψ translation and rotation compliance errors vectors of the end-effector.

δs bodies , δs joints virtual compliance errors vectors due to flexibility of compliant bodies and

joints.

δs MS , δs LR , δs BS , δs M

, δs PJ virtual compliance errors vectors due to flexibility of MS, LRs, ball screws,

motors and passive joints.

δχbodies , δψ bodies & δχjoints , δψ joints virtual translation and rotation compliance errors vectors due to flexibility of

compliant bodies and joints.

δχMS , δψ MS & δχLR , δψ LR & δs BS , δψ BS virtual translation and rotation compliance errors vectors due to flexibility of

the MS, LRs and ball screws.

δχAJ , δψ AJ & δχPJ , δψ PJ virtual translation and rotation compliance errors vectors due to flexibility of

the active and passive joints.

C MS , C LR , C BS , C AJ , C PJ compliance matrices of MS, LRs, ball screws, active and passive joints.

f MS , f LR , m LR overall internal reaction forces/torques vectors of MS and LRs.

f R θP , f R ϕ , f R λ, m R λ overall internal reaction forces/torques vectors of R θ P- and R ϕ- and R λ-joints.

f BS , f Nut , f SB , τM

, f belt overall internal reaction forces/torques vectors of ball screws, ball nuts, sup-

port bearings, resistant torque in motors and tensile forces in the timing

belts.

J w

MS Wrench Jacobian matrix of the compliant MS, WCMJ MS

J fw

LR , J mw

LR Wrench Jacobian matrices of the compliant LRs, WCMJ LR

J w

R θP , J w

R ϕ Wrench Jacobian matrix of compliant R θ P- and R ϕ-joints, WCMJ R θP and

WCMJ R ϕ

J fw

R λ, J mw

R λWrench Jacobian matrix of compliant R λ-joints, WCMJ R λ

J w

BS , J w

Nut , J w

SB , J w

M

, J w

belt Wrench Jacobian matrix of compliant ball screws, WCMJ BS , ball nuts,

WCMJ Nut , support bearings, WCMJ SB , motors, WCMJ M

and timing belt,

WCMJ belt

precise CNC machine tool, stiffness of the robot is considered one of the key design features to improve the accuracy of these

robots [5–10] . From the perspective of error compensation, the compliant errors are the compensable errors and the efficient

methods such as adjusting the actuators or controller inputs as well as direct modification of geometrical parameters of

robot are employed to compensate these errors.

The importance of structural stiffness of the robot on its accuracy has led to widespread researches in this area. In

general, structural stiffness modeling methods of robotic systems can be categorized into three approaches called analytical,

non-analytical and semi-analytical methods. These methods are categorized as illustrated in Fig. 1 .

The analytical methods can be categorized into two approaches such as lumped [11–14] and distributed [5,6,15,16] mod-

eling methods. It is noteworthy that unlike the lumped modeling method, the capabilities of distributed model significantly

reduce the limiting assumptions for calculation of the overall stiffness matrix of robotic system. Furthermore, the Finite

Element Analysis, FEA, method [17–19] can be referred to as a non-analytical method. In the FEA method, all compliant

modules including compliant bodies and compliant joints can be modeled with their true shapes and dimensions. Yet, in

spite of the high accuracy of the FEA to obtain the stiffness model of the robot, this method is a time consuming method

due to re-meshing the finite element model of robot structure and re-computing for each robot configuration throughout its

workspace [10,19,20] .

The methods such as Matrix Structural Analysis, MSA, [21–23] , Virtual Joints Modeling method, VJM, and combined meth-

ods based on FEA [2,24] can be referred to as semi-analytic methods. The MSA method is based on the FEA, with this differ-

ence that the MSA method operates with large compliant elements such as beams, cables, etc. , to reduce the computational

time but the FEA uses small finite elements to design the robot modules [2,25] . This method can lead to obtain an analytical

110 A. Rezaei, A. Akbarzadeh / Mechanism and Machine Theory 126 (2018) 108–140

Fig. 1. The robot structural stiffness analysis methods.

expression for stiffness matrix [26] . In the semi-analytical methods which are based on the FEA, the stiffness matrices of

the robot’s compliant modules which have a complex geometry are usually approximated using FEA-based virtual experi-

ments by a commercial FEA software [27,28] . Klimchik et al. [2,28–31] utilize a FEA-based semi-analytical method and VJM

method to obtain the stiffness matrix of compliant parallel robots considering the passive joints flexibility effects. The VJM

method presents a lumped model of the parallel robot using the localized 6-DOF virtual springs identified using the FEA-

based method to describe the stiffness of compliant bodies and joints. In this method, the links are treated as rigid-bodies

while the joints are considered as flexible modules. In other words, effects of all flexible modules are accumulated in the

joints.

Furthermore, in the design process to select the maximum static load carrying capacity and selecting physical and dimen-

sional specifications of the robot’s components, the stiffness indices can be considered as one of the main criteria to ensure

that the optimal parameters are selected. Since, the stiffness of a robot highly depends on the configuration of robot on its

workspace, selection of standard criteria to evaluate the robot’s stiffness throughout the workspace is one of the methods

to improve the robot performance. Since the stiffness of a robot is limited between maximum and minimum eigenvalues of

its stiffness matrix, to evaluate the robot stiffness throughout the workspace and to obtain the effect of altering the robot

geometrical parameters, the eigenvalues of the stiffness matrix and Kinematic Stiffness Index, KSI, are utilized [ 9,12,32 , 33 ].

In industrial applications, improvement of weight to stiffness ratio of a robot is more demanded. This improves the dynamic

performance of the robot. Courteille et al. [34] investigated on the design optimization of robotic manipulators with respect

to multiple global stiffness indices using a multi-objective genetic algorithm. Also, Zhang [35] utilized two global kineto-

static performance indices called the mean value and the standard deviation of the trace of the generalized stiffness matrix

to optimize the global stiffness matrix of the parallel robots.

This paper is proposed a detailed and comprehensive study using a continuous modeling method based on the energy

method and Castigliano’s 2nd theorem to evaluate the effects of joints and bodies flexibility on position accuracy of a 3-

P RUP parallel manipulator with a specific architecture which is made in FUM Robotics Research Center . Using this method

and concept of WCMJ matrix, the compliance matrix of each flexible body and joint is independently obtained as analytical

expressions. Using the capabilities of this method, the limiting assumptions in the robot’s stiffness modeling process are

significantly reduced. Therefore, this method can be provided a more accurate analytical stiffness model. Using this ana-

lytical model and a computational algorithm, the concept of General Compliance Error Range, GCER, is presented and the

compliance errors boundaries throughout the robot’s workspace are evaluated.

This paper is organized as follows: In Section 2 , the inverse kinematics, InvKin, of the 3- P RUP robot is solved. In Section 3 ,

a continuous stiffness modeling method based on Castigliano’s theorem type II is proposed. In Section 4 , static force analysis

is investigated for each robot’ compliant module i.e. compliant bodies and compliant joints and the concept of Wrench

Compliant Module Jacobian matrix is presented. In Section 5 , the overall compliance matrix of the robot is obtained. In

Section 6 , physical and structural parameters of the robot are identified as well as the passive joints stiffness coefficients

are determined. In Section 7 , the results of analytical model are compared with the results of FEA model. Finally, using the

proposed analytical method, a comprehensive assessment of the robot’s rigidity throughout its workspace is presented.

2. Kinematic model and structural description

The 3- P RUP parallel robot is a symmetric spatial parallel manipulator with three coupled mixed-type DOFs, R2T1, (See

[36,37] for more detail). The robot is composed of:

- Moving platform like a symmetric tripetalous star which is called Moving Star (MS).

- Various tools can be installed to center of the moving star.

- Three identical P RUP legs (1st P-joint in each kinematic chain is an active joint).

- Top and bottom fixed platforms which are connected together by six guide rods.


Fig. 2. Physical model of the 3- P RUP parallel robot.

Fig. 3. A closed-loop vector for i th 3- P RUP leg.

- Each leg is actuated by a servo motor, a gearbox and a ball screw assembly.

The physical model of a 3- P RUP spatial parallel manipulator is illustrated in Fig. 2 and the essential vectors to solve the

kinematic of robot are illustrated in Fig. 3 . These vectors are defined as follows:

- Three position vectors a i locate corners of the fixed base, A i , in {B}.

- Three position vectors q

ac i

specify length of linear rods (LR).

- Three position vector T b i connects point P to the i th i th passive U-joint, U i , in {T}.

- The position vector of end-effector, point P, is given by vector p = { x P y P z P } T . The basic and local coordinate frames, CFs, of the robot are illustrated in Fig. 4 . These frames can be stated as

- Fixed CF {B} is embedded at point O of top fixed triangle �A 1 A 2 A 3 .

- Moving CF {T} is attached to center of the MS, at point P.

- Moving local CF {T 1 i } is attached to rigid body B 1 i and coincident to center of λi R-joint.

- Moving local CF {T } is attached to rigid body B and coincident to center of ϕ R-joint.
2 i 2 i i


Fig. 4. Local CFs and unit vectors for the i th kinematic chain.

- Moving local CF {T 3 i } is attached to i th MS’ branch and coincident to center of θ i R-joint.

In this paper, leading superscripts represent the CFs in which the vectors are referenced. Henceforth, for brevity, the

superscript “B ” denoting CF {B} in which vectors are defined in, is eliminated. As shown in Fig. 4 , λi , ϕi and θ i are the

rotation angles of i th i th passive R- and U-joints about their z-, y- and x-axis in local CFs {T 1 i }, {T 2 i } and {T 3 i }, respectively.

Also, e λi , e ϕi and e θ i represent the unit vectors along rotation axis of R-joints of λi , ϕi and θ i , respectively. Furthermore, unit

vectors e a i , e q i and e b i are defined along the vectors a i , q

ac i

and b i , respectively. Consider Fig. 3 . Three closed vector-loop

equations can be written as

a i e a i + q

ac i e q i = b i e b i + p i = 1 , 2 , 3 (1)

where e b i =

B T

R

T e b i and e q i = { 0 0 1 } T . As illustrated in Fig. 4 , it yields

e a i =

T e b i = { cos ( αi ) sin ( αi ) 0 } T , for αi = 2 ( i − 1 ) π/ 3 i = 1 , 2 , 3 (2)

In this paper, the non-pure coupled mixed-type mode Z θϕ is considered as a principle operational mode for the 3- P RUP

robot (see [36] for more details). Depending on the selected operational mode Z θϕ, the relationship between inputs and

outputs of the inverse kinematic problem can be stated as

Zθϕ mode : Inputs : z p , θ, ϕ and { q

ac 1 , q

ac 2 , q

ac 3 , x p , y p , λ, b 1 , b 2 , b 3 } = f { z p , θ, ϕ } (3)

Upon solving Eq. (3) , kinematic values of all dependent variables of the MS, XY λ, are obtained [36] . Therefore, the rotation

matrix B T

R as well as the rotation values of passive R- and U-joints can be calculated (see Appendix A ).

3. Stiffness modeling of the 3- P RUP parallel manipulator

The existing limitations in lumped stiffness modeling methods require a series of simplifying assumptions used in the

robot stiffness modeling. These simplifications have a significant effect on the accuracy of the analytical model. In this paper,

a continuous modeling method based on calculation of the strain energy of robot’s compliant modules called Castigliano’s

2nd theorem is used. Unlike traditional methods, the capabilities of this method significantly reduce the limiting assump-

tions to obtain a more accurate analytical model of robot’s stiffness. However, due to complexity in the geometry of the

robot’s components, the designers usually use any simplifying assumption to obtain the analytical stiffness model. There-

fore, it is not easy to find an exact analytical stiffness model without using simplifying assumptions. Although, the proposed

method does not essentially require simplifying assumptions, but to reduce the calculating process, the following limiting

assumptions are considered

- Weights of all compliant modules are negligible.

- All joints are frictionless.

- Strain energy due to shear forces is negligible compared to the strain energies due to bending and torsional moments.

- Axial and radial stiffness of bearings are approximated by linear longitudinal springs.

- Torsional stiffness of bearings is approximated by linear torsion springs.

Given that the range of changes in position of the MS’ center of gravity approximately remains about the center of

the robot’s workspace as well as the actuators have only linear motion in vertical direction, it can be concluded that the

effect of MS and LRs gravity forces should be considered as a constant vertical force on each LR. In other words, the effect


Fig. 5. Free body diagram (FBD) for the MS.

of gravitational forces of the robot’s modules throughout the workspace can be approximated by constant forces in the

vertical direction. For this reason, the effect of robot’s components weight can be ignored. Therefore, the proposed list of

assumptions is very realistic and should closely approximate the stiffness of the actual robotic manipulator.

The aim of this paper is obtaining the overall compliance matrix of the 3- P RUP parallel manipulator. The overall compli-

ance matrix contains compliance matrices of both compliant bodies and joints. The MS and LRs are considered as compliant

bodies. Additionally, the influence of all passive and active joints’ flexibilities on overall compliance matrix of the robot is

investigated. The overall compliance of a manipulator is dependent on configuration, joints stiffness properties as well as the

physical and geometrical properties of the robot modules. Furthermore, the relationship between applied external wrench

and the internal force/moment in robot’s compliant modules as well as the reaction forces in all passive and active joints

can be expressed as functions of robot’s configuration. Therefore, Jacobian matrices to express these relations are introduced

called Wrench Compliant Module Jacobian matrices (WCMJ) . These matrices allow us to calculate internal force/moment of

compliant modules as functions of applied external wrench.

4. Static force analysis of the compliant modules

In this section, the WCMJ matrix for each robot’s compliant module is obtained.

4.1. Obtaining internal forces and moments of the MS

As illustrated in Fig. 5 , vector f MS contains all reaction forces between passive P-joints and branches of the MS. For the

i th i th MS branch, the reaction forces are defined in its local CF {T 3 i }. The local CF {T 3 i } is attached to the i th i th MS branch

and its x-axis is along the unit vector T3 i e b i and its z-axis is along z-axis of moving CF {T}.

The relation between f MS and applied external wrench on the MS, w , in matrix form can be written as follow

f MS = J w

MS w (4)

where

f MS =

{f v 31 MS 1 f w 31

MS 1 f v 32 MS 2 f w 32

MS 2 f v 33 MS 3 f w 33

MS 3

}T (5)

w =

{f T ext m

T ext

}T , f ext = { f x f y f z } T m ext = { m x m y m z } T (6)

in which, f v 3 i MS i

and f w 3 i MS i

are values of reaction forces in i th i th passive P-joint and f ext and m ext denote force and moment

vectors defined in {B}. Additionally, J w

MS is called Wrench Compliant Module Jacobian matrix for the MS, WCMJ MS , which

maps w to f MS . As illustrated in Fig. 6 and using WCMJ MS , the internal bending moment in i th i th branch of MS, T3 i τv MS i

and

T3 i τw

MS i , and subsequently the strain energy of the MS can be calculated as function of external wrench. The related

calculations can be found in Appendix B in detail.


Fig. 6. Section view and FBD of the i th branch of MS.

4.2. Obtaining internal forces of the passive RUP joints

The specific passive merged joints RUP are depicted in Fig. 4 . The RUP passive joints set in this parallel robot acts as

a UC passive joints set. It is noteworthy that a multi-DOF joint like a spherical joint can be considered as three one-DOF

revolute joints linked with two movable rigid bodies having zero mass and dimension. Therefore, the RU passive joints

set can be considered as a spherical joint. In RU passive joints set, similar to an S-joint, three rotation axes of R-joints

intersect at one point. Nevertheless, unlike the S-joints, three R-joints’ rotation axes of RU passive joints set do not remain

always perpendicular to each other. It means that, unlike an S-joint, a singular configuration can occur in combined RU

passive joints set. However, because of the specific configuration of the 3- P RUP robot, any types of singularities cannot

occur throughout the robot’s workspace.

As shown in Fig. 3 , the combination of RUP passive joints contains three passive R-joints of λi , ϕi and θ i called R λi , R ϕi

and R θ i -joints and a passive P-joint, P i . For each robot’s leg, the passive P-joint and R θ i -joint are combined as a cylindrical

joint called R θ i P i . In the following, the internal reaction forces of the passive RUP-joints are obtained.

4.2.1. Obtaining internal forces of the passive R θ i P i and R ϕi -joints

The R θ i P i -joint and R ϕi -joint are considered as compliant modules. To calculate the strain energy of R θ i P i - and R ϕi -joints,

their reaction forces should be calculated as functions of external wrench. As shown in Fig. 7 , the reaction forces of i th i th

R θ i P i -joint and i th R ϕi -joint should be obtained in directions of radial and axial deflections of the bearings in the local CF

{T 2 i } as functions of external wrench. Using vector f MS and Eq. (4) , the overall vector of internal reaction forces in combined

passive R θ P-joints and passive R ϕ-joints, f R θP and f R ϕ , can be obtained as function of external wrench in matrix form as

f R θP = J w

R θP w f R ϕ = J w

R ϕ w (7)

In which,

f R θP =

{f v 21 R θP1 f w 21

R θP1 f v 22 R θP2 f w 22

R θP2 f v 23 R θP3 f w 23

R θP3

}T

f R ϕ =

{f v 21 R ϕ1 f w 21

R ϕ1 f v 22 R ϕ2 f w 22

R ϕ2 f v 23 R ϕ3 f w 23

R ϕ3

}T (8)

and matrices J w

R θP and J w

R ϕ are called Wrench Compliant Module Jacobian matrix of passive R θ P- and R ϕ-joints, WCMJ R θP and

WCMJ R ϕ , respectively. These matrices map 6 × 1 vector w to reaction forces of three compliant passive R θ i P i -joints and three

compliant passive R ϕi -joints, respectively. The related calculations can be found in Appendix B in detail.

4.2.2. Obtaining internal forces and moments of the passive R λi -joint

To obtain the strain energy of the compliant passive R λi -joint, its internal reaction forces/moments should be calculated

as a function of external wrench. In R λi -joint, a tilting deflection can occur due to existing a tilting moment. The free body

diagram of this joint is shown in Fig. 8 . The reaction forces/moments in R λi -joint are defined in its local CF {T 1 i }. Using the

vector f R ϕ , the overall vector of internal reaction forces and moments in passive R λ-joints, f R λ and m R λ, can be obtained as


Fig. 7. FBD of i th compliant R θ P- and R ϕ-joint.

Fig. 8. FBD of the i th compliant R ϕ- and R λ-joint.

functions of external wrench in matrix form as

f R λ = J fw

R λw

m R λ = J mw

R λw

(9)

in which

f R λ =

{f u 11 R λ1

f v 11 R λ1

f w 11 R λ1

f u 12 R λ2

f v 12 R λ2

f w 12 R λ2

f u 13 R λ3

f v 13 R λ3

f w 13 R λ3

}T

m R λ =

{m

u 11 R λ1

m

v 11 R λ1

m

u 12 R λ2

m

v 12 R λ2

m

u 13 R λ3

m

v 13 R λ3

}T (10)


Fig. 9. FBD of the i th LR.

and matrices J fw

R λand J mw

R λare 9 × 6 and 6 × 6 matrices called Wrench Compliant Module Force and Moment Jacobian ma-

trices of passive R λ-joints, WCMFJ R λ and WCMMJ R λ, respectively. These matrices map 6 × 1 vector w to reaction forces and

moments for three compliant passive R λ-joints. The related calculations can be found in Appendix B in detail.

4.3. Obtaining internal forces and moments of the LRs

As illustrated in Fig. 9 , to calculate the strain energy due to flexibility of the LRs, the reaction forces/moments in R λi -joint

defined in {T 1i } should be transferred to the local fixed CF {B}. Next, the internal bending moments and internal axial force

in LRs should be calculated as functions of external wrench.

Note that the local fixed CF {B}, attached to R λi -joint, all have the same direction as the base CF {B}. Therefore, the overall

vector of reaction forces and moments between passive R λ-joints and the LRs can be stated as

f LR = J fw

LR

w m LR = J mw

LR w (11)

in which

f LR =

{f x LR 1 f y

LR 1 f z LR 1 f x LR 2 f y

LR 2 f z LR 2 f x LR 3 f y

LR 3 f z LR 3

}T

m LR =

{m

x LR 1 m

y LR 1

m

x LR 2 m

y LR 2

m

x LR 3 m

y LR 3

}T (12)

and matrices J fw

LR and J mw

LR are 9 × 6 and 6 × 6 matrices called Wrench Compliant Module Force and Moment Jacobian matrices

of the LRs, WCMFJ LR , and WCMMJ LR , respectively. These matrices map 6 × 1 vector w to reaction forces and moments of

three LRs. Also, as illustrated in Fig. 10 and using WCMFJ LR and WCMMJ LR , the internal force/moment in i th i th LR, p

ax LR i

, τ x LR i

and τ y LR i

, and subsequently the strain energy of LRs can be calculated as function of external wrench. The related calculations

can be found in Appendix B in detail.

4.4. Obtaining internal forces of the screw shaft, ball nut and support bearings of feed screw system

As illustrated in Fig. 11 , the strain energy of ball screws is a function of internal axial forces in ball screws while the

strain energy due to internal torsional and flexural moments in the ball screws is negligible. Therefore, using the axial force

acting on LRs, the overall vector of reaction forces acting on three ball screws can be obtained in matrix form as

f BS = J w

BS w (13)

where f BS = { f BS 1 f BS 2 f BS 3 } T is a 3 × 1 vector and J w

BS is a 3 × 6 matrix called Wrench Compliant Module Jacobian matrix

of the ball screws, WCMJ BS , which maps 6 × 1 vector w to corresponding reaction forces in three compliant ball screws.

Subsequently, the overall vector of reaction forces acting on three ball nuts and support bearings can be stated in matrix

form as follows

f Nut = J w

Nut w f SB = J w

SB w (14)


Fig. 10. FBD and section view of the i th LR.

Fig. 11. FBD and section view of the i th ball screw and ball nut.

where f Nut = { f Nut 1 f Nut 2 f Nut 3 } T and f SB = { f SB 1 f SB 2 f SB 3 } T are 3 × 1 vectors and J w

Nut and J w

SB are 3 × 6 matrices

called Wrench Compliant Module Jacobian matrix of the ball nuts and support bearings, WCMJ Nut and WCMJ SB , respectively.

These matrices map 6 × 1 vector w to corresponding reaction forces on three compliant ball nuts and support bearings,

respectively. The related calculations can be found in Appendix B in detail.

4.5. Obtaining internal forces of the motor-gearbox assemblies

As illustrated in Fig. 12 , the strain energy due to flexibility of motor-gearbox assemblies can be obtained as a function of

internal motor and ball screw torques.

Therefore, using the axial force acting on LRs, the overall vector of resistant torques in the servo motors can be stated in

matrix form as

τM

= J w

M

w (15)


Fig. 12. FBD of the i th motor-gearbox and ball screw assembly.

where τM

= { τM1 τM2 τM3 } T is a 3 × 1 vector and J w

M

is a 3 × 6 matrix called Wrench Compliant Module Jacobian matrix

of the motors, WCMJ M

, which maps vector w to corresponding reaction torques in three compliant motors. Also, the overall

vector of tensile forces in the timing belts can be obtained as

f belt = J w

belt w (16)

where f belt = { f belt 1 f belt 2 f belt 3 } T is a 3 × 1 vector and J w

belt is a 3 × 6 matrix called Wrench Compliant Module Jacobian

matrix of the timing belts, WCMJ belt , which maps vector w to corresponding tensile forces in the timing belts. The related

calculations can be found in Appendix B in detail.

5. Obtaining the overall compliance matrix of the manipulator

In this section, the strain energies of all compliant modules of the 3- P RUP robot are calculated. The steps of structural

stiffness modeling of a manipulator using the Castigliano’s 2nd theorem are presented as a process diagram in Fig. 13 .

As stated earlier, the robot is comprised of two set of compliant modules i.e. compliant bodies and compliant joints.

Therefore, total strain energy of the robot can be written as

U = U bodies + U joints (17)

where U bodies and U joints are the strain energy of all compliant bodies and joints, respectively. The MS and three LRs are

considered as compliant bodies. Additionally, the compliant joints contain all motors-gearbox assemblies, feed screw systems

and all passive R-, U- and P-joints. Therefore, we can write

U bodies = U MS + U LR + U BS U joints = U PJ + U AJ (18)

where U MS , U LR and U BS are strain energies of the MS, LRs and ball screws, respectively. Also, U PJ and U AJ are strain energies

of all passive joints and motor-gearbox assemblies, respectively. The infinitesimal twist vector of the robot’s end-effector

(Compliance errors twist vector due to flexibility of robot’s modules) can be defined as follow

δs =

{δx δy δz δψ x δψ y δψ z

}T =

{δχT δψ

T }T

(19)

where δψ k is the rotational compliance errors of the end-effector about k -axis of CF {B} due to applied external wrench.

The translational and rotational compliance error vectors, δχ and δψ, can obtain using Castigliano’s theorem as follows

δχ =

∂U

∂ f ext =

∂ U bodies

∂ f ext +

∂ U joints

∂ f ext = δχbodies + δχjoints

δψ =

∂U

∂ m ext =

∂ U bodies

∂ m ext +

∂ U joints

∂ m ext = δψ bodies + δψ joints (20)

in which

δχbodies =

∂ U MS

∂ f +

∂ U LR

∂ f +

∂ U BS

∂ f = δχMS + δχLR + δχBS

ext ext ext


Fig. 13. Steps of the structural stiffness modeling a manipulator using the Castigliano’s 2nd theorem.

δψ bodies =

∂ U MS

∂ m ext +

∂ U LR

∂ m ext +

∂ U BS

∂ m ext = δψ MS + δψ LR + δψ BS

δχjoints =

∂ U PJ

∂ f ext +

∂ U AJ

∂ f ext = δχPJ + δχAJ

δψ joints =

∂ U PJ

∂ m ext +

∂ U AJ

∂ m ext = δψ PJ + δψ AJ (21)

Therefore, the virtual compliance error vectors due to flexibility of the robot’s modules can be written as function of

applied external wrench to the end-effector, w , as follows

δs MS =

{δχT

MS δψ

T MS

}T =

{δχMSx δχMSy δχMSz δψ MSx δψ MSy δψ MSz

}T = C MS w

δs LR =

{δχT

LR δψ

T LR

}T =

{δχLRx δχLRy δχLRz δψ LRx δψ LRy δψ LRz

}T = C LR w

δs BS =

{δχT

BS δψ

T BS

}T =

{δχBSx δχBSy δχBSz δψ BSx δψ BSy δψ BSz

}T = C BS w

δs PJ =

{δχT

PJ δψ

T PJ

}T =

{δχPJx δχPJy δχPJz δψ PJx δψ PJy δψ PJz

}T = C PJ w

δs AJ =

{δχT

AJ δψ

T AJ

}T =

{δχAJx δχAJy δχAJz δψ AJx δψ AJy δψ AJz

}T = C AJ w (22)

where C MS , C LR , C BS , C PJ and C AJ are compliance matrices of the MS, LRs, ball screws, all passive joints and motor-gearbox and

ball screw assemblies, respectively. Therefore, using Eqs. (19) –(21) , the compliance errors twist vector of the end-effector,

δs , can be written as

δs = δs bodies + δs joints = δs MS + δs LR + δs BS + δs PJ + δs AJ (23)

Substituting Eq. (22) into Eq. (23) yield

δs =

(C MS + C LR + C BS + C PJ + C AJ

)w = C w (24)

where C is a 6 × 6 matrix called the overall compliance matrix of 3- P RUP manipulator. Therefore, the overall stiffness matrix

of the robot can be obtained using the Hooke’s law as

w = K δs (25)

Note that vectors w and δs are defined in base CF {B}. Additionally, 6 × 6 matrix K is overall stiffness matrix of the

robot’s structure. Comparing Eqs. (24) and (25) yield

K = C

−1 (26)

Matrix K contains both the effect of stiffness of all flexible passive and active joints and the effect of stiffness of all

flexible bodies. Furthermore, in robotic systems, in addition to physical and geometrical parameters, stiffness is a function

of the robot’s configuration. In this paper, effect of the robot configurations on stiffness matrix of the robot is demonstrated

using concept of Wrench Compliant Module Jacobian Matrices, WCMJ, of the robot’s compliant modules. As stated earlier,

the WCMJ matrices are obtained using force balancing method for each compliant module. Additionally, the InvKin problem

is needed to be solved to obtain values of WCMJ matrices for each robot configuration. In the following subsections, the

compliance matrices of all compliant modules are obtained.


5.1. Compliance matrix of the MS, C MS

The strain energy due to flexibility of the MS, U MS , can be written as a function of internal bending moment in three

branches of the MS as follow

U MS =

3 ∑

i =1

b i ∫ 0

1

2 E MS I MS

((τ v 3 i

MS i

)2 +

(τ w 3 i

MS i

)2 )

d u i 0 ≤ u i ≤ b i (27)

where E MS and I MS are material modulus of elasticity and area moment of inertia for the MS branches, respectively. Addition-

ally, the length of i th i th MS branch, b i , can be determined by solving the InvKin problem at the desired robot configuration.

The virtual infinitesimal translational and rotational vectors due to flexibility of the MS, δχMS and δψ MS , can be calculated

using Castigliano’s theorem and Eq. (21) (see Appendix C ). Subsequently, using the details of related calculations which

are represented in Appendix C , the virtual infinitesimal twist vector due to flexibility of the MS, δs MS , can be obtained in

familiar matrix form as

δs MS = C MS w (28)

where C MS is called compliance matrix of the MS. This matrix can be written as follow

C MS = J w T MS �MS J w

MS (29)

in which, �MS is a 6 × 6 diagonal matrix and can be written as

�MS =

1

3 E MS I MS

diag (b

3 1 , b

3 1 , b

3 2 , b

3 2 , b

3 3 , b

3 3

)(30)

As stated earlier, stiffness matrix of manipulators is a function of both physical/geometrical parameters of robot’s com-

pliant modules as well as configuration of robot. As shown in Eq. (29) , matrix J w

MS represents the effect of the robot con-

figuration on compliance matrix of the MS and matrix �MS represents the effect of geometrical/dimensional properties as

well as mechanical properties of material of the MS on compliance matrix of the MS. Upon solving the InvKin as well as

determining geometrical and material properties of the MS, matrix C MS can be calculated.

5.2. Compliance matrix of the LRs, C LR

The strain energy due to flexibility of three LRs, U LR , can be written as follow

U LR =

3 ∑

i =1

li ∫ 0

( (p

ax LR i

)2

2 A LR E LR

+

1

2 E LR I LR

((τ x

LR i

)2 +

(τ y

LR i

)2 ))

d z i 0 ≤ z i ≤ l i (31)

where A LR , E LR and I LR are cross section area, elastic modulus and area moment of inertia of the LRs, respectively. Addi-

tionally, the length of i th LR, l i , can be determined by solving the InvKin problem at the desired robot configuration as

l i = q ac i

− d i . The virtual infinitesimal translational and rotational vectors due to flexibility of the LRs, δχLR and δψ LR , can be

calculated using Castigliano’s theorem and Eq. (21) (see Appendix C ). Subsequently, using the details of related calculations

which are represented in Appendix C , the virtual infinitesimal twist vector due to flexibility of the LRs, δs LR , can be obtained

in familiar matrix form as

δs LR = C LR w (32)

where C LR is called compliance matrix of the LRs. This matrix can be written as follow

C LR = J fw T LR �f

LR J fw

LR + J mw T LR �m

LR J mw

LR + J mw T LR �fm

LR J fr LR + J fr T LR �fm

LR J mw

LR (33)

where

J fr LR = A J fw

LR (34)

and A is a 6 × 9 matrix as follow

A =

⎡

⎢ ⎢ ⎢ ⎢ ⎣

0 1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0

⎤

⎥ ⎥ ⎥ ⎥ ⎦

(35)

Also, �m

LR and �fm

LR are 6 × 6 diagonal matrices as well as �f LR is a 9 × 9 diagonal matrix which can be written as

follows

�f LR =

1

E LR

diag

(l 3 1

3 I LR

, l 3 1

3 I LR

, l 1

A LR

, l 3 2

3 I LR

, l 3 2

3 I LR

, l 2

A LR

, l 3 3

3 I LR

, l 3 3

3 I LR

, l 3

A LR

)


�m

LR =

1

E LR I LR

diag ( l 1 , l 1 , l 2 , l 2 , l 3 , l 3 ) �fm

LR =

1

2 E LR I LR

diag (l 2 1 , l 2 1 , l 2 2 , l 2 2 , l 2 3 , l 2 3

)(36)

It is noteworthy that matrices J fw

LR , J mw

LR , J fr LR and J mw

LR represent the effect of the robot configuration and matrices �m

LR , �fmLR

and �f LR represent the effect of geometrical/dimensional properties as well as mechanical properties of material of the LRs

on its compliance matrix.

5.3. Compliance matrix of the screw shaft, C BS

The strain energy due to flexibility of three ball screws, U BS , can be written as follow

U BS =

3 ∑

i =1

⎛

⎝

l BS1 i ∫ 0

(p

ax BS 1 i

)2

2A

r BS i

E BS i

d z i +

l BS2 i ∫ 0

(p

ax BS 2 i

)2

2A

r BS i

E BS i

d z i

⎞

⎠ (37)

where A

r BS i

and E BS i are cross section root area and elastic modulus of the i th ball screw, respectively. Additionally, the

lengths of l BS1 i and l BS2 i can be determined by solving the InvKin problem at the desired robot configuration as l BS 1 i =l BS i − q ac

i and l BS 2 i = q ac

i . Note that two sections of each ball screw can be considered as two parallel longitudinal springs.

Therefore, The virtual infinitesimal translational and rotational vectors due to flexibility of the ball screws, δχBS and δψ BS ,

can be calculated using Castigliano’s theorem and Eq. (21) (see Appendix C ). Subsequently, using the details of related

calculations which are represented in Appendix C , the virtual infinitesimal twist vector due to flexibility of the ball screws,

δs BS , can be obtained in familiar matrix form as

δs BS = C BS w (38)

where C BS is called compliance matrix of the ball screws. This matrix can be written as

C BS = J w T BS �BS J w

BS (39)

in which, �BS is a 3 × 3 diagonal matrix and can be written as

�BS = diag

(1

k eq BS 1

, 1

k eq BS 2

, 1

k eq BS 3

)(40)

It is noteworthy that matrices J w

BS and �BS represent the effect of the robot configuration and ball screws stiffness prop-

erties on compliance matrix of the ball screws, respectively.

5.4. Compliance matrix of the passive joints, C PJ

Generally, the radial, axial and tilting deflections in bearings can be occurred due to the joints’ reaction forces/moments.

Therefore, for each passive joint, the radial, axial and tilting stiffness coefficients are considered to obtain the strain energy

due to flexibility of its bearings. To ease, the radial, axial and tilting stiffness of the bearings are considered as longitudinal

and torsional springs, respectively. It is noteworthy that the bearing stiffness coefficient can be defined as the ratio of the

force/moment acting on the bearing to resultant elastic deformation in the bearing. Figs. 14 and 15 show how the directional

equivalent stiffness of passive joints are approximated. Also, as illustrated in Fig. 16 , to calculate the strain energy due to

flexibility of all ball nuts and support bearings, axial rigidity of these compliant modules are considered as longitudinal

springs. If the stiffness coefficients of the bearings are approximated by linear longitudinal and torsional springs, the strain

energy due to flexibility of all R θ P −, R ϕ − and R λ−joints, U R θP , U R ϕ and U R λ, as well as all ball nuts and support bearings,

U Nut and U SB , can be written as

U R θP =

3 ∑

i =1

1

2k rad R θP i

((f v 2 i R θP i

)2 +

(f w 2 i R θP i

)2 )

U R ϕ =

3 ∑

i =1

1

2

( (f v 2 i R ϕi

)2

2k ax R ϕi

+

(f w 2 i R ϕi

)2

2k rad R ϕi

)

U R λ =

3 ∑

i =1

1

2k rad R λi

((f u 1 i R λi

)2 +

(f v 1 i R λi

)2 )

+

(f w 1 i R λi

)2

2k ax R λi

+

1

2k t R λi

((m

u 1 i R λi

)2 +

(m

v 1 i R λi

)2 )

U Nut =

3 ∑

i =1

( f Nut i ) 2

2k ax Nut i

U SB =

3 ∑

i =1

( f SB i ) 2

2k eq SB i

(41)

where k rad R θP i

is equivalent radial stiffness coefficient of i th i th R θ i P i − −−joint and k ax R ϕi

and k rad R ϕi

are the equivalent axial and

radial stiffness coefficients of i th i th R ϕi -joint as well as k ax R λi

, k rad R λi

and k t R λi

are the equivalent axial, radial and tilting stiffness

coefficients of i th i th R λi -joint. Also, k ax Nut i

is the axial rigidity of i th i th ball nut and k eq SB i

is the equivalent axial rigidity of i th i th

support bearing of i th i th feed screw system. Furthermore, the effect of axial rigidity of the ball nut-mounting brackets can

be considered in k ax Nut i

as well as the effect of axial rigidity of the support bearing brackets can be considered in k eq SB i

.


Fig. 14. Equivalent rigidity of i th passive R θ P-joint and i th passive R ϕ-joint.

Fig. 15. Equivalent rigidity of the i th passive R λ-joint.

Fig. 16. Equivalent axial rigidity of the ball nut and support bearings of i th feed-screw system.

In this paper, all mountings brackets are assumed to be rigid. Details of calculation of the stiffness coefficients of support

bearings and ball nut are given in Appendix C . Therefore, the total strain energy due to flexibility of all passive joints, U PJ ,

is expressed as

U PJ = U R θP + U R ϕ + U R λ + U Nut + U SB (42)


The virtual infinitesimal translational and rotational vectors due to flexibility of all passive joints, δχPJ and δψ PJ , can be

calculated using Castigliano’s theorem and Eq. (21) (see Appendix C ). Subsequently, using the details of related calculations

which are represented in Appendix C , the virtual infinitesimal twist vector due to flexibility of all passive joints, δs PJ , can be

obtained in familiar matrix form as

δs PJ = C PJ w (43)

where C PJ is called compliance matrix of the passive joints. This matrix can be written as

C PJ = J w T R θP �R θP J w

R θP + J w T R ϕ �R ϕ J w

R ϕ + J fw T R λ �f

R λ J fw

R λ + J mw T R λ �m

R λ J mw

R λ + J w T Nut �Nut J w

Nut + J w T SB �SB J w

SB (44)

in which, �R θP , �R ϕ and �m

R λare 6 × 6 diagonal matrices and �f

R λis a 9 × 9 diagonal matrix as well as �Nut and �SB are

3 × 3 diagonal matrices which can be written as follows

�R θP = diag

(1

k rad R θP1

, 1

k rad R θP1

, 1

k rad R θP2

, 1

k rad R θP2

, 1

k rad R θP3

, 1

k rad R θP3

)

�R ϕ =

1

2

diag

(1

k ax R ϕ1

, 1

k rad R ϕ1

, 1

k ax R ϕ2

, 1

k rad R ϕ2

, 1

k ax R ϕ3

, 1

k rad R ϕ3

)

�f R λ = diag

(1

k rad R λ1

, 1

k rad R λ1

, 1

k ax R λ1

, 1

k rad R λ2

, 1

k rad R λ2

, 1

k ax R λ2

, 1

k rad R λ3

, 1

k rad R λ3

, 1

k ax R λ3

)

�m

R λ = diag

(1

k t R λ1

, 1

k t R λ1

, 1

k t R λ2

, 1

k t R λ2

, 1

k t R λ3

, 1

k t R λ3

)

�Nut = diag

(1

k ax Nut 1

, 1

k ax Nut 2

, 1

k ax Nut 3

)�SB = diag

(1

k eq SB 1

, 1

k eq SB 2

, 1

k eq SB 3

)(45)

It is noteworthy that matrices J w

R θP , J w

R ϕ , J fw

R λ, J mw

R λ, J w

Nut and J w

SB represent the effect of the robot configuration and matrices

�R θP , �R ϕ , �f R λ

, �m

R λ, �Nut and �SB represent the effect of passive joints stiffness properties on compliance matrix of all

passive joints. Upon solving the InvKin and determining the axial, radial and tilting stiffness coefficients of the bearings of

the passive joints, matrix C PJ can be calculated.

5.5. Compliance matrix of the motor-gearbox assemblies, C AJ

As shown in Fig. 17 , to calculate the total strain energy due to flexibility of motor-gearbox assemblies, stiffness coefficient

of the motors are approximated by torsional springs as well as stiffness coefficient of the timing belts are approximated by

equivalent longitudinal springs. It is noteworthy that the motor-gearbox assemblies are considered as compliant active joint

modules. Therefore, the strain energy of three motor-gearbox assemblies, U AJ , can be written as

U AJ =

3 ∑

i =1

( τM i ) 2

2 k t M i

+

( f belt i ) 2

2 k eq

belt i

(46)

where k t M i

and k eq

belt i are equivalent torsional stiffness coefficient of i th motor and equivalent tensile stiffness coefficient of

i th i th timing belt, respectively.

Therefore, The virtual infinitesimal translational and rotational vectors due to flexibility of the motor-gearbox assemblies,

δχAJ and δψ AJ , can be calculated using Castigliano’s theorem and Eq. (21) (see Appendix C ). Subsequently, using the details

of calculations which are represented in Appendix C , the virtual infinitesimal twist vector due to flexibility of the motor-

gearbox assemblies, δs AJ , can be obtained in familiar matrix form as

δs AJ = C AJ w (47)

where C AJ is called compliance matrix of all active joints which contain all motor-gearbox assemblies. This matrix can be

written as follow

C AJ = J w T M

�M

J w

M

+ J w T belt �belt J w

belt (48)

in which, �M

and �belt are 3 × 3 diagonal matrices and can be written as

�M

= diag

(1

k t M1

, 1

k t M2

, 1

k t M3

)�belt = diag

(1

k eq

belt 1

, 1

k eq

belt 2

, 1

k eq

belt 3

)(49)

It is noteworthy that J w

M

and J w

belt represent the effect of robot configuration on compliance matrix and matrices �M

and

�belt represent the effect of stiffness properties of the motors and timing belts on compliance matrix of the active joints.

In the next section, the physical and structural parameters of the robot are determined and then a comprehensive study on

the compliance errors due to flexibility of the robot’s modules is presented.


Fig. 17. Equivalent rigidity of the i th motor-gearbox assembly.

Table 1

Physical and structural parameters’ values for the 3- P RUP parallel robot.

Body/Joint Para. Values Para. Values Para. Values Para. Values

Moving Star D MS 12 mm A MS 113.1 mm

2 I MS 1017.9 mm

4 E MS 201 kN/mm

2

Linear Rod D LR 20 mm A LR 314.16 mm

2 I LR 7854 mm

4 E LR 201 kN/mm

2

Other dim. a 181 mm d 100 mm D P M 100 mm D P BS 35 mm

Ball screw and

ball nut

D BS 15 mm D r BS 12.5 mm A r BS 176.7 mm

2 K n 680 N/μm

P BS 10 mm l BS 610 mm C a 14.3 kN E BS 206 kN/mm

2

Stiff. Coeff. of

flexible joints

and timing belt

k ax Nut 505 N/μm k eq

SB 34.64 N/μm k eq

belt 18.04 N/μm k t M 6.9 kNm/rad

k rad R λ

96.9 N/μm k ax R λ

46.4 N/μm k t R λ

6.3 kNm/rad k rad R θP

65.2 N/μm

k ax R ϕ 25 N/μm k rad

R ϕ 117.5 N/μm

6. Structural parameters identification

The values for physical and structural parameters of the 3-PRUP robot are shown in Table 1 . Parameter “a” denotes the

length of vector OA i (see Fig. 4 ) and “d” denotes the distance between center of passive R λ-joint and passive R θ P-joint

in each leg (see Fig. 8 ). “D MS ” and “D LR ” are the diameter of each branch of the MS and diameter of each LR as well as

“D BS ” and “D

r BS

” are the screw shaft root diameter (thread minor diameter) and the screw shaft outer diameter, respectively.

Furthermore, “A

r BS ” and “l BS ” are the root cross-sectional area of the screw shaft and the length of the ball screw between

upper and lower support bearings, respectively. Also, “O M

O BS = 100 mm ” is the distance between center of motor’s pulley

and center of the ball screw’s end pulley. The other parameters are defined in the previous sections. In Appendix D , the

details of calculations to obtain the stiffness coefficients of all compliant active and passive joints are presented.

7. Compliance errors evaluation

The compliance errors are one of the most significant types of non-geometrical errors particularly for flexible, low stiff-

ness or heavy manipulators. Therefore, estimation of the contribution of the flexible modules of the robot on its accuracy

over the entire workspace is one of the most important issues in the optimal structural design of the robot. In this section,

a comprehensive study on the compliance errors due to flexibility of the robot’s modules e.g. compliant bodies and joints is

performed. The main goal of this study is to evaluate the influence of compliance errors on the accuracy of the 3- P RUP robot

throughout its workspace as well as determining an allowable range for stiffness values of the robot’s compliant modules

which are effective in the rigidity of the robot.

7.1. Computational algorithm

To evaluate the maximum and minimum deflections of the end-effector over the entire workspace, a different compu-

tational algorithm is used to apply external wrench to the end-effector. For this purpose, using the allowable values for

applied force/torque to the end-effector, a permissible force/torque range is determined such that the compliance errors of

the end-effector do not exceed the acceptable error limit which is defined by the designer. This permissible force/torque

range can be assumed as a cube-shaped space with boundaries as f p ≤ 300(N) and m p ≤ 100(N.m) where f p and m p are the

permissible external force and torque, respectively (see Fig. 18 ). Then, a computational algorithm as shown in Table 2 is

used to calculate the maximum and minimum values of the compliance errors throughout the workspace.

According to Fig. 18 , in each robot’s configuration throughout its workspace, all possible states, including 3 6 = 729

states for applying external force/torque vector to the robot’s end-effector are considered.


Fig. 18. The permissible applied external wrench boundaries.

Table 2

Computational algorithm to obtain the maximum values of the pose errors.

Specify constant inputs (Structural and physical properties)

Z = 300 ( mm )

For θ j and ϕk θ j and ϕk are the rotation angles of MS about x- and y-axes, respectively (Z θϕ operational mode).

− Solve the inverse kinematics and obtain values, θ i , ϕ i , λi and q i for i = 1 , 2 , 3

For i = 1 , 2 , 3 (“i " is the number of robot’s leg)

− Obtain the local rotation matrices.

− Obtain Wrench Jacobian Matrices of compliant joints of i th leg, Eqs. (67) , (70) , (76) , (82) , (96) , (99) , (102)

− Obtain Wrench Jacobian Matrices of compliant bodies of i th leg, Eqs. (58) , (85) , (89) , (93)

− Obtain compliance matrices of compliant bodies and joints of i th leg, Eqs. (29) , (33) , (39) , (44) , (48)

Next i

− Calculate the total compliance matrix of the robot, Eq. (24)

− Calculate the total stiffness matrix of the robot, Eq. (26)

− Specify permissible external wrench applied to the end-effector, f p and m p

For f x , f y , f z = { −300 0 300 } (N) and m x , m y , m z = { −100 0 100 } ( N.m ) − Calculate the compliance error vector δs , Eq. (24)

Next f x , f y , f z and m x , m y , m z

− Determine the maximum compliance error of the robot’s end-effector.

− Calculate the maximum and minimum eigenvalues of the stiffness matrix and calculate the KSI value

Next j & k (next point in Z θϕ workspace)

Calculate minimax and maximax of compliance error of the robot’s end-effector throughout the workspace.

Table 3

Comparison between results of the FEA and analytical method.

All bodies are flexible & all joints are rigid. All bodies and joints are flexible.

Analytical

model

results

× 10 −3

FEA

results

× 10 −3

||Diff.||

%

Analytical

model

results

× 10 −3

FEA

results

× 10 −3

||Diff.||

%

f ext = [ 0 , 0 , −300 ]N δχ (mm) 1.6796 1.6928 0.8% 1.6648 1.6865 1.3%

m ext = [ 0 , 0 , 100 ]N . m δψ(deg) 1.0659 1.0641 0.2% 1.1615 1.1512 0.9%

f ext = [ 30 0 , 30 0 , 30 0 ]N δχ (mm) 5.3094 5.3306 0.4% 5.6919 5.7519 1.1%

m ext = [ 100 , 100 , 100 ]N . m δψ(deg) 1.7484 1.7635 0.9% 1.8199 1.8421 1.2%

f ext = [ 300 , 0 , −300 ]N δχ (mm) 5.5873 5.6108 0.4% 5.9161 5.9607 0.8%

m ext = [ −100 , 100 , 0 ]N . m δψ(deg) 1.8545 1.8634 0.5% 1.8815 1.8997 1.0%

δχ is the linear deformation of end-effector, ‖ δχ‖ . δψ is the angular deformation of end-effector, ‖ δψ‖ .

7.2. Finite element analysis

In this subsection, a finite element commercial software called Ansys software is used to evaluate the correctness of

the results of the analytical method. Since, the effects of bending moments are considered in the analytical method, the

element type BEAM4 (3-D elastic beam) is used for modeling of compliant bodies e.g. LRs and MS. This element has 6-

DOF at each node with tension/compression, torsion and bending capabilities. Also, for modeling of the compliant revolute

joints, the element type COMBIN7 which is defined by three nodes in 3-D space and with joint flexibility, joint friction

torque and rotational viscous friction capabilities, is used. Furthermore, the rigid bodies are modeled using MPC184 element

type. The equivalent stiffness of the i th compliant motor-gearbox assembly, ball-screw, ball nuts and support bearings is

approximated by a longitudinal spring with deformation along z-axis in the fixed CF {B}. The equivalent longitudinal springs

are modeled using COMBIN14 element type. Therefore, a configuration of the robot is selected as z = 0.4(m), θ = −15 (deg)

and ϕ = 10 (deg). In this configuration, three unique force/torque vectors are applied to the end-effector. Results of the FEA

and analytical model as well as comparison of these results for two cases, case#1 “all bodies are flexible & all joints are rigid ”

and case#2 “all bodies and joints are flexible ”, are shown in Table 3 . Also, both deformed and un-deformed shapes of the


Fig. 19. Deformed and un-deformed shapes of the robot in the selected poses.

Fig. 20. Results of InvKin analysis.

Table 4

Inputs/outputs values of InvKin analysis for three robot’s configurations.

Inputs (m or deg) Outputs (m or deg)

Z θ ϕ q 1 λ1 ϕ1 θ1 q 2 λ2 ϕ2 θ2 q 3 λ3 ϕ3 θ3

Ex. #1 0.4 0 0 0.4 0 0 0 0.4 0 0 0 0.4 0 0 0

Ex. #2 0.4 25 15 0.35 3.34 15 25 0.51 −0.34 −28.9 1.31 0.36 −3.01 12.95 −26.1

Ex. #3 0.4 −15 10 0.37 −1.32 10 −15 0.38 1.10 7.7 16.28 0.46 0.22 −17.9 −1.38

robot are illustrated in Fig. 19 . As shown in Table 3 , results of the analytical model and FEA, for all three example, are very

close. Therefore, the proposed analytical method has a good reliability and validity to evaluate the robot’s compliant errors

throughout its workspace.

7.3. Results and discussion

The compliance errors are functions of the structural stiffness of a robot. The structural stiffness of a robot changes

throughout the workspace. Therefore, for 3- P RUP robot, three unique configurations are considered and InvKin is solved.

Configurations of the robot and results of the InvKin are shown in Fig. 20 and Table 4 , respectively.

Next, a permissible external wrench as shown in Fig. 18 and Table 2 is applied to the end-effector. Using Eq. (24) and

computational algorithm shown in Table 2 , the maximum magnitude of linear and angular compliance error vectors, δχmax

and δψ max , as well as the maximum linear and angular compliance errors along x-, y- and z-, ( δx max , δy max , δz max ), and

about x-, y- and z-, ( δθmax , δϕmax , δλmax ), are obtained and a comparison is made between results of the compliance

error analysis for both cases with and without taking into consideration of joints flexibility for three selected configurations

(results are shown in Tables 5 and 6 ).


Table 5

Comparison between results of the stiffness analysis for three configurations.

All bodies are flexible & all joints are rigid. All bodies and joints are flexible. ||Diff.|| ||Diff.|| %

Ex. #1 δχmax (mm) 5.708 6.167 0.459 8%

δψ max (deg) 1.988 2.056 0.068 3%

Ex. #2 δχmax (mm) 11.811 12.258 0.447 4%

δψ max (deg) 3.328 3.436 0.108 3%

Ex. #3 δχmax (mm) 8.815 9.309 0.494 6%

δψ max (deg) 2.651 2.744 0.093 3%

δχmax is the Max. linear deformation of end-effector, ‖ δχ‖ . δψ max is the Max. angular deformation of end-effector, ‖ δψ‖ .

Table 6

Stiffness analysis results for three configurations in various directions.

All bodies are flexible & all joints are rigid. All bodies and joints are flexible. ||Diff.|| ||Diff.|| %

Ex. #1 δx max (mm) 4.023 4.351 0.328 8%

δy max (mm) 3.932 4.260 0.328 8%

δz max (mm) 0.967 0.975 0.007 1%

δθmax (deg) 1.128 1.136 0.009 1%

δϕmax (deg) 1.128 1.136 0.009 1%

δλmax (deg) 1.187 1.282 0.096 8%

Ex. #2 δx max (mm) 9.646 10.035 0.389 4%

δy max (mm) 6.854 7.220 0.366 5%

δz max (mm) 3.501 3.682 0.181 5%

δθmax (deg) 1.851 1.903 0.052 3%

δϕmax (deg) 1.957 2.016 0.058 3%

δλmax (deg) 1.984 2.065 0.081 4%

Ex. #3 δx max (mm) 6.878 7.316 0.437 6%

δy max (mm) 5.697 6.150 0.453 8%

δz max (mm) 2.390 2.505 0.115 5%

δθmax (deg) 1.481 1.530 0.049 3%

δϕmax (deg) 1.502 1.548 0.046 3%

δλmax (deg) 1.683 1.778 0.095 6%

δx max , δy max and δz max are the Max. linear deformation of end-effector along x-, y- and z-, ( δx, δy, δz ).

δθmax , δϕmax and δλmax are the Max. angular deformation of end-effector about x-, y- and z-, ( δθ , δϕ, δλ).

In this case study, the results indicate that if the joints stiffness properly designed, the joints flexibility compared to

flexibility of the bodies does not have significant effects on the end-effector accuracy. For three examples, the contribution

of joints flexibility in the maximum linear and angular end-effector errors is approximately up to 8%.

It is noteworthy that the compliance errors have direct effect on the position accuracy of parallel robots with lightweight

structure which are used in the precise machining process. Robotic system designers are trying to minimize the robot’s end-

effector com pliance errors by optimizing the effective parameters on structural stiffness of the robot. Since, the compliance

errors highly depend on the robot configuration, therefore the influence of these errors on the robot accuracy should be

evaluated entire the workspace. One method used to evaluate the robot’s stiffness is calculating the Kinematic Stiffness

Index, KSI, which is defined as the ratio of minimum to maximum eigenvalues, σ min and σ max , of stiffness matrix. To do

this, a selected z-plane (z = 30 cm) in the robot workspace is considered and σ max , σ min and KSI values are calculated.

Fig. 21 depicts distribution of σ max , σ min and KSI values considering the joints flexibility in plane z = 0.3 (m).

Additionally, using the computational algorithm shown in Table 2 , the maximum magnitude of linear and angular com-

pliance error vectors, δχmax and δψ max , as well as the maximum linear and angular compliance errors along x-, y- and

z-, ( δx max , δy max , δz max ), and about x-, y- and z-, ( δθmax , δϕmax , δλmax ) are obtained for each robot’s configuration in se-

lected z-plane (z = 30 cm) (see Fig. 22 ). According to the physical/structural properties of the robot and permissible applied

external wrench, a General Compliance Error Range, GCER, for the end-effector can be obtained throughout the robot’s

workspace using the computational algorithm shown in Table 2 . The GCER is specified between minimax and maximax

of linear and angular deformations of end-effector. The GCER defines the range of robot’s end-effector accuracy with re-

spect to the compliance errors. For z-plane z = 30 cm, the GCER for linear and angular deformations of the end-effector are

4.47 ≤‖ δχmax . ‖ ≤ 15.72 (mm) and 1.85 ≤‖ δψ max . ‖ ≤ 4.75 (deg.), respectively.

Comparing Fig. 21 with Fig. 22 , it can be concluded that as we get closer to the workspace boundaries, the stiffness

decreases and the compliance errors increases. Conversely, the maximum value of the robot’s stiffness and minimum value

of compliance errors occur at the center of workspace. Therefore, a designer should limit or restrict operation of the robot

around a particular subspace of the whole workspace where the robot’s compliance errors do not exceed the allowable

values.


Fig. 21. Workspace and distribution of Max. and Min. eigenvalues of stiffness matrix as well as KSI values considering the joints flexibilities in section

z = 0.3 (m) of workspace in terms of θ and ϕ.

8. Conclusion

In the present paper, using the Castigliano’s 2nd theorem, a continuous modeling method is proposed to evaluate the

influence of compliance errors on accuracy of a 3- P RUP parallel robot. The main contributions of this paper are the com-

prehensiveness of the proposed methodology for modeling and evaluating the structural stiffness of the robotic systems,

providing a highly accurate analytical method with significant details for stiffness modeling as well as presenting the com-

plex relations in a simplified way. The capabilities of this method significantly reduce the limiting assumptions and this

method gives the designer the ability to obtain the compliance errors of flexible joints and bodies with complex geometries.

Furthermore, bending and torsional moments, shear forces and weight of all robot compliant modules as continuous loads

as well as the nonlinear effects of preloaded joints and joints flexibility on overall compliance errors of the robot can be

considered. Although the proposed method does not essentially require simplifying assumptions, but to reduce the calcula-

tions process, several reasonable limiting assumptions are considered in this paper. To evaluate the effect of each flexible

module on the end-effector accuracy, the compliance matrix of each flexible module is independently obtained using the

concept of WCMJ matrix and physical/structural properties matrix of this module. Also, the overall compliance matrix of the

robot is obtained using the compliance matrices of all the robots’ compliant bodies and joints.

It is noteworthy that to make the results of this article closer to the actual design conditions, the stiffness coefficients

of the flexible components such as bearings of the joints, ball nuts, screw shafts, support bearings, servo motors and tim-

ing belts are obtained using the design specifications provided by the manufacturers. Finally, using the proposed analytical

method and a computational algorithm, a comprehensive assessment for the robot’s rigidity is performed to evaluate the

influence of the joints’ flexibility on the accuracy of the robot. Other contribution of this paper includes obtaining the com-


Fig. 22. Max. linear and angular deformation of end-effector considering the joints flexibilities in section z = 0.3 (m) of workspace in terms of θ and ϕ.

pliance error boundaries as well as defining a General Compliance Error Range, GCER, throughout the robot’s workspace. The

GCER is specified between minimax and maximax of linear and angular deformations of end-effector throughout the robot’s

workspace. It can be concluded that the influence of joints flexibility on the maximum linear and rotation end-effector er-

rors is approximately up to 8% in this robot. In the selected z-plane, z = 30 cm, the GCER for linear and angular deformations

of end-effector are 4.47 ≤‖ δχmax . ‖ ≤ 15.72 (mm) and 1.85 ≤‖ δψ max . ‖ ≤ 4.75 (deg.), respectively. The results indicate that if

the joints stiffness properly designed in this case study, the joints flexibility compared to flexibility of the bodies does not

have significant effects on the end-effector accuracy. In other words, in this robot, the contribution of compliance errors due

to flexibility of the joints from the overall compliance errors due to flexibility of all compliant modules is negligible.

Appendix A

As shown in Fig. 3 , to transfer a vector defined in {T} to {B}, the rotation matrix B T

R can be used as follow

B T R = R x ,θ R y ,ϕ R z ,λ =

[

1 0 0

0 c θ −s θ0 s θ c θ

] [

c ϕ 0 s ϕ

0 1 0

−s ϕ 0 c ϕ

] [

c λ −s λ 0

s λ c λ 0

0 0 1

]

(50)

where c and s stand for cosine and sine, respectively. Also, the angles θ , ϕ and λ represent the Euler angles about x-, y-

and z-axes which describe the orientation of MS. Using the InvKin solution of Eq. (1) , the rotation values of passive R- and

U-joints can be calculated as the following method. As shown in Fig. 4 , the rotation matrix from local moving CF {T } to
1 i


fixed CF {B} can be written as follow

B T1 i R =

B T R

T T3 i R

T3 i T2 i R

T2 i T1 i R =

(R x ,θ R y ,ϕ R z ,λ

)( R z , αi )

(R u2 i, −θ i

)(R v1 i, −ϕi

)i = 1 , 2 , 3 (51)

where αi is constant value and θ and ϕ are specified as input of the InvKin problem and λ is calculated by solving the

InvKin problem as well as the rotation values of i th passive U-joint, θ i and ϕi , are unknown. Furthermore, we know that

R u2 i, −θ i = R

T u2 i, θ i

and R v1 i, −ϕi = R

T v1 i, ϕ i . Therefore, we can write

e q i =

B T1 i R

T1 i e q i i = 1 , 2 , 3 (52)

where T1 i e q i = { 0 0 1 } T . To obtain values of θ i and ϕi , Eq. (52) can be rewritten as

T3 i T R

T B R e q i =

(T3 i T2 i R

T2 i T1 i R

)T1 i e q i i = 1 , 2 , 3 (53)

where T B

R =

B T

R

T = R

T z ,λ

R

T y ,ϕ R

T x ,θ

and

T3 i T

R =

T T3 i

R

T = R z , −αi . Hence, the left hand of Eq. (53) is known. Therefore, using

Eq. (53) , the values for θ i and ϕi can be easily obtained. Furthermore, to calculate the rotation value of i th passive R-joint,

λi , the rotation matrix from {T} to {B} can be written differently as follow

B B i R

B i T1 i R

T1 i T2 i R

T2 i T3 i R

T3 i T R =

B T R i = 1 , 2 , 3 (54)

where

B B i R = R z , αi ,

B i T1 i R = R z , λi ,

T1 i T2 i R = R v1 i, ϕ i ,

T2 i T3 i R = R u2 i, θ i i = 1 , 2 , 3 (55)

Since, the value of λi is only unknown variable in Eq. (54) , this equation can be rewritten as

R z , λi =

B B i R

T B T R

T3 i T R

T T2 i T3 i R

T T1 i T2 i R

T =

(R

T z , αi

)(R x ,θ R y ,ϕ R z ,λ

)( R z , αi )

(R

T u2 i, θ i

)(R

T v1 i, ϕ i

)= L (56)

where L is a known matrix. Therefore, the value of λi can be obtained as follow

λi = sin

−1 (L ( 2 , 1 )

)i = 1 , 2 , 3 (57)

in which, L (2, 1) is the 2nd row and 1st column element of matrix L .

Appendix B

B.1. Calculation of the internal force/moment of the MS

As shown in Fig. 5 , the relationship between reaction forces and external wrench on MS can be written as {f ext

m ext

}+

3 ∑

i =1

{

B T3 i

R

(f v 3 i MS i

T3 i e v + f w 3 i MS i

T3 i e w

)B T3 i

R

(b i f

v 3 i MS i

(T3 i e b i �

T3 i e v )

+ b i f w 3 i MS i

(T3 i e b i �

T3 i e w

))}

= 0 6 × 1 (58)

where, T3 i e b i =

T3 i e u and unit vectors T3 i e u , T3 i e v and

T3 i e w

are along x-, y- and z-axes of frame {T 3 i }, respectively. Also,B T3 i

R =

B T

R

T T3 i

R = ( R x ,θ R y ,ϕ R z ,λ)( R z , αi ) . By simplifying Eq. (58) , f MS is obtained in matrix form as stated in Eq. (4) . Next, as

illustrated in Fig. 6 , the internal bending moment in i th branch of the MS can be calculated as

T3 i τv MS i = f w 3 i

MS i u i

(T3 i e b i �

T3 i e w

)= τ v 3 i

MS i T3 i e v

T3 i τw

MS i = f v 3 i MS i u i

(T3 i e b i �

T3 i e v )

= τ w 3 i MS i

T3 i e w

(59)

To calculate the strain energy due to flexibility of the MS, the internal bending moment should be calculated as function

of external wrench. Using Eq. (4) , we have

τ v 3 i MS i = f w 3 i

MS i u i = u i J w

M S ( 2 i ) × ( 1 ∼6 ) w τ w 3 i

MS i = f v 3 i MS i u i = u i J w

M S ( 2 i −1 ) × ( 1 ∼6 ) w (60)

where J w

M S (k ) × ( 1 ∼6 ) is the k th row of matrix J w

MS . For a given configuration of 3- P RUP manipulator, directions of all unit vectors

are determined by solving the InvKin problem.

B.2. Calculation of the internal forces of the passive R θ i P i - and R ϕi -joints

As shown in Fig. 6 , the reaction force vector of the i th MS branch in its local CF {T 3 i } can be stated as follow

T3 i f MS i =

{f u 3 i MS i

f v 3 i MS i

f w 3 i MS i

}T =

{0 f v 3 i

MS i f w 3 i MS i

}T i = 1 , 2 , 3 (61)

where f u 3 i MS i

= 0 . The reaction force vector of R θ i P i -joint in {T 3 i } is calculated as

T3 i f R θP i =

{0 f v 3 i

R θP i f w 3 i R θP i

}T = − T3 i f MS i i = 1 , 2 , 3 (62)

Accordingly, the reaction force vector of R θ i P i -joint in CF {T 2 i } can be obtained as follow

T2 i f R θP i =

T2 i R

T3 i f R θP i = − T2 i R

T3 i f MS i i = 1 , 2 , 3 (63)
T3 i T3 i


where T2 i T3 i

R = R u2 i, θ i . As shown in Fig. 7 , f u 2 i R θP i

= 0 . Therefore, Eq. (63) can be rewritten as {f v 2 i R θP i

f w 2 i R θP i

}= −

[cos ( θ i ) − sin ( θ i ) sin ( θ i ) cos ( θ i )

] {f v 3 i MS i

f w 3 i MS i

}= J MS i

R θP i

{f v 3 i MS i

f w 3 i MS i

}i = 1 , 2 , 3 (64)

Therefore, combining all the internal reaction forces in the R θ i P i -joint yield

f R θP = J MS R θP f MS (65)

where f v 2 i R θP i

and f w 2 i R θP i

are values of the radial reaction forces in R θ i P i -joint in {T 2 i } and

J MS R θP = diag

({J MS 1 R θP1

J MS 2 R θP2

J MS 3 R θP3

})(66)

Therefore, by substituting f MS from Eq. (4) into Eq. (65) , vector f R θP can be obtained as function of external wrench in

matrix form as stated in Eq. (7) . Furthermore, the WCMJ R θP matrix can be calculated as follow

J w

R θP = J MS R θP J w

MS (67)

Also, as shown in Fig. 7 , the reaction forces for R ϕi -joint can be obtained as

−f v 2 i R ϕi = −f v 2 i R θP i , − f w 2 i R ϕi = −f w 2 i

R θP i i = 1 , 2 , 3 (68)

Therefore, combining all the internal reaction forces in the R ϕi -joint yield

f R ϕ = f R θP (69)

where f v 2 i R ϕi

and f w 2 i R ϕi

are values of the axial and radial reaction forces in R ϕi -joint in {T 2 i }, respectively. Therefore, substituting

f R θP from Eq. (7) into Eq. (69) , vector f R ϕ can be obtained as function of external wrench in matrix form as stated in Eq. (7) .

Furthermore, the WCMJ R ϕ matrix can be calculated as follow

J w

R ϕ = J w

R θP (70)

B.3. Calculation of the internal forces and moments of the passive R λi -joints

To obtain the internal reaction force vector of R λi -joint, the reaction force vector of i th R ϕi -joint can be rewritten as

follow

T2 i f R ϕi =

{f u 2 i R ϕi

f v 2 i R ϕi

f w 2 i R ϕi

}T =

{0 f v 2 i

R ϕi f w 2 i R ϕi

}T = −T2 i T3 i R

T3 i f MS i i = 1 , 2 , 3 (71)

and the reaction force vector of i th R λi -joint in {T 1 i } can be obtained as

T1 i f R λi =

{f u 1 i R λi

f v 1 i R λi

f w 1 i R λi

}T = −T1 i f R ϕi =

T1 i T3 i R

T3 i f MS i i = 1 , 2 , 3 (72)

where

T3 i f MS i =

{0 f v 3 i

MS i f w 3 i MS i

}T =

[0 1 × 6

J w

MS ( ( 2 i −1 ) ∼( 2 i ) ) ×( 1 ∼6 )

]3 × 6

w = J w

MS i w i = 1 , 2 , 3 (73)

Therefore

T1 i f R λi = J fw

R λi w i = 1 , 2 , 3 (74)

in which

J fw

R λi =

T1 i T3 i R J w

MS i i = 1 , 2 , 3 (75)

Therefore, combining the reaction forces in all the R λ-joints in {T 1 i }, the 9 × 1 vector f R λ = { T11 f T R λ1

T12 f T R λ2

T13 f T R λ3

} Tcan be obtained as a function of external wrench as stated in Eq. (9) . Also, the WCMFJ R λ matrix, J fw

R λ, can be stated as follow

J fw

R λ =

[J fw

R λ1

T J fw

R λ2

T J fw

R λ3

T ]T

(76)

As shown in Fig. 8 , the reaction moment vector of R λi -joint can be calculated in CF {T 1 i } as

T1 i m R λi =

{m

u 1 i R λi

m

v 1 i R λi

m

w 1 i R λi

}T = −(d i

T1 i e w

)�

(T1 i f R ϕi

)i = 1 , 2 , 3 (77)

where m

w 1 i R λi

= 0 . We know that

a � b = ( a � I ) b (78)

Therefore, Eq. (77) by using Eqs. (73) and (78) can be rewritten as matrix form as follow

T1 i m R λi = J m

′ w w i = 1 , 2 , 3 (79)
R λi


where

J m

′ w

R λi = d i

(T1 i e w

� I )

T1 i T3 i R J w

MS i i = 1 , 2 , 3 (80)

in which, m

w 1 i R λi

= 0 . Therefore, Eq. (79) can be rewritten as {m

u 1 i R λi

m

v 1 i R λi

}= J m

′ w

R λi ( 1 ∼2 ) ×( 1 ∼6 ) w = J mw

R λi w i = 1 , 2 , 3 (81)

Combining the reaction moments in all the R λ-joints in {T 1 i }, the 6 × 1 vector m R λ can be obtained as a function of

external wrench as stated in Eq. (9) . Also, the WCMMJ R λ matrix, J mw

R λ, can be stated as follow

J mw

R λ =

[J mw

R λ1 T J mw

R λ2 T J mw

R λ3 T ]T

(82)

B.4. Calculation of the internal forces and moments of the LRs

As shown in Fig. 9 , the reaction force vector acting on i th LR can be calculated as follow

f LR i =

{f x LR i

f y LR i

f z LR i

}T = − B T1 i R

T1 i f R λi = −B T3 i R

T3 i f MS i = J fw

LR i w i = 1 , 2 , 3 (83)

where

J fw

LR i = −B T3 i R J w

MS i i = 1 , 2 , 3 (84)

Therefore, combining the reaction forces in all the LRs defined in {B}, the 9 × 1 vector f LR = { f T LR 1

f T LR 2

f T LR 3

} T can be

obtained as a function of external wrench as stated in Eq. (11) . Also, the WCMFJ LR matrix, J fw

LR , can be stated as follow

J fw

LR =

[J fw

LR 1

T J fw

LR 2

T J fw

LR 3

T ]T

(85)

Additionally, as illustrated in Fig. 9 , the reaction moment vector acting on i th LR can be calculated as

m LR i =

{m

x LR i

m

y LR i

m

z LR i

}T = − B T1 i R

T1 i m R λi = J m

′ w

LR i w i = 1 , 2 , 3 (86)

where m

z LR i

= 0 and using Eq. (80) yield

J m

′ w

LR i = − B T1 i R J m

′ w

R λi = −d i

(T1 i e w

� I )

B T3 i R J w

MS i i = 1 , 2 , 3 (87)

in which, m

z LR i

= 0 . Therefore, Eq. (86) can be rewritten as {m

x LR i

m

y LR i

}= J m

′ w

LR i ( 1 ∼2 ) ×( 1 ∼6 ) w = J mw

LR i w i = 1 , 2 , 3 (88)

Combining the reaction moments in all the LRs in {B}, the 6 × 1 vector m LR can be obtained as a function of external

wrench as stated in Eq. (11) . Also, the WCMMJ LR matrix, J mw

LR , can be stated as follow

J mw

LR =

[J mw

LR 1 T J mw

LR 2 T J mw

LR 3 T ]T

(89)

Next, as illustrated in Fig. 10 and using Eq. (11) , the internal bending moment and internal axial force in i th LR should

be calculated as functions of external wrench as follows

p

ax LR i = f z LR i = J fw

LR ( 3 i ) × ( 1 ∼6 ) w

τ x LR i = f y

LR i z i + m

x LR i =

(z i J fw

LR ( 3 i −1 ) × ( 1 ∼6 ) + J mw

LR ( 2 i −1 ) × ( 1 ∼6 )

)w

τ y LR i

= f x LR i z i + m

y LR i

=

(z i J fw

LR ( 3 i −2 ) × ( 1 ∼6 ) + J mw

LR ( 2 i ) × ( 1 ∼6 )

)w i = 1 , 2 , 3 (90)

where J fw

L R (k ) × ( 1 ∼6 ) and J mw

L R (k ) ×( 1 ∼6 ) are the k th row of matrices J fw

LR and J mw

LR , respectively.

B.5. Calculation of the internal force/moment of the feed screw system’s components

As shown in Fig. 11 , the internal axial force in i th ball screw should be calculated as a function of external wrench as

follow

p

ax BS 1 i + p

ax BS 2 i = f BS i = −f z LR i = J w

BS i w i = 1 , 2 , 3 (91)

in which

J w

BS i = −J fw

L R ( 3 i ) × ( 1 ∼6 ) i = 1 , 2 , 3 (92)


Combining the reaction forces in all ball screws, the 3 × 1 vector f BS can be obtained as a function of external wrench as

stated in Eq. (13) . Also, the WCMJ BS matrix, J w

BS , can be stated as follow

J w

BS =

[J w

BS 1 T J w

BS 2 T J w

BS 3 T ]T

(93)

Furthermore, as shown in Fig. 11 , the reaction axial forces of the ball nut and support bearings of i th feed screw system

should be calculated as functions of external wrench as

f Nut i = f z LR i = J w

Nut i w f SB i = f SB 1 i + f SB 2 i = f z LR i = J w

SB i w i = 1 , 2 , 3 (94)

where

J w

Nut i = J fw

L R ( 3 i ) × ( 1 ∼6 ) J w

SB i = J fw

L R ( 3 i ) × ( 1 ∼6 ) i = 1 , 2 , 3 (95)

Combining the resistant forces in all ball nuts and support bearings, the 3 × 1 vectors f Nut and f SB can be obtained as

functions of external wrench as stated in Eq. (14) . Also, the WCMJ Nut and WCMJ SB matrices, J w

Nut and J w

SB , can be stated as

follow

J w

Nut =

[J w

Nut 1 T J w

Nut 2 T J w

Nut 3 T ]T

J w

SB =

[J w

SB 1 T J w

SB 2 T J w

SB 3 T ]T

(96)

B.6. Calculation of the internal force/moment of the motor-gearbox assemblies

As illustrated in Fig. 12 , the i th motor and ball screw’s torques can be obtained as a function of external wrench as follow

τM i = R i τBS i = R i

P BS i ηri

2 π

(f z LR i

)= J w

M i w i = 1 , 2 , 3 (97)

where τ BS i is the resistant torque of the i th ball screw and

J w

M i = R i

P BS i ηri

2 πJ fw

L R ( 3 i ) × ( 1 ∼6 ) i = 1 , 2 , 3 (98)

and τM i is the i th motor torque as well as R i = D

P M i

/ D

P BS i

and P BS i are the pulley ratio of i th gearbox and i th ball screw’s

lead, respectively. Note that D

P M i

and D

P BS i

are the pulley’s diameter of the i th motor and i th ball screw, respectively. Also,

ηri is the reverse efficiency of i th ball screw for its reverse operation when the linear motion of ball nut is converted into

rotation of motor. This coefficient is considered about 80%. Therefore, combining the resistant motor torque in all servo

motors, the 3 × 1 vector τM

can be obtained as a function of external wrench as stated in Eq. (15) . Also, the WCMJ M

matrix,

J w

M

, can be stated as follow

J w

M

=

[J w

M1 T J w

M2 T J w

M3 T ]T

(99)

Furthermore, the resultant tensile force in the timing belt due to the resistant motor torque can be obtained as follow

f belt i = 2 τMi /D

P Mi = R i

P BS i ηri

πD

P M i

( f z LRi ) = J w

belti W (100)

where

J w

belt i = R i

P BS i ηri

πD

P M i

J fw

L R ( 3 i ) × ( 1 ∼6 ) i = 1 , 2 , 3 (101)

Combining the tensile forces in all timing belts, the 3 × 1 vector f belt can be obtained as a function of external wrench

as stated in Eq. (16) . Also, the WCMJ belt matrix, J w

belt , can be stated as follow

J w

belt =

[J w

belt 1 T J w

belt 2 T J w

belt 3 T ]T

(102)

Appendix C

C.1. Calculation of the compliance matrix of the MS, C MS

Using Castigliano’s 2nd theorem and Eqs. (21) and (27) , δχMS and δψ MS vectors can be calculated as

δχMS k =

∂ U MS

∂ f ext k

=

3 ∑

i =1

b i ∫ 0

1

E MS I MS

(τ v 3 i

MS i

∂τ v 3 i MS i

∂ f ext k

+ τ w 3 i MS i

∂τ w 3 i MS i

∂ f ext k

)d u i

δψ MS k =

∂ U MS

∂ m ext k

=

3 ∑

i =1

b i ∫ 0

1

E MS I MS

(τ v 3 i

MS i

∂τ v 3 i MS i

∂ m ext k

+ τ w 3 i MS i

∂τ w 3 i MS i

∂ m ext k

)d u i (103)


where δχMS = δχMS k and δψ MS = δψ MS k for k = 1 , 2 , 3 and δs MS = { δχT MS δψ

T MS

} T is a 6 × 1 vector. Substituting

Eq. (60) into Eq. (103) and factoring w yields

δχMS k =

3 ∑

i =1

b

3 i

3 E MS I MS

(J w

MS ( 2 i ) ( k ) J w

MS ( 2 i ) × ( 1 ∼6 ) + J w

MS ( 2 i −1 ) ( k ) J w

MS ( 2 i −1 ) × ( 1 ∼6 )

)w

δψ MS k =

3 ∑

i =1

b

3 i

3 E MS I MS

(J w

MS ( 2 i ) ( 3+ k ) J w

MS ( 2 i ) × ( 1 ∼6 ) + J w

MS ( 2 i −1 ) ( 3+ k ) J w

MS ( 2 i −1 ) × ( 1 ∼6 )

)w (104)

As shown in Eq. (28) , the relationship between δs MS and C MS is obtained by rewriting Eq. (104) in matrix form.

C.2. Calculation of the compliance matrix of the LRs, C LR

Using Castigliano’s 2nd theorem and Eq. (21) and (31) , vectors δχLR and δψ LR can be calculated as

δχLR k =

∂ U LR

∂ f ext k

=

3 ∑

i =1

li ∫ 0

(p

ax LR i

A LR E LR

∂p

ax LR i

∂ f ext k

+

1

E LR I LR

(τ x

LR i

∂τ x LR i

∂ f ext k

+ τ y LR i

∂τ y LR i

∂ f ext k

))d z i

δψ LR k =

∂ U LR

∂ m ext k

=

3 ∑

i =1

li ∫ 0

(p

ax LR i

A LR E LR

∂p

ax LR i

∂ m ext k

+

1

E LR I LR

(τ x

LR i

∂τ x LR i

∂ m ext k

+ τ y LR i

∂τ y LR i

∂ m ext k

))d z i (105)

where δχLR = δχLR k and δψ LR = δψ LR k for k = 1 , 2 , 3 and δs LR = { δχT LR

δψ

T LR

} T is a 6 × 1 vector. Substituting the internal

forces/moments of LRs as functions of external wrench from Eq. (90) into Eq. (105) and factoring w yields

δχLR k =

3 ∑

i =1

(l i

A LR E LR

J fw

L R ( 3 i ) ( k ) J fw

L R ( 3 i ) ×( 1 ∼6 ) +

l 3 i

3 E LR I LR

(J fw

L R ( 3 i −1 ) ( k ) J fw

L R ( 3 i −1 ) ×( 1 ∼6 )

+ J fw

L R ( 3 i −2 ) ( k ) J fw

L R ( 3 i −2 ) ×( 1 ∼6 )

)+

l 2 i

2 E LR I LR

(J fw

L R ( 3 i −1 ) ( k ) J mw

L R ( 2 i −1 ) ×( 1 ∼6 )

+ J mw

L R ( 2 i −1 ) ( k ) J fw

L R ( 3 i −1 ) ×( 1 ∼6 ) + J fw

L R ( 3 i −2 ) ( k ) J mw

L R ( 2 i ) ×( 1 ∼6 )

+ J mw

L R ( 2 i ) ( k ) J fw

L R ( 3 i −2 ) ×( 1 ∼6 )

)+

l i E LR I LR

(J mw

L R ( 2 i −1 ) ( k ) J mw

L R ( 2 i −1 ) ×( 1 ∼6 )

+ J mw

L R ( 2 i ) ( k ) J fw

L R ( 3 i −2 ) ×( 1 ∼6 )

)+

l i E LR I LR

(J mw

L R ( 2 i −1 ) ( k ) J mw

L R ( 2 i −1 ) ×( 1 ∼6 )

+ J mw

L R ( 2 i )(k ) J mw

L R ( 2 i ) ×( 1 ∼6 )

))w

δψ L R k =

3 ∑

i =1

(l i

A LR E LR

J fw

L R ( 3 i ) ( 3+ k ) J fw

L R ( 3 i ) ×( 1 ∼6 ) +

l 3 i

3 E LR I LR

(J fw

L R ( 3 i −1 ) ( 3+ k ) J fw

L R ( 3 i −1 ) ×( 1 ∼6 )

+ J fw

L R ( 3 i −2 ) ( 3+ k ) J fw

L R ( 3 i −2 ) ×( 1 ∼6 )

)+

l 2 i

2 E LR I LR

(J fw

L R ( 3 i −1 ) ( 3+ k ) J mw

L R ( 2 i −1 ) ×( 1 ∼6 )

+ J mw

L R ( 2 i −1 ) ( 3+ k ) J fw

L R ( 3 i −1 ) ×( 1 ∼6 ) + J fw

L R ( 3 i −2 ) ( 3+ k ) J mw

L R ( 2 i ) ×( 1 ∼6 )

+ J mw

L R ( 2 i ) ( 3+ k ) J fw

L R ( 3 i −2 ) ×( 1 ∼6 )

)+

l i E LR I LR

(J mw

L R ( 2 i −1 ) ( 3+ k ) J mw

L R ( 2 i −1 ) ×( 1 ∼6 )

+ J mw

L R ( 2 i )(3+ k ) J mw

L R ( 2 i ) ×( 1 ∼6 )

))w (106)

As shown in Eq. (32) , the relationship between δs LR and C LR is obtained by rewriting Eq. (106) in matrix form.

C.3. Calculation of the compliance matrix of the screw shaft, C BS

To obtain the equivalent axial rigidity of the screw shaft, Eq. (37) can be rewritten as

U BS =

3 ∑

i =1

( f BS i ) 2

2k eq BS i

(107)

where the axial rigidity of screw shaft in the fixed-fixed configuration can be obtained as

k eq BS i

=

A

r BS i

E BS i l BS i

l l (108)

BS 1 i BS 2 i


in which, k eq BS i

is the equivalent axial rigidity of i th screw shaft. Using Castigliano’s 2nd theorem and Eq. (21) , as well as

substituting Eq. (13) into Eq. (107) and factoring w , the virtual infinitesimal translational and rotational vectors due to

flexibility of the ball screws, δχBS and δψ BS , can be calculated as follows

δχBS k =

∂ U BS

∂ f ext k

=

3 ∑

i =1

f BS i

k eq BS i

∂ f BS i

∂ f ext k

=

3 ∑

i =1

1

k eq BS i

(J w

BS ( i ) ( k ) J w

BS ( i ) ×( 1 ∼6 )

)w

δψ BS k =

∂ U BS

∂ m ext k

=

3 ∑

i =1

f BS i

k eq BS i

∂ f BS i

∂ m ext k

=

3 ∑

i =1

1

k eq BS i

(J w

BS ( i ) ( 3+ k ) J w

BS ( i ) ×( 1 ∼6 )

)w (109)

where δχBS = δχBS k and δψ BS = δψ BS k for k = 1 , 2 , 3 and δs BS = { δχT BS

δψ

T BS

} T is a 6 × 1 vector. As shown in Eq. (38) ,

the relationship between δs BS and C BS is obtained by rewriting Eq. (109) in matrix form.

C.4. Calculation of the stiffness coefficients of support bearings and ball nut

As illustrated in Fig. 16 , the equivalent axial rigidity of support bearings’ set of i th feed screw system can be obtained as

follow

k eq SB i

= k ax SB 1 i + k ax

SB 2 i (110)

where k ax SB 1 i

and k ax SB 2 i

are axial rigidities of support bearings of i th feed screw system. Furthermore, the axial rigidity of

a ball nut can be obtained as a function of preload. It is noteworthy that the preload in feed-screw system can improves

the rigidity and ensures positioning accuracy of a robot by eliminating the axial clearance of the ball screws. Depending

on preloaded or not preloaded ball nut types is used, some manufacturers suggest general equations to calculate the axial

rigidity of the ball nut as follow

k ax Nut = 0 . 8 K n

(f a

n C a

) 1 3

(111)

in which, K n and C a are rigidity value and basic dynamic load rating of the ball nut. These values are specified by man-

ufacturers. The value of K n is obtained from the elastic deformation between screw grooves and balls when an axial load

equivalent to 30% of the basic dynamic load rating is applied. Note that, in this equation, the rigidity is considered roughly

80% of the value in the specification table. Additionally, for not preloaded ball nut type, n = 0 . 3 and f a is axial load applied

to the ball nut. Conversely, for preloaded ball nut type, f a 0 ≤ 1 / 3 f max a where f a 0 is applied preload and f max

a is the maximum

axial load applied to feed-screw system. In other words, the maximum preload should not exceed 10% of C a . Therefore, the

axial rigidity for a preloaded ball nut type can be calculated as

k ax Nut = 0 . 8 K n

(f a 0

0 . 1 C a

) 1 3

(112)

C.5. Calculation of the compliance matrix of the passive joints, C PJ

Using Castigliano’s 2nd theorem and Eq. (21) and Eq. (41) , vectors δχPJ and δψ PJ , can be calculated as

δχPJ k =

∂ U PJ

∂ f ext k

=

3 ∑

i =1

1

k rad R θP i

(f v 2 i R θP i

∂f v 2 i R θP i

∂ f ext k

+ f w 2 i R θP i

∂f w 2 i R θP i

∂ f ext k

)+

f v 2 i R ϕi

2k ax R ϕi

∂f v 2 i R ϕi

∂ f ext k

+

f w 2 i R ϕi

2k rad R ϕi

∂f w 2 i R ϕi

∂ f ext k

+

1

k rad R λi

(f u 1 i R λi

∂f u 1 i R λi

∂ f ext k

+ f v 1 i R λi

∂f v 1 i R λi

∂ f ext k

)+

f w 1 i R λi

k ax R λi

∂f w 1 i R λi

∂ f ext k

+

1

k t R λi

(m

u 1 i R λi

∂m

u 1 i R λi

∂ f ext k

+ m

v 1 i R λi

∂m

v 1 i R λi

∂ f ext k

)+

f Nut i

k ax Nut i

∂ f Nut i

∂ f ext k

+

f SB i

k eq SB i

∂ f SB i

∂ f ext k

δψ PJ k =

∂ U PJ

∂ m ext k

=

3 ∑

i =1

1

k rad R θP i

(f v 2 i R θP i

∂f v 2 i R θP i

∂ m ext k

+ f w 2 i R θP i

∂f w 2 i R θP i

∂ m ext k

)+

f v 2 i R ϕi

2k ax R ϕi

∂f v 2 i R ϕi

∂ m ext k

+

f w 2 i R ϕi

2k rad R ϕi

∂f w 2 i R ϕi

∂ m ext k

+

1

k rad R λi

(f u 1 i R λi

∂f u 1 i R λi

∂ m ext k

+ f v 1 i R λi

∂f v 1 i R λi

∂ m ext k

)+

f w 1 i R λi

k ax R λi

∂f w 1 i R λi

∂ m ext k

+

1

k t R λi

(m

u 1 i R λi

∂m

u 1 i R λi

∂ m ext k

+ m

v 1 i R λi

∂m

v 1 i R λi

∂ m ext k

)+

f Nut i

k ax Nut i

∂ f Nut i

∂ m ext k

+

f SB i

k eq SB i

∂ f SB i

∂ m ext k

(113)

where δχPJ = δχPJ k and δψ PJ = δψ PJ k

for k = 1 , 2 , 3 and δs PJ = { δχT PJ δψ

T PJ } T is a 6 × 1 vector. Substituting the reaction

forces/moments of passive joints as functions of external wrench from Eqs. (7) , (9) and (14) into Eq. (113) and factoring w


yields

δχPJ k =

3 ∑

i =1

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 k rad

R θP i

(J w

R θP ( 2 i −1 ) ( k ) J w

R θP ( 2 i −1 ) ×( 1 ∼6 ) + J w

R θP ( 2 i ) ( k ) J w

R θP ( 2 i ) ×( 1 ∼6 )

)+

1 2k ax

R ϕi

(J w

R ϕ ( 2 i −1 ) ( k ) J w

R ϕ ( 2 i −1 ) ×( 1 ∼6 )

)+

1 2k rad

R ϕi

(J w

R ϕ ( 2 i ) ( k ) J w

R ϕ ( 2 i ) ×( 1 ∼6 )

)+

1 k rad

R λi

(J fw

R λ( 3 i −2 ) ( k ) J fw

R λ( 3 i −2 ) ×( 1 ∼6 ) + J fw

R λ( 3 i −1 ) ( k ) J fw

R λ( 3 i −1 ) ×( 1 ∼6 )

)+

1 k ax

R λi

(J fw

R λ( 3 i ) ( k ) J fw

R λ( 3 i ) ×( 1 ∼6 )

)+

1 k t

R λi

(J mw

R λ ( 2 i −1 ) ( k ) J mw

R λ ( 2 i −1 ) ×( 1 ∼6 ) + J mw

R λ ( 2 i ) ( k ) J mw

R λ ( 2 i ) ×( 1 ∼6 )

)+

1 k ax

Nut i

(J w

Nut ( i ) ( k ) J w

Nut ( i ) ×( 1 ∼6 )

)+

1 k eq

SB i

(J w

SB ( i ) ( k ) J w

SB ( i ) ×( 1 ∼6 )

)

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

w

δψ PJ k =

3 ∑

i =1

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 k rad

R θP i

(J w

R θP ( 2 i −1 ) ( 3+ k ) J w

R θP ( 2 i −1 ) ×( 1 ∼6 ) + J w

R θP ( 2 i ) ( 3+ k ) J w

R θP ( 2 i ) ×( 1 ∼6 )

)+

1 2k ax

R ϕi

(J w

R ϕ ( 2 i −1 ) ( 3+ k ) J w

R ϕ ( 2 i −1 ) ×( 1 ∼6 )

)+

1 2k rad

R ϕi

(J w

R ϕ ( 2 i ) ( 3+ k ) J w

R ϕ ( 2 i ) ×( 1 ∼6 )

)+

1 k rad

R λi

(J fw

R λ( 3 i −2 ) ( 3+ k ) J fw

R λ( 3 i −2 ) ×( 1 ∼6 ) + J fw

R λ( 3 i −1 ) ( 3+ k ) J fw

R λ( 3 i −1 ) ×( 1 ∼6 )

)+

1 k ax

R λi

(J fw

R λ( 3 i ) ( 3+ k ) J fw

R λ( 3 i ) ×( 1 ∼6 )

)+

1 k t

R λi

(J mw

R λ ( 2 i −1 ) ( 3+ k ) J mw

R λ ( 2 i −1 ) ×( 1 ∼6 ) + J mw

R λ ( 2 i ) ( 3+ k ) J mw

R λ ( 2 i ) ×( 1 ∼6 )

)+

1 k ax

Nut i

(J w

Nut ( i ) ( 3+ k ) J w

Nut ( i ) ×( 1 ∼6 )

)+

1 k eq

SB i

(J w

SB ( i ) ( 3+ k ) J w

SB ( i ) ×( 1 ∼6 )

)

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

w (114)

As shown in Eq. (43) , the relationship between δs PJ and C PJ is obtained by rewriting Eq. (114) in matrix form.

C.6. Calculation of the compliance matrix of the motor-gearbox assemblies, C AJ

Using Castigliano’s 2nd theorem and Eq. (21) , as well as substituting Eq. (15) and (16) into Eq. (46) and factoring w ,

vectors δχAJ and δψ AJ , can be calculated as follows

δχAJ k =

∂ U AJ

∂ f ext k

=

3 ∑

i =1

τM i

k t M i

∂ τM i

∂ f ext k

+

f belt i

k eq

belt i

∂ f belt i

∂ f ext k

=

3 ∑

i =1

(1

k t M i

J w

M ( i ) ( k ) J w

M ( i ) ×( 1 ∼6 ) +

1

k eq

belt i

J w

belt ( i ) ( k ) J w

belt ( i ) ×( 1 ∼6 )

)w

δψ AJ k =

∂ U AJ

∂ m ext k

=

3 ∑

i =1

τM i

k t M i

∂ τM i

∂ m ext k

+

f belt i

k eq

belt i

∂ f belt i

∂ m ext k

=

3 ∑

i =1

(1

k t M i

J w

M ( i ) ( 3+ k ) J w

M ( i ) ×( 1 ∼6 ) +

1

k eq

belt i

J w

belt ( i ) ( 3+ k ) J w

belt ( i ) ×( 1 ∼6 )

)w (115)

where δχAJ = δχAJ k and δψ AJ = δψ AJ k

for k = 1 , 2 , 3 and δs AJ = { δχT AJ

δψ

T AJ } T is a 6 × 1 vector. As shown in Eq. (47) , the

relationship between δs AJ and C AJ is obtained by rewriting Eq. (115) in matrix form.

Appendix D

In the following, the stiffness coefficients of all compliant active and passive joints are obtained. To do this, the bearings’

specifications of some manufacturers are used.

D.1. Axial rigidity of the ball nut

As previously mentioned, for the preloaded ball screw type, k ax Nut

is a function of applied preload to the ball nut and basic

dynamic load rating C a . Additionally, to select a permissible value for preload, the criteria of f a 0 ≤ 0.1 C a must be considered.

For the selected ball screw, f a 0 = 0 . 08 C a is considered. Therefore, using Eq. (112) and ball screw specifications which are

expressed in Table 1 , the value of k ax Nut

is equal to 505 N/μm.

D.2. Axial rigidity of the support bearings of screw shaft

Since the axial force is a dominant force on feed-screw system, the angular contact bearings are usually utilized to sustain

this axial load. The ends of screw shafts can be supported as the standard methods such as fixed-free, fixed-supported

and fixed-fixed mounting methods. As shown in Fig. 23 , the fixed-fixed method is utilized to support the ends of screw


Fig. 23. Specification of screw shaft and structure of support bearing units of i th feed-screw system.

Fig. 24. Effect of applying a preload on increasing the bearing stiffness.

Table 7

Dimensional parameters and performance data for duplex mounted angular contact light ball bearing.

Bearing set D (mm) D (mm) C (mm) Contact angle (deg.) Ball QTY

× Dia. (mm)

C a (N) Preload

(N)

Axial

stiffness

(N/μm)

Radial

stiffness

(N/μm)

Support bearings 12 24 6 15 13 × 3.2 3690 15 17.32 76.96

R λ- joint 15 28 7 25 13 × 3.6 4360 45 46.4 96.9

shafts. These support bearings compose of two rows of angular contact ball bearing type with contact angle 15 ° which are

assembled as face-to-face method (DF type). Using this combination, the axial load can be sustained with two bearings.

As stated earlier, applying a preload can enhance the rigidity of support bearings and can reduce noise level. The effect of

preload on increasing stiffness of bearings is depicted in Fig. 24 .

The rigidity of support bearings can be obtained as a function of applied preload. Some manufacturers present the axial

and radial stiffness of support bearings’ set and corresponding applied preload in their specification tables. The dimensional

and performance data for two types of duplex mounted angular contact light ball bearing (DUL) is indicated in Table 7 .

Using Table 7 , the axial stiffness of support bearings of i th

th screw shaft are obtained as k ax SB 1 i

= k ax SB 2 i

= 17 . 32 N/μm using

Eq. (110) , k eq SB i

= 34 . 64 N/μm


Fig. 25. Bearing’s specification of the i th R λi -joint.

Fig. 26. Calculation of L eff for some important bearings combinations.

D.3. Axial, radial and angular (tilting) Rigidity of the R λ-joints

In general, a single ball bearing (deep groove or angular contact ball bearing) is not intended to be loaded with a mo-

ment. Therefore, the moments on a shaft should be withstood by combinations of duplex pairs, triplex set or quadruplex set

of bearings with back-to-back (DB) or face-to-face (DF) or tandem (DT) arrangements. As shown in Fig. 25 , R λ-joints com-

prise of two angular contact ball bearings with contact angle 25 ̊ which are arranged as DB type. The DB type can sustain

relatively large tilting moments. Therefore, in DB type, the tilting deformation is smaller than DF arrangement. As shown in

Table 7 , the axial and radial rigidity of DUL ball bearing which is used for the i th

th R λi -joint can be obtained as k rad R λi

= 96 . 9

N/μm and k ax R λi

= 46 . 4 N/μm.

Note that in a set of bearings comprise of n bearings, the equivalent radial stiffness is sum of radial stiffness of all

bearings. This means that n bearings stiffness act as parallel springs. Furthermore, tilting stiffness of a set of bearings can

be estimated from effective radial stiffness and effective bearing width. This yield

k t R λi ∼=

0 . 25 k rad R λi L eff

2 (116)

where, L eff is the effective bearing width which is obtained as depicted in Fig. 26 . Therefore, for duplex pairs of angle contact

bearing for DB arrangement and considering Table 7 and Fig. 26 , L e f f = 16 . 086 mm and k t R λi

= 6268 . 45 Nm/rad.

D.4. Axial and radial rigidity of the R ϕ-joints and radial rigidity of the R θ P −joints

The deep groove ball bearings have good capacity to carry combined loads as radial (lateral) and axial (thrust) in both

directions. Therefore, a set of two single ultra-light deep groove ball bearings are used to sustain both axial and radial loads

in the R ϕ-joints (see Fig. 27 ). The dimensional and performance data for this bearing is shown in Table 8 . It is noteworthy

that the stiffness coefficient of a bearing is a nonlinear function of applied load on the bearing. Most researches are focused

on obtaining the relationship between the applied external load on the bearing and directional stiffness of the various

bearings [ 38 , 39 ]. In this paper, the results of previous researches are used to estimate the stiffness of deep groove bearings.


Fig. 27. Bearing’s specification of the i th R ϕi –joint and R θ i P i -joint.

Table 8

Dimensional parameters and performance data for single ultra-light deep groove ball bearing and linear-rotary ball bushing bearing.

Bearing set D (mm) D (mm) C (mm)

Dynamic

Load rating

C a (N)

Static Load

rating C 0 (N)

Axial

stiffness

(N/μm)

Radial

stiffness

(N/μm)

R ϕ1&2 - joint 10 26 8 5160 1960 12.5 58.74

R θP - joint 12 21 30 412 598 − 65.2

Fig. 28. Specification of the timing belt.

Therefore, a constant mean value is considered for both axial and radial stiffness coefficients of the deep groove ball bearing.

These values are indicated in Table 8 . Therefore, using Table 8 , the axial and radial stiffness coefficients of the i th

th R ϕ-joint

are obtained as k ax R ϕ 1 i

= k ax R ϕ 2 i

= 12 . 5 N/μm and K

rad R ϕ1 i

= K

rad Rϕ2 i

= 58 . 74 N/μm.

The stiffness of two deep groove bearings acts as parallel springs. Therefore, the equivalent stiffness coefficients of ax-

ial and radial stiffness of i th

th R ϕ-joint are k ax R ϕ i

= 25 N/μm and k rad R ϕ i

= 117 . 48 N/μm. Furthermore, linear-rotary ball bush

bearings are considered for R θ P-joints. This type of linear bushing bearing is used for the cylindrical joints. In this bear-

ing, both linear and rotating motions can occur separately or simultaneously. The specifications of a linear-rotary bearing

are shown in Fig. 27 and Table 8 . Therefore, using Table 8 , the radial stiffness coefficient of i th

th R θ i P i -joint is obtained as

k rad R θP i

= 65 . 2 N/μm.

D.5. Torsional rigidity of the servo motors and tensile stiffness of the timing belt

As stated in [5] , the equivalent torsional stiffness of the servo motor can be considered as k t M

= 6875 . 5 Nm/rad. Addition-

ally, the timing belts are manufactured with thermoplastic polyurethane with high-performance steel cords. The steel cords

provide the high positional accuracy and high structural stiffness for positioning applications in fore/torque transmission sys-

tems of the actuators of the robot. The total stiffness of the timing belt depends on specific stiffness, belt width and length

of the tight and slack side of the belt. For this type of timing belt, the specific stiffness is considered as c sp = 19 . 085 N/μm

using the manufacturer’s specification. Therefore, using Table 1 and Fig. 28 as well as the distance between two pulleys’

centers, the equivalent tensile stiffness coefficient of the timing belt is obtained as k eq

belt = 18 . 04 N/μm.


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