Eindhoven University of Technology MASTER Multi-input ...Eindhoven, December 13, 2004 . Summary Reducing vibrations in precision motion systems can be achieved by adding structural

Eindhoven University of Technology

MASTER

Multi-input feedforward design for vibration reduction in motion systems

Sanders, R.M.W.

Award date:2005

Link to publication

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https://research.tue.nl/en/studentthesis/multiinput-feedforward-design-for-vibration-reduction-in-motion-systems(2c93b674-1b91-45cc-835f-963a32d55347).html

Multi-input feedforward design for vibration reduction in motion

systems Master's thesis

R.M.W. Sanders DCT 2004.124

Supervisor: Prof. Dr. Ir. M. Steinbuch

Coaches: Dr.Ir. M. J.G. v.d. Molengraft Ir. M.G.E. Schneiders

Eindhoven University of Technology Department of Mechanical Engineering Control Systems Technology Group

Eindhoven, December 13, 2004

Summary

Reducing vibrations in precision motion systems can be achieved by adding structural stiffness and thus weight. In motion control, active suppression of resonance vibrations is realized by proper control design. Both feedforward and feedback control techniques are used, both having different approaches toward vibration reduction in flexible systems. Reducing the weight of motion systems will make systems more flexible, but adding more actuators can actively improve the flexible behavior of these motion systems. These systems are called over-actuated motion systems. Over-actuated motion systems are systems with more actuators than actually needed for rigid body movements. Both feedforward and feedback control strategies must be properly designed to correct for the mechanical flexibilities. This thesis focusses on feedforward design for multi-input motion systems in order to reduce vibrations. Various feedforward methods are discussed and their suitability for multi-input flexible motion systems is investigated. A model of a flexible beam motion system is used as a benchmark system to test different strategies. Three feedforward techniques are investigated further: (1) input shaping, (2) jerk derivative feedforward and (3) spatial feedforward. Input shaping is a method to reduce vibrations and is implemented by convolving the desired command signal with a sequence of impulses. The result of the convolution is used to drive the system. Simulation results show that input shaping improves the residual vibration behavior. The number of actuators and trajectory type can affect the input shaping results. Third order reference trajectories combined with input shaping have a better performance compared to second order reference trajectories with input shaping. An increase in the number of actuators also has a positive influence, because delay times can be reduced by adding additional actuators. Jerk derivative feedforward is an extension on classical acceleration feedforward. The derivative of the jerk is used in addition to the conventional acceleration feedforward to compensate for low frequency flexible behavior. Tracking performance can be improved in motion systems with mostly low frequency reference trajectories and relatively high resonance dynamics. The resonance frequencies of the flexible beam used here are too low for jerk derivative feedforward to have results comparable with input shaping. Additional simulations prove that jerk derivative feedforward becomes better compared to input shaping if the frequency of the dominant resonance is higher. Over-actuation makes it possible to avoid excitation of dominant resonances by designing a spatially optimized feedforward. A static feedforward gain vector is designed to establish mode cancelation. Results show cancelation of dominant modes, but high resonances are still present. A integrated method of a spatial feedforward and input shaping is derived to use the benefits of both methods. Significant improvements are made with respect to time-delay and vibration reduction. Real-time experiments are performed on a flexible beam setup. To obtain the modal pa-

rameters of the system and to verify the simulation model, a modal analysis is carried out. Since the flexible beam is fixed to the ground with weak springs, rigid body modes are not completely free of stiffness. Both "rigid body" modes do now posses a low frequency. With- out a correction, residual vibrations are dominated by the low frequent "rigid body" modes. Therefore, input shaping on the setup is implemented to reduce residual vibrations of these modes.

Summary (Dutch)

Het reduceren van trillingen in precisie motion systemen kan worden bereikt door het toevoegen van structurele stijfheid en dus massa. Met behulp van regeltechniek kunnen geschikte rege- laars ontworpen worden om actief resonanties te onderdrukken. Een regelaar kan bestaan uit een voorwaarts-sturing (feedforward), terugkoppeling (feedback) of een combinatie van beide. Voorwaarts-sturingen en terugkoppelingen zijn twee verschillende technieken, die beide op een andere manier trillingen kunnen reduceren. Het reduceren van het gewicht van mechanische systemen maakt deze systemen flexibeler, maar door toevoeging van extra actuatoren kan het stijfheidsgedrag van het systeem actief verbeterd worden. Deze over-geactueerde systemen zijn systemen waarbij meer actuatoren gebruikt worden dan noodzakelijk zijn om het gewenste rigid body gedrag te bewerkstelligen. Voorwaarts-sturingen en terugkoppelingen kunnen zodanig ontworpen worden om trillingen te verminderen ten gevolge van mechanische flexibiliteiten. Dit onderzoek richt zich op het ontwerpen van voorwaarts-sturingen voor flexibele multi-input motion systemen om trillingen te reduceren. Verschillende voorwaarts-sturing technieken worden besproken en hun geschik- theid met betrekking tot multi-input motion systemen wordt onderzocht. Een model van een flexibele balk is gebruikt om verschillende technieken te testen. Drie techniek zijn verder onderzocht: (1) input shaping (2) jerk derivative feedforward en (3) spatial feedforward. Input shaping is een methode om rest trillingen in dynamische systemen te reduceren. Input shaping convolueert impuls reeksen met de oorspronkelijke ingangssignalen. Het resultaat van de convolutie levert nieuwe ingangssignalen op, die systeem resonanties reduceren. Het aantal actuatoren en het referentie pad beinvloeden de verkregen resultaten. Een derde-orde referentie pad levert in combinatie met input shaping een betere onderdrukking van resonanties op, in vergelijking met input shaping en een tweede-order referentiepad. Bij een toenemend aantal actuatoren worden de resultaten ook beter, want de tijdvertragingen worden aanzien- lijk verminderd. Jerk derivative feedforward is een uitbreiding op de traditionele massa voorwaartskoppel- ing. De afgeleide van de jerk wordt toegevoegd aan de massa voorwaarts-sturing om te compenseren voor laag frequent flexibel gedrag. Het volggedrag kan verbeterd worden voor motion systemen met voornamelijk laag frequente referentie trajectories en relatief hoge res- onante dynamica. De resonantie frequenties van de flexibele balk in dit onderzoek hebben te lage resonantie frequenties, waardoor de resultaten van jerk derivative feedforward niet vergeleken worden met de resultaten van bijvoorbeeld input shaping. Additieve simulaties bewijzen dat jerk derivative feedforward beter gaat presteren in vergelijking met input shaping naarmate de resonantie frequenties van de dominante flexibele modes hoger liggen. Over-actuatie maakt het mogelijk om dominante resonanties te vermijden met een spatial geoptimaliseerde feedforward. Een statische versterkingsvector kan ontworpen worden om resonanties te vermijden. Resultaten laten onderdrukking van dominante resonanties zien,

maar hogere orde resonanties blijven aanwezig. Een geconibineerde methode van spatial feedforward en input shaping is afgeleid om ook hogere orde resonanties te onderdrukken. De voordelen van beide technieken worden benut en aanzienlijke verbeteringen worden behaald m.b.t het onderdrukken van resonanties. Een flexibele balk opstelling is gebruikt om experimenten uit te voeren. Een modale analyse is uitgevoerd om de modale parameters te bepalen en om het simulatie model t e verifieren. De flexible balk is met vijf sprieten bevestigd aan de vaste wereld, waardoor het gedrag van de rigid body modes verandert. Beide "rigid body" modes vertonen nu laag-frequent dynamisch gedrag. De rest trillingen in het systeem worden gedomineerd door deze "rigid body" modes. Input shaping is toegepast op de flexibele balk om deze dominante trillingen te onderdrukken.

Contents

Summary i

... Summary (Dutch) 111

1 Introduction 1

2 Feedforward control in motion systems 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Model inversion techniques 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Command shaping techniques 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Recent developments 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Discussion 7

3 Flexible beam test-case 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Flexible beam system 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modal form 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Trajectories 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Second order trajectory 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Third order trajectory 13

4 Multi-input shaping 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Input shaping 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Multi-input shaper design 16 . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Multi-input shaper derivation 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Weighting function 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulation results 19

. . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Multiple-input input shaping 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Trajectory choice 23

5 MIMO jerk derivative feedforward 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Rigid body decoupling 27

. . . . . . . . . . . . . . . . . . . . . . 5.2 Jerk derivative feedforward derivation 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results 30

6 Feedforward design for over-actuated systems 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Spatial feedforward design 35

. . . . . . . . . . . . . . . . 6.2 Spatial feedforward combined with input shaping 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Simulation results 37

7 Experimental implementation 41 7.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.2 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.3 Experimental vibration reduction . . . . . . . . . . . . . . . . . . . . . . . . . 45

. . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Input shaping . two actuators 45 7.3.2 Input shaping . three actuators . . . . . . . . . . . . . . . . . . . . . . 46

8 Conclusions & Recommendations 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.i Conciusions 49

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Bibliography 5 1

A Modal parameters flexible beam 53

B Assembly of system matrices 55

C Actuator specifications 5 7

D Multi-input inverse dynamics feedforward 59 D.l Single input feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 D.2 Multi input feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter 1

Introduction

Control of vibrations in precision motion systems becomes very important as designers at- tempt to obtain high performance. Typical examples of such motion systems can be found in robotics, pick & place systems and wafer scanners. Vibrations in motion systems are traditionally reduced by increasing the structural stiffness of the construction. Higher stiffness results in systems with more weight and increased size. Adding structural stiffness is a passive way of raising performance. This trend is not without costs, because the amount of energy to make movements increases quadratically and subsequently, the temperature losses increase even more. Reducing weight of motion systems will make systems flexible, but adding more actuators can actively improve the stiffness behavior of flexible motion systems. Motion systems with more actuators than strictly needed to perform a rigid body movement are called over-actuated systems. Control engineers have to design proper vibration control strategies to reduce vibrations in flexible motion systems. Even with advanced feedfack controllers, it is difficult to make rapid motion tasks without vibrations. In motion systems, controllers are in general a combination of a feedforward and feedback part to obtain maximum performance. The main difference between both controllers is that a feedforward action is applied before an error has occurred, where a feedback is generated on the basis of a measurement error.

Research to reduce unwanted vibrations started in the late fifties and it is still a well-known research topic. This research focusses on vibration reduction of multiple input flexible motion systems using feedforward control. A flexible beam benchmark system is used to illustrate the different approaches. Different feedforward techniques are discussed and implemented to analyse the reduction of residual vibrations.

Chapter 2 discusses several feedforward techniques, and also focusses on their suitability for residual vibration control of flexible motion systems. In the next chapter, the model and specifications of the flexible beam system are presented. Chapters 4, 5 and 6 will present successively the derivation and simulation results of multi-input shaping, jerk derivative feedforward and spatial feedforward. In Chapter 7, an experimental case study is presented on a flexible beam. Finally, conclusions are drawn and recommendations are given for further research.

2 Introduction

Chapter 2

Feedforward control in motion systems

Suppression of resonances during motion control is important in many mechatronic systems such as flexible robots. The traditional approaches to reduce the effect of resonances focus on either increasing the structural stiffness (which increases the systems size and weight) or use closed-loop control methods, for example adaptive control [HT97]. Feedback methods have the disadvantage that the controller does not react t c resonance disturbances before a control error has occurred. An alternative approach for reducing resonances is the use of a feedforward controller. A pre-specified excitation is designed so that the system moves exactly to the desired end position without exciting resonances. For motion systems in which resonances are excited primarily by actuator inputs, a feedforward technique such as input shaping typically proves to be more convenient and effective than feedback approaches. The design of a feedforward controller arises directly from the model of the controlled system. The better the model represents the behavior of the system, the better the resulting feedforward controller will perform. Since methods in this feedforward category are traditionally considered to be quite sensitive to variations of the system dynamic parameters, much research has been done to increase their robustness features. Several feedforward methods are presented and discussed in this chapter. First, the methodes are classified into model inverse based feedforward (Section 2.1) or command shaping feedforward (Section 2.2). In Section 2.3 three recent developments in feedforward control are presented. Finally, the suitability of each method for feedforward control of flexible motion systems is discussed in Section 2.4.

2.1 Model inversion techniques

Inverse model techniques are mainly designed to improve tracking performance instead of directly suppressing resonances. Acceleration feedforward is the most actual inverse model technique that is used in motion control systems. The acceleration profile of the reference trajectory is multiplied by the mass of the system to obtain the feedforward control signal. The main problem is that settling times are long for flexible motion systems, because low damped resonances are excited.

Another inverse model feedforward technique is computed torque control [Bay88, TBOO,

4 Feedforward control in motion systems

AMTSO]. The desired output trajectory is specified, the resulting input is calculated using the inverted model. Implementing computed torque is not straightforward, since several problems can occur. First of all, inverting the model is not always possible. Moreover, if the trajectory is impossible to follow, no correct input is calculated. Thirdly, a detailed model is required and fourthly, the method is not robust for system parameter errors. Finally, this technique can result in large move time penalties, because the plant inversion process results in an non-causal input.

Iterative Learning Control (ILC) is a data based feedforward technique that also uses inverted system information. ILC uses experimental data to adjust feedforward signals. It can be used for compensating plant dynamics, but also for systematic errors. The error information has to be stored, filtered and added to the previous feedforward. Filtering requires an inverse model. Calculating the inverse model is often complicated, because the inverse is often non-causal or even unstable. However, techniques were developed to calculate a stable estimation of the inverse model [Tom87]. The calculated feedforward is only applicable for the specific motion profile used during off-line learning. Recent studies show that ILC is capable of removing residual vibrations well within half the period of the dominant system vibration [vOBD04].

2.2 Command shaping techniques

Command shaping generally refers to methods of vibration suppression through manipulating desired trajectory based on system information. The earliest form of command shaping is based on high-speed cam profiles as motion templates. These input shapes are generated so as to be continues throughout one cycle. Their smoothness reduces unwanted dynamics by not feeding the system with high frequency inputs. Another early form of command shaping is posicast control [TS58, Coo661. This technique involves breaking a step of a certain magnitude into two separated steps, of which one is delayed in time. This results in a response with reduced settling time, because residual vibrations are canceled by the delayed step. Generally, this method is not used because of robustness problems. Only one system resonance can be canceled with this technique if the resonance is known exactly.

P.H. Meckl and W.P. Seering present another approach, [MS88, MS85] in the eighties. They construct input functions based on either ramped sinusoids or versine functions. This approach involves adding up harmonics of one of these template functions. If all harmonics were included, the input would be a time optimal rectangular input function. The harmonics that have significant spectral energy at the natural frequencies of the system are removed. The modified input approximates the rectangular shape, but does not significantly excite the resonances. In this way, residual vibration amplitudes are minimized with good robustness properties, although move times are at least 20 percent longer than optimal times for most moves [MS88, MS851. Fig. 2.1 illustrates an example of a forcing function for a given input signal. The forcing function is constructed by adding up several ramped sinusoids.

The most common command shaping technique is input shaping, which is still a research topic. This technique started in the early nineties, originally by N.C. Singer and W.P. Seer- ing [SS90]. Implementing this technique is done by convolving a sequence of impulses, known as the input shaper, with the desired system input to obtain a shaped input that is used

2.2 Command s h a ~ i n e techniaues 5

Original input Forcing function input

Figure 2.1: Forcing function illustration

- Shaped Input

I - - - A, Component

Unshaped Input Input Shaper Shaped Input

Figure 2.2: Input shaping process

to drive the system. Fig. 2.2 illustrates input shaping by convolving a desired input with an arbitrary impuls sequence. The time penalties resulting from convolving the input with the shaper equals the length of the impulse sequence. The impulse sequence is derived by solving a set of constraint equations, which represent the residual vibrations of the system. Zero vibration (ZV) constraints cancel vibrations only if the systems natural frequencies and damping coefficients are exact ([SSS95]). To increase robustness under variations of the systems natural frequency and damping, the derivatives of the zero vibration constraints (ZVD) can be set to zero. The process of adding robustness can be extended further by setting the second order derivatives of the zero vibration constraints to zero. By limiting the constraints to a small value, the zero vibration constraint can be enforced to be zero at two frequencies close to the modeling frequency. This alternative constraint description is called extra in- sensitive (EI) and tolerates typically 5% vibration around the modeling frequency ([SSS95]). The difference between both constraint descriptions is visualized in Fig. 2.3. Input shaping is primarily used for single input systems, but L.Y. Pao extended it for multi- input systems [Pao99]. Pao uses the property that multiple inputs can be used to cancel out one resonance. This results in less time delay. Advantages of input shaping are it's simplicity, the optimality of the solution and the ability to be used with arbitrary actuator commands.

Time optimal control approaches have also been used to generate input profiles for flexible systems to reduce resonances. Time-optimal control for motion systems results in series of alternating sign pulses, for which the amplitude of the pulses is equal to the maximum actuator effort. There are three drawbacks of these optimal control approaches. First, com-

6 Feedforward control in motion svstems

0.6 0.7 0.8 0.9 I 1.1 1.2 1.3 1.4

Normalized Frequent y f @aclual/ O S ~ ~ ~ ~ ~ )

Figure 2.3: Robustness of different methods

putation is difficult and each motion of the system requires re-computation of the input. Although simplifkations to this step were made during the late nineties, solving the optimal approach for complex systems is still extremely difficult. Second, the penalty function does not explicitly contain a direct measure of vibration. The tracking error is used in the penalty function. Therefore all errors are packed together. Besides that, optimal control allows also the system to vibrate slightly during the move. Third, the value of optimal input strategies depends on move times, different moves will have different vibration amplitudes. Despite these drawbacks, it has been proven that minimum time zero vibration (ZV) and zero vibration derivative (ZVD) input shapers are equivalent to traditional time optimal control, see [LP03].

2.3 Recent developments

In this section, three recent developments in feedforward control are briefly presented. Namely, jerk derivative feedforward, spatial feedforward and multi-input inverse dynamics feedforward.

To improve tracking, rigid body feedforward control is used in industry. In [BTS04] a jerk derivative feedforward controller is proposed which takes also in account flexible dynamics. Herein, the derivative of the jerk is added to the traditional mass feedforward to compensate for low frequent residual tracking errors due to flexibilities. Jerk derivative feedforward is not designed on exact inversion of specific resonance frequencies but to compensate for the approximately 'static' contribution of flexible modes [Boe04, BTS041.

Over-actuated systems create more freedom in feedforward design compared to traditional actuated systems, [SvdMS04]. Minimal controllability of flexible modes can be created in

2.4 Discussion 7

Global(G) / local(L) performance

L Computed torque ILC Posicast

Method

Mass Feedforward

Forcing functions Input shaping Optimal time Jerk derivative F F

Table 2.1: Feedforward overview

Multiple resonances

-

Robust

n/a

n/a -

-

Spatial feedforward Multi-inmt inverse F F

over-actuated systems. The spatial feedforward technique uses a static gain relation between the actuators to avoid excitation of dominant flexible modes. The modal representation of the system has to be known, because modal gains are used to obtain the static gain relations between the actuators. Increasing the number of actuators will also increase the number of flexible modes that can be included in the design. Robustness against modeling errors is not created with this feedforward method.

Implementation

+

+ + +

Plant inversion techniques have often problems with the determination of a stable and proper transfer functions. Appendix D introduces a case study about multi-input inverse dynamics feedforward. Proper and stable inverses can be determined for multi-input motion systems. The multi-input inverse dynamics feedforward can avoid excitation of flexible modes only locally by adding an extra feedforward term to the original feedforward. This method is not robust to variations in system parameters.

+ +

-

2.4 Discussion

+ + +

+/-

Every feedforward technique is designed for different purposes. Vibration reduction of multi- input flexible motion systems is the goal of this research project. As already mentioned, inverse model methods are not directly designed for global vibration reduction, but results don't have to be worse than the results of vibration reduction techniques. The suitability of each method is discussed with respect to multi-input flexible motion systems. Table 2.1 compares all feedforward techniques of Sections 2.1, 2.2 and Section 2.3 concerning four categories. First, robustness is checked since a feedforward signal has to be robust to model uncertainties, i.e. varying resonance frequencies. Second, the possibility to suppress multiple resonances is verified. Third, the ease of implementation is rated for each technique. Fourth, the methods are divided into global and local performance feedforwards. Global performance considers vibration reduction in the complete system and local performance is only focussed on vibration reduction in one point. Not every feedforward technique creates robustness to uncertainties in resonance frequencies, see Table 2.1. Mass feedforward and computed torque control are not evaluated in this robustness categorie since these methods only compensate for rigid body behavior. Forcing

+/- +

+ +

L L G

+ +

G G G L

+ + 1-

G L

8 Feedforward control in motion systems

functions, input shaping and optimal time control techniques create robustness, but move times with forcing functions are too high compared to input shaping and optimal time control. Mass feedforward and posicast control are not capable to cancel multiple resonance modes, which is required (Table 2.1). Jerk derivative feedforward compensates only for low frequency contributions of flexible modes. Optimal time control and input shaping are adequate techniques with global performance, but the main difference is made with respect to implementation. Input shaping proves to be an effective and easy technique to implement in this research area. Mainly, the ease of implementation compared to the other techniques makes input shaping attractive to use. Besides input shaping (Chapter 4), two other techniques are presented in this report. Chap- ter 5 presents a relatively new technique, MIMO jerk derivative feedforward. This techniques has not the best properties for vibration reduction compared to input shaping, but implementation is interesting since there are little published results with this technique. The second method, spatial feedforward design is presented in Chapter 6 and discusses the benefits of over-actuation in feedforward control. This chapter introduces the spatial feedforward and a combination of input shaping and a spatial feedforward for over-actuated systems. Substantial improvements can be made with respect to the performance of flexible motion systems.

Chapter 3

Flexible beam test-case

Performance of motion systems decreases due to the presence of residual vibrations. Analyzing residual vibrations in this research is done using a flexible beam setup with multiple actuators. In this chapter, the flexible beam setup is presented. Section 3.1 presents the equations of motion of the flexible beam system together with relevant specifications of the beam. A transformation from the equations of motion to the modal form is made in Section 3.2. Several reference trajectories are used to prescribe the displacements of the system. These trajectories are presented in Section 3.3 together with the desired performance specifications of the flexible beam setup.

3.1 Flexible beam system

The test case focusses on moving a flexible beam over a predefined distance r( t) with minimal residual vibrations. An adequate model has to be derived to carry out simulations. Consider a uniform Bernoulli-Euler beam of length L, modulus of elasticity E, density p, cross-sectional area A and moment of inertia I. Suppose that the beam is excited by an external, distributed force f (x, t). The transverse oscillations w(x, t) of the beam are governed by the differential equation, Eq.(3.1). Note that the model doesn't contain gravity. Relevant beam specifications are presented in Table 3.1.

A virtual presentationof the flexible beam is given in Fig. 3.1. Sensors are denoted by 5'1, 52 and S3 and the actuators are denoted by Actl, Act2 and Act3. In practice, fiber optic motion sensors are installed to measure deflections and voice coil actuators are used to drive

Table 3.1: Beam specifications

Value

kg/m3 ~ / m ~

m m m

Beam specifications Density

Modulus of elasticity Length Width Height

Symbol

P E L W H

Units 7800

2.1e11 0.5 0.02 0.002

10 Flexible beam test-case

Figure 3.1: Flexible beam test-case

I ' I ,

the flexible beam system at certain positions. The number of sensors is equal to the number of actuators and each motion sensor is collocated with an actuator.

3.2 Modal form

\

Act 3 z

The model in Eq.(3.1) can be spatially discretized using a finite element formulation. With the finite element formulation, mass and stiffness matrices of each element as well as the assembly of full mass and stiffness matrices of the flexible beam can be derived, see Appendix B. The full mass and stiffness matrices are used to define a Linear Time Invariant system (LTI) of the flexible beam. The Euler-Bernoulli beam can be written in a set of second order differential equations of the following form.

Herein M, B and K are the mass, damping and stiffness matrix of the Euler Bernoulli beam. Adding damping can cause coupling between resonances, but matrix B is chosen "weakly damped" and therefore this coupling can be discarded [dKvCOl]. F and C define the actuator and sensor positions on the beam. The transformation to modal coordinates r7 can be made using Eq.(3.3) and the modal matrix cP. Physically this implies that a motion of the beam can be regarded as a linear combination of the eigenmodes multiplied with the time dependent modal coordinate functions.

A

Eq. 3.2 is transformed into modal coordinates using the transformation relation. The transformed system is shown in Eq.(3.4).

Act 1

Where, I is the identity matrix and A, A are defined as,

Act 2

3.3 Trajectories 11

with wl, . . . , wn are the natural eigenfrequencies and C1,. . . , C, are the damping coefficients of the eigenmodes. For the lowest modes , this yields wl = w2 = Cl = C2 = 0. The input vector is u = [uo,ul , . . . ,u,IT, with m + 1 equal to the number of actuators. I, A and A are all diagonal matrices, which means that the coupled differential equations of Eq.(3.2) are decoupled into a set of n independent second order differential equations. Using the Laplace domain, the system can be rewritten in multiplicative modal form;

Here, @a and @, are input and output transformations and n is the number of eigenmodes. The input/output behavior of the system can now be written in additive modal form, which means a summation of rigid body modes and flexible modes,

" . - - \ -4

Rigid body modes flexible modes

where & and $2, are the ith column out of the corresponding and a s , idem for 4: and 4;. The flexible beam without suspension contains two rigid body modes, a lift mode and a tilt mode, and multiple higher resonances. Only the first five eigenmodes are taken into account during this research. See Fig. 3.2 for the mode shapes of the first five eigenmodes of the flexible beam.

3.3 Trajectories

In the case the beam should be moved to a desired position, with as little residual vibrations as possible according to a predefined reference trajectory. Furthermore, global performance is desired, which means that the whole beam has to be moved to the desired end position with as less residual vibration as possible. In this research two reference trajectories are used and both are briefly discussed in this section. Both trajectories prescribe a displacement from 0 to 2 mm in 0.12 s.

3.3.1 Second order trajectory

A second order reference can be determined by constructing an acceleration profile, integrating it once to obtain a linear velocity profile and integrating it once more to obtain a second order position profile. The acceleration profile of a second order trajectory consists of two constant acceleration phases for a single point to point movement. For a given travel distance (se - xo) , the maximum acceleration amplitude can be determined using Eq.(3.7). Fig. 3.3 shows the position-, velocity- and acceleration profile for a second order reference trajectory.


I I I

resonance 3 resonance 2

Figure 3.2: The first five eigenmodes

Second order trajectory I I

0 0.05 0.1 Ti, 0.15 time [s]

Figure 3.3: Second order trajectory prescription

3.3 Trajectories 13

Third order setpoint and time derivatives

H 1

xo 0 0.05 0.1 Tmp

time [s]

Figure 3.4: Third order trajectory prescription

The input for the system is a force Ft, which is obtained by multiplying the acceleration with the mass of the system, 0.156 kg. The actuator peak force boundary is not exceeded, see Appendix C and therefore the designed input is applicable.

3.3.2 Third order trajectory

A third order reference trajectory is also used to prescribe displacements of the system. Third order reference trajectories require the existence of the derivative of the acceleration. REFS, [vdM] is a third order reference trajectory generator for Matlab. This trajectory generator is used to design a valid third order reference trajectory for the flexible beam system. The third order setpoint and it's derivatives are presented in Fig. 3.4.


Chapter 4

Multi-input shaping

The performance of motion systems can be improved using feedforward techniques. Input shaping is one of these techniques and has proven to be very effective in a wide range of applications. This chapter focusses on the design of multi-input shapers for flexible motion systems. Section 4.1 presents a more detailed explanation about input shaping. Section 4.2 gives the derivation of the input shaping method for multi-input modal systems and Sec- tion 4.3 presents simulation results of multi-input shaping on the flexible beam.

4.1 Input shaping

A combination of an adequate feedback and feedforward design is often used to obtain the desired performance of a motion system. Feedback control methods use physical measurements to reduce tracking errors while feedforward techniques alter the actuator commands based on the trajectory with the same purpose. Input shaping is a feedforward technique that is used to reduce residual vibrations. Input shaping is implemented by convolving a sequence of impulses, an input shaper, with a desired system command to produce a shaped input. This shaped input can be used to drive a open- or closed loop system, which is visualized in a block scheme in Fig. 4.1. This research focusses only on open-loop input shaping design. However, if input shaping accounts for the dynamic characteristics of the closed-loop system, shaped input commands can be given to the closed-loop system as well. Shaped input signals are longer in time than the original unshaped signals, because the input commands are convolved with a sequence of impulses. The time difference between both signals is equal to the shaper length TN. The constraints used to design a shaper only depend on the frequency and damping of the resonance modes of the system. Traditional input shaping cannot cope with multiple inputs, i.e. the results will yield the same input shaper for each input. Taking into account that each

Figure 4.1: Block diagram of a system with a shaper

Input command Shaper

>

System

4.2 Multi-input shaper design 17

flexible poles of the system. Each shaper q, consists of a sequence of impulses and is assumed to be causa1,Eq. (4.5).

Here, b(t) is an impulse function with amplitude a and T is the impulse interval time. The impulse interval time T is assumed to be the same for each impulse in a calculated set of shapers. The number of zeros for each shaper is equal to L, ~ ~ . ( 4 . 6 ) ' and f m is the number of flexible modes that are included in the design.

The conversion of the time domain shaper, Eq.(4.5), into the laplace domain is given in Eq.(4.7).

Q,(S) = ao, + alTePT + . . . + aLTePSLT (4.7)

Eq.(4.7) presents the transfer functions of the input shapers Q, for r = 0,. . . , m. Substituting Eq.(4.7) into Eq.(4.4) yields Eq.(4.8). Solving this equation for the undesired system poles, i.e. undesired modes, will place the zeros of the input shapers on the undesired system poles.

Here si are the system pole locations for i = 1,2,. . . , n. Note that n is the number of undesired flexible modes and not the total number of system modes. Given L, T and Fa,, the impulse amplitudes, ak,, of the complete set of sequences should be calculated, where k = 0,. . . , L. To the set of Eq's.(4.8) extra equality constraints are added to prevent actuator saturation. The sum of the impulse amplitudes of each shaper is fixed to 1. The constraints of Eq.(4.8), along with the additional actuator saturation constraints can be written in matrix form, to give Eq.(4.9). The complex conjugated poles of si are denoted by sa.

herein,

K =

[. . .I: ceil, the nearest integer larger than the argument.

18 Multi-input shaping

time [s]

(a) Impulse sequence

time [s]

(b) Unshaped (. . .) and shaped input(-)

Figure 4.2: Input shaping for one actuator

The number of equations is equal to 2 f m + (m + 1) and the number of unknown impulse amplitudes is (L + l )(m + 1). There are an equal number of equations and unknown variables when L = 2 fm/(m + 1). Typically, the number of unknown variables is equal or greater than the number of equations, L > 2fm/(m + 1). For this reason, the impulse amplitudes, a, are solved using the pseudo inverse of K , (Eq.(4.10)).

The equations are solved for several impulse interval times, T, varied between 0.0001 and 0.1 seconds. If input shapers are allowed to contain negative impulses, [SS94] shows that the time duration of the shapers can be reduced. Negative shapers are disregarded since they have a tendency to excite higher modes of vibration, if the system has multiple modes. Also, negative shapers can cause saturation of the actuators. Therefore, amplitudes of impulses are not allowed to be negative. Valid input shapers have only positive impulse amplitudes and should introduce minimal delay. Thus, the input shaper with minimal delay and positive amplitudes is chosen for implementation. Given the unshaped inputs and input shapers, it is possible to calculate the shaped inputs. In Fig. 4.2, two figures are added to give an indication how an input signal is changed by convolution with an impulse sequence. Fig. 4.2(a) shows an impulse sequence with T=5.9 ms and Fig. 4.2(b) shows the unshaped input and resulting shaped input. Delay is equal to the shaper length TN, which is 11.8 ms. One can also see that the sum of the impulse amplitudes is equal to 1, because the unshaped signal is equal to the shaped signal in the steady-state intervals. Robustness for variations in eigenfrequencies and damping-coefficients can be created by placing multiple complex zeros on each system pole. Thus, the constraint equations presented in Eq.(4.8) have to be extended. Higher order robustness leads to more impulses, which

4.3 Simulation results 19

Table 4.1: Lower and upper boundaries for weighting components

Delay/extra setup time [s] Max residual vibration [m]

generally results in more time delay. However, it also leads to less sensitivity for variations near the corresponding system pole.

4.2.2 Weighting function

Lower boundary 0.005 (LBD) le-9 (LBE)

The case considered here requires the movement of a flexible beam in a predefined time to an end position with minimal residual vibrations. Input shaping reduces residual vibrations, but the setup time increases due to convolution of the input signal with the input shaper. The performance of input shapers is determined by two effects. Firstly, the amount of delay TN and secondly the maximum vibration amplitude after setup time T,, + TN. A weighting function is defined to weight these two effects and to give a quicker insight in the performance of different input shapers. The maximum residual vibration on the beam is obtained by comparing residual vibrations in each point on the beam. The maximum residual vibration is weighed against the delay using Eq.(4.11).

Upper boundary 0.06 (UBD) le-4 (UBE)

Max residual vibration

J = - Izog ( UBE (4.11)

Lower and upper boundaries are chosen for each weighting component, see Table 4.1. These boundaries are chosen based on simulation results and desired performances. The upper and lower boundary values in the weighting function, Eq.(4.11) are used to scale both values in the interval [0, 11. A linear function is chosen for the extra setup time and an exponential function for the maximum residual vibration. A summation of both quantities divided by two gives the resulting weighting value J. A weighting value of 1 represents a very good input shaper and a bad input shaper is represented by an output of 0. If weighting parameters exceed the defined boundaries, Eq.(4.11) cannot be interpreted. A visual interpretation of all possible weighting values as function of the delay and the maximum residual vibration is presented in Fig. 4.3.

4.3 Simulation results

The input shaping simulations are focussed on the influence of the number of actuators and trajectory choice. Resonance frequencies and damping coefficients of the flexible beam are known analytically but these frequencies and damping coefficients will never match exactly with the real system. The robustness properties of the multi-input shaper algorithm should deal with these uncertainties. From simulations it turned out that not every actuator arrange- ment results in valid shaper solutions. The large number of equations makes it difficult to give insight in what might be the reason for this problem. In general, one of the difficulties in designing input shapers for multiple mode models is solving the impulse sequence (Eq. (4.10)).


Maximum residual vibration ,1 x 1Q [m]

Figure 4.3: Weighting function

Numerical approaches are required and convergence is not always guaranteed.

4.3.1 Multiple-input input shaping

Input shaping simulation results for traditional actuation and over-actuation will be compared in this section. Over-actuated systems contain more actuators than rigid body modes. The flexible beam setup contains two rigid body modes, see Fig. 3.2. Suitable actuator configurations to analyse differences between traditional actuation and over-actuation are chosen and visualized in Fig. 4.4. Normalized positions on the beam for 2 and 3 actuators are respectively [O, 11 and [O, 1/2, 11. First, a simulation is done with two actuators and modal frequency and damping disturbances of 5% for resonance 1, 2 and 3. After that, a simulation is done with

Two actuators

Three actuators

/ \

Act 1 1 Act

Figure 4.4: Actuator configurations for 2 and 3 actuators

" Act

A

Act 2

A Act 3

4.3 Simulation results 2 1

time [s]

(a) two actuators

time [s]

(b) three actuators

Figure 4.5: Beam end deviation error

three actuators. Only resonance 1 and 3 are included in the input shaper design. From the simulations it turned out that resonance 2 is not excited at all. This is correct because both actuator configurations cannot excite this asymmetrical mode. All actuators are positioned in or symmetrical around the beam center (Fig. 3.2) and all actuation signals for actuator 1 and 3 are identical. The effective force is equally distributed over the actuators. A second order reference trajectory as described in Section 3.3 is desired to move the beam. The displacement errors in the beam end for a two and three actuator configuration are respectively shown in Fig. 4.5(a) and 4.5(b). This position is chosen, because this is the place where resonance 1 has it's maximum activity. Each figure contains two graphs with the displacement error as a result of input shaping and no input shaping.

Three actuators excite less residual vibrations, as can be seen in Fig. 4.5, compared to two actuators. Without input shaping three actuators reduce the error with 50-60%. The error signal without input shaping in Fig. 4.5(b) is not as smooth as in Fig. 4.5(a), because resonance 3 becomes visible due to a better suppression of resonance 1. The results for both actuator configurations are obviously better with input shaping instead of no input shaping. More precisely, input shaping reduces the maximum vibration after setup-time (Tznp+TN) with two actuators to 5.9x10-~ m and delay TN is 26 ms. Where Tznp is the time length of the unshaped input signal. The maximum residual vibration for two actuators without input shaping is 1 . 4 ~ m. For three actuators with input shaping, the maximum residual vibration is 1 . 1 ~ 1 0 - ~ m and without input shaping it is 6 . 5 ~ 10W6 m. The setup-time with three actuators is much shorter than the setup-time for two actuators. The delay with three actuators is 13 ms. Simulation results are shown together with weighting function outputs in Table 4.2. In Fig. 4.6 and Fig. 4.7 the impulse sequences are presented for both simulations together with the corresponding actuation forces. The configuration with two actuators has exactly the same impulse sequence for both actuators, because both actuators have the same modal contributions with respect to resonance 1 and 3. The modal contributions of the extra actuator (actuator 2) in the over-actuated situation differ from the contributions of actuator 1 and 3 and therefore the impulse sequence is different (Fig. 4.7).

Concluding, three actuators improve residual vibrations substantially without input shaping

Two Actuators

I

I Input shaping / 0.026 / 5.9 x / 0.53 1 Without input shaping

Three Actuators I Delay Is] I Error Iml I Weighting output I - ] I

Delay [s]

- - . , . > - - - .. / Without input shaping I / 6.5 x 1 -

I Input shaping 1 0.013 1 1.1 x I 0.62 I

Error [m] 1.4 x

Table 4.2: Simulation results with different actuator configurations

Weighting output [ - ] -

time [s]

Figure 4.6: Input shaping with two actuators


Actuator 1 and :

0.2

tixe [s]

Actuator 2 0.5 7 1

* -0.04 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25

time [s] time [s]

Figure 4.7: Over-actuated input shaping

compared to two actuators. With input shaping, three actuators have less vibration reduction, but the delay is much lower compared to two actuators. The delay with two actuators is high because the impulse sequence length has to be extended to guarantee an input shaper with positive impulse amplitudes. The number of additional impulses should be two (Eq.(4.6)), but is extended to L=4. Because of the extreme increase of delay, the overall performance of input shaping with three actuators is much better.

4.3.2 Trajectory choice

Excitation of resonances should be avoided as much as possible. The choice of a proper input signal is therefore crucial. Calculating power density spectra gives insight in the energy contents of a signal. High-order reference trajectories have much lower energy content at higher frequencies than low-order trajectories. Therefore, high-order trajectories excite high frequent resonances less than lower order trajectories. In Fig. 4.8 two power spectral density functions are presented for the command inputs of a second and third order reference profile. Both trajectories prescribe a displacement of 2 mm to be reached in 0.12 s, see Section 3.3. The input command for the third order trajectory has less energy contents at higher frequencies. Simulations are done with a second and third order reference trajectory, to illustrate the performance of the input shaper with different reference trajectories. Three actuators are positioned as described in Fig. 4.4. Again, resonances 1 and 3 are included in the input shaper design and modal hequencies and damping disturbances of 5% are added. Now input shaping simulations are carried out with first and second order robustness input shapers. The impuls sequences for both robustness criteria are calculated once. They can be used for both reference trajectories. Simulation results for both trajectories, are presented in Fig. 4.9 and Table 4.3. Fig. 4.9 shows also the beam deviation errors in a close up for each method on the right hand side.


2nd order ref

50 100 150 200

Frequency pz]

Figure 4.8: Power density spectra

Table 4.3: Simulation results with trajectory choice

2nd order ref

3th order ref

Delay

[s] 0.013 0.0128 0.013 0.0128

Error without input shaping [m]

6.7 x

4.1 x l0W7

Robustness order lSt 2nd lSt 2nd

Errorwith input shaping [m]

1.1 x l o r6 9.0 x 5.6 x 10W8 4.5 x l o r8

Weighting output

0.62 0.63 0.75 0.76


- Without input shaping

Input shaping, 2nd order robustness -

Input shaping, 1st order robustness - E - 5 - 5 .*ln ,

time [s] time [s] Figure 4.9:

Beam deviation errors for different trajectories - : 2nd order reference and - - :3th order reference

Fc

B LI a,

G ;= 0 6 .- > 4 E

r

-

g -5 -4 ~4 0 0.05 0.1 0.15 0.2 0.25 0.13 0.19 0.25


Improvements due to a third order reference trajectory are obvious visible in the results without input shaping. Residual vibrations are lowered with approximately 90 percent. The improvements of input shaping are also reasonable. Input shaping with first order robustness reduces the residual vibrations for a second and third order reference trajectory with respectively 83% and 86%. Second order robustness reduces the residual vibration behavior even more. For second order robustness, reduction of residual vibrations for second and third order reference trajectories are respectively 86% and 89%, see Table 4.3. A third order reference trajectory with input shaping performs better than a second order reference trajectory with input shaping. The improvements between lSt and 2nd order robustness are often intuitive. Residual vibrations are lowered but delay increases in general. The weighting function gives a good indication in these situations, but the user can always decide which shaper is suitable. The delay increases normally if robustness increases, because extra impulses are added to the input shaper. The simulations in this section show that it is possible to create more robustness with almost the same amount of delay.

Chapter 5

MIMO jerk derivative feedforward

Feedforward design for motion systems is used to improve tracking performance. An acceleration feedforward is widely used in the industry, because of its simplicity. However, results are limited since especially low frequent errors stay present during jerk phases of motion [Boe04, BTS041. The open loop error during motion is a scaled version of the acceleration profile and the acceleration feedforward cannot compensate for this error. [BTS04] proposed a jerk derivative feedforward, a feedforward technique which compensates for these low frequent errors. The open loop error equation is used to derive an additional term to the traditional acceleration feedforward. This addition is proportional with the fourth derivative of the position profile. A general representation of the new feedforward is presented in Eq.(5.1) and requires the existence of the fourth derivative of the position.

with, F* = 6s'

This chapter presents the implementation of MIMO jerk derivative feedforward on the flexible beam. The model has to be written in the additive modal form and has to be decoupled. De- coupling will be described in Section 5.1. In Section 5.2 MIMO the jerk derivative feedforward is derived and Section 5.3 discusses the simulation results.

5.1 Rigid body decoupling

Implementation of the jerk derivative feedforward requires decoupling of the modal plant into a set of independent SISO systems. Here, rigid body decoupling is discussed by means of actuator and sensor transformations, Ta and T,. This results in two independent SISO systems, namely a translation and rotation degree of freedom. The possibility of using the extra actuator to force mixed modal decoupling and rigid body decoupling is not implemented yet. This point is studied in [Boe04]. For now, the actuators are averaged to realize rigid body motions. It is important that both variables coincide in the center of gravity of the beam. If this is not the case, other techniques, such as gainscheduling, have to be used to map the variables to the center of gravity. Since the variables coincide, the plant can be expressed in degrees of keedom of the center of gravity GCog using Eq.(5.3):

28 MIMO jerk derivative feedforward

Figure 5.1: Rigid body model of the Euier Sernoulli beam

A rigid body model of the beam is shown in Fig. 5.1. Sensor transformations are derived using zcog = Tsy and actuator transformations are derived using F = TaFcog. ResuIting sensor and actuator transformations are given in respectively Eq. (5.4) and Eq. (5.5). The pseudo inverse f is used because Ta and Ts are not square.

T,i

Rigid body decoupling is realized using two inputs: a force F,,, and a torque TZcog, both acting on the center of gravity of the beam. The new outputs are translation z,,, and rotation RZcog of the center of gravity.

5.2 Jerk derivative feedforward derivation

Eq. (5.3) returns a rigid body decoupled plant. This plant is presented in Eq. (3.6) in additive modal form. Using Eq.(5.3) and the representation of the additive modal form, the plant can be written as a summation of two matrices GTb and Gflex.

The rigid body part GTb is a square diagonal matrix with dimension 2 x 2 . Diagonal elements contain the mass m and inertia J of the beam, see Eq.(5.7):

The flexible modes part Gflez is a non-diagonal matrix and can be written into Eq.(5.8),

5.2 Jerk derivative feedforward derivation 29

Bode Magnitude Diagram

10: I o2 I 0' 10~10' ? o2 Frequency (radlsec)

Figure 5.2: Flexible modes transfer functions, GpeX

where, nTb is the number of rigid body modes and n is the total number of modes. The trans- ferfunctions of Gflex are presented in Fig. 5.2 and one can see that for low frequencies Gflex is approximately constant. Eq.(5.8) can be approximated for low frequencies by Eq.(5.9).

A standard mass feedforward can be implemented based on the inverse dynamics of the rigid body plant GTb. The mass feedforward is presented in Eq.(5.10). Herein, KTb is introduced, which is used in further calculations of the jerk derivative feedforward.

During acceleration feedforward tracking errors stay present at jerk phases, because low frequent contributions of Gflex stay present, [Boe04]. Therefore an extra term is added to the acceleration feedforward. The new feedforward contains a term depending on the derivative of the jerk, see Eq.(5.11).


Figure 5.3: Open loop error visualization

KTb is already known and Kflex can be derived from the open loop error equation, see Fig. 5.3 and Eq. (5.12).

eopenloop = ( I - GcogF)~ (5.12)

Substituting Eq.(5.11) and Eq.(5.6) into the open loop error equation is done in four steps from Eq.(5.13) to Eq.(5.16):

Tracking errors during jerk phases are caused by low frequency contributions of resonance modes, [BTS04]. The intention of the additional feedforward is only pointed on the low frequency contributions. Therefore, from Eq.(5.16) fsollows.

An approximation for Gflex can be made, because of the low frequency interest. This results in a constant, real Gflex and Kflex matrix, see Eq.(5.18).

Here, WLF indicates that a low frequent approximation is made with respect to Gpex.

5.3 Results

The existence of the fourth derivative of the position is a requirement for implementation of jerk derivative feedforward. Therefore a trajectory is designed with a fourth order trajectory planner, see [LamOS]. The bounds on the position, velocity, acceleration, jerk and derivative of the jerk have to be specified. The desired end position of 2 x lop3 m is reached in 0.12 s, while actuator peak forces are not violated at the same time. The calculated profiles are presented in Fig. 5.4. The acceleration and derivative of the jerk profiles are now used together with KTb and Kflex to calculate the new feedforward, see Fig. 5.6 for the input signal

5.3 Results 3 1

Fourth order trajectory profiles

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 time [s]

Figure 5.4: Fourth order trajectory planning

3 2 MIMO jerk derivative feedforward

Bode magnitude diagram

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

lo1 lo2 103 Frequency [rad/s]

Figure 5.5: Spectrum of reference trajectory (scaled) and frequency response of the plant

of actuator 1. Simulations are done with three actuators positioned as in Fig. 4.4. In Fig. 5.7, the error profile is presented, measured at the left end of the beam. The error stays after rise time below 4.8 x lop6 m, but a significant low damped vibration remains. This vibration is dominated by resonance 1. A third order reference trajectory performed better without input shaping, see Table 4.3. The reason for these results can be found in the position of the lowest resonance mode, resonance 1. This dominant eigenfrequency lays at approximately 35 Hz, which is in fact too low for jerk derivative feedforward implementation. The jerk derivative feedforward possesses too much energy at this resonance frequency. Successful results require at least 100 dB degradation, [Boe04] of energy contents at resonance frequencies, see Fig. 5.5 for an illustration. Additional simulations prove that the performance of jerk derivative feedforward becomes better compared to input shaping as the resonance frequencies are situated at higher frequencies. To give an indication, jerk derivative feedforward performs better than input shaping with a second order reference trajectory, when the first resonance mode lies above 60 Hz. This switching point on 60 Hz shifts up when a third order reference trajectory is used for input shaping. For this combination of setpoint and system can be concluded that jerk derivative feedforward cannot provide the desired performances.

Jerk derivative feedforward compensates only for low frequent errors by taking the low frequent estimation of Gfl,, Including only low frequent resonance data is not sufjticient for the flexible beam system in this project. A case study is performed in Appendix D to introduce an inverse dynamics feedforward. Herein, full resonance information can be included in the feedforward design to cancel flexible modes exactly, without taking the complete inverse of the system.

5.3 Results 33

-0.081 ' ' I 0 0.04 0.08 0.12 0.16 0.2

time [s]

Figure 5.6: Input signal actuator 1

time [s]

Figure 5.7: Beam deviation error


Chapter 6

Feedforward design for over-actuated systems

In case of over-actuation, there are more inputs (actuators) than rigid body modes. Adding extra actuators to a motion system can improve feedback and feedforward performance, when actuators are positioned correctly [SvdMS04]. The main idea behind over-actuation in motion control is to use actuators for feedback at positions where maximum controllability is required and for feedforward at positions where controllability should be minimized. Controllability is minimal in the nodes of an eigenmode and is maximal in points where excitation of an eigenmode is high (anti-nodes). The spatial feedforward idea is expIained for the over-actuated flexible beam. Derivation of this spatial feedforward method is discussed in Section 6.1. Spatial feedforward and input shaping are different techniques and can be combined. This approach is presented in Sec- tion 6.2. Results of the extended spatial feedforward technique will be discussed and compared to other methods in Section 6.3.

6.1 Spatial feedforward design

The spatial feedforward for over-actuated systems focusses on minimal controllability of dominant flexible modes without designing the feedforward signal itself. A static gain relation, K, between the actuators is derived to avoid excitation of modes. This section will introduce the spatial feedforward for the flexible beam system with three actuators. The spatial feedforward approach is presented in the block scheme of Fig. 6.1. The effective actuation force Ft is premultiplied by K and resulting signals will drive respectively actuators 1, 2 and 3. Vector K is a 3 x 1 vector containing the spatial gains k l , k2 and ks . With three actuators, resonance 1 and the tilt mode can be made uncontrollable if K is designed in a correct way. An expression for the spatial gains will be derived. The modal excitation of the ith eigenmode can be written into Eq.(6.1)

with,

36 Feedforward design for over-actuated systems

Figure 6.1: Visual interpretation with the K-matrix

Vector (Eq. (3.6)) contains the modal input gains of eigenmode i at the positions of the three actuators. The actuator forces are included in vector F. Eq.(6.1) is rewritten into Eq.(6.2) for mode 3 (resonance 1) with one actuation force and spatial gain vector K = [kl k2 k3IT. If the term between the brackets is zero, this mode is not excited a t all.

The modal excitation of rigid body mode 1 (lift mode) and rigid body mode 2 (tilt mode) can be written in the same form as Eq.(6.2), see Eq.(6.3) and Eq.(6.4). But, the term between the brackets has to be equal to 4f for rigid body mode 1, because this mode is the only mode that is expected to be active.

A set of three equations with three unknown variables has to be solved, where kl, k2 and k3 are the unknown variables (Eq.(6.5)). resonance 1 and rigid body mode 2 will now not be excited, if the feedforward signal, Ft is premultiplied by K.

Solving Eq.(6.5) gives the values for kl, k2 and k3, see Eq.(6.6). From Fig. 3.2, it can be seen that the tilt mode, resonance 1 and also resonance 2 cannot be excited with these spatial gains. Resonance 3 will be the first excited resonance.

6.2 Spatial feedforward combined with input shaping 3 7

, - - _ - - - - - - - _ - - - - _ - - I

Yir-tual single input system 1 I

Figure 6.2: Spatial feedforward combined with input shaping

Spatial feedforward combined with input shaping

Resonance 3 is the first active mode due to the design of a spatial feedforward and a proper choice of the actuator positions. In this section, a combination of spatial feedforward and input shaping is presented. Input shaping is added to the spatial feedforward design to cancel resonance 3. Fig. 6.2 shows via block scheme how the combined method has to be implemented. In Chapter 4, input shapers are designed for multiple inputs and multiple modes, but this combined approach considers a virtual system with one input and one undesired resonance mode. Eq.(4.9) is reconstructed and has to be solved with the following expressions for K, a and w.

Here, a contains the impuls amplitudes and Q5 is the modal gain of resonance 3 (mode 5) for the single input system. Q5 is a summation of the multi input modal gains of resonance 3 and the spatial gains kl ,ka and k3, see Eq. (6.lO),

Eq. 4.9 can be solved as discussed in Section 4.2.1, to retrieve the impulse amplitudes a o ~ and a l q .

6.3 Simulation results

This section discusses and compares the results of the following four methods: (1) mass feedforward, (2) input shaping, (3) spatial feedforward and (4) spatial feedforward combined with input shaping. A third order reference trajectory is used to lift the beam in 0.12 s to 2 mm and reducing residual vibrations after setup time with minimal delay is desired for each method. In this section, beam deviation errors and mode activities are considered in the left end of the beam.


Mass feedforward

Table 6.1: Simulation results with discussed methods

Three figures are presented to discuss the differences between the four feedforward methods. First, Fig. 6.3 shows three subplots with the simulation results of input shaping and mass feedforward. These subplots show respectively the modal activities of resonances 1 and 3 and the beam deviation error. The tilt mode and resonance 2 are not presented, because these modes are not excited. Furthermore, Fig. 6.4 presents the simulation results of the spatial feedforward method and the combined spatial feedforward method. Subplots show again the modal activities of resonances 1 and 3 and the beam error. The last figure, Fig. 6.5, shows the residual vibrations due to each method. As can be seen in Fig. 6.5, mass feedforward is not the most effective feedforward to reduce residual vibrations. The residual vibrations excited by the mass feedforward can be reduced by 77% using a spatial feedforward. With respect to tracking, the spatial feedforward reduces the beam error with approximately 85% compared to mass feedforward and input shaping, shown by the beam errors in Fig. 6.3 and Fig. 6.4. Resonance 3 dominates the error behavior

Delay

[s] -

Spatial feedforward Input shaping Combined method

of the spatial feedforward method, because resonance 1 is made uncontrollable. Input shaping is better a t reducing residual vibrations after the setup time. Input shaping reduces the residual vibrations with 77% compared to a spatial feedforward, but input shaping introduces delay, see Fig. 6.5. The main idea behind a combination of spatial feedforward and

1 x 2.3 x 10W8 5.9 x lo-'

-

0.0128 0.0027

input shaping is to use the spatial feedforward to make resonance 1 uncontrollable and input shaping to cancel resonance 3. Designing an input shaper for only one resonance will affect the delay behavior positively. The extra setup time after convoluting the input signal with the input shaper is smaller if only one resonance mode has to be canceled, Fig. 6.5. Results with the combined method show that residual vibrations are again reduced by 75% compared to input shaping and delay is also reduced drastically. The properties of the spatial feedforward during tracking are also maintained, see Fig. 6.4. Table 6.1 summarizes all relevant data from the simulations.

Residual error [m]

4.4 x lo-7 2.7 x 4.2 x 2.5 x lop6

Max tracking error [m]

4.4 x lo-5

0.79 0.94

Weighting value

[-I

6.3 Simulat ion results 39

time [s]

Figure 6.3: (-.) Mass feedforward and (-)input shaping

2 a 5 b 0.05 0.1 0.15 0.2 0.25

time [s]

Figure 6.4: (-) Spatial feedforward and (-.) Spatial feedforward combined with input shaping


- - method (2)

- method (3) - - method (4)

0.12 0.14 0.16 0.18 0.2 0.22 0.24 time [s]

Figure 6.5: Residual vibrations after setup time Method (1): Mass feedforward,

Method (2): Input shaping, Method (3): Spatial feedforward . . -

Method (4): Spatial feedforward with input shaping

Chapter 7

Experimental implementation

This chapter will present a case-study, in which feedforward control is used to reduce residual vibrations on an experimental beam setup. The experimental setup is briefly introduced in Section 7.1. Section 7.2 presents the identification of the system using a modal analysis technique and Section 7.3 will discuss the results of vibration reduction on the flexible beam with respect to traditional actuation and over-actuation.

7 . Experimental set up

In Fig. 7.1 the flexible beam setup is shown. The beam dimensions and material properties are identical to the beam specifications of the theoretical beam, Table 3.1. However, the differences between the real and theoretical flexible beam are visible. The real flexible beam is connected to the ground with five weak springs to fix four degrees of freedom. Translation in z-direction and rotation about the y-axis are the remaining degrees of freedom (Fig.3.1). The flexible beam is actuated by voice coil actuators. Multiple actuators can be placed along the steal beam to perform movements. Displacements of the beam are observed by fiber optic sensors. Fig. 7.2 illustrates the placement of a voice coil actuator and fiber optic sensor.

7.2 Modal analysis

Feedforward methods cannot be applied directly on the real test-setup before the theoretical model is verified. A modal analysis is performed to determine and verify the properties of the system. Parameters of interest are the eigenfrequencies w,, damping coefficients C, and the modal matrix a, i.e. the eigenfunctions. The modal analysis is based on Frequency Response Function (FRF) measurements. Subsequently, the software package ME'scopeVEs4.0 [VibOS] makes a fit of the FRF measurements and calculates the desired system parameters. FRF measurements are carried out under the following conditions. Noise, bandwidth up to 2000 Hz and variance of 0.16 V, is inserted to actuator 3 and the displacement response is measured at 25 equally spaced locations on the beam. No additional feedback control is used to measure FRF's. With 25 measuring points, a good spatial observability of mode shapes is guaranteed for frequencies 5 300 Hz. Since modal frequencies and -damping coefficients are global properties of the beam structure, all FRF measurements posses the same modal frequencies and damping. With the global polynomial curve fit method [MerOS, Vib031 of ME'scope, all FRF's are fitted simultaneously and the global modal frequency, damping

Fiber

Five weak springs

Figure 7.1: Flexible beam test-setup

Voice coil actuator

optic sensor-

Figure 7.2: Sensor and actuator placement

7.2 Modal analysis 43

Frequency [Hz]

Figure 7.3: Magnitudes of the FRF measurements

Table 7.1: Resonance frequencies and damping

Mode Shape / Frequency [Hz] / Damping [%]

coefficient and modal parameters are calculated for each resonance mode. To search for the best fit for each resonance, the number of extra numerator polynomial terms is varied. An overlay of all FRF measurements is shown in Fig. 7.3. For modes shape 1 to 5, the fit values of the eigenfrequency and damping coefficients are presented in Table 7.1. The real and complex modal gains of each mode shape are presented in Appendix A. For the flexible modes, the percentage modal damping is low and therefore, the assumption of modal damping in Eq.(3.4) is correct. All five mode shapes from the fit are visualized in Fig. 7.4. The displacement of the beam in x-direction is fixed by one weak spring. The spring is attached on the right side of the beam and this causes a non-symmetric behavior of the mode shapes. This is clearly visible in mode shape 1. The Modal Assurance Criterion (MAC) is calculated to give an indication wether or not mode shapes are correlated. MAC values between 0 and 0.9 indicate that mode shapes are different [Mer03]. From Table 7.2, one can see that all modes are uncorrelated, because MAC values stay below 0.018 on the cross terms. In theory, two rigid bodies are located at 0 Hz, but the real flexible beam is connected to the ground with weak springs. The consequence is that the rigid body modes are changed to "rigid body" modes with low resonance frequencies of respectively 4.0 and 9.8 Hz.

20.2 1 4.0

44 Experimental implementation

Beam length x, [cm]

Figure 7.4: First five mode shapes of the flexible beam

Table 7.2: MAC values flexible beam

7.3 Ex~erimental vibration reduction 45

0 1 2 3 4 5 6 7 8 9 time [s]

Figure 7.5: Effective input

7.3 Experimental vibration reduction

The modal analysis shows that two "rigid body" modes appear at 4 and 9.8 Hz. These rigid body modes cause dominant vibrations in the response of the system, but the same modes are also needed to perform the motion task of the system. The spatial feedforward of Chapter 6 cannot be applied directly for this reason, because it allows excitation of rigid body modes and does not avoid excitation of residual vibrations. However, input shaping allows excitation of rigid body modes and avoids excitation of residual vibrations. At the moment, no exact compensation is made for the asymmetrical stiffness and therefore the beam will not move completely horizontal. A rough compensation is estimated using the proportions of the modal gains of rigid body mode 1 with respect to the actuator locations. The residual vibrations can still be analyzed, because the asymmetrical stiffness will not affect the design and results of input shaping. The objective with respect to this section is to analyse the reduction of residual vibrations on the flexible beam setup using input shaping. Results will be presented in Sections 7.3.1 and 7.3.2 for respectively two and three actuators. Not every arbitrary input will excite the flexible dynamics sufficiently, especially higher resonance modes. The input during experiments is chosen to add sufficient energy in the system to excite the flexible dynamics, see Fig. 7.5.

7.3.1 Input shaping - two actuators

Two actuators are placed at the normalized beam positions of [0.1, 0.91. Experiments are performed with two sets of input shapers to suppress residual vibrations of rigid body modes 1, 2 and resonance 1, higher modes are hardly or not excited. One set of shapers is designed for first order robustness input shaping and the second set of shapers is designed for second order robustness. Input shapers are calculated using the modal gains and system poles from the modal analysis. Two figures are presented to illustrate the results of residual vibration reduction on the flexible beam. Fig. 7.6 shows the responses in the left end of the beam without input shaping and with first- and second order robustness input shaping. Fig. 7.7 shows the power spectral density functions of the three responses. The delay's introduced by convolution for first- and second order robustness is respectively 0.1995 s and 0.201 s. The

46 Experimental implementation

-0.5' ' ' 0 1 2 3 4 5 6 7 8

time [s]

-0.5' ' ' ' " ' ' ' 0 1 2 3 4 5 6 7 8

time [s]

(a) Without input shaping (b) Input shaping, 1st order robustness

0 1 2 3 4 5 6 7 8 time [s]

( c ) Input shaping, 2nd order robustness

Figure 7.6: Beam responses with two actuators

improvements due to input shaping are substantial. First order robustness reduces residual vibrations with approximately 86% and second order robustness input shaping with 89%.

7.3.2 Input shaping - three actuators

Experiments with three actuators are carried out by placing the actuators a t normalize beam distances of [0.1, 0.5, 0.91. Again two sets of shapers are designed with first- and second order robustness. The results are presented in Fig. 7.8 and Fig. 7.9. Delay times are respectively 0.137 s and 0.254 s and vibration reductions are approximately 58% and 95% for first order robustness and second order robustness shapers. The difference in vibration reduction is now clearly visible between both robustness orders. Adding robustness introduces 85% more delay but gives a much better suppression of residual vibrations. Choosing for first order robustness with less delay and more residual vibrations or choosing for second order robustness with less residual vibrations and more delay depends on the required performance specifications of the motion task. For now, the results are presented to give insight in what is achievable with input shaping on this flexible beam setup. The improvements between two and three actuators are more intuitive. Choosing for first order robustness input shaping with three actuators gives much lower delay, but vibration

7.3 Experimental vibration reduction 47

1 2nd ord rob IS

I 10-1 1 oO I o1 1 02 lo3

Frequency [Hz]

Figure 7.7: Power density spectrum two actuators

suppression is less effective compared to two actuators. Second order robustness input shaping with three actuators gives much better vibration suppression compared to two actuators, but the delay increases.

0 1 2 3 4 5 6 7 8 time [s]

(a) Without input shaping

0 1 2 3 4 5 6 7 % time [s]

(b) Input shaping, 1st order robustness

-0.5l , , . 0 1 2 3 4 5 6 7 %

time [s]

( c ) Input shaping, 2nd order robustness

Figure 7.8: Beam responses with three actuators

-- - 10-1 1 oO lo1 1 02 10"

Frequency [Hz]

Figure 7.9: Power density spectrum three actuators

Chapter 8

Conclusions & Recommendat ions

8.1 Conclusions

For high performance motion control, especially for flexible motion systems, the usefulness of feedforward control is well-known and often implemented. This report discussed three feedforward techniques, multi-input shaping, jerk derivative feedforward and spatial feedforward, to reduce residual vibrations in multi-input flexible motion systems.

Input shaping is a well-known technique in vibration control, due to its ease of implementation and the capabilities of dealing with parameter uncertainties. Simulations show the improvements of input shaping with respect to vibration reduction with multiple actuators and higher order trajectories. Third-order reference trajectories combined with input shaping have a better performance compared to second-order reference trajectories with input shaping. An increasing number of actuators also has a positive influence, because delay times can be reduced by adding multiple actuators. Jerk derivative feedforward is an extension of the classical mass feedforward with a contribution of the jerk derivative. Simulation results with jerk derivative feedforward cannot compete with input shaping results on the benchmark system. Although jerk derivative feedforward gives no satisfying results, additional tests prove that the performance of jerk derivative feedforward becomes better compared to input shaping as the resonance frequencies are located at higher frequencies. The spatial feedforward is an attractive method with respect to over-actuated systems. Ex- citation of flexible modes can be avoided, because extra design freedom in the decoupling is created by the use of extra actuators. Spatial feedforward improves both tracking and the residual vibration behavior, but higher resonance modes are still active. A combination of spatial feedforward and input shaping is constructed to use the benefits of both methods. With this combined method, substantial improvements are made with respect to delay and vibration reduction.

Experiments are performed on a flexible beam setup. The flexible beam differs from the beam model in simulation, since it is asymmetrically fixed to the environment with five weak springs. This results in a non-symmetric system behavior and both rigid body modes at 0 Hz are changed to non-symmetric "rigid body" modes with low resonance frequencies. Reducing residual vibrations is therefore carried out on these "rigid body" modes. Input shaping is

50 Conclusions & Recommendations

implemented to reduce residual vibrations, including the dominant "rigid body" modes. Re- sults are satisfying and show that a maximum reduction of 95% is achievable.

8.2 Recommendat ions

Multi-input shaping is implemented in both simulations and experiments. Here, input shapers contain only positive shaper amplitudes. Negative shapers were disregarded in this research, because they can cause saturation of the actuators. This does not imply that negative shapers cannot improve the present results. Further research with negative input shapers, which cause no actuator saturation could improve the performance. The impulses for each shaper are distributed evenly in time and the impulses of the shapers are synchronous. Issues for future study could be the investigation of unevenly distributed impulses for each shaper and non-synchronous shapers. Unevenly distributed impulses will increase the design freedom substantially. For this reason, choices still have to be made with respect to the constraint equations. The spatial feedforward is implemented in simulations to prevent excitation of the dominant resonances and to preserve pure rigid body motion. The experimental setup shows no pure rigid body behavior, which requires a different approach of the spatial feedforward. Investigate how the spatial feedforward and the combined spatial feedforward have to be interpreted and implemented on the experimental setup is recommended for further research. A stiffness feedforward could be designed to compensate for the low frequent "rigid body" stiffness or a hardware correction can be made. If housed voice coil actuators are used, the system doesn't require weak springs anymore to fix lateral movements. Finally, one improvement to the experimental setup is recommended. The flexible beam is excited with multiple voice coil actuators and the beam is attracted by the magnets inside the voice coil actuators. The choice of a proper beam material, for example brass, can solve this problem.

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Appendix A

Modal parameters flexible beam

This appendix contains the modal parameters of the flexible beam setup. Table A.1 presents the modal parameters of each mode shape. Modal parameters are assigned to points on the flexible beam. Fig. A.l shows the measurement points on the flexible beam. These are called DOF's, a DOF includes a point number and direction. In this case the measuring direction is the z-direction.

lz 2z 32 42 52 62 72 82 92 10z l l z 122 132 142 152 162 172 182 19r 20z 212 222 232 242 252 262 Z ~ o * - - o o * - o ~ * * - o o o * ~ o o * - * ~ - T

272

Figure A. 1 : Measuring points flexible beam

54 Modal parameters flexible beam

DOF Real -98.8 -101 -102 -98.4 -99

-96.7 -93.5 -82.7 -79.9 -87.6 -76.2 -75.3 -66.1 -64.3 -61.6 -59.2 -56.1 -53.8 -51. -47.3 -43.6 -41.6 -36.8 -33.4 -29.6 -26.3 -35.2

e 2 Imag -53.5 -50.3 -45.3 -37.4 -33.1 -25.9 -21.6 -12.6 -7.14 1.6

4.55 10.5 14.1 18.8 24.3 28.5 32.8 37.8 42.6 46.3 51.7 57.5 59.8 66.3 71.4 77.2 64.3

Real e 3 Imag -0.663 -0.58 -0.464 -0.285 -0.162 -0.063 0.0177 0.134 0.215 0.112 0.554 0.637 0.53 0.55 0.508 0.455 0.348 0.259 0.142 0.0205 -0.142 -0.257 -0.363 -0.518 -0.584 -0.624 -0.494

Real )e 4

Imag 0.00943 -0.0193 0.00474 -0.0351 0.00409 0.0221 0.0194

0.00561 0.0139 -0.0282 0.0196 -0.0252 0.00129 0.0017

-0.00142 0.0197

0.00764 -0.0139 -0.00846 0.00252 0.00489 0.0119 0.0146 0.0118 0.0158 0.104

0.00149

Real le 5

Imag -0.443 -0.227 0.0271 0.262 0.416 0.497 0.452 0.309 0.165

0.00253 -0.161 -0.32 -0.371 -0.378 -0.31

-0.194 -0.0283

0.15 0.311 0.414 0.447 0.402 0.275

0.0575 -0.197 -0.434 0.183

Table A.l: Complex modal gains of the flexible beam

Appendix B

Assembly of system matrices

Finite element methods are used to assemble full mass and stiffness matrices of the flexible beam. In this appendix, the mass and stiffness matrix of a beam element are presented together with the assembly of the full matrices. Eq.(B.l) and Eq.(B.2) present respectively the mass and stiffness matrix of a beam element.

1561 S y m m e t r i c

2212 413

541 131' 1561

-1312 -3Z3 -22Z2

S y m m e t r i c

The element matrices of the mass and stiffness will be assembled to yield the full mass and stiffness matrices. This is indicated by Fig B.l and in reference [DH92].

5 6 Assembly of system matrices

Zeros

Zeros element n

El Figure B.l: Matrix assembly

Appendix C

Actuator specifications

This appendix presents the voice coil actuator specifications in Fig C.1. These voice coil actuators of type LA08-10-000A are produced by BE1 Kimco. The actuator specifications can also be found on the online support site of BE1 Kimco, http://beikimco.com/products/pdf/actuators/la0910000a.pdf.

58 Actuator specifications

Figure C.l: Actuator datasheet

Appendix D

Multi-input inverse dynamics feedforward

To achieve exact output tracking, inversion of system dynamics can be used to find ideal inputs. However, plant inversion methods result often in non-causal systems, which are often hard to calculate. Non-causal inputs exist before time is zero. Move time penalties arise, because inputs are shifted in time in order to use these inputs. This appendix will introduce an new inverse dynamics feedforward approach to cancel residual vibrations for multi-input single-output systems. To use this method, system dynamics have to be written in an additive and modal form, Eq.(3.6). Section 3.2 has presented a flexible beam model in additive modal setting, and this model is used to illustrate the implementation of this multi input inverse dynamic feedforward. The inverse method will be applicable for multiple resonance modes. Section D.l presents the single input inverse dynamics feedforward and Section D.2 expands this for multiple inputs.

D.l Single input feedforward

A second order reference trajectory is used for the SISO flexible beam system. Actuator 1 and sensor 1 are collocated in the beam center. For simplicity only three modes are included in feedforward calculations. Adding multiple modes is possible, but does not provide extra insight in the calculations. The total system response in the beam center can be written as a summation of the modal responses, these are respectively the rigid body modes and resonance 1, see Eq.(D.2).

In Fig. D.l the responses of each mode due to a given force profile are visualized. Summation of the responses yields the total system response. The desired response is equal to the response of rigid body mode 1, but resonance 1 is also activated by the current input signal. Rigid body mode 2 is omitted in the further feedforward derivation, because this mode is not excited (Fig. D.l). The new inverse feedforward method adds an extra input signal F* to the original input F to cancel rigid body mode 2 and resonance 1. An expression for the new input signal is derived in this section. The system with undesired dynamics and the new feedforward, F + F* must have the same

60 Multi-input inverse dynamics feedforward

7

0 2 4 Time [s]

, 0-3 Responses

31

XI desired mode

0 0 2 4

I-

&& : undesired s2 mode

0

a: -0.5

- 0.5 0

undesired g o mode

42 420 V)

s2fp03sf X~ $ 4 . 5

Time [s]

Figure D.l: Mode responses

D.2 Multi input feedforward 6 1

- 1 0 ~ " " ' " ' ~ 0 1 2 3 4 5 6 7 8 9 1 0

Time [s]

Figure D.2: Feedforward signals

response as the system with desired dynamics and the original feedforward, see Eq.(D.2).

Factoring out F gives an expression for F*, Eq.(D.3),

Rewriting Eq.(D.3) yields a proper and stable expression for F*

The new feedforward can be calculated with Eq.(D.5), Fig. D.2 shows the old feedforward F , additional feedforward F* and the new feedforward F,,,. F,,, cancels resonance 1 exactly and only the response of rigid body mode 1 is present in the beam center. Local vibration cancelation is obtained in the beam center with this SISO flexible beam system.

Eq.(D.2) can be extended with more resonance modes. The equations become longer when multiple modes are added, but the problem stays proper.

D.2 Multi input feedforward

The single input inverse dynamics feedforward method can be extended for multiple actuators and one output. Derivation of a multi-input inverse dynamics feedforward is presented in this

62 Multi-input inverse dynamics feedforward

section for three actuators and one output. Actuator 1 is used to cancel undesired dynamic behavior. The system response for the left end of the beam can be written in the following

The same steps as in the single input approach can be followed to calculate the new feedforward for the multi-input system. The response of rigid body mode 1 is still the desired system response and rigid body mode 2 and resonance 1 are the undesired modes. Eq.(D.7) shows the equality constraint equation for the multi-input flexible beam system. The actuator inputs for actuator 2 and 3 are unchanged. Only actuator 1 receives an additional feedforward term,

After rewriting, I;;* becomes a summation of proper and stable transfer functions, Eq.(D.8). F;' depends on all actuator inputs. This is comprehensible, because FT has to cancel resonances excited by Fl , F2 and F3.

The new actuator inputs are given in Eq.(D.9) and Fig. D.3.

Not every actuator configuration will lead to a stable transfer function F*, but extra actuators create more freedom in choosing a stable configuration. This method can therefore offer opportunities for over-actuated motion systems. Robustness is one disadvantage of the inverse dynamics feedforward. With an exact model the inverse method works perfect, but modeling errors are always present in real applications. Making this method successful, robustness against model uncertainties has to be created and investigated in future research. In general, feedforward signals are constructed in finite time. The inverse dynamics approach results in feedforward signals with infinite time. This implies that the actuators stay active after the desired setpoint is reached. Further research with respect to these infinite time input signals is required, because this behavior is undesired.

D.2 Multi input feedforward 63

time [s]

Figure D.3: Multi input input signal

Documents

Eindhoven University of Technology MASTER Multi-input ...Eindhoven, December 13, 2004 . Summary Reducing vibrations in precision motion systems can be achieved by adding structural