Synchronization of the CFT-transposers

Citation for published version (APA):Arnolds, M. B. (2004). Synchronization of the CFT-transposers. (DCT rapporten; Vol. 2004.023). Eindhoven:Technische Universiteit Eindhoven.

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Synchronization of the CFT-Transposers

DCT Report no: 2004.23 February 2004

TU/e Internship Report February 2004

Supervisor: Prof. dr. H.Nijmeijer Technische Universiteit Eindhoven

Eindhoven University of Technology Department of Mechanical Engineering Division Dynamical Systems Design Dynamics Control Technology Group

Rescarch on synchronization of the CFT-Transposers has already been per- formed multiple times. Different controllers has been implemented and proved to work. Adaptive controllers have proved to work for the transposers individ- ually and for a kind of Master-Slave synchronization where the slave does not totally depend on the master. In this project, real master-slave synchronization has been investigated using the adaptive controllers.

Abstract i

1 Introduction 1

2 CFT-transposer setup 2

3 CFT-transposer model 3

3.1 Kinematics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 3

3.2 Dynamics . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . 4

3.3 Friction Forces . . . . . . . . . . . . . . . . . . . . .- . . . . . . . . . . 4

4 Controller design 6

4.1 Reference model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.3 Adaptionlaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.4 Measurement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Simulation 9

5.1 Mathematical model . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 9

5.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 11

6 Experiments 13

6.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 13

7 Conclusions and Recommendations 16

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

A Dynamic model of the CFT robot 18

B Estimated friction parameters 24

Chapter 1

Introduction

In the present and future production processes robots play and will play a very im- portant role. Often the manipulators have to work together, for example one robot performs a task which will be followed by another robot performing a second task. Another example is when 2 robots have to work together to perform a task. The first case is a simple "build-a-production-line" problem. The second example is more com- plicated. When 2 robots have to pick up something together they have to do exactly the same thing (including disturbances). That is why they have to be synchronized.

Synchronization can be defined as the mutual time conformity of two or more pro- cesses. Hereby one can distinguish several kinds of synchronization depending on the interconnections. First we have Master-Slave synchronization. The goal of this type is to design a controller for the Slave which synchronize with the output of the Master. Now only the Slave depends on the Master. Every disturbance of the Master has to be followed by the Slave. So this means that the position and velocity error have to be minimized. These errors are defined by

where q, and qm are defined as the position of the slave and the master respectively and where q, and qm their derivatives. Second, we have mutual synchronization in which both (or more) processes depend on each other. So disturbances in one of the processes lead to disturbances in all the robots.

In this project two identical CFT-transposers will be considered in a master-slave syn- chronization problem. This has already been a topic in some other projects where PD-control, Model-based-control and Adaptive control were the main methods to per- form synchronization. In this project, adaptive control will be used. The difference with the previous project with adaptive control is that here the reference trajectory will only be used for the master (subscript ,) robot, and not for the master as well as the slave (subscript ,) like in the previous project. So now the master will receive a reference signal and the slave will receive the true output of the master robot which makes this a real master-slave synchronization problem.

CFT-transposer setup

For this master-slave synchronization project, a multi-robot system will be used. This robot system is an experimental transposer robot designed by the Centre for Manufac- toring Technology (CFT) Philips Laboratory. Two of these robots, identical in design and structure, are located at the DCT-lab.

The CFT robot is an industrial pick-and-place robot which is used f a assembling. It is a Cartesian basic elbow configuration robot. It consists of a two links arm that is placed on a base that can translate and rotate. The tool connected a t the end of the outer link is a kinematically constrained planar support and because of this it is passively actuated and will remain horizontally at all time.

The kinematics of the transposer can be written in Cartesian and in joint space. This is already done in [Rodr2002-11 and [Mans2002]. For this project the joint-space con- figuration will be used.

The robot is actuated by 4 DC brushless servomotors. Although the shaft of the motors and the corresponding h k s are connected by means of belts, the servomotor link pair proved to be stiff enough to be considered as a rigid body. It is also equipped with encoders attached to the motors. The encoders have a resolution of 2000 PPR which leads to an accuracy of f O.5mm in all motion directions.

For the implementation of the controllers and communication with the robots, the whole setup is equipped with a DS1005 dSPACE system. The sampling frequency used during simulations and experiments has been set to 2 kHz.

Figure 2.1: Experimental sctup; two identical CFT-transposcrs

CFT-transposer model

In this chapter the kinematics and dynamics of the CFT-manipulator will briefly be discussed. Briefly, because this is already done in full detail in other reports.

3.1 Kinematics

As mentioned in chapter 2 the joint space configuration will be used for the model of the manipulator. Therefore, seven coordinates are needed. A wire-frame picture of the manipulator is shown in figure 3.1. To obtain the kinematics in the joint space the Denavit-Hartenberg parameters will be used. They are listed in table 3.1.

Table 3.1: Denavit-Hartenberg parameters for the CFT-Transposer (see [Rodr2002-11)

The coordinates ql and q3 are the translations along the zl and z 3 axis respectively. The other coordinates are the rotations along the corresponding z-axis. Because the transposer is kinematically constrained, four coordinates can describe the kinematics in the joint space. First q l which describes the translation of the whole robot. Second, qz describing the rotation of the whole robot. The last two, q4 and q 5 , describe the rotation of the links z 4 and 25 respectively. The three joint-space coordinates that remain can be described as a function of the other ones.

CHAP'TER 3. CFT-TRAATSPOSER MODEL

Figure 3.1: Coordinate-frame figure of the CFT-Transposer

3.2 Dynamics

The dynamics of the transposer can be derived from the Euler-Lagrange approach and the Denavit-Hartenberg parameters. The dynamics are given by:

with index i = m, s and M(q,) the inertia matrix which is symmetric and positive definite. G(q,, q,) represents the Coriolis matrix and the centrifugal torques. G(q) are the conservative forces due to the gravity. T is an input vector of the torques and F(q,) are the friction forces. AIl these matrices are presented in appendix A.

3.3 Friction Forces

The friction forces F(q,) consist mainly of the forces caused by the static forces and forces due to dynamic friction. Because dynamic friction models can not deal with high velocities because of the integration method that is used only the static forces will be considered. To avoid the problem of discontinuity in the Coulomb friction which arises at zero velocity, the model used is based on an approximation like a sigmoid function (see also IRodr2002-11) :

CHAP'TER 3. CFT-TRAATSPOSER. MODEL 5

In (3.5) B, represents the diagonal viscous friction coefficient matrix and the other terms approximate the Coulomb and Stribeck friction effects.

Here we will use an adaptive controller to adapt the friction parameters. However, since a linear parametrization of the friction forces is needed in the adaption law, and the parameters wl and wz are found in the argument of the exponential function, these parametcrs will not be adapted. The values of these non-linear friction parameters and the inertia parameters are obtained from [Rodr2002-11.

Now the kinematics and the dynamics of the CFT-transpose1 are completely described. With this model a controller can be designed.

Chapter 4

Controller design

In this chapter a controller will be designed. This controller should have perfect tracking capacity in order to minimize the error between the master and the reference trajectory and the master and the slave. This means that e, -+ 0 for t + co and eq + 0 for t -+ co. The position error is defined as the difference between the real trajectory and the desired trajectory.

4.1 Reference model

In adaptive control a reference model is used to specify the desired response of the adaptive control system. The master-slave synchronization goal of this project will be applied in two ways. First the master will adapt to the reference trajectory to minimize the tracking error. Simultaneously, the slave will adapt to the masters position. So the master will act as a slave and reference model a t the same time.

4.2 Controller

As already given in chapter 1 the position error and its time derivative are given by:

with; i = m, s and j = r, m where r stands for the reference trajectory. One could also state that:

e, = q - qd (4.3)

6, = q - qd (4.4)

with q the position of the system (master of slave) and qd the desired trajectory (ref- erence or master).

Substitution of these equations into (3.4) will result in the error dynamics equation.

The choice of the controllaw can be based on the 2nd method of Lyapunov. This controi law must be chosen in such a way that e,(t) - 0 is a globally asymptotically stable

CHAPTER 4. CONTROLLER. DERS'IGhT 7

equilibrium point of the closed loop configuration. A suitable Lyapunov candidate can be:

with Kp a constant, symmetric and positive definite matrix. The time-derivative of this Lyapunov candidate is:

With the stability analysis already performed in [Rade2003] the closed loop error dy- namics equation with substitution of the control law becomes:

4.3 Adaption law

In the stability analysis a computed torque control law has been used assuming all the model parameters are known. When the model parameters are unknown, this computed torque control law can not be used. Therefore, it is necessary to use a suitable adaption to estimate the model parameters. In case of the CFT transposer, only the linear friction parameters will be adapted. Substituting the control law based on adapted parameters, the closed loop error dynamics equation becomes:

Now because the unknown parameters appear linearly in equation (4.9) it can be rewrit- ten into

M(q)e, + C(q, q)e , + K,1, + Kpeq = W d e ~

With ee = Q - O and

The goal is to adapt the parameters Q so that e,(t) --t 0 for t -+ oo. The adaption mechanism can be based on the 2nd method of Lyapunov. For example

with = rT and I? > 0 The time derivative becomes

CHAPTER 4. CONTROLLER, DESIGN 8

The first part of (4.12) is semi-negative definite. The second part can be forced to 0 with the parameter adaption law:

do = -I?-' w Z ~ , (4.13)

If the second part is forced to zero, (4.12) is a negative definite function, so dq(t) + 0 for t + 0. Looking at (4.12) this implies that 1s -+ 0. Because of the fact that 2,(t) and es are uniform continuous functions on [0, cc] and their limits for t + cc go to 0, this implies that:

t

lim 1 d s ( ~ ) d r t--tm

exist and are finite. Or, more simple, eq(t) and es(t) become constant if t + 0. In the time limit t + cc, the tracking and the parameter error from equation (4.9) appears to be coupled:

Kpeq + Wdes fo r t + cc (4.14)

So for this time limit, both Wdee and es are constant, resulting in wdes + 0. Therefore, if the time derivative of Wd is persistently exciting, ee + 0.

4.4 Measurement noise

During simulations the positions are "measured" just like in the real setup. The needed velocity will be obtained by numerical differentiation. With the simulations this should not cause any problems but with the experiments it might because of the measurement noise. As can be seen in section 4.3, the adaptation of the parameters consists of multiplication of the velocity with the time-derivative of the position error. This means the velocity appears quadratic in the adaption law. When the measured position signal contains measurement noise the time-derivative, velocity, will contain noise as well. The squared measurement noise will affect the parameter adaption in a negative way.

So the term q2 which appears in the adaption law for the viscous friction parameters has to be replaced. This will be done by replacing the time-derivative of the position error by:

s = d, + he, (4.15)

with A a diagonal matrix given the value 2 on its diagonal (taken from [Rade2003]). So A = diag([2 2 2 21). The advantage of taking this s as the velocity error is that the adaption is now also based on the position error. This results in a more consistent estimation of the friction parameters. So (4.13) can now be rewritten as:

Now the velocity will not appear quadratic anymore in the adaption law. However, the velocity is still obtained by numerical differentiation of the measured position with respect to the time. This velocity also contains measurement noise and cannot be rewritten like is done with the velocity error. Previous solutions have used the velocity of the reference trajectory but this does not satisfy the real master-slave synchroniza- tion, this solution will.

Now that an adaptive controller has been designed, it can be implemented into Mat- lab/Simulink so simulations and, later on, experiments can be performed.

Chapter 5

Simulation

Before implementing the controllers into the real system some simulations have to be performed to check wether the controller works properly and to tune the adaption law. Different from previous projects, here the master robot will adapt to the reference trajectory and the slave will adapt to the position (and velocity) of the master. So there will be no connection of the slave to the reference trajectory which make this a real master-slave behaviour in two situations.

adaption Slave

position / 1 Figure 5.1: Graphical presentation of the adaption scheme

5.1 Mat hematical model

Since it is about simulation, the transposer has to be replaced by a mathematical model of it. [Rodr2002-11 already derived such a mathematical model in his report and so this one will be used. It has the following structure:

Here, the values of the friction parameters have to be specified. In IRodr2002-11 these parameters have already been estimated with the use of a Kalman filter. Assuming this is done properly, they will be used in this project as well. They are listed in appendix B. The initial conditions for these friction parameters are set to be 1.5 times the real values.

In the control law K,, K, and l7 still have to specified. Kp and K, are diagonal matrices. The gains are obtained from [Rade2003] and are tuned by trial and error:

CHAPTER 5. SIMULATION

I Master 1 1 Slave

Table 5.1: Control parameters for Master and Slave

DOF

ql

5.2 Simulation

During this project, onIy two of the four degrees of freedom of the transposer will be considered; the translation ql and the rotation q2. Because of limited calculation

K 15000

capacity, adaption of all four the degrees of freedom for the master as well as the slave

K 20

Kp 60000

becomes too much for the system. So only the degrees of freedom that do not have any

K", 400

kinematically constraints or interconnections will be adapted. Although it is not the complete controller for the transposer, it will give a good picture regarding adaptive master-slave synchronization. The reference signal for the master will be a simple sine function (with a chosen ampiitude and frequency) for both the degrees of freedom.

Before simulations can start, the matrix r from (4.13) has to be specified. r is defined as r= diag([I'l,l, r2,2, . . . r l 2 ,12 ] ) and affects the update speed of the adaption mech- anism. From (4.16), the larger !2 the smaller the update speed of the adaption law. By trial and error the values of this diagonal matrix are obtained. Values already ob- tained in an other project ([Rade2003]) did not work with this approach of master-slave synchronization but have proved to be useful in searching into the right direction.

Master Element I Valuc / Element I Value / Element 1 Value /

Slave I Elemcnt 1 Value / Element 1 Value I Element / Valuc

Table 5.2: I' values after tuning

CHAPTER 5. SIlcrlULATION

5.3 Simulation results

In table 5.2 the results of the I?-tuning are presented. As mentioned in section 5.2 the values of are obtained by trial and error in order to obtain a fast convergence of the synchronization error. During the simulation the adaption mechanism will slowly come up. This is done so the adaption mechanism will not adapt on the initial errors of the system and thus will work against the expectations. The transposers have time now to "settie" before the adaption wiii start. So first it starss surned off and no adaption wiii be active. After 10 seconds, the contribution of the adaption mechanism will be 25% and so until the total contribution will be 55% of the total adaption capacity. This 55% is obtained by trial and error. Searching for the right gains by trial and error means a lot of simulations have been performed. Only in the end satisfying results have been obtained so, naturally, only these results will be presented.

Error Reference Master ql (translation) <:I

X I O - ~ Error Reference Master q2 (rotation) i I

Figure 5.2: Error between the Reference trajectory and the Master

From Figure 5.2 it can be seen that the error for the ql direction adapts in the first 30 seconds to a constant value and the error remains 0.6 mm which can be seen as small. The error of the rotation ( q z ) is pretty large in the beginning but adapts very fast to a much smaller error. At the end of the simulation this error is < 0.5 ~ 1 0 ~ ~ rad.

In Figure 5.3 errors of the slave regarding the master are presented. Here we see the rotation adapting pretty good to the masters position. In the end it is a pretty constant error of < 0.2 X I O - ~ rad. The error in the translation ( q l ) becomes almost constant (f 0.5 mm) after 50 seconds, though it seems still slowly decreasing. The peaks in the error are caused at the moment the velocity changes sign. Probably, a t this moment, the adaption mechanism has a hard time keeping up with the master. This could be explained with the fact that the adaption law uses the velocities. When the position is numerically differentiated to obtain the velocity at the moment this changes sign this could lead to an error.

All the errors obtained from these simulations are smaller or as big as the encoder accuracy of 0.5 mm. This means the controllers behave very well in a simulation and should work properly in experiments.

CHAPTER 5. SIhfULATION

x Error Master Slave q l (translation)

I , '

l ' ! I I I I ' 0 50 100 150 200 250 300 350 400 450 500

tlme(sec)

x 10" Error Master Slave q2 (rotation) I I

Figure 5.3: Error between 1\Iast,er and Slave

Chapter 6

Experiments

In this chapter the results of the experiments that are done at the end of this internship will be presented. These experiments are performed at the end of the internship and will not be too extended. They just give an indication of how the controller would act in the real situation.

6.1 Experimental setup

The CFT transposers are connected to the computer with an DS1005 dSPACE sys- tem. So before experiments can be performed the Simulink model has to be build and compiled to a file the dSPACE-controldesk can read. Like said before in section 5.2 only two degrees of freedom will be investigated. A lay-out has to be designed and connections have to be made to be able to record the signals.

6.2 Experimental results

Before experiments can be performed parameters have to be specified. The initial conditions for the friction parameters will be taken the same as in the experiments, 1.5 times the estimated friction parameters. The values for I? will also be the same as the simulations. In the Figures 6.1 and 6.2 the results of the experiments are presented. The controllers work pretty well although the gains K, and K, have to be tuned again online.

Table 6.1: New control parameters for Slave

DOF

ql

q2 q4

q5

The errors between the reference trajectory, master and slave during the experiments become smaller during the first 30 seconds like in the simulations. At the end of the

Master

1 1

15000 2000 12000 12000

K, 20 50 50 50

Slave Kp

15000 10000 12000 12000

Kv 50 50 30 30

CHAPTER 6. EXPERIMENTS 14

experiments the reference-master error is about < 5 mm for translation and < 0.005 rad for rotation. For the master-slave this error is approximately 10 mm for the translation and f 0.01 rad for the rotation.

Error Reference Master ql (translation)

O 0 2 2

Error Reference Master q2 (rotation)

0 03

Figure 6.1: Error between Reference and Master during an experiment

The fact that the errors during the experiments are larger as the errors during the simulations is not strange. During experiments the robots are dealing with real friction and not a modelled one. Also the presence of measurement noise influences the error. This measurement noise is also the reason that the error between master and slave is larger than the error between reference and master. This is because the reference signal is a perfect signal and the masters position is not. So differentiating this signal with respect to time, will lead to a more disturbed signal which is harder to adapt to. Although it has been tried to avoid this problem (described in section 4.4) it is still not really good. But for now, the results are satisfactory.

CHAPTER 6. EXPERIAdENTS

Error Master Slave ql (translation) I I

Error Master Slave q2 (rotation) 0 06 1

Figure 6.2: Error between Master and Slave during ail experiment

-0 0 4 ~

-0 06-

'

I I I I I I I I I

20 40 60 80 100 120 140 160 180 time

Chapter 7

Conclusions and Recommendat ions

7.1 Conclusions

During this project an adaptive controller has been implemented and tuned in a master- slave synchronization problem where the slave does not depend on the reference velocity but only on the masters position and its derivatives. This causes a lot of problems t o get the simulations and experiments to work properly. Hardware as well as software problems have been there to solve but eventually it works. A problem that occurs during adaption is the fact that the mechanism could adapt to the initial errors of the system and will not adapt to the synchronization error. This problem is solved by letting the adaption mechanism slowly come up. On simulation level, this project results in very small synchronization errors (5 encoder accuracy). Unfortunately, dur- ing experiments, although the synchronization errors are small, they are still too large and the manipulators do not respond very smooth, that especially applies to the slave. This is caused by measurement noise. I have run out of time to investigate the role of observers in this synchronization problem but I believe, if the are tuned well, they will make a difference.

7.2 Recommendat ions

In this project only 2 degrees of freedom were adapted so there are 2 more left to do. Also implementing observers should overcome the problem caused by measurement noise and should improve the performance of the system.

A. Rodriguez, "Modelling and Identification of the CFT-Transposer Robot", 2002, DCT report number: 2002.52

K. Manssouri, "Tuning and Performance of a CFT Master-Slave Robot system", Report Traineeship, 2002, DCT report number: 2002.70

A. Rodriguez, "Synchronization of Mechanical Systems", University Press facilities, Eindhoven, 2002, ISBN: 90-386-2634-7

N.G.M. Rademakers, "Adaptive control of a CFT Master-Slave Robot System", Report Traineeship, 2003, DCT report number: 2003.46

Appendix A

Dynamic model of the CFT robot

,Entries of the inertia matrix M(q)

APPENDIX A. DYNAILUC MODEL OF THE CFT ROBOT 19

APPENDIX A. DYNAMIC MODEL OF THE CFT ROBOT 20

Entries of the Coriolis matrix: C(q,q)

cll = Czl = Cs1 = c41 = 0

APPEhTDIX A. DYNAMIC MODEL OF THE CFT ROBOT 21

APPENDIX A . DYNAMIC MODEL OF THE CFT ROBOT

APPENDIX A. DYNAA/IIC MODEL O F THE CFT ROBOT 23

Entries of the gravity vector: G(q)

Appendix B

Est h a t e d friction parameters

The physical parameters of the CFT-transposer robots present in the DCT-lab, are estimated using an Extended Kalman Filter and least square methods. This estimation is already performed by Rodriquez in ???. The results are presented in table B for the master and B for the slave.

APPENDIX B. ESTIMATED FRICTION PARAMETERS

Table B.l: Estimated parameters for the CFT transposer Master robot

Parameter description value 147.0161

2.i448 -0.6363

0.5931

APPENDIX B. ESTIMATED FRICTION PARAMETERS

Table R.2: Estimated parameters for the CFT transposer Slave robot

Parameter 01

description value m~ + m2 121.3049

n o i nn m 2 1 2 C 2 U.31U I