The 3rd Joint International Conference on Multibody System DynamicsThe 7th Asian Conference on Multibody Dynamics

June 30-July 3, 2014, BEXCO, Busan, Korea

Experimental Studies of Control Conceptsfor a Parallel Manipulator with Flexible Links

Markus Burkhardt∗, Robert Seifried†, Peter Eberhard∗

∗ Institute of Engineering and Computational MechanicsUniversity of Stuttgart

Pfaffenwaldring 9, 70569 Stuttgart, Germany[markus.burkhardt, peter.eberhard]@itm.uni-stuttgart.de

†Institute of Vehicle Technology - Dynamical Systems GroupUniversity of Siegen

Paul-Bonatz-Straße 9-11, 57068 Siegen, Germanyrobert.seifried@uni-siegen.de

ABSTRACT

Control of flexible multibody systems, such as flexible manipulators, is a challenging task. This is especiallytrue if end-effector trajectory tracking is aspired. On the one hand, these systems require a large number ofgeneralized coordinates to describe their dynamical behavior accurately. On the other hand, only a smallsubset of these values can be measured or reconstructed on-the-fly. Hence, it is difficult, if not nearlyimpossible, to use a state controller. In addition, flexible systems are underactuated, i.e. they possess lesscontrol inputs than generalized coordinates. In case of a non-collocated output controller, which is thecase for end-effector trajectory tracking, the closed loop of the system might loose passivity and is non-minimum phase. In order to achieve end-effector trajectory tracking, exact and approximate feed-forwardcontrols can be applied. In this work, two different versions of such concepts are compared experimentally.These model-based concepts are computed off-line and they supply, next to the required input values, aC1-continuous solution of the complete state vector which can be used for feedback control. If the systemis non-minimum phase, a two-sided boundary value problem has to be solved and the solution includes apre-actuation as well as a post-actuation phase. While the exact method incorporates all dynamical effectsof the flexible multibody system, the approximate concepts neglect certain implications, for example thedynamical effects due to the flexibility. In addition to the presentation of the theoretical basics of the controlapproaches and the underlying models, this contribution addresses some of the crucial obstacles, whichhave to be overcome for the operation of the test bench, e.g., signal conditioning, state reconstruction andfriction compensation. Since the installed sensors do not allow the direct measurement of the end-effectorposition, image tracking is used to judge the quality of the different control approaches.

1 INTRODUCTION

In this contribution, different control approaches for end-effector trajectory tracking of a parallel manip-ulator with highly flexible links are compared experimentally. The considered approaches are based ondifferent model-based feed-forward control designs [1] which are supported by additional feedback con-trollers. In the following sections, the experimental setup is presented and the forward dynamics as wellas the inverse dynamics of the model are derived. Conclusively, a comparison of the different control ap-proaches is carried out.

2 A PARALLEL MANIPULATOR WITH FLEXIBLE LINKS

For the experimental studies, a custom-made parallel manipulator with flexible links is available. Thistest stand can be used for arbitrary control tasks, because the hardware as well as the software is highlyadjustable. In the following, a short overview of the available hardware components and the process controlis given.

2.1 Experimental setup

The experimental platform, which can be seen in Fig. 1, consists of two identical linear direct drives whichare mounted on a granite plate. Two revolute joints, each located on top of one direct drive, define therotational axis of two highly flexible links with different length. An additional revolute joint connects themiddle of the long link with the end of the short link. An additional mass which is attached to the free endof the long link defines the end-effector point. Hence, this parallel manipulator permits a non-redundantmovement of the end-effector in the horizontal plane. The drive positions are measured with optical encodes.In addition, five full bridges with four strain gauges each are attached to the elastic links. Three of thesefive bridges can be employed simultaneously to capture the elastic deformations.

Figure 1: Experimental setup of the parallel manipulator and the control box.

2.2 Process control

The process control is carried out with a Speedgoat performance real-time target machine running Math-works’ xPCtarget kernel, which is called Simulink Real-Time since Matlab R2014a. Figure 2 shows theflowchart of the process control. The embedded system implements a rudimentary safety logic as well asthe observers and controllers needed for the control tasks. The current controllers are realized in hardwareby means of two distinct Kollmorgen AKD servo drives, which offer high bandwidth torque-loops. Thecommunication between the embedded system and the servo drives is carried out with digital and analogI/O-lines as well as with a serial bus according to the CANopen protocol [3]. The human-machine interfaceis realized in software as a graphical user interface (GUI). The GUI is based on C/C++ and heavily relieson the application framework Qt [4]. It communicates with the embedded system via a proprietary TCP/IP-based protocol which is provided as a C-API by xPCtarget. The openly published communication protocolModbus [5] is used to interact with the servo drives. The file management is realized with the HierarchicalData Format (HDF5), which allows an efficient data exchange with Matlab. The target machine runs amultirate model to accommodate the different tasks and communication types. The basic sampling rate of1.666 kHz is used for the analog I/O-lines, the control loops and the CANopen process data objects (PDO).This rather slow sampling rate is due to the limited number of messages that can be transmitted within acertain time interval. The digital I/O-lines and the safety logic are processed with a sampling rate of 500 Hz.These different sample times are executed in parallel on designated cores.

hdf5

dlm

graphical user interface

teststand

amplifier

serial busanalog

digital

file exchangeencoder

embeddedsystem

servodrives

power

Figure 2: Flowchart of the process control.

3 FORWARD AND INVERSE DYNAMICS

The control concepts, that is used in this work, requires a detailed model of the parallel manipulator. Dueto the fact that the time constant of the electrical part is very small compared to the mechanical part, thedynamics of the electrical part can be neglected. This assumption [6] is valid for many electro-mechanicalsystem and is adopted in this work. Therefore, it is only necessary to state the forward dynamics and theinverse dynamics of the mechanical subsystem.

3.1 Forward dynamics

The mechanical part of the manipulator is modeled as a flexible multibody system using the floating frameof reference approach. This approach is suitable for bodies that undergo a large working motion and expe-rience only small elastic deformations. The large nonlinear motion of the body is described with absolutecoordinates of a reference frame, whereas the deformation is described relative to this moving referenceframe. For the integration into the multibody system, the absolute coordinates of the reference frame areexpressed in minimal coordinates. In this contribution, the equations of motion are derived by applying theNewton-Euler equations together with D’Alembert’s principle. For systems with a tree-like topology and nfdegrees of freedom, this formalism yields the equations of motion in terms of a minimal set of generalizedcoordinates y ∈ Rnf which are linearly independent. However, if there are kinematic loops in the system,it might be impossible or disadvantageous to find an appropriate set of independent coordinates. In thiscase, the kinematic loops have to be cut open. This yields additional nc degrees of freedom, that have to beeliminated after the equations of motion are stated. Therefore, the equations of motion are stated in terms ofthe dependent coordinates z ∈ Rnf+nc . Finally, algebraic constraint equations c(t, z) ∈ Rnc and Lagrangemultipliers λ ∈ Rnc are used to close the kinematic loops again. Hence, the set of differential-algebraicequations which describes the constrained dynamics of a general multibody system can be stated as

M(z)z = f(z, z) +B(z)u+C(z)Tλ , (1a)0 = c(z) . (1b)

The dynamics equations (1a) of the open-loop system are characterized by the generalized mass matrixM , which is symmetric positive definite, and the generalized force vector f , which contains the cen-trifugal, Coriolis and gyroscopic forces as well as those applied forces, which are not regarded as controlinputs u. For input-affine systems, the input matrix B is introduced. The Jacobian matrix of the con-straints C = ∂c/∂z associates the Lagrange multipliers with the system dynamics. All computations arecarried out with the research code Neweul-M2 [7], which derives the equations of motion symbolically.

s1

s2

α2

α1

end-effector

0 0.2 0.4 0.6 0.8 1

0

2

·10−2

position on beam axis x [m]

bend

ing

mod

esφi(x

)[m

]

2.496 Hz20.42 Hz37.48 Hz71.18 Hz110.7 Hz894.7 Hz

Figure 3: Parallel manipulator with flexible links (left) and mode shapes of the long link (right).

The topology of the parallel manipulator and the mode shapes of the isolated long arm are shown inFig. 3. The elastic body is obtained by a linear FE-model and reduced with a hybrid model-order reduc-tion, which combines the classical Craig-Bampton approach with modern Gramian-matrix-based reductiontechniques [8]. Afterward, Rayleigh damping is added. The short arm is regarded as a rigid body. Hence,the vector of dependent coordinates z consists of the drive positions s1 and s2, the angles α1 and α2 andthe elastic coordinates qe ∈ R6, which are assigned to the shown mode shapes. The kinematic loop due tothe joint connecting the links is stated as the difference of the two position vectors rj,1 and rj,2. Hence, thetwo-dimensional constraint equations c on position, velocity, and acceleration level are expressed as

c(z) = rj,2(z)− rj,1(z) , (2a)

c(z, z) =∂rj,2∂z

z − ∂rj,1∂z

z = (Jj,2 − Jj,1) z = Cz , (2b)

c(z, z, z) = (Jj,2 − Jj,1) z +

(∂Jj,2z

∂z− ∂Jj,1z

∂z

)z = Cz + c′′ . (2c)

Next to the reaction forces, which are provided by the Lagrange multipliers, the applied forces acting onthe drives have to be taken into account. On the one hand, the driving forces arising from the product ofthe force constants Km,i and the currents ii of the motors have to be considered. These currents are thecomponents of the input vector u. On the other hand, there are friction forces, which antagonize the controlinputs.

3.2 Inverse dynamics

Based on the proposed dynamical model of the parallel manipulator, the inverse dynamics can be statedin a straight-forward way. By using the concept of servo constraints [9], the forward dynamics can beextended with additional ns constraint equations s describing the trajectory-tracking problem. In contrastto classical constraints, the control inputs u instead of the Lagrange multipliers are used to enforce the servoconstraints s ∈ Rns .

The solution of the so-called internal dynamics, which describe the dynamics that fulfills both the classi-cal and the servo constraints, provides the feed-forward control. Depending on the stability of the internaldynamics, which depends on the stability of the system dynamics as well as on the input-output combina-tion [2], a two-point boundary value has to be solved to obtain a bounded solution. A detailed descriptionof this approach can be found in [10]. Due to the separation of the overall dynamics in two parts, namelythe driven and the internal dynamics, it is not necessary for the solution of the boundary-value problem tospecify the friction forces which act on the drives, since they do not affect the internal dynamics. The dryfriction for example, which has to be modeled as a non-smooth force and, therefore, impedes the numerical

solution, can be neglected in the solution phase. Instead, these friction forces are computed algebraically ina post-processing step or online in the form of a friction compensation [11].

The inverse dynamics of the parallel manipulator can be stated as a set of differential-algebraic equationsaccording to

M(z)z = f(z, z) +Bu+C(z)Tλ , (3a)0 = c(z) , (3b)0 = s(t, z) . (3c)

For end-effector trajectory tracking, the servo constraints can be expressed as the difference between theactual end-effector position in the plane ree and the desired position rdes yielding the servo constraints onposition, velocity, and acceleration level

s(t, z) = ree(z)− rdes(t) , (4a)

s(t, z, z) =∂ree∂z

z − rdes = Jeez − rdes , (4b)

s(t, z, z, z) = Jeez +∂Jeez

∂zz − rdes = Sz + s′′ . (4c)

Based on the constraint equations on acceleration level, the system inputs and the Lagrange multipliers canbe expressed as [

λu

]= −

[CM−1CT CM−1BSM−1CT SM−1B

]−1 [CM−1f + c′′

SM−1f + s′′

], (5)

if the inverse of this matrix exists. In this case, the differentiation index of the servo control problem is threeand there remain nf − ns free coordinates. In nonlinear control theory, this correlates with a relative degreeof two, see [12]. In the next step, the driven dynamics is eliminated and the internal dynamics is stated asa set of ordinary differential equations for the independent states. For the flexible manipulator, a possiblebasis for the independent states consists of the elastic coordinates and their derivatives. The dependentcoordinates, which still appear in the dynamics, have to be computed with a root search.

Based on the provided end-effector trajectories and the solution of the internal dynamics, it is possible toprovide the trajectories of the dependent coordinates z∗, their time derivatives z∗ as well as for the systeminputs u∗ and the Lagrange multipliers λ∗ that fulfill both types of constraints on a fixed time interval.These form the feed-forward control. Therefore, this approach can be regarded as dynamical trajectoryplanning. For the application on the real-time target system, the internal dynamics are solved off-line andthe computed trajectories are interpolated with a constant sample time of 0.6 ms, which corresponds to thesampling time of the control loops, and are exported as HDF5-files.

4 CONTROL CONCEPT AND SIGNAL CONDITIONING

For the stabilization of the precomputed solution, two distinct cascade controllers for the drive positions areimplemented. These well-proven controllers offer the possibility to feed-forward the positions, velocities aswell as the currents, which accords with the requirements of the overall control concept. Figure 4 shows thetopology of the control loop. The outer loop is a proportional gain, which amplifies the position error andfeeds the middle loop. The velocity loop is carried out as a PI-controller and the velocity is feed-forwarded.The inner loop, which consists of the electrical subsystem Σe and a proportional gain, is realized in theservo controllers. The states needed for the feedback controller are the drive positions si and velocitiessi. The positions of the mechanical subsystem Σm are acquired from the servo drives and transmitted asCANopen PDOs. In order to reconstruct the velocities, a Kalman filter (KF) [13] is applied. The underlyingmodel of the filter is a zero-mean first-order Markov model [14], which is borrowed from target trackingapplications, see [15]. The main advantage of this filter is its simplicity and the consequential efficiency inreal-time applications.

In addition, the assumptions of the filter regarding the smoothness of the trajectories consort with a feed-

∫Iv

PvPp Pi Σe Σm

KF

Keposition s∗ velocity s∗ current i∗

position s

filtered velocity ˙s

filtered position s

i

velocity s

----

in hardware

desired desired desired

Figure 4: Control loop

forward control. The discrete-time state space representation of the model is

xk+1 =

1 T 1α2 (−1 + αT + e−αT )

0 1 1α (1− e−αT )

0 0 e−αT

xk + uk , uk ∼ N (0,Qk) , (6)

yk =[1 0 0

]xk + vk , vk ∼ N (0, Rk) , (7)

in which the discrete state vectorxk includes the position, velocity, and acceleration of the drive. The systemdynamics, which consists of two integrators and stable pole associated with the acceleration, is driven by thediscrete time white noise sequence uk. Since the position signals are acquired from an optical encoder andprocessed digitally, the measurement noise covariance Rk of the output yk is almost negligible. Therefore,the main purpose of the filter is the robust reconstruction of the velocity. These are the only signals that areneeded for the feedback controllers. At this point, the available strain gauges are not included in the controlconcept.

5 EXPERIMENTAL STUDIES

Based on the presented models and control approaches, two different implementations are benchmarked.The first implementation is based on the full dynamical model of the elastic system and the second oneis based on the equivalent rigid model. While the first concept requires the solution of a boundary valueproblem, see Section 3.2, the second concept can be solved algebraically, see e.g. [16]. For both scenarios,the end-effector trajectory is supposed to cover a distance of 0.282 m within 0.5 s following a straight line.The used feedback controllers are parametrized identically and in both cases, the positions, velocities andcurrents are feed-forwarded. The friction compensation adds, in dependency of the motion of the drives, aconstant current to the precomputed inputs.

5.1 Image tracking

In order to compare the two implementations, the position of the end-effector point is tracked with a camera.To obtain the best possible contrast, all reflecting parts of the end-effector are covered with felt. Thedesignated point and an additional point are highlighted with reflector foil. The distance between theseemphasized dots is used for the conversion from pixels to meters. Compared to a designated measure, thesetwo points remain in the image plane, which avoids additional conversions. The used camera is capable torecord 60 frames per seconds with a resolution of 1280× 720 pixels.

Figure 5 shows the results of the image tracking for the feed-forward control implementation based on theelastic model. The colored areas are proportional to the brightness of the image. A darker color in theLight-Bertlein color map indicates that the area was exposed for a longer period of time than a lighter color,i.e. slow motion. It is easy to see that the end-effector point follows a straight line with small deviations.The right side of the image also reveals a small oscillation at the end of the trajectory.

Figure 5: Brightness distribution of all frames merged into one figure for the elastic feed-forward control.

Figure 6: Brightness distribution of all frames merged into one figure for the rigid feed-forward control.

In contrast, Fig. 6 shows the feed-forward control based on the rigid model. This implementation showsimmediately a comparatively big deviation from the desired trajectory. In addition, this solution shows alarge overshoot at the end of the trajectory yielding a large oscillation.

5.2 Post-processing

Based on these images, the trajectory of the end-effector is identified. In a first step, the positions of theend-effector point and the additional marker are obtained frame-by-frame. Due to the fact that multiplepixels are exposed by the light reflected by the foil, it is possible to interpret the brightness of the frame as atwo-dimensional non-normalized probability distribution. In order to improve the accuracy, it is importantto filter the image to remove noise and to isolate the two peaks from each other. Otherwise, the size of theimage would have a great influence on the expected value. The filtering is done with a sliding-neighborhoodfilter. The expected values of these distributions, which equals the components x and y of the position vectorrpix of the designated markers in the horizontal plane, can be obtained by

p(i, j) =b(i, j)∑

i∈M∑j∈N b(i, j)

, (8a)

x =∑i∈M

∑j∈N

ip(i, j) , (8b)

y =∑i∈M

∑j∈N

jp(i, j) . (8c)

Here, the normalized probability p(i, j) of the i, j-th pixel is obtained from the brightness b(i, j), whichranges from 0 to 255. Due to the filtering, the summation can be performed on the subsets M and N ,which are the neighborhoods of the maximum value of the brightness distribution. Outside of these sets, thebrightness is zero.

In a next step, the conversion from pixels to meters is carried out with the conversion factor

fc =|∆||∆pix|

, (9)

where ∆ label the difference of the position vectors of the two designated markers. The values with theindex pix are measured in pixels and the index-free values are measured in meters. After the conversion ofthe reconstructed position vectors, the data is resampled using spline interpolation, aligned with the desiredvalues and cropped. In a last step, the measured and processed trajectories are rotated and translated in sucha way, that the initial conditions of the desired and measured trajectories match.

5.3 Interpretation of the experimental results

The processed measurement data sets permit to benchmark the different feed-forward control implemen-tations. To compare the two approaches, the deviations of the end-effector from the desired trajectory areregarded. Figure 7 shows the tracking errors of both approaches over time. Due to the fact that the post-processing was carried out in such way that the initial conditions of the end-effector match the desiredvalues, there are no deviations at t = 0. Consistent with Figs. 5 and 6, the trajectory of the feed-forward

approach, which is based on the elastic model, shows a much smaller error than the second approach whichis based on the equivalent rigid model. The maximum errors of the flexible and the rigid feed-forwardimplementation is 1.3 cm and 10 cm, respectively.

0 0.1 0.2 0.3 0.4 0.5

0

2

4

6

8

10·10−2

time [s]

devi

atio

n[m

] elasticrigid

Figure 7: Reconstructed deviation of the two implementations.

The errors shown in Fig. 7 do not only results from inaccurate models. In fact, the control deviations can notbe regarded as negligible. Therefore, it is reasonable to investigate the tracking errors of the drive positions.Figure 8 shows the position-tracking errors of the first drive for both approaches. Both graphs exhibit errorsin the initial conditions as well as in the movement. These errors are mainly caused by the friction forces,which act on the drives and, in case of the rigid feed-forward-control, by elastic deformations, that affectthe drive positions during the movement. Due to the feedback controllers and the friction compensation,this tracking error is stabilized. These errors in the drive positions are, in case of the flexible feed-forwardcontrol implementation, the main source of errors in the end-effector tracking. It should also be noted,that at this point the elastic deformation is not included in the feedback control. Thus, errors in the elasticdeformation cannot be compensated yet.

0 0.1 0.2 0.3 0.4 0.5

−0.5

0

0.5

1·10−2

time [s]

devi

atio

n[m

] elasticrigid

Figure 8: Drive tracking errors of the two implementations.

6 CONCLUSIONS

In this contribution, experimental studies regarding two different feed-forward control designs for a parallelmanipulator with flexible links are carried out. After a short presentation of the invoked hardware andsoftware, the forward and inverse dynamics of the flexible multibody system are stated. A quick introductionto the used control concept leads to the discussion of the experimental results. Here, the main idea of imagetracking is revised and the two implementations are compared. The feed-forward control approach, which isbased on a complex flexible multibody system and obtained by a boundary-value problem, shows obviousimprovements in the end-effector deviations compared to the approach, which is based on an equivalentrigid model. Even though there remain deviations in the drive positions as well as in the end-effectortracking, this approach is a promising alternative to classical control concepts.

Acknowledgment

The authors would like to thank the German Research Foundation (DFG) for their financial support ofthe project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University ofStuttgart and the project SE 1685/3-1.

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