Trajectory Planning and Control of Parallel Manipulators

Yunjiang Lou, Fang Feng, and Michael Yu Wang

Abstract— Kinematics based trajectory planning for parallelmanipulators has been studied extensively, e.g., to determinea singularity-free path. Dynamics based trajectory planning,however, gains little concern. In order to fully exploit capacityof parallel robots, it is inevitable to conduct motion planningconsidering both kinematic and dynamic constraints. In thispaper, three dynamics based trajectory planning methods,the time-optimal, the jerk-bounded, and the cubic polynomialplanning, are investigated and implemented. The Orthopod,a 3-DoF purely translational parallel manipulator, is used asthe experimental plant. The computed-torque control scheme isapplied to follow the computed trajectory. In stead of manuallytuning, control parameters are determined by formulating anoptimization problem. Optimal parameters are thus obtainedsystematically. This study provides a quantitative comparisonof the three planning methods and an engineering method toobtain optimal control parameters. The experimental resultsprovide a guideline to choose an appropriate planning methodfor practical industrial applications.

Index Terms— parallel manipulators, trajectory planning,time-optimal, dynamics-based

I. INTRODUCTION

There are generally two types of motion planning schemes,the kinematics based and the dynamics based motion plan-ning, respectively. For kinematics based motion planning, thepath (position, velocity, acceleration, etc) is determined withrespect to its kinematic constraints only. While a dynamicsbased motion planning determines all path information basedon not only its kinematic constraints but also its dynamicconstraints. In order to fully exploit dynamic capabilities ofa manipulator, it is inevitable to include dynamics in motionplanning. In this paper, we deal with motion planning forparallel manipulators.

Dynamics based motion planning involves determinationof path information while optimizing some desired objective,e.g., actuation force, Cartesian force, operating time. In [1],the objectives are to maximize the structure stiffness andto minimize actuation force while satisfying constraints ofworkspace and singularity. Oen and Wang [2] investigateddynamic trajectory planning problems for a Gough-Stewartplatform based machine tool. The distribution of the cuttingforce of the tool bit along a specified machining contour andthe distribution of actuator forces are optimized, respectively.

This work was supported by National Science Foundation of China(NSFC) under Grant 50705015.

Yunjiang Lou and Fang Feng are with the Division of Control andMechatronics Engineering, Shenzhen Graduate School, Harbin Institute ofTechnology, Shenzhen, China. Michael Yu Wang is with the Departmentof Mechanical and Automation Engineering, Chinese University of HongKong, Shatin, NT, Hong Kong. He is also with the Division of Controland Mechatronics Engineering, Shenzhen Graduate School, Harbin Insti-tute of Technology, Shenzhen, China. (Email: louyj@hitsz.edu.cn, feng-fang0717@hotmail.com, yuwang@mae.cuhk.edu.hk)

In [3], the objective is to minimize actuating effort andreactions to generate a singularity-free trajectory.

In order to improve productivity in industry, time-optimaltrajectory planning was proposed and studied extensively.The objective is to minimize operating time or maximize ve-locity capacity of the manipulators while satisfying all otherkinematic and/or dynamic constraints. Since the mid 1980s,powerful and attractive algorithms for time-optimal trajectoryplanning for industrial serial manipulators are known [4][5].Little concern was put on time-optimal motion planningof parallel manipulators. Actually, the dynamics and themodeling of parallel manipulators are categorically differentthan serial ones [6]. Abdellatif and Heimann [7] proposeda time-optimal trajectory planning for the PaLiDa parallelmanipulator. In this paper, a systematic modeling is presentedfor parallel manipulators. For comparison purpose, threetrajectory planning methods, time-optimal, jerk-bounded, andcubic polynomial methods, are investigated by simulationand experiments.

Dynamics and control of parallel manipulators have beenstudied extensively [8][9][6]. Computed-torque control [10]is widely used in various applications. A usual way todetermine the control parameters is to tune the parametersmanually so that satisfactory performance appears. For acomplex system having a large number of control parametersit is difficult to tune control parameters to obtain gooddynamic performance. Furthermore, it is unknown how goodthe performance is. Few researchers touched how to deter-mine optimal control parameters. In this paper, we proposeto formulate the optimal control parameter determinationproblem as an optimization problem. Dynamic criteria, e.g.,overshooting, settling time, and steady error, are taken asobjective or constraints. The optimal control parameters canthus be obtained by solving the optimization problem.

The paper is organized as follows. In Section II, A sys-tematic modeling of parallel manipulators is presented. Thetime-optimal trajectory generation is discussed in Section III.In Section IV, control algorithms is discussed. In SectionV, the time-optimal, jerk-bounded, and cubic polynomialplanning are experimentally implemented. Discussions onexperimental results are given. At last, a conclusion is drawnin section VI.

II. DYNAMIC MODELING

For a normally-actuated parallel manipulator, a tree sys-tem, which is also called a reduced system, is obtained bycutting the end-effector [11]. By regarding all the jointsin the tree system as actuated ones, the dynamics of thetree system can be obtained by either the Newton-Euler

2009 IEEE International Conference on Control and AutomationChristchurch, New Zealand, December 9-11, 2009

ThMT3.5

978-1-4244-4707-7/09/$25.00 ©2009 IEEE 1013

Fig. 1. A prototype of the Orthopod parallel manipulator

Fig. 2. A kinematic schematic of the Orthopod parallel manipulator

formulation or the direct Lagrangian approach [10]. Fol-lowing the Lagrange-D’Alembert formulation, a normally-actuated parallel manipulator can be viewed as its corre-sponding open-chain tree system constrained by the loop-closure equations, which are obtained by equating pairwisethe end-effector positions from each of the subchains. Bymathematic manipulations, a dynamic model in Cartesiancoordinate frame can be obtained in the following form [6].

M(x)x +C(x, x)x + Rx+ G(x) = F, (1)

where M(x) is the mass matrix , C(x, x)x is the Coriolis forceterm and R is the viscous friction coefficient matrix, and G(x)is the gravitational force vector. Here, x represents the gener-alized coordinate in task-space and F is the generalized forcevector. In this paper, we focus the study to the Orthopod, anovel 3-DoF purely translational parallel manipulator [12].Fig. 1 shows a developed prototype of the manipulator. Asimplified schematic for the Orthopod with three identicalsubchains is shown in Fig. 2.

To integrate actuator constraints into the planning algo-rithms, a transformation of the dynamics in the space ofactuated joints is necessary. It is possible to transform thegeneralized rigid-body force vector F into actuator forcevector τ by using the Jacobian of the manipulator F = JT τ ,where J is the Jacobian relating the joint velocity to Cartesianvelocity. We define ρ the generalized coordinate in jointspace. Readily, the dynamic model in joint space is obtainedin (2).

M(ρ)ρ +C(ρ , ρ)ρ + Rρ + G(ρ) = τ, (2)

where

M = J−T MJ−1;

C = J−T (MJ−1 +CJ−1);

R = J−T RJ−1;

G = J−T G.

A geometric path may be described generally as a re-lationship between coordinates x and a path coordinate λwith x = f (λ ) , 0≤ λ ≤ λmax. For parallel manipulators it iseasy to transform the path geometry to joint space, since theinverse kinematic problem is simple to solve. The relationρ = f (λ ) is obtained analytically for every λ ∈ [0,λmax], withλmax corresponding to the path end. The actuation dynamicsto the one dimensional motion with respect to λ can bederived with the use of

ρ =d fdλ

λ ;

ρ =d2 fdλ 2 λ 2 +

d fdλ

λ . (3)

This yields

m(λ )λ + c(λ )λ 2 + r(λ )λ + g(λ ) = τ(λ ). (4)

The path coefficients are obtained from the dynamic modelwith

m(λ ) = M( f (λ ))d fdλ

(λ );

c(λ ) = M( f (λ ))d2 fdλ 2 (λ )+

d fdλ

(λ )TC(λ )d fdλ

(λ );

r(λ ) = Rd fdλ

(λ );

g(λ ) = G( f (λ )).

With the dynamics along a prescribed trajectory (4), wecan do planning issue.

III. TRAJECTORY PLANNING

In this section, formulations of time-optimal, jerk-bounded, and cubic polynomial motion planning are dis-cussed.

A. Formulation of time-optimal planning

By assuming that a desired geometric path was prescribedand is given in a parametric form x = f (λ ) and ρ = f (λ ),the formulation of time-optimal planning for parallelmanipulators is not different than any other type ofmanipulators. As it was demonstrated by Shin and McKayin [4], minimizing trajectory time can be formulated inmaximizing path velocity and therefore maximizing pathacceleration by maximizing the actuation forces. Let n bethe degree of freedom for robot and λ the velocity alongthe path. The actuation constraints are given as follows.

τmini ≤ τ ≤ τmax

i ,∀i = 1,2, · · ·n (5)

1014

and the boundary conditions

f (λ = 0) = f0;

f (λ = λmax) = fmax;

λ (λ = 0) = λ0;

λ (λ = λmax) = λend .

Combining (4) and (5) yields a set of double inequalitiescorresponding the n actuators of the manipulator

τ imin ≤ miλ + ciλ

2 + riλ + gi ≤ τ imax. (6)

For mi �= 0, a set of 2n simple inequalities is obtained

LBi ≤ λ ≤UBi, (7)

where

LBi =τ i

min(mi > 0)+ τ imax(mi < 0)− ciλ

2− riλ −gi

mi;

UBi =τ i

max(mi > 0)+ τ imin(mi < 0)− ciλ

2− riλ −gi

mi.

The expression (mi > 0) is equal to one if mi(λ ) > 0, zerootherwise. The region within all constraints defined by allactuators is called ”region of admissible motion” or simply”admissible region”, because only motion with pairs (λ , λ )within this area could be executed by the manipulator. Therelevant parabolic constraints are depicted and evaluated withthe full dynamic model. For a nominal actuated parallelmanipulator with identical subchains, the nominal inertiaare usually equal. However, the dynamics is configuration,velocity, acceleration dependent. When the manipulator is ata different configuration in the workspace with a differentacceleration, using (4) and (5), we compute a different max-imum reachable speed. Each position λ is usually associatedwith a maximum possible path velocity λmax, which is theoutermost point of the admissible region and is characterizedby a single possible acceleration λ (λmax). Collecting λmax

for every 0 ≤ λ ≤ λmax yields the boundary curve, alsocalled maximum velocity curve or simply maximum curve[4][5][13].

B. Trajectory construction of time-optimal planning

The time-optimal velocity profile is constructed basedon the resulting maximum curve. The aim of time-optimaltrajectory planning is to determine the maximum velocityprofile along a prescribed path that observes all given dy-namic and kinematic robot constraints. The problem canbe described mathematically as an optimal control problemaiming to minimize the operating time

T =

∫ t f

0dt =

∫ λmax

0

dtdλ

dλ =

∫ λmax

0

1

λdλ . (8)

The time-optimal velocity profile is constructed based on theresulting maximum curve

φ(λ ) = {(λ , λ )⊂ℜ2,0≤ λ ≤ λmax, λ (λ ) = λmax(λ )}. (9)

As shown in the experiment part later, the drawback oftime-optimal trajectories is an increasing tracking error due

to model errors. This runs a risk that the manipulator willtemporarily exceed the drive torque limitations. In practical,robot drives are not saturated when it works, because thissituation is bad for manipulator. Therefore, the drive torquelimitations should be chosen carefully in practical so that themanipulator will not exceed torque limitations.

C. The cubic-polynomial planning

In the cubic-polynomial planning, the trajectory isparametrized by time t

λ (t) = a1t3 + a2t2 + a3t + a4, (10)

where ai, i = 1, · · · ,4 are coefficients determined by startingand end conditions. The cubic-polynomial planning is actu-ally the ”S-profile” of speed in motion control text. Clearly,the acceleration λ = 6a1t +2a2 is an affine function of t andis continuous on t. Constantinescu and Croft [14] used thecubic-polynomial planning to generate a smooth trajectoryby imposing limits on jerk, the third derivative of λ withrespect to time t.

D. The jerk-bounded planning

In cubic-polynomial planning, the jerk,...λ = 6a1, is con-

stant. In switch points from an acceleration section to aconstant-speed section, or from a constant-speed sectionto a deceleration section in speed profile, the jerk is dis-continuous, i.e., from 6a1 to 0, or from 0 to −6a1. Thediscontinuity of jerk may lead to vibration that deterioratesmanipulator performance. Therefore, jerk continuity is de-sirable to improve control performance of a manipulator. In[15], Macfarlane and Croft proposed a fifth-order (quintic)polynomial to plan a given path.

λ (t) = a1t5 + a2t4 + a3t3 + a4t2 + a5t + a6 (11)

In this case, the jerk...λ = 60a1t2 +24a2t +6a3 is a quadratic

function of t, which is continuous and second-order differen-tiable. Therefore, it will not produce severe vibration. Usingbounded jerk limits and actuator torque limits, we can derivetime-optimal planning with a path parametrized by a quinticpolynomial.

IV. CONTROLLER DESIGN

Because the dynamics of a parallel manipulator has thesame form as that of a parallel manipulator, the extension ofexisting control schemes designed for serial robots to closed-chain mechanism is natural, i.e., PD control, augmentedPD control, computed torque control, etc. Independent jointcontrollers (of PD or PID type) are very computationallyefficient and easy to implement. In this paper, the computedtorque control scheme that is shown in Fig. 3 was used fortrajectory tracking. This scheme utilizes nonlinear feedbackto decouple the manipulator. The control law is taken as

τ = M(ρ)(ρd− kve− kpe)+C(ρ , ρ)ρ + Rρ + G(ρ), (12)

where e = ρ−ρd is the tracking error, kp and kv are constantfeedback gain matrices. By substituting (12) to (2), we obtainerror dynamics of the manipulator as follows.

e + kve+ kpe = 0 (13)

1015

Fig. 3. A block diagram of the computed torque control scheme

The stability issue of the computed torque scheme is dis-cussed in [10].

In order to obtain optimal dynamic performance, it isnecessary to determine the parameters in matrix kv and kp

optimally. Generally, kv and kp are taken as diagonal ma-trices, kv = diag{kv1,kv2,kv3} and kp = diag{kp1,kp2,kp3}.However, there are coupling effects among the dynamics ac-tually. Especially it is not the case for parallel manipulators.Therefore, kv and kp should be regarded as a generic 3×3matrix.

In order to find the optimal control parameters, a stepinput is given and the response is investigated. The settlingtime ts reflecting response speed is taken as the objective.The overshooting, δ , and the steady error ess are taken asconstraints. Actuation saturation is also considered to reflectreal limits on actuators. Finally, we obtain a optimizationproblem as follows.

Problem 1: Find optimal control parameters kv and kp

such that

minkv,kp

ts

subject to δ < δ0;

ess < e0;

F < Fmax;

where δ0 is the largest allowable overshooting, e0 is thelargest allowable steady-state error, and Fmax is largest ap-plicable actuation force.By solving the optimization problem above, the optimalparameters kv and kp can thus be obtained. Since the opti-mization involves solving dynamic equations, we apply a fastdirect search algorithm for global optimization, the differen-tial Evolution (DE) [16], to solve the problem numerically.

V. EXPERIMENTS AND DISCUSSIONS

In experiment, the time-optimal, the cubic polynomial,and the jerk-bounded (quintic polynomial) algorithms wareapplied to the Orthopod for a desired motion of a circulartrajectory. It is a horizontal circular arc with a radius of 5cm in the center of the workspace. The actuator forces werelimited to the range of 250 N. The admissible region for eachactuator is computed in phase plane of joint states, see Fig.4.

Fig. 4. The velocity boundary of the time-optimal trajectory

Fig. 5. The actuation force plot for the time-optimal trajectory

Trajectories were generated off-line used the three plan-ning methods off-line in MATLAB�. The maximum ac-celeration and velocity for the time-optimal planning werecomputed as 2.1832 m/s2 and 0.3162 m/s, respectively. Themaximum acceleration and velocity were 3.1795 m/s2 and1.9739 m/s2, 0.3299 m/s and 0.3141 m/s for jerk-boundedand cubic polynomial planning, respectively. For all threeplanning methods, actuator limits are applied.

The performance of the three trajectory planning methodswill be evaluated by comparing the practical actuated torque,running time and the tracking accuracy of the end-effector.The tracking accuracy was evaluated by the root square mean

error (RSME), RSME=√

∑Err2

n . The experimental results fortest motions are presented in Table I.

The success of the trajectory planning can be verifiedby investigating the actuator forces during the time-optimalmotion. The time history of all forces is shown in Fig. 5, Fig.6, and Fig. 7 for these three methods. The results show thatat every time, at least one actuation force is maximized fortime-optimal planning. In all three figures, the No. 3 actuator

1016

Fig. 6. The actuation force plot for jerk-bounded planning

Fig. 7. The actuation force plot for cubic polynomial planning

forces keep constant since the trajectory is in XY plane andthe No. 3 actuator, which is along Z axis, needs only tocounteract gravitation force.

For the cycle time, the time-optimal planning takes leasttime, the cubic polynomial planning takes second leasttime, while the jerk-bounded planning takes longest time tofinish a circle. It is obvious, from Table I, that the time-optimal planning improve the time-efficiency tremendouslycompared with the other two planning methods. For thetracking error issue, the jerk-bounded planning presentedbest performance, the cubic polynomial gave the second best,while the time-optimal planning algorithm leaded to worsttracking accuracy. Fig. 8, Fig. 9, and Fig. 10 show tracking

TABLE I

A PERFORMANCE COMPARISON OF THREE PLANNING METHODS

Algorithm RMSE Peak acc(m/s2)/vel (m/s) cycle time (s)Time-optimal 0.0465 2.183/0.3162 230×0.005Jerk-bounded 0.0332 3.179/0.3292 322×0.005Cubic polynomial 0.0375 1.974/0.3141 300×0.005

Fig. 8. Tracking error for the circular test trajectory: the time-optimalplanning

Fig. 9. Tracking error for the circular test trajectory: the jerk-boundedplanning

error history of the time-optimal, the jerk-bounded, and thecubic polynomial planning methods, respectively.

VI. CONCLUSIONS

In order to fully exploit capacity of parallel robots andmanipulators, it is necessary to do motion planning whileconsidering both kinematic and dynamic constraints. In thispaper, we investigated dynamics based trajectory planningmethods, the time-optimal, the jerk-bounded, and the cubicpolynomial planning. The three methods were implementedto the Orthopod, a 3-DoF purely translational parallel ma-nipulator. Torque limits were applied for all three imple-mentations. Using a circular motion as a test trajectory,experiments show that the time-optimal planning takes leasttime while has worst tracking accuracy. On the contrary, thejerk-bounded planning takes longest time while gives besttracking accuracy. The cubic polynomial planning presentsin-between performance in both cycle time and the tracking

1017

Fig. 10. Tracking error for the circular test trajectory: the cubic polynomial

accuracy. This study provides a quantitive comparison of thethree planning methods. Based on the results, we are able tochoose a appropriate planning method for practical industrialapplications.

REFERENCES

[1] S. Pugazhenthi, T. Nagarajan, and M. Singaperumal. Optimal trajec-tory planning for a hexapod machine tool during contour machining.Proceedings of the Institution of Mechanical Engineers, Part C:Journal of Mechanical Engineering Science, 216:1247–1257, 2002.

[2] K.-T. Oen and L.-C. T. Wang. Optimal dynamic trajectory planningfor linearly actuated platform type parallel manipulators having taskspace redundant degree of freedom. Mechanism and Machine Theory,42:727–750, 2007.

[3] C. Chen and H. Chi. Singularity-free trajectory planning of platform-type parallel manipulators for minimum actuating effort and reactions.Robotica, 26(3):371–384, 2008.

[4] K. G. Shin and N. D. McKay. Minimum-time control of roboticmanipulators with geometric path constraints. IEEE Transactions onAutomatic Control, 30(6):531–541, 1985.

[5] F. Pfeifer and R. Johanni. A concept for manipulator trajectoryplanning. IEEE Journal of Robotics and Automation, 3(2):115–123,1987.

[6] H. Cheng, Y.K. Yiu, and Z.X. Li. Dynamics and control of redun-dantly actuated parallel manipulators. IEEE/ASME Transactions onMechatronics, 8(4):483–491, 2003.

[7] H. Abdellatif and B. Heimann. Adapted time-optimal trajectoryplanning for parallel manipulators with full dynamic modelling. InProceedings of IEEE International Conference on Robotics and Au-tomation, pages 411–416, 2005.

[8] Y. Nakamura and M. Ghodoussi. Dynamics computation of closed-link robot mechanisms with nonredundant and redundant actuators.IEEE Transactions on Robotics and Automation, 5(3):294–302, 1989.

[9] T. Ropponen. Actuator redundancy in a closed-chain robot mecha-nism. PhD thesis, Helsinki University of Technology, Espoo, Finland,1993.

[10] R. Murray, Z.X. Li, and S. Sastry. A Mathematical Introduction toRobotic Manipulation. CRC Press, 1994.

[11] Y. K. Yiu. Geometry, Dynamics and control of parallel manipulators.PhD thesis, The Hong Kong University of Science and Technology,2002.

[12] Y. J. Lou, J. G. Li, J. B. Shi, and Z. X. Li. Development of a novel3-dof purely translational parallel mechanism. In Proceedings of IEEEInternational Conference on Robotics and Automation, pages 169–174,2007.

[13] J.-J. E. Slotine and H. S. Yang. Improving the efficiency of time-optimal path-following algorithms. IEEE Transactions on Roboticsand Automation, 5(1):118–124, 1989.

[14] D. Constantinescu and A. Croft. Smooth and time-optimal trajectoryplanning for industrial manipulators along specified paths. Journal ofRobotic Systems, 17(5):233–249, 2000.

[15] S. Macfarlane and E. A. Croft. Jerk-bounded manipulator trajectoryplanning: design for real-time applications. IEEE Transactions onRobotics and Automation, 19(1):42–52, 2003.

[16] K. Price, R. Storn, and J. Lampinen. Differential Evolution - APractical Approach to Global Optimization. Springer, Berlin, 1stedition, 2005.

1018