Dynamic Model of a 7-DOF Whole Arm Manipulator and

Dynamic Model of a 7-DOF Whole Arm Manipulator and Validationfrom Experimental Data

Zaira Pineda Rico, Andrea Lecchini-Visintini and Rodrigo Quian QuirogaDepartment of Engineering, University of Leicester, University Road, Leicester, U.K.

Keywords: Whole Arm Manipulator Model, Friction Model, Friction Identification.

Abstract: The present paper describes the design of the dynamic model of a 7 degrees of freedom whole arm manipulatorimplemented in SimMechanics. The friction phenomena of the manipulator is identified, represented througha fitted model and included in the system model with the aim of increment the accuracy of the model withrespect to the real system. The characteristics of the model make it suitable to test and design control strategiesfor motion and friction compensation in MATLAB/Simulink.

1 INTRODUCTION

The computation of the dynamic model of a robot ma-nipulator plays an important role in simulation of mo-tion, analysis of the manipulator’s structure and de-sign of optimal control algorithms. The inclusion ofthe effects of friction in a mechanical system modelwhen simulating control strategies, helps to improvethe performance of the controller to be implementedin the real system (Kostic et al., 2004; Indri, 2006;Bompos et al., 2007). Most of the whole arm manip-ulators mathematical models are based in the com-putation of multi-links serial robot’s mathematicalequations. These equations are obtained using theNewton-Euler recursive method to calculate the Cori-olis, centrifugal and inertial forces observed when theend-effector is in motion. Moreover, a Jacobian ap-proach may be implemented in parallel for mappingbetween Cartesian and joint space, in order to mini-mize singularity conditions that increase the computa-tion load of the control algorithm (Lau and Wai, 2002;Sousa et al., 2009). This methodology involves bothoperational and joint forces.

In most cases these mathematical models areimplemented using high level computing languagesas MATLAB (Corke, 1996), C/C++ or Fortran.Nevertheless, in order to avoid significant com-putation time, some authors have found in MAT-LAB/SimMechanics a comfortable tool to design me-chanical systems used for experimental verification.The capability of this tool yields appropriate resultswhen working with joints with 1 DOF and whenall the manipulator’s inertial parameters are known

(MathWorks, 2011).Section 2 offers a brief description of the real ma-

nipulator system. Section 3 gives insights on the de-sign of the model and shows the importance of exper-imental data in the development of the friction model.In Section 4 the response of the real system is com-pared to that of the model when simulating some ex-periments. Finally, section 5 presents some conclu-sions related to this work.

2 THE REAL SYSTEM

The real system is a 7-degrees-of-freedom whole armmanipulator (WAM) fromBarrett Technology Inc. Itis a joint torque controlled manipulator equipped withconfigurable PD/PID control and gravity compensa-tion. The information related to the joints configu-ration, joint motor drives and the body part masses,centre of gravity and inertia matrix is provided by themanufacturer in the WAM ARM User’s Manual (Bar-rett Technology Inc, 2008). Figure 1 shows the con-figuration and attached frames of the 7-DOF systemwith a grasper. All the joints of the manipulator are1 DOF revolute joints. An image of the real systemis shown in Figure 2, consisting in a Barret 7-DOFWhole Arm Manipulator with an attached BH8-seriesBarrettHand.

For the development of this project the 7-DOF ma-nipulator is configured with joint PD control and grav-ity compensation given by Equation (1), where thejoint torqueτ is expressed as the sum of the difference

217Pineda Rico Z., Lecchini-Visintini A. and Quian Quiroga R..Dynamic Model of a 7-DOF Whole Arm Manipulator and Validation from Experimental Data.DOI: 10.5220/0004013002170222In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 217-222ISBN: 978-989-8565-22-8Copyright c 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Z4X3,X4

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Xtool

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Figure 1: WAM 7-DOF Denavit-Hartenberg architecturewith attached frames as shown in (Barrett Technology Inc,2008).

Figure 2: The Barret 7-DOF Whole Arm Manipulator withthe BH8-series BarrettHand.

between the reference and measured position, namelyposition error ˜q, multiplied by a constant proportionalgainKP, the derivative of the error̃̇q multiplied by arespective derivative gainKD and the compensationfor gravityg, which is a function of the joint position.

τ = KPq̃+KD ˙̃q+ g (1)

The WAM joint PD control block diagram is shownin Figure 3. The control loop uses the reference sig-nal r(t), namely a joint trajectory, and the measuredpositiony(t) to compute the present error. Then thederivative of the error is calculated, and the gravitycompensation is added to conform the joint torqueu(t) to be applied to the manipulator’s joint.

_+

lIr(t) lIy(t)

P+

lu(t)

K P

K sD

++

IIg( )y(t)

+ie(t)

Figure 3: Configuration of the joint PD control with grav-ity compensation. r(t) is the reference trajectory,e(t) isthe response error,P represents the manipulator,g(y(t)) isthe gravity compensation,KD andKP are the derivative andproportional gains, respectively.

3 THE MODEL

The 7-DOF WAM model is built by four blocks:

• Seven modules that compute the joint referencetrajectories.

• The dynamic model of the system.

• Fitted friction models for each joint of the manip-ulator.

• Joint PD controllers configured and tuned as thoseimplemented in the real system.

The dynamic model of the manipulator is based onthe configuration and physical characteristics of thereal system, the parameters of the friction model aredetermined through the identification of the joint fric-tions whilst the joint trajectory and joint controller areemulations of those existent on the real system.

3.1 Trajectory Generation

The reference signal used to perform every joint ro-tation is a linear segment parabolic blend (LSPB) tra-jectory defined as follows. Up to a timetc the trajec-tory is parabolic with linear velocity, attc the trajec-tory chances to linear with constant velocity and zeroacceleration, finally after a time(t f − tc) the trajec-tory changes to parabolic again and the velocity de-creases linearly until reaching zero. The motion pro-duced when applying this velocity profile to any jointis translated in a rotation from the initial joint posi-tion qi of the manipulator toq f radians. Equation (2)defines the described trajectory, whereqi is the initialposition,q f is the final position reached in a timet f ,q̇c = q̈ctc is the value of the constant velocity exhib-ited from timetc to time (t f − tc) and ¨qc is the valueof the desired constant acceleration.

q(t) =

qi +12 q̈ct2 0≤ t ≤ tc

12(q f +qi − q̇ct f )+ q̇ct tc < t ≤ t f − tcq f −

12 q̈c(t f − t)2 t f − tc < t ≤ t f

(2)

ICINCO�2012�-�9th�International�Conference�on�Informatics�in�Control,�Automation�and�Robotics

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Figure 4: Example of a trapezoidal velocity profile whenthe values of the variables are set as followsqi = −0.55,q f =−0.05 rads, ¨qc = 0.083 rads/s,t f = 7s andtc = 1.

An example of the trajectory given by Equation (2)can be seen in Figure 4 where a rotation of 0.5 rads isperformed.

In order to generate the LSPB joint trajectory, thereal system sets the value of the time of changetc as1, and the velocity ˙qc and acceleration ¨qc are calculatedusing Equations (3) and (4), taking into account the mea-sured values of the initial and the final joint positionqi

andq f .In simulations, the joint trajectory is emulated by the

model considering the experimental values of the initialjoint positionqi, the final joint positionq f and the exe-cution timet f . The value oftc is set as 1 and the velocityand acceleration are also calculated using Equations (3)and (4). Finally, taking all the parameters previously es-timated, the joint trajectory is generated using Equation(2).

q̇c =4(q f −qi)

t2f − (−2(tc −0.5t f ))2

(3)

q̈c = q̇c (4)

3.2 The Dynamic Model

The dynamic model equation for a robot manipulator isgenerally written in the form (Siciliano et al., 2009):

B(q)q̈+C(q, q̇)q̇+G(q)+F = τ (5)

whereq, q̇, q̈ are the position, velocity and accelerationvectors of all the joints that shape the manipulator,B(q)is the inertial matrix,C(q, q̇)q̇ is the Coriolis-Centrifugalmatrix, G(q) is the gravity vector andF is the vector offriction, which is usually obtained using a fitted frictionmodel.

The dynamic model of the WAM Arm was designedand simulated using MATLAB/SimMechanics, and mostof the inertial data provided by the manufacturer had tobe adapted according to the inertial reference system.

The MATLAB/SimMechanics toolbox is a block dia-gram modelling environment like Simulink, created byMATLAB, to design and simulate mechanical systems.The toolbox contains several modules that represent par-ticular bodies and which inertial properties can be spec-ified by the user.

A great advantage of using SimMechanics in mod-elling mechanical systems is that the toolbox is promptto be used with Simulink so that control routines can beadded with ease in order to analyse the behaviour of thesystem’s dynamics under motion constraints.

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Figure 5: Register of position and torque in the Barret WAMwhen rotating joint 1 by 0.5 rads.

3.3 The Friction Model

Friction phenomena in robot manipulators may affectsthe accuracy of the system in position control and whenmoving the manipulator at very low velocities. Mostof the friction models employed when modelling posi-tion controlled mechanisms, include several componentsas the sliding friction (or Coulomb friction), the break-down friction (or stiction) and the viscous friction (Bonaand Indri, 2005). In most cases if the manipulator is ex-pected to displace at medium or medium-high velocities,the friction can be expressed mathematically as a func-tion of the joint velocity considering the effects of theviscous friction and the Coulomb friction only (Guranet al., 2001; Lewis et al., 2004; Kelly et al., 2005; Si-ciliano et al., 2009). This simplified friction model isknown as classic friction model:

F = Fcsign(v)+σ2v (6)

whereFc in the parameter for Coulomb friction,σ2 isthe viscous damping coefficient andv is the velocity ofthe system. The value of the parameters for the classicfriction model can be estimated from experimental data

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m]

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Friction-Velocity Map of Joint 1


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Estimation

Figure 6: Friction velocity maps corresponding to each joint of the 7 DOF manipulator. The measured data is obtained afterrotating each joint of the manipulator at different constant velocities whilst the estimated value is computed using a fittedclassic friction model.

through the execution of joint rotations at constant veloc-ity, this process in not straight forward and is explainedin detail in section 3.3.1.

3.3.1 Friction Identification

The procedure for the identification of the friction modelparameters for each joint of the 7-DOF whole arm ma-nipulator follows several steps. First a closed loop PDcontrol with gravity compensation is used to move thelink following a trapezoidal velocity profile at differentpositive and negative velocities, the measurements of po-sition in addition with the corresponding sampling timeare employed to compute the velocity of the link in eachexperiment. Afterwards, the values of torque and veloc-ity during the constant velocity stage of the trajectory areaveraged and used to construct the friction-velocity map,which is basically a torque versus velocity plot. Finally,the parameters for the friction model are estimated tofit the friction-velocity map by applying a least-squaresminimization method (Canudas de Wit and Lischinsky,1997; Canudas de Wit et al., 1995; Olsson et al., 1998;Johnson and Lorenz, 1992) using

n

∑i=1

[F(vi)− F̂(vi)]2 (7)

whereF(vi) is the measured torque at certain velocityvi (i.e the friction force), andF̂(vi) is the value com-puted by the friction model represented in Equation (6).Figure 5 shows an example of the measured position andtorque in the manipulator when rotating joint 1. The datasegment in the inset of the figure corresponds to the lin-ear change in the position which is used to compute thejoint velocity. The averages of the computed velocityvi

and measured torqueF(vi), respectively, yield to orderedpairs[F(vi),vi] that shape the friction velocity map.

The obtained friction model parameters for each

Table 1: Friction model parameters for the seven joints ofthe robot manipulator.V +/V− stands for positive and neg-ative velocities respectively.

Joint Coulomb friction Viscous frictionFc σ2

(V + /V−) (V + /V−)1 4.4748/ 4.3609 1.3348/ 0.82662 2.6385/ 3.5643 3.5572/-2.39513 1.7399/ 2.3529 0.3944/ 1.20754 1.5414/ 0.7342 -2.7045/-1.09695 0.2798/ 0.1172 -0.1972/ 1.46706 0.4834/ 0.4417 0.8291/ 0.18367 0.0538/ 0.1370 1.0689/ 0.7954

joint of the 7-DOF whole arm manipulator are listed inTable 1. There can be noted that the first joint is themost affected by the Coulomb frictionFc whilst the 3DOF that shape the wrist of the manipulator are the leastinfluenced. These parameters were used to estimate thejoint friction velocity maps plotted in Figure 6, where thevelocity maps obtained using real data are also shown.

The measured friction suited perfectly the shape ofthe classic friction model, which reinforces the initialchoice of the friction model for the development of thisproject. In general, the estimation of friction is very ac-curate for all the joints, and it is suitable enough to beconsidered when compensating the friction phenomenamanifested in the real manipulator.

4 SIMULATIONS

Several joint rotational motion experiments were per-formed with the aim of evaluate the accuracy of modelwith respect to the real manipulator. First, a LSPB joint


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Figure 7: Real and simulated joint error of the 7 DOF manipulator when executing a specific trajectory. Every link is rotatedone at a time and the error is calculated using the reference position minus the actual position.

trajectory with a rotation of 0.5 rad at a velocity of 0.083rads/sec was used as a reference in the real system, andthe joint position was measured while the manipulatorwas in motion. Subsequently, the joint position errorwas calculated to be used as a performance measure-ment. Later, in simulation, the same joint trajectory wasgenerated using experimental data as described in Sec-tion 3.1. The response of the model was recorded andthe model joint position error was calculated.

The position error was chosen as a performance mea-surement in order to have a closer comparison of the re-sponse of both the real manipulator and the model whenexecuting the exact same trajectory. As mentioned ear-lier, the real system uses a PD control with gravity com-pensation, and therefore, throughout the simulations aPD control was used and zero gravity was assumed.

During the first set of simulations every link was ro-tated one at a time. The second set consisted in one simu-lation only, where all the joints were rotated at the sametime. This gave a total of eight simulations performedby the model, to be compared to the behaviour of thereal system when executing the exact same trajectory.The graphs in Figure 7 show a comparison of the posi-tion error obtained in the first set of simulations. Thispractice helped to prove the veracity of the model withfriction joint per joint and showed that, regardless of thereal manipulator is exposed to transducer noise and sen-sor errors, the response of the model is accurate.

The second set of simulations was more an evalua-tion of reliability, due to in real exercise the manipulatormight operate in Cartesian space and all the joints wouldrotate at the same time. The response of the model dur-ing this simulation was good as expected despite of theclear difference between the simulated and the real po-sition error of the third joint. Figure 8 shows the resultsof the second set of simulations. The difference in errorsbetween model and real system are expected due to the

disturbances at which the real system is exposed, such asnonlinearities caused by the joint motors and the inertiaof the system. Other sources of disturbance to be con-sidered are the joint actuators and the scheme used forgravity compensation.

5 CONCLUSIONS

The identification of friction phenomena in the manip-ulator is helpful to observe the effects that friction hasin the overall performance of the system, so a compen-sation technique, if necessary according to the applica-tion task of the manipulator, may be implemented on thecurrent joint PD/PID control. On the other hand, the in-clusion of a friction model in the dynamic model of themechanical system establish a good platform for simu-lations in order to observe and analyse the response ofthe manipulator to different control strategies in a morerealistic scenario.

The presented dynamic model with friction is promptto be used as a reference in the design of joint controlstrategies implemented in Simulink. By taking advan-tage of the fact that real joint trajectory is completelyemulated in our simulations by setting up the desired ro-tation angle and execution time, a close comparison ofthe simulation results with respect to experimental datais also possible.

The development of a dynamic model with frictionis extremely useful for observing and improving the per-formance of the manipulator by analysing the effective-ness of the on system control. In certain cases, sometypes of stimuli utilised in controller tuning techniquesare not feasible to be implemented in experiments whilstthe model can be easily manipulated in simulations.

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Figure 8: Real and simulated joint error of the 7 DOF manipulator when executing a specific trajectory. All the joints arerotating at the same time and the error is calculated using the reference position minus the actual position.

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Canudas de Wit, C. and Lischinsky, P. (1997). Adaptivefriction compensation with partially known dynamicfriction model. InInternational Journal of AdaptiveControl and Signal Processing.

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Corke, P. (1996). A robotics toolbox for matlab. InIEEERobotics and Automation Magazine.

Guran, A., Pfeiffer, F., and Popp, K. (2001). Dynamics withfriction. In Series on Stability, Vibration and Controlof systems.

Indri, M. (2006). Control of manipulators subject to un-known friction. In45th IEEE Conference on Decisionand Control.

Johnson, C. and Lorenz, R. (1992). Experimental identifi-cation of friction and its compensation in precise, po-sition controlled mechanisms. InIEEE Transactionson Industry and Applications.

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Kostic, D., de Jager, B., Steinbuch, M., and Hensen, R.(2004). Modeling and identification for high perfor-mance robot control: An rrr-robotic arm case study.In IEEE Transactions on Control System Technology.

Lau, H. and Wai, L. (2002). A jacobian-based redun-dant control strategy for the 7-DOF WAM. InSev-

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Lewis, F., Dawson, D., and Abdallah, C. (2004).RobotManipulator Control: Theory and Practice. MarcelDekker, second edition.

MathWorks (2011).MATLAB/SimMechanics.Olsson, H., Astrom, K., Canudas de Wit, C., Gafvert, M.,

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Dynamic Model of a 7-DOF Whole Arm Manipulator and