Abstract— This paper presents an efficient control

scheme for compliant motion control of kinematically redundant manipulators and its evaluation using an experimental 7 degrees-of-freedom manipulator, REDIESTRO (a Redundant Dextrous Isotropically Enhanced, Seven Turning-pair RObot). A Transpose Jacobian based Hybrid Impedance Control (TJ-HIC) scheme is proposed that provides a unified approach for combining compliant motion control, redundancy resolution, and user defined secondary tasks in a single methodology. A Cartesian control scheme at the acceleration level is extended for compliant motion control of redundant manipulators. The manipulator is directly controlled in task space using the transpose Jacobian relationship, without solving the inverse kinematics problem and without computing the manipulator’s inverse Jacobian. The TJ-HIC is implemented for real-time control of REDIESTRO. Simulation and experimental results validate the TJ-HIC scheme, and demonstrate its capabilities for performing various tasks.

I. INTRODUCTION

There are a number of practical applications in robotics where, it is necessary to control not only the position but also the force of interaction between a manipulator’s end-effector and its environment. In the last few decades, there has been considerable work on developing control strategies suitable for industrial contact-type tasks. Among these, two approaches are worth noting: hybrid position/force control and impedance control. Raibert and Craig [1] proposed a hybrid position-force control scheme. Yoshikawa [2],[3], and McClamroch and Wang [4] proposed a method based on a constrained dynamic model of a manipulator. Khatib [5] proposed an operational space approach for force control. Hogan [6] proposed a fundamental theory of impedance control which showed that command and control of a vector such as position or force is not enough to control the dynamic interaction between a manipulator and its environment. This highlights the main problem of hybrid position-force control and its inability to take into account the manipulator’s impedance. The impedance control scheme overcomes this problem, but it ignores the distinction between position and force controlled subspace. Therefore, Anderson and Spong [7] proposed a Hybrid Impedance Control (HIC) scheme. Liu and Goldenburg [8]

∗ This research was supported by the Natural Sciences & Engineering Research Council (NSERC) of Canada under Grant RGPIN1345.

introduced a robust HIC method in 1991. All these methods were developed for non-redundant manipulators. A force control scheme for redundant manipulators is presented in [9] which decouples the motion of the manipulator into task-space motion and internal motion while providing for the selection of the dynamic characteristics for the motions. Hattori and Ohnishi [10] describe a decentralized compliant motion control scheme for redundant manipulators based on the concept of virtual impedance. The manipulator is divided into several subsystems each of which performs autonomously using virtual impedance and information from the end-effector subsystem. Simulation and experimental results are given for a redundant planar manipulator. Extended impedance control methods are discussed in [11] and [12]. Shadpey and Patel [13] introduced the robust Augmented Hybrid Impedance Control (AHIC) scheme by extending the Configuration Control approach at the acceleration level. These schemes can be considered as multi-purpose algorithms since different additional tasks can be incorporated without modifying the schemes and the control laws. Several experimental results have been reported in the last decade on force or compliant motion control of redundant manipulator systems, e.g., see [9],[10],[14], and [15]. Most of these have used planar 3 or 4 DOF manipulators to illustrate their control schemes. The notable exceptions are [13]-[15] where experimental results are presented for the 7-DOF Robotics Research Arm.

This paper is concerned with developing efficient techniques for position and force control of redundant manipulators that exploit the dexterity resulting from kinematic redundancy. We propose a Transpose-Jacobian based Hybrid Impedance Control (TJ-HIC) scheme for force and compliant motion control. Our work extends the approach presented for non-redundant manipulators in [19],[20] by combining HIC [7],[8] with Configuration Control [16] The scheme has the following key features:

(a) The control problem is formulated directly in task space; so the inverse kinematics problem is replaced by a transformation using the transpose Jacobian matrix.

(b) The control scheme does not require exact knowledge of the manipulator’s dynamics.

Transpose Jacobian based Hybrid Impedance Control of Redundant Manipulators∗

M. Shah, and R.V. Patel, Department of Electrical & Computer Engineering,

The University of Western Ontario, London, Ontario, Canada, N6A 5B9

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WB1.1

0-7803-9354-6/05/$20.00 ©2005 IEEE 1367

(a) The control concept based on the use of an artificial potential energy function is used to allow the implementation of any other upper-level control algorithm.

II.ALGORITHM DEVELOPMENT

The TJ-HIC scheme is a combination of two control loops. The outer-loop generates a desired Cartesian acceleration trajectory, and combines force and position trajectories in real-time. The inner-loop generates the desired input torque commands for the actuator which make the end-effector track the desired outer-loop trajectory. Fig 1 shows the general block diagram of the TJ-HIC scheme.

q

eF

rX

ddd XXX ,,

eF

rr XX ,

Y~

τ

dF

rX

=ZX

Y

dZ

TJ

dK q

=d

rd Z

XY

q

+

++

+

Fig. 1 Block diagram of the proposed TJ-HIC scheme

II.1 OUTER-LOOP DESIGN The outer-loop consists of the following modules: (i) The

Hybrid Impedance Control Module (HICM) (ii) The Cartesian Trajectory Generation Module (CTGM) and (iii) The Forward Kinematics Module (FKM).

II.1.1 Hybrid Impedance Control Module (HICM) The HICM uses the concept of hybrid impedance control [7],[8] to combine impedance control and hybrid force/position control into one scheme. The task space is split into two subspaces: (1) position-controlled and, (2) force-controlled. In the force-controlled subspace, the desirable dynamics are represented by

(1)edrdrd FFXBXM −=−+

where dF and eF are the desired and an environment

contact forces; dM and dB are the desired mass and

damping parameters; rX and rX are vector of the reference acceleration and velocity trajectories. The response of the dynamic system can be changed by changing the desired inertia and damping. Before the end-effector makes contact with environment, the interaction force

0=eF . If we assume a constant commanded force dF , the equation becomes

(2))(1drddr FXBMX +−= −

The impedance model of the compliant environment can be described by the following mathematical model.

(3))()()( edrddrddrd FXXKXXBXXM −=−+−+−

where dK is the desired stiffness; rX is vector of the reference position trajectories. We can define the objective of the hybrid impedance control loop by combining (1) and (3) and obtain the reference acceleration trajectory as

(4))](

)()([1

ddrd

drddedr

XSXXSK

XSXBFSIFMX

+−

−−−−+−= −

where S is the diagonal selection matrix with a diagonal term equal to 1 for a position-controlled direction and 0 for a force-controlled direction. The reference hybrid impedance trajectory rX is defined such that it satisfies (4) with known initial conditions

(5a))()()()(

ed

drddrddrd

FFSI

XXSKXSXBXSXM

−=−−−+−+−

with )0()0( XX r = , (5b))0()0( XXr =

The problem is now to provide a control law which achieves tracking of the acceleration trajectory rXgenerated from the outer-loop.

II.1.2 Cartesian Trajectory Generation Module (CTGM) The CTGM generates reference position, velocity and

acceleration trajectories in Cartesian space The user needs to specify the initial, intermediate and final positions (and orientations) of the end-effector, leaving the CTGM to generate appropriate position, velocity and acceleration profiles. It should be noted that the desired initial position (and orientation) of the end-effector should match with the position (and orientation) described by the joint angles in the FKM to avoid any sudden movement that may cause instability during initialization of the tracking maneuver.

II.1.3 Forward Kinematics Module (FKM)This module determines the end-effector position and

orientation, the linear and angular velocities, and the Jacobian matrices. The kinematic description of the REDIESTRO is given in [24]. The position error is given by

(6)ppe dp −=

where p and dp are )13( × vectors of the current and desired positions of the end-effector with respect to the base frame; The orientation error is calculated using an “equivalent angle-axis” representation described in [17],[18].

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II.2 INNER LOOP DESIGN The inner-loop consists of the Augmented Transpose

Jacobian Controller Module (ATJCM). The ATJCM controls the Cartesian position and generates appropriate joint torques to track the outer-loop trajectory.

II.2.1 Augmented Transpose Jacobian Controller Module (ATJCM).

The control module is based on an artificial potential function technique [19], [20]. Let the position and orientation of the end-effector with respect to the fixed reference base frame be denoted by the )1( ×m vector X ,which is related to the )1( ×n joint displacement vector qby

(7))(qfX =

where )(qf represents the )1( ×m forward kinematic vector function. Since the manipulator is kinematically redundant, in general, there exists an infinite number of joint trajectories for a given end-effector trajectory. To resolve redundancy, we use the Configuration Control concept [16] and define an additional )1( ×r task vector

(8))(qgZ =

where Tr qgqgqgqg )](),...,(),([)( 21= denotes the

user-defined additional tasks and r is the “degree-of-redundancy” of the manipulator. The desired user-defined additional tasks can be specified by an )1( ×r reference

vector )(tZd . The task requirements are combined to form

an )1( ×n vector Y as

(9))()(

==qgqf

ZX

Y

with the desired task requirements being given by the )1( ×n vector

(10)=d

dd Z

XY

The corresponding augmented )( nn × Jacobian matrix is given by

(11))()(

)(

)(

)( =

∂∂

∂∂

=qJ

qJ

qqg

qqf

qJc

e

where )(qJ e is the )( nm × Jacobian matrix

corresponding to the primary task and )(qJ c is the )1( ×rJacobian matrix corresponding to the additional or secondary tasks. We assume that the secondary task description is such that the augmented Jacobian )(qJ has full rank. The

kinematic representation of the manipulator in (9) is therefore no longer redundant; i.e., the dimension of the augmented task space and joint space are equal. Boundedness of the augmented Jacobian matrix can be shown as discussed in [21], [22].

Following [19]-[22] the transpose Jacobian-based controller for a redundant manipulator can be defined as

(12))()~()( qgqKyUKqJ vpT +−∇=τ

where q and q )( nR∈ are the joint position and velocity

vectors respectively; YYy d −=~ is the Cartesian tracking

error; )(qg is the gravity compensation term; pK and

dK are )( nn × constant positive-definite matrices for the

task space error and joint space velocity; )~(yU is an )1( ×n continuously differentiable positive-definite

function called the artificial potential energy function;)~(yU∇ is the gradient of the potential function )~(yU

with respect to y~ . A new positive-definite artificial potential function )~(yU , which possesses a minimum point at 0~ =y [24] , is suggested for REDIESTRO

(13)2

~}2log)log(

)1{log(12)~(

)~(

7

1

)~(

−−

−+=

−

=

−

ye

eKyU

y

i

y

ipi

i

i

σ

σ

σ

where piK and iσ are positive constants. The control law can be described using this potential function as follows

(14))()1()1()( ~

~

qgqKee

KqJ vy

y

pT +−

+−= −

−

σ

σ

τ

II.3 Alternatives for Additional Task: Posture Control The presence of redundant degrees of freedom in a

manipulator results in “self motion”. The problem of posture control is to use self-motion for the purpose of adjusting a manipulator’s configuration while the end-effector tracks a desired trajectory. We can define )( mn − additional tasks

or kinematic constraints ]...,[)( ,2,1 mnq −= φφφφ , which

specify the desired posture. Once )(tφ and )(tdφ have been defined, we can utilize the TJ-HIC scheme.

III. SIMULATION AND EXPERIMENT

III.1 Simulation Results In the simulations, the main task consists of position and

force trajectories defined in the constraint frame. The desired impedance values shown in Table 1 were selected based on stability considerations described in [23].

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TABLE 1CONTROL PARAMETERS USED FOR THE SIMULATION TASK

Type S M(Kg)

B(Nsec/m)

K(N/m)

Fd (N)

Surface K(N/m)

Non -Contact 1 477 1100 1100 --- ---

Contact 0 5 450 -- -20 10000

The environment was modeled as a spring with stiffness N/m10000=cK . The desired force trajectory was

specified as N20−=dF . The posture control task was

chosen to control the movement of joint 2q of REDIESTRO. The desired posture was chosen as the initial home position, i.e., 02 =dq (radians). The additional kinematic constraint is defined by

(15))( 2qq =φ

The desired trajectory for the additional task was chosen as

(16)0)( 2 == dd qqφ

The augmented Jacobian is defined as

(17)∂∂=

q

JJ

e

φ

where q∂

∂φ = [ 0 1 0 0 0 0 0]

A surface cleaning task with posture control as an additional task was incorporated in the ATJCM. The manipulator’s end-effector was required to approach the surface along the force-controlled direction and reach the desired force Nfd 0.20−= in 10s. Fig. 2 shows the simulation results for the desired and actual force trajectories. Initially when the end-effector is in free-space, the actual force is zero at the end-effector. At t=10s the force reaches -20 N. Figs. 3 and 4 show the position errors and the joint 2q profile.

III.2 Experimental Evaluation Experiments were performed to evaluate the performance

of the proposed TJ-HIC scheme for compliant motion and force control of REDIESTRO. Fig. 5 shows the REDIESTRO manipulator set up for the surface cleaning task. Two experiments were carried out at two different desired forces; Nfd 10−= and N20− . The following local gains were selected.

}22,,22,15,8,,8{}515,,2025,35,35,35,{

}50450,450,450,650,650,,650{

diag

diagK

diagK

d

p

==

=

σ

In the first experiment, the manipulator approaches the surface along the force-controlled direction and then maintains a desired force Nfd 10−= , while keeping a fixed position and orientation along the position-controlled directions. Fig. 6 shows the force trajectory with a fixed position and orientation. After a smooth transient, the desired force is reached and maintained. The results show that steady-state error in the force trajectory is small ( N05.0± ). Fig. 7(a) and 7(b) shows the position and the orientation errors.

In the second experiment, the surface cleaning task was carried out at Nfd 20−= with posture control as an additional task. The task was performed in two segments. In the first segment, the manipulator approaches the surface along the force-controlled direction, while keeping a fixed position and orientation along the position-controlled directions. In the second segment, the manipulator tracks the desired position trajectory along the position-controlled direction, while keeping the desired force along the force-controlled direction. Fig. 8 shows the force tracking for the task. Figs. 9 and 10 show the position errors and joint 2 profile. The performance degrades during segment 2, when the eraser pad starts to move on the surface. The error in the force-controlled direction during the commanded trajectory is due to unmodeled dynamics such as joint friction, and joint and link flexibility. Though the position-controlled and force-controlled directions are decoupled, unmodeled dynamics act as coupling that can cause performance degradation in force tracking.

IV. CONCLUSION

A transpose Jacobian based Hybrid Impedance Control scheme was proposed and implemented to achieve force and compliant motion control of redundant manipulators. The simulation and experimental results carried out for a 7-DOF manipulator show that the scheme is capable of achieving this objective while also performing user defined additional tasks. A key feature of the controller is the use of an artificial potential function. It is possible to use various kinds of artificial potential functions based on the nature of each task.

REFERENCES

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[2] T. Yoshikawa, “Dynamic hybrid position/force control of robot manipulators: Description of hand constraint and calculation of joint driving force,” IEEE Journal of Robotics and Automation, pp. 386-392, 1987.

[3] T. Yoshikawa, “Force control of robot manipulators,” IEEE Int. Conference on Robotics and Automation, San Francisco, pp 220-226, 2000.

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[4] N.H. McClamroch, D. Wang, “Feedback stabilization and tracking of constrained robots,” IEEE Transaction on Automatic Control, Vol. 33, pp. 419-426, 1988.

[5] O. Khatib, “A unified approach for motion and force control of robot manipulators: The operational space formulation,” IEEE Journal of Robotics and Automation, Vol. 3, pp. 43-53, 1987.

[6] N. Hogan, “Impedance Control: an approach to manipulation,” ASME, J. of Dynamic Systems, Measurement and Control, Vol. 107, pp. 1-24, March 1985.

[7] R.J Anderson and M.W. Spong, “Hybrid impedance control of robotic manipulators,” IEEE Journal of Robotics and Automation, Vol. 4, No. 5, p. 549-556, Oct 1988.

[8] G.J. Liu and A.A. Goldenberg, “Robust hybrid impedance control of robot manipulators,” IEEE Intl. Conf. on Robotics and Automation,Sacramento, CA, pp. 287-292, 1991.

[9] B. Nemec and L. Zlajpah, “Force control of redundant robots in unstructured environments,” IEEE Trans. on Industrial Electronics, vol. 49, no. 1, pp. 233-240, 2002.

[10] H. Hattori and K. Ohnishi, “A realization of compliant motion by decentralized control in redundant manipulators,” Proc. IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, Como, Italy, pp. 799-803, 2001.

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[12] W.S. Newman and M.E. Dohring, “Augmented impedance control: An approach to compliant control of kinematically redundant manipulators,” Proc. IEEE Int. Conf. on Robotics and Automation, pp. 30-35, 1991.

[13] F. Shadpey and R.V. Patel, “Compliant motion control with self-motion stabilization for kinematically redundant manipulators,” 3rd IASTED Intl. Conf. on Robotics and Manufacturing, pp. 392-396, June 1995.

[14] H. Seraji, D. Lim, and R. Steele, “Experiments in contact control”, Journal of RoboticSystems, vol. 13, no. 2, pp. 53-73, 1996.

[15] H. Seraji and R. Steele, “Nonlinear contact control for space station dexterous arms”, Proc. IEEE Int. Conf. on Robotics and Automation,Leuven, Belgium, pp. 899-906, 1998.

[16] H. Seraji, “Configuration control of redundant manipulators: Theory and implementation,” IEEE Journal of Robotics and Automation, vol. 5, pp. 472-490, 1989.

[17] J.J. Craig, “Introduction to Robotics: Mechanics and Control,” Addison-Wesley, Second Edition, 1989

[18] J.Y.S. Luh, M.W. Walker and R.P.C. Paul, “Resolved acceleration control of mechanical manipulators,” IEEE Transactions on Automatic Control, Vol. AC-25, No. 3, pp. 468-474, June 1980.

[19] F. Miyazaki and S. Arimoto, “Sensory feedback for robot manipulators,” Journal of Robotic Systems, 2, No. 1, pp. 53-71, 1985.

[20] R. Kelly and A. Coello, “Analysis and experimentation of transpose Jacobian-based Cartesian regulators,” Robotica, Vol. 17, pp. 303-312, 1999.

[21] C.C. Cheah, M. Hirano, S. Kawamura and S. Arimoto, “Approximate Jacobian control for robotics with uncertain kinematics and dynamics,” IEEE Transactions on Robotics and Automation, Vol. 19, No. 4, pp. 692-700, August 2003

[22] J C.C. Cheah, S. Kawamura and S. Arimoto, “Stability of hybrid position and force control for robotic manipulator with kinematics and dynamics uncertainties,” Automatica, Vol. 39, pp. 847-855, 2003

[23] F. Shadpey, “Force control and collision avoidance strategies for kinematically redundant manipulators,” Ph. D. Thesis, Concordia University, June 1997.

[24] M. Shah, “Force and position control of kinematically redundant manipulators,” M.E.Sc. Thesis, The University of Western Ontario, June 2004.

0 50 100 150-25

-20

-15

-10

-5

0

Time (S)

Forc

e (N

)

actualdesired

Fig. 2 Force trajectories with posture control as the additional task

0 50 100 150-2

-1

0

1

2x 10

-4

Time (S)

Posi

tion

erro

r (M

)

XY

Fig. 3 Position errors (X and Y directions) with posture control as the additional task

0 50 100 150-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-3

Time (S)

q2 (R

adia

ns)

Fig. 4 Joint angle (q2) profile with posture control as the additional task

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Fig. 5 Hardware setup for REDIESTRO used in the surface-cleaning task

0 10 20 30 40 50 60 70 80

-12

-10

-8

-6

-4

-2

0

Time (S)

Forc

e (N

)

Fig. 6 Force tracking without moving on the surface

0 10 20 30 40 50 60 70 80-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Time (S)

Position

error

(M)

XY

Fig. 7(a) Position errors without moving on the surface

0 10 20 30 40 50 60 70 80-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Time (S)

Orie

ntation erro

r (Ra

dian

s)

alphabetagama

Fig. 7(b) Orientation errors without moving on the surface

0 20 40 60 80 100 120 140

-25

-20

-15

-10

-5

0

Time (S)

For

ce (

N)

Fig. 8 Force tracking for the surface-cleaning task

0 20 40 60 80 100 120 140 160 180-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (S)

Pos

ition

err

or (

M)

y

x

Fig. 9 Position errors during the surface-cleaning task

0 50 100 150-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-3

Time (S)

Join

t q2

(Rad

ians

)

Fig. 10 Joint profile (q2) with posture control as the additional task

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