Kurfess T.R. Robotics and Automation Handbook

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-20 Robotics and Automation Handbook

−2

×10

−3

(rad)

−2

0246

−5

Time (s)

(rad)

×10

−3

×10

−3

FIGURE 15.21 Position errors in the ﬁrst experiment achieved with the PD deﬁned by Equation (15.36) (grey) and

with the µ-synthesis controller (black).

authority of the drives. The position errors achieved with the PDs (15.36) are presentedin Figure 15.21 with

the grey lines. Because of the loweropen-loop gains within the bandwidth, the errors achieved with the PDs

(15.32) were even higher, so there is no need to plot them here. The errors achieved using the µ-synthesis

controllers are shown in Figure 15.21 with the black lines. The given plots conﬁrm that the µ-synthesis

controllers realize the reference motion more accurately. They better reduce the tracking errors because

of the higher gains at low frequencies. Integral action of the µ-synthesis controllers especially contributes

to the higher gains. The error ranges, given in Table 15.2, verify the superior tracking performance with

the µ-synthesis controllers. In joints 1 and 2, these controllers achieve the errors within the range [−10

−3

] radians. The range of the error in joint 3 is twice that of the ﬁrst two joints. Such accuracy can be

considered as very good for a direct-drive robot.

To evaluate if the reduction of the tracking error was below the prescribed one, we found the ratio

between the spectra (determined by fast Fourier transform) of the error e

and of the reference q

r,1

.The

ratio is plotted on the left in Figure 15.22 (solid line), together with inverse of the weighting function

(dashed line). Apparently, the curve corresponding to the ratio is always below the weighting. Having

in mind the relation S

= e

r,1

, it follows that condition (15.19) is satisﬁed, i.e., the speciﬁed error

reduction is realized. It appears that robust performance control of the RRR robot was realized in the

experiment.

The contribution of parasitic effects to the tracking performance can be evaluated from the cumulative

power spectra (CPSs) of the tracking errors. On the right in Figure 15.22 we show the CPSs of the errors

in the ﬁrst robot joint achieved with both PDs and with the µ-synthesis controller. We may observe that

most of the signals’ energy lies within the bandwidth of the joint references (up to 5 Hz). Since above the

bandwidth none of the CPSs abruptly changes its slope, it appears that inﬂuence of the parasitic effects on

the trajectory tracking is eliminated, i.e., the design rule r.4 from subsection 15.3.2 is met with all feedback

controllers. Inspection of the given plots reveals the lowest energy content of the CPS related with the

µ-synthesis controller. This can be quantitatively veriﬁed by comparing the ﬁnal CPS values in all joints,

presented in Table 15.3. The ﬁnal CPSs represent the variances (squares of the standard deviations) of

TABLE 15.2 Ranges of the Joint Errors (in ×10

−3

radians)

i PD Equation (15.32) PD Equation (15.36) µ

1[−5.3, 4.3] [−2.5, 2.7] [−1.0, 1.0]

2[−3.9, 2.9] [−1.7, 2.8] [−1.0, 0.87]

3[−10.2, 17.7] [−4.1, 7.0] [−2.0, 2.2]

Motion Control by Linear Feedback Methods 15

-21

TABLE 15.3 Variances of the Joint Errors (in ×10

−6

rad

)

i PD Equation (15.32) PD Equation (15.36) µ

1 7.19 1.57 0.11

2 3.08 0.87 0.14

3 46.84 5.59 0.72

1102030

−70

−60

−50

−40

−30

−20

−10

Frequency (Hz)

Magnitude (dB)

×10

−6

CPS

0.5

1.5

CPS

0.5

CPS

Frequency (Hz)

×10

−6

×10

−7

FIGURE 15.22 Results obtained for joint 1 in the ﬁrst experiment: (left) the ratio between spectra of the position

error achieved with the µ-synthesis controller and of the position reference (solid) is below the prescribed weighting

1/W

(dashed); (right) the CPSs of the position errors achieved with PD (15.32) (top), PD (15.36) (middle), and

µ-synthesis controller (bottom).

the position errors. As obvious from the table, the variances achieved with the µ-synthesis controllers are

dramatically lower than the variances obtained with the conventional feedbacks.

In the second experiment the RRR robot performed the writing task [23], also considered in the chapter

on modeling and identiﬁcation. For convenience, the reference tip path and the corresponding joint

motions are shown in Figure 15.23. The Cartesian position errors achieved with the PD (15.36) and with

the µ-synthesis controller are shown in Figure 15.24 with the grey and black lines, respectively. The latter

0369

−1.5

−0.75

0.75

1.5

2.25

Time (s)

Angles (rad)

−0.3 0.25

0.35

0.85

(m)

z (m)

FIGURE 15.23 The writing task as the second experiment: (left) the tip path; (right) the joint motions.

-22 Robotics and Automation Handbook

−6

×10

−4

(m)

−8

0369

−2.1

1.7

Time (s)

×10

−4

×10

−3

(m)

FIGURE 15.24 Cartesian errors in the second experiment achieved with the PD deﬁned by Equation (15.36) (grey)

and with the µ-synthesis controller (black).

controller performed better than the former one along each direction. It achieved errors (black lines) less

than 0.5 mm, which is very accurate for a direct-drive robot.

15.5 Conclusions

Various aspects of manipulator motion control with linear feedback are considered in this chapter: per-

formance criteria, robustness issues, feedback control designs, etc. It is noted that the decentralized con-

ventional control is generally applicable, but its effects on motion performance can be limited if dynamic

couplings between the manipulator joints are emphasized. To enhance the performance, a model-based

compensation of the nonlinear dynamics should be applied. Two compensation approaches are consid-

ered — computed-torque and feedback linearization. The latter one facilitates an experimental approach

to the feedback control design. In this approach, the manipulator dynamics that remains after feedback

linearization is characterized from several frequency response functions identiﬁed directly on the system.

The identiﬁed dynamics are then used for feedback control designs. Several feedback control designs are

considered, with the emphasis on the tuning parts of these designs. Practical demonstration of the control

designs is done for an experimental direct-drive robotic manipulator. The presented experimental results

show superior performance of the advanced control design over the conventional ones.

References

[1] Fu,K.S., Gonzales, R.C., and Lee, C.S.G., Robotics: Control,Sensing, Vision, and Intelligence, McGraw-

Hill, London, 1987.

[2] Sciavicco, L. and Siciliano, B., Modeling and Control of Robot Manipulators, McGraw-Hill, London,

1996.

[3] Slotine, J.J.E. and Li, W., Applied Nonlinear Control, Prentice Hall, Upper Saddle River, NJ, 1991.

[4] Franklin, G.F., Powell, J.D., and Emami-Naeini, A., Feedback Control of Dynamic Systems, Prentice

Hall, London, 2001.

[5] Asada, H. and Youcef-Toumi, K., Direct-Drive Robots: Theory and Practice, MIT Press, London, 1987.

[6] Chen, Y., Replacing a PID controller by a lag-lead compensator for a robot — a frequency response

approach, IEEE Trans. Rob. Autom., 5, 174, 1989.

[7] Sage, H.G., de Mathelin, M.F., and Ostertag, E., Robust control of robot manipulators: a survey, Int.

J. Control, 72, 1498, 1999.

Motion Control by Linear Feedback Methods 15

-23

[8] Steinbuch, M. and Norg, M.L., Advanced motion control, Europ. J. Control, 4, 278, 1998.

[9] Van Beek, B. and De Jager, B., RRR-robot design: basic outlines, servo sizing, and control, in Proc.

IEEE Int. Conf. Control Appl., Hartford, U.S.A., 1997, 36.

[10] Van Beek, B. and De Jager, B., An experimental facility for nonlinear robot control, in Proc. IEEE

Int. Conf. Control Appl., Hawaii, U.S.A., 1999, 668.

[11] Vukobratovi

c, M. and Potkonjak, V., Dynamics of Manipulation Robots: Theory and Application,

Springer-Verlag, Berlin, 1982.

[12] Nijmeijer, H. and van der Schaft, A., Nonlinear Dynamical Control Systems, Springer-Verlag, Berlin,

1991.

[13] Paden, B. and Panja, R., Globally asymptotically stable PD+ controller for robot manipulators, Int.

J. Control, 47, 1697, 1988.

[14] Sanathanan, C.K. and Koerner, J., Transfer function synthesis as a ratio of two complex polynomials,

IEEE Trans. Autom. Control, 8, 56, 1963.

[15] Koot, M., Kosti

c, D., De Jager, B., and Steinbuch, M., A systematic approach to improverobot motion

control, in Proc. Conf. on Mechantronics, Enschede, The Netherlands, 2002, 1377.

[16] Kosti

c, D., De Jager, B., and Steinbuch, M., Experimentally supported control design for a direct

drive robot, in Proc. IEEE Int. Conf. Control Appl., Glasgow, Scotland, 2002, 186.

[17] Kwakernaak, H., Robust control and H

∞

-optimization-tutorial paper, Automatica, 29, 255, 1993.

[18] Kwakernaak, H., MATLAB Marcos for Polynomial H

∞

Control System Optimization, Memorandum

No. 881, Department of Applied Mathematics, University of Twente, Enschede, The Netherlands,

1990.

[19] Zhou, K., Doyle, J.C., and Glover, K., Robust and Optimal Control, Prentice Hall, Upper Saddle River,

NJ, 1996.

[20] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and Smith, R., µ-Analysis and Synthesis Toolbox, The

MathWorks, Natick, MA, 1998.

[21] Anderson, B.D.O. and Moore, J.B., Optimal Control: Linear Quadratic Methods, Prentice Hall,

Englewood Cliffs, NJ, 1990.

[22] B

elanger, P.R., Dobrovolny, P., Helmy, A., and Zhang, X., Estimation of angular velocity and accel-

eration from shaft-encoder measurements, Int. J. Rob. Res., 17, 1225, 1998.

[23] Potkonjak, V., Tzafestas, S., Kostic, D., Djordjevic, G., and Rasic, M., Illustrating man-machine

motion analogy in robotics — the handwriting problem, IEEE Rob. Autom. Mag., 10, 35, 2003.

Force/Impedance

Control for Robotic

Manipulators

Siddharth P. Nagarkatti

MKS Instruments, Wilmington, MA

Darren M. Dawson

Clemson University

16.1 Introduction

16.2 Dynamic Model

Joint-Space Model

•

Task-Space Model and Environmental

Forces

16.3 Stiffness Control

Controller Design

16.4 Hybrid Position/Force Control

Controller Design

16.5 Hybrid Impedance Control

Types of Impedance

•

Duality Principle

•

Controller Design

16.6 Reduced Order Position/Force Control

Holonomic Constraints

•

Reduced Order Model

•

Reduced

Order Controller Design

16.7 Background and Further Reading

16.1 Introduction

The convergence of the ﬁelds of robotics and automationin many industrial applications is well established.

During the execution of many industrial automation tasks, the robot manipulator is often in contact with

its environment (either directly or indirectly via an end-effector payload). For purely positioning tasks

such as spray painting that have negligible force interaction with the environment, controlling the manip-

ulator end-effector position results in satisfactory performance. However, in applications such as polishing

or grinding, the manipulator end effector will experience interaction forces from the environment. Fur-

thermore, due to contact with the environment, the motion of the end effector in certain directions is

restricted. In the ﬁeld of robotics, this resulting motion is often referred to as constrained motion or

compliant motion. This chapter focuses on the control strategies of manipulators for constrained motion.

While there are many techniques that may be used to design controllers for constrained motion, a

popular active compliant approach is the use of force control. The fundamental philosophy is to regulate

the contact force of the environment. This is often supplemented by a position control objective that

regulates the orientation and the location of the end effector to a desired conﬁguration in its environment.

For example, in the case of grinding applications, the motion of the manipulator arm is constrained by

the grinding surface. It can easily be seen that it is vital to control not only the position of the manipulator

end effector to ensure contact with the grinding surface but also the interaction forces to ensure sufﬁcient

contact force to enable grinding action.

-2 Robotics and Automation Handbook

Another example that illustrates the need for combining force and position controllers is drawing on

a blackboard with chalk. If a robot manipulator is required to draw multiple instances of one shape

(e.g., a circle) on the blackboard, the end-effector position must follow a speciﬁc (in this case, circular)

trajectory while ensuring that the chalk applies a certain amount of force on the blackboard. We know

that if the force is too small, the drawing may not be visible; however, if the force is too large, the chalk will

break. Moreover, as multiple instances of shapes are drawn, the length of the chalk reduces. Thus, one can

appreciate complexities and intricacies of simple tasks such as drawing/writing using chalk. This example

clearly illustrates the need for desired force trajectory in addition to a desired position trajectory. Another

example that motivates the use of a position/force control strategy is the pick-and-place operations using

grippers for glass tubes. Clearly, if the gripper closes too tight, the glass may shatter, and if it is too lose, it

is likely the glass may slip. The above examples establish the motivation for position and force control.

In many applications, the position/force control objectives are considerably more intertwined than in

the examples discussed thus far. Consider the example of a circular diamond-tipped saw cutting a large

block of metal, wherein the saw moves from one end to the other. The motion from one end to the other

manifests itself as the position control objective, whereas cutting the block of metal without the saw blade

binding is the force control objective. The speed of cutting (position control objective) depends on many

parameters, such as geometric dimensions and material composition. For example, for a relatively softer

metal such as aluminum, the saw can move safely from one end to the other much faster than in the case

of a relatively harder material such as steel. It is obvious that the cutting speed would be much faster for

thinner blocks of metal.

To achieve control objectives such as those discussed above, the controller must be designed to regulate

the dynamic behavior between the force exerted on the environment (force control objective) and the end

effector motion (position control objective). This approach forms the basis of impedance control.

In this chapter, we present the following four control strategies that control the force exerted by the

end effector on its environment in conjunction with the typical end effector position control for an n-link

robot manipulator:

Stiffness control

Hybrid position/force control

Hybrid impedance control

Reduced order position/force control

Furthermore, for each section, an example for a two-degree-of-freedom (2-DOF) Cartesian manipulator

is also presented.

Remark 16.1 It is important to note that for all the aforementioned strategies, the desired velocity

and force trajectories should be consistent with the environmental model [1]. If such is not the case, the

trajectories may be modiﬁed by techniques such as “kinestatic ﬁltering” (see [1] for more details).

16.2 Dynamic Model

This section presents the formulation of the manipulator dynamics that facilitates the design of strategies

to control position and force. First, the widely known joint-space model is presented. Next, we deﬁne some

notation for the contact forces (sometimes referred to as interactive forces) exerted by the manipulator on

the environment by employing a coordinate transformation.

16.2.1 Joint-Space Model

The dynamics of motion for an n-link robotic manipulator [2] are constructed in the Euler-Lagrange form

as follows:

M(q)

q + V

(q,

q + N(q,

q) + τ

= τ (16.1)

Force/Impedance Control for Robotic Manipulators 16

-3

where

N(q,

q) = F (

q) + G(q)

(16.2)

M(q) ∈

n×n

denotes the inertia matrix, V

(q,

q) ∈

n×n

contains the centripetal and Coriolis terms,

F (

q) ∈

contains the static (Coulomb) and dynamic (e.g., viscous) friction terms, G(q) ∈

is the

gravity vector, τ

(t) ∈

represents the joint-space end-effector forces exerted on the environment by the

end-effector, q(t) ∈

represents the joint-space variable vector, and τ (t) ∈

is the torque input vector.

16.2.2 Task-Space Model and Environmental Forces

In order to facilitate the design of force controllers, the forces are commonly transformed into the task-

space via a Jacobian matrix by using a coordinate transformation [2]. Note that this Jacobian matrix is

deﬁned in terms of the task-space coordinate system and dependent on the robot application. Speciﬁcally,

depending on the type of application, the axes (directions) of force control and motion may vary and are

captured by the deﬁnition of the Jacobian.

Toward formulating the task-space dynamics, we deﬁne a task-space vector x ∈

as follows:

x = h(q)

(16.3)

where h(q) is obtained from the manipulator kinematics and the joint and task-space relationships. The

derivative of the task-space vector is deﬁned as

x = J (q)

(16.4)

where the task-space Jacobian matrix J (q) ∈

n×n

is deﬁned as [2]

J (q) =



I 0

0 T



∂h(q)

∂q

(16.5)

with I being the identity matrix, 0 being the zero matrix, and the transformation matrix T is used to

convert joint velocities to derivatives of roll, pitch, and yaw angles associated with end-effector orientation.

It should be noted that the joint-space representation of the force exerted on the environment (deﬁned

ﬁrst in Equation (16.1)) can be rewritten as

M(q)

q + V

(q,

q + N(q,

q) + J

(q) f = τ (16.6)

where f (t) ∈

denotes the vector of forces and torques exerted on the environment in task-space.

Example 16.1 Task-Space Formulation for a Slanted Surface

The following example presents the task-space formulation of the dynamics for a 2-DOF Cartesian ma-

nipulator (i.e., both joints are prismatic) moving along a slanted surface as shown in Figure 16.1. To that

end, the objectives are to formulate the manipulator dynamics and to decompose the forces exerted on

the slanted surface into tangential and normal components. The motion component of the dynamics can

easily be formulated from the unconstrained robot model. Thus, after removing the surface, and hence

the interaction forces f

and f

, the manipulator dynamics are constructed as follows:

q + F (

q) + G = τ

(16.7)

where

q =





, M =



0 m

+ m



, F =



(

)

(

)



, τ =





, G =



+ m



(16.8)

-4 Robotics and Automation Handbook

45°

− 3

FIGURE 16.1 Manipulator moving along a slanted surface [4].

Note that the explicit structure of the friction vector F is beyond the scope of the material presented in this

chapter.

The next step is to incorporate the contact forces. To that end, let x ∈

denote the task-space vector

deﬁned as

x =





(16.9)

where u represents the normal distance to the surface and v represents the tangential distance to the

surfaceofaﬁxed coordinate system. As discussed in Equation (16.3), the representation of the joint-space

coordinates into task-space is given by

x = h(q)

(16.10)

where h(q) is constructed based on the system geometry as follows:

h(q) =

√



−q



(16.11)

Based on Equation (16.5) and Equation (16.11), the task-space Jacobian matrix can be constructed as

follows:

J =

∂h(q)

∂q

√



1 −1



(16.12)

It should be noted that the matrix T in Equation (16.5) is an identity matrix because the end-effector

angles are irrelevant to this problem. Following the procedure outlined in the previous section, the robot

manipulator equation can be formulated as follows:

q + F (

q) + G + J

f = τ (16.13)

where

f =





(16.14)

Remark 16.2 It should be noted that the normal force f

and the tangential force f

in Figure 16.1 are

drawn with respect to the task-space coordinate system.

Force/Impedance Control for Robotic Manipulators 16

-5

16.3 Stiffness Control

Most early robot manipulators used in industrial automation were required to perform positional tasks

(e.g., spray painting). As a result, the robot manipulators were designed to be very rigid, and the control

strategies of choice were simple position controllers that achieved satisfactory positioning accuracy. How-

ever, in applications such as sanding and grinding, a rigid (or “stiff”) manipulator does not lend itself

favorably to the force control objective. If, however, the “stiffness” of the manipulator can be controlled

[3], the force control objective can be more easily accomplished. This section presents the stiffness control

formulation for an n-DOF robot manipulator and is followed by an example for a 2-DOF Cartesian robot.

16.3.1 Controller Design

The stiffness control objective for an n–link is formulated in this section. First, the force exerted by the

manipulator on the environment is deﬁned as

f ≡ K

(x − x

)

(16.15)

where K

∈

n×n

is a diagonal, positive semideﬁnite,

constant matrix used to denote the environmental

stiffness with x

∈

being the task-space vector denoting static location of the environment.

A proportional-derivative type of controller can be designed as follows to achieve the multi-dimensional

stiffness control objective:

τ = J

(q)(−K

x + K

x) + N(q,

(16.16)

where the task-space tracking error is deﬁned as

x = x

− x

(16.17)

with x

denoting the desired constant end-effector position. After formulating the closed-loop error system

and performing a comprehensive stability analysis [4], it can be shown that for the ith diagonal element

of matrices K

and K

lim

t→∞

≈ K

− x

) in the constrained directions

(16.18)

lim

t→∞

= x

in the unconstrained directions

The result in Equation (16.18) can be interpreted as follows. In the unconstrained task-space direction,

the desired steady-state setpoint is reached and the position control objective is achieved. In the constrained

task-space direction (with the assumption that K

 K

for force control), the force control objective

is achieved and K

may be interpreted as the manipulator stiffness in the task-space direction.

Example 16.2 Stiffness Control of a Cartesian Manipulator

The following example presents the design of a stiffness control strategy for the 2-DOF Cartesian manip-

ulator shown in Figure 16.1. The control objective is to move the end effector to a desired ﬁnal position

while exerting a desired normal force f

. In this example, the surface friction f

and the joint friction

are assumed to be negligible, and the normal force f

is assumed to satisfy the following relationship:

= k

(u − u

) (16.19)

A diagonal element of the matrix K

is zero if the manipulator is not constrained in the corresponding task-space

direction.

-6 Robotics and Automation Handbook

To accomplish the control objective, the stiffness control algorithm is designed as follows:

τ = J

(q)(−K

x + K

x) +G(q)

(16.20)

where

x =



− u

− v



(16.21)

denotes the desired normal position, the quantities τ , J (q), and G(q) retain their deﬁnitions of

Example 1, and the gain matrices can simply be selected as K

= k

I and K

= k

I with k

and k

being positive constants. It is important to select k

such that k

 k

as required by the stiffness control

formulation. To satisfy the control objective, the desired normal position u

needstobespeciﬁed. To that

end, by using the ﬁrst condition in Equation (16.18), we can solve for u

from the following equation:

= k

− u

) (16.22)

16.4 Hybrid Position/Force Control

An important disadvantage of the stiffness controller discussed in the previous section is that the desired

manipulator position and the desired forceexerted on the environmentare constants. That is, the controller

is restricted to a constant setpoint regulation objective. In applications such as polishing and deburring, the

end effector must track a prescribed path while tracking some desired force trajectory on the environment.

In such a scenario, the stiffness controller will not achieve satisfactory results.

To that end, another approach that simultaneously achievesposition and force tracking objectives is used.

The hybrid position/force controller developed in [5, 6] decouples the position and force control problems

into subtasks via a task-space formulation. This formulation is critical in determining the directions in

which force or position should be controlled. Once these subtasks are identiﬁed, separate position and

force controllers can be developed. In this section, we ﬁrst present a hybrid position/force controller for an

n-link manipulator and then discuss this control strategy with respect to the 2-DOF Cartesian manipulator

shown in Figure 16.1.

16.4.1 Controller Design

The following hybrid position/force controller ﬁrst uses feedback-linearization to globally linearize the

robot manipulator dynamics and then employs linear controllers to track the desired position and force

trajectories. To that end, by using the task-space transformation of Equation (16.3) to decompose the

normal and tangential surface motions and after some mathematical manipulation, the robot dynamics

can be expressed as

M(q)J

−1

(q)(

x −

J (q)

q) + V

(q,

q + N(q,

q) + J

(q) f = τ (16.23)

Based on the structure of Equation (16.23) and the control objectives, a feedback-linearizing controller

for the above system can be constructed as follows:

τ = M(q)J

−1

(q)(

y −

J (q)

q) + V

(q,

q + N(q,

q) + J

(q) f (16.24)

where

y ∈

denotes the linear position and force control strategies. Note that from Equation (16.23)

and Equation (16.24), we have

x =

(16.25)