Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 10 pps

• For any selection of the symmetric positive definite matrices K p and K v,the origin of the closed-loop equation of robots with the PD control lawwith compensation, expressed in terms o

• For any selection of the symmetric positive definite matrices K p and K v,the origin of the closed-loop equation of robots with the PD control lawwith compensation, expressed in terms of the state vector

˜

q T ˙˜q TT

, isglobally uniformly asymptotically stable Therefore, the PD control lawwith compensation satisfies the motion control objective, globally This

• For any choice of the symmetric positive definite matrices K p and K v, theorigin of the closed-loop equation of a robot with the PD+ control law,expressed in terms of the state vector

˜

q T ˙˜q TT

, is globally uniformlyasymptotically stable Therefore, PD+ control satisfies the motion control

Bibliography

The structure of the PD control law with compensation has been proposedand studied in

• Slotine J J., Li W., 1987 “On the adaptive control of robot manipulators”,

The International Journal of Robotics Research, Vol 6, No 3, pp 49–59

• Slotine J J., Li W., 1988 “Adaptive manipulator control: A case study”,

IEEE Transactions on Automatic Control, Vol AC-33, No 11, November,

pp 995–1003

• Slotine J J., Li W., 1991, “Applied nonlinear control”, Prentice-Hall.

The Lyapunov function (11.3) for the analysis of global uniform asymptoticstability for the PD control law with compensation was proposed in

• Spong M., Ortega R., Kelly R., 1990, “Comments on “Adaptive ulator control: A case study”, IEEE Transactions on Automatic Control,

manip-Vol 35, No 6, June, pp.761–762

• Egeland O., Godhavn J M., 1994, “A note on Lyapunov stability for an adaptive robot controller”, IEEE Transactions on Automatic Control, Vol.

39, No 8, August, pp 1671–1673

The structure of the PD+ control law was proposed in

• Koditschek D E., 1984, “Natural motion for robot arms”, Proceedings

of the IEEE 23th Conference on Decision and Control, Las Vegas, NV.,December, pp 733–735

260 11 PD+ Control and PD Control with Compensation

• Paden B., Panja R., 1988, “Globally asymptotically stable PD+ controller for robot manipulators”, International Journal of Control, Vol 47, No 6,

˜

q ˙˜ qT

.Verify that the origin is its unique equilibrium point

2 Consider PD+ control for the ideal pendulum presented in Example 11.2

is a globally asymptotically stable equilibrium point

2 This, together with PD control with compensation were the first controls withrigorous proofs of global uniform asymptotic stability proposed for the motioncontrol problem

3 Consider the model of the pendulum from Example 3.8 and illustrated inFigure 3.13,

• v is the armature voltage (input)

• q is the angular position of the pendulum with respect to the vertical

(output),

and the rest of the parameters are constants related to the electrical andmechanical parts of the system and which are positive and known

b) Verify that the origin is an equilibrium and propose a Lyapunov tion to demonstrate its stability

func-c) Could it be possible to show as well that the origin is actually globallyasymptotically stable?

4 Consider the control law

Hint: Use Property 4.2.

5 Verify Equation (11.6) by use of (11.5)

3Notice that since the task here is position control, in this case the controller issimply of type PD with gravity compensation

• sampling of the joint position q (and of the velocity ˙q);

• computation of the control action τ from the control law;

• the ‘order’ to apply this control action is sent to the actuators.

In certain applications where it is required that the robot realize repetitivetasks at high velocity, the previous stages must be executed at a high cadence

The bottleneck in time-consumption terms, is the computation of the control

action τ Naturally, a reduction in the time for computation of τ has the

ad-vantage of a higher processing frequency and hence a larger potential for theexecution of ‘fast’ tasks This is the main reason for the interest in controllersthat require “little” computing power In particular, this is the case for con-trollers that use information based on the desired positions, velocities, and

pe-riodic functions of time and moreover they are known once the task has beenspecified Once the processing frequency has been established, the terms inthe control law that depend exclusively on the form of these functions, may

be computed and stored in memory, in a look-up table During computation

of the control action, these precomputed terms are simply collected out of

memory, thereby reducing the computational burden

In this chapter we consider two control strategies which have been gested in the literature and which make wide use of precomputed terms intheir respective control laws:

sug-• feedforward control;

• PD control plus feedforward.

Each of these controllers is the subject of a section in the present chapter

12.1 Feedforward Control

Among the conceptually simplest control strategies that may be used to trol a dynamic system we find the so-called open-loop control, where thecontroller is simply the inverse dynamics model of the system evaluated alongthe desired reference trajectories

con-For the case of linear dynamic systems, this control technique may beroughly sketched as follows Consider the linear system described by

˙

x = Ax + u

i {A} have negative real part,

˜

x = x d − x, the control problem consists in designing a controller that allows

solution to this control problem using the inverse dynamic model approach

the system to control, and then solving for u, i.e.

know from linear systems theory that since the eigenvalues of the matrix A

In robot control, this strategy provides the supporting arguments to the

following argument If we apply a torque τ at the input of the robot, the behavior of its outputs q and ˙q is governed by (III.1), i.e.

d dt

If we wish that the behavior of the outputs q and ˙q be equal to that

leads to the equation of the feedforward controller, given by

τ = M(q )¨q + C(q , ˙q ) ˙q + g(q ) (12.2)

12.1 Feedforward Control 265

Notice that the control action τ does not depend on q nor on ˙q, that is,

it is an open loop control Moreover, such a controller does not possess anydesign parameter As with any other open-loop control strategy, this approachneeds the precise knowledge of the dynamic system to be controlled, that is,

of the dynamic model of the manipulator and specifically, of the structure

of the matrices M (q), C(q, ˙q) and of the vector g(q) as well as knowledge

of their parameters (masses, inertias etc.) For this reason it is said that thefeedforward control is (robot-) ‘model-based’ The interest in a controller ofthis type resides in the advantages that it offers in implementation Indeed,

the control action τ according to Equation (12.2) This motivates the qualifier

“feedforward” in the name of this controller

Nonetheless, one should not forget that a controller of this type has the

intrinsic disadvantages of open-loop control systems, e.g lack of robustness

with respect to parametric and structural uncertainties, performance dation in the presence of external perturbations, etc In Figure 12.1 we presentthe block-diagram corresponding to a robot under feedforward control

Figure 12.1.Block-diagram: feedforward control

The behavior of the control system is described by an equation obtained

by substituting the equation of the controller (12.2) in the model of the robot(III.1), that is

equa-tion but in general, it is not the only one This is illustrated in the followingexamples.

Example 12.1 Consider the model of an ideal pendulum of length l

with mass m concentrated at the tip and subject to the action of gravity g Assume that a torque τ is applied at the rotating axis

ml2q + mgl sin(q) = τ¨

The feedforward controller (12.2), reduces to

The following example presents the study of the feedforward control of a DOF Cartesian robot The dynamic model of this manipulator is an innocuouslinear system

3-12.1 Feedforward Control 267

Example 12.2 Consider the 3-DOF Cartesian robot studied in

Exam-ple 3.4 (see page 69) and shown in Figure 3.5 Its dynamic model isgiven by

Notice that the dynamic model is characterized by a linear

control is given by

d dt

but it is not isolated Consequently, this equilibrium (and actuallyany other) may not be asymptotically stable even locally Moreover,due to the linear nature of the equation that characterizes the controlsystem, it may be shown that in this case any equilibrium point is

The previous examples makes it clear that multiple equilibria may coexistfor the differential equation that characterizes the behavior of the controlsystem Moreover, due to the lack of design parameters in the controller, it isimpossible to modify either the location or the number of equilibria, and even

1Here we write “closed-loop” in quotes since as a matter of fact the control system

in itself is a system in open loop

less, their stability properties, which are determined only by the dynamics ofthe manipulator Obviously, a controller whose behavior in robot control hasthese features is of little utility in real applications As a matter of fact, itsuse may yield catastrophic results in certain applications as we show in thefollowing example.

Figure 12.2.Diagram of the Pelican prototype

Example 12.3 Consider the 2-DOF prototype robot studied in Chapter

5, and shown in Figure 12.2

Consider the feedforward control law (12.2) on this robot The

(5.7) and whose graph is depicted in Figure 5.5 (cf page 129).

The initial conditions for positions and velocities are chosen as

˙

Figure 12.3 presents experimental results; it shows the components

Naturally, this behavior is far from satisfactory

♦

12.2 PD Control plus Feedforward 269

−0.5

0.0

0.5

1.0

1.5

2.0 [rad]

˜

q1

˜

q2

t [s]

Figure 12.3.Graphs of position errors ˜q1 and ˜q2

So far, we have presented a series of examples that show negative features

of the feedforward control given by (12.2) Naturally, these examples might discourage a formal study of stability of the origin as an equilibrium of the differential equation which models the behavior of this control system Moreover, a rigorous generic analysis of stability or instability seems to

be an impossible task While we presented in Example 12.2 the case when the origin of the equation which characterizes the control system is unstable, Problem 12.1 addresses the case in which the origin is a stable equilibrium The previous observations make it evident that feedforward control, given

by (12.2), even with exact knowledge of the model of the robot, may be inadequate to achieve the motion control objective and even that of position control Therefore, we may conclude that, in spite of the practical motivation

to use feedforward control (12.2) should not be applied in robot control Feedforward control (12.2) may be modified by the addition, for example,

of a feedback Proportional–Derivative (PD) term

τ = M(q d)¨q d + C(q d , ˙q d ) ˙q d + g(q d ) + K p˜q + K v ˙˜q (12.4)

respectively The controller (12.4) is now a closed-loop controller in view of

controller (12.4) is studied in the following section

12.2 PD Control plus Feedforward

The wide practical interest in incorporating the smallest number of computa-tions in real time to implement a robot controller has been the main motiva-tion for the PD plus feedforward control law, given by

τ = K p q + K˜ v ˙˜q + M(q d)¨q d + C(q d , ˙q d ) ˙q d + g(q d ), (12.5)

q d −q stands for the position error The term ‘feedforward’ in the name of the

controller results from the fact that the control law uses the dynamics of therobot evaluated explicitly at the desired motion trajectory In the control law

(12.5), the centrifugal and Coriolis forces matrix, C(q, ˙q), is assumed to be

computed via the Christoffel symbols (cf Equation 3.21) This allows one to

2M (q)˙ − C(q, ˙q) is skew-symmetric, a property which

is fundamental to the stability analysis of the closed-loop control system

It is assumed that the manipulator has only revolute joints and that theupper-bounds on the norms of desired velocities and accelerations, denoted as

The PD control law plus feedforward given by (12.5) may be regarded as

a generalization of the PD control law with gravity precompensation (8.1).Figure 12.4 shows the block-diagram corresponding to the PD control lawplus feedforward

τ

¨

q d M (q d)

Figure 12.4.Block-diagram: PD control plus feedforward

Reported experiences in the literature of robot motion control using thecontrol law (12.5) detail an excellent performance actually comparable with

the performance of the popular computed-torque control law, which is

pre-sented in Chapter 10 Nevertheless, these comparison results may be ing since good performance is not only due to the controller structure, butalso to appropriate tuning of the controller gains

mislead-The dynamics in closed loop is obtained by substituting the control action

τ from (12.5) in the equation of the robot model (III.1) to get

M (q)¨ q + C(q, ˙q) ˙q + g(q) = K p q + K˜ v ˙˜q + M(q d)¨q d + C(q d , ˙q d ) ˙q d + g(q d )

(12.6)

12.2 PD Control plus Feedforward 271

The closed-loop Equation (12.6) may be written in terms of the statevector

the number of equilibria of the system in closed loop, i.e.(12.7), depends on

12.2.1 Unicity of the Equilibrium

equilibrium (the origin) for the closed-loop Equation (12.7)

For the case of robots having only revolute joints and with a sufficiently

˜

q T ˙˜q TT

unique equilibrium of the closed-loop Equation (12.7) Indeed, the equilibria

solution of

proportional gain to ensure unicity of the equilibrium are presented next Tothat end define

p h(t, ˜q ∗ , 0)

unique

and Coriolis forces C(q, ˙q), and to the vector of gravitational torques g(q)

respectively, such that

*

positive definite matrix, we get

k(x) − k(y)
≤ λ 1

min{K p }

k g + k M
¨q d
M+ k C2
˙q d
2M
x − y

Finally, invoking the contraction mapping theorem (cf Theorem 2.1 on

page 26), we conclude that

λmin{K p } > k g + k M
¨q d
M+ k C2
˙q d
2M (12.9)

for the origin of the state space to be the unique equilibrium of the closed-loop

system, i.e Equation (12.7).

As has been shown before, the PD control law plus feedforward, (12.5),reduces to control with desired gravity compensation (8.1) in the case when

8.2 that the corresponding closed-loop equation had a unique equilibrium if

λmin{K p } > k g It is interesting to remark that when q dis constant we recoverfrom (12.9), the previous condition for unicity

12.2 PD Control plus Feedforward 273

Example 12.4 Consider the model of an ideal pendulum of length l

with mass m concentrated at its tip and subject to the action of gravity

g Assume that a torque τ is applied at the axis of rotation, that is

Example 8.1 shows the case when the previous equation has three

We stress that according to Theorem 2.6, if there exist more thanone equilibrium, then none of them may be globally uniformly asymp-

12.2.2 Global Uniform Asymptotic Stability

In this section we present the analysis of the closed-loop Equation (12.6) orequivalently, of (12.7) In this analysis we establish conditions on the design

the origin of the state space corresponding to the closed-loop equation We

Before studying the stability of the origin

chosen sufficiently “large” in the sense that

Lyapunov Function Candidate