Robot Manipulator Control Theory and Practice - Frank L.Lewis Part 8 pptx

In fact, the state-feedback controller may be used to define an appropriate output Ke such that the input-output closed-loop linear systems KsI-A+BK-1B is strictly positive real SPR.. wh

of q for robots with prismatic joints, and that (5.2.10) is satisfied by

so that a=(µ2

-µ1)/(µ2+µ1)) [Spong and Vidyasagar 1987] Finally, (5.2.10) is a result of the

properties of the Coriolis and centripetal terms discussed in Section 3.3

We will give different representative designs of the feedback-linearizationapproach, starting with controllers whose behavior is studied using Lyapunovstability theory

Lyapunov Designs

Static feedback compensators have been extensively used starting with theworks of [Freund 1982] and [Tarn et al 1984] Consider the controllerintroduced in (4.4.13):

(5.2.12)

such that

(5.2.13)

It can be seen that by placing the poles of Ac sufficiently far in the left

half-plane, the robust stability of the closed- loop system in the presence of  isguaranteed This was shown true in [Arimoto and Miyazaki 1985] for thecase where as described in Theorem 4.4.1 and Example4.4.3 It was also shown true for the trajectory-following problem assumingthat in [Dawson et al 1990] as described in Theorem 4.4.2.

There are as many robust controllers designed using Lyapunov stabilityconcepts as there are ways of choosing Lyapunov function candidates, and

of designing the gain K to guarantee that the Lyapunov function candidate is

decreasing along the trajectories of (5.2.13) To decrease the asymptotictrajectory error, however, excessively large gains may be required (see Example4.4.3) We therefore choose to use the passivity theorem and a choice of the

gain matrix K that renders the linear part of the closed-loop system SPR As

described in Section 2.11, an output may be chosen to make the closed-loopsystem SPR; therefore allowing large passive uncertainties in the knowledge

of M(q) In fact, the state-feedback controller may be used to define an appropriate output Ke such that the input-output closed-loop linear systems K(sI-A+BK)-1B is strictly positive real (SPR) Consider the following closed-

loop linear system:

It may then be shown using Theorem 2.11.5 that this system is SPR if

(5.2.15)with the choice of

(5.2.19)The next theorem presents sufficient conditions for the uniform boundedness

of the trajectory error

THEOREM 5.2–1: The closed-loop system given by (5.2.13) will be uniformly

bounded if

and

where K v=2aI and Kp=4aI.

Proof:

Consider the closed-loop system given by (5.2.8), with the controller (5.2.12), and choose the following Lyapunov function candidate:

(1)

where is the Lyapunov function corresponding to the SPR system (5.2.14) Then if ≥0, we have that V>0 This condition is satisfied for ≥µ2I Then differentiate to find

The error will be bounded by a term that goes to zero as a increases (see

Theorem 2.10.3 and its proof in [Dawson et al 1990] for details) Thisanalysis then allows  to be arbitrarily large as long as ≥µ2I, as shown in the next example In fact, if N were known, global asymptotic stability is

assured from the passivity theorem since in that case =0 The controller is

It is instructive to study (6) and try to understand the contribution ofeach of its terms The following choices will help satisfy (6)

1 Large gains Kp and Kv which correspond to a large a.

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summarized in Table 5.2.1

2 A good knowledge of N which translates into small i’s.

3 A large µ1 or a large inertia matrix M(q).

4 A trajectory with a small c, this a small desired acceleration d

Figure 5.2.2: K p =50, K v=25 (a) errors of joint 1; (b) errors of joint 2; (c) torques of joints 1 and 2.

The following example illustrates the sufficiency of condition (6) and of the

effects of larger gains Kp and Kv.

where

EXAMPLE 5.2–1: Static Controller (Lyapunov Design)

In all our examples in this chapter we use the two-link revolute-joint robotfirst described in Chapter 3, Example 3.2.2, whose dynamics are repeatedhere:

(2)

The parameters m1=1kg, m2=1kg, a1=1m, a2=1m, and g=9.8 m/s2 are given.Let the desired trajectory used in all examples throughout this chapter bedescribed by

Let =0 then

or that

Then use =6I and a=172 to satisfy (6) In fact, these values will lead to a

larger controller gains than are actually needed Suppose instead that we let

6^I, =0, and that

Note that this is basically a computed-torque-like PD controller A simulation

of the robot’s trajectory is shown in Figure 5.2.2 We also start our simulation

at =0 The effect of increasing the gains is shown in Figure 5.2.3,which corresponds to the controller

(4)

Note that at least initially, more torque is required for the higher-gains case(compare Figs 5.2.2c and 5.2.3c) but that the errors magnitude is greatlyreduced by expanding more effort

There are other proofs of the uniform boundedness of these static controllers

In particular, the results in [Dawson et al 1990] provide an explicit expressionfor the bound on e in terms of the controller gains In the interest of brevityand to present different designs, we choose to limit our development to onecontroller in this section

As discussed in Section 4.4, a residual stead-state error may be presenteven when using an exact computed-torque controller if disturbances arepresent A common cure for this problem (and one that will eliminate constantdisturbances) is to introduce integral feedback as done in Section 4.4 Such acontroller may again be used within a robust controller framework and willlead to similar improvements if the integrator windup problem is avoided(see Section 4.4)

In the next section we show the stability of static controllers similar to theones designed here and use input-output stability methods to design moregeneral dynamic compensators

Input-Output Designs

In this section we group designs that show the stability of the trajectoryerror using input-output methods In particular, we present controllers thatshow ∞ and 2 stability of the error We divide this section into a subsectionthat deals with static controllers such as the ones described previously,

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the error signals was shown using a static controller The norms used in(5.2.8)–(5.2.10) are then ∞ norms The development of this controller startswith assumptions (5.2.8), (5.2.9), and a modification of (5.2.10) to

(5.2.20)

This assumption is justified by the fact that N is composed of gravity and

velocity-dependent terms which may be bounded independent from the

position error e [see (5.1.1)] We shall also assume that =0 Let usthen choose the state-feedback controller (5.2.12) repeated here forconvenience:

(5.2.21)The corresponding input-output differential equation

(5.2.26)Consider then the following inequalities:

and using (5.2.8)–(5.2.11), we have that

Note that (2) reduces to

2 A good knowledge of N, which will translate into small i’s.

3 A large µ1 or a large inertia matrix M(q).

4 A trajectory with a small d

5 Robots whose inertia matrix M(q) does not vary greatly throughout its workspace (i.e µ1≈µ2)), so that a is small Note that a small a is needed

to guarantee that at least < 1 in (5.2.28) This will translate into

the severe requirement that the matrix M should be close to the inertia matrix M(q) in all configurations of the robot.

The controller is summarized in Table 5.2.2

These observations are similar to those made after inequality (6) and areillustrated in the next example

EXAMPLE 5.2–2: Static Controller (Input-Output Design)

Consider the nonlinear controller (5.2.6), where

(1)

Therefore,

(2)

Condition (1) is then satisfied if kv>720 This of course is a large bound that

can be improved by choosing a better A simulation of the closed-loop

behavior for kp=225 and kv=30 is shown in Figure 5.2.5 The errorsmagnitudes are much smaller than those achieved with the PD controllers ofExample 5.2.1 with a comparable control effort This improvement came

with the expense of knowing the inertia matrix M(q) as seen in (1).

Dynamic Controllers

The controllers discussed so far are static controllers in that they do nothave a mechanism of storing previous state information In Chapter 4 and inthis chapter, these controllers could operate only on the current position andvelocity errors In this section we present three approaches to show therobustness of dynamic controllers based on the feedback-linearization

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2 Good knowledge of N, resulting in a small 1.

3 Small 1 due to a large gain of the compensator C.

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4 2 close to 1, which may also be obtained with a large-gain compensator

C.

Note that in the limit, and as the gain of C(s) becomes infinitely large, 1

goes to zero This will then transform condition (1) to

(5.2.34)

It is also seen from (5.2.33)–(5.2.34) that increasing the gain k of C(s) will

decrease 1, therefore decreasing ||e||∞ A particular compensator may now

be obtained by choosing the parameter Q(s) to satisfy other design criteria,

such as suppressing the effects of  One can, for example, recover Graig’s

compensator, by choosing C(s)=-K so that the control effort is given by

u=Ke (5.2.35)

Then note that conditions (5.2.28) and (1) are identical if 2=0 and

2=11k p+12k v Also note from (2) that a smaller d results in a smaller tracking

error In fact, if e=0 and 0=0, the asymptotic stability of the error may beshown Finally, note that the presence of bounded disturbance will make thebound on the error e larger but will not affect the stability condition (1).This controller is summarized in Table 5.2.3

The factorization approach gives the family of all one-degree-of-freedom

stabilizing compensators C(s) The design methodology is illustrated for the

two-link robot in the next example

EXAMPLE 5.2–3: Dynamic Controller (Input-Output Design)

Let Gv (s) of Example 4.2.1 be factored as

(1)

where N(s), D(s), N(s), and D(s) are matrices of stable rational functions.

We can then find

(2)

As k increases, the disturbance rejection property of the controller is enhanced

at the expense of higher gains as seen from the expression of C(s) A simulation

of this controller for k=225 is shown in Figure 5.2.6 The following

observations are in order: The trajectory errors are smaller than any of theprevious controllers while the torque efforts are comparable In addition,the complexity of the controller is acceptable since the dynamics of the robotare not used in implementing the control

Figure 5.2.6: (a) errors of joint 1; (b) errors of joint 2; (c) torques of joints 1 and 2.

As it was discussed in [Craig 1988] and presented in Theorem 5.2.2,including the more reasonable quadratic bound will not destroy the ∞

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stability result of [Spong and Vidyasagar 1987] It was shown in [Beckerand Grimm 1988], however, that the 2 stability of the error cannot beguaranteed unless the problem is reformulated and more assumptions aremade It effect, the error will be bounded, but it may or may not have a finiteenergy In particular, noisy measurements are no longer tolerated for 2

stability to hold We next present an extension of the ∞ stability result thatapplies to dynamical compensators similar to the one described in Theorem5.2.3 but without the requirement that 2=0

THEOREM 5.2–4: The error system of (5.2.30) is ∞ bounded if

1 A large µ1 due to a large M(q).

2 A small 1 and a 2 close to 1, which will result from a large-gain

compensator C.

3 Small i’s, which will result from a good knowledge of N.

4 A small c due to a small d

Note that Craig’s conditions in Theorem 5.2.2 are recovered if 1=max 11,

12 and 2k=11k p+12k v.

On the other hand, assuming that d =0 and 2=0, the 2 stability of e was

shown in [Becker and Grimm 1988] if

(5.2.36)

controller is summarized in Table 5.2.4

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is due to the fact that the velocity terms are truly negligent in this particularapplication Such terms will, however, make a more vital contribution infaster trajectories.

Figure 5.2.7: (a) errors of joint 1; (b) errors of joint 2; (c) torques of joints 1 and 2.

Two-Degree-of-Freedom Design.

It is well known that the two-DOF structure is the most general linearcontroller structure The two-DOF design allows us simultaneously to specifythe desired response to a command input and guarantee the robustness of

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the closed-loop system This design was briefly discussed in Chapter 2,Example 2.11.4 It is in a different spirit from the other design of this chapter,because it relies on classical frequency-domain SISO concepts The generalstructure is shown in Figure 5.2.8 A two-DOF robust controller was designedand simulated in [Sugie et al 1988] and will be presented next Let the plant

be given by (5.2.5) and consider the following factorization:

THEOREM 5.2–5: Consider the two-DOF structure of Figure 5.2.8 Let

K1(S) be a stable system and K2(s) be a compensator to stabilize G(s) Then the controller

will lead to the closed-loop system

q=K1v (2) and the closed-loop error system (5.2.13) will be ∞ stable.

Note from (2) that K1(s) is used to obtain the desired closed-loop transfer

function It should then be stable, and to guarantee a zero steady-state error,

we choose v=qd and make sure that the dc gain K1(0)=1 Finally, we would

like K1(S) to be exactly proper (i.e., zero relative degree) K2(s), on the other hand, will assure the robustness of the closed-loop system Therefore, K2(s) should stabilize G(s) and provide suitable stability margins It should contain

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settled down to its final value qd and therefore the controller (3) becomes

equivalent to a PID compensator [see Chapter 4, equation (4.4.35)] It seems

that a different structure for k1 and K2 is warranted because in the meantime,the two-DOF controller preforms rather poorly This is a characteristic ofthe example rather than an inherent flow in the twoDOF methodology As amatter of fact, this structure has shown better performance than the one-DOF PID compensator in [Sugie et al 1988] for a set-tracking case Thereader is encouraged to work the problems at the end of the chapter related

to this design in order to compare the performance of one- and two-DOFdesigns

We have this presented a large sample of controllers that are more or lesscomputed-torque based We have shown using different stability argumentsthat the computed-torque structure is inherently robust and that byincreasing the gains on the outer-loop linear compensator, the position andvelocity errors tend to decrease in the norm This class of compensatorsconstitutes by far the most common structure used by roboticsmanufacturers and is the simplest to implement and study There are morecompensators that would fit into this structure while appealing to someclassical control applications The PD and PID compensators may bereplaced with the lead-lag compensators These are especially appealingwhen only position measurements are available Such designs are discussed

in [Chen 1989] in the discrete-time case There is also some work being done

in the nonlinear observer area which is directly relevant to this problem[Canudas de Wit and Fixot 1991] We refer the reader to the observabilitydiscussion in Section 2.11 We also suggest some of the problems at the end

of this chapter, which discuss further modification of the linearization designs

feedback-䊏