Robot Manipulator Control Theory and Practice - Frank L.Lewis Part 9 ppt

Robust Control of Robotic Manipulators 310Choose the Lyapunov function Note that in the equations above, the matrix B is defined as in 5.2.3, the i’s are defined as in 5.2.10, and the ma

Robust Control of Robotic Manipulators 308

Robust Control of Robotic Manipulators 310

Choose the Lyapunov function

Note that in the equations above, the matrix B is defined as in (5.2.3), the

i’s are defined as in (5.2.10), and the matrix P is the symmetric, definite solution of the lyapunov equation (5.3.12), where Q is symmetric and positive-definite matrix and A c is given in (5.2.14).

The expression of P in (5.3.14) may therefore be used in the expression of

v r in (2) This design is summarized in Table 5.3.4.

Robust Control of Robotic Manipulators 312

Upon closer examination of Spong’s controller in Theorem 5.3.4, it

becomes clear that vr depends on the servo gains Kp and Kv through p This

might obscure the effect of adjusting the servo gains and may be avoided

The same trajectory is followed by the two-link robot as shown in Figure5.3.10 Note that although the trajectory errors seem to be diverging, theyare indeed ultimately bounded and may be shown to be so by running thesimulation for a long time

Figure 5.3.10: (a) errors of joint 1; (b) errors of joint 1; (c) torques of joints 1 and 2

䊏

as described in [Dawson et al 1990]

THEOREM 5.3–5: The trajectory error e is uniformly ultimately bounded

(UUB) with the controller

(1)

Figure 5.3.11: (a) errors of joint 1; (b) errors of joint 1; (c) torques of joints 1 and 2

5.3 Nonlinear Controllers

Robust Control of Robotic Manipulators 314

Note that p no longer contains the servo gains and, as such, one may adjust

K p and Kv without tampering with the auxiliary control vr As was also shown

in [Dawson et al 1990], if the initial error e(0)=0 and by choosing Kv=2Kp =k v I,

the tracking error may be bounded by the following, which shows the directeffect of the control parameters on the tracking error:

(5.3.15)

Finally, note that if e(0)=e.(0)=0, the uniform boundedness of e(t) may be

deduced This controller is given in Table 5.3.5

EXAMPLE 5.3–6: Saturation Controller 2

In this example, let

Robust Control of Robotic Manipulators 316

5.4 Dynamics Redesign

In this section we present two other approaches to design robust controllers.The first starts with the mechanical design of the robot and proposes todesign robots such that their dynamics are simple and decoupled It thensolves the robust controller problem by eliminating its causes The secondapproach may be recast into one of the approaches discussed previously, but

it presents such a novel way to looking at the problem that we decided toinclude it separately

Decoupled Designs

It was shown throughout the previous chapters that the controller complexity

is directly dependent on that of the robot dynamics Thus it would makesense to design robots such that they have simple dynamics making theircontrol much easier This approach is advocated in [Asada and Youcef-Toumi1987] In fact, it is shown that certain robotic structures will have a decoupled

dynamical structures resulting in a decoupled set of n SISO nonlinear systems

which are easier controlled than the one MIMO nonlinear system Thedecoupling is achieved by modifying the dimensions and mass properties

of the arm to cancel out the velocity-dependent terms and decouple theinertia matrix An illustrative example of such robots is given in the nextexample

EXAMPLE 5.4–1: Decoupled Design

Consider the robot described in Figure 5.4.1 This mechanism is known asthe five-bar linkage and its dynamics are described when the followingcondition holds:

by

Note that the inertia matrix is decoupled and position independent Thecontroller given by

Robust Control of Robotic Manipulators 318

Some standard robotic structures may also be decoupled by design Studieshave been carried out to partially or totally decouple robots up to six links.The interested reader is referred to [Yang and Tzeng 1986], [Asada andYoucef-Toumi 1987], and [Kazerooni 1989] for good discussions of thistopic

Imaginary Robot Concept

The decoupled design alternative is very useful if the control engineer hasaccess to, or can modify, the robot design at an early stage It is morereasonable, however, to assume that the robot has already been constructed

to satisfy may mechanical requirements before the control law is actuallyimplemented Thus a dynamics redesign is difficult if not impossible The

imaginary robot concept is presented as an alternative robust design

methodology [Gu and Loh 1988] The development of this approach isdescribed next Consider an output function of the robot given by

(5.4.1)

so that

(5.4.2)and

(5.4.3)

The generalized output y may denote the coordinates of the end effector

of the robot or the trajectory joint error qd -q The imaginary robot concept

attempts to simplify the design of the control law for the physical robot, bycontrolling an “imaginary” robot that is close to the actual robot This choice

of the controller is shown to achieve the global stability of an imaginary

robot whose joint positions are described by the components of the vector y The methodology starts by decomposing M(q) as follows:

(5.4.4)

and then using the controller

(5.4.5)(5.4.6)

Since M(q) is unknown, however, the actual M~ is not available Theresulting controller is then simpler and may be applied to the physical robot

to lead acceptable, if not optimal behavior

The following theorem illustrates a controller to guarantee the boundedness

of the error

THEOREM 5.4–1: Let

(1)(2)

(3)

Proof:

more detail and for illustrative examples.

The controller is shown in Table 5.4.1

䊏

Table 5.4.1: Computed-Torque-Like Robot Controllers.

5.4 Dynamics Redesign

Robust Control of Robotic Manipulators 320

5.5 Summary

The design of robust motion controllers of rigid robots was reviewed Theremain designs were identified and explained All controllers were robust withrespect to a range of uncertain parameters and will guarantee theboundedness of the position-tracking error In the presence of disturbancetorques, a bounded error is best achievable outcome The question of whichrobust control method to choose is difficult to answer analytically, but thefollowing guidelines are suggested The linear-multivariable approach isuseful when linear performance specifications (percent overshoot, dampingratio, etc.) are available The one-DOF dynamic compensators performedrather well, with little or no knowledge of the robot dynamics They may,however, result in high-gain control laws in the attempt to achieverobustness The passive controllers are easy to implement but do notprovide easily quantifiable performance measures The modified variable-structure controllers seem to preform well when using the physics of therobot without excessive torque effort The saturation controllers are mostuseful when a short transient error can be tolerated, but ultimately, the errorwill have to be bounded

A common thread throughout this chapter has been the fact that a gain controller will guarantee the robustness of the closed-loop system Thechallenge is, however, to guarantee the robust stability of the robot withoutrequiring excessive torques The robustness of the motion controllers whennonzero initial errors or disturbances are present was also verified throughsome of the examples and is discussed in the problems at the end of thechapter It is useful to note that although the robot’s dynamics are highlynonlinear, most successful controllers have exploited their physics and theirvery special structure In the next chapter we describe the design of adaptivecontrollers in the case of uncertain dynamical description of the robots

[Abdallah et al 1991] C.T.Abdallah and D.Dawson and P.Dorato and M Jamshidi.

“Survey of Robust Control for Rigid Robots” IEEE Control Syst Mag., vol.

11, number 2, pp 24–30, 1991.

[Anderson 1989] R.J.Anderson “A Network Approach to Force Control in Robotics and Teleoperation” Ph.D Thesis, Departemnt of Electrical & Computer

Engineering University of Illinois at Urbana-Champaign, 1989.

[Arimoto and Miyazaki 1985] S.Arimoto and F.Miyazaki “Stabiliy and Robustness

of PID Feedback Control for Robot Manipulators of Sensory Capability” Proc Third Int Symp Robot Res., Gouvieux, France July, 1985.

[Asada and Youcef-Toumi 1987] H.Asada and K.Youcef-Toumi “Direct-Drive Robots:

Theory and Practice” MIT Press, Cambridge, MA, 1987.

[Becker and Grimm 1988] N.Becker and W.M.Grimm “On L2 and L8 Stability

Approaches for the Robust Control of Robot Manipulators” IEEE Trans Autom Control, vol 33, number 1, pp 118–122, January, 1988.

[Canudas de Wit and Fixot 1991] C.Canudas de Wit and N.Fixot “Robot Control

via Robust Estimated State Feedback” IEEE Trans Autom Control, vol 36,

number 12, pp 1497–1501, December, 1991.

[Chen 1989] Y.Chen “Replacing a PID Controller by a Lag-Lead Compensator for a

Robot: A Frequency Response Approach” IEEE Trans Robot Autom vol 5,

number 2, pp 174–182, April, 1989.

[Chen et al 1990] Y-F Chen and T.Mita and S.Wahui “A New and simple Algorithm

for Sliding Mode Control of Robot Arms” IEEE Trans Autom Control, vol.

35, number 7, pp 828–829, 1990.

[Corless 1989] M.Corless “Tracking Controllers for Uncertain Systems: Application

to a Manutec R3 Robot” J Dyn Syst Meas Control vol 111, pp 609–618,

December, 1989.

[Craig 1988] J.J.Craig “Adaptive Control of Mechanical Manipulators” Addis

on-323 REFERENCES

[Vidyasagar 1985] M.Vidyasagar “Control Systems Synthesis: A Factorization

Approach” MIT Press, Cambridge, MA 1985.

[Yang and Tzeng 1986] D.C-H.Yang and S.W.Tzeng “Simplification and Linearization

of Manipulator Dynamics by the Design of Inertia Distribution” Int J Rob Res vol 5, number 3, pp 120–128, 1986.

[Yeung and Chen 1988] K.S.Yeung and Y.P.Chen “A New Controller Design for

Manipulators Using the Theory of Variable Structure Systems” IEEE Trans Autom Control, vol 33, number 2, pp 200–206, February, 1988.

[Young 1978] K-K.D.Young “Controller Design for a Manipulator Using theory of

Variale Structure Systems” IEEE Trans Syst Man Cybern vol 8, number 2,

pp 210–218, February, 1978.

REFERENCES 324

5.2–1 WE consider the three-axis SCARA robot shown in Figure 5.5.1,

where all links are assumed to be thin, homogeneous rods of mass

m i and length ai , i=1, 2, 3 Then the dynamics are given by:

Figure 5.5.1: Three-axis SCARA robot

PROBLEMS

Section 5.2

325 REFERENCES

and let

Note that the first two joints are decoupled from the last one Find , i ,

and µi defined in (5.2.8)–(5.2.11), assuming that m12(q)=m21(q)=0 and that

5.2–2 Choose a desired set point q 1d =45 deg, q 2d=90 deg, and for

the robot in Problem 5.2–1 Design an SPR controller as described

in (5.2.14)–(5.2.19) Also, find a value of a to satisfy Theorem 5.2.1

5.2–3 Let q 1d =10 sin t, q 2d =10 cos t, and for the robot in Problem

5.2–1

1 Design an SPR controller as described in (5.2.14)–(5.2.19)

2 Let the desired trajectory now be q 1d =sin t, q 2d =cos t, and

Study the effect that decreasing qd has on a and on the trajectory

error

5.2–4 Choose a value of kv that satisfies condition (5.2.28) for the robot in

Problem 5.2–1 and the trajectory in Problem 5.2–3(b) What happens

to kv and to the trajectory errors if m1, m2, m3 increase to 20?

5.2–5 Design a dynamic controller similar to that of Example 5.2.3 for the

robot in Problem 5.2–1 and the trajectory in Problem 5.2–3(a)

Compare the performance for k=10 and k=50.

REFERENCES 326

5.2–6 Find the value of k that satisfies inequality (1) for you design in

Problem 5.2–5 What happens to your conditions if both the velocityterms and gravity terms are available to feedback (i.e., i=0)?

5.2–7 Consider the robot in Problem 5.2–1 and the set point of Problem

5.2–2 Also, assume that all i=0 so that velocity and gravity terms

are available to feedback Find a gain k to satisfy (5.2.36) and

implement the resulting controller

5.2–8 Repeat Problem 5.2–7 with the trajectory of Problem 5.2–3(a)

5.2–9 Design a controller similar to the one in Example 5.2.5 for the robot

of Problem 5.2–1 to follow the trajectory of Problem 5.2–3(a).Choose the same parameters used in that example and compare theresulting behavior to a set of parameters of your choosing

Section 5.3

5.3–1 Design a controller similar to the one in Example 5.3.1 for the robot

of Problem 5.2–1 to follow the trajectory of Problem 5.2–3(a).Choose the same parameters used in that example and compare theresulting behavior to a set of parameters of your choosing

5.3–2 Consider the robot of Problem 5.2–1 with the desired set point of

Problem 5.2–2 Design a variable-structure controller as described

in Theorem 5.3.2 You may want to start your design with theparameter values in Example 5.3.2

5.3–3 Repeat Problem 5.3–2 for the trajectory described in Problem 5.2–

3(a)

5.3–4 Consider the robot of Problem 5.2–1 with the desired set point of

Problem 5.2–2 Design a variable-structure controller as described

in Theorem 5.3.3 You may want to start your design with theparameter values in Example 5.3.3

5.3–5 Repeat Problem 5.3–4 for the trajectory described in Problem 5.2–

3(a)

5.3–6 Consider the robot of Problem 5.2–1 with the desired set point of

Problem 5.2–2 Design a saturation-type controller as described inTheorem 5.3.4 You may want to start your design with theparameter values in Example 5.3.5

327 REFERENCES

5.3–7 Repeat Problem 5.3–6 for the trajectory described in Problem 5.2–

3(a)

5.3–8 Consider the robot of Problem 5.2–1 with the desired set point of

Problem 5.2–2 Design a saturation-type controller as described inTheorem 5.3.5 You man want to start your design with theparameter values in Example 5.3.6

5.3–9 Repeat Problem 5.3–8 for the trajectory described in Problem 5.2–

3(a)

to motivate the formulation of the controllers Some issues regarding parameter error convergence, persistency of excitation, and robustness are also discussed.

6.1 Introduction

The problem of designing adaptive control laws for rigid-robot manipulatorsthat ensure asymptotic trajectory tracking has interested researchers for manyyears The development of effective adaptive controllers represents animportant step toward high-speed/precision robotic applications Even in awell-structured industrial facility, robots may face uncertainty regarding theparameters describing the dynamic properties of the grasp load (e.g., unknownmoments of inertia) Since these parameters are difficult to compute ormeasure, they limit the potential for robots to manipulate accurately objects

of considerable size and weight It has recently been recognized that theaccuracy of conventional approaches in high-speed applications is greatlyaffected by parametric uncertainties

To compensate for this parametric uncertainty, many researchers haveproposed adaptive strategies for the control of robotic manipulators An

Adaptive Control of Robotic Manipulators 330

extracting information from the tracking error Therefore, adaptive controllerscan give consistent performance in the face of load variations

It is only recently that adaptive control results have included rigorousproofs for global convergence of the tracking error Now that the existence

of globally convergent adaptive control laws has been established, it is difficult

to justify control schemes based on approximate models, local linearizationtechniques, or slowly time varying assumptions In the control literaturethere also seems to be no general agreement as to what constitutes an adaptivecontrol algorithm; therefore, in this chapter, the discussion will be limited tocontrol schemes that explicitly incorporate parameter estimation in the controllaw

Approximate Computed-Torque Controller

Of course, in reality, we never have exact knowledge of the robot model due

to many problems associated with model formulation Two commonuncertainties that do not allow exact model cancellation in roboticapplications are unknown link masses due to payload disturbances andunknown friction coefficients One way of dealing with these types of

parametric uncertainties would be to use the computed-torque controller

given in Chapter 3 with some fixed estimate of the unknown parameters inplace of the actual parameters This approximate computed-torque controllerwould have the form

(6.2.2)loads improves with time because the adaptation mechanism continues