Feedback linearization
The last two chapters of the book are concerned with state feedback control of wheeled mobile robots.
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Feedback linearization The last two chapters of the book are concerned with state feedback control of wheeled mobile robots. The most challenging issue from a theoretical viewpoint is to find feedback control laws that can stabilize the robot about an equilibrium point. The reason is that a nonholonomic mobile robot cannot be stabilized by a smooth state feedback, as we have anticipated in Section 7.4.4. It is therefore necessary to find more clever solutions involving nonstationary (time varying) and/or singular feedback controls. This issue will be further discussed in the next chapter. In this chapter, we will not be concerned with the stabilization problem about an equilibrium point (which is a regulation problem) but with another control problem, possibly more important in practice; namely, stable tracking of a reference motion (this is called also stabilization about a trajectory). Interestingly enough, the tracking problem is easier to solve than the regulation problem for wheeled mobile robots. Our purpose in the present chapter is to investigate the solvability of tracking problems for mobile robots by means of smooth static and dynamic feedback linearization. In particular, we will elucidate the connection between the intrinsic structural mobility of the robots and their feedback linearizability. We will address two basic tracking problems; namely, point tracking and posture tracking that will be described in the next section. By means of static feedback linearization, it is shown how to solve the point and posture tracking problems for omnidirectional robots and the point tracking problem only for robots having a restricted mobility. Then, it is shown how the posture tracking problem can be solved for all types of robots by dynamic feedback linearization, albeit with minor singularities. Design
C. C. de Wit et al. (eds.), Theory of Robot Control © Springer-Verlag London Limited 1996
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CHAPTER 8. FEEDBACK LINEARIZATION
specifications that guarantee the avoidance of singularities are also given.
8.1
Feedback control problems
Let us now formulate the two main feedback control problems that will be considered in this chapter; namely, posture tracking and point tracking.
8.1.1
Posture tracking
The problem is to find a state feedback controller that can achieve tracking, with stability, of a given reference moving posture ~r(t) which will be assumed to be twice differentiable. More precisely, the objective is to find a state feedback control law v such that: • the tracking error ~(t) = ~(t) - ~r(t) and the control v are bounded for all t; • the
tracking error asymptotically - ~r(t)) = 0;
converges
to
zero,
i.e.,
limt-+oo(~(t)
• if ~(O)
= ~r(O), then ~(t) = ~r(t) for all t.
In other words, this can be seen as the problem of tracking the posture of a virtual reference robot of the same type. For omnidirectional robots, this problem can be solved by a smooth static linearizing state feedback as we have seen in Section 7.4.4. For restricted mobility robots we will show that this problem can b
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