Nonlinear feedback control
The previous chapter has been devoted to solving point and posture tracking problems by state feedback linearization for the five generic types of wheeled mobile robots. However, as it has been already mentioned, feedback linearization through regular con
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Nonlinear feedback control The previous chapter has been devoted to solving point and posture tracking problems by state feedback linearization for the five generic types of wheeled mobile robots. However, as it has been already mentioned, feedback linearization through regular controllers has serious limitations for control of mobile robots. In particular, it does not allow a robot to be stabilized about a fixed point in the configuration space. In this chapter we present several advanced nonlinear feedback control design methods allowing us to solve various control problem; namely, posture tracking, path following, and posture stabilization. In order to keep the exposition as clear and pedagogical as possible, we will limit ourselves to unicycle robots of Type (2,0), as represented in Section 7.3.3. Indeed, this type of mobile robots is sufficient to capture the underlying nonholonomy property of restricted mobility robots which is the core of the difficulties involved in the control problems discussed in this chapter.
9.1
Unicycle robot
In order to make the notations more convenient and consistent with those which are most often used in the literature, the posture coordinates (x, y, 0) of a Type (2,0) robot are redefined according to Fig. 9.1. The posture kinematic model of a unicycle robot is then described by :i;
= vcosO
C. C. de Wit et al. (eds.), Theory of Robot Control © Springer-Verlag London Limited 1996
332
CHAPTER 9. NONLINEAR FEEDBACK CONTROL
a
x
Figure 9.1: Redefinition of posture coordinates.
iJ
= vsinO
(9.1)
O=w
where the two velocity control inputs are the linear velocity v and the angular velocity w. Notice that, differently from the model in (7.23), 0 indicates the angle between the direction of v and axis Xb. We will sometimes use the compact notation
.i = G(z)u where
9.1.1
z=(i)
G(z)
=(
(9.2)
COSo
Si~O
u
= (:).
Model transformations
Control design may in some cases be facilitated by a preliminary change of state coordinates which transforms the model equations of the robot into a simpler "canonical" form. For instance, the following change of coordinates:
( ~~) (C?~O0 - Si~O0 ~) =
X3
sm
cos
0
(:)
(9.3)
0
together with the change of inputs (9.4)
333
9.1. UNICYCLE ROBOT
transforms system (9.1) into
:i:t =
UI
X2
=
U2
X3
=
X2UI·
(9.5)
This system belongs to the more general class of so-called chained systems characterized by equations in the form Xl = X2 = X3 =
UI U2 X2UI·
(9.6)
Chained systems are of particular interest in the field of mobile robotics because the modelling equations of several nonholonomic systems (e.g., unicycle-type and car-like vehicles pulling trailers) can locally be transformed into this form. An alternative local change of coordinates is
X2
= X = tan(}
X3
= Y
Xl
(9.7)
associated with the change of control inputs UI
= cos(} v
U2
=
1
~(}w.
(9.8)
cos
This yields the same system (9.5). However, the transformation presents in this case a singularity when cos () = o. It is thus only valid in domains where () E (
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