Robot Motion Planning and Control - J.P. Laumond Part 11 ppt

242 A De Luca, G Oriolo and C Samson Fig, 37 Point stabilization with nonsmooth time-varying feedback (I): ~ (tad) vs time (sec) Z o 2o 4o m m lOO izo 14o Fig 38 Point stabilization with nonsmooth time-varying feedback (I): ¢ (red) vs time (see) Ii ) iiiiIlllii i o ~o 4o eo m loo Izo i,lo Fig 39 Point stabilization with nonsmooth time-varying feedback (I): vl (m/sec) vs, time (sec) Feedback Control of a Nonholonomic Car-Like Robot 243 ! -2 Fig 40 Point stabilization with nonsmooth time-varying feedback (I): v2 (rad/sec) vs time (sec) i i i i i ! Fig 41 Point stabilization with nonsmooth time-varying feedback (II): cartesian motion 2O 40 e3 m ~o t~ Fig 42 Point stabilization with nonsmooth time-varying feedback (II): x (m) vs time (sec) 244 A De Luca, G Oriolo and C Samson i i i .i .i .i i t ~= i i ! i i i i i ! Fig 43 Point stabilization with nonsmooth time-varying feedback (II): y (m) vs time (sec) i i i i i i | o.ii , , ,,| , Fig 44 Point stabilization with nonsmooth time-varying feedback (II): (rad) vs time (see) t ! oJ *, ~ ! ! ! ! i ! ! i i ~ i / Fig 45 Point stabilization with nonsmooth time-varying feedback (II):¢ (rad) vs time (sec) Feedback C o n t r o l 2.'.- of a Nonholonomic Robot 245 , ! ? 1: i o~ i i : o ~ Car-Like i i ; i i i i ? i ~ :~ ~2o -1 s Fig 46 Point stabilization with nonsmooth time-varying feedback (II): vl (m/sec) vs time (see) Q .,212211'i i122112112112121112121112121212211211211111121121Ziil;i211111212i21121121211111 ~~ ; ,o ® ,,o ,&' ,~o Fig 4? Point stabilization with nonsmooth time-varying feedback (II): v2 (rad/sec) vs time (sec) 246 4.3 A De Luca, G Oriolo and C Samson A b o u t e x p o n e n t i a l convergence The peculiar convergence behavior of both presented stabilizing methods deserves some comments We have already pointed out in Sect 2.2 that the failure of the linear controllability test for the car-like robot indicates that smooth exponential stability in the sense of Lyapunov cannot be obtained Recall that (local) exponential stability means that the system trajectories X(t) satisfy the following inequality HX(t)H to, (90) with K, A positive real numbers and B a neighborhood of the origin The practical significance of this relationship is twofold: (i) small initial errors cannot produce arbitrarily large transient deviations since IIX(t)ll < KIIX(to)l I, and (ii) all solutions converge to zero exponentially While it is still unclear whether both properties can be simultaneously achieved for nonholonomic systems, one can still design a control law that guarantees at least one of the two In the case of smooth time-varying feedback laws, such as the one presented in Sect 4.1, it may be easily verified that tlX(t)l I _< KllX(to)ll, VX(to), Vt _to, (91) holds for some positive constant K However, when using the control law w2 of Prop 4.2, convergence to zero of ]lZ[I (and hence, of IIXIf) cannot be exponential In fact, if this were the case, ul would itself converge to zero exponentially, and thus the integral fto lul('r)]dT would not diverge This is in contradiction with the fact that divergence of this integral is necessary for the asymptotic convergence of tIZ2ll to zero As a matter of fact, it is only possible to show that [[X(t)[] _< K[IX(to)[lP(t), with p(O) = 1, thin p(t) = 0, (92) where p(t) is a decreasing function whose convergence rate is strictly less than exponential This theoretical expectation is confirmed by the simulations results of Sect 4.1 In particular, it has been observed [41] that smooth time-varying feedback control applied to a unicycle yields a convergence rate slower than t -1/2 for most initial configurations, a fact that can be proven using center manifold theory On the other hand, existing nonsmooth feedback laws for nonholonomic systems not guarantee uniform boundedness of the transient error ratio IIX(t)ll/llx(to)ll For example, the piecewise-continuous time-invariant feedback law proposed in [8] for the stabilization of a unicycle yields Hx(t)l[