Tham khảo tài liệu 'robot motion planning and control - . laumond part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Feedback Control of a Nonholonomic Car-Like Robot 217 with the definitions X sign ui - and u2 . dxi U1 Equation 58 is a linear time-invariant system an equivalent input-output representation of which is x 2 1 signal n-1U2- Such system is controllable and admits an exponentially stable linear feedback in the form n 1 2 2 -sign -1 2 kiX l 59 i l where the gains ki 0 are chosen so as to satisfy the Hurwitz stability criterion. Hence the time-varying control 2 2 t Ui t u 2 X2 60 globally asymptotically stabilizes the origin x2 0. The above approach provides a solution to the path following problem. Consider in particular the case of a car-like robot. We have seen at the end of Sect. that the system equations 14 in path coordinates can be transformed in chained form. By reordering the variables as in eq. 55 Xi represents the arc length 8 along the path X2 is the distance d between the car and the path while X3 and Xi are related to the car steering angle Ộ and to the relative orientation 3p between the path and the car. Path following requires zeroing the X2 X3 and Xi variables . x2 0 independently from Xi- Then for any piecewise-continuous bounded and strictly positive or negative lit eq. 59 is particularized as 2 X2 X3 X4 -sign ux fciX2 k2 sign i x3 3X4 - Using eq. 60 the path following feedback control law is U2 X2 X3 X4 t -fcl ul t X2 - 2Ul t X3 - 3 1 01x4 which can be compared with eq. 30 to appreciate the analogy. Such an approach was originally proposed in 42 for the path following of a unicycle. Rom the above developments it is clear that it can be applied to any mobile robot which can be converted into chained form. 218 A. De Luca G. Oriolo and c. Samson Skew-symmetric chained forms and Lyapunov control design We show now by introducing a modified chained form that it is possible to stabilize globally the origin x2 0 under more general hypotheses namely that 1 t I ủi t are bounded and Ui t does not asymptotically tend to zero. An important difference .
