TAILIEUCHUNG - Robot Motion Planning and Control - J.P. Laumond Part 9

Tham khảo tài liệu 'robot motion planning and control - . laumond part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 192 A. De Luca G. Oriolo and c. Samson where we dropped the dependence on Ơ for compactness. The derivation of the reference inputs that generate a desired cartesian trajectory of the car-like robot can also be performed for the 2 4 chained form. In fact with the reference system given by id di ĩd2 z d2_. 26 Xd4 from the output trajectory 16 and the change of coordinates 8 we easily obtain Xdl t 2 d 0 Xd2 t ỹd t xd t - ỹd t xd t x3d t Zd3 i ỳd f xd t Xdi t Vdtt and Wrfi t Xd t Udỉơ ỹd i id t -ã d í ỉ d í id O - 3yd t id t xd i 3ỹd t ã t xịự . To work out an example for this case consider a sinusoidal trajectory stretching along the X axis and starting from the origin at time t0 0 Xd t t yd t - Asinut. 27 The feedforward commands for the chained-form representation are given by Udi t 1 Udĩịt Aw3coswt while its initial state should be set at Zdi 0 0 r í2 ơ 0 Xds p Aw 4 0 0. We note that if the change of coordinates 8 is used there is an asymmetric singularity in the state and input trajectory when id t 0 for some t to. This coincides with the situation 0d t 7r 2 where the chained-form transformation is not defined. On the other hand if the chained form comes from the model in path variables 14 through the change of coordinates 15 the state and input trajectory needed to track the reference output trajectory s sd t d dd t 0 i to Feedback Control of a Nonholonomic Car-Like Robot 193 are simply obtained as Xdi t Sd t Xd2 t - 0 Xd3 t 0 Xdi t 0 and Sd i Ud2 t 0 without any singularity. Similar developments can be repeated more in general . for the case of a nonholonomic mobile robot with N trailers. In fact once the position of the last trailer is taken as the system output it is possible to compute the evolution of the remaining state variables as well as of the system inputs as functions of the output trajectory . of the output and its derivatives up to a certain order . Not surprisingly the same is true for the chained form 7 by defining xi xn as system outputs.

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