Tham khảo tài liệu 'multiagent-systems 2010 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Control Analysis and Feedback Techniques for Multi Agent Robots 113 cos0 - sin0 z i Sin0_ i . cos0 _ 29 Substituting the value of i from Eq. 27 and 0 a the following is obtained z a cosẽ d - sin0_ a cos0 -dsin0 a _sin0_ _ cos0 _ sin0 dcos0 .a. 30 From Eq. 30 it can be observed that the decoupling matrix multiplied with the modified input a a is nonsingular provided that i 0. Let z r so the inputs can be obtained as a cos0 sin0 -dsin0 -1 M cos0 L- sin0 d sin0 cos 0 i M 31 _a_ dcos0_ lr2 J Lr2 J Substituting the values for original inputs the resulting dynamic compensator and the inputs are V Ẹ r2cos0- r1 sin0 a --------i---------- 32 i r1 cos 0 r2 sin 0 As one integrator Ị. was added hence the order of the dynamic compensator is one. The new coordinates can be written as z1 x z 2 y 33 z 3 x icos0 z 4 y i sin 0 The extended system of Eq. 33 is fully linearized in a controllable form. The decoupled chain of input output integrators can be written as z1 r1 z2 r2 34 Assuming that the robot must follow a smooth output trajectory Xd t yXOF. The globally exponentially stabilizing feedback law for the trajectory is given as r xd t kpi xd t - x kdi xd t - x r2 yd t kp 2 yd t - y kd 2 y d t - y 35 114 Multiagent Systems with PD gains chosen as kpi 0 kdi 0 for i 1 2. The values of x and y can be computed from Eq. 33 as a function of the robot state and the compensator state y The values of the feedback gains are chosen such that the polynomial expressed by Eq. 36 is Hurwitz. 22 kdlẰ kpi i 1 2 36 6. Feedback strategies for the follower robots In this section the feedback strategies for the follower robots are presented. The follower robots follow the leader robot with a desired distance and angle. The feedback laws as presented for both of the formation control strategies. Feedback strategy for separation bearing controller The kinematic model for the follower robot using the separation bearing controller was expressed in Eq. 7 . The kinematic model can be written in .
