By Francis K. Mason
London released WW II - Air
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If motion planning is to be performed in a cluttered environment such trajectory may be unacceptable. In that case, one should consider using the local Lie-algebraic method. It is worth noticing that the energy of controls increases signiﬁcantly as smoother controls are needed. Fig. 3 Task 2: no restrictions on controls for local-global planning; a trajectory in the conﬁguration space; b controls (Lie-algebraic method: the ﬁrst two intervals; the Newton algorithm: last interval); c robot motions performing the ﬁrst sub-task; d robot movements on the second sub-task 34 I.
The paper is organized as follows. In Sect. 2 two components of a hybrid method of motion planning are recalled, namely the Newton algorithm of motion planning and the Lie algebraic method. A separate paragraph is devoted to making controls continuous at the time point where switching between the methods is executed. In Sect. 3 simulations of the hybrid method are presented using the model of a planar double pendulum placed atop of a free-floating base. Section 4 summarizes the paper. Hybrid Method of Motion Planning for Driftless Systems 27 2 Compounds of the Hybrid Method In this section two components of the hybrid method will be recalled.
When the integral gains reach some values, they have a stabilizing effect on the system and the tracking error starts decreasing. However, if, for some reasons, the tracking error does not reach zero, the integral gains would continue to increase. Furthermore, every sudden change in the desired trajectory would result in greater values of these adaptive gains, which could eventually lead to instability. With this adjustment, the integral control gains are obtained as a ﬁrst-order ﬁltering of the tracking error, and cannot diverge, unless the tracking errors diverge.