Time Optimal Trajectories for Bounded Velocity Differential Drive Vehicles

Abstract

This paper presents the time optimal trajectories for differential drive vehicles in the unobstructed plane. The wheel angular velocities are bounded, but may be discontinuous. The paper proves the existence of optimal controls, derives the structure of optimal trajectories, and develops an algorithm for producing a time optimal trajectory between any two configurations. Every nontrivial optimal trajectory is composed of straight segments alternating with turns about the robot's center. Optimal trajectories may have as many as five actions, but four actions are sufficient-for every optimal trajectory of five actions, there is an equally fast trajectory with four actions.

A suboptimal and analytical solution to mobile robot trajectory generation amidst moving obstacles

“…Apparently, by minimizing J L2 k , we will be able to obtain the near shortest trajectory through finding the analytical solution for c In this paper, we take into account both minimum-energy related performance index and shortest-path related performance index in a unified way by employing J k in (10). Weights w 1 and w 2 are used to more flexibly adjust the preference for different performance requirements.…”

“…and ω 2 J L2 k , we first consider those two terms separately, then combine their solutions into (10).…”

“…In [8] Newton algorithm and Jacobian matrix are used to solve energy optimization without obstacles. Moreover, the planning problem can be directly considered as an optimal control problem using Hamiltonian equations and Pontriagin's maximum principle in [9] [10]. Input parameterizations are used in [11][12] to design steering input and represent the trajectories by sinusoidal, polynomial, or piecewise constant functions to smooth control inputs.…”

System Theory and Analytical Techniques

“…In recent years, two more families of optimal curves have been determined [8,26]. Recall the differential-drive system from Section 13.1.2, which appears in many mobile robot systems.…”

“…An alternative criterion is the total amount of wheel rotation; this leads to an alternative family of optimal curves [26]. It was shown in [8] that only the four motion primitives shown in Figure 15.11 are needed to express time-optimal paths for the differential-drive robot. Each primitive corresponds to holding one action variable fixed at its limit for an interval of time.…”