Time-Optimal Path Planning Along Specified Trajectories
Time-optimal motion planning along specified paths is a well-understood problem in robotics, for which well-established methods exist for some standard effects, such as actuator force limits, maximal path velocity, or sliding friction. This paper describe
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Time-Optimal Path Planning Along Specified Trajectories Francisco Geu Flores and Andre´s Kecskeme´thy
Abstract Time-optimal motion planning along specified paths is a wellunderstood problem in robotics, for which well-established methods exist for some standard effects, such as actuator force limits, maximal path velocity, or sliding friction. This paper describes some extensions of the classical methods which consider, on the one hand side, additional non linear constraints such as sticking friction, acceleration limits at the end-effector, as well power limits for the overall system, and on the other, general paths featuring smooth interpolation of angular acceleration as well as arbitrary multibody systems comprising multiple loops. The methods are illustrated with two applications from robotics and the mining industry.
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Introduction
The problem of computing the time-optimal motion of a manipulator along a prescribed spatial path under forces, velocities and acceleration constraints has been thoroughly studied in the past. The basic idea of most solution algorithms is based on the pioneer work by Dubowsky, Bobrow and Gibson [1] as well as Shin and McKay [7], and the modifications proposed by Pfeiffer and Johanni [5] and Shiller and Lu [6]. It consists in mapping both the multibody differential equations and the system constraints to a one-dimensional motion along the prescribed path in order to define the maximally allowed accelerations at each point on the path, and to seek, by means of forwards and backwards integrations, the optimal set of points where the optimal acceleration must switch from a maximum to a minimum or vice versa.
F. Geu Flores (*) • A. Kecskeme´thy Universita¨t Duisburg-Essen, Lotharstr. 1, Duisburg 47057, Germany e-mail: [email protected]; [email protected] H. Gattringer and J. Gerstmayr (eds.), Multibody System Dynamics, Robotics and Control, DOI 10.1007/978-3-7091-1289-2_1, # Springer-Verlag Wien 2013
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F. Geu Flores and A. Kecskeme´thy
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Messner, Gattringer and Bremer [4] extend these ideas to handle jerk constraints. Moreover, by applying methods of optimal control, they avoid searching for the switching points with backwards and forwards integrations, thus rendering very efficient, online-suitable code. However, their method is restricted to non-singular arcs, as well as systems for which the state of vanishing velocity is always feasible, thus limiting both search space and application domain. Although such cases are very special, they do happen in practice, two corresponding examples being shown in this paper. A further recent approach is the reformulation of the problem as a convex optimal control problem by Verscheure, Demeulenaere, Swevers, De Schutter and Diehl [9]. This approach yields a very efficient and stable offline procedure. However, by considering only constraints that are linear in €s and s_2, the application domain is additionally limited with respect to [7]. In summary, research on optimal-time path planning can be viewed as being focus
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