The Planning of Robotic Optimal Motions in the Presence of Obstacles

Abstract

An approach to the planning of optimal robotic motions in the pres ence of obstacles is proposed. It is based on the use of nonclassical formulation of Pontryagin's maximum principle, which makes it possible to handle efficiently the state constraints resulting from the robotic tasks to be performed. The convergence properties of the algorithm are examined. A computer example involving a pla nar redundant manipulator of three revolute kinematic pairs, which performs two tasks in a two-dimensional work space including ob stacles, is given. A comparison of the proposed approach with the well-known method of penalty function is made.

The planning of optimal motions of non-holonomic systems

“…Consequently, from the obvious inequality (δv k * is the solution of optimal control problem (19), (20) for sufficiently large k)…”

“…The optimization processes known from the literature [19,20] project infinite-dimensional control space into a finite-dimensional one and then apply techniques of a linear programming problem to find δv k what causes a drastic increase of the amount of numerical computations and results only in nearoptimal solutions. Furthermore, the algorithms from [19,20] search for the solution of control problem (1)-(5) in a class of quasi-constant (discontinuous) controls. Consequently, they are not suitable for non-holonomic systems particularly of the first order, in which the controls (system velocities) have to be continuous.…”

“…Consequently, they are not suitable for non-holonomic systems particularly of the first order, in which the controls (system velocities) have to be continuous. Moreover, by transforming state inequality constraints (4) to equivalent scalar integral form χ , t))dt, control scheme (17)- (22) avoids the troublesome process of computing the Gateaux differentials which were used in works [14,19,20]. To numerically find Gateaux differentials, one has to solve a number of Cauchy's problems.…”

“…This approach, which consists in transforming the state constraints into control-dependent ones, in contrast to the interior penalty function method, does not require an initial admissible system trajectory (whose determination may be very troublesome in practice). In contrast to optimization algorithms known from the literature [19,20], which project infinite-dimensional control space into a finite-dimensional one and then apply techniques of liner programming problems, thus resulting only in near-optimal trajectories generated by discontinuous controls, the method proposed herein provides continuous solutions in an infinite-dimensional control space. As opposed to the most of the existing control algorithms, which provide only sub-optimal or near-optimal solutions and are based on maximizing the Hamiltonian [44,47], the control schemes offered in our study directly minimize the performance index.…”

Lagrangian Jacobian Motion Planning: A Parametric Approach

“…As has been said, the driving force behind this paper is the calculus of variations. Alternatively, a non-classical formulation of the Pontryagin maximum principle could be employed [15].…”