A class of non-linear time scaling functions for smooth time optimal control along specified paths

Computing time optimal motions along specified paths forms an integral part of the solution methodology for many motion planning problems. Conventionally, this optimal control problem is solved considering piece-wise constant parametrization for the control input which leads to convexity and sparsity in the optimization structure. However, it also results in discontinuous control trajectory which is difficult to track. Thus, in this paper we revisit this time optimal control problem with the primary motivation of ensuring a high degree of smoothness in the resulting motion profile. In particular, we solve it with continuity constraints in control and higher order motion derivatives like jerk, snap etc. It is clear that such constraints would necessitate the use of time varying control inputs over the commonly used piece-wise constant form. The primary contribution of the current work lies in the introduction of a C^∞ class of time scaling functions represented as parametric exponentials. This in turn allows us to represent time varying control inputs as products of parametric exponential and a polynomial functions. We present the motivation behind adopting such representation of time scaling function over more common polynomial forms, both from mathematical as well as implementation standpoint. We also show that the proposed representation of time scaling function and control input leads to a very simple optimization structure where most of the constraints are linear. The non-linearity has a quasi-convex structure which can be reformulated into a simple difference of convex form. Thus, the resulting optimization can be efficiently solved through sequential convex programming where, at each iteration, the constraints in difference of convex form are further simplified to more conservative linear constraints.