Optimal Splines for Rigid Motion Systems: A Convex Programming Framework

Abstract

This paper develops a general framework to synthesize optimal polynomial splines for rigid motion systems driven by cams or servomotors. This framework is based on numerical optimization and has three main characteristics: (i) spline knot locations are optimized through an indirect approach based on providing a large number of fixed, uniformly distributed candidate knots; (ii) in order to efficiently solve the corresponding large-scale optimization problem to global optimality, only design objectives and constraints are allowed that result in convex programs and (iii) one-norm regularization is used as an effective tool for selecting the better (that is, having fewer active knots) solution if many equally optimal solutions exist. The framework is developed and validated based on a double dwell benchmark problem for which an analytical solution exists.

Optimal input design for flat systems using B-splines

“…2, a closer inspection reveals that some conservatism is introduced by (5e), related to the approach from II-B. However, the relative difference between the optimal time by gridding (3e, 3f), as in [3], and our approach is only 0.005 %. Contrary to [3], constraints are guaranteed to be satisfied between candidate knots without any postanalysis.…”

“…However, the relative difference between the optimal time by gridding (3e, 3f), as in [3], and our approach is only 0.005 %. Contrary to [3], constraints are guaranteed to be satisfied between candidate knots without any postanalysis. Also, the flat system approach allows us to formulate constraints on the states and outputs elegantly, without having to introduce additional variables and integrate the differential equations.…”

“…This requires treating the knots as variables, resulting in a highly nonlinear and nonconvex optimization problem [4]. To overcome the nonlinearity, [3] propose to optimize the spline knot locations indirectly by supplying many candidate knot locations (n = 500 à 1000) and using a regularization to favor solutions with few active 1 knots when many equally optimal solutions exist. In this work we apply this indirect approach and use B-splines as a basis to represent polynomial splines, which allows us to deal with constraints (3e,3f) in an elegant manner (see Section II-B).…”

“…However, the authors consider the knot locations to be fixed, resulting in a suboptimal solution. In this paper we improve upon [2] by also optimizing over the spline knots using an indirect approach as in [3]. In addition, we formulate the problem as a linear program for improved numerical efficiency at the cost of introducing conservatism.…”

Integrating structural and input design of a 2-DOF high-speed parallel manipulator: A flexible model-based approach

“…A general framework to synthesize optimal polynomial splines for rigid motion systems has been proposed in [16,17]. This framework has been extended in [12], by the inclusion of discrete-time linear time-invariant state-space system models in the convex optimization framework.…”