Fast Approximation of Over-Determined Second-Order Linear Boundary Value Problems by Cubic and Quintic Spline Collocation

• Main Authors: Philipp Seiwald, Daniel J. Rixen
• Format: Article
• Language: English
• Published: MDPI AG 2020-06-01
• Series: Robotics
• Subjects: cubic; quintic; spline; collocation; second-order; over-determined
• Online Access: https://www.mdpi.com/2218-6581/9/2/48

We present an efficient and generic algorithm for approximating second-order linear boundary value problems through spline collocation. In contrast to the majority of other approaches, our algorithm is designed for over-determined problems. These typically occur in control theory, where a system, e.g., a robot, should be transferred from a certain initial state to a desired target state while respecting characteristic system dynamics. Our method uses polynomials of maximum degree three/five as base functions and generates a cubic/quintic spline, which is <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="script">C</mi> <mn>2</mn> </msup> </semantics> </math> </inline-formula>/<inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="script">C</mi> <mn>4</mn> </msup> </semantics> </math> </inline-formula> continuous and satisfies the underlying ordinary differential equation at user-defined collocation sites. Moreover, the approximation is forced to fulfill an over-determined set of two-point boundary conditions, which are specified by the given control problem. The algorithm is suitable for time-critical applications, where accuracy only plays a secondary role. For consistent boundary conditions, we experimentally validate convergence towards the analytic solution, while for inconsistent boundary conditions our algorithm is still able to find a “reasonable” approximation. However, to avoid divergence, collocation sites have to be appropriately chosen. The proposed scheme is evaluated experimentally through comparison with the analytical solution of a simple test system. Furthermore, a fully documented C++ implementation with unit tests as example applications is provided.