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

Abstract

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 C 2 / C 4 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.

Quintic Spline Collocation for Real-Time Biped Walking-Pattern Generation with variable Torso Height

“…In contrast to [4], we propose collocation based on quintic splines, which can by construction include boundary conditions up to the second-order derivative, thus removing the need for modifying the EoMs or adding virtual control points. This algorithm is a combination and extension of [4], [31], [33] and is derived in detail in [5], where we also show that, for sufficiently smooth ODEs, using quintic splines is in general more efficient than the cubic counterpart.…”

“…The main advantage of this method is that there is no restriction as to the shape of the ZMP or vertical torso motion, such that it can be used to incorporate kinematic constraints as shown in [6] and adapted in this paper. For the sake of brevity, we do not list methods for spline collocation here, but instead refer to [5], which collects literature related to our proposed algorithm. Another extension to the LIPM is made in [18], where an additional flywheel together with actuation constraints are used to compute a region of socalled Capture Points (CPs), i. e. predicted footholds which bring the robot to a complete stop.…”

“…The control points are then interpolated using a C 4 -smooth quintic spline respecting initial and end conditions to connect trajectories in a C 2 -smooth manner. For interpolation, we use an efficient algorithm proposed in [5]. The resulting vertical RMT motion z t (t) for stepping up and down a platform of 12.5 cm height is shown in Figure 7.…”

“…In particular, a lower count of (expensive) collocation sites leads to similar results. As explained in [5], it is difficult to quantify the approximation quality. This is due to the fact that we try to approximate a solution which actually does not exist.…”

“…Additionally, runtime is reduced, since fewer collocation sites are necessary to obtain the same approximation quality, cf. [5]. Furthermore, we avoid manually tuned parameters of the torso height optimization in [6], by following a simpler, pure geometric approach to incorporate kinematic limits.…”