Time-optimal parking and flying: Solving path following problems efficiently

Abstract

Path following deals with the problem of following a geometric path without any preassigned timing information and constitutes an important step in solving the general motion planning problem. The current paper considers path following for differentially flat systems. In this case the dynamics of the system can be projected along the path to a single input system, resulting in a free end-time optimal control problem. We propose to rewrite the problem in terms of the velocity along the path and the path itself. This way, we arrive at a fixed end-time optimal control problem that can be solved efficiently by interiorpoint solvers. Two challenging examples, a truck-trailer parking simulation and a quadrotor mission, illustrate the efficiency of the problem formulation and the practicality of the developed software.

Optimal Tube Following for Robotic Manipulators

“…. Now, by using the same transformation of variables as in (Verscheure et al, 2009a;Van Loock et al, 2013a) we transform the problem from a time t dependent problem into a path s dependent problem where we use s as an independent variable instead of time t.…”

“…Common practice is to transform the Cartesian path into a joint path using the inverse kinematics. Path following is often considered to be the low level stage in a decoupled motion planning approach (Bobrow et al, 1985;Shin and Mckay, 1985;Van Loock et al, 2013a), since the motion planning problem (path planning and following) is difficult and highly complex to solve in its entirety (von Stryk and Bulirsch, 1992;Diehl et al, 2005). First, a high level path planner determines a geometric path, ignoring the system dynamics but taking into account geometric path constraints.…”

“…Second, an optimal trajectory along the geometric path is determined that takes the system dynamics and limitations into account. Since the dynamics along a geometric path can be described by a scalar path coordinate s and its time derivatives (Bobrow et al, 1985;Shin and Mckay, 1985;Van Loock et al, 2013a), the decoupled approach simplifies the motion planning problem to great extent. Furthermore, the path following problem in joint space for a robotic manipulator with simplified constraints can be cast as a convex optimization problem (Verscheure et al, 2009a,b).…”

Time-Optimal Path Following for Robots With Convex–Concave Constraints Using Sequential Convex Programming

“…Many industrial robot tasks, such as welding, glueing, laser cutting and milling can be cast as path following problems. In addition, path following is often considered to be the low level stage in a decoupled motion planning approach [1]- [3], since the motion planning problem is difficult and highly complex to solve in its entirety [4], [5]. First, a high level planner determines a geometric path ignoring the system dynamics but taking into account geometric path constraints.…”

Iterative learning control for optimal path following problems

“…In [1], the optimal path following problem for the specific model structure of robotic manipulators was reformulated as a convex optimization problem through a nonlinear change of variables. This ap-proach is extended in [11], [12] for the more general case of differentially flat systems [13], however, the resulting problem is no longer convex.…”