Time-Optimization of Trajectories Using Zero-Clamped Cubic Splines and Their Analytical Gradients

Modern flexible production systems benefit from collaborative robot systems that support to teach robot configurations by hand to quickly implement collision-free motions. However, appropriate interpolation schemes that allow a fast and smooth motion through this sequence of waypoints is often not part of robot control systems. Instead, only point-to-point motions or a set of predefined curves like circular motions are supported. We propose an approach to improve the efficiency of finding a time parameterization with cubic splines that computes minimum trajectory durations while respecting actuator limits and $C^2$-continuity. Compared to the existing method, our contribution consists in the analytical gradients of the underlying nonlinear optimization problem and thus we are able to propose an efficient solution approach using state-of-the-art optimization solvers. We validate our implementation in simulation and in experiments, and benchmark it with the three trajectory generation methods implemented in MoveIt!.

I. Introduction

Modern robotic systems more and more operate in dynamic and unknown environments. Within applications like autonomous driving or in flexible production systems, either sampling-based path planning approaches compute sequences of feasible waypoints or, in case of collaborative robots, operators teach collision-free configurations along the desired path. Within the commonly used decoupled planning approach, these path waypoints are parameterized with time information in a subsequent step to build multiple waypoint trajectories. Trajectory computation is one of the main modules in autonomous motion planning platforms and is well-studied. Besides online approaches based on potential fields, various offline methods for planning point-to-point trajectories have been proposed. In case of multiple waypoint trajectories, many approaches are based on spline interpolation. In contrast to other common techniques like linear velocity profiles with parabolic blending [1], splines ensure a smooth motion without discontinuities in the acceleration profile which reduces stress to the motors. Other approaches use online tracking of subsequent waypoints, e.g. based on potential fields or PID control. These local approaches require appropriate tuning of control gains and switching points in order to avoid overshoots and in order to ensure a continuous motion without stopping at intermediate waypoints. The offline computation of the global trajectory ensures smoothness and allows to compute the global minimal time parameterization of the trajecotry since modifications in one spline segment effect all other segments. The state-of-the-art open source motion planning framework MoveIt! [2], widely used in the robotics community, provides three different trajectory generation approaches. However, there is an official statement about a bug in the default planning approach [3] and, further, there are user complaints about constraint violations and jerky motions in general [4].