Time-optimal parabolic interpolation with velocity, acceleration, and minimum-switch-time constraints

Graphical Abstract

Abstract

Time-optimal trajectories with bounded velocities and accelerations are known to be parabolic, i.e. piecewise constant in acceleration. An important characteristic of this class of trajectories is the distribution of the switch points – the time instants when the acceleration of any robot joint changes. When integrating parabolic trajectory generation into a motion planning pipeline, especially one that involves a shortcutting procedure, resulting trajectories usually contain a large number of switch points with a dense distribution. This high frequency acceleration switching intensifies joint motor wear as well as hampers the robot performance. In this paper, we propose an algorithm for planning parabolic trajectories subject to both physical bounds, i.e. joint velocity and acceleration limits, and the minimum-switch-time constraint. The latter constraint ensures that the time duration between any two consecutive switch points is always greater than a given minimum value. Analytic derivations are given, as well as comparisons with other methods to demonstrate the efficiency of our approach.

Keywords:

• Parabolic trajectories; shortcutting; switch time

Notes

No potential conflict of interest was reported by the authors.

1 Some planners, such as RRT* [29] optimize path quality during the planning itself, as opposed to the post-processing approach advocated here. However, such planners have large execution overheads and are seldom used in practical industrial settings.

2 There are cases where the magnitude of velocity is maximum at one end. In these cases, the velocity profiles have two ramps, one ramp having zero acceleration. In this paper, we consider these cases as special cases of and .

3 The calculations in those papers are actually similar. They differ from one another in some minor details and applications. In the sequel, when we discuss velocity profile interpolation without the minimum-switch-time constraint, we will primarily address the work in [6] since the authors also integrated this interpolation method into a trajectory smoothing (shortcutting) process.

4 The term similarity here refers to similarity in the geometric sense. In particular, a PP2-velocity profile can be seen as a mirror image along a horizontal axis of the corresponding PP1-profile. Figure 3 shows a PP2-profile and its corresponding PP1-profile. Therefore, the procedure for the case PP2 is essentially the same as the procedure for the case PP1.

5 PyIpopt is a freely available Python optimization module; see https://github.com/xuy/pyipopt.