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On trajectory design from motion primitives for near time-optimal transitions for systems with oscillating internal dynamics

Thomas Auer, Frank Woittennek

TL;DR

The paper tackles the challenge of designing near time-optimal trajectories for linear kinematic systems with oscillatory internal dynamics under velocity, acceleration, and jerk constraints. It introduces the OCP-J approach, which constructs full trajectories from precomputed jerk segments that can overlap, enabling faster rest-to-rest transitions than traditional ZV shaping while keeping computation light. The method accounts for system damping and demonstrates improved transition times with acceptable sensitivity to parameter uncertainty, validated by laboratory experiments on a PLC. By enabling rapid recomputation of jerk segments, OCP-J supports online parameter updates and robust performance in production environments. Overall, the work provides a practical, PLC-friendly framework that achieves meaningful speedups over conventional shaping methods and offers a clear path for further enhancements in multi-mode systems and online adaptation.

Abstract

An efficient approach to compute near time-optimal trajectories for linear kinematic systems with oscillatory internal dynamics is presented. Thereby, kinematic constraints with respect to velocity, acceleration and jerk are taken into account. The trajectories are composed of several motion primitives, the most crucial of which is termed jerk segment. Within this contribution, the focus is put on the composition of the overall trajectories, assuming the required motion primitives to be readily available. Since the scheme considered is not time-optimal, even decreasing particular constraints can reduce the overall transition time, which is analysed in detail. This observation implies that replanning of the underlying jerk segments is required as an integral part of the motion planning scheme, further insight into which has been analysed in a complementary contribution. Although the proposed scheme is not time-optimal, it allows for significantly shorter transition times than established methods, such as zero-vibration shaping, while requiring significantly lower computational power than a fully time-optimal scheme.

On trajectory design from motion primitives for near time-optimal transitions for systems with oscillating internal dynamics

TL;DR

The paper tackles the challenge of designing near time-optimal trajectories for linear kinematic systems with oscillatory internal dynamics under velocity, acceleration, and jerk constraints. It introduces the OCP-J approach, which constructs full trajectories from precomputed jerk segments that can overlap, enabling faster rest-to-rest transitions than traditional ZV shaping while keeping computation light. The method accounts for system damping and demonstrates improved transition times with acceptable sensitivity to parameter uncertainty, validated by laboratory experiments on a PLC. By enabling rapid recomputation of jerk segments, OCP-J supports online parameter updates and robust performance in production environments. Overall, the work provides a practical, PLC-friendly framework that achieves meaningful speedups over conventional shaping methods and offers a clear path for further enhancements in multi-mode systems and online adaptation.

Abstract

An efficient approach to compute near time-optimal trajectories for linear kinematic systems with oscillatory internal dynamics is presented. Thereby, kinematic constraints with respect to velocity, acceleration and jerk are taken into account. The trajectories are composed of several motion primitives, the most crucial of which is termed jerk segment. Within this contribution, the focus is put on the composition of the overall trajectories, assuming the required motion primitives to be readily available. Since the scheme considered is not time-optimal, even decreasing particular constraints can reduce the overall transition time, which is analysed in detail. This observation implies that replanning of the underlying jerk segments is required as an integral part of the motion planning scheme, further insight into which has been analysed in a complementary contribution. Although the proposed scheme is not time-optimal, it allows for significantly shorter transition times than established methods, such as zero-vibration shaping, while requiring significantly lower computational power than a fully time-optimal scheme.

Paper Structure

This paper contains 34 sections, 42 equations, 29 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Previous research in this topic
  3. Contribution of this paper to the field
  4. Organisation of paper
  5. Preliminaries
  6. Mathematical model
  7. Solution of the slider dynamics
  8. Optimal control formulation
  9. Method explained: $\mathtt{OCP}\text{-}\mathtt{{J}}$
  10. Preprocessing of jerk segments
  11. Case 1
  12. Case 2
  13. Case 3
  14. Violation of jerk constraints
  15. Optimizing the full trajectory
  16. ...and 19 more sections

Figures (29)

  • Figure 1: Symbolic representation of one axis of the pick-and-place machine.
  • Figure 2: Trajectories: $\mathtt{S}\text{-}\mathtt{{Curve}}$, $\mathtt{ZV}$ and $\mathtt{OCP}\text{-}\mathtt{{S}}$ for a comparison of transition times for the motion distance $z\! \left( t_\text{f,t} \right) = 300mm$.
  • Figure 3: Two $\mathtt{OCP}\text{-}\mathtt{{J}}$ trajectories illustrating the method (picture taken from Auer2023IFAC).
  • Figure 4: Trajectories for multiple distances and regions associated with different cases.
  • Figure 5: $\mathtt{OCP}\text{-}\mathtt{{J}}$ trajectory for Case 1.
  • ...and 24 more figures