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Calculation of time-optimal motion primitives for systems exhibiting oscillatory internal dynamics

Thomas Auer, Frank Woittennek

TL;DR

A numerical calculation method enabling fast and reliable calculation by explicitly evaluating the optimality conditions that arise for the problem, and further by reducing the evaluation of these conditions to a line-search problem on a bounded interval.

Abstract

An algorithm for planning near time-optimal trajectories for systems with an oscillatory internal dynamics has been developed in previous work. It is based on assembling a complete trajectory from motion primitives called jerk segments, which are the time-optimal solution to an optimization problem. To achieve the shortest overall transition time, it is advantageous to recompute these segments for different acceleration levels within the motion planning procedure. This publication presents a numerical calculation method enabling fast and reliable calculation. This is achieved by explicitly evaluating the optimality conditions that arise for the problem, and further by reducing the evaluation of these conditions to a line-search problem on a bounded interval. This reduction guarantees, that a valid solution if found after a fixed number of computational steps, making the calculation time constant and predictable. Furthermore, the algorithm does not rely on optimisation algorithms, which allowed its implementation on a laboratory system for measurements with the purpose of validating the approach.

Calculation of time-optimal motion primitives for systems exhibiting oscillatory internal dynamics

TL;DR

A numerical calculation method enabling fast and reliable calculation by explicitly evaluating the optimality conditions that arise for the problem, and further by reducing the evaluation of these conditions to a line-search problem on a bounded interval.

Abstract

An algorithm for planning near time-optimal trajectories for systems with an oscillatory internal dynamics has been developed in previous work. It is based on assembling a complete trajectory from motion primitives called jerk segments, which are the time-optimal solution to an optimization problem. To achieve the shortest overall transition time, it is advantageous to recompute these segments for different acceleration levels within the motion planning procedure. This publication presents a numerical calculation method enabling fast and reliable calculation. This is achieved by explicitly evaluating the optimality conditions that arise for the problem, and further by reducing the evaluation of these conditions to a line-search problem on a bounded interval. This reduction guarantees, that a valid solution if found after a fixed number of computational steps, making the calculation time constant and predictable. Furthermore, the algorithm does not rely on optimisation algorithms, which allowed its implementation on a laboratory system for measurements with the purpose of validating the approach.

Paper Structure

This paper contains 25 sections, 1 theorem, 64 equations, 22 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution of this paper
  4. Organisation of paper
  5. Model and mathematical background
  6. Method explained: $\mathtt{OCP}\text{-}\mathtt{{J}}$
  7. Efficient algorithm to calculate the jerk segments
  8. Optimality conditions
  9. Introduction of graphical approach
  10. Calculating the switching-times of the jerk segment
  11. Algorithm to calculate the jerk segments numerically
  12. Discussion of time-optimality of jerk segments
  13. Edge-case
  14. Comparison to other approaches
  15. Measurements
  16. ...and 10 more sections

Key Result

Theorem 1

The optimal control problem eq:optim_ctrl_problem possesses a solution $\left(\bm{x},u,t_\text{f}\right)$, satisfying the following properties Moreover, with eq:equation_lambda_5n, the switching points $t_2,\dots,t_{n-1}$ correspond to the zeros of $\lambda_{5,\text{n}}\! \left( t \right)$, and Finally, the analytical solutions for the system state variables to such an input are given according

Figures (22)

  • Figure 1: Symbolic representation of one axis of the pick-and-place machine.
  • Figure 2: Two $\mathtt{OCP}\text{-}\mathtt{{J}}$ trajectories demonstrating how a full trajectory is assembled from the jerk segments (picture taken from tau_ocpJ_assembly_part1).
  • Figure 3: Showing the overlapping without adjustment of $a_\text{max}$ for a transition distance of $z\! \left( t_\text{f,t} \right)=1.5mm$ to show the advantage of recalculation of jerk segments in order to reduce the transition time. Picture taken from tau_ocpJ_assembly_part1.
  • Figure 4: Results for $\lambda_\text{5,n}\! \left( t \right)$ for the two jerk profiles required in the trajectories shown in \ref{['fig:two_OcpJ_trajectories_for_demo_of_method']}. Included are the acceleration profiles $\ddot{z}\! \left( t \right)$ of the jerk segments for comparison.
  • Figure 5: Jerk segment to show the acceleration $\ddot{z}$, jerk $z^{\left(3\right)}$ and coefficients $a_1,\dots,a_4$.
  • ...and 17 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Conjecture 1