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Analysis of solutions to the time-optimal planning and execution problem

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Abstract

This paper analyses solutions to the time-optimal planning and execution (TOPE) problem, in which the aim is to minimise the total time required for an agent to achieve its objectives. The TOPE process provides a means of adjusting system parameters in real-time to achieve this aim. Prior work by the authors showed that agent-based planning systems employing the TOPE process can yield better performance than existing techniques, provided that a key estimation step can be run sufficiently fast and accurately. This paper describes several real-time implementations of this estimation step. A Monte-Carlo analysis compares the performance of TOPE systems using these implementations against existing state-of-the-art planning techniques. It is shown that the average case performance of the TOPE systems is significantly better than the existing methods. Since the TOPE process can be added to an existing system without modifying the internal processes, these results suggest that similar performance improvement may be obtained in a multitude of robotics applications.

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Notes

  1. A full derivation may be found in [1] and [2].

  2. The ‘highest’ level of a hierarchical planning system is the level of control furthest from that which directly commands the agent’s actuators. This is also commonly referred to as the ‘global’ level.

  3. The selection of optimal parameters was done by running the comparison methods with every possible combination of parameter values. For dynamic state spaces in general it is impossible to know the best parameter values in advance. Nonetheless, the results showed that if the possible combinations of parameter sets were the same as those available to the TOPE process, then even the optimal static choice was inferior to the TOPE process’ ability to change the values dynamically.

  4. This commitment process was first described in [3] and ensures that updated plans always start from a future position of the agent, meaning that backtracking is guaranteed never to occur.

  5. Although this treatment affects the expected execution time of the planner’s output, in practice these areas are always revealed by the simulated agent’s sensors before they are ever traversed. As a result, the measured execution time is always correct, despite the planners treating unknown regions optimistically. It is also relevant to point out that the combination of this cost value for unknown regions, and the heuristic function used, tends to reduce the computational time required by the planning algorithms. This is because the heuristic cost is closer to the true cost, which limits the number of states explored by a graph search algorithm [31, 35].

  6. It should be noted that given these constraints, it is possible for a particular baseline technique to fail entirely if the start and goal points are blocked by a closed ring of obstacle cells in the baseline technique’s fixed cell size cost map. This point itself is indicative of the benefits of TOPE techniques that allow adjustment of system parameters, over techniques that rely on a fixed parameter set and might therefore fail unnecessarily.

  7. Memoization refers to recording the output of a function given a particular input, such that this output can be rapidly retrieved when the same input is used again, rather than recalculating the result. With reference to plans, this would imply that for a given parameter set, the entire plan could be stored and reused if the same parameters were selected again. Since the agent’s state will differ between TOPE estimation iterations, as will the planning state space, this is not possible within the TOPE process.

  8. The learning-discount parameter is functionally equivalent to the ‘temperature’ used in SA.

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Acknowledgments

This work is supported in part by the Australian Research Council (ARC) Centre of Excellence programme, funded by the ARC and the New South Wales State Government. The authors are grateful to the Rio Tinto Centre for Mine Automation for the use of \(\sim \)70,000 hours of CPU time required to perform this analysis.

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Authors and Affiliations

  1. The Australian Centre for Field Robotics, The University of Sydney, Sydney, NSW, 2006, Australia

    Thomas Allen & Steven Scheding

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Correspondence to Thomas Allen.

Appendix A: Tabulated results of the Monte-Carlo analysis

Appendix A: Tabulated results of the Monte-Carlo analysis

This appendix tabulates (Tables 1, 2, 3, 4, 5, 6, 7, 8) the results of the Monte-Carlo analysis of TOPE estimators described in Sect. 6.1. Two tables are shown for each of the four scenarios. The first shows the percentage of runs in which each TOPE estimator outperforms each baseline technique. The second shows the inverse; the percentage of runs in which each baseline technique outperforms each TOPE estimator.

Table 1 Percentage of runs in which each comparison method beat each baseline for a Monte-Carlo simulation over real-world cost map data
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Table 2 Percentage of runs in which each baseline beat each comparison method for a Monte-Carlo simulation over real-world cost map data
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Table 3 Percentage of runs in which each comparison method beat each baseline for randomised fractal terrain data with a roughness parameter of \(H=0.2\)
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Table 4 Percentage of runs in which each baseline beat each comparison method for randomised fractal terrain data with a roughness parameter of \(H=0.2\)
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Table 5 Percentage of runs in which each comparison method beat each baseline for randomised fractal terrain data with a roughness parameter of \(H=0.5\)
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Table 6 Percentage of runs in which each baseline beat each comparison method for randomised fractal terrain data with a roughness parameter of \(H=0.5\)
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Table 7 Percentage of runs in which each comparison method beat each baseline for randomised fractal terrain data with a roughness parameter of \(H=0.8\)
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Table 8 Percentage of runs in which each baseline beat each comparison method for randomised fractal terrain data with a roughness parameter of \(H=0.8\)
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Allen, T., Scheding, S. Analysis of solutions to the time-optimal planning and execution problem. Intel Serv Robotics 5, 245–258 (2012). https://doi.org/10.1007/s11370-012-0116-0

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