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Research Papers

Minimum-Time Trajectories for Steered Agent With Constraints on Speed, Lateral Acceleration, and Turning Rate

[+] Author and Article Information
William Lewis Scott

Mem. ASME
Department of Aerospace Engineering,
Institute for Systems Research,
University of Maryland,
College Park, MD 20742
e-mail: wlscott@umd.edu

Naomi Ehrich Leonard

Professor
Fellow ASME
Department of Mechanical and
Aerospace Engineering,
Princeton University,
Princeton, NJ 08544
e-mail: naomi@princeton.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received November 14, 2016; final manuscript received January 25, 2018; published online March 28, 2018. Editor: Joseph Beaman.

J. Dyn. Sys., Meas., Control 140(7), 071017 (Mar 28, 2018) (10 pages) Paper No: DS-16-1554; doi: 10.1115/1.4039283 History: Received November 14, 2016; Revised January 25, 2018

We present time-optimal trajectories for a steered agent with constraints on speed, lateral acceleration, and turning rate for the problem of reaching a point on the plane in minimum time with free terminal heading angle. Both open-loop and state-feedback forms of optimal controls are derived through application of Pontryagin's minimum principle. We apply our results for the single agent to solve a multi-agent coverage problem in which each agent has constraints on speed, lateral acceleration, and turning rate.

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References

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Scott, W. L. , and Leonard, N. E. , 2014, “ Dynamics of Pursuit and Evasion in a Heterogeneous Herd,” IEEE Conference on Decision and Control (CDC), Los Angeles, CA, Dec. 15–17, pp. 2920–2925.
Soueres, P. , and Boissonnat, J.-D. , 1998, “ Optimal Trajectories for Nonholonomic Mobile Robots,” Robot Motion Planning and Control, Springer, Berlin, pp. 93–170. [CrossRef]
Sussmann, H. J. , and Tang, G. , 1991, “ Shortest Paths for the Reeds-Shepp Car: A Worked out Example of the Use of Geometric Techniques in Nonlinear Optimal Control,” Rutgers Cent. Syst. Control Tech. Rep., 10, pp. 1–71. http://www-personal.acfr.usyd.edu.au/spns/motion/SussmannTang1991.pdf
Balkcom, D. J. , and Mason, M. T. , 2002, “ Extremal Trajectories for Bounded Velocity Mobile Robots,” IEEE International Conference on Robotics and Automation (ICRA), Washington, DC, May 11–15, pp. 1747–1752.
Tan, H. , and Wilson, A. M. , 2011, “ Grip and Limb Force Limits to Turning Performance in Competition Horses,” Proc. R. Soc. London B: Biol. Sci., 278(1715), pp. 2105–2111. [CrossRef]
Hudson, P. E. , Corr, S. A. , and Wilson, A. M. , 2012, “ High Speed Galloping in the Cheetah (Acinonyx jubatus) and the Racing Greyhound (Canis familiaris): Spatio-Temporal and Kinetic Characteristics,” J. Exp. Biol., 215(14), pp. 2425–2434. [CrossRef] [PubMed]
Steenkamp, N. F. , and Patel, A. , 2016, “ Minimum Time Sprinting From Rest in a Planar Quadruped,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, South Korea, Oct. 9–14, pp. 3866–3871.
Balkcom, D. J. , and Mason, M. T. , 2002, “ Time Optimal Trajectories for Bounded Velocity Differential Drive Vehicles,” Int. J. Rob. Res., 21(3), pp. 199–217. [CrossRef]
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Chitsaz, H. , LaValle, S. M. , Balkcom, D. J. , and Mason, M. T. , 2009, “ Minimum Wheel-Rotation Paths for Differential-Drive Mobile Robots,” Int. J. Rob. Res., 28(1), pp. 66–80. [CrossRef]
Scott, W. L. , and Leonard, N. E. , 2016, “ Time-Optimal Trajectories for Steered Agent With Constraints on Speed and Turning Rate,” ASME Paper No. DSCC2016-9892.
Cortes, J. , Martinez, S. , Karatas, T. , and Bullo, F. , 2004, “ Coverage Control for Mobile Sensing Networks,” IEEE Trans. Rob. Autom., 20(2), pp. 243–255. [CrossRef]
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Figures

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Fig. 1

Admissible control inputs for steered agent with limits on speed, turning rate, and lateral acceleration as described in Sec. 2

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Fig. 2

Illustration of system state relative to the η = 0 line (shown dashed) at points where a switching function ϕi reaches zero. Left: trajectory ending in forward segment (β1=0). Right: trajectory ending in fast turn segment.

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Fig. 3

Partition of the plane based on the optimal trajectory type to reach each point in minimum time for an agent starting at the origin with heading in the direction of the positive x-axis and v¯=1, ω¯=1, and μ=0.5. Destinations with y > 0 use left turn controls and those with y < 0 use right turn controls. For those destinations along the positive x-axis, no turning is necessary. For those on the negative x-axis, left and right turning trajectories take the same amount of time.

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Fig. 4

Minimum-time trajectories for an agent starting at the origin with heading in the direction of the positive x-axis with v¯=1 and ω¯=1. Trajectories' segments are colored by type: blue is slow turn, green is fast turn, and red is forward motion.

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Fig. 5

Minimum time to reach surfaces for an agent starting at the origin with heading in the direction of the positive x-axis with v¯=1 and ω¯=1, for different values of the lateral acceleration constraint μ. Isochron (equal time-to-reach) lines are drawn at 1 s interval.

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Fig. 6

Control switching regions for destination relative to agent at the origin with heading in the direction of the positive x-axis with v¯=1 and ω¯=1. The lateral acceleration parameter μ is varied, from left, μ=0.1, 0.5, and 0.9. Destinations in the upper half plane (y > 0) are optimally reached by left turn maneuvers and in the lower half plane by right turn maneuvers.

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Fig. 7

Trajectory-type partition (top) and optimal trajectories (bottom) for extreme values of μ for an agent starting at the origin with heading in the direction of the positive x-axis with v¯=1 and ω¯=1. Left: μ≥v¯ω¯, so fast and slow turns are equivalent, both having radius of b. Right: μ = 0, with only rotation and forward motion possible. Trajectories' segments are colored by type: magenta is turn and red is forward.

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Fig. 8

Simulation of coverage algorithm for N = 9 agents in a 20 by 20 square domain. All agents have motion constraints v¯=1, ω¯=1,  and μ=0.5. Agents are depicted as black circles with a line in the direction of their heading. Colored regions denote the dominance region for each agent. White lines are time-to-reach isochrons in 1 s interval. Left: at initial time t = 0, agents are randomly placed near the center of the domain with random headings. Initial worst-case time-to-reach is V(Q(0))≈13.37 s. Right: configuration at the completion of the coverage algorithm after eight timesteps with Δt=1 s. At the final time t = 8 s, the worst-case time-to-reach has been reduced to V(Q(8))≈6.79 s. The lower bound on attainable V for this system from Theorem 4 is V≥4.52.

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