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Technical Brief

Steering a Chaplygin Sleigh Using Periodic Impulses

[+] Author and Article Information
Phanindra Tallapragada

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: ptallap@clemson.edu

Vitaliy Fedonyuk

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: vfedony@g.clemson.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 28, 2015; final manuscript received January 17, 2017; published online March 27, 2017. Assoc. Editor: Arend L. Schwab.

J. Comput. Nonlinear Dynam 12(5), 054501 (Mar 27, 2017) (5 pages) Paper No: CND-15-1463; doi: 10.1115/1.4036117 History: Received December 28, 2015; Revised January 17, 2017

The control of the motion of nonholonomic systems is of practical importance from the perspective of robotics. In this paper, we consider the dynamics of a cartlike system that is both propelled forward by motion of an internal momentum wheel. This is a modification of the Chaplygin sleigh, a canonical nonholonomic system. For the system considered, the momentum wheel is the sole means of locomotive thrust as well the only control input. We first derive an analytical expression for the change in the heading angle of the sleigh as a function of its initial velocity and angular velocity. We use this solution to design an open-loop control strategy that changes the orientation of sleigh to any desired angle. The algorithm utilizes periodic impulsive torque inputs via the motion of the momentum wheel.

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References

Chaplygin, S. A. , 1911, “ On the Theory of Motion of Nonholonomic Systems. The Theorem on the Reducing Multiplier,” Mat. Sb., XXVIII, pp. 303–314.
Osborne, J. M. , and Zenkov, D. V. , 2005, “ Steering the Chaplygin Sleigh by a Moving Mass,” American Control Conference (ACC), Seville, Spain, Dec. 15, pp. 1114–1118.
Fairchild, M. J. , Hassing, P. M. , Kelly, S. D. , Pujari, P. , and Tallapragada, P. , 2011, “ Single-Input Planar Navigation Via Proportional Heading Control Exploiting Nonholonomic Mechanics and Vortex Shedding,” ASME Paper No. DSCC2011-6191.
Kelly, S. D. , Fairchild, M. J. , Hassing, P. M. , and Tallapragada, P. , 2012, “ Proportional Heading Control for Planar Navigation: The Chaplygin Beanie and Fishlike Robotic Swimming,” American Control Conference (ACC), Montreal, QC, Jun 27–29, pp. 4885–4890.
Caratheodory, C. , 1933, “ Der schlitten,” J. Appl. Math. Mech., 13, pp. 71–76.
Neimark, J. I. , and Fufaev, N. A. , 1972, Dynamics of Nonholonomic Systems, AMS, Providence, RI.
Murray, R. M. , and Sastry, S. S. , 1990, “ Steering Nonholonomic Systems Using Sinusoids,” 29th IEEE Conference of Decision and Control (CDC), Dec. 5–7, pp. 2097–2101.
Murray, R. , and Sastry, S. S. , 1993, “ Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEE Trans. Autom. Control, 38(5), pp. 700–716. [CrossRef]
Fedonyuk, V. , and Tallapragada, P. , 2016, “ Stick-Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint,” ASME J. Comput. Nonlinear Dyn., 12(5), p. 031021.
Bloch, A. M. , Krishnaprasad, P. S. , Marsden, J. E. , and Murray, R. M. , 1996, “ Nonholonomic Mechanical Systems With Symmetry,” Arch. Ration. Mech. Anal., 136(1), pp. 21–99. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Chaplygin sleigh with a balanced rotor. The rotor is placed at distance of b from the rear contact. The centers of mass of both sleigh and the momentum wheel coincide at (x, y). The no-slip constraint is enforced at point P. The picture on the right shows an illustration of a physical cart to realize the Chaplygin sleigh with castors at the front.

Grahic Jump Location
Fig. 2

The asymptotic value of Δθ as a function of ux(0) and ω(0). The data correspond to K=2.

Grahic Jump Location
Fig. 3

Executing a 60 deg turn—the initial conditions are ux(0) = 1 and ω(0) = 0. The number of impulsive inputs necessary is N = 7.

Grahic Jump Location
Fig. 4

Executing a 180 deg turn—the initial conditions are ux(0) = 5 and ω(0) = 0. Number of impulsive inputs, N = 83.

Grahic Jump Location
Fig. 5

Parallel translation—the sleigh moves parallel to itself, by first turning by 30 deg and then by −30deg

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