0
Research Papers

Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint

[+] Author and Article Information
Vitaliy Fedonyuk

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

Phanindra Tallapragada

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: ptallap@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 18, 2015; final manuscript received November 29, 2016; published online January 19, 2017. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 12(3), 031021 (Jan 19, 2017) (8 pages) Paper No: CND-15-1442; doi: 10.1115/1.4035407 History: Received December 18, 2015; Revised November 29, 2016

The Chaplygin sleigh is a canonical problem of mechanical systems with nonholonomic constraints. Such constraints often arise due to the role of a no-slip requirement imposed by friction. In the case of the Chaplygin sleigh, it is well known that its asymptotic motion is that of pure translation along a straight line. Any perturbations in angular velocity decay and result in an increase in asymptotic speed of the sleigh. Such motion of the sleigh is under the assumption that the magnitude of friction is as high as necessary to prevent slipping. We relax this assumption by setting a maximum value to the friction. The Chaplygin sleigh is then under a piecewise-smooth nonholonomic constraint and transitions between “slip” and “stick” modes. We investigate these transitions and the resulting nonsmooth dynamics of the system. We show that the reduced state space of the system can be partitioned into sets of distinct dynamics and that the stick–slip transitions can be explained in terms of transitions of the state of the system between these sets.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Chaplygin, S. A. , 2008, “ On the Theory of the Motion of Nonholonomic Systems: The Reducing Multiplier Theorem,” Regular Chaotic Dyn., 13(4), pp. 369–376. [CrossRef]
Neimark, J. I. , and Fufaev, N. A. , 1972, Dynamics of Nonholonomic Systems, AMS, Providence, RI.
Bloch, A. M. , 2003, Nonholonomic Mechanics and Control, Springer Verlag, New York.
O'Reilly, O. M. , 1996, “ The Dynamics of Rolling Disks and Sliding Disks,” Nonlinear Dyn., 10(3), pp. 287–305. [CrossRef]
Osborne, J. M. , and Zenkov, D. V. , 2005, “ Steering the Chaplygin Sleigh by a Moving Mass,” 44th IEEE Conference on Decision and Control, Dec. 15.
Marsden, J. E. , Bloch, A. M. , and Zenkov, D. V. , 2009, “ Quasivelocities and Stabilization of Relative Equilibria of Underactuated Nonholonomic Systems,” Conference on Control and Decision, Shanghai, China, pp. 3335–3340.
Dear, T. , Kelly, S. D. , Travers, M. , and Choset, H. E. , 2013, “ Mechanics and Control of a Terrestrial Vehicle Exploiting a Nonholonomic Constraint for Fishlike Locomotion,” ASME Paper No. DSCC2013-3941.
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,” Energy 50(1), p. 2.
Polllard, B. , and Tallapragada, P. , 2016, “ An Aquatic Robot Propelled by an Internal Rotor,” IEEE/ASME Trans. Mechatronics, PP(99), pp. 657–662.
Tallapragada, P. , 2015, “ A Swimming Robot With an Internal Rotor as a Nonholonomic System,” American Control Conference, pp. 657–662.
Tallapragada, P. , and Kelly, S. D. , 2016, “ Integrability of Velocity Constraints Modeling Vortex Shedding in Ideal Fluids,” ASME J. Comput. Nonlinear Dyn., 12(2), p. 021008. [CrossRef]
Hamerlain, F. , Achour, K. , Floquet, T. , and Perruquetti, W. , 2005, “ Higher Order Sliding Mode Control of Wheeled Mobile Robots in the Presence of Sliding Effects,” 44th IEEE Conference on Decision and Control and the European Control Conference, Dec. 15, pp. 1959–1963.
Wright, C. , Johnson, A. , Peck, A. , McCord, Z. , Naaktgeboren, A. , Gianfortoni, P. , Gonzalez-Rivero, M. , Hatton, R. , and Chose, H. , 2007, “ Design of a Modular Snake Robot,” 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, Oct. 29–Nov. 2, pp. 2609–2614.
Sidek, N. , and Sarkar, N. , 2008, “ Dynamic Modeling and Control of Nonholonomic Mobile Robot With Lateral Slip,” Third International Conference on Systems, Apr. 13–18, pp. 35–40.
Bazzi, S. , Shammas, E. , and Asmar, D. , 2014, “ A Novel Method for Modeling Skidding for Systems With Nonholonomic Constraints,” Nonlinear Dyn., 76(2), pp. 1517–1528. [CrossRef]
Goodwine, B. , and Burdick, J. , 1997, “ Trajectory Generation for Kinematic Legged Robots,” IEEE International Conference on Robotics and Automation, Apr. 25, pp. 2689–2696.
Li, Z. , and Canny, J. , 1990, “ Motion of Two Rigid Bodies With Rolling Constraints,” IEEE Trans. Rob. Autom., 6(1), pp. 62–72. [CrossRef]
Fedonyuk, V. , and Tallapragada, P. , 2015, “ The Stick-Slip Motion of a Chaplygin Sleigh With a Piecewise Smooth Nonholonomic Constraint,” ASME Paper No. DSCC2015-9820.
Corradini, M. L. , 1999, “ Robust Stabilization of a Mobile Robot Violating the Nonholonomic Constraint Via Quasi-Sliding Modes,” American Control Conference, June 2–4, pp. 3935–3939.
Motte, I. , and Campion, G. , 2000, “ A Slow Manifold Approach for the Control of Mobile Robots Not Satisfying the Kinematic Constraints,” IEEE Trans. Rob. Autom., 16(6), pp. 875–880. [CrossRef]
Low, C. B. , and Wang, D. , 2008, “ GPS-Based Tracking Control for a Car-Like Wheeled Mobile Robot With Skidding and Slipping,” IEEE/ASME Trans. Mechatronics, 13(4), pp. 480–484. [CrossRef]
Wang, D. , and Low, C. B. , 2008, “ Modeling and Analysis of Skidding and Slipping in Wheeled Mobile Robots: Control Design Perspective,” IEEE/ASME Trans. Mechatronics, 24(3), pp. 676–687.
Saito, M. , Fukaya, M. , and Iwasaki, T. , 2002, “ Serpentine Locomotion With Robotic Snakes,” IEEE Control Syst. Mag., 22(1), pp. 64–81. [CrossRef]
Ma, S. , Ohmameuda, Y. , and Inoue, K. , 2004, “ Dynamic Analysis of 3-Dimensional Snake Robots,” Intelligent Robots and Systems Conference (IROS), Sept. 28–Oct. 2, pp. 767–762.
Transeth, A. A. , Van De Wouw, N. , Pavlov, A. , Hespanha, J. P. , and Pettersen, K. Y. , 2007, “ Tracking Control for Snake Robot Joints,” 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, Oct. 29–Nov. 2, pp. 3539–3546.
Transeth, A. A. , Leine, R. I. , Glocker, C. , Pettersen, K. Y. , and Liljebck, P. , 2008, “ Snake Robot Obstacle-Aided Locomotion: Modeling, Simulations, and Experiments,” IEEE Trans. Rob., 24(1), pp. 88–104. [CrossRef]
Transeth, A. A. , Leine, R. I. , Glocker, C. , and Pettersen, K. Y. , 2008, “ 3-D Snake Robot Motion: Nonsmooth Modeling, Simulations, and Experiments,” IEEE Trans. Rob., 24(2), pp. 361–376. [CrossRef]
Transeth, A. A. , Pettersen, K. Y. , and Liljebck, P. , 2009, “ A Survey on Snake Robot Modeling and Locomotion,” Robotica, 27(7), pp. 999–1015. [CrossRef]
Liljebck, P. , Pettersen, K. Y. , Stavdahl, O. , and Gravdahl, J. T. , 2013, Snake Robots—Modelling, Mechatronics, and Control, Springer-Verlag, London.
Ruina, A. , 1998, “ Nonholonomic Stability Aspects of Piecewise Nonholonomic Systems,” Rep. Math. Phys., 42(1–2), pp. 91–100. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) Chaplygin sleigh with a balanced rotor. The center of mass of the arrangement is at a distance b from the rear contact. 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

Phase space plot of the Chaplygin sleigh for parameter values m = 1, b = 1, and I = 1. The energies for the trajectories from the innermost trajectory outward are KE = [0.125, 0.5, 1.125, 2].

Grahic Jump Location
Fig. 3

Cart parameters for initial conditions of ux(0)=5 and ω(0)=2. Required friction for slip is assumed to be infinite.

Grahic Jump Location
Fig. 4

Phase space portrait of the reduced equations of the Chaplygin sleigh. The black curves (the rectangular hyperbolas) denote the critical friction, Fc. The shaded region is the set of initial conditions, which leads to pure nonholonomic motion.

Grahic Jump Location
Fig. 5

S1→S2→S4 transition from stick to slip and then back to stick mode of motion. Initial conditions are ux(0)=0 and ω(0)=2 (starting in S1). (a) The green (in plane trajectory) and red (out of the plane trajectory) represent the stick and slip motion respectively. (b) The dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the blue dashed–dotted curve represents the normal velocity of point P. (c) and (d) show ω(t) and ux(t), respectively. In (e), the black line is the total kinetic energy and the dashed curve is the translational kinetic energy.

Grahic Jump Location
Fig. 6

S1→S2→S1→S2→S4 transition. Initial conditions are (ω(0)=3.3,ux(0)=0)∈S1). (a) The green (in plane trajectory) and red (out of the plane trajectory) represent the stick and slip motion respectively. (b) The dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the blue dashed–dotted curve represents the normal velocity of point P. (c) and (d) show ω(t) and ux(t), respectively.

Grahic Jump Location
Fig. 7

Trajectories of the cart for (ux(0),ω(0))=(0,5.6) (shown purple) and (ux(0),ω(0))=(0,5.7) (shown red). In both cases, the no-slip trajectory confined to the plane uy = 0 is shown in green.

Grahic Jump Location
Fig. 8

Number of slip phases against ω(0) for ux(0)=0

Grahic Jump Location
Fig. 9

Normalized change in kinetic energy for a large range of initial conditions, (ux(0),ω(0))

Grahic Jump Location
Fig. 10

Plot showing the zero contour of the change in kinetic energy surface. Green (shaded) region represents positive change, and white (unshaded) region represents negative change.

Grahic Jump Location
Fig. 11

(a) Values of (ux(0),Δω) for which Δθ is positive (blue squares) and negative (red asterisks), respectively. The ΔKE=0 curve is also pictured in black. (b) The intersection of the ΔKE=0 curve with the Δθ=0 curve is shown. Blue (region 4) marks initial conditions where 10−5<ΔKE<5(10−5). Similarly, cyan (region 5) corresponds to |ΔKE|<10−5 and for initial conditions marked green (region 6) −5(10−5)<ΔKE<−10−5. For red (region 1) initial conditions 5(10−5)<Δθ<5(10−4), for yellow (region 3) |Δθ|<5(10−5), and for purple (region 2) −5(10−4)<Δθ<−5(10−5).

Grahic Jump Location
Fig. 12

Parallel translation for the case where ΔKE=0 and Δθ=0. Initial conditions are ux(0)=1.539646 and Δω=3.020854. In (c), the dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the dashed–dotted curve represents thenormal velocity of point P. In (f), the black line represents the total kinetic energy, and the blue-dashed curve represents the translational kinetic energy.

Grahic Jump Location
Fig. 13

Perturbation is applied periodically such that the cart avoids a rectangular obstacle. In (a), the path of the cart is shown, and ω is shown in (b).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In