Research Papers

Motion Planning for an Underactuated Planar Robot in a Viscous Environment

[+] Author and Article Information
Elie Shammas

Assistant Professor
Department of Mechanical Engineering,
American University of Beirut,
Beirut 11-0236, Lebanon
e-mail: es34@aub.edu.lb

Daniel Asmar

Assistant Professor
Department of Mechanical Engineering,
American University of Beirut,
Beirut 11-0236, Lebanon
e-mail: da20@aub.edu.lb

Manuscript received May 29, 2013; final manuscript received December 27, 2014; published online April 2, 2015. Assoc. Editor: Jozsef Kovecses.

J. Comput. Nonlinear Dynam 10(5), 051002 (Sep 01, 2015) (11 pages) Paper No: CND-13-1159; doi: 10.1115/1.4029509 History: Received May 29, 2013; Revised December 27, 2014; Online April 02, 2015

In this paper, we solve the motion planning problem for a class of underactuated multibodied planar mechanical systems. These systems interact with the environment via viscous frictional forces. The motion planning problem is solved by specifying the location of friction pads on the robot as well as by specifying the input of the actuated degrees of freedom. Moreover, through the proposed novel motion planning analysis, we identify the simplest planar swimming robot, the two-link swimmer.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Laumond, J.-P., 1987, “Finding Collision-Free Smooth Trajectories for a Non-Holonomic Mobile Robot,” 10th International Joint Conference on Artificial (IJCAI), pp. 1120–1123.
Laumond, J.-P., Jacobs, P. E., Taix, M., and Murray, R. M., 1994, “A Motion Planner for Nonholonomic Mobile Robots,” IEEE Trans. Rob. Autom., 10(5), pp. 577–593. [CrossRef]
Li, Z., and Canny, J., 1990, “Motion of Two Rigid Bodies With Rolling Constraint,” IEEE Trans. Rob. Autom., 6(1), pp. 62–72. [CrossRef]
Laumond, J.-P., 1986, “Feasible Trajectories for Mobile Robots With Kinematic and Environment Constraints,” Intelligent Autonomous Systems, International Conference, North-Holland Publishing Co., Amsterdam, The Netherlands, pp. 346–354.
Murray, R. M., and Sastry, S. S., 1993, “Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEE Trans. Autom. Control, 38(5), pp. 700–716. [CrossRef]
Barraquand, J., and Latombe, J.-C., 1991, “Robot Motion Planning: A Distributed Representation Approach,” Int. J. Rob. Res., 10(6), pp. 628–649. [CrossRef]
Canny, J., 1988, The Complexity of Robot Motion Planning, MIT Press, Cambridge, MA.
Lafferriere, G., and Sussmann, H. J., 1993, “A Differential Geometric Approach to Motion Planning,” Nonholonomic Motion Planning, Springer, New York, pp. 235–270.
Bloch, A. M., 2003, Nonholonomic Mechanics and Control, Vol. 24, Springer, New York. [CrossRef]
Choset, H. M., 2005, Principles of Robot Motion: Theory, Algorithms, and Implementation, MIT Press, Cambridge, MA.
Bullo, F., 2005, Geometric Control of Mechanical Systems, Vol. 49, Springer, New York. [CrossRef]
Tsakiris, D. P., 1995, “Motion Control and Planning for Nonholonomic Kinematic Chains,” DTIC Document, Maryland Univ., College Park Inst. for Systems Research, Technical Report No. ISR-PHD-95-4.
Krishnaprasad, P., and Tsakiris, D. P., 2001, “Oscillations, SE (2)-Snakes and Motion Control: A Study of the Roller Racer,” Dyn. Syst.: Int. J., 16(4), pp. 347–397. [CrossRef]
Bloch, A. M., Krishnaprasad, P., Marsden, J. E., and Murray, R. M., 1996, “Nonholonomic Mechanical Systems With Symmetry,” Arch. Ration. Mech. Anal., 136(1), pp. 21–99. [CrossRef]
Lafferriere, G., and Sussmann, H., 1990, “Motion Planning for Controllable Systems Without Drift: A Preliminary Report,” Rutgers University Systems and Control Center, Report No. SYCON-90-04.
Sussmann, H. J., 1991, “Local Controllability and Motion Planning for Some Classes of Systems With Drift,” 30th IEEE Conference, Brighton, Dec. 11–13, pp. 1110–1114. [CrossRef]
Ladd, A. M., and Kavraki, L. E., 2005, “Motion Planning in the Presence of Drift, Underactuation and Discrete System Changes,” Robotics: Science and Systems, pp. 233–240.
Mason, R., and Burdick, J., 1999, “Propulsion and Control of Deformable Bodies in an Ideal Fluid,” IEEE International Conference on Robotics and Automation, Detroit, MI, Vol. 1, pp. 773–780. [CrossRef]
Kanso, E., Marsden, J. E., Rowley, C. W., and Melli-Huber, J., 2005, “Locomotion of Articulated Bodies in a Perfect Fluid,” J. Nonlinear Sci., 15(4), pp. 255–289. [CrossRef]
Kanso, E., and Marsden, J., 2005, “Optimal Motion of an Articulated Body in a Perfect Fluid,” 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference (CDC-ECC’05), Dec. 12–15, pp. 2511–2516. [CrossRef]
Melli, J. B., Rowley, C. W., and Rufat, D. S., 2006, “Motion Planning for an Articulated Body in a Perfect Planar Fluid,” SIAM J. Appl. Dyn. Syst., 5(4), pp. 650–669. [CrossRef]
Melsaac, K., and Ostrowski, J., 1999, “A Geometric Approach to Anguilliform Locomotion: Modelling of an Underwater Eel Robot,” IEEE International Conference on Robotics and Automation, Vol. 4, pp. 2843–2848.
Chernous’ko, F., 1999, “The Wavelike Motion of a Multilink System on a Horizontal Plane,” J. Appl. Math. Mech., 64(4), pp. 497–508. [CrossRef]
Chernous’ko, F., 2001, “Controllable Motions of a Two-Link Mechanism Along a Horizontal Plane,” J. Appl. Math. Mech., 65(4), pp. 565–577. [CrossRef]
Chernous’ko, F., 2005, “Modelling of Snake-Like Locomotion,” Appl. Math. Comput., 164(2), pp. 415–434. [CrossRef]
Chernous’ko, F., and Shunderyuk, M. M., 2010, “The Influence of Friction Forces on the Dynamics of a Two-Link Mobile Robot,” Appl. Math. Comput., 74, pp. 13–23.
Figurina, T., 2004, “Quasi-Static Motion of a Two-Link System Along a Horizontal Plane,” Multibody Syst. Dyn., 11(3), pp. 251–272. [CrossRef]
Burton, L. J., Hatton, R. L., Choset, H., and Hosoi, A. E., 2010, “Two Link Swimming Using Buoyant Orientation,” Phys. Fluids, 22(9), p. 091703. [CrossRef]
Melli, J., and Rowley, C., 2010, “Models and Control of Fish-Like Locomotion,” Exp. Mech., 50(9), pp. 1355–1360. [CrossRef]
Babikian, S., Shammas, E., and Asmar, D., 2012, “Motion Planning for a Two-Link Planar Robot in a Viscous Environment,” IEEE International Conference on Intelligent Robots and Systems (IROS), Vilamoura, Oct. 7–12, pp. 888–895. [CrossRef]
Greenwood, D., 1988, Principles of Dynamics, Prentice-Hall, Upper Saddle River, NJ.
Fiedler, M., 1986, Special Matrices and Their Applications in Numerical Mathematics, 1st ed., Springer, New York.
Bloch, A., Krishnaprasad, P., Marsden, J., and Ratiu, T., 1996, “The Euler-Poincaré Equations and Double Bracket Dissipation,” Commun. Math. Phys., 175(1), pp. 1–42. [CrossRef]
Ostrowski, J., 1998, “Reduced Equations for Nonholonomic Mechanical Systems With Dissipative Forces,” Rep. Math. Phys., 42(1–2), pp. 185–209. [CrossRef]
Murray, R. N., Li, Z., and Sastry, S., 1994, A Mathematical Introduction to Robotics Manipulation, CRC Press, Boca Raton, FL.
Shammas, E. A., 2006, “Generalized Motion Planning for Underactuated Mechanical Systems,” Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA.
Shammas, E. A., Choset, H., and Rizzi, A. A., 2007, “Towards a Unified Approach to Motion Planning for Dynamic Underactuated Mechanical Systems With Non-Holonomic Constraints,” Int. J. Rob. Res., 26(10), pp. 1075–1124. [CrossRef]


Grahic Jump Location
Fig. 1

A two-link underactuated planar system

Grahic Jump Location
Fig. 2

An allowable input, ri = fa,b,A,B,s(t) − fa,b,A,B,s(t + κ)

Grahic Jump Location
Fig. 3

Numerical simulation for the two inputs, φ1(t)=f0.5,5.5,0,2,5(t) and its reverse φ¯1(t)=-f0.5,5.5,0,2,5(t)

Grahic Jump Location
Fig. 4

Same signed regions for the push term Γp

Grahic Jump Location
Fig. 5

The evolution of y, φ1, the push term, and the drift term for the two-link system

Grahic Jump Location
Fig. 6

Net motion along the y of the two-link system, ti ⇔ (t = i sec)

Grahic Jump Location
Fig. 7

Simulation using webots depicting motion along y

Grahic Jump Location
Fig. 8

The drift terms of the two-link system

Grahic Jump Location
Fig. 9

The evolution of the degrees of freedom for the two-link system

Grahic Jump Location
Fig. 10

Net motion along the x direction of the two-link system

Grahic Jump Location
Fig. 11

Simulation using webots depicting motion along x

Grahic Jump Location
Fig. 12

Motion of the proposed two-link robot in a viscous environment (dashed lines) versus the motion of the roller racer (solid lines). (a) Motion along the y axis. (b) Motion along the x axis.




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