0
Research Papers

A Piecewise-Linear Approximation of the Canonical Spring-Loaded Inverted Pendulum Model of Legged Locomotion

[+] Author and Article Information
Zhuohua Shen

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47906
e-mail: shen38@purdue.edu

Justin Seipel

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47906
e-mail: jseipel@purdue.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 15, 2014; final manuscript received January 15, 2015; published online June 30, 2015. Assoc. Editor: Jozsef Kovecses.

J. Comput. Nonlinear Dynam 11(1), 011007 (Jan 01, 2016) (9 pages) Paper No: CND-14-1187; doi: 10.1115/1.4029664 History: Received August 15, 2014; Revised January 15, 2015; Online June 30, 2015

Here, we introduce and analyze a novel approximation of the well-established and widely used spring-loaded inverted pendulum (SLIP) model of legged locomotion, which has made several validated predictions of the center-of-mass (CoM) or point-mass motions of animal and robot running. Due to nonlinear stance equations in the existing SLIP model, many linear-based systems theories, analytical tools, and corresponding control strategies cannot be readily applied. In order to provide a significant simplification in the use and analysis of the SLIP model of locomotion, here we develop a novel piecewise-linear, time-invariant approximation. We show that a piecewise-linear system, with the only nonlinearity due to the switching event between stance and flight phases, can predict all the bifurcation features of the established nonlinear SLIP model over the entire three-dimensional model parameter space. Rather than precisely fitting only one particular solution, this approximation is made to quantitatively approximate the entire solution space of the SLIP model and capture all key aspects of solution bifurcation behavior and parametric sensitivity of the original SLIP model. Further, we provide an entirely closed-form solution for the stance trajectory as well as the system states at the end of stance, in terms of common functions that are easy to code and compute. Overall, the closed-form solution is found to be significantly faster than numerical integration when implemented using both matlab and c++. We also provide a closed-form analytical stride map, which is a Poincaré return section from touchdown (TD) to next TD event. This is the simplest closed-form approximate stride mapping yet developed for the SLIP model, enabling ease of analysis and numerical coding, and reducing computational time. The approximate piecewise-linear SLIP model presented here is a significant simplification over previous SLIP-based models and could enable more rapid development of legged locomotion theory, numerical simulations, and controllers.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Blickhan, R., 1989, “The Spring–Mass Model for Running and Hopping,” J. Biomech., 22(11–12), pp. 1217–1227. [CrossRef] [PubMed]
Blickhan, R., and Full, R., 1993, “Similarity in Multilegged Locomotion: Bouncing Like a Monopode,” J. Comp. Physiol., A, 173(5), pp. 509–517. [CrossRef]
Holmes, P., Full, R., Koditschek, D., and Guckenheimer, J., 2006, “The Dynamics of Legged Locomotion: Models, Analyses, and Challenges,” SIAM Rev., 48(2), pp. 207–304. [CrossRef]
Dalleau, G., Belli, A., Bourdin, M., and Lacour, J., 1998, “The Spring–Mass Model and the Energy Cost of Treadmill Running,” Eur. J. Appl. Physiol. Occup. Physiol., 77(3), pp. 257–263. [CrossRef] [PubMed]
McMahon, T., 1984, “Mechanics of Locomotion,” Int. J. Rob. Res., 3(2), pp. 4–28. [CrossRef]
Raibert, M., 1986, Legged Robots That Balance, Massachusetts Institute of Technology, Cambridge, MA.
McGeer, T., 1990, “Passive Bipedal Running,” Proc. R. Soc. London, Ser. B, 240(1297), pp. 107–134. [CrossRef]
McGeer, T., 1990, “Passive Dynamic Walking,” Int. J. Rob. Res., 9(2), pp. 62–82. [CrossRef]
Alexander, R., 1995, “Simple Models of Human Movement,” ASME Appl. Mech. Rev., 48(8), pp. 461–470. [CrossRef]
Schwind, W., 1998, “Spring Loaded Inverted Pendulum Running: A Plant Model,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.
Farley, C., Blickhan, R., Saito, J., and Taylor, C., 1991, “Hopping Frequency in Humans: A Test of How Springs Set Stride Frequency in Bouncing Gaits,” J. Appl. Physiol., 71(6), pp. 2127–32. [PubMed]
He, J., Kram, R., and McMahon, T., 1991, “Mechanics of Running Under Simulated Low Gravity,” J. Appl. Physiol., 71(3), pp. 863–870. [PubMed]
Farley, C., Glasheen, J., and McMahon, T., 1993, “Running Springs: Speed and Animal Size,” J. Exp. Biol., 185, pp. 71–86. [PubMed]
Ferris, D., and Farley, C., 1997, “Interaction of Leg Stiffness and Surface Stiffness During Human Hopping,” J. Appl. Physiol., 82(1), pp. 15–22. [PubMed]
Ferris, D., Louie, M., and Farley, C., 1998, “Running in the Real World: Adjusting Leg Stiffness for Different Surfaces,” Proc. R. Soc. London, Ser. B, 265(1400), pp. 989–994. [CrossRef]
Ferris, D., Liang, K., and Farley, C., 1999, “Runners Adjust Leg Stiffness for Their First Step on a New Running Surface,” J. Biomech., 32(8), pp. 787–794. [CrossRef] [PubMed]
Lee, C., and Farley, C., 1998, “Determinants of the Center of Mass Trajectory in Human Walking and Running,” J. Exp. Biol., 201(21), pp. 2935–2944. [PubMed]
Geyer, H., Seyfarth, A., and Blickhan, R., 2006, “Compliant Leg Behavior Explains Basic Dynamics of Walking and Running,” Proc. R. Soc. London, Ser. B, 273(1603), pp. 2861–2867. [CrossRef]
Jun, J., and Clark, J., 2009, “Dynamic Stability of Variable Stiffness Running,” Proceedings of the 2009 IEEE international Conference on Robotics and Automation, Kobe, Japan, May 12–17, pp. 3985–3990.
Seyfarth, A., Geyer, H., Gunther, M., and Blickhan, R., 2002, “A Movement Criterion for Running,” J. Biomech., 35(5), pp. 649–655. [CrossRef] [PubMed]
Altendorfer, R., Koditschek, D., and Holmes, P., 2003, “Towards a Factored Analysis of Legged Locomotion Models,” Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Taipei, Taiwan, Sept. 14–19, pp. 37–44.
Seipel, J., and Holmes, P., 2006, “Three Dimensional Translational Dynamics and Stability of Multi-Legged Runners,” Int. J. Rob. Res., 25(9), pp. 889–902. [CrossRef]
Grimmer, S., Ernst, M., Gunther, M., and Blickhan, R., 2008, “Running on Uneven Ground: Leg Adjustment to Vertical Steps and Self-Stability,” J. Exp. Biol., 211(Pt. 18), pp. 2989–3000. [CrossRef] [PubMed]
Full, R., and Koditschek, D., 1999, “Templates and Anchors: Neuromechanical Hypotheses of Legged Locomotion on Land,” J. Exp. Biol., 202(Pt. 23), pp. 3325–3332. [PubMed]
Schmitt, J., 2006, “A Simple Stabilizing Control for Sagittal Plane Locomotion,” ASME J. Comput. Nonlinear Dyn., 1(4), pp. 348–357. [CrossRef]
Schmitt, J., and Clark, J., 2009, “Modeling Posture-Dependent Leg Actuation in Sagittal Plane Locomotion,” Bioinspiration Biomimetics, 4(4), p. 046005. [CrossRef] [PubMed]
Tamaddoni, S., Jafari, F., and Medhdari, A., 2010, “Biped Hopping Control on Spring Loaded Inverted Pendulum,” Int. J. Humanoid Rob., 7(2), pp. 263–280. [CrossRef]
Kimura, Y., Oh, S., and Hori, Y., 2012, “Leg Space Observer on Biarticular Actuated Two-Link Manipulator for Realizing Spring Loaded Inverted Pendulum Model,” 2012 IEEE International Workshop on Advanced Motion Control, Sarajevo, Bosnia and Herzegovina, Mar. 25–27, pp. 1–6. [CrossRef]
Piovan, G., and Byl, K., 2012, “Enforced Symmetry of the Stance Phase for the Spring-Loaded Inverted Pendulum,” 2012 IEEE International Conference on Robotics and Automation, St. Paul, MN, May 14–18, pp. 1908–1914. [CrossRef]
Sato, A., and Buehler, M., 2004, “A Planar Hopping Robot With One Actuator: Design, Simulation, and Experimental Results,” Intelligent Robots and Systems, 2004, Sendai, Japan, 28 Sept.–2 Oct., pp. 3540–3545.
Seyfarth, A., Geyer, H., and Herr, H., 2003, “Swing-Leg Retraction: A Simple Control Model for Stable Running,” J. Exp. Biol., 206(Pt. 15), pp. 2547–2555. [CrossRef] [PubMed]
Blum, Y., Lipfert, S., Rummel, J., and Seyfarth, A., 2010, “Swing Leg Control in Human Running,” Bioinspiration Biomimetics, 5(2), p. 026006. [CrossRef] [PubMed]
Kim, K., Kwon, O., Yeon, J., and Park, J. H., 2006, “Elliptic Trajectory Generation for Galloping Quadruped Robots,” IEEE International Conference on Robotics and Biomimetics, ROBIO '06, Kunming, China, Dec. 17–20, pp. 103–108.
Andrews, B., Miller, B., Schmitt, J., and Clark, J., 2011, “Running Over Unknown Rough Terrain With a One-Legged Planar Robot,” Bioinspiration Biomimetics, 6(2), p. 026009. [CrossRef] [PubMed]
Poulakakis, I., and Grizzle, J., 2009, “The Spring Loaded Inverted Pendulum as the Hybrid Zero Dynamics of an Asymmetric Hopper,” IEEE Trans. Autom. Control, 54(8), pp. 1779–1793. [CrossRef]
Saranli, U., Buehler, M., and Koditschek, D., 2001, “RHex: A Simple and Highly Mobile Hexapod Robot,” Int. J. Rob. Res., 20(1), pp. 616–631. [CrossRef]
Altendorfer, R., Koditschek, D., Komsuoglu, H., Buehler, M., Moore, N., and Mcmordie, D., 2001, “Evidence for Spring Loaded Inverted Pendulum Running in a Hexapod Robot,” Exp. Rob. VII, 271, pp. 291–302. [CrossRef]
Komsuoglu, H., Majumdar, A., Aydin, Y., and Koditschek, D., 2010, “Characterization of Dynamic Behaviors in a Hexapod Robot,” International Symposium on Experimental Robotics, pp. 667–684. [CrossRef]
Komsuoglu, H., 2009, “Dynamic Legged Mobility—An Overview,” International Joint Robotics Conference and Workshop, p. 1.
Koepl, D., and Hurst, J., 2011, “Force Control for Planar Spring–Mass Running,” Intelligent Robots and Systems (IROS), San Francisco, CA, Sept. 25–30, pp. 3758–3763. [CrossRef]
Mordatch, I., Lasa, M., and Hertzmann, A., 2010, “Robust Physics-Based Locomotion Using Low-Dimensional Planning,” ACM Trans. Graphics, 29(3), p. 71. [CrossRef]
Kenwright, B., Davison, R., and Morgan, G., 2011, “Dynamic Balancing and Walking for Real-Time 3D Characters,” 4th International Conference on Motion in Games, pp. 63–73. [CrossRef]
Kenwright, B., 2012, “Responsive Biped Character Stepping: When Push Comes to Shove,” Cyberworlds 2012, Darmstadt, Sept. 25–27, pp. 151–156. [CrossRef]
Kenwright, B., and Huang, C., 2013, “Beyond Keyframe Animations: A Controller Character-Based Stepping Approach,” 2013 ACM SIGGRAPH Asia Technical Briefs, pp. 10:1–10:4. [CrossRef]
Kwon, T., and Hodgins, J., 2010, “Control Systems for Human Running Using an Inverted Pendulum Model and a Reference Motion Capture Sequence,” 2010 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 129–138.
Seipel, J., and Holmes, P., 2007, “A Simple Model for Clock-Actuated Legged Locomotion,” Regular Chaotic Dyn., 12(5), pp. 502–520. [CrossRef]
Spence, A., Revzen, S., Seipel, J., Mullens, C., and Full, R., 2010, “Insects Running on Elastic Surfaces,” J. Exp. Biol., 213(11), pp. 1907–1920. [CrossRef] [PubMed]
Maus, H., Rummel, J., and Seyfarth, A., 2008, “Stable Upright Walking and Running Using a Simple Pendulum Based Control Scheme,” International Conference of Climbing and Walking Robots 2008, pp. 623–629. [CrossRef]
Shen, Z. H., and Seipel, J. E., 2012, “A Fundamental Mechanism of Legged Locomotion With Hip Torque and Leg Damping,” Bioinspiration Biomimetics, 7(4), p. 046010. [CrossRef] [PubMed]
Sreenath, K., Park, H., Poulakakis, I., and Grizzle, J., 2013, “Embedding Active Force Control Within the Compliant Hybrid Zero Dynamics to Achieve Stable, Fast Running on MABEL,” Int. J. Rob. Res., 32(3), pp. 324–345. [CrossRef]
Larson, P., and Seipel, J., 2012, “Analysis of a Spring-Loaded Inverted Pendulum Locomotion Model With Radial Forcing,” ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, pp. 877–883.
Ankaralı, M., and Saranli, U., 2010, “Stride-to-Stride Energy Regulation for Robust Self-Stability of a Torque-Actuated Dissipative Spring–Mass Hopper,” Chaos, 20(3), p. 033121. [CrossRef] [PubMed]
Ghigliazza, R., Altendorfer, R., Holmes, P., and Koditschek, D., 2003, “A Simply Stabilized Running Model,” SIAM J. Appl. Dyn. Syst., 2(2), pp. 187–218. [CrossRef]
Arslan, O., Ankaralı, M., and Morgül, O., 2010, “Approximate Analytic Solutions to Non-Symmetric Stance Trajectories of the Passive Spring-Loaded Inverted Pendulum With Damping,” Nonlinear Dyn., 62(4), pp. 729–742. [CrossRef]
Geyer, H., Seyfarth, A., and Blickhan, R., 2005, “Spring–Mass Running: Simple Approximate Solution and Application to Gait Stability,” J. Theor. Biol., 232(3), pp. 315–328. [CrossRef] [PubMed]
Schwind, W., and Koditschek, D., 2000, “Approximating the Stance Map of a 2-DOF Monoped Runner,” J. Nonlinear Sci., 10(5), pp. 533–568. [CrossRef]
Robilliard, J., and Wilson, A., 2005, “Prediction of Kinetics and Kinematics of Running Animals Using an Analytical Approximation to the Planar Spring–Mass System,” J. Exp. Biol., 208(Pt. 23), pp. 4377–4389. [CrossRef] [PubMed]
Arslan, O., Saranli, U., and Morgül, O., 2009, “An Approximate Stance Map of the Spring Mass Hopper With Gravity Correction for Nonsymmetric Locomotions,” IEEE International Conference on Robotics and Automation, ICRA '09, Kobe, Japan, May 12–17, pp. 2388–2393. [CrossRef]
Altendorfer, R., Koditschek, D., and Holmes, P., 2004, “Stability Analysis of Legged Locomotion Models by Symmetry-Factored Return Maps,” Int. J. Rob. Res., 23, pp. 10–11. [CrossRef]
Pedotti, A., 1977, “A Study of Motor Coordination and Neuromuscular Activities in Human Locomotion,” Biol. Cybern., 26(1), pp. 53–62. [CrossRef] [PubMed]
Lulic, T., and Muftic, O., 2002, “Trajectory of the Human Body Mass Center During Walking at Different Speed,” International Design Conference, pp. 797–802.
Strogatz, S., 2001, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry: Engineering (Studies in Nonlinearity), Westview Press, Boulder, CO.

Figures

Grahic Jump Location
Fig. 1

The SLIP model. The parameters m, k, and β stand for body mass, leg stiffness, and landing angle, respectively. The CoM position during stance is characterized by leg length l and leg angle θ. Here, TD and LO stand for touchdown and liftoff, respectively.

Grahic Jump Location
Fig. 2

A comparison of a numerical solution of the exact stance equations and a closed-form solution of the approximate stance equations. (a) A comparison when both the numerical and closed-form solutions have the same fixed parameters: (k˜,g˜, β) = (20, 0.66, 72 deg). The fixed point value δ* for both cases is 0.1392 and 0.0971, respectively. (b) A comparison between exact and approximated stance solutions. The parameter and fixed point value of the exact solution is the same as in panel (a) (k˜,g˜, β, δ*) = (20, 0.66, 72 deg, 0.1392), while (k˜,g˜, β, δ*) = (18.74, 0.69, 72 deg, 0.1047) for the approximated case.

Grahic Jump Location
Fig. 3

A comparison of the time required to compute a numerically integrated solution and an evaluation of a closed-form solution. (a) The time to compute 1000 stance solutions for both the exact SLIP and closed-form approximation cases. (b) The time to compute the state of the system at LO, 1000 different times. (a) Trajectory generation, approximation is 249 times faster and (b) and LO state prediction, approximation is 4106 times faster.

Grahic Jump Location
Fig. 4

Periodic locomotion solutions (fixed points of the periodic stride map) versus dimensionless stiffness for the (a) piecewise-linear SLIP model and (b) numerically integrated exact SLIP model. Solid lines are partially asymptotically stable fixed point values, and dashed lines are unstable fixed point values. The leg landing angle β and the dimensionless gravity g˜ are fixed to be 72 deg and 0.46, respectively. (a) Approximate and (b) exact.

Grahic Jump Location
Fig. 5

Families of fixed point solutions as the dimensionless gravity is changed. For (a) the piecewise-linear SLIP model and (b) the exact SLIP model. Here, the leg landing angle is β = 72 deg. Solid lines are partially asymptotically stable fixed points, and dashed lines are unstable fixed points. (a) Approximate and (b) exact.

Grahic Jump Location
Fig. 6

Families of fixed point solutions as the leg landing angle β is changed. For (a) the piecewise-linear SLIP model and (b) the exact SLIP model. Here, the dimensionless gravity g˜ = 0.46. Solid lines are partially asymptotically stable fixed points, and dashed lines are unstable fixed points. (a) Approximate and (b) exact.

Grahic Jump Location
Fig. 7

For varying dimensionless gravity g˜, a plot of the eigenvalues approximating stability of locomotion for (a) the piecewise-linear SLIP model and (b) the exact SLIP model. Here, the leg landing angle is β = 72 deg. The shaded region stands for the stable region where the eigenvalue magnitude is less than one. (a) Approximate and (b) exact.

Grahic Jump Location
Fig. 8

For varying leg landing angle β, a plot of the eigenvalues approximating stability of locomotion for (a) the piecewise-linear SLIP model and (b) the exact SLIP model. Here, the dimensionless gravity g˜=0.46. The shaded region stands for the stable region where the eigenvalue magnitude is less than one. (a) Approximate and (b) exact.

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