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.
Skip Nav Destination
Article navigation
January 2016
Research-Article
A Piecewise-Linear Approximation of the Canonical Spring-Loaded Inverted Pendulum Model of Legged Locomotion
Zhuohua Shen,
Zhuohua Shen
School of Mechanical Engineering,
e-mail: shen38@purdue.edu
Purdue University
,West Lafayette, IN 47906
e-mail: shen38@purdue.edu
Search for other works by this author on:
Justin Seipel
Justin Seipel
School of Mechanical Engineering,
e-mail: jseipel@purdue.edu
Purdue University
,West Lafayette, IN 47906
e-mail: jseipel@purdue.edu
Search for other works by this author on:
Zhuohua Shen
School of Mechanical Engineering,
e-mail: shen38@purdue.edu
Purdue University
,West Lafayette, IN 47906
e-mail: shen38@purdue.edu
Justin Seipel
School of Mechanical Engineering,
e-mail: jseipel@purdue.edu
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. Jan 2016, 11(1): 011007 (9 pages)
Published Online: January 1, 2016
Article history
Received:
August 15, 2014
Revision Received:
January 15, 2015
Online:
June 30, 2015
Citation
Shen, Z., and Seipel, J. (January 1, 2016). "A Piecewise-Linear Approximation of the Canonical Spring-Loaded Inverted Pendulum Model of Legged Locomotion." ASME. J. Comput. Nonlinear Dynam. January 2016; 11(1): 011007. https://doi.org/10.1115/1.4029664
Download citation file:
Get Email Alerts
Cited By
Reduced-Order Modeling and Optimization of a Flapping-Wing Flight System
J. Comput. Nonlinear Dynam (February 2025)
Nonlinearity Measure for Nonlinear Dynamic Systems Using a Multi-Model Framework
J. Comput. Nonlinear Dynam
Generation of a Multi-wing Hyperchaotic System with a Line Equilibrium and its Control
J. Comput. Nonlinear Dynam
Bifurcation analysis and control of traffic flow model considering the impact of smart devices for drivers
J. Comput. Nonlinear Dynam
Related Articles
Global Analysis of Gravity Gradient Satellite's Pitch Motion in an Elliptic Orbit
J. Comput. Nonlinear Dynam (November,2015)
A Detailed Look at the SLIP Model Dynamics: Bifurcations, Chaotic Behavior, and Fractal Basins of Attraction
J. Comput. Nonlinear Dynam (August,2019)
Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems
J. Comput. Nonlinear Dynam (July,2016)
Search for Initial Conditions for Sustained Hopping of Passive Springy-Leg Offset-Mass Hopping Robot
J. Dyn. Sys., Meas., Control (July,2007)
Related Proceedings Papers
Related Chapters
Dynamic Behavior in a Singular Delayed Bioeconomic Model
International Conference on Instrumentation, Measurement, Circuits and Systems (ICIMCS 2011)
Dynamic Simulations to Become Expert in Order to Set Fuzzy Rules in Real Systems
International Conference on Advanced Computer Theory and Engineering, 4th (ICACTE 2011)
Stable Analysis on Speed Adaptive Observer in Low Speed Operation
International Conference on Instrumentation, Measurement, Circuits and Systems (ICIMCS 2011)