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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.

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Figures

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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