Research Papers

A Detailed Look at the SLIP Model Dynamics: Bifurcations, Chaotic Behavior, and Fractal Basins of Attraction

[+] Author and Article Information
Petr Zaytsev

Robotics and Motion Laboratory ME,
University of Michigan,
Ann Arbor, MI 48109;
Institute for Nonlinear Mechanics,
University of Stuttgart,
Stuttgart 70569, Germany
e-mail: petr.zaytsev@inm.uni-stuttgart.de

Tom Cnops

Robotics and Motion Laboratory ME,
University of Michigan,
Ann Arbor, MI 48109
e-mail: tcnops@umich.edu

C. David Remy

Robotics and Motion Laboratory ME,
University of Michigan,
Ann Arbor, MI 48109;
Institute for Nonlinear Mechanics,
University of Stuttgart,
Stuttgart 70569, Germany
e-mail: david.remy@inm.uni-stuttgart.de

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 20, 2018; final manuscript received April 4, 2019; published online May 13, 2019. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 14(8), 081002 (May 13, 2019) (11 pages) Paper No: CND-18-1416; doi: 10.1115/1.4043453 History: Received September 20, 2018; Revised April 04, 2019

This paper provides a comprehensive numerical analysis of a simple 2D model of running, the spring-loaded inverted pendulum (SLIP). The model consists of a point-mass attached to a massless spring leg; the leg angle at touch-down is fixed during the motion. We employ numerical continuation methods combined with extensive simulations to find all periodic motions of this model, determine their stability, and compute the basins of attraction of the stable solutions. The result is a detailed and complete analysis of all possible SLIP model behavior, which expands upon and unifies a range of prior studies. In particular, we demonstrate and explain the following effects: (i) saddle-node bifurcations, which lead to two distinct solution families for a range of energies and touch-down angles; (ii) period-doubling (PD) bifurcations which lead to chaotic behavior of the model; and (iii) fractal structures within the basins of attraction. In contrast to prior work, these effects are found in a single model with a single set of parameters while taking into account the full nonlinear dynamics of the SLIP model.

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Grahic Jump Location
Fig. 1

The SLIP model has a point mass m at the hip and a massless elastic leg with stiffness k and rest length 0. Only planar running forward is considered here in which the motion alternates between the flight and stance phases. The leg is assumed to instantaneously swing to a given angle α in the beginning of flight; α is fixed throughout the motion. A step is defined to start at flight apex and end at the next apex. For a given α and energy level E, the state at apex is described by the height ya; the apex speed x˙a is a function of ya and E.

Grahic Jump Location
Fig. 2

A continuation algorithm is used to iteratively find all periodic motions for a given angle-of-attack α. These are all combinations of the initial apex height y0 and energy E that satisfy the periodicity condition (8). At each iteration, given some known solutions, a new solution is found numerically such that: (i) constraint (8) is satisfied; (ii) the distance to the previous solution is d; (iii) the new solution is in the direction “away” from the second last solution. The linear extrapolation of the last two solutions is used as the initial guess for solving this problem.

Grahic Jump Location
Fig. 3

Period-1 solutions of the SLIP model. The surface in (a) represents all period-1 solutions defined by the apex height y0, energy E, and angle-of-attack α; only solutions with E  < 40 are shown. The horizontal curves along the surface show solutions for several different angles α. Solid lines and lighter area in the middle of the surface indicate stable solutions and dashed lines and darker part of the surface unstable solutions. Vertical projections of the fixed-angle curves onto the height-energy plane are shown together in (b). For small angles α < 0.32, solutions are unique for each energy level and the branches are bounded from above by the energy of a fully compressed spring Esp. For medium angles, 0.32 < α < 1.11, two distinct solutions exist for a range of energies: “H-solutions” (high) with larger apex heights and “F-solutions” (fast) with lower heights but larger speeds. H-solution branches are bounded from above by Esp, while F-branches extend to arbitrarily large energies; the H- and F-branches connect at saddle-node bifurcation points (H, I, and J). For large angles α > 1.11, solutions are unique for each energy; the branches are bounded by Esp from below and unbounded from above.

Grahic Jump Location
Fig. 4

Comparison of H- and F-solutions. The figure displays the dynamics of different energy components during one step for two different period-1 solutions with the angle-of-attack α = 0.4 and energy E  = 5: an F-solution with the apex height y0*=0.92 and an H-solution with y0*=3.83 (see Fig. 3(b)). The H-solution has a pronounced anti-phase interchange between the CoM kinetic energy (solid line) and the spring elastic energy (dashed line) during stance, and a long flight phase. The F-solution's stance dynamics are dominated by the kinetic energy, the CoM moves nearly horizontally, and the flight phase is short. H- and F-solutions become more similar at smaller energy levels closer to the turning points, such as point H in Fig. 3(b).

Grahic Jump Location
Fig. 5

Period-2 solutions for selected angles-of-attack. Each thick curve shows all solutions (the apex height y0 and energy E) of the model with a period of 2 steps and a given angle α. The upper and lower parts of each curve correspond to two subsequent apex states along the period-2 trajectories. Period-2 solutions exist for 0.32 < α < 1.17 and originate from PD bifurcations along the corresponding period-1 solution curves (shown by thinner lines). For angles α < 1.13, period-2 branches terminate at solution points where the apex height is equal to the touch-down height ytd=cos α. Most period-2 solutions are unstable (dashed lines); stable solutions (solid lines) are found only near α = 1.11 and full-compression energy Esp where chaotic behavior emerges.

Grahic Jump Location
Fig. 6

PD cascades and chaos. The figure on the left shows periodic solutions of the model for the angle-of-attack α = 1.11 and a small range of energy levels just above the energy of full-compression Esp. Shown are period-1 (thick, lightest color), period-2, period-4, and period-8 (thin, darkest color) solution branches; higher period solutions also exist but are not shown here. Every higher period branch emerges as a PD bifurcation from a lower period solution (e.g., period-2 solutions emerge at the bifurcation point L). Stable solutions are shown by solid lines and unstable by dashed lines. The zoom-in on the right explores the boxed area on the left and shows the traces of nonfailed trajectories, disregarding transient behavior. For larger energies, these are along the stable periodic solutions. For energies within the range 12.627  < E  <12.635, the motions cease to be periodic and the model exhibits chaotic behavior.

Grahic Jump Location
Fig. 7

Basins of attraction and dynamics with the angle-of-attack α = 0.4. The figure illustrates behavior of the model for small energy levels for the angle α = 0.4 (see also Fig. 3(b)). (a) The thick black lines represent stable (solid lines) and unstable (dashed lines) periodic solutions. Blue colors outside the region in the middle show the apex heights y0 and energies E that lead to failure, with lighter blue indicating more steps before failure. From red points in the middle, the model converges to a stable period-1 F-solution, with darker red indicating more steps before approximately reaching the solution. Thus, the full red region is the set of all basins of attraction of the stable solutions for the given α. The zoom-in in the top-right shows in more detail the area where unstable period-2 solutions exist. The Poincaré-map plots at the bottom illustrate the behavior of the model inside the basins at different energy levels. (b) At lower energies, trajectories asymptotically converge to a stable solution. (c) At higher energies, the Poincaré map has a local minimum below the diagonal. This results in oscillations of the converging motions around the stable solution and also in a trajectory that reaches the stable solution exactly. Such trajectories for different energies are shown in (a) by the sequence of green lines converging at the stable point at E = 2.37; tracked backward in time, they asymptotically approach the unstable H-solution at larger heights. (d) For a range of basins at the highest energies, the Poincaré map crosses below the touch-down height y1=cos α, thus causing some trajectories to fail. Integrated backward in time, such trajectories create gaps in the basins of attraction that become arbitrarily fine near the unstable H-solution. This results in a fractal-like structure of the basins of attraction. The plot (e) summarizes possible behavior of the model and the corresponding Poincaré-map graphs.

Grahic Jump Location
Fig. 8

Basins of attraction and Poincaré maps for selected angles-of-attack. This figure is analogous to Fig. 7 and shows basins of attraction, model dynamics, and Poincaré maps for three different angles α. ((a), (d)) For α = 0.7, the set of basins and model dynamics are similar to those in Fig. 7 for α = 0.4. The large-height H-solutions do not exist for energies above Esp = 12.5. ((b), (e)) For α = 0.9, Poincaré maps do not cross below the touch-down height y1=cos α for energies E < Esp and so the gaps and the fractal structures in the basins disappear. The sequence of preimages of the stable solutions (shown by the green line through the basins) remains inside the set of basins. ((c), (f)) For α = 1.15, large-height H-solutions cease to exist. All nonfailed trajectories are trapped between the “loop” of unstable period-2 solutions and converge to stable period-1 solutions oscillating around them.

Grahic Jump Location
Fig. 9

Periodic solutions of the constrained model. The figure shows effects of actuator limitations on periodic motions of the model. The surface is a copy of that in Fig. 3(a) at a slightly different view angle; it shows all period-1 solutions of the unconstrained model, with the lighter area corresponding to stable solutions. Along the surface are level lines of four constraint functions (12): minimum leg length in stance (dashed blue lines), maximum leg compression rate (solid red), maximum swing leg acceleration (dash-dotted green), and maximum leg angle (dotted black). The small dark-red area on the surface near the origin shows feasible solutions for a sample set of constraint bounds: all such solutions are at small energy levels and are unstable H-solutions.



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