A Simple Stabilizing Control for Sagittal Plane Locomotion

[+] Author and Article Information
John Schmitt

Department of Mechanical Engineering,  Oregon State University, Corvallis, OR 97331schmitjo@engr.orst.edu

J. Comput. Nonlinear Dynam 1(4), 348-357 (May 10, 2006) (10 pages) doi:10.1115/1.2338650 History: Received November 15, 2005; Revised May 10, 2006

The spring loaded inverted pendulum model has been shown to accurately model sagittal plane locomotion for a variety of legged animals and has been used as a target for control for higher dimensional robotic implementations. Tuned appropriately, the model exhibits passively stable periodic gaits using either fixed leg touch-down angle or swing-leg retraction leg touch-down protocols. In this work, we examine the performance of the model when model parameters are set to values characteristic of an insect, in particular the cockroach Blaberus discoidalis. While body motions and forces exhibited during a stride are shown to compare well with those observed experimentally, almost all of the resulting periodic gaits are unstable. We therefore develop and analyze a simple adaptive control scheme that improves periodic gait stability properties. Since it is unlikely that neural reflexes can act quickly enough during a stride to effect control, control is applied once per stance phase through appropriate choice of the leg touch-down angle. The control law developed is novel since it achieves gait stabilization solely through a judicious combination of leg lift-off and touch-down angles, instead of utilizing all of the system positions and velocities in full-state feedback control. Implementing the control law improves the stability properties of a large number of periodic gaits and enables movement between stable periodic gaits by changing a single parameter.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

SLIP point mass model formulation, illustrating the coordinate system and relevant quantities through a full stride. Each leg, represented by an elastic spring, is assumed massless for this model. Relevant quantities include: velocity (v), heading angle defined from the horizontal axis (δ), leg angle (β), leg length (ζ), and angle of the leg with respect to the vertical axis (ψ). Superscripts TD and LO denote touch-down and lift-off events, while subscripts indicate the particular stance phase. Fore-aft motion occurs in the positive y direction.

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Figure 2

Fore-aft and vertical velocity and force profiles for a single left or right stride (stance plus flight phase) of the SLIP model, with vnTD=0.24m∕s, δnTD=0.0757rad, βnTD=1.2rad, k̃=31.25 and g̃=2.55. Plus signs in the figures denote the start and end of the stance phase.

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Figure 3

Point mass SLIP gait families (vnTD,δnTD) for βnTD=1.1−1.3 with the fixed angle leg reset protocol. Gait stability is indicated by dashed and solid lines for unstable and stable gaits, respectively. Model parameters used in calculating the gaits are similar to those of Blaberus discoidalis, as described in the text. Gait families for increasing βnTD are obtained as one moves from right to left in the plot, and the thicker dashed line indicates the gait family for the nominal leg touch-down angle, βnTD=1.2.

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Figure 4

(a) SLIP periodic gait family and stability for β=1.2, with other parameters set to those of Blaberus discoidalis, as described in the text. (b) Gait family eigenvalues for the controlled system with c=0.7 (dotted), c=0.5 (dashed), c=0.3 (chain-dotted), c=0.1 (solid). (c) Eigenvalue variation for analytical (dashed) and numerical results (solid) as c varies from 0 to 1 for the periodic gait (∗) identified in (a). (d) Comparison of analytical (dashed) and numerical (solid) eigenvalues for the gait family shown in (a) and c=0.1.

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Figure 5

Re-stabilization of the SLIP model to a periodic gait (vnTD,δnTD)=(0.2797,0.25) in response to a perturbation of 0.05 in δnTD for c=0.4 and βdesTD=1.2, with parameters set to values similar to the cockroach Blaberus discoidalis. The periodic orbit corresponds to fore-aft and vertical velocities of 0.271 and −0.0692m∕s respectively. Plus signs (+) denote the start and end of the stance phase in the first three panels while stars (∗) denote the start of the stance phase in the final panel.

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Figure 6

Periodic gaits and stability for the controlled SLIP model as a function of c, for a constant energy surface corresponding to a touch-down velocity vnTD=0.25m∕s and leg touch-down angle βnTD=βdesTD=1.2rad. All other parameters are similar to those of Blaberus discoidalis, as described in the text. (a) Periodic gaits for c=0.0−1.0, with dotted and solid lines denoting unstable and stable periodic gaits, respectively. (b) Eigenvalue variation for the periodic gaits of panel (a). The horizontal plane indicates ∣λ∣=1.

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Figure 7

Controlled SLIP gait families for βnTD=1.1−1.3 for (a) c=0.7, (b) c=0.5, (c) c=0.3 and (d) c=0.1. Gait stability is indicated by dotted and solid lines for unstable and stable gaits, respectively. Model parameters used in calculating the gaits are similar to those of Blaberus discoidalis, as described in the text.




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