Research Papers

A Nonlinear Leg Damping Model for the Prediction of Running Forces and Stability

[+] Author and Article Information
Ian Abraham

Department of Mechanical Engineering,
Rutgers University,
Piscataway Township, NJ 08854
e-mail: ianabraham21@gmail.com

ZhuoHua Shen

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: Shen38@purdue.edu

Justin Seipel

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: jseipel@purdue.edu

Manuscript received March 8, 2014; final manuscript received October 2, 2014; published online April 2, 2015. Assoc. Editor: José L. Escalona.

J. Comput. Nonlinear Dynam 10(5), 051008 (Apr 02, 2015) (8 pages) Paper No: CND-14-1068; doi: 10.1115/1.4028751 History: Received March 08, 2014

Despite the neuromechanical complexity underlying animal locomotion, the steady-state center-of-mass motions and ground reaction forces of animal running can be predicted by simple spring-mass models such as the canonical spring-loaded inverted pendulum (SLIP) model. Such SLIP models have been useful for the fields of biomechanics and robotics in part because ground reaction forces are commonly measured and readily available for comparing with model predictions. To better predict the stability of running, beyond the canonical conservative SLIP model, more recent extensions have been proposed and investigated with hip actuation and linear leg damping (e.g., hip-actuated SLIP). So far, these attempts have gained improved prediction of the stability of locomotion but have led to a loss of the ability to accurately predict ground reaction forces. Unfortunately, the linear damping utilized in current models leads to an unrealistic prediction of damping force and ground reaction force with a large nonzero magnitude at touchdown (TD). Here, we develop a leg damping model that is bilinear in leg length and velocity in order to yield improved damping force and ground reaction force prediction. We compare the running ground reaction forces, small and large perturbation stability, parameter sensitivity, and energetic cost resulting from both the linear and bilinear damping models. We found that bilinear damping helps to produce more realistic, smooth vertical ground reaction forces, thus fixing the current problem with the linear damping model. Despite large changes in the damping force and power loss profile during the stance phase, the overall dynamics and energetics on a stride-to-stride basis of the two models are largely the same, implying that the integrated effect of damping over a stride is what matters most to the stability and energetics of running. Overall, this new model, an actuated SLIP model with bilinear damping, can provide significantly improved prediction of ground reaction forces as well as stability and energetics of locomotion.

Copyright © 2015 by ASME
Topics: Stability , Damping
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Grahic Jump Location
Fig. 1

The actuated-SLIP model. A stride includes stance from TD to LO, and then flight until the next TD. CoM velocity is measured at every TD event.

Grahic Jump Location
Fig. 2

A free body diagram of the (a) main body during stance, (b) leg during stance, (c) main body in flight

Grahic Jump Location
Fig. 5

The vertical ground reaction force of human runners [29] is compared with that of the linear and bilinear cases of damping in the hip-SLIP model. The following human-representative parameters are used for the comparison: τ˜ = 0.4, β = 67 deg, krel = 20, v˜n = 1.4344. For the linear damping case, ζ = 0.4657 and δ = 0.1446. For the bilinear damping case, ζ = 10.32 and δ = 0.1266. In this figure, all three forces are normalized by body weight.

Grahic Jump Location
Fig. 8

Here, we show the speed of all stable periodic hip-SLIP locomotion solutions found for varying levels of torque, for the cases (a) linear leg damping and (b) bilinear leg damping. Fixed point solutions are searched for within the range of δn from −30 deg to the fixed touch down angle β and v˜n from 0 to 5.

Grahic Jump Location
Fig. 9

Here, we show the maximum eigenvalue magnitude of hip-SLIP with (a) linear damping and (b) bilinear damping as relative stiffness krel changes. In the top row, the parameters that are held constant are β = 60 deg and v˜n = 1.4344. Each branch for plots (a) and (b) represents distinct incremental τ˜ values of 0.5, 0.8, and 1.2. In the row below, the parameters that are held constant are τ˜ = 1 and v˜n = 1.4344. Each branch for plots (a) and (b) represents distinct incremental β values of 60, 65, 70 deg.

Grahic Jump Location
Fig. 6

Here, we compare how bilinear damping changes the maximum eigenvalue of hip-SLIP for each fixed point solution as hip torque is varied from 0 (SLIP) all the way up to τ = 2, though values above τ = 0.5 are not likely to be commonly seen in animal locomotion. We compare the eigenvalues of hip-SLIP with bilinear damping to that of hip-SLIP with linear damping. We fix the parameters v˜n = 1.4344 and krel = 12.7004, while varying τ˜ and searching for fixed point solutions and solving for the eigenvalues via small perturbation analysis.

Grahic Jump Location
Fig. 7

Here, we look at the effects that bilinear damping has on the basin of attraction for hip-SLIP for four different levels of hip actuation: (a) τ˜ = 0 (equivalent to the canonical SLIP model), (b) τ˜ = 0.10, (c) τ˜ = 0.40, and (d) τ˜ = 1.0. The constant parameters for the (top row) hip-SLIP with linear damping are krel = 12.7004, ζ = 0.3678, and β = 60 deg. The parameters for the (bottom row) hip-SLIP with bilinear damping are krel = 12.7004, ζ = 4.817, and β = 60 deg.

Grahic Jump Location
Fig. 3

Here, we show the trajectories of hip-SLIP with linear damping and bilinear damping. The parameters used for hip-SLIP with bilinear damping are τ˜ = 0.40, krel = 20, v˜n = 1.4344, δ = 0.1266, β = 67 deg, and ζ = 10.32 and for hip-SLIP with linear damping, τ˜ = 0.40, krel = 20, v˜n = 1.4344, δ = 0.1449, β = 67 deg, and ζ = 0.4657. The open circles superimposed on the trajectories indicate TD and LO events.

Grahic Jump Location
Fig. 4

Here, we show leg length, leg velocity, damping force, and the power lost through damping for hip-SLIP with both linear and bilinear damping. The parameters used for hip-SLIP with bilinear damping are τ˜ = 0.40, krel = 20, v˜n = 1.4344, δ = 0.1266, β = 67 deg, and ζ = 10.32 and for hip-SLIP with linear damping, τ˜ = 0.40, krel = 20, v˜n = 1.4344, δ = 0.1449, β = 67 deg, and ζ = 0.4657.

Grahic Jump Location
Fig. 10

Here, we determine how bilinear damping effects the cost of transport of hip-SLIP. For hip-SLIP, we fixed krel = 20, β = 1.1694, and varied τ˜. The region in gray shows the biologically relevant region for human locomotion.




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