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Research Papers

Stability of Human Balance With Reflex Delays Using Galerkin Approximations

[+] Author and Article Information
Zaid Ahsan, Akash Subudhi

Department of Mechanical
and Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Kandi, Sangareddy 502285,
Telangana, India

Thomas K. Uchida

Department of Bioengineering,
James H. Clark Center,
Stanford University,
318 Campus Drive,
Stanford, CA 94305

C. P. Vyasarayani

Department of Mechanical
and Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Kandi, Sangareddy 502285,
Telangana, India
e-mail: vcprakash@iith.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 28, 2015; final manuscript received October 27, 2015; published online December 10, 2015. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 11(4), 041009 (Dec 10, 2015) (9 pages) Paper No: CND-15-1080; doi: 10.1115/1.4031979 History: Received March 28, 2015; Revised October 27, 2015

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.

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Figures

Grahic Jump Location
Fig. 1

Single-pendulum model of a human during stance (sagittal plane). The body is represented by an inverted pendulum of mass m and length ℓ with orientation θ relative to vertical. Passive muscle torques generated at the ankle are modeled as a torsional spring of stiffness kt and a torsional dashpot with damping coefficient bt.

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Fig. 2

Stability regions for the single-pendulum human stance model (case 1) using PD (open circles) and PDA (filled circles) controllers, with (a) N = 2, (b) N = 3, and (c) N = 5 terms in the series solution. The shaded areas are the analytical stability regions reported by Insperger et al. [12]. The approximate Galerkin method converges to the analytical solution at N = 5. The parameters used in the simulation are Ka=54 N m s2 rad−1 and those listed for case 1 in Table 1.

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Fig. 3

Stability regions for the single-pendulum human stancemodel (case 2) using PD (open circles) and PDA (filledcircles) controllers. The shaded areas are the analytical stability regions reported by Insperger et al. [12]. The approximate Galerkin method converges to the analytical solution at N = 5. The parameters used in the simulation are Ka=57.51 N m s2 rad−1 and those listed for case 2 in Table 1.

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Fig. 4

Double-pendulum model of a human during stance (sagittal plane). The lumped thigh and shank segment have mass m1, length 2ℓ1, and orientation θ1 relative to vertical; the knee is assumed to remain locked. The head, arms, and trunk are represented by a pendulum of mass m2 and length 2ℓ2 with orientation θ2 relative to vertical. Passive muscle torques generated at the ankle and hip are modeled as torsional springs of stiffness k1 and k2, respectively, and torsional dashpots with damping coefficients b1 and b2.

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Fig. 5

Stability regions obtained for the double-pendulum human stance model (22) using Galerkin and analytical approaches. The parameter values are (a) kp2=1813, kd2=300,ka1=ka2=10, (b) kp1=1813, kd1=300, ka1=ka2=10; and (c) kd1=kd2=300, ka1=ka2=10. Dots indicate stable points obtained using the Galerkin method and the solid line indicates the analytical stability boundary.

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Fig. 6

Stability regions obtained for the double-pendulum human stance model (22) using Galerkin and analytical approaches and the following parameters: (a) kd1 = kd2 = 300; (b) kp2 = 1813, kd2 = 300; (c) kp1 = 1813, kd2 = 300; (d) kp1 = 1813, kd1 = 300; and (e) kp1 = 1813, kp2 = 2000. The acceleration gains are ka1 = 5 and ka2 = 10 in all cases. Stable points obtained using the Galerkin method are indicated with dots (PDA) and circles (PD); the analytical stability boundaries are indicated with dashed (PDA) and solid (PD) lines.

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Fig. 7

Stability regions obtained for the double-pendulum human stance model (22) using Galerkin and analytical approaches and the following parameters: (a) kd1 = 300, kd2 = 150; (b) kp2 = 1813, kd2 = 300; (c) kp1 = 1813, kd2 = 300; (d) kp1 = 1813, kd1 = 300; and (e) kp1 = 1813, kp2 = 2000. The acceleration gains are ka1 = 10 and ka2 = 5 in all cases. Stable points obtained using the Galerkin method are indicated with dots (PDA) and circles (PD); the analytical stability boundaries are indicated with dashed (PDA) and solid (PD) lines.

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Fig. 8

Stability regions obtained for the double-pendulum human stance model (22) using Galerkin and analytical approaches and the following position (τp), velocity (τv), and acceleration (τa) time delays: (a) τp = 0.05, τv = 0.01, τa = 0.09; (b) τp = 0.09, τv = 0.01, τa = 0.05 ; and (c) τp = 0.09, τv = 0.05, τa = 0.01. The remaining parameters are the same as those used to generate Fig. 7(d). Stable points obtained using the Galerkin method are indicated with dots (PDA) and circles (PD); the analytical stability boundaries are indicated with dashed (PDA) and solid (PD) lines.

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Fig. 9

Stability margins obtained for the double-pendulum human stance model using Galerkin (shaded) and analytical (line) approaches with the following parameters: (a)kp1 = 1813, kd1 = 300, ka1 = 10, ka2 = 5, τp = 0.05, τv = 0.09, τa = 0.01 and (b) kp2 = 1813,kd1 = 300, ka1 = 10, ka2 = 5, τp = τv = τa = 0.05. Arrows indicate the direction of increasing ω.

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Fig. 10

Possible locations of roots of the transcendental equation (29) for a given value of ω

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Fig. 11

Stability regions obtained for the double-pendulum human stance model using the Galerkin method and the following parameter values: (a) kp1 = 1813, kd1 = 300, kd2 = 200,ka1 = 5, ka2 = 10 and (b) kp1 = 1813, kp2 = 2000, kd1 = 300, ka1 = 5, ka2 = 10. Stability regions are shown for PDA (shaded) and PD (dots) controllers.

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