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Research Papers

# 3D Bipedal Model With Holonomic Constraints for the Decoupled Optimal Controller Design of the Biomechanical Sit-to-Stand Maneuver

[+] Author and Article Information
Asif Mughal

University of Arkansas at Little Rock, 2801 S. University Avenue, ETAS 383B, Little Rock, AR 72204asifmahmoodmughal@gmail.com

Kamran Iqbal

University of Arkansas at Little Rock, 2801 S. University Avenue, ETAS 383B, Little Rock, AR 72204kxiqbal@ualr.edu

J Biomech Eng 132(4), 041010 (Mar 19, 2010) (9 pages) doi:10.1115/1.4000992 History: Received May 23, 2008; Revised September 28, 2009; Posted January 13, 2010; Published March 19, 2010; Online March 19, 2010

## Abstract

Human voluntary movements are complex physical phenomena due to the complex control mechanism for coordination of limbs in the presence of physiological constraints. In this study, we propose a nonlinear human bipedal model with thirteen generalized coordinates to model sit-to-stand (STS) transfer. The model has three position based holonomic constraints and at the first stage, we decouple the translational variables (constrained system) from rotational variables (unconstrained systems). The unconstrained rotational degrees consist of seven sagittal and three frontal plane angles, which are controlled through their respective joint torques. We further decouple these angles in sagittal and frontal plane systems for a better control strategy. In this scheme, there are three decoupled controllers working together to stabilize the nonlinear model for a STS maneuver while satisfying the holonomic constraints. We adopt $H∞$ and $H2$ controller designs for feedback torques in sagittal and frontal planes, respectively, and provide simulation results to show the improvement in the angular profiles. We further adopt this modeling strategy to study and analyze the neuromuscular disorders by decoupling healthy and neurodeficient extremities. Our study indicates that the decoupling of the bipedal model improves the controllability of the system and produces better angular profiles for a bipedal STS maneuver. This modeling scheme is useful for analysis of neuromuscular disorders and other relevant physiological motor control models.

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## Figures

Figure 1

Bipedal rigid body model, with sagittal plane angles θ1–θ6 and θ9, as well as frontal plane angles φ7–φ8 and φ10. The sagittal plane angles are shown along the x-axis and frontal plane angles are shown along the z-axis. A small arrow in the angle points toward its placement.

Figure 2

Bipedal model with a weld joint in the right foot and a 6DOF joint in the left foot with different connection trees of (a) rotational and (b) translational state variables to the ground

Figure 3

Simulation block for the bipedal STS movement

Figure 4

Movement profiles joint angles in radians for 4 s (solid lines) and their reference trajectories (dashed lines). R denotes right, L denotes left, A denotes the ankle angle, K denotes the knee angle, P denotes the pelvic-hip joint angle, H denotes the HAT-pelvic joint angle, S denotes the sagittal plane, and F denotes the frontal plane.

Figure 5

Joint torques (N m) of the bipedal model for STS. R denotes right, L denotes left, A denotes the ankle angle, K denotes the knee angle, P denotes the pelvic-hip joint angle, H denotes the HAT-pelvic joint angle, S denotes the sagittal plane, and F denotes the frontal plane.

Figure 6

Left foot position (mm) in the x, y, and z directions (solid lines) with reference trajectories (dashed lines) and constraint (m) violations during movement

Figure 7

Bipedal models with angles of neurodeficient extremities are shown in circles for sagittal and frontal planes

Figure 8

Angular profiles (radians) with decoupling of healthy and neurodeficient systems

Figure 9

Joint torques (N m) for decoupled healthy and neurodeficient systems regulated through H∞ and LQR designs, respectively

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