Technical Brief

An Inverse Dynamics Optimization Formulation With Recursive B-Spline Derivatives and Partition of Unity Contacts: Demonstration Using Two-Dimensional Musculoskeletal Arm and Gait

[+] Author and Article Information
Yujiang Xiang

School of Mechanical and Aerospace Engineering,
Oklahoma State University,
Stillwater, OK 74078
e-mail: yujiang.xiang@okstate.edu

1Corresponding author.

Manuscript received September 2, 2018; final manuscript received December 19, 2018; published online January 31, 2019. Assoc. Editor: Guy M. Genin.

J Biomech Eng 141(3), 034503 (Jan 31, 2019) (6 pages) Paper No: BIO-18-1395; doi: 10.1115/1.4042436 History: Received September 02, 2018; Revised December 19, 2018

In this study, an inverse dynamics optimization formulation and solution procedure is developed for musculoskeletal simulations. The proposed method has three main features: high order recursive B-spline interpolation, partition of unity, and inverse dynamics formulation. First, joint angle and muscle force profiles are represented by recursive B-splines. The formula for high order recursive B-spline derivatives is derived for state variables calculation. Second, partition of unity is used to handle the multicontact indeterminacy between human and environment during the motion. The global forces and moments are distributed to each contacting point through the corresponding partition ratio. Third, joint torques are inversely calculated from equations of motion (EOM) based on state variables and contacts to avoid numerical integration of EOM. Therefore, the design variables for the optimization problem are joint angle control points, muscle force control points, knot vector, and partition ratios for contacting points. The sum of muscle stress/activity squared is minimized as the cost function. The constraints are imposed for human physical constraints and task-based constraints. The proposed formulation is demonstrated by simulating a trajectory planning problem of a planar musculoskeletal arm with six muscles. In addition, the gait motion of a two-dimensional musculoskeletal model with sixteen muscles is also optimized by using the approach developed in this paper. The gait optimal solution is obtained in about 1 min central processing unit (CPU) time. The predicted kinematics, kinetics, and muscle forces have general trends that are similar to those reported in the literature.

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Fig. 1

Two-dimensional musculoskeletal arm (a) and gait (b) models

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Fig. 2

Arm joint angle and torque profiles

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Fig. 3

Arm muscle-tendon forces

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Fig. 4

Arm motion trajectory (a) and EOM constraint (b)

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Fig. 5

Gait joint angle, torque, and GRF profiles

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Fig. 6

Gait muscle-tendon activities (fi(t)/fimax [23])

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Fig. 7

Gait motion trajectory



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