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Research Papers

Nonlinear Optimal Control of Planar Musculoskeletal Arm Model With Minimum Muscles Stress Criterion

[+] Author and Article Information
Mojtaba Sharifi

Department of Mechanical Engineering,
Sharif University of Technology,
Azadi Avenue,
Tehran 11155-9567, Iran
e-mail: mojtaba_sharifi@mech.sharif.edu

Hassan Salarieh

Department of Mechanical Engineering,
Sharif University of Technology,
Azadi Avenue,
Tehran 11155-9567, Iran
e-mail: salarieh@sharif.edu

Saeed Behzadipour

Department of Mechanical Engineering,
Sharif University of Technology,
Azadi Avenue,
Tehran 11155-9567, Iran
e-mail: behzadipour@sharif.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 9, 2015; final manuscript received July 28, 2016; published online September 9, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 12(1), 011014 (Sep 09, 2016) (10 pages) Paper No: CND-15-1126; doi: 10.1115/1.4034392 History: Received May 09, 2015; Revised July 28, 2016

In this paper, the optimal performance of a planar humanlike musculoskeletal arm is investigated during reaching movements employing an optimal control policy. The initial and final states (position and velocity) are the only known data of the response trajectory. Two biomechanical objective functions are taken into account to be minimized as the central nervous system (CNS) strategy: (1) a quadratic function of muscle stresses (or forces), (2) total time of movement plus a quadratic function of muscle stresses. A two-degress of freedom (DOF) nonlinear musculoskeletal arm model (for planar movements) with six muscle actuators and four state variables is used in order to evaluate the proposed optimal policy, while the constraints of the arm motion and muscle forces are considered mathematically. The nonlinear differential equations of this optimal control problem with the first objective function are solved using the method of variation of extremals (VE). For the second objective function, a modified version of the VE method is employed. Accordingly, the optimal total time of the motion is predicted via the second objective function in addition to the optimal trajectory and forces that are also predicted using the first objective function. The influence of the motion time duration on the optimal trajectory is shown and discussed. Finally, the obtained optimal trajectories are compared to the experimental trajectories of the human arm movements.

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Figures

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Fig. 1

Planar musculoskeletal arm model with two DOFs: θ1 shoulder angle and θ1 elbow angle, and six muscles: q1 anterior deltoid, q2 posterior deltoid, q3 brachialis, q4 lateral triceps, q5 long head of biceps, and q6 long head of triceps

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Fig. 2

The optimal trajectories of x1=θ1, x2=θ2, x3=θ˙1, and x4=θ˙2 employing the minimum muscle stresses (first criterion) and tf=2  s

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Fig. 3

The optimal muscle forces using the minimum muscle stresses (first criterion) and tf=2  s

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Fig. 4

The shoulder and elbow torques produced by the optimal muscle forces using the first criterion (minimum muscle stresses), with tf=2  s

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Fig. 6

The optimal muscle forces using the minimum time plus muscle stresses (second criterion) with a cost factor of ρ=7600 for the time in Eq. (24)

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Fig. 5

The optimal trajectories of x1=θ1, x2=θ2, x3=θ˙1, and x4=θ˙2 employing the minimum time plus muscle stresses (second criterion) with a cost factor of ρ=7600 for the time in Eq. (24)

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Fig. 7

The optimal trajectories of x1=θ1, x2=θ2, x3=θ˙1, and x4=θ˙2 employing the minimum time plus muscle stresses (second criterion) with a cost factor of ρ=150,000 for the time in Eq. (24)

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Fig. 8

The optimal muscle forces using the minimum time plus muscle stresses (second criterion) with a cost factor of ρ=150,000 for the time in Eq. (24)

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Fig. 9

The shoulder and elbow torques generated by the optimal muscle forces using the second criterion (minimum time plus muscle stresses) with a cost factor of ρ=150,000 for the time in Eq. (24)

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Fig. 10

The obtained optimal trajectories for the musculoskeletal arm's end-point in the Cartesian space (x−y) using the minimum time and muscle stresses (second criterion) with tf=1.997  s and tf=0.994  s

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Fig. 11

The obtained optimal trajectory of the arm's end-point in the Cartesian space (x−y) together with the experimental trajectory [30] with x0=[20deg,  90deg,  0deg/s,  0deg/s]T, xf=[5deg,  25deg,  0deg/s,  0deg/s]T, and tf=0.72 s

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Fig. 12

The obtained optimal trajectory of the arm's end-point in the Cartesian space (x−y) together with the experimental trajectory [30] with x0=[65deg,  90deg,  0deg/s,  0deg/s]T, xf=[5deg,  25deg,  0deg/s,  0deg/s]T, and tf=0.83 s

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Fig. 13

The obtained optimal trajectory of the arm's end-point in the Cartesian space (x−y) together with the experimental trajectory [30] with x0=[65deg,  90deg,  0deg/s,  0deg/s]T, xf=[50deg,  25deg,  0deg/s,  0deg/s]T, and tf=0.72 s

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