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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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References

Li, W. , 2006, “ Optimal Control for Biological Movement Systems,” Ph.D. thesis, University of California, San Diego, CA.
Stroeve, S. , 1998, “ Neuromuscular Control Model of the Arm Including Feedback and Feedforward Components,” Acta Psychol., 100(1–2), pp. 117–131.
Park, H. , and Durand, D. , 2008, “ Motion Control of Musculoskeletal Systems With Redundancy,” Biol. Cybern., 99(6), pp. 503–516. [PubMed]
Blana, D. , Kirsch, R. , and Chadwick, E. , 2009, “ Combined Feedforward and Feedback Control of a Redundant, Nonlinear, Dynamic Musculoskeletal System,” Med. Biol. Eng. Comput., 47(5), pp. 533–542. [PubMed]
Tahara, K. , Luo, Z.-W. , Arimoto, S. , and Kino, H. , 2005, “ Sensory-Motor Control Mechanism for Reaching Movements of a Redundant Musculo-Skeletal Arm,” J. Rob. Syst., 22(11), pp. 639–651.
Wada, Y. , and Kawato, M. , 1993, “ A Neural Network Model for Arm Trajectory Formation Using Forward and Inverse Dynamics Models,” Neural Networks, 6(7), pp. 919–932.
Mansouri, M. , and Reinbolt, J. A. , 2012, “ A Platform for Dynamic Simulation and Control of Movement Based on OpenSim and MATLAB,” J. Biomech., 45(8), pp. 1517–1521. [PubMed]
Praagman, M. , Chadwick, E. K. J. , Van Der Helm, F. C. T. , and Veeger, H. E. J. , 2006, “ The Relationship Between Two Different Mechanical Cost Functions and Muscle Oxygen Consumption,” J. Biomech., 39(4), pp. 758–765. [PubMed]
Tsirakos, D. , Baltzopoulos, V. , and Bartlett, R. , 1997, “ Inverse Optimization: Functional and Physiological Considerations Related to the Force-Sharing Problem,” Crit. Rev. Biomed. Eng., 25(4–5), pp. 371–407. [PubMed]
Crowninshield, R. D. , and Brand, R. A. , 1981, “ A Physiologically Based Criterion of Muscle Force Prediction in Locomotion,” J. Biomech., 14(11), pp. 793–801. [PubMed]
Prilutsky, B. I. , 2000, “ Coordination of Two- and One-Joint Muscles: Functional Consequences and Implications for Motor Control,” Mot. Control, 4(1), pp. 1–44.
Todorov, E. , 2004, “ Optimality Principles in Sensorimotor Control,” Nat. Neurosci., 7(9), pp. 907–915. [PubMed]
Hatze, H. , 1976, “ The Complete Optimization of a Human Motion,” Math. Biosci., 28(1), pp. 99–135.
Nelson, W. L. , 1983, “ Physical Principles for Economies of Skilled Movements,” Biol. Cybern., 46(2), pp. 135–147. [PubMed]
Oğuztöreli, M. N. , and Stein, R. B. , 1983, “ Optimal Control of Antagonistic Muscles,” Biol. Cybern., 48(2), pp. 91–99. [PubMed]
Kim, H. J. , Wang, Q. , Rahmatalla, S. , Swan, C. C. , Arora, J. S. , Abdel-Malek, K. , and Assouline, J. G. , 2008, “ Dynamic Motion Planning of 3D Human Locomotion Using Gradient-Based Optimization,” ASME J. Biomech. Eng., 130(3), p. 031002.
Shourijeh, M. S. , and McPhee, J. , 2014, “ Forward Dynamic Optimization of Human Gait Simulations: A Global Parameterization Approach,” ASME J. Comput. Nonlinear Dyn., 9(3), p. 031018.
Neptune, R. R. , 1999, “ Optimization Algorithm Performance in Determining Optimal Controls in Human Movement Analyses,” ASME J. Biomech. Eng., 121(2), pp. 249–252.
Mughal, A. , and Iqbal, K. , 2010, “ 3D Bipedal Model With Holonomic Constraints for the Decoupled Optimal Controller Design of the Biomechanical Sit-to-Stand Maneuver,” ASME J. Biomech. Eng., 132(4), p. 041010.
Hogan, N. , 1984, “ An Organizing Principle for a Class of Voluntary Movements,” J. Neurosci., 4(11), pp. 2745–2754. [PubMed]
Flash, T. , and Hogan, N. , 1985, “ The Coordination of Arm Movements: An Experimentally Confirmed Mathematical Model,” J. Neurosci., 5(7), pp. 1688–1703. [PubMed]
Rosenbaum, D. A. , Loukopoulos, L. D. , Meulenbroek, R. G. J. , Vaughan, J. , and Engelbrecht, S. E. , 1995, “ Planning Reaches by Evaluating Stored Postures,” Psychol. Rev., 102(1), pp. 28–67. [PubMed]
Uno, Y. , Kawato, M. , and Suzuki, R. , 1989, “ Formation and Control of Optimal Trajectory in Human Multijoint Arm Movement,” Biol. Cybern., 61(2), pp. 89–101. [PubMed]
Nakano, E. , Imamizu, H. , Osu, R. , Uno, Y. , Gomi, H. , Yoshioka, T. , and Kawato, M. , 1999, “ Quantitative Examinations of Internal Representations for Arm Trajectory Planning: Minimum Commanded Torque Change Model,” J. Neurophysiol., 81(5), pp. 2140–2155. [PubMed]
Ben-Itzhak, S. , and Karniel, A. , 2007, “ Minimum Acceleration Criterion With Constraints Implies Bang-Bang Control as an Underlying Principle for Optimal Trajectories of Arm Reaching Movements,” Neural Comput., 20(3), pp. 779–812.
Shourijeh, M. S. , and McPhee, J. , 2013, “ Optimal Control and Forward Dynamics of Human Periodic Motions Using Fourier Series for Muscle Excitation Patterns,” ASME J. Comput. Nonlinear Dyn., 9(2), p. 021005.
Todorov, E. , and Li, W. , 2005, “ A Generalized Iterative LQG Method for Locally-Optimal Feedback Control of Constrained Nonlinear Stochastic Systems,” 2005 American Control Conference, Hilton Portland & Executive Tower, Portland, OR, June 8–10, Vol. 1, pp. 300–306.
Li, W. , and Todorov, E. , 2007, “ Iterative Linearization Methods for Approximately Optimal Control and Estimation of Non-Linear Stochastic System,” Int. J. Control, 80(9), pp. 1439–1453.
Kirk, D. E. , 2004, Optimal Control Theory: An Introduction, Dover Publications, Mineola, NY.
Suzuki, M. , Yamazaki, Y. , Mizuno, N. , and Matsunami, K. , 1997, “ Trajectory Formation of the Center-of-Mass of the Arm During Reaching Movements,” Neuroscience, 76(2), pp. 597–610. [PubMed]
Veeger, H. E. J. , Yu, B. , An, K.-N. , and Rozendal, R. H. , 1997, “ Parameters for Modeling the Upper Extremity,” J. Biomech., 30(6), pp. 647–652. [PubMed]
Pigeon, P. , Yahia, L. H. , and Feldman, A. G. , 1996, “ Moment Arms and Lengths of Human Upper Limb Muscles as Functions of Joint Angles,” J. Biomech., 29(10), pp. 1365–1370. [PubMed]
Holzbaur, K. R. S. , Murray, W. M. , Gold, G. E. , and Delp, S. L. , 2007, “ Upper Limb Muscle Volumes in Adult Subjects,” J. Biomech., 40(4), pp. 742–749. [PubMed]
Murray, W. M. , Buchanan, T. S. , and Delp, S. L. , 2000, “ The Isometric Functional Capacity of Muscles That Cross the Elbow,” J. Biomech., 33(8), pp. 943–952. [PubMed]
Arjmand, N. , and Shirazi-Adl, A. , 2006, “ Model and In Vivo Studies on Human Trunk Load Partitioning and Stability in Isometric Forward Flexions,” J. Biomech., 39(3), pp. 510–521. [PubMed]
Davis, J. R. , and Mirka, G. A. , 2000, “ Transverse-Contour Modeling of Trunk Muscle–Distributed Forces and Spinal Loads During Lifting and Twisting,” Spine, 25(2), pp. 180–189. [PubMed]
Mcgill, S. M. , and Norman, R. W. , 1986, “ Partitioning of the L4-L5 Dynamic Moment Into Disc, Ligamentous, and Muscular Components During Lifting,” Spine, 11(7), pp. 666–678. [PubMed]

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. 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. 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. 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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