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.

References

References
1.
Li
,
W.
,
2006
, “
Optimal Control for Biological Movement Systems
,”
Ph.D. thesis
, University of California, San Diego, CA.
2.
Stroeve
,
S.
,
1998
, “
Neuromuscular Control Model of the Arm Including Feedback and Feedforward Components
,”
Acta Psychol.
,
100
(
1–2
), pp.
117
131
.
3.
Park
,
H.
, and
Durand
,
D.
,
2008
, “
Motion Control of Musculoskeletal Systems With Redundancy
,”
Biol. Cybern.
,
99
(
6
), pp.
503
516
.
4.
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
.
5.
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
.
6.
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
.
7.
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
.
8.
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
.
9.
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
.
10.
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
.
11.
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
.
12.
Todorov
,
E.
,
2004
, “
Optimality Principles in Sensorimotor Control
,”
Nat. Neurosci.
,
7
(
9
), pp.
907
915
.
13.
Hatze
,
H.
,
1976
, “
The Complete Optimization of a Human Motion
,”
Math. Biosci.
,
28
(
1
), pp.
99
135
.
14.
Nelson
,
W. L.
,
1983
, “
Physical Principles for Economies of Skilled Movements
,”
Biol. Cybern.
,
46
(
2
), pp.
135
147
.
15.
Oğuztöreli
,
M. N.
, and
Stein
,
R. B.
,
1983
, “
Optimal Control of Antagonistic Muscles
,”
Biol. Cybern.
,
48
(
2
), pp.
91
99
.
16.
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
.
17.
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
.
18.
Neptune
,
R. R.
,
1999
, “
Optimization Algorithm Performance in Determining Optimal Controls in Human Movement Analyses
,”
ASME J. Biomech. Eng.
,
121
(
2
), pp.
249
252
.
19.
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
.
20.
Hogan
,
N.
,
1984
, “
An Organizing Principle for a Class of Voluntary Movements
,”
J. Neurosci.
,
4
(
11
), pp.
2745
2754
.
21.
Flash
,
T.
, and
Hogan
,
N.
,
1985
, “
The Coordination of Arm Movements: An Experimentally Confirmed Mathematical Model
,”
J. Neurosci.
,
5
(
7
), pp.
1688
1703
.
22.
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
.
23.
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
.
24.
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
.
25.
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
.
26.
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
.
27.
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
.
28.
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
.
29.
Kirk
,
D. E.
,
2004
,
Optimal Control Theory: An Introduction
,
Dover Publications
,
Mineola, NY
.
30.
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
.
31.
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
.
32.
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
.
33.
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
.
34.
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
.
35.
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
.
36.
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
.
37.
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
.
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