Inverse dynamics combined with a constrained static optimization analysis has often been proposed to solve the muscular redundancy problem. Typically, the optimization problem consists in a cost function to be minimized and some equality and inequality constraints to be fulfilled. Penalty-based and Lagrange multipliers methods are common optimization methods for the equality constraints management. More recently, the pseudo-inverse method has been introduced in the field of biomechanics. The purpose of this paper is to evaluate the ability and the efficiency of this new method to solve the muscular redundancy problem, by comparing respectively the musculo-tendon forces prediction and its cost-effectiveness against common optimization methods. Since algorithm efficiency and equality constraints fulfillment highly belong to the optimization method, a two-phase procedure is proposed in order to identify and compare the complexity of the cost function, the number of iterations needed to find a solution and the computational time of the penalty-based method, the Lagrange multipliers method and pseudo-inverse method. Using a 2D knee musculo-skeletal model in an isometric context, the study of the cost functions isovalue curves shows that the solution space is 2D with the penalty-based method, 3D with the Lagrange multipliers method and 1D with the pseudo-inverse method. The minimal cost function area (defined as the area corresponding to 5% over the minimal cost) obtained for the pseudo-inverse method is very limited and along the solution space line, whereas the minimal cost function area obtained for other methods are larger or more complex. Moreover, when using a 3D lower limb musculo-skeletal model during a gait cycle simulation, the pseudo-inverse method provides the lowest number of iterations while Lagrange multipliers and pseudo-inverse method have almost the same computational time. The pseudo-inverse method, by providing a better suited cost function and an efficient computational framework, seems to be adapted to the muscular redundancy problem resolution in case of linear equality constraints. Moreover, by reducing the solution space, this method could be a unique opportunity to introduce optimization methods for a posteriori articulation of preference in order to provide a palette of solutions rather than a unique solution based on a lot of hypotheses.

References

References
1.
Fleming
,
B. C.
, and
Beynnon
,
B. D.
, 2004, “
in vivo Measurement of Ligament/Tendon Strains and Forces: A Review
,”
Ann. Biomed. Eng.
,
32
(
3
), pp.
318
328
.
2.
Erdemir
,
A.
,
McLean
,
S.
,
Herzog
,
W.
, and
van den Bogert
,
A. J.
, 2007, “
Model-Based Estimation of Muscle Forces Exerted during Movements
,”
Clin. Biomech.
,
22
, pp.
131
154
.
3.
Zajac
,
F. E.
, 2002, “
Understanding Muscle Coordination of the Human Leg with Dynamical Simulations
,”
J. Biomech.
,
35
(
8
), pp.
1011
1018
.
4.
Andersen
,
M. S.
,
Damsgaard
,
M.
, and
Rasmussen
,
J.
, 2009, “
Kinematic Analysis of Over-Determinate Biomechanical Systems
,”
Comput. Methods Biomech. Biomed. Eng.
,
12
(
4
), pp.
371
384
.
5.
Fraysse
,
F.
,
Dumas
,
R.
,
Cheze
,
L.
, and
Wang
,
X.
, 2009, “
Comparison of Global and Joint-to-Joint Methods for Estimating the Hip Joint Load and the Muscle Forces during Walking
,”
J. Biomech.
,
42
(
14
), pp.
2357
2362
.
6.
Bernstein
,
N.
, 1967,
The Coordination and Regulation of Movements
,
Pergamon Press
,
Oxford, UK
.
7.
Crowninshield
,
R. D.
, and
Brand
,
R. A.
, 1981, “
A Physiologically Based Criterion of muscle Force Prediction in Locomotion
,”
J. Biomech.
,
14
, pp.
793
801
.
8.
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
.
9.
Anderson
,
F. C.
, and
Pandy
,
M. G.
, 2001, “
Static and Dynamic Optimization Solutions for Gait are Practically Equivalent
,”
J. Biomech.
,
34
, pp.
153
161
.
10.
Delp
,
S. L.
,
Anderson
,
F. C.
,
Arnold
,
A. S.
,
Loan
,
J. P.
,
Habib
,
A.
,
John
,
C. T.
,
Guendelman
,
E.
, and
Thelen
,
D. G.
, 2007, “
Opensim: Open-Source Software to Create And Analyze Dynamic Simulations of Movement
,”
IEEE Trans. Biomed. Eng.
,
54
(
11
), pp.
1940
1950
.
11.
Delp
,
S. L.
,
Loan
,
J. P.
,
Hoy
,
M. G.
,
Zajac
,
F. E.
,
Topp
,
E. L.
, and
Rosen
,
J. M.
, 1990, “
An Interactive Graphics-Based Model of the Lower Extremity to Study Orthopaedic Surgical Procedures
,”
IEEE Trans. Biomed. Eng.
,
37
(
8
), pp.
757
767
.
12.
Lenaerts
,
G.
,
De Groote
,
F.
,
Demeulenaere
,
B.
,
Mulier
,
M.
,
Van der Perre
,
G.
,
Spaepen
,
A.
, and
Jonkers
,
I.
, 2008, “
Subject-Specific Hip Geometry Affects Predicted Hip Joint Contact Forces during Gait
,”
J. Biomech.
,
41
(
6
), pp.
1243
1252
.
13.
Bertsekas
,
D. P.
, 1996,
Constrained Optimization and Lagrange Multiplier Methods.
Athena Scientific
,
Nashua, NH
.
14.
Nocedal
,
J.
, and
Wright
,
S. J.
, 2006,
Numerical Optimization
,
2nd ed.
,
Springer
,
Berlin
.
15.
Terrier
,
A.
,
Aeberhard
,
M.
,
Michellod
,
Y.
,
Mullhaupt
,
P.
,
Gillet
,
D.
,
Farron
,
A.
, and
Pioletti
,
D.
, 2010, “
A Musculoskeletal Shoulder Model Based on Pseudo-Inverse and Null-Space Optimization
,”
Med. Eng. Phys.
,
32
(
9
), pp.
1050
1056
.
16.
Smidt
,
G. L.
, 1973, “
Biomechanical Analysis of Knee Flexion and Extension
,”
J. Biomech.
,
6
(
1
), pp.
79
92
.
17.
Dumas
,
R.
,
Moissenet
,
F.
,
Gasparutto
,
X.
, and
Cheze
,
L.
, 2012, “
Influence of the Joint Models on the Lower Limb Musculo-Tendon and Contact Forces during Gait
,”
Proc. Inst. Mech. Eng., Part H: J. Eng. Med.
,
226
(
2
), pp.
146
160
.
18.
Moissenet
,
F.
,
Cheze
,
L.
, and
Dumas
,
R.
, 2012, “
Anatomical Kinematic Constraints: Consequences on Muscular Forces and Joint Reactions
,”
Multibody Syst. Dyn.
, available online.
19.
Cereatti
,
A.
,
Camomilla
,
V.
, and
Cappozzo
,
A.
, 2004, “
Estimation of the Centre of Rotation: a Methodological Contribution
,”
J. Biomech.
,
37
(
3
), pp.
413
416
.
20.
Reinbolt
,
J. A.
,
Schutte
,
J. F.
,
Fregly
,
B. J.
,
Koh
,
B. I.
,
Haftka
,
R. T.
,
George
,
A. D.
, and
Mitchell
,
K. H.
, 2005, “
Determination of Patient-Specific Multi-Joint Kinematic Models through Two-Level Optimization
,”
J. Biomech.
,
38
(
3
), pp.
621
626
.
21.
Feikes
,
J.
,
O’Connor
,
J.
, and
Zavatsky
,
A.
, 2003, “
A Constraint-Based Approach to Modelling the Mobility of the Human Knee Joint
,”
J. Biomech.
,
36
(
1
), pp.
125
129
.
22.
Di Gregorio
R.
,
Parenti-Castelli
,
V.
,
O’Connor
,
J.
, and
Leardini
,
A.
, 2007, “
Mathematical Models of Passive Motion at the Human Ankle Joint by Equivalent Spatial Parallel Mechanisms
,”
Med. Biol. Eng. Comput.
,
45
(
3
), pp.
305
313
.
23.
Lenaerts
,
G.
,
Bartels
,
W.
,
Gelaude
,
F.
,
Mulier
,
M.
,
Spaepen
,
A.
,
Van der Perre
,
G.
, and
Jonkers
,
I.
, 2009, “
Subject-Specific Hip Geometry and Hip Joint Centre Location Affects Calculated Contact Forces at the Hip during Gait
,”
J. Biomech.
,
42
, pp.
1246
1251
.
24.
Fregly
,
B. J.
,
Besier
,
T. F.
,
Lloyd
,
D. G.
,
Delp
,
S. L.
,
Banks
,
S. A.
,
Pandy
,
M. G.
, and
D’Lima
D. D.
, 2011, “
Grand Challenge Competition to Predict in vivo Knee Loads
,”
J. Orthop. Res.
, conditionally accepted.
25.
De Luca
,
C. J.
,
Gilmore
,
L. D.
,
Kuznetsov
,
M.
, and
Roy
,
S. H.
, 2010, “
Filtering the surface EMG Signal: Movement Artifact and Baseline Noise Contamination
,”
J. Biomech.
,
43
(
8
), pp.
1573
1579
.
26.
De Luca
,
S. J.
, 1997, “
The Use of Surface Electromyography in Biomechanics
,”
J. Appl. Biomech.
,
13
, pp.
135
163
.
27.
Biggs
,
M. C.
, 1975, “
Constrained Minimization using Recursive Quadratic Programming
,”
Towards Global Optimization
,
North-Holland
,
Amsterdam
, pp.
341
349
.
28.
Han
,
S. P.
, 1977, “
A Globally Convergent Method for Nonlinear Programming
,”
J. Optim. Theory Appl.
,
22
, pp.
297
309
.
29.
Powell
,
M. J. D.
, 1978, “
The Convergence of Variable Metric Methods for Nonlinearly Constrained Optimization Calculations
,”
Nonlinear Programming
,
3rd ed.
,
Academic Press
,
New York
.
30.
Powell
,
M. J. D.
, 1978, “
A Fast Algorithm for Nonlinearly Constrained Optimization Calculations
,”
Numerical Analysis, Lecture Notes in Mathematics
, Vol.
630
,
Springer Verlag
,
Berlin
.
31.
Bean
,
J. C.
, and
Chaffin
,
D. B.
, 1988, “
Biomechanical Model Calculation of Muscle Contraction Forces - A Double Linear Programming Method
,”
J. Biomech.
,
21
(
1
), pp.
59
66
.
32.
Marler
,
R. T.
, and
Arora
,
J. S.
, 2004, “
Survey of Multi-Objective Optimization Methods for Engineering
,”
Struct. Multidiscip. Optim.
,
26
, pp.
369
395
.
33.
Martelli
,
S.
,
Taddei
,
F.
,
Cappello
,
A.
,
Van Sint Jan
,
S.
,
Leardini
,
A.
, and
Viceconti
,
M.
, 2011, “
Effect of Sub-Optimal Neuromotor Control on the Hip Joint Load during Level Walking
,”
J. Biomech.
,
44
, pp.
1716
1721
.
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