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March  2021, 13(1): 1-23. doi: 10.3934/jgm.2020032

Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model

Bernard Bonnard 1, and  Jérémy Rouot 2,,

1. 

McTAO team, INRIA Sophia Antipolis, 2004 Route des Lucioles, 06902 Valbonne, France
2. 

L@bISEN, Yncréa Ouest, 20 rue Cuirassé Bretagne, 29200 Brest, France

* Corresponding author: Jérémy Rouot

Dedicated to Professor Tony Bloch on the occasion of his 65th birthday

Received  May 2020 Revised  September 2020 Published  March 2021 Early access  November 2020

Fund Project: This research paper benefited from the support of the FMJH Program PGMO and from the support of EDF, Thales, Orange and the authors are partially supported by the Labex AMIES

A recent force-fatigue parameterized mathematical model, based on the seminal contributions of V. Hill to describe muscular activity, allows to predict the muscular force response to external electrical stimulation (FES) and it opens the road to optimize the FES-input to maximize the force response to a pulse train, to track a reference force while minimizing the fatigue for a sequence of pulse trains or to follow a reference joint angle trajectory to produce motion in the non-isometric case. In this article, we introduce the geometric frame to analyze the dynamics and we present Pontryagin types necessary optimality conditions adapted to digital controls, used in the experiments, vs permanent control and which fits in the optimal sampled-data control frame. This leads to Hamiltonian differential variational inequalities, which can be numerically implemented vs direct optimization schemes.

Keywords: Biomechanics force-fatigue models, geometric optimal control, sampled-data control problem, Pontryagin type necessary conditions, Hamiltonian differential variational inequality.
Mathematics Subject Classification: 49K15, 93B07, 92B05.
Citation: Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032
References:
[1]

A. A. Agrachëv and R. V. Gamkrelidze, Symplectic geometry for optimal control, in Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., 133, Dekker, New York, 1990,263-277.

[2]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation, in progress.

[3]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations, J. Optim. Theory Appl., 184 (2020), 581-602.  doi: 10.1007/s10957-019-01599-4.

[4]

T. Bakir, B. Bonnard and S. Othman, Predictive control based on nonlinear observer for muscular force and fatigue model, 2018 Annual American Control Conference (ACC), Milwaukee, WI, 2018, 2157-2162. doi: 10.23919/ACC.2018.8430962.

[5]

T. Bakir, B. Bonnard and J. Rouot, A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model, Netw. Heterog. Media, 14 (2019), 79-100.  doi: 10.3934/nhm.2019005.

[6] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946. 
[7]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematics & Applications, 40, Springer-Verlag, Berlin, 2003.

[8]

B. Bonnard and I. Kupka, Generic properties of singular trajectories, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 167-186.  doi: 10.1016/S0294-1449(97)80143-6.

[9]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathematics & Applications, 43, Springer-Verlag, Berlin, 2004.

[10]

N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, in Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-89394-3.

[11]

L. Bourdin and G. Dhar, Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times, Math. Control Signals Systems, 31 (2019), 503-544.  doi: 10.1007/s00498-019-00247-6.

[12]

L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales, Math. Control Relat. Fields, 6 (2016), 53-94.  doi: 10.3934/mcrf.2016.6.53.

[13]

P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188. 

[14]

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optim. Methods Softw., 27 (2012), 177-196.  doi: 10.1080/10556788.2011.593625.

[15]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅰ. Predicting the effect of stimulation frequency and pattern on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 48-58.  doi: 10.1109/TNSRE.2002.1021586.

[16]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅱ. Predicting the effect of resting times on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 59-67.  doi: 10.1109/TNSRE.2002.1021587.

[17]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains, J. Appl. Physiol., 88 (2000), 917-925.  doi: 10.1152/jappl.2000.88.3.917.

[18]

B. D. Doll, N. A. Kirsch and N. Sharma, Optimization of a stimulation train based on a predictive model of muscle force and fatigue, IFAC-PapersOnLine, 48 (2015), 338-342.  doi: 10.1016/j.ifacol.2015.10.162.

[19]

R. V. Gamkrelidze, Discovery of the maximum principle, J. Dynam. Control Systems, 5 (1999), 437-451.  doi: 10.1023/A:1021783020548.

[20]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963.

[21]

R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki, The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66 (2012), 427-438.  doi: 10.1007/s00407-012-0098-5.

[22]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/034.

[23]

R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Automatic Control, AC-22 (1977), 728-740.  doi: 10.1109/tac.1977.1101601.

[24]

A. Isidori, Non-Linear Control Systems, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-1-84628-615-5.

[25]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Robert E. Kreiger Publishing Co., Inc., Melbourne, FL, 1986.

[26]

M. S. Marion, Predicting Fatigue During Electrically Stimulated Non-Isometric Contractions, Ph.D thesis, University of California Davis, 2010.

[27]

M. S. Marion, A. S. Wexler and M. L. Hull, Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration, J. Neuroengineering Rehab., 10 (2013). doi: 10.1186/1743-0003-10-13.

[28]

V. Renault, M. Thieullen and E. Trélat, Minimal time spiking in various ChR2-controlled neuron models, J. Math. Biol., 76 (2018), 567-608.  doi: 10.1007/s00285-017-1101-1.

[29]

H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Interdisciplinary Applied Mathematics, 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[30]

H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations, 12 (1972), 95-116.  doi: 10.1016/0022-0396(72)90007-1.

[31]

R. Vinter, Optimal Control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000.

show all references

References:
[1]

A. A. Agrachëv and R. V. Gamkrelidze, Symplectic geometry for optimal control, in Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., 133, Dekker, New York, 1990,263-277.

[2]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation, in progress.

[3]

T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations, J. Optim. Theory Appl., 184 (2020), 581-602.  doi: 10.1007/s10957-019-01599-4.

[4]

T. Bakir, B. Bonnard and S. Othman, Predictive control based on nonlinear observer for muscular force and fatigue model, 2018 Annual American Control Conference (ACC), Milwaukee, WI, 2018, 2157-2162. doi: 10.23919/ACC.2018.8430962.

[5]

T. Bakir, B. Bonnard and J. Rouot, A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model, Netw. Heterog. Media, 14 (2019), 79-100.  doi: 10.3934/nhm.2019005.

[6] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946. 
[7]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematics & Applications, 40, Springer-Verlag, Berlin, 2003.

[8]

B. Bonnard and I. Kupka, Generic properties of singular trajectories, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 167-186.  doi: 10.1016/S0294-1449(97)80143-6.

[9]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathematics & Applications, 43, Springer-Verlag, Berlin, 2004.

[10]

N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, in Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-89394-3.

[11]

L. Bourdin and G. Dhar, Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times, Math. Control Signals Systems, 31 (2019), 503-544.  doi: 10.1007/s00498-019-00247-6.

[12]

L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales, Math. Control Relat. Fields, 6 (2016), 53-94.  doi: 10.3934/mcrf.2016.6.53.

[13]

P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188. 

[14]

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optim. Methods Softw., 27 (2012), 177-196.  doi: 10.1080/10556788.2011.593625.

[15]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅰ. Predicting the effect of stimulation frequency and pattern on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 48-58.  doi: 10.1109/TNSRE.2002.1021586.

[16]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, A predictive fatigue model. Ⅱ. Predicting the effect of resting times on fatigue, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 10 (2002), 59-67.  doi: 10.1109/TNSRE.2002.1021587.

[17]

J. Ding, A. S. Wexler and S. A. Binder-Macleod, Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains, J. Appl. Physiol., 88 (2000), 917-925.  doi: 10.1152/jappl.2000.88.3.917.

[18]

B. D. Doll, N. A. Kirsch and N. Sharma, Optimization of a stimulation train based on a predictive model of muscle force and fatigue, IFAC-PapersOnLine, 48 (2015), 338-342.  doi: 10.1016/j.ifacol.2015.10.162.

[19]

R. V. Gamkrelidze, Discovery of the maximum principle, J. Dynam. Control Systems, 5 (1999), 437-451.  doi: 10.1023/A:1021783020548.

[20]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963.

[21]

R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki, The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66 (2012), 427-438.  doi: 10.1007/s00407-012-0098-5.

[22]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/034.

[23]

R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Automatic Control, AC-22 (1977), 728-740.  doi: 10.1109/tac.1977.1101601.

[24]

A. Isidori, Non-Linear Control Systems, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-1-84628-615-5.

[25]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Robert E. Kreiger Publishing Co., Inc., Melbourne, FL, 1986.

[26]

M. S. Marion, Predicting Fatigue During Electrically Stimulated Non-Isometric Contractions, Ph.D thesis, University of California Davis, 2010.

[27]

M. S. Marion, A. S. Wexler and M. L. Hull, Predicting non-isometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration, J. Neuroengineering Rehab., 10 (2013). doi: 10.1186/1743-0003-10-13.

[28]

V. Renault, M. Thieullen and E. Trélat, Minimal time spiking in various ChR2-controlled neuron models, J. Math. Biol., 76 (2018), 567-608.  doi: 10.1007/s00285-017-1101-1.

[29]

H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Interdisciplinary Applied Mathematics, 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[30]

H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations, 12 (1972), 95-116.  doi: 10.1016/0022-0396(72)90007-1.

[31]

R. Vinter, Optimal Control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000.

Figure 1.  Extremal of the Ding et al. model defined on $ [0,T] $, $ T = 100ms $, with four sampling times and maximizing the final force
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Figure 2.  Sensitivity analysis. Time evolution of the Jacobi fields component $ \delta F(\cdot) $ according to Definition 2.7 and associated to the trajectory of the force-fatigue model
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Figure 3.  Sensitivity analysis. Time evolution of the Jacobi fields component $ \delta F(\cdot) $ where their initializations are $ 10\% $ of the physical values $ A_{\text{rest}},K_{m,\text{rest}},\tau_{1,\text{rest}} $ associated to the trajectory of the force-fatigue model
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Figure 4.  Time evolution on $ [0,T],\, T = 0.4s $ of the state of the Ding et al. model computed with the indirect approach maximizing $ \varphi(x(T)) = F(T) $ with $ n = 6 $, $ I_m = 20 $ms. The optimal value is the same as the optimal value computed with the direct method (see Fig.5). The model parameters are given in Table 1
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Figure 5.  Time evolution on $ [0,T],\, T = 0.4s $ of the normalized state of the Ding et al. model computed with the direct approach maximizing $ \varphi(x(T)) = F(T) $ with $ n = 6 $, $ I_m = 20 $ms. The model parameters are given in Table 1
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Figure 6.  Time evolution on $ [0,T],\, T = 0.9s $ of the normalized state of the Marion et al. model computed with the direct approach maximizing $ \varphi(x(T)) = F(T)/F_{ext}+A_{90}(T)/A_{90,0} $ with $ n = 10 $, $ I_m = 20 $ms. The model parameters are given in Table 2
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Figure 7.  Time evolution on $ [0,T],\, T = 0.9s $ of the normalized state of the Marion et al. model with the direct method when considering $ t_i = T - (n-i+1)I_m,\ i = 1,\dots,n $, $ n = 10 $, $ I_m = 20ms $. The model parameters are given in Table 2
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Table 2.  Numerical values for parameters of the Marion et al. model
Variable Value Unit Variable Value Unit
$ F_{load} $ 45.4 N $ F_M $ 247.5 N
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 99.4 s
$ A_{90,0} $ 2100 N s−1 $ K_{m,0} $ 3.52×-1
$ \tau_{1,0} $ 3.61×-2 s $ \tau_2 $ 5.21×-2 s
$ a $ 4.49×-4 deg−2 $ b $ 3.44×-2 deg−1
$ v_1 $ 3.71×-1 N deg−2 $ v_2 $ 2.29×-2 deg−1
$ \ell = L/I $ 9.85 kg m−1 $ \alpha_A $ -4.02×-1 s−2
$ \alpha_{K_m} $ 1.36×-5 s−1 N−1 $ \alpha_{\tau_1} $ 2.93×-5 N−1
$ \beta_{\tau_1} $ 8.54×-7 s deg−1 N−1 $ R_0 $ 1.143
$ \beta_{90} $ 0 deg−1 s−1 $ \beta_{K_m} $ 0 deg−1 N−1
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Variable Value Unit Variable Value Unit
$ F_{load} $ 45.4 N $ F_M $ 247.5 N
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 99.4 s
$ A_{90,0} $ 2100 N s−1 $ K_{m,0} $ 3.52×-1
$ \tau_{1,0} $ 3.61×-2 s $ \tau_2 $ 5.21×-2 s
$ a $ 4.49×-4 deg−2 $ b $ 3.44×-2 deg−1
$ v_1 $ 3.71×-1 N deg−2 $ v_2 $ 2.29×-2 deg−1
$ \ell = L/I $ 9.85 kg m−1 $ \alpha_A $ -4.02×-1 s−2
$ \alpha_{K_m} $ 1.36×-5 s−1 N−1 $ \alpha_{\tau_1} $ 2.93×-5 N−1
$ \beta_{\tau_1} $ 8.54×-7 s deg−1 N−1 $ R_0 $ 1.143
$ \beta_{90} $ 0 deg−1 s−1 $ \beta_{K_m} $ 0 deg−1 N−1
Table 1.  Numerical values for parameters of the Ding et al. model
Variable Value Unit Variable Value Unit
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 127 s
$ A_{rest} $ 3.009 kN s−1 $ K_{m,rest} $ 1.03×-1
$ \tau_{1,rest} $ 5.1×-2 s $ \tau_2 $ 1.24×-1 s
$ \alpha_A $ -4×-1 s−2 $ \alpha_{K_m} $ 1.9×-2 s−1 kN−1
$ \alpha_{\tau_1} $ 2.1×-2 kN−1 $ R_0 $ 2
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Variable Value Unit Variable Value Unit
$ \tau_c $ 20×-3 s $ \tau_{fat} $ 127 s
$ A_{rest} $ 3.009 kN s−1 $ K_{m,rest} $ 1.03×-1
$ \tau_{1,rest} $ 5.1×-2 s $ \tau_2 $ 1.24×-1 s
$ \alpha_A $ -4×-1 s−2 $ \alpha_{K_m} $ 1.9×-2 s−1 kN−1
$ \alpha_{\tau_1} $ 2.1×-2 kN−1 $ R_0 $ 2
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