A bilevel optimization approach to obtain optimal cost functions for human arm movements

Sebastian Albrecht; Marion  Leibold; Michael Ulbrich

doi:10.3934/naco.2012.2.105

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2012, 2(1): 105-127. doi: 10.3934/naco.2012.2.105

A bilevel optimization approach to obtain optimal cost functions for human arm movements

Sebastian Albrecht ^1, , Marion Leibold ^2, and Michael Ulbrich ^3,

1.	Chair for Mathematical Optimization, Department of Mathematics, Technische Universität München (TUM), Boltzmannstr. 3, 85747 Garching b. München, Germany
2.	Institute of Automatic Control Engineering, Technische Universität München (TUM), Arcisstr. 21, 80290 München, Germany
3.	Chair for Mathematical Optimization, Department of Mathematics, Technische Universität München (TUM), Boltzmannstr. 3, 85747 Garching b. München, Germany

Received April 2011 Revised August 2011 Published March 2012

Abstract
Related Papers

Using a bilevel optimization approach, we investigate the question how humans plan and execute their arm motions. It is known that human motions are (approximately) optimal for suitable and unknown cost functions subject to the dynamics. We investigate the following inverse problem: Which cost function out of a parameterized family (e.g., convex combinations of functions suggested in the literature) reproduces recorded human arm movements best? The lower level problem is an optimal control problem governed by a nonlinear model of the human arm dynamics. The approach is analyzed for a dynamical 3D model of the human arm. Furthermore, results for a two-dimensional experiment with human probands are presented.

Keywords: human arm motions, optimal control, Bilevel optimization, direct discretization method..

Mathematics Subject Classification: 90C90, 49N45, 49M2.

Citation: Sebastian Albrecht, Marion Leibold, Michael Ulbrich. A bilevel optimization approach to obtain optimal cost functions for human arm movements. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 105-127. doi: 10.3934/naco.2012.2.105

References:

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[3]	S. Albrecht, C. Weber, M. Sobotka, A. Peer, M. Buss and M. Ulbrich, Optimization criteria for human trajectory formation in dynamic virtual environments, in "Haptics: Generating and Perceiving Tangible Sensations," Lecture Notes in Computer Science, 6192 (2010), 257-262. doi: 10.1007/978-3-642-14075-4_37. Google Scholar
[4]	C. G. Atkeson and J. M. Hollerbach, Kinematic features of unrestrained vertical arm movements, J. Neurosci., 5 (1985), 2318-2330. Google Scholar
[5]	J. F. Bard, An algorithm for solving the general bilevel programming problem, Math. Oper. Res., 8 (1983), 260-272. doi: 10.1287/moor.8.2.260. Google Scholar
[6]	J. F. Bard, Convex two-level optimization, Math. Program., 40 (1988), 15-27. doi: 10.1007/BF01580720. Google Scholar
[7]	J. F. Bard, "Practical Bilevel Optimization: Algorithms and Applications," Kluwer Academic Publishers, Dordrecht, 1998. Google Scholar
[8]	A. Billard and M. J. Mataric, Learning human arm movements by imitation: evaluation of a biologically inspired connectionist architecture, Robot. Auton. Syst., 37 (2001), 145-160. doi: 10.1016/S0921-8890(01)00155-5. Google Scholar
[9]	J. Bracken and J.T. McGill, Mathematical programs with optimization problems in the constraints, Oper. Res., 21 (1973), 37-44. doi: 10.1287/opre.21.1.37. Google Scholar
[10]	W. Candler and R. Norton, Multilevel Programming, World Bank Development Research Center, Departml Working Paper, 20 (1977). Google Scholar
[11]	R. F. Chandler, C. E. Claser, J. T. McConville, H. M. Reynolds and J. W. Young, Investigation of inertial properties of the human body, AMLR Technical Report, Wright-Patterson Air Force Base, OH, (1975). Google Scholar
[12]	B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization, Ann. Oper. Res., 153 (2007), 235-256. doi: 10.1007/s10479-007-0176-2. Google Scholar
[13]	B. Corteville, E. Aertbelien, H. Bruyninckx, J. De Schutter and H. Van Brussel, Human-inspired robot assistant for fast point-to-point movements, in "Proc. IEEE ICRA," (2007), 3639-3644. Google Scholar
[14]	S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems, Optimization, 25 (1992), 341-354. doi: 10.1080/02331939208843831. Google Scholar
[15]	S. Dempe, "Foundations of Bilevel Programming," Kluwer Academic Publishers, Dordrecht, 2002. Google Scholar
[16]	S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. Google Scholar
[17]	P. Deuflhard and F. Bornemann, "Scientific Computing with Ordinary Differential Equations," Springer, Berlin, 2002. Google Scholar
[18]	M. Diehl, and D.B. Leineweber and A.A.S. Schäfer, "MUSCOD-II User's Manual," IWR-Preprint, University of Heidelberg, Germany, 2001. Google Scholar
[19]	T. A. Edmunds and J. F. Bard, Algorithms for nonlinear bilevel mathematical programs, IEEE Trans. Syst. Man Cyb., 21 (1991), 83-89. doi: 10.1109/21.101139. Google Scholar
[20]	R. Featherstone, "Robot Dynamics Algorithms," Kluwer Academic Publishers, Dordrecht, 1987. Google Scholar
[21]	T. Flash and N. Hogan, The coordination of arm movements: an experimentally confirmed mathematical model, J. Neurosci., 5 (1985), 1688-1703. Google Scholar
[22]	M. Gleicher, Retargetting motion to new characters, in "Proc. Int. Conf. on Computer Graphics and Interactive Techniques," (1998), 33-42. Google Scholar
[23]	E. Hairer, S. P. N{\o}rsett and G. Wanner, "Solving Ordinary Differential Equations," Springer, New York, 1993. Google Scholar
[24]	C. M. Harris and D. M. Wolpert, Signal-dependend noise determines motor planning, Nature, 394 (1998), 780-784. doi: 10.1038/29528. Google Scholar
[25]	H. Hatze, Neuromusculoskeletal control systems modeling - a critical survey of recent developments, IEEE Trans. Autom. Control, 25 (1980), 375-385. doi: 10.1109/TAC.1980.1102380. Google Scholar
[26]	A. V. Hill, The heat of shortening and the dynamic constants of muscle, Proc. Roy. Soc. B, 126 (1938), 136-195. doi: 10.1098/rspb.1938.0050. Google Scholar
[27]	K. W. Lilly, "Efficient Dynamic Simulation of Robotic Mechanisms," Kluwer Academic Publishers, Dordrecht, 1993. Google Scholar
[28]	C. K. Liu, A. Hertzmann and Z. Popovi\'c, Learning physics-based motion style with nonlinear inverse optimization, ACM Trans. Graph., 24 (2005), 1071-1081. doi: 10.1145/1073204.1073314. Google Scholar
[29]	Y. Maeda, T. Hara and T. Arai, Human-robot cooperative manipulation with motion estimation, in "Proc. IEEE IROS", 4 (2001), 2240-2245. Google Scholar
[30]	P. Marcotte and G. Savard, Bilevel programming: a combinatorial perspective, in "Graph Theory and Combinatorial Optimization" (eds. D. Avis, A. Hertz, and O. Marcotte), Springer, New York (2005), 191-217. doi: 10.1007/0-387-25592-3_7. Google Scholar
[31]	K. Mombaur, A. Truong and J. P. Laumond, From human to humanoid locomotion - an inverse optimal control approach, Auton. Robot., 28 (2010), 369-383. doi: 10.1007/s10514-009-9170-7. Google Scholar
[32]	P. Morasso, Spatial control of arm movements, Exp. Brain Res., 42 (1981), 223-227. doi: 10.1007/BF00236911. Google Scholar
[33]	P. Morasso, Three dimensional arm trajectories, Biol. Cybern., 48 (1983), 187-194. doi: 10.1007/BF00318086. Google Scholar
[34]	E. Nakano, H. Imamizu, R. Osu, Y. Uno, H. Gomi, T. Yoshioka and M. Kawato, Quanitative examinations of internal representations for arm trajectory planning: Minimum commanded torque change model, J. Neurophysiol., 81 (1999), 2140-2155. Google Scholar
[35]	W. L. Nelson, Physical principles for economies of skilled movements, Biol. Cybern., 46 (1983), 135-147. doi: 10.1007/BF00339982. Google Scholar
[36]	A. Y. Ng and S. Russell, Algorithms for inverse reinforcement learning, in "Proc. Int. Conf. on Machine Learning," (2000), 663-670. Google Scholar
[37]	J. Nocedal and S. J. Wright, "Numerical Optimization," Second Edition, Springer, New York, 2006. Google Scholar
[38]	J. V. Outrata, On the numerical solution of a class of Stackelberg problems, Z. Oper. Res., 34 (1990), 255-277. Google Scholar
[39]	J. V. Outrata, M. Kocvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints," Kluwer Academic Publishers, Dordrecht, 1998. Google Scholar
[40]	M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives, Cambridge DAMTP Report NA2009/06, University of Cambridge, Cambridge, 2009. Google Scholar
[41]	D. Ramachandran and E. Amir, Bayesian inverse reinforcement learning, in "Proc. Int. Conf. on Artificial Intelligence," (2007), 2586-2591. Google Scholar
[42]	N. D. Ratliff, J. A. Bagnell and M. A. Zinkevich, Maximum margin planning, in "Proc. Int. Conf. on Machine Learning," (2006), 729-736. Google Scholar
[43]	S. Russell, Learning agents for uncertain environments (extended abstract), in "Proc. Int. Conf. on Computational Learning Theory," (1998), 101-103. Google Scholar
[44]	S. Schaal, Is imitation learning the route to humanoid robots?, Trends in Cognitive Sciences, 3 (1999), 233-242. doi: 10.1016/S1364-6613(99)01327-3. Google Scholar
[45]	K. Shimizu and M. Lu, A global optimization method for the Stackelberg problem with convex functions via problem transformation and concave programming, IEEE Trans. Syst. Man Cyb., 25 (1995), 1635-1640. doi: 10.1109/21.478449. Google Scholar
[46]	C. Smith and H. I. Christensen, A minimum jerk predictor for teleoperation with variable time delay, in "Proc. IEEE IROS," (2009), 5621-5627. Google Scholar
[47]	S. Stroeve, Impedance characteristics of neuromusculoskeletal model of the human arm, Biol. Cybern., 81 (1999), 475-494. doi: 10.1007/s004220050577. Google Scholar
[48]	W. Suleiman, E. Yoshida, F. Kanehiro, J. P. Laumond and A. Monin, On human motion imitation by humanoid robot, in "Proc. IEEE ICRA," (2008), 2697-2704. Google Scholar
[49]	E. Todorov, Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system, Neural. Comput., 17 (2005), 1084-1108. doi: 10.1162/0899766053491887. Google Scholar
[50]	E. Todorov, Optimality principles in sensorimotor control, Nature Neurosci., 7 (2004), 907-915. doi: 10.1038/nn1309. Google Scholar
[51]	Y. Uno, M. Kawato and R. Suzuki, Formation and control of optimal trajectory in human multijoint arm movement, Biol. Cybern., 61 (1989), 89-101. doi: 10.1007/BF00204593. Google Scholar
[52]	Y. Uno, R. Suzuki and M. Kawato, Minimum muscle-tension-change model which reproduces human arm movement, in "Proc. 4th Symp. Biol. Physiol. Eng.," (1989), 299-302. Google Scholar
[53]	L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: a bibliography review, J Global. Optim., 5 (1994), 291-306. doi: 10.1007/BF01096458. Google Scholar
[54]	A. Wächter and L.T. Biegler, Line search filter methods for nonlinear programming: motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. Google Scholar
[55]	A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. Google Scholar
[56]	D. A. Winter, "Biomechanics and Motor Control of Human Movement," Third Edition, Wiley, New York, 2005. Google Scholar
[57]	J. M. Winters and L. Stark, Analysis of fundamental human movement patterns through the use of in-depth antagonistic muscle models, IEEE Trans. Biomedical Eng., 10 (1985), 826-839. doi: 10.1109/TBME.1985.325498. Google Scholar
[58]	J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optimization, 33 (1995), 9-27. doi: 10.1080/02331939508844060. Google Scholar
[59]	T. Yoshikawa, "Foundations of Robotics: Analysis and Control," MIT Press, Cambridge, 1990. Google Scholar

show all references

References:

[1]	P. Abbeel and A. Y. Ng, Apprenticeship learning via inverse reinforcement learning, in "Proc. Int. Conf. on Machine Learning," (2004), 1-8. Google Scholar
[2]	W. Abend, E. Bizzi and P. Morasso, Human arm trajectory formation, Brain, 105 (1982), 331-348. doi: 10.1093/brain/105.2.331. Google Scholar
[3]	S. Albrecht, C. Weber, M. Sobotka, A. Peer, M. Buss and M. Ulbrich, Optimization criteria for human trajectory formation in dynamic virtual environments, in "Haptics: Generating and Perceiving Tangible Sensations," Lecture Notes in Computer Science, 6192 (2010), 257-262. doi: 10.1007/978-3-642-14075-4_37. Google Scholar
[4]	C. G. Atkeson and J. M. Hollerbach, Kinematic features of unrestrained vertical arm movements, J. Neurosci., 5 (1985), 2318-2330. Google Scholar
[5]	J. F. Bard, An algorithm for solving the general bilevel programming problem, Math. Oper. Res., 8 (1983), 260-272. doi: 10.1287/moor.8.2.260. Google Scholar
[6]	J. F. Bard, Convex two-level optimization, Math. Program., 40 (1988), 15-27. doi: 10.1007/BF01580720. Google Scholar
[7]	J. F. Bard, "Practical Bilevel Optimization: Algorithms and Applications," Kluwer Academic Publishers, Dordrecht, 1998. Google Scholar
[8]	A. Billard and M. J. Mataric, Learning human arm movements by imitation: evaluation of a biologically inspired connectionist architecture, Robot. Auton. Syst., 37 (2001), 145-160. doi: 10.1016/S0921-8890(01)00155-5. Google Scholar
[9]	J. Bracken and J.T. McGill, Mathematical programs with optimization problems in the constraints, Oper. Res., 21 (1973), 37-44. doi: 10.1287/opre.21.1.37. Google Scholar
[10]	W. Candler and R. Norton, Multilevel Programming, World Bank Development Research Center, Departml Working Paper, 20 (1977). Google Scholar
[11]	R. F. Chandler, C. E. Claser, J. T. McConville, H. M. Reynolds and J. W. Young, Investigation of inertial properties of the human body, AMLR Technical Report, Wright-Patterson Air Force Base, OH, (1975). Google Scholar
[12]	B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization, Ann. Oper. Res., 153 (2007), 235-256. doi: 10.1007/s10479-007-0176-2. Google Scholar
[13]	B. Corteville, E. Aertbelien, H. Bruyninckx, J. De Schutter and H. Van Brussel, Human-inspired robot assistant for fast point-to-point movements, in "Proc. IEEE ICRA," (2007), 3639-3644. Google Scholar
[14]	S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems, Optimization, 25 (1992), 341-354. doi: 10.1080/02331939208843831. Google Scholar
[15]	S. Dempe, "Foundations of Bilevel Programming," Kluwer Academic Publishers, Dordrecht, 2002. Google Scholar
[16]	S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. Google Scholar
[17]	P. Deuflhard and F. Bornemann, "Scientific Computing with Ordinary Differential Equations," Springer, Berlin, 2002. Google Scholar
[18]	M. Diehl, and D.B. Leineweber and A.A.S. Schäfer, "MUSCOD-II User's Manual," IWR-Preprint, University of Heidelberg, Germany, 2001. Google Scholar
[19]	T. A. Edmunds and J. F. Bard, Algorithms for nonlinear bilevel mathematical programs, IEEE Trans. Syst. Man Cyb., 21 (1991), 83-89. doi: 10.1109/21.101139. Google Scholar
[20]	R. Featherstone, "Robot Dynamics Algorithms," Kluwer Academic Publishers, Dordrecht, 1987. Google Scholar
[21]	T. Flash and N. Hogan, The coordination of arm movements: an experimentally confirmed mathematical model, J. Neurosci., 5 (1985), 1688-1703. Google Scholar
[22]	M. Gleicher, Retargetting motion to new characters, in "Proc. Int. Conf. on Computer Graphics and Interactive Techniques," (1998), 33-42. Google Scholar
[23]	E. Hairer, S. P. N{\o}rsett and G. Wanner, "Solving Ordinary Differential Equations," Springer, New York, 1993. Google Scholar
[24]	C. M. Harris and D. M. Wolpert, Signal-dependend noise determines motor planning, Nature, 394 (1998), 780-784. doi: 10.1038/29528. Google Scholar
[25]	H. Hatze, Neuromusculoskeletal control systems modeling - a critical survey of recent developments, IEEE Trans. Autom. Control, 25 (1980), 375-385. doi: 10.1109/TAC.1980.1102380. Google Scholar
[26]	A. V. Hill, The heat of shortening and the dynamic constants of muscle, Proc. Roy. Soc. B, 126 (1938), 136-195. doi: 10.1098/rspb.1938.0050. Google Scholar
[27]	K. W. Lilly, "Efficient Dynamic Simulation of Robotic Mechanisms," Kluwer Academic Publishers, Dordrecht, 1993. Google Scholar
[28]	C. K. Liu, A. Hertzmann and Z. Popovi\'c, Learning physics-based motion style with nonlinear inverse optimization, ACM Trans. Graph., 24 (2005), 1071-1081. doi: 10.1145/1073204.1073314. Google Scholar
[29]	Y. Maeda, T. Hara and T. Arai, Human-robot cooperative manipulation with motion estimation, in "Proc. IEEE IROS", 4 (2001), 2240-2245. Google Scholar
[30]	P. Marcotte and G. Savard, Bilevel programming: a combinatorial perspective, in "Graph Theory and Combinatorial Optimization" (eds. D. Avis, A. Hertz, and O. Marcotte), Springer, New York (2005), 191-217. doi: 10.1007/0-387-25592-3_7. Google Scholar
[31]	K. Mombaur, A. Truong and J. P. Laumond, From human to humanoid locomotion - an inverse optimal control approach, Auton. Robot., 28 (2010), 369-383. doi: 10.1007/s10514-009-9170-7. Google Scholar
[32]	P. Morasso, Spatial control of arm movements, Exp. Brain Res., 42 (1981), 223-227. doi: 10.1007/BF00236911. Google Scholar
[33]	P. Morasso, Three dimensional arm trajectories, Biol. Cybern., 48 (1983), 187-194. doi: 10.1007/BF00318086. Google Scholar
[34]	E. Nakano, H. Imamizu, R. Osu, Y. Uno, H. Gomi, T. Yoshioka and M. Kawato, Quanitative examinations of internal representations for arm trajectory planning: Minimum commanded torque change model, J. Neurophysiol., 81 (1999), 2140-2155. Google Scholar
[35]	W. L. Nelson, Physical principles for economies of skilled movements, Biol. Cybern., 46 (1983), 135-147. doi: 10.1007/BF00339982. Google Scholar
[36]	A. Y. Ng and S. Russell, Algorithms for inverse reinforcement learning, in "Proc. Int. Conf. on Machine Learning," (2000), 663-670. Google Scholar
[37]	J. Nocedal and S. J. Wright, "Numerical Optimization," Second Edition, Springer, New York, 2006. Google Scholar
[38]	J. V. Outrata, On the numerical solution of a class of Stackelberg problems, Z. Oper. Res., 34 (1990), 255-277. Google Scholar
[39]	J. V. Outrata, M. Kocvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints," Kluwer Academic Publishers, Dordrecht, 1998. Google Scholar
[40]	M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives, Cambridge DAMTP Report NA2009/06, University of Cambridge, Cambridge, 2009. Google Scholar
[41]	D. Ramachandran and E. Amir, Bayesian inverse reinforcement learning, in "Proc. Int. Conf. on Artificial Intelligence," (2007), 2586-2591. Google Scholar
[42]	N. D. Ratliff, J. A. Bagnell and M. A. Zinkevich, Maximum margin planning, in "Proc. Int. Conf. on Machine Learning," (2006), 729-736. Google Scholar
[43]	S. Russell, Learning agents for uncertain environments (extended abstract), in "Proc. Int. Conf. on Computational Learning Theory," (1998), 101-103. Google Scholar
[44]	S. Schaal, Is imitation learning the route to humanoid robots?, Trends in Cognitive Sciences, 3 (1999), 233-242. doi: 10.1016/S1364-6613(99)01327-3. Google Scholar
[45]	K. Shimizu and M. Lu, A global optimization method for the Stackelberg problem with convex functions via problem transformation and concave programming, IEEE Trans. Syst. Man Cyb., 25 (1995), 1635-1640. doi: 10.1109/21.478449. Google Scholar
[46]	C. Smith and H. I. Christensen, A minimum jerk predictor for teleoperation with variable time delay, in "Proc. IEEE IROS," (2009), 5621-5627. Google Scholar
[47]	S. Stroeve, Impedance characteristics of neuromusculoskeletal model of the human arm, Biol. Cybern., 81 (1999), 475-494. doi: 10.1007/s004220050577. Google Scholar
[48]	W. Suleiman, E. Yoshida, F. Kanehiro, J. P. Laumond and A. Monin, On human motion imitation by humanoid robot, in "Proc. IEEE ICRA," (2008), 2697-2704. Google Scholar
[49]	E. Todorov, Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system, Neural. Comput., 17 (2005), 1084-1108. doi: 10.1162/0899766053491887. Google Scholar
[50]	E. Todorov, Optimality principles in sensorimotor control, Nature Neurosci., 7 (2004), 907-915. doi: 10.1038/nn1309. Google Scholar
[51]	Y. Uno, M. Kawato and R. Suzuki, Formation and control of optimal trajectory in human multijoint arm movement, Biol. Cybern., 61 (1989), 89-101. doi: 10.1007/BF00204593. Google Scholar
[52]	Y. Uno, R. Suzuki and M. Kawato, Minimum muscle-tension-change model which reproduces human arm movement, in "Proc. 4th Symp. Biol. Physiol. Eng.," (1989), 299-302. Google Scholar
[53]	L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: a bibliography review, J Global. Optim., 5 (1994), 291-306. doi: 10.1007/BF01096458. Google Scholar
[54]	A. Wächter and L.T. Biegler, Line search filter methods for nonlinear programming: motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. Google Scholar
[55]	A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. Google Scholar
[56]	D. A. Winter, "Biomechanics and Motor Control of Human Movement," Third Edition, Wiley, New York, 2005. Google Scholar
[57]	J. M. Winters and L. Stark, Analysis of fundamental human movement patterns through the use of in-depth antagonistic muscle models, IEEE Trans. Biomedical Eng., 10 (1985), 826-839. doi: 10.1109/TBME.1985.325498. Google Scholar
[58]	J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optimization, 33 (1995), 9-27. doi: 10.1080/02331939508844060. Google Scholar
[59]	T. Yoshikawa, "Foundations of Robotics: Analysis and Control," MIT Press, Cambridge, 1990. Google Scholar

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A bilevel optimization approach to obtain optimal cost functions for human arm movements

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A bilevel optimization approach to obtain optimal cost functions for human arm movements

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