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

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

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.
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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P. Abbeel and A. Y. Ng, Apprenticeship learning via inverse reinforcement learning,, in, (2004), 1.

[2]

W. Abend, E. Bizzi and P. Morasso, Human arm trajectory formation,, Brain, 105 (1982), 331. doi: 10.1093/brain/105.2.331.

[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, 6192 (2010), 257. doi: 10.1007/978-3-642-14075-4_37.

[4]

C. G. Atkeson and J. M. Hollerbach, Kinematic features of unrestrained vertical arm movements,, J. Neurosci., 5 (1985), 2318.

[5]

J. F. Bard, An algorithm for solving the general bilevel programming problem,, Math. Oper. Res., 8 (1983), 260. doi: 10.1287/moor.8.2.260.

[6]

J. F. Bard, Convex two-level optimization,, Math. Program., 40 (1988), 15. doi: 10.1007/BF01580720.

[7]

J. F. Bard, "Practical Bilevel Optimization: Algorithms and Applications,", Kluwer Academic Publishers, (1998).

[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. doi: 10.1016/S0921-8890(01)00155-5.

[9]

J. Bracken and J.T. McGill, Mathematical programs with optimization problems in the constraints,, Oper. Res., 21 (1973), 37. doi: 10.1287/opre.21.1.37.

[10]

W. Candler and R. Norton, Multilevel Programming,, World Bank Development Research Center, 20 (1977).

[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, (1975).

[12]

B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization,, Ann. Oper. Res., 153 (2007), 235. doi: 10.1007/s10479-007-0176-2.

[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, (2007), 3639.

[14]

S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems,, Optimization, 25 (1992), 341. doi: 10.1080/02331939208843831.

[15]

S. Dempe, "Foundations of Bilevel Programming,", Kluwer Academic Publishers, (2002).

[16]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints,, Optimization, 52 (2003), 333. doi: 10.1080/0233193031000149894.

[17]

P. Deuflhard and F. Bornemann, "Scientific Computing with Ordinary Differential Equations,", Springer, (2002).

[18]

M. Diehl, and D.B. Leineweber and A.A.S. Schäfer, "MUSCOD-II User's Manual,", IWR-Preprint, (2001).

[19]

T. A. Edmunds and J. F. Bard, Algorithms for nonlinear bilevel mathematical programs,, IEEE Trans. Syst. Man Cyb., 21 (1991), 83. doi: 10.1109/21.101139.

[20]

R. Featherstone, "Robot Dynamics Algorithms,", Kluwer Academic Publishers, (1987).

[21]

T. Flash and N. Hogan, The coordination of arm movements: an experimentally confirmed mathematical model,, J. Neurosci., 5 (1985), 1688.

[22]

M. Gleicher, Retargetting motion to new characters,, in, (1998), 33.

[23]

E. Hairer, S. P. N{\o}rsett and G. Wanner, "Solving Ordinary Differential Equations,", Springer, (1993).

[24]

C. M. Harris and D. M. Wolpert, Signal-dependend noise determines motor planning,, Nature, 394 (1998), 780. doi: 10.1038/29528.

[25]

H. Hatze, Neuromusculoskeletal control systems modeling - a critical survey of recent developments,, IEEE Trans. Autom. Control, 25 (1980), 375. doi: 10.1109/TAC.1980.1102380.

[26]

A. V. Hill, The heat of shortening and the dynamic constants of muscle,, Proc. Roy. Soc. B, 126 (1938), 136. doi: 10.1098/rspb.1938.0050.

[27]

K. W. Lilly, "Efficient Dynamic Simulation of Robotic Mechanisms,", Kluwer Academic Publishers, (1993).

[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. doi: 10.1145/1073204.1073314.

[29]

Y. Maeda, T. Hara and T. Arai, Human-robot cooperative manipulation with motion estimation,, in, 4 (2001), 2240.

[30]

P. Marcotte and G. Savard, Bilevel programming: a combinatorial perspective,, in, (2005), 191. doi: 10.1007/0-387-25592-3_7.

[31]

K. Mombaur, A. Truong and J. P. Laumond, From human to humanoid locomotion - an inverse optimal control approach,, Auton. Robot., 28 (2010), 369. doi: 10.1007/s10514-009-9170-7.

[32]

P. Morasso, Spatial control of arm movements,, Exp. Brain Res., 42 (1981), 223. doi: 10.1007/BF00236911.

[33]

P. Morasso, Three dimensional arm trajectories,, Biol. Cybern., 48 (1983), 187. doi: 10.1007/BF00318086.

[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.

[35]

W. L. Nelson, Physical principles for economies of skilled movements,, Biol. Cybern., 46 (1983), 135. doi: 10.1007/BF00339982.

[36]

A. Y. Ng and S. Russell, Algorithms for inverse reinforcement learning,, in, (2000), 663.

[37]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Second Edition, (2006).

[38]

J. V. Outrata, On the numerical solution of a class of Stackelberg problems,, Z. Oper. Res., 34 (1990), 255.

[39]

J. V. Outrata, M. Kocvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints,", Kluwer Academic Publishers, (1998).

[40]

M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives,, Cambridge DAMTP Report NA2009/06, (2009).

[41]

D. Ramachandran and E. Amir, Bayesian inverse reinforcement learning,, in, (2007), 2586.

[42]

N. D. Ratliff, J. A. Bagnell and M. A. Zinkevich, Maximum margin planning,, in, (2006), 729.

[43]

S. Russell, Learning agents for uncertain environments (extended abstract),, in, (1998), 101.

[44]

S. Schaal, Is imitation learning the route to humanoid robots?,, Trends in Cognitive Sciences, 3 (1999), 233. doi: 10.1016/S1364-6613(99)01327-3.

[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. doi: 10.1109/21.478449.

[46]

C. Smith and H. I. Christensen, A minimum jerk predictor for teleoperation with variable time delay,, in, (2009), 5621.

[47]

S. Stroeve, Impedance characteristics of neuromusculoskeletal model of the human arm,, Biol. Cybern., 81 (1999), 475. doi: 10.1007/s004220050577.

[48]

W. Suleiman, E. Yoshida, F. Kanehiro, J. P. Laumond and A. Monin, On human motion imitation by humanoid robot,, in, (2008), 2697.

[49]

E. Todorov, Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system,, Neural. Comput., 17 (2005), 1084. doi: 10.1162/0899766053491887.

[50]

E. Todorov, Optimality principles in sensorimotor control,, Nature Neurosci., 7 (2004), 907. doi: 10.1038/nn1309.

[51]

Y. Uno, M. Kawato and R. Suzuki, Formation and control of optimal trajectory in human multijoint arm movement,, Biol. Cybern., 61 (1989), 89. doi: 10.1007/BF00204593.

[52]

Y. Uno, R. Suzuki and M. Kawato, Minimum muscle-tension-change model which reproduces human arm movement,, in, (1989), 299.

[53]

L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: a bibliography review,, J Global. Optim., 5 (1994), 291. doi: 10.1007/BF01096458.

[54]

A. Wächter and L.T. Biegler, Line search filter methods for nonlinear programming: motivation and global convergence,, SIAM J. Optim., 16 (2005), 1.

[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.

[56]

D. A. Winter, "Biomechanics and Motor Control of Human Movement,", Third Edition, (2005).

[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. doi: 10.1109/TBME.1985.325498.

[58]

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems,, Optimization, 33 (1995), 9. doi: 10.1080/02331939508844060.

[59]

T. Yoshikawa, "Foundations of Robotics: Analysis and Control,", MIT Press, (1990).

show all references

References:
[1]

P. Abbeel and A. Y. Ng, Apprenticeship learning via inverse reinforcement learning,, in, (2004), 1.

[2]

W. Abend, E. Bizzi and P. Morasso, Human arm trajectory formation,, Brain, 105 (1982), 331. doi: 10.1093/brain/105.2.331.

[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, 6192 (2010), 257. doi: 10.1007/978-3-642-14075-4_37.

[4]

C. G. Atkeson and J. M. Hollerbach, Kinematic features of unrestrained vertical arm movements,, J. Neurosci., 5 (1985), 2318.

[5]

J. F. Bard, An algorithm for solving the general bilevel programming problem,, Math. Oper. Res., 8 (1983), 260. doi: 10.1287/moor.8.2.260.

[6]

J. F. Bard, Convex two-level optimization,, Math. Program., 40 (1988), 15. doi: 10.1007/BF01580720.

[7]

J. F. Bard, "Practical Bilevel Optimization: Algorithms and Applications,", Kluwer Academic Publishers, (1998).

[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. doi: 10.1016/S0921-8890(01)00155-5.

[9]

J. Bracken and J.T. McGill, Mathematical programs with optimization problems in the constraints,, Oper. Res., 21 (1973), 37. doi: 10.1287/opre.21.1.37.

[10]

W. Candler and R. Norton, Multilevel Programming,, World Bank Development Research Center, 20 (1977).

[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, (1975).

[12]

B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization,, Ann. Oper. Res., 153 (2007), 235. doi: 10.1007/s10479-007-0176-2.

[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, (2007), 3639.

[14]

S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems,, Optimization, 25 (1992), 341. doi: 10.1080/02331939208843831.

[15]

S. Dempe, "Foundations of Bilevel Programming,", Kluwer Academic Publishers, (2002).

[16]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints,, Optimization, 52 (2003), 333. doi: 10.1080/0233193031000149894.

[17]

P. Deuflhard and F. Bornemann, "Scientific Computing with Ordinary Differential Equations,", Springer, (2002).

[18]

M. Diehl, and D.B. Leineweber and A.A.S. Schäfer, "MUSCOD-II User's Manual,", IWR-Preprint, (2001).

[19]

T. A. Edmunds and J. F. Bard, Algorithms for nonlinear bilevel mathematical programs,, IEEE Trans. Syst. Man Cyb., 21 (1991), 83. doi: 10.1109/21.101139.

[20]

R. Featherstone, "Robot Dynamics Algorithms,", Kluwer Academic Publishers, (1987).

[21]

T. Flash and N. Hogan, The coordination of arm movements: an experimentally confirmed mathematical model,, J. Neurosci., 5 (1985), 1688.

[22]

M. Gleicher, Retargetting motion to new characters,, in, (1998), 33.

[23]

E. Hairer, S. P. N{\o}rsett and G. Wanner, "Solving Ordinary Differential Equations,", Springer, (1993).

[24]

C. M. Harris and D. M. Wolpert, Signal-dependend noise determines motor planning,, Nature, 394 (1998), 780. doi: 10.1038/29528.

[25]

H. Hatze, Neuromusculoskeletal control systems modeling - a critical survey of recent developments,, IEEE Trans. Autom. Control, 25 (1980), 375. doi: 10.1109/TAC.1980.1102380.

[26]

A. V. Hill, The heat of shortening and the dynamic constants of muscle,, Proc. Roy. Soc. B, 126 (1938), 136. doi: 10.1098/rspb.1938.0050.

[27]

K. W. Lilly, "Efficient Dynamic Simulation of Robotic Mechanisms,", Kluwer Academic Publishers, (1993).

[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. doi: 10.1145/1073204.1073314.

[29]

Y. Maeda, T. Hara and T. Arai, Human-robot cooperative manipulation with motion estimation,, in, 4 (2001), 2240.

[30]

P. Marcotte and G. Savard, Bilevel programming: a combinatorial perspective,, in, (2005), 191. doi: 10.1007/0-387-25592-3_7.

[31]

K. Mombaur, A. Truong and J. P. Laumond, From human to humanoid locomotion - an inverse optimal control approach,, Auton. Robot., 28 (2010), 369. doi: 10.1007/s10514-009-9170-7.

[32]

P. Morasso, Spatial control of arm movements,, Exp. Brain Res., 42 (1981), 223. doi: 10.1007/BF00236911.

[33]

P. Morasso, Three dimensional arm trajectories,, Biol. Cybern., 48 (1983), 187. doi: 10.1007/BF00318086.

[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.

[35]

W. L. Nelson, Physical principles for economies of skilled movements,, Biol. Cybern., 46 (1983), 135. doi: 10.1007/BF00339982.

[36]

A. Y. Ng and S. Russell, Algorithms for inverse reinforcement learning,, in, (2000), 663.

[37]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Second Edition, (2006).

[38]

J. V. Outrata, On the numerical solution of a class of Stackelberg problems,, Z. Oper. Res., 34 (1990), 255.

[39]

J. V. Outrata, M. Kocvara and J. Zowe, "Nonsmooth Approach to Optimization Problems with Equilibrium Constraints,", Kluwer Academic Publishers, (1998).

[40]

M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives,, Cambridge DAMTP Report NA2009/06, (2009).

[41]

D. Ramachandran and E. Amir, Bayesian inverse reinforcement learning,, in, (2007), 2586.

[42]

N. D. Ratliff, J. A. Bagnell and M. A. Zinkevich, Maximum margin planning,, in, (2006), 729.

[43]

S. Russell, Learning agents for uncertain environments (extended abstract),, in, (1998), 101.

[44]

S. Schaal, Is imitation learning the route to humanoid robots?,, Trends in Cognitive Sciences, 3 (1999), 233. doi: 10.1016/S1364-6613(99)01327-3.

[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. doi: 10.1109/21.478449.

[46]

C. Smith and H. I. Christensen, A minimum jerk predictor for teleoperation with variable time delay,, in, (2009), 5621.

[47]

S. Stroeve, Impedance characteristics of neuromusculoskeletal model of the human arm,, Biol. Cybern., 81 (1999), 475. doi: 10.1007/s004220050577.

[48]

W. Suleiman, E. Yoshida, F. Kanehiro, J. P. Laumond and A. Monin, On human motion imitation by humanoid robot,, in, (2008), 2697.

[49]

E. Todorov, Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system,, Neural. Comput., 17 (2005), 1084. doi: 10.1162/0899766053491887.

[50]

E. Todorov, Optimality principles in sensorimotor control,, Nature Neurosci., 7 (2004), 907. doi: 10.1038/nn1309.

[51]

Y. Uno, M. Kawato and R. Suzuki, Formation and control of optimal trajectory in human multijoint arm movement,, Biol. Cybern., 61 (1989), 89. doi: 10.1007/BF00204593.

[52]

Y. Uno, R. Suzuki and M. Kawato, Minimum muscle-tension-change model which reproduces human arm movement,, in, (1989), 299.

[53]

L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: a bibliography review,, J Global. Optim., 5 (1994), 291. doi: 10.1007/BF01096458.

[54]

A. Wächter and L.T. Biegler, Line search filter methods for nonlinear programming: motivation and global convergence,, SIAM J. Optim., 16 (2005), 1.

[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.

[56]

D. A. Winter, "Biomechanics and Motor Control of Human Movement,", Third Edition, (2005).

[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. doi: 10.1109/TBME.1985.325498.

[58]

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems,, Optimization, 33 (1995), 9. doi: 10.1080/02331939508844060.

[59]

T. Yoshikawa, "Foundations of Robotics: Analysis and Control,", MIT Press, (1990).

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