Abstract
Numerical optimal control (the approximation of an optimal trajectory using numerical iterative algorithms) is a promising approach to compute the control of complex dynamical systems whose instantaneous linearization is not meaningful. Aside from the problems of computation cost, these methods raise several conceptual problems, like stability, robustness, or simply understanding of the nature of the obtained solution. In this chapter, we propose a rewriting of the Differential Dynamic Programing solver. Our variant is more efficient and numerically more interesting. Furthermore, it draws some interesting comparisons with the classical inverse formulation: in particular, we show that inverse kinematics can be seen as singular case of it, when the preview horizon collapses.
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Notes
- 1.
The problem we name inverse geometry is sometimes referred as inverse kinematics, the second being referred as differential (or closed-loop) inverse kinematics. We use ‘geometry’ when only static postures are implied and keep the word ‘kinematics’ when a motion is explicitly implied.
References
Alamir, M.: Stabilization of Nonlinear Systems Using Receding-Horizon Control Schemes. Springer, Lecture Notes in Control and Information Sciences (2006)
Biegler, L.: Nonlinear programming: concepts, algorithms, and applications to chemical processes. SIAM, Philadelphia (2010)
Das, H., Slotine, J.J., Sheridan, T.: Inverse kinematic algorithms for redundant systems. In: IEEE International Conference on Robotics and Automation (ICRA’88), pp. 43–48. Philadelphia (1988)
Deo, A., Walker, I.: Robot subtask performance with singularity robustness using optimal damped least squares. In: IEEE ICRA, pp. 434–441. Nice (1992)
Escande, A., Mansard, N., Wieber, P.B.: Hierarchical quadratic programming. Int. J. Robot. Res. (2012) (in press)
Goldfarb, D.: A family of variable-metric methods derived by variational means. Mathematics of computation 24(109), 23–26 (1970)
Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. John Hopkins University Press, Baltimore (1996)
Jacobson, D.H., Mayne, D.Q.: Differential Dynamic Programming. Elsevier, Amsterdam (1970)
Khatib, O.: A unified approach for motion and force control of robot manipulators: the operational space formulation. Int. J. Robot. Res. 3(1), 43–53 (1987)
Leineweber, D.B., Schäfer, A., Bock, H.G., Schlöder, J.P.: An efficient multiple shooting based reduced sqp strategy for large-scale dynamic process optimization: part II: software aspects and applications. Comput. Chem. Eng. 27(2), 167–174 (2003)
McCarthy, J.: Introduction to Theoretical Kinematics. MIT Press, Cambridge (1990)
Mombaur, K.: Using optimization to create self-stable human-like running. Robotica 27(03), 321 (2008). doi:10.1017/S0263574708004724
Mordatch, I., Todorov, E., Popović, Z.: Discovery of complex behaviors through contact-invariant optimization. In: ACM SIGGRAPH’12. Los Angeles (2012)
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)
Pantoja, D.O.: Differential dynamic programming and newton’s method. Int. J. Control 47(5), 1539–1553 (1988). doi:10.1080/00207178808906114. http://www.tandfonline.com/doi/abs/10.1080/00207178808906114
Schulman, J., Lee, A., Awwal, I., Bradlow, H., Abbeel, P.: Finding locally optimal, collision-free trajectories with sequential convex optimization. In: Robotics: Science and Systems (2013)
Tassa, Y., Erez, T., Todorov, E.: Synthesis and stabilization of complex behaviors through online trajectory optimization. In: IROS’12, Portugal
Tassa, Y., Erez, T., Todorov, E.: Synthesis and stabilization of complex behaviors through online trajectory optimization. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS’12), pp. 4906–4913 (2012). doi:10.1109/IROS.2012.6386025
Tassa, Y., Mansard, N., Todorov, E.: Control-Limited Differential Dynamic Programming (Under Review)
Todorov, E., Li, W.: A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems. In: Proceedings of the American Control Conference (ACC’05), pp. 300–306. Portland (2005). doi:10.1109/ACC.2005.1469949
Whitney, D.: Resolved motion rate control of manipulators and human prostheses. IEEE Trans. Man Mach. Syst. 10(2), 47–53 (1969)
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Geoffroy, P., Mansard, N., Raison, M., Achiche, S., Todorov, E. (2014). From Inverse Kinematics to Optimal Control. In: Lenarčič, J., Khatib, O. (eds) Advances in Robot Kinematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06698-1_42
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DOI: https://doi.org/10.1007/978-3-319-06698-1_42
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