Abstract
We present Extended LQR, a novel approach for locally-optimal control for robots with non-linear dynamics and non-quadratic cost functions. Our formulation is conceptually different from existing approaches, and is based on the novel concept of LQR-smoothing, which is an LQR-analogue of Kalman smoothing. Our approach iteratively performs both a backward Extended LQR pass, which computes approximate cost-to-go functions, and a forward Extended LQR pass, which computes approximate cost-to-come functions. The states at which the sum of these functions is minimal provide an approximately optimal sequence of states for the control problem, and we use these points to linearize the dynamics and quadratize the cost functions in the subsequent iteration. Our results indicate that Extended LQR converges quickly and reliably to a locally-optimal solution of the non-linear, non-quadratic optimal control problem. In addition, we show that our approach is easily extended to include temporal optimization, in which the duration of a trajectory is optimized as part of the control problem. We demonstrate the potential of our approach on two illustrative non-linear control problems involving simulated and physical differential-drive robots and simulated quadrotor helicopters.
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References
Bar-Shalom, Y., Li, R., Kirubarajan, T.: Estimation with Applications to Tracking and Navigation, Wiley-Interscience, New Jersey (2004)
Bell, B.: The iterated Kalman smoother as a Gauss-Newton method. SIAM J. Optim. 4(3), 626–636 (1994)
Betts, J.: Practical methods for optimal control and estimation using nonlinear programming, vol. 19, SIAM (2009)
Bertsekas, D.: Dynamic Programming and Optimal Control. Athena Scientific, Belmont (2001)
A. Björck. Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)
Chen, M.S., Kao, C.Y.: Control of linear time-varying systems using forward Riccati equation. J. Dyn. Syst. Meas. Control 119(3), 536540 (1997)
Fujita, Y., Nakamura, Y., Shiller, Z.: Dual Dijkstra search for paths with different topologies. In: Proceedings of the IEEE International Conference on Robotics and Automation (2003)
Higham, N.: Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl. 103, 103–118 (1988)
Jacobsen, D., Mayne, D.: Differential Dynamic Programming. Elsevier, New York (1970)
Karaman, S., Frazzoli, E.: Sampling-based algorithms for optimal motion planning. Int. J. Robot. Res. 30(7), 846–894 (2011)
Lavalle, S.: Planning Algorithms. Cambridge University Press, Cambridge (2006)
Li, W., Todorov, E.: Iterative linear-quadratic regulator design for nonlinear biological movement systems. In: Proceedings of the International Conference on Informatics in Control, Automation and Robotics (2004)
Nocedal, J., Wright, S.: Numerical Optimization. Springer Science+ Business Media, Germany (2006)
Rauch, H., Tung, F., Striebel, C.: Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8), 1445–1450 (1965)
Rawlik, K., Toussaint, M., Vijayakumar, S.: An approximate inference approach to temporal optimization in optimal control. In: Proceedings of the Advances in Neural Information Processing Systems, pp. 2011–2019 (2010)
Rawlik, K., Toussaint, M., Vijayakumar, S.: On stochastic optimal control and reinforcement learning by approximate inference. In: Proceedings of the Robotics Science and Systems Conference (R:SS 2012), Sydney, Australia (2012)
Schulman, J., Ho, J., Lee, A., Awwal, I., Bradlow, H., Abbeel, P.: Finding locally optimal, collisi-on-free trajectories with sequential convex optimization. In: Robotics: Science and Systems (2013)
Tedrake, R., Manchester, I., Tobenkin, M., Roberts, J.: LQR-trees: Feedback motion planning via sums-of-squares verification. Int. J. Robot. Res. 29(8), 1038–1052 (2010)
Theodorou, E., Tassa, Y., Todorov, E.: Stochastic differential dynamic programming. Proceedings of the American Control Conference (2010)
Todorov, E.: General duality between optimal control and estimation. In: Proceedings of the IEEE Conference on Decision and Control (2008)
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 (2005)
Toussaint, M.: Robot trajectory optimization using approximate inference. In: Proceedings of the International Conference on Machine Learning (2009)
van den Berg, J., Patil, S., Alterovitz, R.: Motion planning under uncertainty using iterative local optimization in belief space. Int. J. Robot. Res. 31(11), 1263–1278 (2012)
Weiss, A., Kolmanovsky, I., Bernstein, D.: Forward-integration Riccati-based output-feedback control of linear time-varying systems. In: American Control Conference (2012)
Whittle, P.: Risk-sensitive linear/quadratic/Gaussian control. Adv. Appl. Prob. 13(4), 764–777 (1981)
Yakowitz, S.: Algorithms and computational techniques in differential dynamic programming. Control Dyn. Syst. 31, 75–91 (1989)
Zucker, M., Ratliff, N., Dragan, A., Pivtoraiko, M., Klingensmith, M., Dellin, C., Bagnell, J., Srinivasa, S.: CHOMP: Covariant Hamiltonian optimization for motion planning. Int. J. Robot. Res. (2013)
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van den Berg, J. (2016). Extended LQR: Locally-Optimal Feedback Control for Systems with Non-Linear Dynamics and Non-Quadratic Cost. In: Inaba, M., Corke, P. (eds) Robotics Research. Springer Tracts in Advanced Robotics, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-28872-7_3
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