A Family of Iterative Gauss-Newton Shooting Methods for Nonlinear Optimal Control

This paper introduces a family of iterative algorithms for unconstrained nonlinear optimal control. We generalize the well-known iLQR algorithm to different multiple shooting variants, combining advantages like straightforward initialization and a closed-loop forward integration. All algorithms have similar computational complexity, i.e. linear complexity in the time horizon, and can be derived in the same computational framework. We compare the full-step variants of our algorithms and present several simulation examples, including a high-dimensional underactuated robot subject to contact switches. Simulation results show that our multiple shooting algorithms can achieve faster convergence, better local contraction rates and much shorter runtimes than classical iLQR, which makes them a superior choice for nonlinear model predictive control applications.

[1]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[2]  Moritz Diehl,et al.  The Lifted Newton Method and Its Application in Optimization , 2009, SIAM J. Optim..

[3]  Jonas Buchli,et al.  An efficient optimal planning and control framework for quadrupedal locomotion , 2016, 2017 IEEE International Conference on Robotics and Automation (ICRA).

[4]  Jonas Buchli,et al.  Automatic Differentiation of Rigid Body Dynamics for Optimal Control and Estimation , 2017, Adv. Robotics.

[5]  Ferdinando Cannella,et al.  Design of HyQ – a hydraulically and electrically actuated quadruped robot , 2011 .

[6]  Peter Fankhauser,et al.  ANYmal - a highly mobile and dynamic quadrupedal robot , 2016, 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[7]  Robert J. Gelinas,et al.  Sensitivity analysis of ordinary differential equation systems—A direct method , 1976 .

[8]  Pierre-Brice Wieber,et al.  Fast Direct Multiple Shooting Algorithms for Optimal Robot Control , 2005 .

[9]  D. Mayne A Second-order Gradient Method for Determining Optimal Trajectories of Non-linear Discrete-time Systems , 1966 .

[10]  Roland Siegwart,et al.  Fast nonlinear Model Predictive Control for unified trajectory optimization and tracking , 2016, 2016 IEEE International Conference on Robotics and Automation (ICRA).

[11]  Yuval Tassa,et al.  Synthesis and stabilization of complex behaviors through online trajectory optimization , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[12]  Jonas Buchli,et al.  Trajectory Optimization Through Contacts and Automatic Gait Discovery for Quadrupeds , 2016, IEEE Robotics and Automation Letters.

[13]  Jonas Buchli,et al.  The control toolbox — An open-source C++ library for robotics, optimal and model predictive control , 2018, 2018 IEEE International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR).

[14]  Athanasios Sideris,et al.  A Riccati approach for constrained linear quadratic optimal control , 2011, Int. J. Control.

[15]  James E. Bobrow,et al.  An efficient sequential linear quadratic algorithm for solving nonlinear optimal control problems , 2005, Proceedings of the 2005, American Control Conference, 2005..

[16]  Olivier Stasse,et al.  Whole-body model-predictive control applied to the HRP-2 humanoid , 2015, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[17]  Marco Hutter,et al.  Whole-Body Nonlinear Model Predictive Control Through Contacts for Quadrupeds , 2017, IEEE Robotics and Automation Letters.

[18]  M. Diehl,et al.  Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations , 2000 .

[19]  Jonas Buchli,et al.  Efficient kinematic planning for mobile manipulators with non-holonomic constraints using optimal control , 2017, 2017 IEEE International Conference on Robotics and Automation (ICRA).

[20]  Stefan Schaal,et al.  Risk sensitive nonlinear optimal control with measurement uncertainty , 2016, ArXiv.

[21]  E. Todorov,et al.  A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems , 2005, Proceedings of the 2005, American Control Conference, 2005..

[22]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .