We consider the optimal control of mechanical systems on Lie groups and develop numerical methods that exploit the structure of the state space and preserve the system motion invariants. Our approach is based on a coordinate-free variational discretization of the dynamics that leads to structure-preserving discrete equations of motion. We construct necessary conditions for optimal trajectories that correspond to discrete geodesics of a higher order system and develop numerical methods for their computation. The resulting algorithms are simple to implement and converge to a solution in very few iterations. A general software implementation is provided and applied to two example systems: an underactuated boat and a satellite with thrusters.

Discrete Mechanics and Optimal Control (DMOC) is newly developed methodology to solve the optimal control problems of mechanical systems. It is based on a direct discretization of the Lagrange-d'Alembert principle. The constraints are modeled as Euler-Langrange equations for discrete states. With the application of DMOC, a new real-time trajectory generation approach is proposed. The formulated problem is solved in the open source free nonlinear programming solver IPOPT (Interior Point OPTimizer). Superior performance of DMOC is verified through a standard forced VanDerPol Oscillator example. It is shown DMOC is much faster than NTG (Nonlinear Trajectory Generation) in computation time, it is 1.31% of the CPU time. Finally, for real-time trajectory generation, a dynamical glider example is shown as one of its possible application fields.

The problem of showing that Lagrangian Coherent Structures (LCS) are useful in determining near optimal trajectories for autonomous underwater vehicles (AUVs) known as gliders is investigated. This paper extends our preliminary results in couple ways. First, the ocean current flows are modeled by 3D B-spline functions in which the input variables are latitude, longitude and time, and the output variable is the ocean current velocity in 2D. The 3D (2D and time-varying) B-spline model of the ocean current is utilized in the Nonlinear Trajectory Generation (NTG) algorithm to find the optimal trajectory of the glider. The trajectories found using the 2D and time-varying B-spline ocean flows model are compared with the trajectories from 2D B-spline model in which the time is assumed to be constant in the ocean current model. In the second part of the paper, the dynamical glider model is established and controlled by gyroscopic forces rather than the simplified kinematic glider model as in the previous work. Finally, numerical solutions of several scenarios and animations of glider trajectories are presented. The results show that the 2D and time-varying B-spline ocean model not only can make the whole trajectory generating process much easier, but also the glider can reach the same destination in a comparable time and with much less energy than it does with the previous 2D ocean current model for both the kinematic and dynamic glider models. Both of the kinematic and dynamic glider optimal trajectories, successfully generated with 2D and time varying ocean current B-spline models, are shown to correspond to LCS.

Obtains feedback stabilization of an inverted pendulum on a rotor arm by the "method of controlled Lagrangians". This approach involves modifying the Lagrangian for the uncontrolled system so that the Euler-Lagrange equations derived from the modified or "controlled" Lagrangian describe the closed-loop system. For the closed-loop equations to be consistent with available control inputs, the modifications to the Lagrangian must satisfy "matching" conditions. The pendulum on a rotor arm requires an interesting generalization of our earlier approach which was used for systems such as a pendulum on a cart.

The control of nonholonomic and underactuated systems with symmetry is illustrated by the problem of controlling a bicycle. We derive a controller which, using steering and rear-wheel torque, causes a model of a riderless bicycle to recover its balance from a near fall as well as converge to a time parameterized path in the ground plane. Our construction utilizes new results for both the derivation of equations of motion for nonholonomic systems with symmetry, as well as the control of underactuated robotic systems.