Optimal Trajectory Generation via Double Generating Functions and Application to Biped Robots

When a robot is walking in a complex environment, it needs to adjust its step length and walking speed for each step (very probably). This implies that the initial position and/or velocity, the designed terminal position and/or velocity, and the walking time period for each step are very often different. From this view point, the optimal gait generation problem for a biped robot is equivalent to a family of optimal control problems parameterized by the boundary conditions. The finite time optimal control problem with fixed initial and terminal state values can be deduced to a two-point boundary-value problem (TPBVP) for ordinary differential equations (ODEs) with respect to a Hamiltonian system. The shooting method is a conventional method for TPBVP. Since the basic principle of the shooting method is computing the trajectory repeatedly so that the exact one satisfying the boundary values is obtained. It needs to solve the TPBVP (or run the program) again if we change the desired boundary values. Also, the model predictive control (MPC) can be applied to solve finite time optimal control problems. However, since MPC is based on iterative, finite horizon optimization of a dynamic model, the iterative optimization increases the online computational burden. Recently, a method based on single generating function is proposed to solve the optimal control problems with general types of boundary conditions. This method allows one to obtain a family of optimal trajectories for different boundary conditions by integrating the system dynamic equation with a family of optimal control inputs. In this thesis, for a finite time optimal control problem with fixed boundary condition, the optimal trajectories are given as functions of the boundary values of the sate and the endpoints of the time interval by using double generating functions. The expressions of the optimal trajectories are given in the form of the Taylor series with respect to the boundary values of the state (the initial value and the terminal value of the state). The coefficients of the Taylor series iv Abstract can be calculated off-line and are same for any boundary condition. Therefore, the on-demand computation time for different boundary conditions decreases greatly compared with the conventional method, e.g., the shooting method. Since the finite time optimal control problem can be reduced to a TPBVP for a Hamiltonian system, the solution of the optimal control problem is derived from double generating functions by canonical transformations for the Hamiltonian systems. The generating functions are solutions of Hamilton-Jacobi equations (HJEs). This method is called the double generating functions method. For Linear-Quadratic (LQ) optimal control problems, the generating functions are quadratic with respect to the state and the initial/terminal value of the costate (or the costate and the initial/terminal value of the state). Double generating functions generate a parametrization of optimal trajectories for different boundary conditions and different time periods. It is very convenient to generate optimal trajectories for different boundary conditions. Furthermore, it is proved that a pair of generating functions with the same time direction causes numerical instability in calculating optimal trajectories. For nonlinear optimal control problems, it is difficult to solve HJE for generating functions. A systematic algorithm is proposed to obtain an approximate solution to HJE. This algorithm allows one to calculate the Taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ODEs recursively. Furthermore, the coefficients of the Taylor series expansion of the generating function can be computed exactly under a certain technical condition. The double generating functions method is extended to the nonlinear case. The optimal trajectories and inputs are given in the forms of the Taylor series with respect to the boundary values of the state. Also, a systematic procedure with an appropriate data structure is developed to compute the coefficients of the Taylor series solution recursively up to any order off-line. Lastly, the double generating functions method is applied to the on-demand optimal gaits generation of a compass biped robot walking on the level ground. It is employed to generate reference optimal trajectories and inputs considering the energy consumption by linearizing the compass biped robot. The simulation result shows that the double generating functions method might be useful to optimal gaits generation for biped robots.