On The Finite-time Stability Analysis And Gait Optimal Control For Dynamic Bipedal Robot

The dynamic bipedal robots have possessed the advantages of low energy consumption and humanoid walking. The dynamic bipedal robots which convert kinetic energy and potential energy in the process of the movement into drive energy can walk stable periodic motion under the working conditions and multi-tasking in the little energy. Recently, the theory of dynamic walking has been widely applied in the development bipedal robot prototype. However, the theory of analysis and synthesis for dynamic bipedal robots is relatively lag behind the development of the prototype. It is far from achieving a perfect situation about gait optimization control and robust stability for dynamic bipedal robots. Especially, the control theory and approach of the finite-time stability, robust stability and gait optimization and nonlinear numerical optimization problems for dynamic bipedal robot need to be further research. In this paper, we have mainly studied four key problems for the dynamic bipedal robots, for instance, the problem of finite-time stability, robust stability and gait optimization control and nonlinear numerical optimization problems. Furthermore, we have enriched the stability theory for a class of impulsive hybrid dynamic system represented by dynamic bipedal robots. It can establish the unified framework of finite-time stability analysis for impulsive hybrid dynamical systems. What’s more, it can enhance the efficiency of dynamic walking for the biped robots. It has achieved dynamical walking gait in low energy consumption, high efficiency, anthropomorph gait. The concrete research contents and innovation works include the following several aspects:First, the rapidly convergence nonlinear numerical optimal approaches are developed for the nonlinear optimal problems, for instance, nonlinear conjugate gradient method, spectral conjugate gradient method and trust region-SQP method. Two modified three-term type conjugate gradient algorithms and modified spectral conjugate gradient methods which satisfy both the descent condition and the Dai-Liao type conjugacy condition are presented for unconstrained optimization problems. Moreover, these properties are independent of the step length line search rules. Under some mild conditions, the given methods are global convergence, which is independent of the Wolfe line search for general functions. The numerical experiments show that the proposed methods are very robust and efficient. Subsequently, inequality constrained programming problems are discussed, based on a combination technique of a trust region method and an SQP method, a new feasible algorithm is proposed. A “compression”technique is used such that search direction is feasible for convex polyhedron based on the QP subproblem. Therefore, the QP subproblems are always consistent and the direction of the QP subproblems is always a feasible direction. We use high order revised direction to avoid Maratos effect. If the search directions satisfy the trust region search rules, the high order directions will not solved by the algorithm, therefore, it can simple the structure of the algorithm and enhance the computational efficiency. The numerical results show that the algorithm is effective. These algorithms established theoretical basis and algorithm framework for optimal trajectories and control signals and robust controller of bipedal walking robots.Second, a novel finite-time stability is proposed using control Lyapunov function, hybrid zero dynamics and Poincare map method for hybrid dynamical systems with impulsive effects problems. Assume that the periodic orbit of continuous dynamics lies in an invariant manifold that is contained in the zero set of out put function. Furthermore, a universal finite-time stabilizing controller is redesigned for hybrid systems. The primary result of this paper is that the periodic orbit of nonlinear systems convergence to the zero dynamics surface in finite-time through a variant of control Lyapunov functions. It can be used to guarantee that the continuous dynamics of hybrid systems is finite-time stability. Finite-time stability analysis of discrete dynamics of hybrid systems is proposed using Poincare fixed point theorem. Due to the control Lyapunov function and the fixed point theorem, the flow of continuous dynamics can transverse to the switching surface of discrete dynamics in finite-time. According to the finite-time stability, the kinematical and dynamical mechanism of bipedal walking robot is analyzed. Therefore, it can unify framework for the hybrid dynamical systems with impulsive effects problems.Third, a finite-time stabilizing optimal robust controller is proposed for robust stability problems of bipedal walking robots. For parameter perturbation problems, an optimal robust controller is constructed by the finite-time stabilizing control Lyapunov function. According to the robust controller, the robust stability of bipedal walking robots is analyzed. Combined finite-time stabilizing control Lyapunov function and torque saturation conditions, the optimal robust controller problems can be transformed into searching for the solutions of the nonlinear optimization problem with equality constraints. An online convex algorithm is developed for the optimal robust controller of nonlinear optimization problem with equality constraints. The bipedal walking robot can highly efficient and stable walking.Fourth, two nonlinear optimal methods are developed for periodic gait optimization problem of the bipedal walking robot. Based on the discrete mechanics and optimal control, functional extremum problems can be transformed into searching for the solutions of the nonlinear optimization problem with equality constraints. On the one hand, a class of global and feasible sequential quadratic programming algorithm(FSQPA) is proposed. The optimal controls and trajectories are solved by the modified FSQPA. The algorithm can rapidly converge to a stable gait cycle by selecting an appropriate initial gait, otherwise, the algorithm only needs one step correction which generates a stable gait cycle. Under appropriate conditions, we provide a rigorous proof of global convergence and well-defined properties for the FSQPA. Numerical results show that the algorithm is feasible and effective. Meanwhile, it reveals the movement mechanism in the process of bipedal dynamic walking, which is the velocity oscillations. Furthermore, we overcome the oscillatory behavior via the FSQPA, which makes the bipedal robot walk efficiently and stably on the even terrain. The main result is illustrated on a hybrid model of a compass-like robot through simulations and is utilized to achieve bipedal locomotion via FSQPA. To demonstrate the effectiveness of the high dimensional bipedal robot systems, we will conduct numerical simulations on the model of robot with nonlinear, hybrid and underactuated dynamics. Numerical simulation results show that the FSQPA is feasible and effective. On the other hand, a class of smoothing penalty function method is developed for periodic gait optimization problem of bipedal walking robot. The optimal control strategy and trajectory are solved by a new smoothing exact penalty function algorithm. The algorithm can quickly converge to a stable gait cycle independent the selection of the initial gait, otherwise, the algorithm only needs one step correction and then generate a stable gait cycle. Numerical simulation results show that the algorithm is feasible and effective. It can simple the structure of the algorithm and enhance the computational efficiency. The algorithm makes the bipedal robot walk efficiently and stably on the even terrain.Finally, the main content of this dissertation is summarized, and further works are discussed.