Balance control of constrained bipedal standing and stability analysis using the concept of Lyapunov exponents

Balancing control of a constrained biped during standing is extremely challenging in that (1) the constraints between the feet and the ground impose bounds on the control torque, which makes balance controller design challenging, and (2) the lack of a single quantitative criterion and an effective tool for stability analysis prohibits the stability analysis of such highly nonlinear systems. There are two objectives of this thesis. The first objective is to systematically study balancing control of a simplified biped using the modeling approach, which includes (i) investigating the effects of the constraints between the feet and the ground during standing, (ii) designing a balancing controller satisfying the constraints, and (iii) analyzing the stability of the constrained bipedal control system using the concept of Lyapunov exponents, where the exponents are calculated from the mathematical model. The results show that the constraints between the feet and the ground have significant effects on balancing control design and thus they must be satisfied. The stability analysis reveals that the stability region determined using the concept of Lyapunov exponents is reasonably close to the one from previous work where the stability was defined based on clinical observations.; This thesis deals with a stability analysis of a constrained biped during standing. The problem is inherently difficult because the corresponding system is highly nonlinear. There is no simple methodology to determine the stability of the system. Lyapunov's stability theory has very limited applicability. Although Lyapunov exponents calculated from either the mathematical model or a time series can be used to characterize the type of stability of the potentially-stable system under consideration, the existing analytical and numerical techniques have been developed for chaotic systems for which at least one exponent is positive. Since in potentially stable systems the largest exponent is either negative or zero, the existing techniques to compute the exponents are inaccurate. The second objective of this thesis is to develop a method for calculating negative Lyapunov exponents using a time series so that the stability of potential stable engineering systems can be studied using the concept of Lyapunov exponents. The balancing control system of the biped is used as an example to demonstrate the efficacy of the proposed method. The time series is generated by computer simulations from the mathematical model. The results show that: (1) The Lyapunov exponents calculated are more accurate. For nonlinear mapping, the minimum average relative error is 6.74%. For linear mapping, the minimum average relative error is 6.84%; (2) The calculated exponents' spectrum is not sensitive to the values of time lag, Tlag, and evolving time, Tevol, while, using linear mapping, the calculated exponents' spectrum is extremely sensitive to above key parameters; (3) No spurious Lyapunov exponents are generated. Some ground work has been laid on applying the concept of Lyapunov exponents to the analysis of stable systems.; The proposed method for calculating Lyapunov exponents based on time series is both systematic and constructive. It has a great potential for obtaining negative Lyapunov exponents and has significant practical applications in engineering systems. Furthermore, the method is not restricted to bipedal robotic systems. It can be used to general nonlinear potentially stable systems, especially for practical engineering systems.