Dynamically Stable 3D Quadrupedal Walking with Multi-Domain Hybrid System Models and Virtual Constraint Controllers

Hybrid systems theory has become a powerful approach for designing feedback controllers that achieve dynamically stable bipedal locomotion, both formally and in practice. This paper presents an analytical framework 1) to address multi-domain hybrid models of quadruped robots with high degrees of freedom, and 2) to systematically design nonlinear controllers that asymptotically stabilize periodic orbits of these sophisticated models. A family of parameterized virtual constraint controllers is proposed for continuous-time domains of quadruped locomotion to regulate holonomic and nonholonomic outputs. The properties of the Poincare return map for the full-order and closed-loop hybrid system are studied to investigate the asymptotic stabilization problem of dynamic gaits. An iterative optimization algorithm involving linear and bilinear matrix inequalities is then employed to choose stabilizing virtual constraint parameters. The paper numerically evaluates the analytical results on a simulation model of an advanced 3D quadruped robot, called GR Vision 60, with 36 state variables and 12 control inputs. An optimal amble gait of the robot is designed utilizing the FROST toolkit. The power of the analytical framework is finally illustrated through designing a set of stabilizing virtual constraint controllers with 180 controller parameters.Comment: American Control Conference 201

Exponential stabilization of driftless nonlinear control systems using homogeneous feedback

This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers

3 sampled-data control of nonlinear systems

This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research

Stabilizing Randomly Switched Systems

This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control designComment: 22 pages. Submitte

Synthesis of Minimal Error Control Software

Software implementations of controllers for physical systems are at the core of many embedded systems. The design of controllers uses the theory of dynamical systems to construct a mathematical control law that ensures that the controlled system has certain properties, such as asymptotic convergence to an equilibrium point, while optimizing some performance criteria. However, owing to quantization errors arising from the use of fixed-point arithmetic, the implementation of this control law can only guarantee practical stability: under the actions of the implementation, the trajectories of the controlled system converge to a bounded set around the equilibrium point, and the size of the bounded set is proportional to the error in the implementation. The problem of verifying whether a controller implementation achieves practical stability for a given bounded set has been studied before. In this paper, we change the emphasis from verification to automatic synthesis. Using synthesis, the need for formal verification can be considerably reduced thereby reducing the design time as well as design cost of embedded control software. We give a methodology and a tool to synthesize embedded control software that is Pareto optimal w.r.t. both performance criteria and practical stability regions. Our technique is a combination of static analysis to estimate quantization errors for specific controller implementations and stochastic local search over the space of possible controllers using particle swarm optimization. The effectiveness of our technique is illustrated using examples of various standard control systems: in most examples, we achieve controllers with close LQR-LQG performance but with implementation errors, hence regions of practical stability, several times as small.Comment: 18 pages, 2 figure