This paper uses geometric methods to study basic problems in the mechanics and control of locomotion. We consider in detail the case of "undulatory locomotion," in which net motion is generated by coupling internal shape changes with external nonholonomic constraints. Such locomotion problems have a natural geometric interpretation as a connection on a principal fiber bundle. The properties of connections lead to simplified results for studying both dynamics and issues of controllability for locomotion systems. We demonstrate the utility of this approach using a novel "Snakeboard" and a multi-segmented serpentine robot which is modeled after Hirose's Active Cord Mechanism.

This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, x = X 1 (x)u 1 + \Delta \Delta \Delta + Xm (x)um ; x 2 R n : Such systems arise when modeling mechanical systems with nonholonomic constraints. In engineering applications it is often required to maintain the mechanical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equilibrium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a non-standard dilation. Even though the approximation can be given a coordinate-free interpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous structure in the close...

This thesis deals with the use of logic-based switching in the control of imprecisely modeled nonlinear systems. Each control system considered consists of a continuous-time dynamical process to be controlled, a family of candidate controllers, and an event-driven switching logic. The need for switching arises when no single candidate controller is capable, by itself, of guaranteeing good performance when connected with a poorly modeled process. In this thesis we develop provably correct switching strategies capable of determining in real-time which candidate controller should be put in feedback with a process so as to achieve a desired closed-loop performance. The resulting closed-loop systems are hybrid in the sense that in each case, continuous dynamics interact with event-driven logic. In the process of designing these switching algorithms, we develop several tools for the analysis and synthesis o...

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A constructive method for time-varying stabilization of smooth driftless controllable systems is developed. It provides time-varying homogeneous feedback laws that are continuous and smooth away from the origin. These feedbacks make the closed-loop system globally exponentially asymptotically stable if the control system is homogeneous with respect to a family of dilations and, using local homogeneous approximation of control systems, locally exponentially asymptotically stable otherwise. The method uses some known algorithms that construct oscillatory control inputs to approximate motion in the direction of iterated Lie brackets that we adapt to the closed-loop context.

This paper demonstrates how to stabilize a nonholonomic integrator using a hybrid control law employing switching and logic. Results concerning asymptotic stability and exponentially fast convergence to the origin are derived. The methodology used seems to be generalizable to a larger class of control problems related to nonholonomic systems.

Systems in canonical power form have recently been used to model the kinematic equations of nonholonomic mechanical systems. In [14, 15], McCloskey and Murray have had the idea of using the properties of homogeneous systems to derive exponentially stabilizing continuous time-periodic feedbacks for this class of systems. Motivated by this work, the present study extends a control design method previously proposed by Samson to the design of such homogeneous feedbacks. The approach here followed has the advantage of yielding simple and direct stability proofs. Homogeneity-related results needed for time-varying exponential stabilization are also provided.

It is known that the kinematic model of several nonholonomic systems can be converted into a chained form control system. Asymptotical stabilization of any equilibrium point of this system cannot be achieved by means of a continuous pure state feedback, but can be obtained by using a time-varying continuous feedback [20]. In the present paper, a backstepping technique is used to derive explicit time-varying feedbacks that ensure exponential stability of the closed-loop system. Two classes of control laws are proposed, with one of them involving a dynamic extension of the original chained system. Like in other recent studies on the same topic, exponential convergence is obtained by using the properties associated with homogeneous systems. The control laws so obtained are continuous in both the state and time variables. A complementary and novel feature of the proposed control design technique lies in the estimation of a lower bound of the asymptotical rate of convergence as a function ...

Abstract: This paper considers the task of trajectory stabilization for a fish-like robot by means of feedback. We use oscillatory control inputs and apply correction signals at the endpoints of each periodic input signal. Such a strategy can be proven to cause the system to converge to a desired trajectory. We present a specific model of a planar carangiform fish, and verify the stabilization results with simulations and with experiment on a planar robotic fish system that is propelled using carangiform-like movements. 1

In this note a continuous feedback control law with time-periodic terms is derived for the control of nonholonomic systems in power form. The control law is derived by Lyapunov design from a homogeneous Lyapunov function. Global asymptotic stability is shown by applying the principle of invariance for time-periodic systems. Exponential convergence follows since the vector fields are homogeneous of degree zero. I. INTRODUCTION Systems in power form have in the recent years been used to model the kinematic equations of nonholonomic mechanical systems as e.g. nonholonomic wheeled vehicles [1]. Two types of control laws have been derived to solve this problem which can not be solved with static state feedback, namely piecewise continuous and time-varying control laws. In this note time-varying control will be addressed. Asymptotic stability with exponential convergence for systems in power form has been achieved using timevarying control laws by Pomet and Samson [2], M'Closkey and Murray ...