Kevin S. Galloway

Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics

This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models-systems with impulse effects-through control Lyapunov functions. The periodic orbit is assumed to lie in a C1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions.

Control lyapunov functions and hybrid zero dynamics

Hybrid zero dynamics extends the Byrnes-Isidori notion of zero dynamics to a class of hybrid models called systems with impulse effects. Specifically, given a smooth submanifold that is contained in the zero set of an output function and is invariant under both the continuous flow of the system with impulse effects as well as its reset map, the restriction dynamics is called the hybrid zero dynamics. Prior results on the stabilization of periodic orbits of the hybrid zero dynamics have relied on input-output linearization of the transverse variables. The principal result of this paper shows how control Lyapunov functions can be used to exponentially stabilize periodic orbits of the hybrid zero dynamics, thereby significantly extending the class of stabilizing controllers. An illustration of this result on a model of a bipedal walking robot is provided.

Preliminary walking experiments with underactuated 3D bipedal robot MARLO

This paper reports on an underactuated 3D bipedal robot with passive feet that can start from a quiet standing position, initiate a walking gait, and traverse the length of the laboratory (approximately 10 m) at a speed of roughly 1 m/s. The controller was developed using the method of virtual constraints, a control design method first used on the planar point-feet robots Rabbit and MABEL. For the preliminary experiments reported here, virtual constraints were experimentally tuned to achieve robust planar walking and then 3D walking. A key feature of the controller leading to successful 3D walking is the particular choice of virtual constraints in the lateral plane, which implement a lateral balance control strategy similar to SIMBICON. To our knowledge, MARLO is the most highly underactuated bipedal robot to walk unassisted in 3D.

Torque Saturation in Bipedal Robotic Walking Through Control Lyapunov Function-Based Quadratic Programs

This paper presents a novel method to address the actuator saturation for nonlinear hybrid systems by directly incorporating user-defined input bounds in a controller design. In particular, we consider the application of bipedal walking and show that our method [based on a quadratic programming (QP) implementation of a control Lyapunov function (CLF)-based controller] enables a gradual performance degradation while still continuing to walk under increasingly stringent input bounds. We draw on our previous work, which has demonstrated the effectiveness of the CLF-based controllers for stabilizing periodic gaits for biped walkers. This paper presents a framework, which results in more effective handling of control saturations and provides a means for incorporating a whole family of user-defined constraints into the online computation of a CLF-based controller. This paper concludes with an experimental validation of the main results on the bipedal robot MABEL, demonstrating the usefulness of the QP-based CLF approach for real-time robotic control.

Geometry of cyclic pursuit

Pursuit strategies (formulated using constant-speed particle models) provide a means for achieving cohesive behavior in systems of multiple mobile agents. In the present paper, we explore an n-agent cyclic pursuit scheme (i.e. agent i pursues agent i+1, modulo n) in which each agent employs a constant bearing pursuit strategy. We demonstrate the existence of an invariant submanifold, and state necessary and sufficient conditions for the existence of rectilinear and circling relative equilibria on that submanifold. We present a full analysis of steady-state solutions and stability characteristics for two-particle “mutual CB pursuit” and then outline steps to extend the nonlinear stability analysis to the many particle case.

Symmetry and reduction in collectives: cyclic pursuit strategies

We specify and analyse models that capture the geometry of purposeful motion of a collective of mobile agents, with a focus on planar motion, dyadic strategies and attention graphs which are static, directed and cyclic. Strategies are formulated as constraints on joint shape space and are implemented through feedback laws for the actions of individual agents, here modelled as self-steering particles. By reduction to a labelled shape space (using a redundant parametrization to account for cycle closure constraints) and a further reduction through time rescaling, we characterize various special solutions (relative equilibria and pure shape equilibria) for cyclic pursuit with a constant bearing (CB) strategy. This is accomplished by first proving convergence of the (nonlinear) dynamics to an invariant manifold (the CB pursuit manifold), and then analysing the closed-loop dynamics restricted to the invariant manifold. For illustration, we sketch some low-dimensional examples. This formulation—involving strategies, attention graphs and sensor-driven steering laws—and the resulting templates of collective motion, are part of a broader programme to interpret the mechanisms underlying biological collective motion.

Cyclic pursuit in three dimensions

Pursuit strategies for interacting particles and feedback laws to execute them are formulated in three dimensions, focusing on constant bearing (CB) pursuit - a case of interest in biology. In the analysis of such laws for the setting of n particles engaged in cyclic pursuit, we reveal interesting invariant manifold dynamics and associated explicit integrability properties, as well as conditions for special solutions such as relative equilibria.

Motion camouflage in a stochastic setting

Recent work has formulated 2- and 3-dimensional models and steering control laws for motion camouflage, a stealthy pursuit strategy observed in nature. Here we extend the model to encompass the use of a high-gain pursuit law in the presence of sensor noise as well as in the case when the evaders steering is driven by a stochastic process, demonstrating (in the planar setting) that motion camouflage is still accessible (in the mean) in finite time. We also discuss a family of admissible stochastic evader controls, laying out the groundwork for a future game-theoretic study of optimal evasion strategies.

Station keeping through beacon-referenced cyclic pursuit

This paper investigates a modification of cyclic CB pursuit in a multi-agent system in which each agent pays attention to a neighbor and a beacon. The problem admits shape equilibria with collective circling about the beacon, with the circling radius and angular separation of agents determined by choice of parameters in the feedback law. Stability of circling shape equilibria is shown for a 2-agent system, and the results are demonstrated on a collective of mobile robots tracked by a motion capture system.

Stability and pure shape equilibria for beacon-referenced cyclic pursuit

Cyclic pursuit systems provide a means to generate useful global behaviors in a collective of autonomous agents based on dyadic pursuit interactions between neighboring agents in a cycle graph. Here we consider a modified version of the cyclic pursuit framework in which a stationary beacon provides an additional reference for the agents in the system. Building on the framework proposed in our previous work, we derive necessary conditions for stability of circling equilibria in the n-agent system. Furthermore, we employ a change of variables to reveal the existence of a family of invariant manifolds related to spiral motions which maintain the formation shape up to geometric similarity.