Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.

Hybrid dynamical systems

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Biomechanics And Motor Control Of Human Movement

Bipedal locomotion is among the most difficult challenges in control engineering. Most books treat the subject from a quasi-static perspective, overlooking the hybrid nature of bipedal mechanics. Feedback Control of Dynamic Bipedal Robot Locomotion is the first book to present a comprehensive and mathematically sound treatment of feedback design for achieving stable, agile, and efficient locomotion in bipedal robots.In this unique and groundbreaking treatise, expert authors lead you systematically through every step of the process, including:Mathematical modeling of walking and running gaits in planar robotsAnalysis of periodic orbits in hybrid systemsDesign and analysis of feedback systems for achieving stable periodic motionsAlgorithms for synthesizing feedback controllersDetailed simulation examplesExperimental implementations on two bipedal test bedsThe elegance of the authors' approach is evident in the marriage of control theory and mechanics, uniting control-based presentation and mathematical custom with a mechanics-based approach to the problem and computational rendering. Concrete examples and numerous illustrations complement and clarify the mathematical discussion. A supporting Web site offers links to videos of several experiments along with MATLAB® code for several of the models. This one-of-a-kind book builds a solid understanding of the theoretical and practical aspects of truly dynamic locomotion in planar bipedal robots.

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Feedback Control of Dynamic Bipedal Robot Locomotion

Technological advancements have led to the development of numerous wearable robotic devices for the physical assistance and restoration of human locomotion. While many challenges remain with respect to the mechanical design of such devices, it is at least equally challenging and important to develop strategies to control them in concert with the intentions of the user. This work reviews the state-of-the-art techniques for controlling portable active lower limb prosthetic and orthotic (P/O) devices in the context of locomotive activities of daily living (ADL), and considers how these can be interfaced with the user’s sensory-motor control system. This review underscores the practical challenges and opportunities associated with P/O control, which can be used to accelerate future developments in this field. Furthermore, this work provides a classification scheme for the comparison of the various control strategies. As a novel contribution, a general framework for the control of portable gait-assistance devices is proposed. This framework accounts for the physical and informatic interactions between the controller, the user, the environment, and the mechanical device itself. Such a treatment of P/Os – not as independent devices, but as actors within an ecosystem – is suggested to be necessary to structure the next generation of intelligent and multifunctional controllers. Each element of the proposed framework is discussed with respect to the role that it plays in the assistance of locomotion, along with how its states can be sensed as inputs to the controller. The reviewed controllers are shown to fit within different levels of a hierarchical scheme, which loosely resembles the structure and functionality of the nominal human central nervous system (CNS). Active and passive safety mechanisms are considered to be central aspects underlying all of P/O design and control, and are shown to be critical for regulatory approval of such devices for real-world use. The works discussed herein provide evidence that, while we are getting ever closer, significant challenges still exist for the development of controllers for portable powered P/O devices that can seamlessly integrate with the user’s neuromusculoskeletal system and are practical for use in locomotive ADL.

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Control strategies for active lower extremity prosthetics and orthotics: a review

Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers

Recent powered (or robotic) prosthetic legs independently control different joints and time periods of the gait cycle, resulting in control parameters and switching rules that can be difficult to tune by clinicians. This challenge might be addressed by a unifying control model used by recent bipedal robots, in which virtual constraints define joint patterns as functions of a monotonic variable that continuously represents the gait cycle phase. In the first application of virtual constraints to amputee locomotion, this paper derives exact and approximate control laws for a partial feedback linearization to enforce virtual constraints on a prosthetic leg. We then encode a human-inspired invariance property called effective shape into virtual constraints for the stance period. After simulating the robustness of the partial feedback linearization to clinically meaningful conditions, we experimentally implement this control strategy on a powered transfemoral leg. We report the results of three amputee subjects walking overground and at variable cadences on a treadmill, demonstrating the clinical viability of this novel control approach.

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Virtual constraint control of a powered prosthetic leg: From simulation to experiments with transfemoral amputees

Control systems for powered prosthetic legs typically divide the gait cycle into several periods with distinct controllers, resulting in dozens of control parameters that must be tuned across users and activities. To address this challenge, this paper presents a control approach that unifies the gait cycle of a powered knee–ankle prosthesis using a continuous, user-synchronized sense of phase. Virtual constraints characterize the desired periodic joint trajectories as functions of a phase variable across the entire stride. The phase variable is computed from residual thigh motion, giving the amputee control over the timing of the prosthetic joint patterns. This continuous sense of phase enabled three transfemoral amputee subjects to walk at speeds from 0.67 to 1.21 m/s and slopes from $-2.5^{\circ }$ to +9.0 $^{\circ }$ . Virtual constraints based on task-specific kinematics facilitated normative adjustments in joint work across walking speeds. A fixed set of control gains generalized across these activities and users, which minimized the configuration time of the prosthesis.

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Continuous-Phase Control of a Powered Knee–Ankle Prosthesis: Amputee Experiments Across Speeds and Inclines

The ankle contributes the majority of mechanical power during walking and is a frequently studied joint in biomechanics. Specifically, researchers have extensively investigated the torque-angle relationship for the ankle during dynamic tasks, such as walking and running. The slope of this relationship has been termed the “quasi-stiffness.” However, over time, researchers have begun to interchange the concepts of quasi-stiffness and stiffness. This is an especially important distinction as researchers currently begin to investigate the appropriate control systems for recently developed powered prosthetic legs. The quasi-stiffness and stiffness are distinct concepts in the context of powered joints, and are equivalent in the context of passive joints. The purpose of this paper is to demonstrate the difference between the stiffness and quasi-stiffness using a simple impedance-controlled inverted pendulum model and a more sophisticated biped walking model, each with the ability to modify the trajectory of an impedance controller's equilibrium angle position. In both cases, stiffness values are specified by the controller and the quasi-stiffness are shown during a single step. Both models have widely varying quasi-stiffness but each have a single stiffness value. Therefore, from this simple modeling approach, the differences and similarities between these two concepts are elucidated.

The Difference Between Stiffness and Quasi-Stiffness in the Context of Biomechanical Modeling

Understanding Zeno phenomena plays an important role in understanding hybrid systems. A natural - and intriguing - question to ask is: what happens after a Zeno point? Inspired by the construction of Filippov (1988), we propose a method for extending Zeno executions past a Zeno point for a class of hybrid systems: Lagrangian hybrid systems. We argue that after the Zeno point is reached, the hybrid system should switch to a holonomically constrained dynamical system, where the holonomic constraints are based on the unilateral constraints on the configuration space that originally defined the hybrid system. These principles are substantiated with a series of examples.

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Is there life after Zeno? Taking executions past the breaking (Zeno) point

The phase of human gait is difficult to quantify accurately in the presence of disturbances. In contrast, recent bipedal robots use time-independent controllers relying on a mechanical phase variable to synchronize joint patterns through the gait cycle. This concept has inspired studies to determine if human joint patterns can also be parameterized by a mechanical variable. Although many phase variable candidates have been proposed, it remains unclear which, if any, provide a robust representation of phase for human gait analysis or control. In this paper we analytically derive an ideal phase variable (the hip phase angle) that is provably monotonic and bounded throughout the gait cycle. To examine the robustness of this phase variable, ten able-bodied human subjects walked over a platform that randomly applied phase-shifting perturbations to the stance leg. A statistical analysis found the correlations between nominal and perturbed joint trajectories to be significantly greater when parameterized by the hip phase angle (0.95+) than by time or a different phase variable. The hip phase angle also best parameterized the transient errors about the nominal periodic orbit. Finally, interlimb phasing was best explained by local (ipsilateral) hip phase angles that are synchronized during the double-support period.

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Robert D. Gregg

Papers

Virtual constraint control of a powered prosthetic leg: From simulation to experiments with transfemoral amputees

Continuous-Phase Control of a Powered Knee–Ankle Prosthesis: Amputee Experiments Across Speeds and Inclines

The Difference Between Stiffness and Quasi-Stiffness in the Context of Biomechanical Modeling

Is there life after Zeno? Taking executions past the breaking (Zeno) point

A Robust Parameterization of Human Gait Patterns Across Phase-Shifting Perturbations