Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking

This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered jumps in the state evolution, potentially accompanied by mode switching in the control vector field as well. For a given trajectory with state jumps, we show how to construct an approximation of a nearby perturbed trajectory corresponding to a small variation of the initial condition and input. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal to those of the nominal one. The main contribution of this work is the development of a notion of error to clarify in which sense the approximate trajectory is, at each instant of time, a firstorder approximation of the perturbed trajectory. This notion of error naturally finds application in the (local) tracking problem of a time-varying reference trajectory of a hybrid system. To illustrate the possible use of this new error definition in the context of trajectory tracking, we outline how the standard linear trajectory tracking control for nonlinear systems -based on linear quadratic regulator (LQR) theory to compute the optimal feedback gain- could be generalized for hybrid systems

Analysis of collocated feedback controllers for four-bar planar mechanisms with joint clearances

International audienceThis article presents an analysis of two-dimensional four-bar mechanisms with joint clearance, when one joint is actuated by collocated open-loop or state feedback controllers (proportional-derivative, state feedback linearization, passivity-based control). The study is led with numerical simulations obtained with a projected Moreau-Jean's event-capturing algorithm. The contact/impact model uses kinematic coefficients of restitution, and Coulomb's friction. The focus is put on how much the performance deteriorates when clearances are added in the joints. It is shown that collocated feedback controllers behave in a very robust way

Passivity-based switching control of flexible-joint complementarity mechanical systems

In this study one considers the tracking control problem of a class of nonsmooth Lagrangian systems with flexible joints and subject to frictionless unilateral constraints. The task under consideration consists of a succession of free-motion and constrained-motion phases. A particular attention is paid to impacting and detachment phases. A passivity-based switching controller that allows to extend the stability analysis described in our previous works to the case of systems with lumped flexibilities, is proposed. Numerical tests show the effectiveness of the controller

On the discretisation of actuation in locomotion: Impulse- and shape-based modelling for hopping robots

In an age where computers challenge the smartest human beings in cognitive tasks, the conspicuous discrepancy between robot and animal locomotion appears paradoxical. While animals can move around autonomously in complex environments, today’s robots struggle to independently operate in such surroundings. There are many reasons for robots’ inferior performance, but arguably the most important one is our missing understanding of complexity. This thesis introduces the notion of discrete actuation for the study of locomotion in robots and animals. The actuation of a system with discrete actuation is restricted to be applied at a finite number of instants in time and is impulsive. We find that, despite their simplicity, such systems can predict various experimental observations and inspire novel technologies for robot design and control. We further find that, through the study of discrete actuation, causal relationships between actuation and resulting behaviour are revealed and become quantifiable, which relates the findings presented in this thesis to the broader concepts of complexity, self-organisation, and self-stability. We present four case studies in Chapters 3-6 which demonstrate how the concept of discrete actuation can be employed to understand the physics of locomotion and to facilitate novel robot technologies. We first introduce the impulsive eccentric wheel model which is a discretely actuated system for the study of hopping locomotion. We find that the model predicts robot hopping trajectories and animal related hopping characteristics by reducing the dynamics of hopping–usually described by hybrid differential equations–to analytic maps. The reduction of complexity of the model equations reveals the underlying physics of the locomotion process, and we identify the importance of robot shape and mass distribution for the locomotion performance. As a concrete application of the model, we compare the energetics of hopping and rolling locomotion in environments with obstacles and find when it is better to hop than to roll, based on the fundamental physical principles we discover in the model analysis. The theoretical insights of this modelling approach enable new actuation techniques and design for robots which we display in Robbit; a robot that uses strictly convex foot shapes and rotational impulses to induce hopping locomotion. We show that such systems outperform hopping with non-strictly convex shapes in terms of energy effective and robust locomotion. A system with discrete actuation motivates the exploitation of shape and the environment to improve locomotion dynamics, which reveals advantageous effect of inelastic impacts between the robot foot and the environment. We support this idea with experimental results from the robot CaneBot which can change its foot shape to induce timed impacts with the environment. Even though inelastic impacts are commonly considered detrimental for locomotion dynamics, we show that their appropriate control improves the locomotion speed considerably. The findings presented in this thesis show that discrete actuation for locomotion inspires novel ways to appreciate locomotion dynamics and facilitates unique control and design technologies for robots. Furthermore, discrete actuation emphasises the definition of causality in complex systems which we believe will bring robots closer to the locomotion behaviour of animals, enabling more agile and energy effective robots

Observer-based Synchronization of Multi-agent Systems Using Intermittent Output Measurements

The problem of synchronizing multiple continuous-time linear time-invariant systems connected over a complex network, with intermittently available measurements of their outputs, is considered. To solve this problem, we propose a distributed observer-based feedback controller that utilizes a local hybrid observer to estimate neighboring states only from output measurements at such potentially nonperiodic isolated event times. Due to the inherent continuous and discrete dynamics emerging from coupling the impulsive measurement updates and the interconnected networked systems, we use hybrid systems to model and analyze the resulting closed-loop system. The problem of synchronization and state estimation is then recast as a set stabilization problem, and, utilizing a Lyapunov-based analysis for hybrid systems, we provide sufficient conditions for global exponential stability of the synchronization and zero estimation error set. A numerical example is provided to illustrate the results

Semi-analytical method to study piecewise linear oscillators

Acknowledgements The authors would like to thank the Balseiro Institute and the National Commission of Atomic Energy for the support. In particular, financial support for the Invited Professor Programme of the Balseiro Institute, which made international collaboration possible.Peer reviewedPublisher PD

Feedback Control of a Bipedal Walker and Runner with Compliance.

This dissertation contributes to the theoretical foundations of robotic bipedal locomotion and advances the experimental state of the art as well. On the theoretical side, a mathematical formalism for designing provably stable, walking and running gaits in bipedal robots with compliance is presented. A key contribution is a novel method of force control in robots with compliance. The theoretical work is validated experimentally on MABEL, a planar bipedal testbed that contains springs in its drivetrain for the purpose of enhancing both energy efficiency and agility of dynamic locomotion. While the potential energetic benefits of springs are well documented in the literature, feedback control designs that effectively realize this potential are lacking. The methods of virtual constraints and hybrid zero dynamics, originally developed for rigid robots with a single degree of underactuation, are extended and applied to MABEL, which has a novel compliant transmission and multiple degrees of underactuation. A time-invariant feedback controller is designed such that the closed-loop system respects the natural compliance of the open-loop system and realizes exponentially stable walking gaits. A second time-invariant feedback controller is designed such that the closed-loop system not only respects the natural compliance of the open-loop system, but also enables active force control within the compliant hybrid zero dynamics and results in exponentially stable running gaits. Several experiments are presented that highlight different aspects of MABEL and the feedback design method, ranging from basic elements such as stable walking, robustness under perturbations, energy efficient walking to a bipedal robot walking speed record of 1.5 m/s (3.4 mph), stable running with passive feet and with point feet. On MABEL, the full hybrid zero dynamics controller is implemented and was instrumental in achieving rapid walking and running, leading upto a kneed bipedal running speed record of 3.06 m/s (6.8 mph).Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/89801/1/koushils_1.pd

Expert-in-the-Loop Multilateral Telerobotics for Haptics-Enabled Motor Function and Skills Development

Among medical robotics applications are Robotics-Assisted Mirror Rehabilitation Therapy (RAMRT) and Minimally-Invasive Surgical Training (RAMIST) that extensively rely on motor function development. Haptics-enabled expert-in-the-loop motor function development for such applications is made possible through multilateral telerobotic frameworks. While several studies have validated the benefits of haptic interaction with an expert in motor learning, contradictory results have also been reported. This emphasizes the need for further in-depth studies on the nature of human motor learning through haptic guidance and interaction. The objective of this study was to design and evaluate expert-in-the-loop multilateral telerobotic frameworks with stable and human-safe control loops that enable adaptive “hand-over-hand” haptic guidance for RAMRT and RAMIST. The first prerequisite for such frameworks is active involvement of the patient or trainee, which requires the closed-loop system to remain stable in the presence of an adaptable time-varying dominance factor. To this end, a wave-variable controller is proposed in this study for conventional trilateral teleoperation systems such that system stability is guaranteed in the presence of a time-varying dominance factor and communication delay. Similar to other wave-variable approaches, the controller is initially developed for the Velocity-force Domain (VD) based on the well-known passivity assumption on the human arm in VD. The controller can be applied straightforwardly to the Position-force Domain (PD), eliminating position-error accumulation and position drift, provided that passivity of the human arm in PD is addressed. However, the latter has been ignored in the literature. Therefore, in this study, passivity of the human arm in PD is investigated using mathematical analysis, experimentation as well as user studies involving 12 participants and 48 trials. The results, in conjunction with the proposed wave-variables, can be used to guarantee closed-loop PD stability of the supervised trilateral teleoperation system in its classical format. The classic dual-user teleoperation architecture does not, however, fully satisfy the requirements for properly imparting motor function (skills) in RAMRT (RAMIST). Consequently, the next part of this study focuses on designing novel supervised trilateral frameworks for providing motor learning in RAMRT and RAMIST, each customized according to the requirements of the application. The framework proposed for RAMRT includes the following features: a) therapist-in-the-loop mirror therapy; b) haptic feedback to the therapist from the patient side; c) assist-as-needed therapy realized through an adaptive Guidance Virtual Fixture (GVF); and d) real-time task-independent and patient-specific motor-function assessment. Closed-loop stability of the proposed framework is investigated using a combination of the Circle Criterion and the Small-Gain Theorem. The stability analysis addresses the instabilities caused by: a) communication delays between the therapist and the patient, facilitating haptics-enabled tele- or in-home rehabilitation; and b) the integration of the time-varying nonlinear GVF element into the delayed system. The platform is experimentally evaluated on a trilateral rehabilitation setup consisting of two Quanser rehabilitation robots and one Quanser HD2 robot. The framework proposed for RAMIST includes the following features: a) haptics-enabled expert-in-the-loop surgical training; b) adaptive expertise-oriented training, realized through a Fuzzy Interface System, which actively engages the trainees while providing them with appropriate skills-oriented levels of training; and c) task-independent skills assessment. Closed-loop stability of the architecture is analyzed using the Circle Criterion in the presence and absence of haptic feedback of tool-tissue interactions. In addition to the time-varying elements of the system, the stability analysis approach also addresses communication delays, facilitating tele-surgical training. The platform is implemented on a dual-console surgical setup consisting of the classic da Vinci surgical system (Intuitive Surgical, Inc., Sunnyvale, CA), integrated with the da Vinci Research Kit (dVRK) motor controllers, and the dV-Trainer master console (Mimic Technology Inc., Seattle, WA). In order to save on the expert\u27s (therapist\u27s) time, dual-console architectures can also be expanded to accommodate simultaneous training (rehabilitation) for multiple trainees (patients). As the first step in doing this, the last part of this thesis focuses on the development of a multi-master/single-slave telerobotic framework, along with controller design and closed-loop stability analysis in the presence of communication delays. Various parts of this study are supported with a number of experimental implementations and evaluations. The outcomes of this research include multilateral telerobotic testbeds for further studies on the nature of human motor learning and retention through haptic guidance and interaction. They also enable investigation of the impact of communication time delays on supervised haptics-enabled motor function improvement through tele-rehabilitation and mentoring