Applied Safety Critical Control

There is currently a clear gap between control-theoretical results and the reality of robotic implementation, in the sense that it is very difficult to transfer analytical guarantees to practical ones. This is especially problematic when trying to design safety-critical systems where failure is not an option. While there is a vast body of work on safety and reliability in control theory, very little of it is actually used in practice where safety margins are typically empiric and/or heuristic. Nevertheless, it is still widely accepted that a solution to these problems can only emerge from rigorous analysis, mathematics, and methods. In this work, we therefore seek to help bridge this gap by revisiting and expanding existing theoretical results in light of the complexity of hardware implementation. To that end, we begin by making a clear theoretical distinction between systems and models, and outline how the two need to be related for guarantees to transfer from the latter to the former. We then formalize various imperfections of reality that need to be accounted for at a model level to provide theoretical results with better applicability. We then discuss the reality of digital controller implementation and present the mathematical constraints that theoretical control laws must satisfy for them to be implementable on real hardware. In light of these discussions, we derive new realizable set-invariance conditions that, if properly enforced, can guarantee safety with an arbitrary high levels of confidence. We then discuss how these conditions can be rigorously enforced in a systematic and minimally invasive way through convex optimization-based Safety Filters. Multiple safety filter formulations are proposed with varying levels of complexity and applicability. To enable the use of these safety filters, a new algorithm is presented to compute appropriate control invariant sets and guarantee feasibility of the optimization problem defining these filters. The effectiveness of this approach is demonstrated in simulation on a nonlinear inverted pendulum and experimentally on a simple vehicle. The aptitude of the framework to handle a system's dynamics uncertainty is illustrated by varying the mass of the vehicle and showcasing when safety is conserved. Then, the aptitude of this approach to provide guarantees that account for controller implementation's constraints is illustrated by varying the frequency of the control loop and again showcasing when safety is conserved. In the second part of this work, we revisit the safety filtering approach in a way that addresses the scalability issues of the first part of this work. There are two main approaches to safety-critical control. The first one relies on computation of control invariant sets and was presented in the first part of this work. The second approach draws from the topic of optimal control and relies on the ability to realize Model-Predictive-Controllers online to guarantee the safety of a system. In that online approach, safety is ensured at a planning stage by solving the control problem subject for some explicitly defined constraints on the state and control input. Both approaches have distinct advantages but also major drawbacks that hinder their practical effectiveness, namely scalability for the first one and computational complexity for the second one. We therefore present an approach that draws from the advantages of both approaches to deliver efficient and scalable methods of ensuring safety for nonlinear dynamical systems. In particular, we show that identifying a backup control law that stabilizes the system is in fact sufficient to exploit some of the set-invariance conditions presented in the first part of this work. Indeed, one only needs to be able to numerically integrate the closed-loop dynamics of the system over a finite horizon under this backup law to compute all the information necessary for evaluating the regulation map and enforcing safety. The effect of relaxing the stabilization requirements of the backup law is also studied, and weaker but more practical safety guarantees are brought forward. We then explore the relationship between the optimality of the backup law and how conservative the resulting safety filter is. Finally, methods of selecting a safe input with varying levels of trade-off between conservativeness and computational complexity are proposed and illustrated on multiple robotic systems, namely: a two-wheeled inverted pendulum (Segway), an industrial manipulator, a quadrotor, and a lower body exoskeleton.</p

Towards Variable Assistance for Lower Body Exoskeletons

This letter presents and experimentally demonstrates a novel framework for variable assistance on lower body exoskeletons, based upon safety-critical control methods. Existing work has shown that providing some freedom of movement around a nominal gait, instead of rigidly following it, accelerates the spinal learning process of people with a walking impediment when using a lower body exoskeleton. With this as motivation, we present a method to accurately control how much a subject is allowed to deviate from a given gait while ensuring robustness to patient perturbation. This method leverages control barrier functions to force certain joints to remain inside predefined trajectory tubes in a minimally invasive way. The effectiveness of the method is demonstrated experimentally with able-bodied subjects and the Atalante lower body exoskeleton

Stabilization of Exoskeletons through Active Ankle Compensation

This paper presents an active stabilization method for a fully actuated lower-limb exoskeleton. The method was tested on the exoskeleton ATALANTE, which was designed and built by the French start-up company Wandercraft. The main objective of this paper is to present a practical method of realizing more robust walking on hardware through active ankle compensation. The nominal gait was generated through the hybrid zero dynamic framework. The ankles are individually controlled to establish three main directives; (1) keeping the non-stance foot parallel to the ground, (2) maintaining rigid contact between the stance foot and the ground, and (3) closing the loop on pelvis orientation to achieve better tracking. Each individual component of this method was demonstrated separately to show each component's contribution to stability. The results showed that the ankle controller was able to experimentally maintain static balance in the sagittal plane while the exoskeleton was balanced on one leg, even when disturbed. The entire ankle controller was then also demonstrated on crutch-less dynamic walking. During testing, an anatomically correct manikin was placed in the exoskeleton, in lieu of a paraplegic patient. The pitch of the pelvis of the exoskeleton-manikin system was shown to track the gait trajectory better when ankle compensation was used. Overall, active ankle compensation was demonstrated experimentally to improve balance in the sagittal plane of the exoskeleton manikin system and points to an improved practical approach for stable walking

A Scalable Safety Critical Control Framework for Nonlinear Systems

There are two main approaches to safety-critical control. The first one relies on computation of control invariant sets and is presented in the first part of this work. The second approach draws from the topic of optimal control and relies on the ability to realize Model-Predictive-Controllers online to guarantee the safety of a system. In the second approach, safety is ensured at a planning stage by solving the control problem subject for some explicitly defined constraints on the state and control input. Both approaches have distinct advantages but also major drawbacks that hinder their practical effectiveness, namely scalability for the first one and computational complexity for the second. We therefore present an approach that draws from the advantages of both approaches to deliver efficient and scalable methods of ensuring safety for nonlinear dynamical systems. In particular, we show that identifying a backup control law that stabilizes the system is in fact sufficient to exploit some of the set-invariance conditions presented in the first part of this work. Indeed, one only needs to be able to numerically integrate the closed-loop dynamics of the system over a finite horizon under this backup law to compute all the information necessary for evaluating the regulation map and enforcing safety. The effect of relaxing the stabilization requirements of the backup law is also studied, and weaker but more practical safety guarantees are brought forward. We then explore the relationship between the optimality of the backup law and how conservative the resulting safety filter is. Finally, methods of selecting a safe input with varying levels of trade-off between conservatism and computational complexity are proposed and illustrated on multiple robotic systems, namely: a two-wheeled inverted pendulum (Segway), an industrial manipulator, a quadrotor, and a lower body exoskeleton

Feedback Control of an Exoskeleton for Paraplegics: Toward Robustly Stable Hands-free Dynamic Walking

This manuscript presents control of a high-DOF fully actuated lower-limb exoskeleton for paraplegic individuals. The key novelty is the ability for the user to walk without the use of crutches or other external means of stabilization. We harness the power of modern optimization techniques and supervised machine learning to develop a smooth feedback control policy that provides robust velocity regulation and perturbation rejection. Preliminary evaluation of the stability and robustness of the proposed approach is demonstrated through the Gazebo simulation environment. In addition, preliminary experimental results with (complete) paraplegic individuals are included for the previous version of the controller.Comment: Submitted to IEEE Control System Magazine. This version addresses reviewers' concerns about the robustness of the algorithm and the motivation for using such exoskeleton

A Scalable Safety Critical Control Framework for Nonlinear Systems

There are two main approaches to safety-critical control. The first one relies on computation of control invariant sets and is presented in the first part of this work. The second approach draws from the topic of optimal control and relies on the ability to realize Model-Predictive-Controllers online to guarantee the safety of a system. In the second approach, safety is ensured at a planning stage by solving the control problem subject for some explicitly defined constraints on the state and control input. Both approaches have distinct advantages but also major drawbacks that hinder their practical effectiveness, namely scalability for the first one and computational complexity for the second. We therefore present an approach that draws from the advantages of both approaches to deliver efficient and scalable methods of ensuring safety for nonlinear dynamical systems. In particular, we show that identifying a backup control law that stabilizes the system is in fact sufficient to exploit some of the set-invariance conditions presented in the first part of this work. Indeed, one only needs to be able to numerically integrate the closed-loop dynamics of the system over a finite horizon under this backup law to compute all the information necessary for evaluating the regulation map and enforcing safety. The effect of relaxing the stabilization requirements of the backup law is also studied, and weaker but more practical safety guarantees are brought forward. We then explore the relationship between the optimality of the backup law and how conservative the resulting safety filter is. Finally, methods of selecting a safe input with varying levels of trade-off between conservatism and computational complexity are proposed and illustrated on multiple robotic systems, namely: a two-wheeled inverted pendulum (Segway), an industrial manipulator, a quadrotor, and a lower body exoskeleton