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The control system in surgical robots must ensure patient safety and real time control. As such, all the uncertainties which could appear should be considered into an extended model of the plant. After such an uncertain plant is formed, an adequate controller which ensures a minimum set of performances for each situation should be computed. As such, the continuous-time robust control paradigm is suitable for such scenarios. However, the problem is generally solved only for linear and time invariant plants. The main focus of the current paper is to include m-link serial surgical robots into Robust Control Framework by considering all nonlinearities as uncertainties. Moreover, the paper studies an incipient problem of numerical implementation of such control structures.

We consider distributionally robust optimal control of stochastic linear systems under signal temporal logic (STL) chance constraints when the disturbance distribution is unknown. By assuming that the underlying predicate functions are Lipschitz continuous and the noise realizations are drawn from a distribution having a concentration of measure property, we first formulate the underlying chance-constrained control problem as stochastic programming with constraints on expectations and propose a solution using a distributionally robust approach based on the Wasserstein metric. We show that by choosing a proper Wasserstein radius, the original chance-constrained optimization can be satisfied with a user-defined confidence level. A numerical example illustrates the efficacy of the method., Comment: 8 pages and 1 fiugre

Many optimal and robust control problems are nonconvex and potentially nonsmooth in their policy optimization forms. In Part II of this paper, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our ECL offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. Therefore, no spurious stationarity exists in the set of non-degenerate policies. This ECL framework can cover many benchmark control problems, including state feedback linear quadratic regulator (LQR), dynamic output feedback linear quadratic Gaussian (LQG) control, and $\mathcal{H}_\infty$ robust control. ECL can also handle a class of distributed control problems when the notion of quadratic invariance (QI) holds. We further show that all static stabilizing policies are non-degenerate for state feedback LQR and $\mathcal{H}_\infty$ control under standard assumptions. We believe that the new ECL framework may be of independent interest for analyzing nonconvex problems beyond control.

This paper presents a novel Wasserstein distributionally robust control and state estimation algorithm for partially observable linear stochastic systems, where the probability distributions of disturbances and measurement noises are unknown. Our method consists of the control and state estimation phases to handle distributional ambiguities of system disturbances and measurement noises, respectively. Leveraging tools from modern distributionally robust optimization, we consider an approximation of the control problem with an arbitrary nominal distribution and derive its closed-form optimal solution. We show that the separation principle holds, thereby allowing the state estimator to be designed separately. A novel distributionally robust Kalman filter is then proposed as an optimal solution to the state estimation problem with Gaussian nominal distributions. Our key contribution is the combination of distributionally robust control and state estimation into a unified algorithm. This is achieved by formulating a tractable semidefinite programming problem that iteratively determines the worst-case covariance matrices of all uncertainties, leading to a scalable and efficient algorithm. Our method is also shown to enjoy a guaranteed cost property as well as a probabilistic out-of-sample performance guarantee. The results of our numerical experiments demonstrate the performance and computational efficiency of the proposed method.

End-to-end engineering design pipelines, in which designs are evaluated using concurrently defined optimal controllers, are becoming increasingly common in practice. To discover designs that perform well even under the misspecification of system dynamics, such end-to-end pipelines have now begun evaluating designs with a robust control objective in place of the nominal optimal control setup. Current approaches of specifying such robust control subproblems, however, rely on hand specification of perturbations anticipated to be present upon deployment or margin methods that ignore problem structure, resulting in a lack of theoretical guarantees and overly conservative empirical performance. We, instead, propose a novel methodology for LQR systems that leverages conformal prediction to specify such uncertainty regions in a data-driven fashion. Such regions have distribution-free coverage guarantees on the true system dynamics, in turn allowing for a probabilistic characterization of the regret of the resulting robust controller. We then demonstrate that such a controller can be efficiently produced via a novel policy gradient method that has convergence guarantees. We finally demonstrate the superior empirical performance of our method over alternate robust control specifications in a collection of engineering control systems, specifically for airfoils and a load-positioning system.

We introduce a novel method for safe mobile robot navigation in dynamic, unknown environments, utilizing onboard sensing to impose safety constraints without the need for accurate map reconstruction. Traditional methods typically rely on detailed map information to synthesize safe stabilizing controls for mobile robots, which can be computationally demanding and less effective, particularly in dynamic operational conditions. By leveraging recent advances in distributionally robust optimization, we develop a distributionally robust control barrier function (DR-CBF) constraint that directly processes range sensor data to impose safety constraints. Coupling this with a control Lyapunov function (CLF) for path tracking, we demonstrate that our CLF-DR-CBF control synthesis method achieves safe, efficient, and robust navigation in uncertain dynamic environments. We demonstrate the effectiveness of our approach in simulated and real autonomous robot navigation experiments, marking a substantial advancement in real-time safety guarantees for mobile robots., Comment: Submitted to the International Journal of Robotics Research (IJRR). Project page: https://existentialrobotics.org/DR_Safe_Navigation_Webpage

Learning-based techniques have become popular in both model predictive control (MPC) and reinforcement learning (RL). Probabilistic ensemble (PE) models offer a promising approach for modelling system dynamics, showcasing the ability to capture uncertainty and scalability in high-dimensional control scenarios. However, PE models are susceptible to mode collapse, resulting in non-robust control when faced with environments slightly different from the training set. In this paper, we introduce the $\textbf{u}$ncertainty-$\textbf{d}$riven rob$\textbf{u}$st $\textbf{c}$ontrol (UDUC) loss as an alternative objective for training PE models, drawing inspiration from contrastive learning. We analyze the robustness of UDUC loss through the lens of robust optimization and evaluate its performance on the challenging Real-world Reinforcement Learning (RWRL) benchmark, which involves significant environmental mismatches between the training and testing environments.

The paper presents a robust control technique that combines the Control Lyapunov function and Hamilton-Jacobi Reachability to compute a controller and its Region of Attraction (ROA). The Control Lyapunov function uses a linear system model with an assumed additive uncertainty to calculate a control gain and the level sets of the ROA as a function of the uncertainty. Next, Hamilton-Jacobi reachability uses the nonlinear model with the modeled uncertainty, which need not be additive, to compute the backward reachable set (BRS). Finally, by juxtaposing the level sets of the ROA with BRS, we can calculate the worst-case additive disturbance and the ROA of the nonlinear model. We illustrate our approach on a 2D quadcopter tracking trajectory and a 2D quadcopter with height and velocity regulation in simulation.

Real-time navigation in a priori unknown environment remains a challenging task, especially when an unexpected (unmodeled) disturbance occurs. In this paper, we propose the framework Safe Returning Fast and Safe Tracking (SR-F) that merges concepts from 1) Robust Control Lyapunov-Value Functions (R-CLVF), and 2) the Fast and Safe Tracking (FaSTrack) framework. The SR-F computes an R-CLVF offline between a model of the true system and a simplified planning model. Online, a planning algorithm is used to generate a trajectory in the simplified planning space, and the R-CLVF is used to provide a tracking controller that exponentially stabilizes to the planning model. When an unexpected disturbance occurs, the proposed SR-F algorithm provides a means for the true system to recover to the planning model. We take advantage of this mechanism to induce an artificial disturbance by ``jumping'' the planning model in open environments, forcing faster navigation. Therefore, this algorithm can both reject unexpected true disturbances and accelerate navigation speed. We validate our framework using a 10D quadrotor system and show that SR-F is empirically 20\% faster than the original FaSTrack while maintaining safety., Comment: 6 pages, 4 figures, 1 table, 2 algorithms. Submitted to LCSS on 03/06

As uncertainty considerations become increasingly important aspects of concurrent plant and control optimization, it is imperative to identify and compare the impact of uncertain control co-design (UCCD) formulations on their associated solutions. While previous work developed the theory for various UCCD formulations, their implementation, along with an in-depth discussion of the structure of UCCD problems, implicit assumptions, method-dependent considerations, and practical insights, is currently missing from the literature. Therefore, in this study, we address some of these limitations by focusing on UCCD formulations, with an emphasis on optimal control structures, and uncertainty propagation techniques. Specifically, we propose three optimal control structures for UCCD problems: (i) open-loop multiple-control (OLMC), (ii) multi-stage control (MSC), and (iii) open-loop single-control (OLSC). Stochastic in expectation UCCD (SE-UCCD) and worst-case robust UCCD (WCR-UCCD) formulations, which are motivated by probabilistic and crisp representations of uncertainties, respectively, are implemented for a simplified strain-actuated solar array case study. Solutions to the OLMC SE-UCCD problem are obtained using two uncertainty propagation techniques: generalized Polynomial Chaos expansion (gPC) and Monte Carlo simulation (MCS). The OLMC and MSC WCR-UCCD problems are solved by leveraging the structure of the linear program, leading to polytopic uncertainties. To highlight the importance of uncertainty in early-stage design, the closed-loop reference-tracking response of the systems is also investigated. Insights from such studies underscore the role of the control structure in managing the trade-offs between risk and performance, as well as meeting problem requirements. The results also emphasize the benefits of efficient uncertainty propagation techniques for dynamic optimization problems., Comment: 18 pages and 11 figures

The discovery of linear embedding is the key to the synthesis of linear control techniques for nonlinear systems. In recent years, while Koopman operator theory has become a prominent approach for learning these linear embeddings through data-driven methods, these algorithms often exhibit limitations in generalizability beyond the distribution captured by training data and are not robust to changes in the nominal system dynamics induced by intrinsic or environmental factors. To overcome these limitations, this study presents an adaptive Koopman architecture capable of responding to the changes in system dynamics online. The proposed framework initially employs an autoencoder-based neural network that utilizes input-output information from the nominal system to learn the corresponding Koopman embedding offline. Subsequently, we augment this nominal Koopman architecture with a feed-forward neural network that learns to modify the nominal dynamics in response to any deviation between the predicted and observed lifted states, leading to improved generalization and robustness to a wide range of uncertainties and disturbances compared to contemporary methods. Extensive tracking control simulations, which are undertaken by integrating the proposed scheme within a Model Predictive Control framework, are used to highlight its robustness against measurement noise, disturbances, and parametric variations in system dynamics., Comment: Corrected the title

The abstraction of dynamical systems is a powerful tool that enables the design of feedback controllers using a correct-by-design framework. We investigate a novel scheme to obtain data-driven abstractions of discrete-time stochastic processes in terms of richer discrete stochastic models, whose actions lead to nondeterministic transitions over the space of probability measures. The data-driven component of the proposed methodology lies in the fact that we only assume samples from an unknown probability distribution. We also rely on the model of the underlying dynamics to build our abstraction through backward reachability computations. The nondeterminism in the probability space is captured by a collection of Markov Processes, and we identify how this model can improve upon existing abstraction techniques in terms of satisfying temporal properties, such as safety or reach-avoid. The connection between the discrete and the underlying dynamics is made formal through the use of the scenario approach theory. Numerical experiments illustrate the advantages and main limitations of the proposed techniques with respect to existing approaches., Comment: 6 pages

Discrete-time robust optimal control problems generally take a min-max structure over continuous variable spaces, which can be difficult to solve in practice. In this paper, we extend the class of such problems that can be solved through a previously proposed local reduction method to consider those with existence constraints on the uncountable variables. We also consider the possibility of non-unique trajectories that satisfy equality and inequality constraints. Crucially, we show that the problems of interest can be cast into a standard semi-infinite program and demonstrate how to generate optimal uncertainty scenario sets in order to obtain numerical solutions. We also include examples on model predictive control for obstacle avoidance with logical conditions, control with input saturation affected by uncertainty, and optimal parameter estimation to highlight the need for the proposed extension. Our method solves each of the examples considered, producing violation-free and locally optimal solutions., Comment: 6 pages, 1 figure. To be published in the proceedings of the 8th IFAC Conference on Nonlinear Model Predictive Control (NMPC 2024)

The Koopman operator approach for data-driven control design of a nonlinear system is on the rise because of its capability to capture the behaviours of global dynamics. However, the measurement noises of inputs and outputs will bias the Koopman model identification and cause model mismatch from the actual nonlinear dynamics. The current work evaluates the bounds of the noise-induced model bias of the Koopman operator model and proposes a data-driven robust dual-loop control framework (Koopman based robust control-KROC) for the biased model. First, the model mismatch is found bounded under radial basis functions (RBF) and the bounded noises, and the bound of model mismatch is assessed. Second, the pitfalls of linear quadratic Gaussian (LQG) control based on the biased Koopman model of Van Der Pol oscillator are shown. Motivated from the pitfalls, the dual-loop control is proposed, which consist of an observer-based state-feedback control based on the nominal Koopman model and an additional robust loop to compensate model mismatch. A linear matrix inequality (LMI) is derived, which can guarantee robust stability and performance under bounded noises for the finite-dimensional Koopman operator model. Finally, the proposed framework is implemented to a nonlinear Van Der Pol oscillator to demonstrate enhanced control performance by the dual-loop robust control.

Control Lyapunov Functions (CLFs) have been extensively used in the control community. A well-known drawback is the absence of a systematic way to construct CLFs for general nonlinear systems, and the problem can become more complex with input or state constraints. Our preliminary work on constructing Control Lyapunov Value Functions (CLVFs) using Hamilton-Jacobi (HJ) reachability analysis provides a method for finding a non-smooth CLF. In this paper, we extend our work on CLVFs to systems with bounded disturbance and define the Robust CLVF (R-CLVF). The R-CLVF naturally inherits all properties of the CLVF; i.e., it first identifies the "smallest robust control invariant set (SRCIS)" and stabilizes the system to it with a user-specified exponential rate. The region from which the exponential rate can be met is called the "region of exponential stabilizability (ROES)." We provide clearer definitions of the SRCIS and more rigorous proofs of several important theorems. Since the computation of the R-CLVF suffers from the "curse of dimensionality," we also provide two techniques (warmstart and system decomposition) that solve it, along with necessary proofs. Three numerical examples are provided, validating our definition of SRCIS, illustrating the trade-off between a faster decay rate and a smaller ROES, and demonstrating the efficiency of computation using warmstart and decomposition., Comment: 13 pages, 5 figures

Model uncertainty poses a significant challenge to the implementation of safety-critical control systems. With this as motivation, this paper proposes a safe control design approach that guarantees the robustness of nonlinear feedback systems in the presence of matched or unmatched unmodelled system dynamics and external disturbances. Our approach couples control barrier functions (CBFs) with a new uncertainty/disturbance estimator to ensure robust safety against input and state-dependent model uncertainties. We prove upper bounds on the estimator's error and estimated outputs. We use an uncertainty estimator-based composite feedback control law to adaptively improve robust control performance under hard safety constraints by compensating for the matched uncertainty. Then, we robustify existing CBF constraints with this uncertainty estimate and the estimation error bounds to ensure robust safety via a quadratic program (CBF-QP). We also extend our method to higher-order CBFs (HOCBFs) to achieve safety under unmatched uncertainty, which causes relative degree differences with respect to control input and disturbance. We assume the relative degree difference is at most one, resulting in a second-order cone (SOC) condition. The proposed robust HOCBFs method is demonstrated in a simulation of an uncertain elastic actuator control problem. Finally, the efficacy of our method is experimentally demonstrated on a tracked robot with slope-induced matched and unmatched perturbations.

This paper presents a hierarchical control framework that enables robust quadrupedal locomotion on a dynamic rigid surface (DRS) with general and unknown vertical motions. The key novelty of the framework lies in its higher layer, which is a discrete-time, provably stabilizing footstep controller. The basis of the footstep controller is a new hybrid, time-varying, linear inverted pendulum (HT-LIP) model that is low-dimensional and accurately captures the essential robot dynamics during DRS locomotion. A new set of sufficient stability conditions are then derived to directly guide the controller design for ensuring the asymptotic stability of the HT-LIP model under general, unknown, vertical DRS motions. Further, the footstep controller is cast as a computationally efficient quadratic program that incorporates the proposed HT-LIP model and stability conditions. The middle layer takes the desired footstep locations generated by the higher layer as input to produce kinematically feasible full-body reference trajectories, which are then accurately tracked by a lower-layer torque controller. Hardware experiments on a Unitree Go1 quadrupedal robot confirm the robustness of the proposed framework under various unknown, aperiodic, vertical DRS motions and uncertainties (e.g., slippery and uneven surfaces, solid and liquid loads, and sudden pushes).

This paper proposes a safety-critical controller for dynamic and uncertain environments, leveraging a robust environment control barrier function (ECBF) to enhance the robustness against the measurement and prediction uncertainties associated with moving obstacles. The approach reduces conservatism, compared with a worst-case uncertainty approach, by incorporating a state observer for obstacles into the ECBF design. The controller, which guarantees safety, is achieved through solving a quadratic programming problem. The proposed method's effectiveness is demonstrated via a dynamic obstacle-avoidance problem for an autonomous vehicle, including comparisons with established baseline approaches.

This article investigates the problem of controlling linear time-invariant systems subject to time-varying and a priori unknown cost functions, state and input constraints, and exogenous disturbances. We combine the online convex optimization framework with tools from robust model predictive control to propose an algorithm that is able to guarantee robust constraint satisfaction. The performance of the closed loop emerging from application of our framework is studied in terms of its dynamic regret, which is proven to be bounded linearly by the variation of the cost functions and the magnitude of the disturbances. We corroborate our theoretical findings and illustrate implementational aspects of the proposed algorithm by a numerical case study of a tracking control problem of an autonomous vehicle., Comment: 16 pages

Aerial manipulator, which is composed of an UAV (Unmanned Aerial Vehicle) and a multi-link manipulator and can perform aerial manipulation, has shown great potential of applications. However, dynamic coupling between the UAV and the manipulator makes it difficult to control the aerial manipulator with high performance. In this paper, system modeling and control problem of the aerial manipulator are studied. Firstly, an UAV dynamic model is proposed with consideration of the dynamic coupling from an attached manipulator, which is treated as disturbance for the UAV. In the dynamic model, the disturbance is affected by the variable inertia parameters of the aerial manipulator system. Then, based on the proposed dynamic model, a disturbance compensation robust $H_{\infty}$ controller is designed to stabilize flight of the UAV while the manipulator is in operation. Finally, experiments are conducted and the experimental results demonstrate the feasibility and validity of the proposed control scheme.