Joel W. Burdick

Matanya B. Horowitz, Anil Damle, Joel W. Burdick

The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finite-horizon, average cost, and first-exit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively.

Prithvi Akella, Anushri Dixit, Mohamadreza Ahmadi et al.

The dramatic increase of autonomous systems subject to variable environments has given rise to the pressing need to consider risk in both the synthesis and verification of policies for these systems. This paper aims to address a few problems regarding risk-aware verification and policy synthesis, by first developing a sample-based method to bound the risk measure evaluation of a random variable whose distribution is unknown. These bounds permit us to generate high-confidence verification statements for a large class of robotic systems. Second, we develop a sample-based method to determine solutions to non-convex optimization problems that outperform a large fraction of the decision space of possible solutions. Both sample-based approaches then permit us to rapidly synthesize risk-aware policies that are guaranteed to achieve a minimum level of system performance. To showcase our approach in simulation, we verify a cooperative multi-agent system and develop a risk-aware controller that outperforms the system's baseline controller. We also mention how our approach can be extended to account for any $g$-entropic risk measure - the subset of coherent risk measures on which we focus.

Prithvi Akella, Skylar X. Wei, Joel W. Burdick et al.

Recent advances in safety-critical risk-aware control are predicated on apriori knowledge of the disturbances a system might face. This paper proposes a method to efficiently learn these disturbances online, in a risk-aware context. First, we introduce the concept of a Surface-at-Risk, a risk measure for stochastic processes that extends Value-at-Risk -- a commonly utilized risk measure in the risk-aware controls community. Second, we model the norm of the state discrepancy between the model and the true system evolution as a scalar-valued stochastic process and determine an upper bound to its Surface-at-Risk via Gaussian Process Regression. Third, we provide theoretical results on the accuracy of our fitted surface subject to mild assumptions that are verifiable with respect to the data sets collected during system operation. Finally, we experimentally verify our procedure by augmenting a drone's controller and highlight performance increases achieved via our risk-aware approach after collecting less than a minute of operating data.

Pol Mestres, Shima Sadat Mousavi, Pio Ong et al.

This paper studies the efficient implementation of safety filters that are designed using control barrier functions (CBFs), which minimally modify a nominal controller to render it safe with respect to a prescribed set of states. Although CBF-based safety filters are often implemented by solving a quadratic program (QP) in real time, the use of off-the-shelf solvers for such optimization problems poses a challenge in applications where control actions need to be computed efficiently at very high frequencies. In this paper, we introduce a closed-form expression for controllers obtained through CBF-based safety filters. This expression is obtained by partitioning the state-space into different regions, with a different closed-form solution in each region. We leverage this formula to introduce a resource-aware implementation of CBF-based safety filters that detects changes in the partition region and uses the closed-form expression between changes. We showcase the applicability of our approach in examples ranging from aerospace control to safe reinforcement learning.

David E. J. van Wijk, Tamas G. Molnar, Samuel Coogan et al.

Guaranteeing the safety of controllers is vital for real-world applications, but is markedly difficult when the states are not perfectly known and when the control inputs are bounded. Backup control barrier functions (bCBFs) use predictions of the flow under a prescribed controller to achieve safety in the presence of bounded inputs and perfect state information. However, when only an estimate of the true state is known, this flow may not be precisely computed, as the initial condition is unknown. Furthermore, the true flow evolves using feedback from the estimated state, thus introducing coupling between known and unknown flows. To address these challenges, we propose a technique that leverages an uncertainty envelope centered around the estimated flow and show that ensuring the safety of this envelope guarantees that the true state satisfies the safety constraints. Additionally, we show that in the presence of state uncertainty, using the resulting Output Feedback Backup Control Barrier Functions (O-bCBFs), there always exists a feasible control input that can guarantee the safety of the true state, even in the presence of input constraints.

Meg Wilkinson, Gilbert Bahati, Ryan M. Bena et al.

Collision avoidance for robotic manipulators requires enforcing full-body safety constraints in high-dimensional configuration spaces. Control Barrier Function (CBF) based safety filters have proven effective in enabling safe behaviors, but enforcing the high number of constraints needed for safe manipulation leads to theoretic and computational challenges. This work presents a framework for full-body collision avoidance for manipulators in dynamic environments by leveraging 3D Poisson Safety Functions (PSFs). In particular, given environmental occupancy data, we sample the manipulator surface at a prescribed resolution and shrink free space via a Pontryagin difference according to this resolution. On this buffered domain, we synthesize a globally smooth CBF by solving Poisson's equation, yielding a single safety function for the entire environment. This safety function, evaluated at each sampled point, yields task-space CBF constraints enforced by a real-time safety filter via a multi-constraint quadratic program. We prove that keeping the sample points safe in the buffered region guarantees collision avoidance for the entire continuous robot surface. The framework is validated on a 7-degree-of-freedom manipulator in dynamic environments.

Pio Ong, David E. J. van Wijk, Massimiliano de Sa et al.

Control barrier functions (CBFs) provide a principled framework for enforcing safety in control systems -- yet the certified safe operating region in practice is often conservative, especially under input bounds. In many applications, multiple smaller safe sets can be certified independently, e.g., around distinct equilibria with different stabilizing controllers. This paper proposes a framework for uniting such regions into a single certified safe set using \emph{combinatorial CBFs}. We refine the combinatorial CBF framework by introducing an auxiliary variable that enables logical compositions of individual CBFs. In the proposed framework, we show that such compositions yield a \emph{generalized combinatorial CBF} under a condition termed \emph{conjunctive compatibility}. Building on this result, we extend the framework to enable the aggregation of multiple implicit safe sets generated by the backup CBF framework. We show that the resulting CBF-based quadratic program yields a continuous safety filter over the aggregated safe region. The approach is demonstrated on two spacecraft safety problems, safe attitude control and safe station keeping, where multiple certified safe regions are combined to expand the operational envelope.

Masih Haseli, Jorge Cortés, Joel W. Burdick

This paper extends the forward-backward consistency index, originally introduced in Koopman modeling of systems without input, to the setting of control systems, providing a closed-form computable measure of accuracy for data-driven models associated with the Koopman Control Family (KCF). Building on a forward-backward regression perspective, we introduce the control forward-backward consistency matrix and demonstrate that it possesses several favorable properties. Our main result establishes that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of the control consistency index, defined as the maximum eigenvalue of the consistency matrix. This provides a sharp, closed-form computable error bound for finite-dimensional KCF models. We further specialize this bound to the widely used lifted linear and bilinear models. We also discuss how the control consistency index can be incorporated into optimization-based modeling and illustrate the methodology via simulations.

David E. J. van Wijk, Dohyun Lee, Ersin Das et al.

Guaranteeing the safety of nonlinear systems with bounded inputs remains a key challenge in safe autonomy. Backup control barrier functions (bCBFs) provide a powerful mechanism for constructing controlled invariant sets by propagating trajectories under a pre-verified backup controller to a forward invariant backup set. While effective, the standard bCBF method utilizes the same backup controller for both set expansion and safety certification, which can restrict the expanded safe set and lead to conservative dynamic behavior. In this study, we generalize the bCBF framework by separating the set-expanding controller from the verified backup controller, thereby enabling a broader class of expansion strategies while preserving formal safety guarantees. We establish sufficient conditions for forward invariance of the resulting implicit safe set and show how the generalized construction recovers existing bCBF methods as special cases. Moreover, we extend the proposed framework to parameterized controller families, enabling online adaptation of the expansion controller while maintaining safety guarantees in the presence of input bounds.

Stella Kombo, Masih Haseli, Skylar Wei et al.

Autonomous systems often must predict the motions of nearby agents from partial and noisy data. This paper asks and answers the question: "can we learn, in real-time, a nonlinear predictive model of another agent's motions?" Our online framework denoises and forecasts such dynamics using a modified sliding-window Hankel Dynamic Mode Decomposition (Hankel-DMD). Partial noisy measurements are embedded into a Hankel matrix, while an associated Page matrix enables singular-value hard thresholding (SVHT) to estimate the effective rank. A Cadzow projection enforces structured low-rank consistency, yielding a denoised trajectory and local noise variance estimates. From this representation, a time-varying Hankel-DMD lifted linear predictor is constructed for multi-step forecasts. The residual analysis provides variance-tracking signals that can support downstream estimators and risk-aware planning. We validate the approach in simulation under Gaussian and heavy-tailed noise, and experimentally on a dynamic crane testbed. Results show that the method achieves stable variance-aware denoising and short-horizon prediction suitable for integration into real-time control frameworks.

Kazuya Echigo, David E. J. van Wijk, Pol Mestres et al.

Safety-critical control systems, such as spacecraft performing proximity operations, must provide formal safety guarantees despite stochastic uncertainties from state estimation and unmodeled dynamics. Although Control Barrier Functions (CBFs) have been extended to stochastic systems, existing approaches typically face a trade-off between the tightness of probabilistic guarantees and computational tractability. This paper presents a particle-based probabilistic CBF framework that overcomes this limitation by exploiting the sub-Gaussian structure of the barrier function increment under Gaussian uncertainties. We establish that Gaussian uncertainties propagating through Lipschitz-continuous control-affine dynamics preserve sub-Gaussianity of the barrier function increment, with explicit tail bounds. Leveraging this structure, we derive finite-sample bounds on the approximation error between particle-based Conditional Value at Risk (CVaR) estimates and ground-truth probabilistic constraints; applying this yields a tractable optimization problem formulation with finite-sample safety certificates. We show through numerical experiments how the proposed approach provides tight yet provably valid probabilistic safety guarantees.

Vince Kurtz, Joel W. Burdick

Generative control policies have recently unlocked major progress in robotics. These methods produce action sequences via diffusion or flow matching, with training data provided by demonstrations. But existing methods come with two key limitations: they require expert demonstrations, which can be difficult to obtain, and they are limited to relatively slow, quasi-static tasks. In this paper, we leverage a tight connection between sampling-based predictive control and generative modeling to address each of these issues. In particular, we introduce generative predictive control, a supervised learning framework for tasks with fast dynamics that are easy to simulate but difficult to demonstrate. We then show how trained flow-matching policies can be warm-started at inference time, maintaining temporal consistency and enabling high-frequency feedback. We believe that generative predictive control offers a complementary approach to existing behavior cloning methods, and hope that it paves the way toward generalist policies that extend beyond quasi-static demonstration-oriented tasks.

Carl Folkestad, Skylar X. Wei, Joel W. Burdick

Nonlinear dynamical effects are crucial to the operation of many agile robotic systems. Koopman-based model learning methods can capture these nonlinear dynamical system effects in higher dimensional lifted bilinear models that are amenable to optimal control. However, standard methods that lift the system state using a fixed function dictionary before model learning result in high dimensional models that are intractable for real time control. This paper presents a novel method that jointly learns a function dictionary and lifted bilinear model purely from data by incorporating the Koopman model in a neural network architecture. Nonlinear MPC design utilizing the learned model can be performed readily. We experimentally realized this method on a multirotor drone for agile trajectory tracking at low altitudes where the aerodynamic ground effect influences the system's behavior. Experimental results demonstrate that the learning-based controller achieves similar performance as a nonlinear MPC based on a nominal dynamics model in medium altitude. However, our learning-based system can reliably track trajectories in near-ground flight regimes while the nominal controller crashes due to unmodeled dynamical effects that are captured by our method.

Carl Folkestad, Joel W. Burdick

Koopman-based learning methods can potentially be practical and powerful tools for dynamical robotic systems. However, common methods to construct Koopman representations seek to learn lifted linear models that cannot capture nonlinear actuation effects inherent in many robotic systems. This paper presents a learning and control methodology that is a first step towards overcoming this limitation. Using the Koopman canonical transform, control-affine dynamics can be expressed by a lifted bilinear model. The learned model is used for nonlinear model predictive control (NMPC) design where the bilinear structure can be exploited to improve computational efficiency. The benefits for control-affine dynamics compared to existing Koopman-based methods are highlighted through an example of a simulated planar quadrotor. Prediction error is greatly reduced and closed loop performance similar to NMPC with full model knowledge is achieved.

Mohamadreza Ahmadi, Anushri Dixit, Joel W. Burdick et al.

We consider the stochastic shortest path planning problem in MDPs, i.e., the problem of designing policies that ensure reaching a goal state from a given initial state with minimum accrued cost. In order to account for rare but important realizations of the system, we consider a nested dynamic coherent risk total cost functional rather than the conventional risk-neutral total expected cost. Under some assumptions, we show that optimal, stationary, Markovian policies exist and can be found via a special Bellman's equation. We propose a computational technique based on difference convex programs (DCPs) to find the associated value functions and therefore the risk-averse policies. A rover navigation MDP is used to illustrate the proposed methodology with conditional-value-at-risk (CVaR) and entropic-value-at-risk (EVaR) coherent risk measures.

Richard Cheng, Richard M. Murray, Joel W. Burdick

When autonomous robots interact with humans, such as during autonomous driving, explicit safety guarantees are crucial in order to avoid potentially life-threatening accidents. Many data-driven methods have explored learning probabilistic bounds over human agents' trajectories (i.e. confidence tubes that contain trajectories with probability $δ$), which can then be used to guarantee safety with probability $1-δ$. However, almost all existing works consider $δ\geq 0.001$. The purpose of this paper is to argue that (1) in safety-critical applications, it is necessary to provide safety guarantees with $δ< 10^{-8}$, and (2) current learning-based methods are ill-equipped to compute accurate confidence bounds at such low $δ$. Using human driving data (from the highD dataset), as well as synthetically generated data, we show that current uncertainty models use inaccurate distributional assumptions to describe human behavior and/or require infeasible amounts of data to accurately learn confidence bounds for $δ\leq 10^{-8}$. These two issues result in unreliable confidence bounds, which can have dangerous implications if deployed on safety-critical systems.

Yidan Qin, Max Allan, Yisong Yue et al.

Surgical state estimators in robot-assisted surgery (RAS) - especially those trained via learning techniques - rely heavily on datasets that capture surgeon actions in laboratory or real-world surgical tasks. Real-world RAS datasets are costly to acquire, are obtained from multiple surgeons who may use different surgical strategies, and are recorded under uncontrolled conditions in highly complex environments. The combination of high diversity and limited data calls for new learning methods that are robust and invariant to operating conditions and surgical techniques. We propose StiseNet, a Surgical Task Invariance State Estimation Network with an invariance induction framework that minimizes the effects of variations in surgical technique and operating environments inherent to RAS datasets. StiseNet's adversarial architecture learns to separate nuisance factors from information needed for surgical state estimation. StiseNet is shown to outperform state-of-the-art state estimation methods on three datasets (including a new real-world RAS dataset: HERNIA-20).

Anushri Dixit, Mohamadreza Ahmadi, Joel W. Burdick

We consider the problem of risk-sensitive motion planning in the presence of randomly moving obstacles. To this end, we adopt a model predictive control (MPC) scheme and pose the obstacle avoidance constraint in the MPC problem as a distributionally robust constraint with a KL divergence ambiguity set. This constraint is the dual representation of the Entropic Value-at-Risk (EVaR). Building upon this viewpoint, we propose an algorithm to follow waypoints and discuss its feasibility and completion in finite time. We compare the policies obtained using EVaR with those obtained using another common coherent risk measure, Conditional Value-at-Risk (CVaR), via numerical experiments for a 2D system. We also implement the waypoint following algorithm on a 3D quadcopter simulation.

Kejun Li, Maegan Tucker, Erdem Bıyık et al.

Characterizing what types of exoskeleton gaits are comfortable for users, and understanding the science of walking more generally, require recovering a user's utility landscape. Learning these landscapes is challenging, as walking trajectories are defined by numerous gait parameters, data collection from human trials is expensive, and user safety and comfort must be ensured. This work proposes the Region of Interest Active Learning (ROIAL) framework, which actively learns each user's underlying utility function over a region of interest that ensures safety and comfort. ROIAL learns from ordinal and preference feedback, which are more reliable feedback mechanisms than absolute numerical scores. The algorithm's performance is evaluated both in simulation and experimentally for three non-disabled subjects walking inside of a lower-body exoskeleton. ROIAL learns Bayesian posteriors that predict each exoskeleton user's utility landscape across four exoskeleton gait parameters. The algorithm discovers both commonalities and discrepancies across users' gait preferences and identifies the gait parameters that most influenced user feedback. These results demonstrate the feasibility of recovering gait utility landscapes from limited human trials.

Yidan Qin, Seyedshams Feyzabadi, Max Allan et al.

This paper presents a technique to concurrently and jointly predict the future trajectories of surgical instruments and the future state(s) of surgical subtasks in robot-assisted surgeries (RAS) using multiple input sources. Such predictions are a necessary first step towards shared control and supervised autonomy of surgical subtasks. Minute-long surgical subtasks, such as suturing or ultrasound scanning, often have distinguishable tool kinematics and visual features, and can be described as a series of fine-grained states with transition schematics. We propose daVinciNet - an end-to-end dual-task model for robot motion and surgical state predictions. daVinciNet performs concurrent end-effector trajectory and surgical state predictions using features extracted from multiple data streams, including robot kinematics, endoscopic vision, and system events. We evaluate our proposed model on an extended Robotic Intra-Operative Ultrasound (RIOUS+) imaging dataset collected on a da Vinci Xi surgical system and the JHU-ISI Gesture and Skill Assessment Working Set (JIGSAWS). Our model achieves up to 93.85% short-term (0.5s) and 82.11% long-term (2s) state prediction accuracy, as well as 1.07mm short-term and 5.62mm long-term trajectory prediction error.

Richard Cheng, Mohammad Javad Khojasteh, Aaron D. Ames et al.

Robots operating in real world settings must navigate and maintain safety while interacting with many heterogeneous agents and obstacles. Multi-Agent Control Barrier Functions (CBF) have emerged as a computationally efficient tool to guarantee safety in multi-agent environments, but they assume perfect knowledge of both the robot dynamics and other agents' dynamics. While knowledge of the robot's dynamics might be reasonably well known, the heterogeneity of agents in real-world environments means there will always be considerable uncertainty in our prediction of other agents' dynamics. This work aims to learn high-confidence bounds for these dynamic uncertainties using Matrix-Variate Gaussian Process models, and incorporates them into a robust multi-agent CBF framework. We transform the resulting min-max robust CBF into a quadratic program, which can be efficiently solved in real time. We verify via simulation results that the nominal multi-agent CBF is often violated during agent interactions, whereas our robust formulation maintains safety with a much higher probability and adapts to learned uncertainties

Mohamadreza Ahmadi, Andrew Singletary, Joel W. Burdick et al.

Multi-agent partially observable Markov decision processes (MPOMDPs) provide a framework to represent heterogeneous autonomous agents subject to uncertainty and partial observation. In this paper, given a nominal policy provided by a human operator or a conventional planning method, we propose a technique based on barrier functions to design a minimally interfering safety-shield ensuring satisfaction of high-level specifications in terms of linear distribution temporal logic (LDTL). To this end, we use sufficient and necessary conditions for the invariance of a given set based on discrete-time barrier functions (DTBFs) and formulate sufficient conditions for finite time DTBF to study finite time convergence to a set. We then show that different LDTL mission/safety specifications can be cast as a set of invariance or finite time reachability problems. We demonstrate that the proposed method for safety-shield synthesis can be implemented online by a sequence of one-step greedy algorithms. We demonstrate the efficacy of the proposed method using experiments involving a team of robots.

Maegan Tucker, Myra Cheng, Ellen Novoseller et al.

Optimizing lower-body exoskeleton walking gaits for user comfort requires understanding users' preferences over a high-dimensional gait parameter space. However, existing preference-based learning methods have only explored low-dimensional domains due to computational limitations. To learn user preferences in high dimensions, this work presents LineCoSpar, a human-in-the-loop preference-based framework that enables optimization over many parameters by iteratively exploring one-dimensional subspaces. Additionally, this work identifies gait attributes that characterize broader preferences across users. In simulations and human trials, we empirically verify that LineCoSpar is a sample-efficient approach for high-dimensional preference optimization. Our analysis of the experimental data reveals a correspondence between human preferences and objective measures of dynamicity, while also highlighting differences in the utility functions underlying individual users' gait preferences. This result has implications for exoskeleton gait synthesis, an active field with applications to clinical use and patient rehabilitation.

Yidan Qin, Sahba Aghajani Pedram, Seyedshams Feyzabadi et al.

Many tasks in robot-assisted surgeries (RAS) can be represented by finite-state machines (FSMs), where each state represents either an action (such as picking up a needle) or an observation (such as bleeding). A crucial step towards the automation of such surgical tasks is the temporal perception of the current surgical scene, which requires a real-time estimation of the states in the FSMs. The objective of this work is to estimate the current state of the surgical task based on the actions performed or events occurred as the task progresses. We propose Fusion-KVE, a unified surgical state estimation model that incorporates multiple data sources including the Kinematics, Vision, and system Events. Additionally, we examine the strengths and weaknesses of different state estimation models in segmenting states with different representative features or levels of granularity. We evaluate our model on the JHU-ISI Gesture and Skill Assessment Working Set (JIGSAWS), as well as a more complex dataset involving robotic intra-operative ultrasound (RIOUS) imaging, created using the da Vinci Xi surgical system. Our model achieves a superior frame-wise state estimation accuracy up to 89.4%, which improves the state-of-the-art surgical state estimation models in both JIGSAWS suturing dataset and our RIOUS dataset.

Mohamadreza Ahmadi, Rangoli Sharan, Joel W. Burdick

Partially observable Markov decision processes (POMDPs) provide a modeling framework for autonomous decision making under uncertainty and imperfect sensing, e.g. robot manipulation and self-driving cars. However, optimal control of POMDPs is notoriously intractable. This paper considers the quantitative problem of synthesizing sub-optimal stochastic finite state controllers (sFSCs) for POMDPs such that the probability of satisfying a set of high-level specifications in terms of linear temporal logic (LTL) formulae is maximized. We begin by casting the latter problem into an optimization and use relaxations based on the Poisson equation and McCormick envelopes. Then, we propose an stochastic bounded policy iteration algorithm, leading to a controlled growth in sFSC size and an any time algorithm, where the performance of the controller improves with successive iterations, but can be stopped by the user based on time or memory considerations. We illustrate the proposed method by a robot navigation case study.

H. J. Terry Suh, Xiaobin Xiong, Andrew Singletary et al.

Hybrid locomotion, which combines multiple modalities of locomotion within a single robot, enables robots to carry out complex tasks in diverse environments. This paper presents a novel method for planning multi-modal locomotion trajectories using approximate dynamic programming. We formulate this problem as a shortest-path search through a state-space graph, where the edge cost is assigned as optimal transport cost along each segment. This cost is approximated from batches of offline trajectory optimizations, which allows the complex effects of vehicle under-actuation and dynamic constraints to be approximately captured in a tractable way. Our method is illustrated on a hybrid double-integrator, an amphibious robot, and a flying-driving drone, showing the practicality of the approach.

Ellen R. Novoseller, Yibing Wei, Yanan Sui et al.

In preference-based reinforcement learning (RL), an agent interacts with the environment while receiving preferences instead of absolute feedback. While there is increasing research activity in preference-based RL, the design of formal frameworks that admit tractable theoretical analysis remains an open challenge. Building upon ideas from preference-based bandit learning and posterior sampling in RL, we present DUELING POSTERIOR SAMPLING (DPS), which employs preference-based posterior sampling to learn both the system dynamics and the underlying utility function that governs the preference feedback. As preference feedback is provided on trajectories rather than individual state-action pairs, we develop a Bayesian approach for the credit assignment problem, translating preferences to a posterior distribution over state-action reward models. We prove an asymptotic Bayesian no-regret rate for DPS with a Bayesian linear regression credit assignment model. This is the first regret guarantee for preference-based RL to our knowledge. We also discuss possible avenues for extending the proof methodology to other credit assignment models. Finally, we evaluate the approach empirically, showing competitive performance against existing baselines.

Richard Cheng, Abhinav Verma, Gabor Orosz et al.

Dealing with high variance is a significant challenge in model-free reinforcement learning (RL). Existing methods are unreliable, exhibiting high variance in performance from run to run using different initializations/seeds. Focusing on problems arising in continuous control, we propose a functional regularization approach to augmenting model-free RL. In particular, we regularize the behavior of the deep policy to be similar to a policy prior, i.e., we regularize in function space. We show that functional regularization yields a bias-variance trade-off, and propose an adaptive tuning strategy to optimize this trade-off. When the policy prior has control-theoretic stability guarantees, we further show that this regularization approximately preserves those stability guarantees throughout learning. We validate our approach empirically on a range of settings, and demonstrate significantly reduced variance, guaranteed dynamic stability, and more efficient learning than deep RL alone.

Richard Cheng, Gabor Orosz, Richard M. Murray et al.

Reinforcement Learning (RL) algorithms have found limited success beyond simulated applications, and one main reason is the absence of safety guarantees during the learning process. Real world systems would realistically fail or break before an optimal controller can be learned. To address this issue, we propose a controller architecture that combines (1) a model-free RL-based controller with (2) model-based controllers utilizing control barrier functions (CBFs) and (3) on-line learning of the unknown system dynamics, in order to ensure safety during learning. Our general framework leverages the success of RL algorithms to learn high-performance controllers, while the CBF-based controllers both guarantee safety and guide the learning process by constraining the set of explorable polices. We utilize Gaussian Processes (GPs) to model the system dynamics and its uncertainties. Our novel controller synthesis algorithm, RL-CBF, guarantees safety with high probability during the learning process, regardless of the RL algorithm used, and demonstrates greater policy exploration efficiency. We test our algorithm on (1) control of an inverted pendulum and (2) autonomous car-following with wireless vehicle-to-vehicle communication, and show that our algorithm attains much greater sample efficiency in learning than other state-of-the-art algorithms and maintains safety during the entire learning process.

Mohamadreza Ahmadi, Andrew Singletary, Joel W. Burdick et al.

A multi-agent partially observable Markov decision process (MPOMDP) is a modeling paradigm used for high-level planning of heterogeneous autonomous agents subject to uncertainty and partial observation. Despite their modeling efficiency, MPOMDPs have not received significant attention in safety-critical settings. In this paper, we use barrier functions to design policies for MPOMDPs that ensure safety. Notably, our method does not rely on discretization of the belief space, or finite memory. To this end, we formulate sufficient and necessary conditions for the safety of a given set based on discrete-time barrier functions (DTBFs) and we demonstrate that our formulation also allows for Boolean compositions of DTBFs for representing more complicated safe sets. We show that the proposed method can be implemented online by a sequence of one-step greedy algorithms as a standalone safe controller or as a safety-filter given a nominal planning policy. We illustrate the efficiency of the proposed methodology based on DTBFs using a high-fidelity simulation of heterogeneous robots.

Yanan Sui, Vincent Zhuang, Joel W. Burdick et al.

Enforcing safety is a key aspect of many problems pertaining to sequential decision making under uncertainty, which require the decisions made at every step to be both informative of the optimal decision and also safe. For example, we value both efficacy and comfort in medical therapy, and efficiency and safety in robotic control. We consider this problem of optimizing an unknown utility function with absolute feedback or preference feedback subject to unknown safety constraints. We develop an efficient safe Bayesian optimization algorithm, StageOpt, that separates safe region expansion and utility function maximization into two distinct stages. Compared to existing approaches which interleave between expansion and optimization, we show that StageOpt is more efficient and naturally applicable to a broader class of problems. We provide theoretical guarantees for both the satisfaction of safety constraints as well as convergence to the optimal utility value. We evaluate StageOpt on both a variety of synthetic experiments, as well as in clinical practice. We demonstrate that StageOpt is more effective than existing safe optimization approaches, and is able to safely and effectively optimize spinal cord stimulation therapy in our clinical experiments.

Yanan Sui, Kun ho Kim, Joel W. Burdick

Spinal cord stimulation has enabled humans with motor complete spinal cord injury (SCI) to independently stand and recover some lost autonomic function. Quantifying the quality of bipedal standing under spinal stimulation is important for spinal rehabilitation therapies and for new strategies that seek to combine spinal stimulation and rehabilitative robots (such as exoskeletons) in real time feedback. To study the potential for automated electromyography (EMG) analysis in SCI, we evaluated the standing quality of paralyzed patients undergoing electrical spinal cord stimulation using both video and multi-channel surface EMG recordings during spinal stimulation therapy sessions. The quality of standing under different stimulation settings was quantified manually by experienced clinicians. By correlating features of the recorded EMG activity with the expert evaluations, we show that multi-channel EMG recording can provide accurate, fast, and robust estimation for the quality of bipedal standing in spinally stimulated SCI patients. Moreover, our analysis shows that the total number of EMG channels needed to effectively predict standing quality can be reduced while maintaining high estimation accuracy, which provides more flexibility for rehabilitation robotic systems to incorporate EMG recordings.

Kun Li, Joel W. Burdick

This work handles the inverse reinforcement learning (IRL) problem where only a small number of demonstrations are available from a demonstrator for each high-dimensional task, insufficient to estimate an accurate reward function. Observing that each demonstrator has an inherent reward for each state and the task-specific behaviors mainly depend on a small number of key states, we propose a meta IRL algorithm that first models the reward function for each task as a distribution conditioned on a baseline reward function shared by all tasks and dependent only on the demonstrator, and then finds the most likely reward function in the distribution that explains the task-specific behaviors. We test the method in a simulated environment on path planning tasks with limited demonstrations, and show that the accuracy of the learned reward function is significantly improved. We also apply the method to analyze the motion of a patient under rehabilitation.

Kun Li, Joel W. Burdick

This works handles the inverse reinforcement learning problem in high-dimensional state spaces, which relies on an efficient solution of model-based high-dimensional reinforcement learning problems. To solve the computationally expensive reinforcement learning problems, we propose a function approximation method to ensure that the Bellman Optimality Equation always holds, and then estimate a function based on the observed human actions for inverse reinforcement learning problems. The time complexity of the proposed method is linearly proportional to the cardinality of the action set, thus it can handle high-dimensional even continuous state spaces efficiently. We test the proposed method in a simulated environment to show its accuracy, and three clinical tasks to show how it can be used to evaluate a doctor's proficiency.

Kun Li, Joel W. Burdick

This paper introduces a new method for inverse reinforcement learning in large-scale and high-dimensional state spaces. To avoid solving the computationally expensive reinforcement learning problems in reward learning, we propose a function approximation method to ensure that the Bellman Optimality Equation always holds, and then estimate a function to maximize the likelihood of the observed motion. The time complexity of the proposed method is linearly proportional to the cardinality of the action set, thus it can handle large state spaces efficiently. We test the proposed method in a simulated environment, and show that it is more accurate than existing methods and significantly better in scalability. We also show that the proposed method can extend many existing methods to high-dimensional state spaces. We then apply the method to evaluating the effect of rehabilitative stimulations on patients with spinal cord injuries based on the observed patient motions.

Kun Li, Joel W. Burdick

This paper develops an online inverse reinforcement learning algorithm aimed at efficiently recovering a reward function from ongoing observations of an agent's actions. To reduce the computation time and storage space in reward estimation, this work assumes that each observed action implies a change of the Q-value distribution, and relates the change to the reward function via the gradient of Q-value with respect to reward function parameter. The gradients are computed with a novel Bellman Gradient Iteration method that allows the reward function to be updated whenever a new observation is available. The method's convergence to a local optimum is proved. This work tests the proposed method in two simulated environments, and evaluates the algorithm's performance under a linear reward function and a non-linear reward function. The results show that the proposed algorithm only requires a limited computation time and storage space, but achieves an increasing accuracy as the number of observations grows. We also present a potential application to robot cleaners at home.

Kun Li, Yanan Sui, Joel W. Burdick

This paper develops an inverse reinforcement learning algorithm aimed at recovering a reward function from the observed actions of an agent. We introduce a strategy to flexibly handle different types of actions with two approximations of the Bellman Optimality Equation, and a Bellman Gradient Iteration method to compute the gradient of the Q-value with respect to the reward function. These methods allow us to build a differentiable relation between the Q-value and the reward function and learn an approximately optimal reward function with gradient methods. We test the proposed method in two simulated environments by evaluating the accuracy of different approximations and comparing the proposed method with existing solutions. The results show that even with a linear reward function, the proposed method has a comparable accuracy with the state-of-the-art method adopting a non-linear reward function, and the proposed method is more flexible because it is defined on observed actions instead of trajectories.

Kun Li, Joel W. Burdick

This paper develops a method to use RGB-D cameras to track the motions of a human spinal cord injury patient undergoing spinal stimulation and physical rehabilitation. Because clinicians must remain close to the patient during training sessions, the patient is usually under permanent and transient occlusions due to the training equipment and the movements of the attending clinicians. These occlusions can significantly degrade the accuracy of existing human tracking methods. To improve the data association problem in these circumstances, we present a new global feature based on the geodesic distances of surface mesh points to a set of anchor points. Transient occlusions are handled via a multi-hypothesis tracking framework. To evaluate the method, we simulated different occlusion sizes on a data set captured from a human in varying movement patterns, and compared the proposed feature with other tracking methods. The results show that the proposed method achieves robustness to both surface deformations and transient occlusions.

Yanan Sui, Yisong Yue, Joel W. Burdick

We consider sequential decision making under uncertainty, where the goal is to optimize over a large decision space using noisy comparative feedback. This problem can be formulated as a $K$-armed Dueling Bandits problem where $K$ is the total number of decisions. When $K$ is very large, existing dueling bandits algorithms suffer huge cumulative regret before converging on the optimal arm. This paper studies the dueling bandits problem with a large number of arms that exhibit a low-dimensional correlation structure. Our problem is motivated by a clinical decision making process in large decision space. We propose an efficient algorithm CorrDuel which optimizes the exploration/exploitation tradeoff in this large decision space of clinical treatments. More broadly, our approach can be applied to other sequential decision problems with large and structured decision spaces. We derive regret bounds, and evaluate performance in simulation experiments as well as on a live clinical trial of therapeutic spinal cord stimulation. To our knowledge, this marks the first time an online learning algorithm was applied towards spinal cord injury treatments. Our experimental results show the effectiveness and efficiency of our approach.

Yanan Sui, Vincent Zhuang, Joel W. Burdick et al.

The dueling bandits problem is an online learning framework for learning from pairwise preference feedback, and is particularly well-suited for modeling settings that elicit subjective or implicit human feedback. In this paper, we study the problem of multi-dueling bandits with dependent arms, which extends the original dueling bandits setting by simultaneously dueling multiple arms as well as modeling dependencies between arms. These extensions capture key characteristics found in many real-world applications, and allow for the opportunity to develop significantly more efficient algorithms than were possible in the original setting. We propose the \selfsparring algorithm, which reduces the multi-dueling bandits problem to a conventional bandit setting that can be solved using a stochastic bandit algorithm such as Thompson Sampling, and can naturally model dependencies using a Gaussian process prior. We present a no-regret analysis for multi-dueling setting, and demonstrate the effectiveness of our algorithm empirically on a wide range of simulation settings.

Tiffany A. Huang, Matanya B. Horowitz, Joel W. Burdick

In this work, we present a method to perform Model Predictive Control (MPC) over systems whose state is an element of $SO(n)$ for $n=2,3$. This is done without charts or any local linearization, and instead is performed by operating over the orbitope of rotation matrices. This results in a novel MPC scheme without the drawbacks associated with conventional linearization techniques. Instead, second order cone- or semidefinite-constraints on state variables are the only requirement beyond those of a QP-scheme typical for MPC of linear systems. Of particular emphasis is the application to aeronautical and vehicular systems, wherein the method removes many of the transcendental trigonometric terms associated with these systems' state space equations. Furthermore, the method is shown to be compatible with many existing variants of MPC, including obstacle avoidance via Mixed Integer Linear Programming (MILP).

Matanya B. Horowitz, Joel W. Burdick

This paper presents a new methodology to craft navigation functions for nonlinear systems with stochastic uncertainty. The method relies on the transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear partial differential equation. This approach allows for optimality criteria to be incorporated into the navigation function, and generalizes several existing results in navigation functions. It is shown that the HJB and that existing navigation functions in the literature sit on ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. In particular, it is shown that under certain criteria the optimal navigation function is related to Laplace's equation, previously used in the literature, through an exponential transform. Further, analytical solutions to the HJB are available in simplified domains, yielding guidance towards optimality for approximation schemes. Examples are used to illustrate the role that noise, and optimality can potentially play in navigation system design.

Matanya B. Horowitz, Nikolai Matni, Joel W. Burdick

This paper proposes a new method for rigid body pose estimation based on spectrahedral representations of the tautological orbitopes of $SE(2)$ and $SE(3)$. The approach can use dense point cloud data from stereo vision or an RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data. The method is a convex relaxation of the classical pose estimation problem, and is based on explicit linear matrix inequality (LMI) representations for the convex hulls of $SE(2)$ and $SE(3)$. Given these representations, the relaxed pose estimation problem can be framed as a robust least squares problem with the optimization variable constrained to these convex sets. Although this formulation is a relaxation of the original problem, numerical experiments indicate that it is indeed exact - i.e. its solution is a member of $SE(2)$ or $SE(3)$ - in many interesting settings. We additionally show that this method is guaranteed to be exact for a large class of pose estimation problems.