Maegan Tucker

We characterize a notion of system comparison, termed as $(T_e,γ,δ)$-similarity, for non-deterministic nonlinear systems. Building on a similar notion recently proposed for stable linear systems, the proposed notion characterizes the dissimilarity between the outputs, measured using the $L_2$ norm, of two nonlinear dynamical systems in terms of their inputs and disturbances. By establishing a relationship between $(T_e,γ,δ)$-similarity and differential dissipativity, we establish equivalence between $(T_e,γ,δ)$-similarity of nonlinear systems and the $(T_e,γ,δ)$-similarity of their differential dynamics. We characterize the $(T_e,γ,δ)$-similarity for nonlinear systems as a Linear Matrix Inequality feasibility problem and also provide necessary and sufficient conditions for solving this feasibility problem. We demonstrate the utility of the proposed notion through its use in two applications: (i) robust hierarchical control applied to a planar aircraft and (ii) the improvement (or design) of abstract models applied to the Moore-Greitzer model and an electronic circuit.

We characterize a notion of system comparison, termed as $(T_e,γ,δ)$-similarity, for non-deterministic nonlinear systems. Building on a similar notion recently proposed for stable linear systems, the proposed notion characterizes the dissimilarity between the outputs, measured using the $L_2$ norm, of two nonlinear dynamical systems in terms of their inputs and disturbances. By establishing a relationship between $(T_e,γ,δ)$-similarity and differential dissipativity, we establish equivalence between $(T_e,γ,δ)$-similarity of nonlinear systems and the $(T_e,γ,δ)$-similarity of their differential dynamics. We characterize the $(T_e,γ,δ)$-similarity for nonlinear systems as a Linear Matrix Inequality feasibility problem and also provide necessary and sufficient conditions for solving this feasibility problem. We demonstrate the utility of the proposed notion through its use in two applications: (i) robust hierarchical control applied to a planar aircraft and (ii) the improvement (or design) of abstract models applied to the Moore-Greitzer model and an electronic circuit.

Multi-objective reinforcement learning (MORL) is a powerful tool to learn Pareto-optimal policy families across conflicting objectives. However, unlike traditional RL algorithms, existing MORL algorithms do not effectively leverage large-scale parallelization to concurrently simulate thousands of environments, thus facing vastly increased computation time. Ultimately, this has limited the application of MORL towards complex multi-objective robotics problems. To address these challenges, we present 1) MORLAX, a new GPU-native, fast MORL algorithm, and 2) MO-Playground, apip-installable playground of GPU-accelerated multi-objective environments. Together, MORLAX and MO-Playground approximate Pareto sets within minutes, offering 26-271x speed-ups compared to legacy CPU-based approaches and up to 19x speed-ups over prior GPU-based approaches whilst learning superior Pareto front hypervolumes. We demonstrate MO-Playground's versatility by implementing a custom BRUCE humanoid robot environment and learning Pareto-optimal locomotion policies across 6 practical objectives in simulation, such as smoothness, efficiency and arm swinging.

Efficient learning of user preferences is crucial for many modern decision making systems but typically requires costly labeled data. Active learning reduces this cost, yet standard methods are computationally expensive due to pool-based evaluation. Further, most methods assume all query feedback is equally reliable, ignoring that pairwise queries between nearly identical or entirely dissimilar items yield ambiguous, low-confidence responses. To address the issue of feedback reliability, we introduce a novel confidence aware response model that explicitly accounts for these ambiguous comparisons. To overcome the computational bottleneck of pool-based evaluation, we propose an active query synthesis framework, Info-Synth that generates optimal queries by maximizing a mutual information-based objective within a continuous space. Moreover, we propose two strategies, Pair M-dist and Pair Opt-dist, that extend Info-Synth to select effective queries even when restricted to finite query pools. We demonstrate our framework's versatility and performance across synthetic preference learning, constrained text summary datasets, and subjective, continuous-space controller gain tuning for a simulated mobile robot.

Efficient learning of user preferences is crucial for many modern decision making systems but typically requires costly labeled data. Active learning reduces this cost, yet standard methods are computationally expensive due to pool-based evaluation. Further, most methods assume all query feedback is equally reliable, ignoring that pairwise queries between nearly identical or entirely dissimilar items yield ambiguous, low-confidence responses. To address the issue of feedback reliability, we introduce a novel confidence aware response model that explicitly accounts for these ambiguous comparisons. To overcome the computational bottleneck of pool-based evaluation, we propose an active query synthesis framework, Info-Synth that generates optimal queries by maximizing a mutual information-based objective within a continuous space. Moreover, we propose two strategies, Pair M-dist and Pair Opt-dist, that extend Info-Synth to select effective queries even when restricted to finite query pools. We demonstrate our framework's versatility and performance across synthetic preference learning, constrained text summary datasets, and subjective, continuous-space controller gain tuning for a simulated mobile robot.

Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map captures the evolution of the hybrid system on a guard surface, reducing the stability analysis of a periodic orbit to that of a discrete-time system. While linearization provides local stability information, assessing robustness to disturbances requires identifying invariant sets of the state space under the return dynamics. However, computing such invariant sets is computationally difficult, especially when system dynamics are only available through forward simulation. In this work, we propose an algorithmic framework leveraging sampling-based optimization to compute a finite-step invariant ellipsoid around a nominal periodic orbit using sampled evaluations of the return map. The resulting solution is accompanied by probabilistic guarantees on finite-step invariance satisfying a user-defined accuracy threshold. We demonstrate the approach on two low-dimensional systems and a compass-gait walking model.

For hybrid systems exhibiting periodic behavior, analyzing the invariant set containing the limit cycle is a natural way to study the robustness of the closed-loop system. However, computing these sets can be computationally expensive, especially when applied to contact-rich cyber-physical systems such as legged robots. In this work, we extend existing methods for overapproximating reachable sets of continuous systems using parametric embeddings to compute a forward-invariant set around the nominal trajectory of a simplified model of a bipedal robot. Our three-step approach (i) computes an overapproximating reachable set around the nominal continuous flow, (ii) catalogs intersections with the guard surface, and (iii) passes these intersections through the reset map. If the overapproximated reachable set after one step is a strict subset of the initial set, we formally verify a forward invariant set for this hybrid periodic orbit. We verify this condition on the bipedal walker model numerically using immrax, a JAX-based library for parametric reachable set computation, and use it within a bi-level optimization framework to design a tracking controller that maximizes the size of the invariant set.

For hybrid systems exhibiting periodic behavior, analyzing the invariant set containing the limit cycle is a natural way to study the robustness of the closed-loop system. However, computing these sets can be computationally expensive, especially when applied to contact-rich cyber-physical systems such as legged robots. In this work, we extend existing methods for overapproximating reachable sets of continuous systems using parametric embeddings to compute a forward-invariant set around the nominal trajectory of a simplified model of a bipedal robot. Our three-step approach (i) computes an overapproximating reachable set around the nominal continuous flow, (ii) catalogs intersections with the guard surface, and (iii) passes these intersections through the reset map. If the overapproximated reachable set after one step is a strict subset of the initial set, we formally verify a forward invariant set for this hybrid periodic orbit. We verify this condition on the bipedal walker model numerically using immrax, a JAX-based library for parametric reachable set computation, and use it within a bi-level optimization framework to design a tracking controller that maximizes the size of the invariant set.

Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map captures the evolution of the hybrid system on a guard surface, reducing the stability analysis of a periodic orbit to that of a discrete-time system. While linearization provides local stability information, assessing robustness to disturbances requires identifying invariant sets of the state space under the return dynamics. However, computing such invariant sets is computationally difficult, especially when system dynamics are only available through forward simulation. In this work, we propose an algorithmic framework leveraging sampling-based optimization to compute a finite-step invariant ellipsoid around a nominal periodic orbit using sampled evaluations of the return map. The resulting solution is accompanied by probabilistic guarantees on finite-step invariance satisfying a user-defined accuracy threshold. We demonstrate the approach on two low-dimensional systems and a compass-gait walking model.

Multi-objective reinforcement learning (MORL) is a powerful tool to learn Pareto-optimal policy families across conflicting objectives. However, unlike traditional RL algorithms, existing MORL algorithms do not effectively leverage large-scale parallelization to concurrently simulate thousands of environments, resulting in vastly increased computation time. Ultimately, this has limited MORL's application towards complex multi-objective robotics problems. To address these challenges, we present 1) MORLAX, a new GPU-native, fast MORL algorithm, and 2) MO-Playground, a pip-installable playground of GPU-accelerated multi-objective environments. Together, MORLAX and MO-Playground approximate Pareto sets within minutes, offering 25-270x speed-ups compared to legacy CPU-based approaches whilst achieving superior Pareto front hypervolumes. We demonstrate the versatility of our approach by implementing a custom BRUCE humanoid robot environment using MO-Playground and learning Pareto-optimal locomotion policies across 6 realistic objectives for BRUCE, such as smoothness, efficiency and arm swinging.

We present the KinoDynamic Motion Retargeting (KDMR) framework, a novel approach for humanoid locomotion that models the retargeting process as a multi-contact, whole-body trajectory optimization problem. Conventional kinematics-based retargeting methods rely solely on spatial motion capture (MoCap) data, inevitably introducing physically inconsistent artifacts, such as foot sliding and ground penetration, that severely degrade the performance of downstream imitation learning policies. To bridge this gap, KDMR extends beyond pure kinematics by explicitly enforcing rigid-body dynamics and contact complementarity constraints. Further, by integrating ground reaction force (GRF) measurements alongside MoCap data, our method automatically detects heel-toe contact events to accurately replicate complex human-like contact patterns. We evaluate KDMR against the state-of-the-art baseline, GMR, across three key dimensions: 1) the dynamic feasibility and smoothness of the retargeted motions, 2) the accuracy of GRF tracking compared to raw source data, and 3) the training efficiency and final performance of downstream control policies trained via the BeyondMimic framework. Experimental results demonstrate that KDMR significantly outperforms purely kinematic methods, yielding dynamically viable reference trajectories that accelerate policy convergence and enhance overall locomotion stability. Our end-to-end pipeline will be open-sourced upon publication.

We present the KinoDynamic Motion Retargeting (KDMR) framework, a novel approach for humanoid locomotion that models the retargeting process as a multi-contact, whole-body trajectory optimization problem. Conventional kinematics-based retargeting methods rely solely on spatial motion capture (MoCap) data, inevitably introducing physically inconsistent artifacts, such as foot sliding and ground penetration, that severely degrade the performance of downstream imitation learning policies. To bridge this gap, KDMR extends beyond pure kinematics by explicitly enforcing rigid-body dynamics and contact complementarity constraints. Further, by integrating ground reaction force (GRF) measurements alongside MoCap data, our method automatically detects heel-toe contact events to accurately replicate complex human-like contact patterns. We evaluate KDMR against the state-of-the-art baseline, GMR, across three key dimensions: 1) the dynamic feasibility and smoothness of the retargeted motions, 2) the accuracy of GRF tracking compared to raw source data, and 3) the training efficiency and final performance of downstream control policies trained via the BeyondMimic framework. Experimental results demonstrate that KDMR significantly outperforms purely kinematic methods, yielding dynamically viable reference trajectories that accelerate policy convergence and enhance overall locomotion stability. Our end-to-end pipeline will be open-sourced upon publication.

Multi-objective reinforcement learning (MORL) is a powerful tool to learn Pareto-optimal policy families across conflicting objectives. However, unlike traditional RL algorithms, existing MORL algorithms do not effectively leverage large-scale parallelization to concurrently simulate thousands of environments, resulting in vastly increased computation time. Ultimately, this has limited MORL's application towards complex multi-objective robotics problems. To address these challenges, we present 1) MORLAX, a new GPU-native, fast MORL algorithm, and 2) MO-Playground, a pip-installable playground of GPU-accelerated multi-objective environments. Together, MORLAX and MO-Playground approximate Pareto sets within minutes, offering 25-270x speed-ups compared to legacy CPU-based approaches whilst achieving superior Pareto front hypervolumes. We demonstrate the versatility of our approach by implementing a custom BRUCE humanoid robot environment using MO-Playground and learning Pareto-optimal locomotion policies across 6 realistic objectives for BRUCE, such as smoothness, efficiency and arm swinging.

Preferential Bayesian optimization (PBO) is a framework for optimizing a decision-maker's latent preferences over available design choices. While preferences often involve multiple conflicting objectives, existing work in PBO assumes that preferences can be encoded by a single objective function. For example, in robotic assistive devices, technicians often attempt to maximize user comfort while simultaneously minimizing mechanical energy consumption for longer battery life. Similarly, in autonomous driving policy design, decision-makers wish to understand the trade-offs between multiple safety and performance attributes before committing to a policy. To address this gap, we propose the first framework for PBO with multiple objectives. Within this framework, we present dueling scalarized Thompson sampling (DSTS), a multi-objective generalization of the popular dueling Thompson algorithm, which may be of interest beyond the PBO setting. We evaluate DSTS across four synthetic test functions and two simulated exoskeleton personalization and driving policy design tasks, showing that it outperforms several benchmarks. Finally, we prove that DSTS is asymptotically consistent. As a direct consequence, this result provides, to our knowledge, the first convergence guarantee for dueling Thompson sampling in the PBO setting.

Successfully achieving bipedal locomotion remains challenging due to real-world factors such as model uncertainty, random disturbances, and imperfect state estimation. In this work, we propose a novel metric for locomotive robustness – the estimated size of the hybrid forward invariant set associated with the step-to-step dynamics. Here, the forward invariant set can be loosely interpreted as the region of attraction for the discrete-time dynamics. We illustrate the use of this metric towards synthesizing nominal walking gaits using a simulation-in-the-loop learning approach. Further, we leverage discrete-time barrier functions and a sampling-based approach to approximate sets that are maximally forward invariant. Lastly, we experimentally demonstrate that this approach results in successful locomotion for both flat-foot walking and multi-contact walking on the Atalante lower-body exoskeleton.

Input-to-State Stability (ISS) is fundamental in mathematically quantifying how stability degrades in the presence of bounded disturbances. If a system is ISS, its trajectories will remain bounded, and will converge to a neighborhood of an equilibrium of the undisturbed system. This graceful degradation of stability in the presence of disturbances describes a variety of real-world control implementations. Despite its utility, this property requires the disturbance to be bounded and provides invariance and stability guarantees only with respect to this worst-case bound. In this work, we introduce the concept of “ISS in probability (ISSp)” which generalizes ISS to discrete-time systems subject to unbounded stochastic disturbances. Using tools from martingale theory, we provide Lyapunov conditions for a system to be exponentially ISSp, and connect ISSp to stochastic stability conditions found in literature. We exemplify the utility of this method through its application to a bipedal robot confronted with step heights sampled from a truncated Gaussian distribution.

Selecting robot design parameters can be challenging since these parameters are often coupled with the performance of the controller and, therefore, the resulting capabilities of the robot. This leads to a time-consuming and often expensive process whereby one iterates between designing the robot and manually evaluating its capabilities. This is particularly challenging for bipedal robots, where it can be difficult to evaluate the behavior of the system due to the underlying nonlinear and hybrid dynamics. Thus, in an effort to streamline the design process of bipedal robots, and maximize their performance, this paper presents a systematic framework for the co-design of humanoid robots and their associated walking gaits. To this end, we leverage the framework of hybrid zero dynamic (HZD) gait generation, which gives a formal approach to the generation of dynamic walking gaits. The key novelty of this paper is to consider both virtual constraints associated with the actuators of the robot, coupled with design virtual constraints that encode the associated parameters of the robot to be designed. These virtual constraints are combined in an HZD optimization problem which simultaneously determines the design parameters while finding a stable walking gait that minimizes a given cost function. The proposed approach is demonstrated through the design of a novel humanoid robot, ADAM, wherein its thigh and shin are co-designed so as to yield energy efficient bipedal locomotion.

Successfully achieving bipedal locomotion remains challenging due to real-world factors such as model uncertainty, random disturbances, and imperfect state estimation. In this work, we propose a novel metric for locomotive robustness -- the estimated size of the hybrid forward invariant set associated with the step-to-step dynamics. Here, the forward invariant set can be loosely interpreted as the region of attraction for the discrete-time dynamics. We illustrate the use of this metric towards synthesizing nominal walking gaits using a simulation-in-the-loop learning approach. Further, we leverage discrete-time barrier functions and a sampling-based approach to approximate sets that are maximally forward invariant. Lastly, we experimentally demonstrate that this approach results in successful locomotion for both flat-foot walking and multi-contact walking on the Atalante lower-body exoskeleton.

Selecting robot design parameters can be challenging since these parameters are often coupled with the performance of the controller and, therefore, the resulting capabilities of the robot. This leads to a time-consuming and often expensive process whereby one iterates between designing the robot and manually evaluating its capabilities. This is particularly challenging for bipedal robots, where it can be difficult to evaluate the behavior of the system due to the underlying nonlinear and hybrid dynamics. Thus, in an effort to streamline the design process of bipedal robots, and maximize their performance, this paper presents a systematic framework for the co-design of humanoid robots and their associated walking gaits. To this end, we leverage the framework of hybrid zero dynamic (HZD) gait generation, which gives a formal approach to the generation of dynamic walking gaits. The key novelty of this paper is to consider both virtual constraints associated with the actuators of the robot, coupled with design virtual constraints that encode the associated parameters of the robot to be designed. These virtual constraints are combined in an HZD optimization problem which simultaneously determines the design parameters while finding a stable walking gait that minimizes a given cost function. The proposed approach is demonstrated through the design of a novel humanoid robot, ADAM, wherein its thigh and shin are co-designed so as to yield energy efficient bipedal locomotion.

The ability to generate robust walking gaits on bipedal robots is key to their successful realization on hard-ware. To this end, this work extends the method of Hybrid Zero Dynamics (HZD) – which traditionally only accounts for locomotive stability via periodicity constraints under perfect impact events – through the inclusion of the saltation matrix with a view toward synthesizing robust walking gaits. By jointly minimizing the norm of the extended saltation matrix and the torque of the robot directly in the gait generation process, we demonstrate that the synthesized gaits are more robust than gaits generated with either term alone; these results are shown in simulation and on hardware for the AMBER-3M planar biped and the Atalante lower-body exoskeleton (both with and without a human subject). The end result is experimental validation that combining saltation matrices with HZD methods produces more robust bipedal walking in practice.

Input-to-State Stability (ISS) is fundamental in mathematically quantifying how stability degrades in the presence of bounded disturbances. If a system is ISS, its trajectories will remain bounded, and will converge to a neighborhood of an equilibrium of the undisturbed system. This graceful degradation of stability in the presence of disturbances describes a variety of real-world control implementations. Despite its utility, this property requires the disturbance to be bounded and provides invariance and stability guarantees only with respect to this worst-case bound. In this work, we introduce the concept of ``ISS in probability (ISSp)'' which generalizes ISS to discrete-time systems subject to unbounded stochastic disturbances. Using tools from martingale theory, we provide Lyapunov conditions for a system to be exponentially ISSp, and connect ISSp to stochastic stability conditions found in literature. We exemplify the utility of this method through its application to a bipedal robot confronted with step heights sampled from a truncated Gaussian distribution.

Uneven terrain necessarily transforms periodic walking into a non-periodic motion. As such, traditional stability analysis tools no longer adequately capture the ability of a bipedal robot to locomote in the presence of such disturbances. This motivates the need for analytical tools aimed at generalized notions of stability -- robustness. Towards this, we propose a novel definition of robustness, termed \emph{$δ$-robustness}, to characterize the domain on which a nominal periodic orbit remains stable despite uncertain terrain. This definition is derived by treating perturbations in ground height as disturbances in the context of the input-to-state-stability (ISS) of the extended Poincaré map associated with a periodic orbit. The main theoretic result is the formulation of robust Lyapunov functions that certify $δ$-robustness of periodic orbits. This yields an optimization framework for verifying $δ$-robustness, which is demonstrated in simulation with a bipedal robot walking on uneven terrain.

Lower-limb prostheses aim to restore ambulatory function for individuals with lower-limb amputations. While the design of lower-limb prostheses is important, this paper focuses on the complementary challenge — the control of lower-limb prostheses. Specifically, we focus on powered prostheses, a subset of lower-limb prostheses, which utilize actuators to inject mechanical power into the walking gait of a human user. In this paper, we present a review of existing control strategies for lower-limb powered prostheses, including the control objectives, sensing capabilities, and control methodologies. We separate the various control methods into three main tiers of prosthesis control: high-level control for task and gait phase estimation, mid-level control for desired torque computation (both with and without the use of reference trajectories), and low-level control for enforcing the computed torque commands on the prosthesis. In particular, we focus on the high- and mid-level control approaches in this review. Additionally, we outline existing methods for customizing the prosthetic behavior for individual human users. Finally, we conclude with a discussion on future research directions for powered lower-limb prostheses based on the potential of current control methods and open problems in the field.

Uneven terrain necessarily transforms periodic walking into a non-periodic motion. As such, traditional stability analysis tools no longer adequately capture the ability of a bipedal robot to locomote in the presence of such disturbances. This motivates the need for analytical tools aimed at generalized notions of stability – robustness. Towards this, we propose a novel definition of robustness, termed <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula><i>-robustness</i>, to characterize the domain on which a nominal periodic orbit remains stable despite uncertain terrain. This definition is derived by treating perturbations in ground height as disturbances in the context of the input-to-state-stability (ISS) of the extended Poincaré map associated with an orbit. The main theoretic result is the formulation of robust Lyapunov functions that certify <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-robustness of periodic orbits. This yields an optimization framework for verifying <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-robustness, which is demonstrated in simulation with a bipedal robot walking on uneven terrain.