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 the use of discrete-time barrier functions to certify hybrid forward invariance of reduced step-to-step dynamics. The size of these invariant sets can then be used as a metric for locomotive robustness. We demonstrate an application of this metric towards synthesizing robust nominal walking gaits using a simulation-in-the-loop approach. This procedure produces reference motions with step-to-step dynamics that are maximally forward-invariant with respect to the reduced representation of choice. The results demonstrate robust 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.

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{$\delta$-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 Poincar\'{e} map associated with a periodic orbit. The main theoretic result is the formulation of robust Lyapunov functions that certify $\delta$-robustness of periodic orbits. This yields an optimization framework for verifying $\delta$-robustness, which is demonstrated in simulation with a bipedal robot walking on uneven terrain.

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.

The ability to generate robust walking gaits on bipedal robots is key to their successful realization on hardware. 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.