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This study introduces a robust planning framework that utilizes a model predictive control (MPC) approach, enhanced by incorporating signal temporal logic (STL) specifications. This marks the first-ever study to apply STL-guided trajectory optimization for bipedal locomotion, specifically designed to handle both translational and orientational perturbations. Existing recovery strategies often struggle with reasoning complex task logic and evaluating locomotion robustness systematically, making them susceptible to failures caused by inappropriate recovery strategies or lack of robustness. To address these issues, we design an analytical robustness metric for bipedal locomotion and quantify this metric using STL specifications, which guide the generation of recovery trajectories to achieve maximum locomotion robustness. To enable safe and computational-efficient crossed-leg maneuver, we design data-driven self-leg-collision constraints that are $1000$ times faster than the traditional inverse-kinematics-based approach. Our framework outperforms a state-of-the-art locomotion controller, a standard MPC without STL, and a linear-temporal-logic-based planner in a high-fidelity dynamic simulation, especially in scenarios involving crossed-leg maneuvers. Additionally, the Cassie bipedal robot achieves robust performance under horizontal and orientational perturbations such as those observed in ship motions. These environments are validated in simulations and deployed on hardware. Furthermore, our proposed method demonstrates versatility on stepping stones and terrain-agnostic features on inclined terrains.

Simulation-to-reality (sim-to-real) transfer is a fundamental problem for robot learning. Domain Randomization, which adds randomization during training, is a powerful technique that effectively addresses the sim-to-real gap. However, the noise in observations makes learning significantly harder. Recently, studies have shown that employing a teacher-student learning paradigm can accelerate training in randomized environments. Learned with privileged information, a teacher agent can instruct the student agent to operate in noisy environments. However, this approach is often not sample efficient as the experience collected by the teacher is discarded completely when training the student, wasting information revealed by the environment. In this work, we extend the teacher-student learning paradigm by proposing a sample efficient learning framework termed Learn to Teach (L2T) that recycles experience collected by the teacher agent. We observe that the dynamics of the environments for both agents remain unchanged, and the state space of the teacher is coupled with the observation space of the student. We show that a single-loop algorithm can train both the teacher and student agents under both Reinforcement Learning and Inverse Reinforcement Learning contexts. We implement variants of our methods, conduct experiments on the MuJoCo benchmark, and apply our methods to the Cassie robot locomotion problem. Extensive experiments show that our method achieves competitive performance while only requiring environmental interaction with the teacher.

This study investigates formal-method-based trajectory optimization (TO) for bipedal locomotion, focusing on scenarios where the robot encounters external perturbations at unforeseen times. Our key research question centers around the assurance of task specification correctness and the maximization of specification robustness for a bipedal robot in the presence of external perturbations. Our contribution includes the design of an optimization-based task and motion planning framework that generates optimal control sequences with formal guarantees of external perturbation recovery. As a core component of the framework, a model predictive controller (MPC) encodes signal temporal logic (STL)-based task specifications as a cost function. In particular, we investigate challenging scenarios where the robot is subjected to lateral perturbations that increase the risk of failure due to leg self-collision. To address this, we synthesize agile and safe crossed-leg maneuvers to enhance locomotion stability. This work marks the first study to incorporate formal guarantees offered by STL into a TO for perturbation recovery of bipedal locomotion. We demonstrate the efficacy of the framework via perturbation experiments in simulations.

Enabling bipedal walking robots to learn how to maneuver over highly uneven, dynamically changing terrains is challenging due to the complexity of robot dynamics and interacted environments. Recent advancements in learning from demonstrations have shown promising results for robot learning in complex environments. While imitation learning of expert policies has been well-explored, the study of learning expert reward functions is largely under-explored in legged locomotion. This paper brings state-of-the-art Inverse Reinforcement Learning (IRL) techniques to solving bipedal locomotion problems over complex terrains. We propose algorithms for learning expert reward functions, and we subsequently analyze the learned functions. Through nonlinear function approximation, we uncover meaningful insights into the expert's locomotion strategies. Furthermore, we empirically demonstrate that training a bipedal locomotion policy with the inferred reward functions enhances its walking performance on unseen terrains, highlighting the adaptability offered by reward learning.

This study proposes a novel planning framework based on a model predictive control formulation that incorporates signal temporal logic (STL) specifications for task completion guarantees and robustness quantification. This marks the first-ever study to apply STL-guided trajectory optimization for bipedal locomotion push recovery, where the robot experiences unexpected disturbances. Existing recovery strategies often struggle with complex task logic reasoning and locomotion robustness evaluation, making them susceptible to failures caused by inappropriate recovery strategies or insufficient robustness. To address this issue, the STL-guided framework generates optimal and safe recovery trajectories that simultaneously satisfy the task specification and maximize the locomotion robustness. Our framework outperforms a state-of-the-art locomotion controller in a high-fidelity dynamic simulation, especially in scenarios involving crossed-leg maneuvers. Furthermore, it demonstrates versatility in tasks such as locomotion on stepping stones, where the robot must select from a set of disjointed footholds to maneuver successfully.

This study proposes a hierarchically integrated framework for safe task and motion planning (TAMP) of bipedal locomotion in a partially observable environment with dynamic obstacles and uneven terrain. The high-level task planner employs linear temporal logic (LTL) for a reactive game synthesis between the robot and its environment and provides a formal guarantee on navigation safety and task completion. To address environmental partial observability, a belief abstraction is employed at the high-level navigation planner to estimate the dynamic obstacles' location. Accordingly, a synthesized action planner sends a set of locomotion actions to the middle-level motion planner, while incorporating safe locomotion specifications extracted from safety theorems based on a reduced-order model (ROM) of the locomotion process. The motion planner employs the ROM to design safety criteria and a sampling algorithm to generate non-periodic motion plans that accurately track high-level actions. At the low level, a foot placement controller based on an angular-momentum linear inverted pendulum model is implemented and integrated with an ankle-actuated passivity-based controller for full-body trajectory tracking. To address external perturbations, this study also investigates safe sequential composition of the keyframe locomotion state and achieves robust transitions against external perturbations through reachability analysis. The overall TAMP framework is validated with extensive simulations and hardware experiments on bipedal walking robots Cassie and Digit designed by Agility Robotics., Comment: 22 pages, 18 figures

In this paper, we examine the problem of push recovery for bipedal robot locomotion and present a reactive decision-making and robust planning framework for locomotion resilient to external perturbations. Rejecting perturbations is an essential capability of bipedal robots and has been widely studied in the locomotion literature. However, adversarial disturbances and aggressive turning can lead to negative lateral step width (i.e., crossed-leg scenarios) with unstable motions and self-collision risks. These motion planning problems are computationally difficult and have not been explored under a hierarchically integrated task and motion planning method. We explore a planning and decision-making framework that closely ties linear-temporal-logic-based reactive synthesis with trajectory optimization incorporating the robot's full-body dynamics, kinematics, and leg collision avoidance constraints. Between the high-level discrete symbolic decision-making and the low-level continuous motion planning, behavior trees serve as a reactive interface to handle perturbations occurring at any time of the locomotion process. Our experimental results show the efficacy of our method in generating resilient recovery behaviors in response to diverse perturbations from any direction with bounded magnitudes.

Asynchronous Advantage Actor Critic (A3C) is an effective Reinforcement Learning (RL) algorithm for a wide range of tasks, such as Atari games and robot control. The agent learns policies and value function through trial-and-error interactions with the environment until converging to an optimal policy. Robustness and stability are critical in RL; however, neural network can be vulnerable to noise from unexpected sources and is not likely to withstand very slight disturbances. We note that agents generated from mild environment using A3C are not able to handle challenging environments. Learning from adversarial examples, we proposed an algorithm called Adversary Robust A3C (AR-A3C) to improve the agent's performance under noisy environments. In this algorithm, an adversarial agent is introduced to the learning process to make it more robust against adversarial disturbances, thereby making it more adaptive to noisy environments. Both simulations and real-world experiments are carried out to illustrate the stability of the proposed algorithm. The AR-A3C algorithm outperforms A3C in both clean and noisy environments.

Our goal is to send legged robots into challenging, unstructured terrains that wheeled systems cannot traverse. Moreover, precise estimation of the robot's position and orientation in rough terrain is especially difficult. To address this problem, we introduce a new state estimation algorithm which we term Contact-Centric Leg Odometry (COCLO). This new estimator uses a Square Root Unscented Kalman Filter (SR-UKF) to fuse multiple proprioceptive sensors available on a legged robot. In contrast to IMU-centric filtering approaches, COCLO formulates prediction and measurement models according to the contact status of legs. Additionally, COCLO has an indirect measurement model using joint velocities to estimate the robot body velocity. In rough terrain, when IMUs suffer from large amounts of noise, COCLO's contact-centric approach outperforms previous IMU-centric methods. To demonstrate improved state estimation accuracy, we compare COCLO with Visual Inertial Navigation System (VINS), a state-of-the-art visual inertial odometry, in three different environments: flat ground, ramps, and stairs. COCLO achieves better estimation precision than VINS in all three environments and is robust to unstable motion. Finally, we also show that COCLO and a modified VINS can work in tandem to improve each other's performance.

This paper reports the design and implementation of a three-link brachiation robot. The robot is able to travel along horizontal monkey bars using continuous arm swings. We build a full order dynamics model for the robot and formulate each cycle of robot swing motion as an optimal control problem. The iterative Linear Quadratic Regulator (iLQR) algorithm is used to find the optimal control strategy during one swing. We select suitable robot design parameters by comparing the cost of robot motion generated by the iLQR algorithm for different robot designs. In particular, using this approach we show the importance of having a body link and low inertia arms for efficient brachiation. Further, we propose a trajectory tracking controller that combines a cascaded PID controller and an input-output linearization controller to enable the robot to track desired trajectory precisely and reject external disturbance during brachiation. Experiments on the simulated robot and the real robot demonstrate that the robot can robustly swing between monkey bars with same or different spacing of handholds.