This paper extends the family of gap-based local planners to unknown dynamic environments through generating provably collision-free properties for hierarchical navigation systems. Existing perception-informed local planners that operate in dynamic environments rely on emergent or empirical robustness for collision avoidance as opposed to performing formal analysis of dynamic obstacles. In addition to this, the obstacle tracking that is performed in these existent planners is often achieved with respect to a global inertial frame, subjecting such tracking estimates to transformation errors from odometry drift. The proposed local planner, dynamic gap, shifts the tracking paradigm to modeling how the free space, represented as gaps, evolves over time. Gap crossing and closing conditions are developed to aid in determining the feasibility of passage through gaps, and a breadth of simulation benchmarking is performed against other navigation planners in the literature where the proposed dynamic gap planner achieves the highest success rate out of all planners tested in all environments.

This paper proposes an optimization-based task and motion planning framework, named “Logic Network Flow”, to integrate signal temporal logic (STL) specifications into efficient mixed-binary linear programmings. In this framework, temporal predicates are encoded as polyhedron constraints on each edge of the network flow, instead of as constraints between the nodes as in the traditional Logic Tree formulation. Synthesized with Dynamic Network Flows, Logic Network Flows render a tighter convex relaxation compared to Logic Trees derived from these STL specifications. Our formulation is evaluated on several multi-robot motion planning case studies. Empirical results demonstrate that our formulation outperforms Logic Tree formulation in terms of computation time for several planning problems. As the problem size scales up, our method still discovers better lower and upper bounds by exploring fewer number of nodes during the branch-and-bound process, although this comes at the cost of increased computational load for each node when exploring branches.

Humanoid robots offer significant advantages for search and rescue tasks, thanks to their capability to traverse rough terrains and perform transportation tasks. In this study, we present a task and motion planning framework for search and rescue operations using a heterogeneous robot team composed of humanoids and aerial robots. We propose a terrain-aware Model Predictive Controller (MPC) that incorporates terrain elevation gradients learned using Gaussian processes (GP). This terrain-aware MPC generates safe navigation paths for the bipedal robots to traverse rough terrain while minimizing terrain slopes, and it directs the quadrotors to perform aerial search and mapping tasks. The rescue subjects' locations are estimated by a target belief GP, which is updated online during the map exploration. A high-level planner for task allocation is designed by encoding the navigation tasks using syntactically cosafe Linear Temporal Logic (scLTL), and a consensus-based algorithm is designed for task assignment of individual robots. We evaluate the efficacy of our planning framework in simulation in an uncertain environment with various terrains and random rescue subject placements.

We propose an integrated planning framework for quadrupedal locomotion over dynamically changing, unforeseen terrains. Existing approaches either rely on heuristics for instantaneous foothold selection–compromising safety and versatility–or solve expensive trajectory optimization problems with complex terrain features and long time horizons. In contrast, our framework leverages reactive synthesis to generate correct-by-construction controllers at the symbolic level, and mixed-integer convex programming (MICP) for dynamic and physically feasible footstep planning for each symbolic transition. We use a high-level manager to reduce the large state space in synthesis by incorporating local environment information, improving synthesis scalability. To handle specifications that cannot be met due to dynamic infeasibility, and to minimize costly MICP solves, we leverage a symbolic repair process to generate only necessary symbolic transitions. During online execution, re-running the MICP with real-world terrain data, along with runtime symbolic repair, bridges the gap between offline synthesis and online execution. We demonstrate, in simulation, our framework’s capabilities to discover missing locomotion skills and react promptly in safety-critical environments, such as scattered stepping stones and rebars.

This study addresses the challenge of social bipedal navigation in a dynamic, human-crowded environment, a research area largely underexplored in legged robot navigation. We present a zonotope-based framework that couples prediction and motion planning for a bipedal ego-agent to account for bidirectional influence with the surrounding pedestrians. This framework incorporates a Social Zonotope Network (SZN), a neural network that predicts future pedestrian reachable sets and plans future socially acceptable reachable set for the ego-agent. SZN generates the reachable sets as zonotopes for efficient reachability-based planning, collision checking, and online uncertainty parameterization. Locomotion-specific losses are added to the SZN training process to adhere to the dynamic limits of the bipedal robot that are not explicitly present in the human crowds data set. These loss functions enable the SZN to generate locomotion paths that are more dynamically feasible for improved tracking. SZN is integrated with a Model Predictive Controller (SZN-MPC) for footstep planning for our bipedal robot Digit. SZN-MPC solves for collision-free trajectory by optimizing through SZN’s gradients. Our results demonstrate the framework’s effectiveness in producing a socially acceptable path, with consistent locomotion velocity, and optimality. The SZN-MPC framework is validated with extensive simulations and hardware experiments. Note to Practitioners—This paper is motivated by the challenge of navigating bipedal robots through dynamic, human-crowded environments in a socially acceptable manner. Existing methods for social navigation often only address obstacle avoidance and are demonstrated on a robot with simple dynamics. This paper proposes the Social Zonotope Network (SZN), a novel neural network that couples pedestrian future trajectory prediction and robot motion planning to facilitate socially aware navigation for bipedal robots such as Digit, designed by Agility Robotics. The social behaviors are learned from real open-sourced pedestrian data using the SZN, which outputs the future predictions as reachable sets for each agent in the environment. The SZN is then integrated into a trajectory optimization problem that takes into account personal space preferences and bipedal robot capabilities to design trajectories that are both collision-free and socially acceptable. This work also highlights the computational efficiency of the SZN design that makes it suitable for real-time integration with motion planners. The framework is validated through extensive simulations and hardware experiments. From a practical standpoint, this research provides a framework that can be applied to bipedal robots to improve automation in human-populated environments such as hospitals, shopping centers, and airports. The framework’s ability to automatically adapt to surrounding human movement helps minimize disruptions and ensures that the robot’s presence is not a hindrance to the flow of human traffic. Future work will focus on outdoor deployment, which will require onboard perception capabilities to detect surrounding pedestrians.

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 stability metric for bipedal locomotion and quantify this metric using STL specifications, which guide the generation of recovery trajectories to achieve maximum robustness degree. 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.

This study focuses on a layered, experience-based, multi-modal contact planning framework for agile quadrupedal locomotion over a constrained rebar environment. To this end, our hierarchical planner incorporates locomotion-specific modules into the high-level contact sequence planner and performs kinodynamically-aware trajectory optimization as the low-level motion planner. Through quantitative analysis of the experience accumulation process and experimental validation of the kinodynamic feasibility of the generated locomotion trajectories, we demonstrate that the planning heuristic of experience offers an effective way of providing candidate footholds for a legged contact planner. Additionally, we introduce a guiding torso path heuristic at the global planning level to enhance the navigation success rate in the presence of environmental obstacles. Our results indicate that the torso-path guided experience accumulation requires significantly fewer offline trials to successfully reach the goal compared to regular experience accumulation. Finally, our planning framework is validated in both dynamics simulations and real hardware implementations on a quadrupedal robot provided by Skymul Inc.

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 due to 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.

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.

We study the problem of bipedal robot navigation in complex environments with uncertain and rough terrain. In particular, we consider a scenario in which the robot is expected to reach a desired goal location by traversing an environment with uncertain terrain elevation. Such terrain uncertainties induce not only untraversable regions but also robot motion perturbations. Thus, the problems of terrain mapping and locomotion stability are intertwined. We evaluate three different kernels for Gaussian process (GP) regression to learn the terrain elevation. We also learn the motion deviation resulting from both the terrain as well as the discrepancy between the reduced-order Prismatic Inverted Pendulum Model used for planning and the full-order locomotion dynamics. We propose a hierarchical locomotion-dynamics-aware sampling-based navigation planner. The global navigation planner plans a series of local waypoints to reach the desired goal locations while respecting locomotion stability constraints. Then, a local navigation planner is used to generate a sequence of dynamically feasible footsteps to reach local waypoints. We develop a novel trajectory evaluation metric to minimize motion deviation and maximize information gain of the terrain elevation map. We evaluate the efficacy of our planning framework on Digit bipedal robot simulation in MuJoCo.i

This paper presents an incremental replanning algorithm, dubbed LTL-D*, for temporal-logic-based task planning in a dynamically changing environment. Unexpected changes in the environment may lead to failures in satisfying a task specification in the form of a Linear Temporal Logic (LTL). In this study, the considered failures are categorized into two classes: (i) the desired LTL specification can be satisfied via replanning, and (ii) the desired LTL specification is infeasible to meet strictly and can only be satisfied in a "relaxed" fashion. To address these failures, the proposed algorithm finds an optimal replanning solution that minimally violates desired task specifications. In particular, our approach leverages the D* Lite algorithm and employs a distance metric within the synthesized automaton to quantify the degree of the task violation and then replan incrementally. This ensures plan optimality and reduces planning time, especially when frequent replanning is required. Our approach is implemented in a robot navigation simulation to demonstrate a significant improvement in the computational efficiency for replanning by two orders of magnitude.

This study addresses the challenge of bipedal navigation in a dynamic human-crowded environment, a research area that remains largely underexplored in the field of legged navigation. We propose two cascaded zonotope-based neural networks: a Pedestrian Prediction Network (PPN) for pedestrians’ future trajectory prediction and an Ego-agent Social Network (ESN) for ego-agent social path planning. Representing future paths as zonotopes allows for efficient reachability-based planning and collision checking. The ESN is then integrated with a Model Predictive Controller (ESN-MPC) for footstep planning for our bipedal robot Digit designed by Agility Robotics. ESN-MPC solves for a collision-free optimal trajectory by optimizing through the gradients of ESN. ESN-MPC optimal trajectory is sent to the low-level controller for full-order simulation of Digit. The overall proposed framework is validated with extensive simulations on randomly generated initial settings with varying human crowd densities.

Safe quadrupedal navigation through unknown environments is a challenging problem. This paper proposes a hierarchical vision-based planning framework (GPF-BG) integrating our previous Global Path Follower (GPF) navigation system and a gap-based local planner using Bézier curves, so called$B$ézier Gap (BG). This BG-based trajectory synthesis can generate smooth trajectories and guarantee safety for point-mass robots. With a gap analysis extension based on non-point, rectangular geometry, safety is guaranteed for an idealized quadrupedal motion model and significantly improved for an actual quadrupedal robot model. Stabilized perception space improves performance under oscillatory internal body motions that impact sensing. Simulation-based and real experiments under different benchmarking configurations test safe navigation performance. GPF-BG has the best safety outcomes across all experiments.

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 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 model is designed by partitioning the environment into multiple belief regions and employed at the high-level navigation planner to estimate the dynamic obstacles' location. This additional location information of dynamic obstacles offered by belief abstraction enables less conservative long-horizon navigation actions beyond guaranteeing immediate collision avoidance. 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 nonperiodic 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 the 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.

The force modulation of robotic manipulators has been extensively studied for several decades. However, it is not yet commonly used in safety-critical applications due to a lack of accurate interaction contact modeling and weak performance guarantees—a large proportion of them concerning the modulation of interaction forces. This study presents a high-level framework for simultaneous trajectory optimization and force control of the interaction between a manipulator and soft environments, which is prone to external disturbances. Sliding friction and normal contact force are taken into account. The dynamics of the soft contact model and the manipulator are simultaneously incorporated in a trajectory optimizer to generate desired motion and force profiles. A constrained optimization framework based on the alternative direction method of multipliers has been employed to efficiently generate real-time optimal control inputs and high-dimensional state trajectories in a model-predictive control fashion. The experimental validation of the model performance is conducted on a soft substrate with known material properties using a Cartesian space force control mode. Results show a comparison of ground truth and real-time model-based contact force and motion tracking for multiple Cartesian motions in the valid range of the friction model. It is shown that a contact-model-based motion planner can compensate for frictional forces and motion disturbances and improve the overall motion and force tracking accuracy. The proposed high-level planner has the potential to facilitate the automation of medical tasks involving the manipulation of compliant, delicate, and deformable tissues.

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.

Estimation of center of mass (CoM) and center of pressure (CoP) is critical for lower limb exoskeletons, prostheses, and legged robots. To meet the demand in these fields, this study presents a novel CoM and CoP estimation method for human walking through a wearable visual odometry (VO) device. This method is named VO-based estimation of CoM and CoP (VOECC). The methodology of VOECC is that the VO provides CoM trajectory estimation and the inherent walking dynamics model is exploited as prior knowledge for CoP trajectory estimation during human walking. Gait cycle is estimated based on the frequency analysis of the CoM trajectory, which is cropped into segments. Each segment mainly includes a half gait cycle. The segments are designed to be sliding to mitigate the disturbance of double-stance phase. For each segment, a quadratic programming (QP) problem is formulated to fit the CoM measurement with the theoretical walking dynamics model. The solution to this QP problem is an optimal gait parameters estimation, including CoP. Based on this solution, the human walking model with the CoM trajectory and CoP excursion is reconstructed. VOECC is evaluated experimentally where human walks on level ground and upstairs with VO device attached in front of the chest. The ground truth of CoM and CoP position is directly measured by the motion capture system and fully instrumented treadmill, respectively, and compared with the VOECC results. The proposed method is demonstrated to be effective in terms of wearable and extensible functionalities compared with the existing methods. Root-mean-squared errors between the CoP measured by fully instrumented treadmill and the CoP estimated by VOECC are evaluated and compared. This method has the potential to be extensible in lower limb rehabilitation, prosthetic, and legged locomotion fields.Note to Practitioners—This article addresses the problem of estimating center of mass (CoM) and center of pressure (CoP) trajectories using a minimum number of wearable sensors and reliable algorithms during human daily walking. Estimation of CoM and CoP trajectories is critical for lower limb exoskeletons, prostheses, and legged robots. In this study, a novel method named VO-based estimation of CoM and CoP (VOECC) is presented that utilizes a walking model as prior knowledge and integrates it with visual odometry data, which estimates the trajectory of the wearable visual device. Compared with the inertia measurement unit (IMU)-based method, VOECC only uses one wearable visual device and thus significantly reduces the cost and system complexity. In addition, the VOECC outperforms the motion capture system and force plate since it is not limited to space constraints and has the potential to be applicable for daily life locomotion tasks. VOECC is wearable and untethered and therefore can be directly amounted on lower limb exoskeletons, prostheses, and legged robots. In the experiments, motion capture system and force plates are used to measure CoM and CoP as ground truth to demonstrate the effectiveness of the proposed VOECC. Practical limitations include failure from fast turning and synchronization of multichannel sensors. These limitations will be addressed as our future research directions.

In this paper, we study the problem of dynamic interaction between a robot and an unknown object (e.g., catching a ball, or handing off an object during locomotion). In particular, we propose a method for estimating the inertial parameters of an object during dynamic interaction, while minimally altering the trajectory of the object – a minimal interaction approach. Our method combines trajectory estimation (e.g., using standard methods from computer vision) with a model-based estimator that exploits the robot’s known dynamic model. We first develop the method for a generalized three-dimensional problem, and then evaluate the method for the case of an object moving along a linear trajectory. We present experimental results obtained using a KUKA iiwa 7 interacting with rolling balls of varying mass. Our experiments demonstrate that the mass of the objects can be accurately estimated at the moment of impact when accurate object trajectory estimates are available, and that significant improvement can be obtained by incorporating force measurements at the contact point while following the object.

Reinforcement learning with safety constraints is promising for autonomous vehicles, of which various failures may result in disastrous losses. In general, a safe policy is trained by constrained optimization algorithms, in which the average constraint return as a function of states and actions should be lower than a predefined bound. However, most existing safe learning-based algorithms capture states via multiple high-precision sensors, which complicates the hardware systems and is power-consuming. This article is focused on safe motion planning with the stability guarantee for autonomous vehicles with limited size and power. To this end, the risk-identification method and the Lyapunov function are integrated with the well-known soft actor-critic (SAC) algorithm. By borrowing the concept of Lyapunov functions in the control theory, the learned policy can theoretically guarantee that the state trajectory always stays in a safe area. A novel risk-sensitive learning-based algorithm with the stability guarantee is proposed to train policies for the motion planning of autonomous vehicles. The learned policy is implemented on a differential drive vehicle in a simulation environment. The experimental results show that the proposed algorithm achieves a higher success rate than the SAC.

Force modulation of robotic manipulators has been extensively studied for several decades but is not yet commonly used in safety-critical applications due to a lack of accurate interaction contact modeling and weak performance guarantees - a large proportion of them concerning the modulation of interaction forces. This study presents a high-level framework for simultaneous trajectory optimization and force control of the interaction between manipulator and soft environments. Sliding friction and normal contact force are taken into account. The dynamics of the soft contact model and the manipulator dynamics are simultaneously incorporated in a trajectory optimizer to generate desired motion and force profiles. A constrained optimization framework based on Differential Dynamic Programming and Alternative Direction Method of Multipliers has been employed to generate optimal control inputs and high-dimensional state trajectories. Experimental validation of the model performance is conducted on a soft substrate with known material properties using a Cartesian space force control mode. Results show a comparison of ground truth and predicted model based contact force states for multiple Cartesian motions and the validity range of the friction model. The proposed high-level planning has the potential to be leveraged for medical tasks involving manipulation of compliant, delicate, and deformable tissues.

Whole-body operational space controllers (WBOSCs) are versatile and well suited for simultaneously controlling motion and force behaviors, which can enable sophisticated modes of locomotion and balance. In this paper, we formulate a WBOSC for point-foot bipeds with series-elastic actuators (SEA) and experiment with it using a teen-size SEA biped robot. Our main contributions are on devising a WBOSC strategy for point-foot bipedal robots, 2) formulating planning algorithms for achieving unsupported dynamic balancing on our point-foot biped robot and testing them using a WBOSC, and 3) formulating force feedback control of the internal forces-corresponding to the subset of contact forces that do not generate robot motions-to regulate contact interactions with the complex environment. We experimentally validate the efficacy of our new whole-body control and planning strategies via balancing over a disjointed terrain and attaining dynamic balance through continuous stepping without a mechanical support.

Legged robots naturally exhibit continuous and discrete dynamics when maneuvering over level-ground and uneven terrains. In recent years, numerous studies have focused on locomotion hybrid dynamics. However, locomotion on more challenging terrains such as split wedges in Figure 1 has rarely been explored, let alone its hybrid dynamics. In this study, we specifically focus on a two-phase hybrid automaton formulation for this highly steep wedge locomotion. This automaton incorporates both multi-contact and flight single contact phase motions. To dynamically balance and jump upwards on this wedge, an aperiodic phase space planning is used for trajectory generations. Three control strategies are employed simultaneously: internal force control, linear and angular momentum control. Finally, simulation results are shown to verify our strategy's effectiveness.

This brief is concerned with the robust stability problem for a class of discrete-time uncertain Markovian jumping neural networks with defective statistics of modes transitions. The parameter uncertainties are considered to be norm-bounded, and the stochastic perturbations are described in terms of Brownian motion. Defective statistics means that the transition probabilities of the multimode neural networks are not exactly known, as assumed usually. The scenario is more practical, and such defective transition probabilities comprise three types: known, uncertain, and unknown. By invoking the property of the transition probability matrix and the convexity of uncertain domains, a sufficient stability criterion for the underlying system is derived. Furthermore, a monotonicity is observed concerning the maximum value of a given scalar, which bounds the stochastic perturbation that the system can tolerate as the level of the defectiveness varies. Numerical examples are given to verify the effectiveness of the developed results.