Legged Locomotion Research Articles

Reachability Map for Diverse and Energy Efficient Stepping of Humanoids

In legged locomotion, the relationship between different gait behaviors and energy consumption must consider the full-body dynamics and the robot control as a whole, which cannot be captured by simple models. This work studies the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">totality</i> of robot dynamics and whole-body optimal control as a coupled system to investigate energy consumption during balance recovery. We developed a two-phase nonlinear optimization pipeline for dynamic stepping, which generates reachability maps showing complex energy-stepping relations. We optimize gait parameters to search all reachable locations and quantify the energy cost during dynamic transitions, which allows studying the relationship between energy consumption and stepping locations given different initial conditions. We found that to achieve efficient actuation, the stepping location and timing can have simple approximations close to the underlying optimality, resulting in optimal step positions with a 10.9% lower energy cost than those generated by linear inverted pendulum model. Despite the complexity of this nonlinear process, we found that near-minimal effort stepping locations are within a region of attractions, rather than a narrow solution space suggested by a simple model. This provides new insights into the nonuniqueness of near-optimal solutions in robot motion planning and control, and the diversity of stepping behavior in humans.

Optimizing and designing a leg shape to increase robustness of a running robot on rough terrain

The superior ability of dynamic legged locomotion in traversing rough terrain relative to wheeled or tracked mechanisms comes with the cost of fragile stability. Simple control methods that use only a few basic detection sensors and apply a single controller help robots keep their balance when traversing unforeseen rough terrain. Exploiting multiple controllers simultaneously, such as the free leg length and stiffness in our hopping monopod, can further improve robustness but is often mechanically hard to implement. This work demonstrates that a curved leg shape can improve the robustness of a robot to perturbations in both terrain levels and initial horizontal velocity without complicating the control scheme. Our work develops spring loaded inverted pendulum (SLIP) based models that manifest the coupling of the leg’s parameters and capture the rolling motion. We use these models to find an optimal combination of parameters that maximizes a measure for long-term stability−reaching a desired relative height above terrain. We demonstrate that when traversing unknown rough terrain, such optimal coupling can increase robustness to perturbations in the initial horizontal velocity by 93% relative to the optimal conventional SLIP model. We further demonstrate our results in experiments.

Intersegmental coordination of the central pattern generator via interleaved electrical and chemical synapses in zebrafish spinal cord.

A significant component of the repetitive dynamics during locomotion in vertebrates is generated within the spinal cord. The legged locomotion of mammals is most likely controled by a hierarchical, multi-layer spinal network structure, while the axial circuitry generating the undulatory swimming motion of animals like lamprey is thought to have only a single layer in each segment. Recent experiments have suggested a hybrid network structure in zebrafish larvae in which two types of excitatory interneurons (V2a-I and V2a-II) both make first-order connections to the brain and last-order connections to the motor pool. These neurons are connected by electrical and chemical synapses across segments. Through computational modeling and an asymptotic perturbation approach we show that this interleaved interaction between the two neuron populations allows the spinal network to quickly establish the correct activation sequence of the segments when starting from random initial conditions, as needed for a swimming spurt, and to reduce the dependence of the intersegmental phase difference (ISPD) of the oscillations on the swimming frequency. The latter reduces the frequency dependence of the waveform of the swimming motion. In the model the reduced frequency dependence is largely due to the different impact of chemical and electrical synapses on the ISPD and to the significant spike-frequency adaptation that has been observed experimentally in V2a-II neurons, but not in V2a-I neurons. Our model makes experimentally testable predictions and points to a benefit of the hybrid structure for undulatory locomotion that may not be relevant for legged locomotion.

CPG-based Motion Planning of Hybrid Underwater Hexapod Robot for Wall Climbing and Transition

Most of the existing underwater legged robots are capable of moving on small-angled slopes, but few of them can climb the large-angled slope or transition from one plane to another, such as transition from horizontal plane to vertical plane. In this paper, we propose a motion planning method of a hybrid underwater hexapod robot (HUHR) driven by six C-shape legs and eight thrusters. By analyzing the relationship between rotation and displacement of the hip joint, we establish a single-leg kinematic model. By analyzing the force at the touchpoint, we propose a locomotion mechanism to ensure no slip of the C-shape leg. Based on the central pattern generator (CPG) and tripod gait, we design an aperiodic mapping between the oscillator outputs and the desired rotation angles of hip joints. Overall, a gait planning and control method for our robot is proposed to realize continuous legged locomotion from one plane to another, including directional climbing and transition between them. Furthermore, the effectiveness of the proposed method has been verified on HUHR.

Multi-scopic neuro-cognitive adaptation for legged locomotion robots

Dynamic locomotion is realized through a simultaneous integration of adaptability and optimality. This article proposes a neuro-cognitive model for a multi-legged locomotion robot that can seamlessly integrate multi-modal sensing, ecological perception, and cognition through the coordination of interoceptive and exteroceptive sensory information. Importantly, cognitive models can be discussed as micro-, meso-, and macro-scopic; these concepts correspond to sensing, perception, and cognition; and short-, medium-, and long-term adaptation (in terms of ecological psychology). The proposed neuro-cognitive model integrates these intelligent functions from a multi-scopic point of view. Macroscopic-level presents an attention mechanism with short-term adaptive locomotion control conducted by a lower-level sensorimotor coordination-based model. Macrosopic-level serves environmental cognitive map featuring higher-level behavior planning. Mesoscopic level shows integration between the microscopic and macroscopic approaches, enabling the model to reconstruct a map and conduct localization using bottom-up facial environmental information and top-down map information, generating intention towards the ultimate goal at the macroscopic level. The experiments demonstrated that adaptability and optimality of multi-legged locomotion could be achieved using the proposed multi-scale neuro-cognitive model, from short to long-term adaptation, with efficient computational usage. Future research directions can be implemented not only in robotics contexts but also in the context of interdisciplinary studies incorporating cognitive science and ecological psychology.

Variable stiffness locomotion with guaranteed stability for quadruped robots traversing uneven terrains.

Quadruped robots are widely applied in real-world environments where they have to face the challenges of walking on unknown rough terrains. This paper presents a control pipeline that generates robust and compliant legged locomotion for torque-controlled quadruped robots on uneven terrains. The Cartesian motion planner is designed to be reactive to unexpected early and late contacts using the estimated contact forces. Moreover, we present a novel scheme of optimal stiffness modulation that aims to coordinate desired compliance and tracking performance. It optimizes joint stiffness and contact forces coordinately in a quadratic programming (QP) formulation, where the constraints of non-slipping contacts and torque limits are imposed as well. In addition, the issue of stability under variable stiffness control is solved by imposing a tank-based passivity constraint explicitly. We finally validate the proposed control pipeline on our quadruped robot CENTAURO in experiments on uneven terrains and, through comparative tests, demonstrate the improvements of the variable stiffness locomotion.

Tracked Locomotion Systems for Ground Mobile Robots: A Review

The paper discusses the state-of-the-art of locomotion systems for ground mobile robots comprising tracks. Tracked locomotion, due to the large contact surface with the ground, is particularly suitable for tackling soft, yielding, and irregular terrains, but is characterized by lower speed and energy efficiency than wheeled locomotion, and lower obstacle-climbing capability than legged locomotion. Therefore, in recent years academic and industrial researchers have designed a wide variety of hybrid solutions, combining tracks with legs and wheels. The paper proposes three possible parallel taxonomies, based on body architecture, track profile, and track type, to help designers select the most suitable architecture on the basis of the operative necessities. Moreover, modeling, simulation, and design methodologies for tracked ground mobile robots are recalled.

Robust Walking Based on MPC With Viability Guarantees

Model predictive control (MPC) has shown great success for controlling complex systems, such as legged robots. However, when closing the loop, the performance and feasibility of the finite horizon optimal control problem (OCP) solved at each control cycle are not guaranteed anymore. This is due to model discrepancies, the effect of low-level controllers, uncertainties, and sensor noise. To address these issues, we propose a modified version of a standard MPC approach used in legged locomotion with viability (weak forward invariance) guarantees. In this approach, instead of adding a (conservative) terminal constraint to the problem, we propose to use the measured state projected to the viability kernel in the OCP solved at each control cycle. Moreover, we use past experimental data to find the best cost weights, which measure a combination of performance, constraint satisfaction robustness, or stability (invariance). These interpretable costs measure the tradeoff between robustness and performance. For this purpose, we use Bayesian optimization to systematically design experiments that help efficiently collect data to learn a cost function leading to robust performance. Our simulation results with different realistic disturbances (i.e., external pushes, unmodeled actuator dynamics, and computational delay) show the effectiveness of our approach to create robust controllers for humanoid robots.

Invariant Filtering for Legged Humanoid Locomotion on a Dynamic Rigid Surface

State estimation for legged locomotion over a dynamic rigid surface (DRS), which is a rigid surface moving in the world frame (e.g., ships, aircraft, and trains), remained an underexplored problem. This article introduces an invariant extended Kalman filter that estimates the robot’s pose and velocity during DRS locomotion by using common sensors of legged robots [e.g., inertial measurement units (IMUs), joint encoders, and RDB-D camera]. A key feature of the filter lies in that it explicitly addresses the nonstationary surface–foot contact point and the hybrid robot behaviors. Another key feature is that in the absence of IMU biases, the filter satisfies the attractive group affine and invariant observation conditions, and is thus provably convergent for the deterministic continuous phases. The observability analysis is performed to reveal the effects of DRS movement on the state observability, and the convergence property of the hybrid deterministic filter system is examined for the observable state variables. Experiments of a Digit humanoid robot walking on a pitching treadmill validate the effectiveness of the proposed filter under large estimation errors and moderate DRS movement. The video of the experiments can be found at: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://youtu.be/ScQIBFUSKzo</uri> .

Does the Heel’s Dissipative Energetic Behavior Affect Its Thermodynamic Responses During Walking?

Most of the terrestrial legged locomotion gaits, like human walking, necessitate energy dissipation upon ground collision. In humans, the heel mostly performs net-negative work during collisions, and it is currently unclear how it dissipates that energy. Based on the laws of thermodynamics, one possibility is that the net-negative collision work may be dissipated as heat. If supported, such a finding would inform the thermoregulation capacity of human feet, which may have implications for understanding foot complications and tissue damage. Here, we examined the correlation between energy dissipation and thermal responses by experimentally increasing the heel’s collisional forces. Twenty healthy young adults walked overground on force plates and for 10 min on a treadmill (both at 1.25 ms−1) while wearing a vest with three different levels of added mass (+0%, +15%, & +30% of their body mass). We estimated the heel’s work using a unified deformable segment analysis during overground walking. We measured the heel’s temperature immediately before and after each treadmill trial. We hypothesized that the heel’s temperature and net-negative work would increase when walking with added mass, and the temperature change is correlated with the increased net-negative work. We found that walking with +30% added mass significantly increased the heel’s temperature change by 0.72 ± 1.91 (p = 0.009) and the magnitude of net-negative work (extrapolated to 10 min of walking) by 326.94 ± 379.92 J (p = 0.005). However, we found no correlation between the heel’s net-negative work and temperature changes (p = 0.277). While this result refuted our second hypothesis, our findings likely demonstrate the heel’s dynamic thermoregulatory capacity. If all the negative work were dissipated as heat, we would expect excessive skin temperature elevation during prolonged walking, which may cause skin complications. Therefore, our results likely indicate that various heat dissipation mechanisms control the heel’s thermodynamic responses, which may protect the health and integrity of the surrounding tissue. Also, our results indicate that additional mechanical factors, besides energy dissipation, explain the heel’s temperature rise. Therefore, future experiments may explore alternative factors affecting thermodynamic responses, including mechanical (e.g., sound & shear-stress) and physiological mechanisms (e.g., sweating, local metabolic rate, & blood flow).

3D-SLIP model based dynamic stability strategy for legged robots with impact disturbance rejection

Inspired by biomechanical studies, the spring-loaded inverted pendulum model is an effective behavior model to describe the running movement of animals and legged robots in the sagittal plane. However, when confronted with external lateral disturbances, the model has to move out of the 2-D plane and be extended to 3-D locomotion. With the degree of freedom increasing, the computational complexity is higher and the real-time control is more and more difficult, especially when considering the complex legged model. Here, we construct a control strategy based on the classical Raibert controller for legged locomotion under lateral impact disturbances. This strategy, named 3D-HFC, is composed of three core modules: touchdown angle control, body attitude angle control and energy compensation. The energy loss in each step is taken into consideration, and the real-time measured energy loss of the current step is adopted to predict that of the next step. We demonstrate the efficiency of the proposed control strategy on a simulated 3D-SLIP lower order model and a simulated running quadruped, which are perturbed by different impact forces. Furthermore, a quadruped bionic prototype named MBBOT was set up, on which lateral impact experiments were designed and implemented. Both simulation and experimental results show that the proposed approach can realize the impact disturbance rejection.

Learning Free Gait Transition for Quadruped Robots Via Phase-Guided Controller

Gaits and transitions are key components in legged locomotion. For legged robots, describing and reproducing gaits as well as transitions remain longstanding challenges. Reinforcement learning has become a powerful tool to formulate controllers for legged robots. Learning multiple gaits and transitions, nevertheless, is related to the multi-task learning problems. In this work, we present a novel framework for training a simple control policy for a quadruped robot to locomote in various gaits. Four independent phases are used as the interface between the gait generator and the control policy, which characterizes the movement of four feet. Guided by the phases, the quadruped robot is able to locomote according to the generated gaits, such as walk, trot, pacing and bounding, and to make transitions among those gaits. More general phases can be used to generate complex gaits, such as mixed rhythmic dancing. With the control policy, the Black Panther robot, a medium-dog-sized quadruped robot, can perform all learned motor skills while following the velocity commands smoothly and robustly in natural environment.

BirdBot achieves energy-efficient gait with minimal control using avian-inspired leg clutching.

Designers of legged robots are challenged with creating mechanisms that allow energy-efficient locomotion with robust and minimalistic control. Sources of high energy costs in legged robots include the rapid loading and high forces required to support the robot's mass during stance and the rapid cycling of the leg's state between stance and swing phases. Here, we demonstrate an avian-inspired robot leg design, BirdBot, that challenges the reliance on rapid feedback control for joint coordination and replaces active control with intrinsic, mechanical coupling, reminiscent of a self-engaging and disengaging clutch. A spring tendon network rapidly switches the leg's slack segments into a loadable state at touchdown, distributes load among joints, enables rapid disengagement at toe-off through elastically stored energy, and coordinates swing leg flexion. A bistable joint mediates the spring tendon network's disengagement at the end of stance, powered by stance phase leg angle progression. We show reduced knee-flexing torque to a 10th of what is required for a nonclutching, parallel-elastic leg design with the same kinematics, whereas spring-based compliance extends the leg in stance phase. These mechanisms enable bipedal locomotion with four robot actuators under feedforward control, with high energy efficiency. The robot offers a physical model demonstration of an avian-inspired, multiarticular elastic coupling mechanism that can achieve self-stable, robust, and economic legged locomotion with simple control and no sensory feedback. The proposed design is scalable, allowing the design of large legged robots. BirdBot demonstrates a mechanism for self-engaging and disengaging parallel elastic legs that are contact-triggered by the foot's own lever-arm action.

Real-time adaptive leg-stiffness for roll compensation via magnetorheological control in a legged robot

Over the recent few decades, the evolving research-field of legged robotics has seen various mechanical and control-based developments. Inspired by biological species, a significant adaptation in modern mechanical leg designs has been the implementation of adjustable stiffness, shifting from what were previously simple linkages to more-complex variable stiffness actuators. Physiological studies previously demonstrated leg-stiffness modulation was not only a common trait in multiple biological locomotors, but also played a key role in disturbance recovery for humans. Guided by this, recent robotics research has shown that this can also be applied to legged robots to achieve similar locomotion adaptations, albeit often limited by the tuning time of leg stiffness in such circumstances. This study proposes real-time adaptive stiffness robot legs which are governed by fast-response magnetorheological fluid dampers, enabling stiffness adjustment upon a single step. Through experimental characterisation and model validation, these legs are shown to achieve a maximum stiffness shift of 114%. Enabled by real-time control during locomotion, improved performance and roll-angle stability is experimentally demonstrated for a bipedal robot test platform. Such improvement to locomotion is found through typical legged locomotion scenarios, with the platform encountering: obstacles, valleys, and coronal gradients in a comprehensive series of experiments.

Integration of prioritized impedance controller in improved hierarchical operational-space torque control frameworks for legged locomotion robots

Integration of prioritized impedance controller in improved hierarchical operational-space torque control frameworks for legged locomotion robots

Control of Thruster-Assisted, Bipedal Legged Locomotion of the Harpy Robot.

Fast constraint satisfaction, frontal dynamics stabilization, and avoiding fallovers in dynamic, bipedal walkers can be pretty challenging. The challenges include underactuation, vulnerability to external perturbations, and high computational complexity that arise when accounting for the system full-dynamics and environmental interactions. In this work, we study the potential roles of thrusters in addressing some of these locomotion challenges in bipedal robotics. We will introduce a thruster-assisted bipedal robot called Harpy. We will capitalize on Harpy’s unique design to propose an optimization-free approach to satisfy gait feasibility conditions. In this thruster-assisted legged locomotion, the reference trajectories can be manipulated to fulfill constraints brought on by ground contact and those prescribed for states and inputs. Unintended changes to the trajectories, especially those optimized to produce periodic orbits, can adversely affect gait stability and hybrid invariance. We will show our approach can still guarantee stability and hybrid invariance of the gaits by employing the thrusters in Harpy. We will also show that the thrusters can be leveraged to robustify the gaits by dodging fallovers or jumping over large obstacles.

Undulatory gait planning method of multi-legged robot with passive-spine

In this paper, we propose a new gait planning method for a multi-legged robot which has only 1 degree-of-freedom in each leg and has a passive body joint between two body segments. We firstly introduce the Finite State Machine (FSM) to the undulatory gait planning method of the 2n-legged robot. Then, the undulatory gait sequence for straight line motion is achieved by undulations motion. The idea that legged locomotion is achievable by less actuation of 2n-legged robot as well as the gait planning methods are verified finally by simulations and experiments.

Online Learning of Unknown Dynamics for Model-Based Controllers in Legged Locomotion

The performance of a model-based controller can severely suffer when its model inaccurately represents the real world dynamics. We propose to learn a time-varying, locally linear residual model along the robot's current trajectory, to compensate for the prediction errors of the controller's model. Supervised learning is performed online, as the robot is running in the unknown environment, using data collected from its immediate past. We theoretically investigate our method in its general formulation, then apply it to a bipedal controller derived from the full-order dynamics of virtual constraints, and a quadrupedal controller derived from a simplified model of contact forces. For a biped in simulation, our method consistently outperforms the baseline and a recent learning-based method. We also experiment with a 12 kg quadruped in simulation and real world, where the baseline fails to walk with 10 kg of payload but our method succeeds.

Quadrupedal Human-Assistive Robotic Platform (Q-HARP): Design, Control, and Preliminary Testing

Abstract With the rapid expansion of older adult populations around the world, mobility impairment is becoming an increasingly challenging issue. For the assistance of individuals with mobility impairments, there are two major types of tools in the current practice, including the passive (unpowered) walking aids (canes, walkers, rollators, etc.) and wheelchairs (powered and unpowered). Despite their extensive use, there are significant weaknesses that affect their effectiveness in daily use, especially when challenging uneven terrains are encountered. To address these issues, the authors developed a novel robotic platform intended for the assistance of mobility-challenged individuals. Unlike the existing assistive robots serving similar purposes, the proposed robot, namely, quadrupedal human-assistive robotic platform (Q-HARP), utilizes legged locomotion to provide an unprecedented potential to adapt to a wide variety of challenging terrains, many of which are common in people’s daily life (e.g., roadside curbs and the few steps leading to a front door). In this paper, the design of the robot is presented, including the overall structure of the robot and the design details of the actuated robotic leg joints. For the motion control of the robot, a joint trajectory generator is formulated, with the purpose of generating a stable walking gait to provide reliable support to its human user in the robot’s future application. The Q-HARP robot and its control system were experimentally tested, and the results demonstrated that the robot was able to provide a smooth gait during walking.

Autonomous speed adaptation by a muscle-driven hind leg robot modeled on a cat without intervention from brain

This study aims to design a nervous system model to drive the realistic muscle-driven legs for the locomotion of a quadruped robot. We evaluate our proposed nervous system model with a hind leg simulated model and robot. We apply a two-level central pattern generator for each leg, which generates locomotion rhythms and reproduces cat-like leg trajectories by driving different sets of the muscles at any timing during one cycle of moving the leg. The central pattern generator receives sensory feedback from leg loading. A cat simulated model and a robot with two hind legs, each with three joints driven by six muscle models, are controlled by our nervous system model. Even though their hind legs are forced backward at a wide range of speeds, they can adapt to the speed variation by autonomously adjusting its stride and cyclic duration without changing any parameters or receiving any descending inputs. In addition to the autonomous speed adaptation, the cat hind leg robot switched from a trot-like gait to a gallop-like gait while speeding up. These features can be observed in existing animal locomotion tests. These results demonstrate that our nervous system is useful as a valid and practical legged locomotion controller.