Dynamic Legged Locomotion Research Articles

A Real-Time Planning and Control Framework for Robust and Dynamic Quadrupedal Locomotion

Legged locomotion poses significant challenges due to its nonlinear, underactuated and hybrid dynamic properties. These challenges are exacerbated by the high-speed motion and presence of aerial phases in dynamic legged locomotion, which highlights the requirement for online planning based on current states to cope with uncertainty and disturbances. This article proposes a real-time planning and control framework integrating motion planning and whole-body control. In the framework, the designed motion planner allows a wider body rotation range and fast reactive behaviors based on the 3-D single rigid body model. In addition, the combination of a Bézier curve based trajectory interpolator and a heuristic-based foothold planner helps generate continuous and smooth foot trajectories. The developed whole-body controller uses hierarchical quadratic optimization coupled with the full system dynamics, which ensures tasks are prioritized based on importance and joint commands are physically feasible. The performance of the framework is successfully validated in experiments with a torque-controlled quadrupedal robot for generating dynamic motions.

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.

Learning Spring Mass Locomotion: Guiding Policies With a Reduced-Order Model

In this letter, we describe an approach to achieve dynamic legged locomotion on physical robots which combines existing methods for control with reinforcement learning. Specifically, our goal is a control hierarchy in which highest-level behaviors are planned through reduced-order models, which describe the fundamental physics of legged locomotion, and lower level controllers utilize a learned policy that can bridge the gap between the idealized, simple model and the complex, full order robot. The high-level planner can use a model of the environment and be task specific, while the low-level learned controller can execute a wide range of motions so that it applies to many different tasks. In this letter, we describe this learned dynamic walking controller and show that a range of walking motions from reduced-order models can be used as the command and primary training signal for learned policies. The resulting policies do not attempt to naively track the motion (as a traditional trajectory tracking controller would) but instead balance immediate motion tracking with long term stability. The resulting controller is demonstrated on a human scale, unconstrained, untethered bipedal robot at speeds up to 1.2 m/s. This letter builds the foundation of a generic, dynamic learned walking controller that can be applied to many different tasks.

Granular Jamming Feet Enable Improved Foot-Ground Interactions for Robot Mobility on Deformable Ground

Recent studies on dynamic legged locomotion have focused on incorporating passive compliant elements into robot legs which can help with energy efficiency and stability, enabling them to work in wide range of environments. In this work, we present the design and testing of a soft robotic foot capable of active stiffness control using granular jamming. This foot is designed and tested to be used on soft, flowable ground such as sand. Granular jamming feet enable passive foot shape change when in contact with the ground for adaptability to uneven surfaces, and can also actively change stiffness for the ability to apply sufficient propulsion forces. We seek to study the role of shape change and stiffness change in foot-ground interactions during foot-fall impact and shear. We have measured the acceleration during impact, surface traction force, and the force to pull the foot out of the medium for different states of the foot. We have demonstrated that the control of foot stiffness and shape using the proposed foot design leads to improved locomotion, specifically a 52% reduced foot deceleration at the joints after impact, 63% reduced depth of penetration in the sand on impact, higher shear force capabilities for a constant depth above the ground, and 98% reduced pullout force compared to a rigid foot.

Centroidal-momentum-based trajectory generation for legged locomotion

This paper presents a trajectory optimization framework for planning dynamic legged locomotion based on a robot’s centroidal momentum (CM), which is the aggregation of all the links’ momenta at the robot’s Center of Mass (CoM). This new framework is built around CM dynamic model driven by Ground Reaction Forces (GRFs) parameterized with Bézier polynomials. Due to the simple form of CM dynamics, the closed-form solution of the robot’s CM can be obtained by directly integrating the Bézier polynomials of GRFs. The CM can be also calculated from the robot’s generalized coordinates and velocities using Centroidal Momentum Matrices (CMM). For dynamically feasible motions, these CM values should match, thereby providing equality constraints for the proposed trajectory optimization framework. Direct collocation methods are utilized to obtain feasible GRFs and joint trajectories simultaneously under kinematic and dynamic constraint. With the closed-form solutions of CM due to the parameterization of GRFs in the formulation, numerical error induced by collocation methods in the solution of trajectory optimization can be reduced, which is crucial for reliable tracking control when applied to real robotic systems. Using the proposed framework, jumping trajectories of legged robots are obtained in the simulation. Experimental validation of the algorithm is performed on a planar robot testbed, proving the effectiveness of the proposed method in generating dynamic motions of the legged robots.

The Soft-Landing Problem: Minimizing Energy Loss by a Legged Robot Impacting Yielding Terrain

Enabling robots to walk and run on yielding terrain is vital to endeavors ranging from disaster response to extraterrestrial exploration. While dynamic legged locomotion on rigid ground is challenging enough, yielding terrain presents additional challenges such as ground deformation which dissipates energy. In this paper, we examine the soft-landing problem: given some impact momentum, bring the robot to rest while minimizing foot penetration depth. To gain insight into properties of penetration depth-minimizing control policies, we formulate a constrained optimal control problem and obtain a bang-bang open-loop force profile. Motivated by examples from biology and recent advances in legged robotics, we also examine impedance-control solutions to the soft-landing problem. Through simulations and experiments, we find that optimal impedance reduces penetration depth nearly as much as the open-loop force profile, while remaining robust to model uncertainty. Lastly, we discuss the relevance of this work to minimum-cost-of-transport locomotion for several actuator design choices.

A non-periodic planning and control framework of dynamic legged locomotion

This study proposes an integrated planning and control framework for achieving three-dimensional robust and dynamic legged locomotion over uneven terrain. The proposed framework is composed of three hierarchical layers. The high-level layer is a state-space motion planner designing highly dynamic locomotion behaviors based on a reduced-order robot model. This motion planner incorporates two robust bundles, named as invariant and recoverability bundles, which quantify analytical state-space deviations for robust planning design. The low-level layer is a model-based trajectory tracking controller capable of robustly realizing the planned locomotion behaviors. This controller is synthesized based on full-order hybrid dynamic modeling, model-based state feedback control, and Lyapunov stability analysis. The planning and control layers are concatenated by a middle-level trajectory generator that produces nominal behaviors for a full-order robot model. The proposed framework is validated through flat and uneven terrain walking simulations of a three-dimensional bipedal robot.

Mechanical design optimization for a five-link walking bipedal robot

Hybrid zero dynamics is an established theoretical framework that allows to perform dynamic legged locomotion by enforcing virtual constraints through feedback controllers. One of the major functions of the framework is finding virtual constraints that result in the most efficient locomotion in terms of energy expenditure. This paper argues that the problem of reducing such transportation costs requires the optimization of mechanical parameters along with gait parameters. Our study showed that a simultaneous optimization of mechanical and gait parameters results, on average, in threefold reduction in the energy consumption for the whole range of achievable velocities.

Design, Simulation and Fabrication of Quadrupedal Robot Integrated using Five-Jointed Legs with Suspension Spring

In this paper, a complete description of the design process for a four-legged locomotion robot or also known as quadrupedal robot will be presented. The quadrupedal robot is purposely designed as the Messenger Robot 2 (MR2) to participate in Robocon 2019. To overcome the challenges in Robocon 2019, each leg of the quadrupedal robot is designed with five joints integrated with a compression spring at the foot for suspension. The quadrupedal robot consists of a total sixteen standard servomotors where groups of four servos actuate leg joints of the quadrupedal robot. Furthermore, there are an additional three servomotors, where one servomotor is a joint at each front leg to allow the robot to rotate its orientation, and the last servo for an extension mechanism system. Finally, the simulation and experimental results demonstrated that the quadrupedal robot achieves a stable walking motion with the fastest locomotion of two legs contacting the ground at half walking cycle. In the future, the legged mechanism of the quadrupedal robot will be further improved and optimized toward the generalization of the dynamic legged locomotion in other challenging terrains.

Improving Legged Robot Hopping by Using Coupling-Based Series Elastic Actuation

This paper investigates the use of discrete couplings in series elastic actuation in order to increase variability of dynamic legged robot locomotion. The couplings are capable of engaging and disengaging the connection between an actuation element and a mechanical spring in real time such that the series elasticity could be used to load, store, and retain energy, and release it in the necessary timings. Such a mechanism is particularly beneficial for hopping or running locomotion, because it allows preloading of spring in flight phase. Through a series of simulated and physical robot experiments, this paper explains how such variability of actuation dynamics can improve performance of hopping locomotion under the limitation of actuator forces, as well as the controllability of behaviors. Compared to the conventional hopping robots, the proposed control strategy makes locomotion robots capable of higher hopping without larger actuation forces, and changing behaviors for more aggressive manoeuvres.

Morphological and control criteria for self-stable underwater hopping

This paper presents the self-stabilisation features of a hopping gait during underwater legged locomotion. We used a bio-inspired fundamental model of this gait, the underwater spring-loaded inverted pendulum model, to numerically derive quantitative (dimension of the basin of attraction, Floquet multipliers, mean horizontal speed) and qualitative (shape of the basin) features which characterise the self-stability of the system. Furthermore, we compared the results obtained with a terrestrial self-stable running model (i.e. the spring-loaded inverted pendulum with swing-leg retraction) to highlight the role of water-related components in relation to dynamic legged locomotion. The analysis revealed fundamental morphological and actuation parameters that could be used to design self-stabilising underwater hopping machines, as well as elucidating their role with respect to stability and speed. Underwater hopping is a simple and reliable locomotion, as it does not require complex control feedback to reject significant disturbances. Thanks to its high self-stabilising property, underwater hopping appears to be a reliable alternative locomotion for underwater robots.

Analysis of Foot Slippage Effects on an Actuated Spring-Mass Model of Dynamic Legged Locomotion

The classical model of spring-loaded inverted pendulum (SLIP) and its extensions have been widely accepted as a simple description of dynamic legged locomotion at various scales in humans, legged robots and animals. Similar to the majority of models in the literature, the SLIP model assumes ideal sticking contact of the foot. However, there are practical scenarios of low ground friction that causes foot slippage, which can have a significant influence on dynamic behaviour. In this work, an extension of the SLIP model with two masses and torque actuation is considered, which accounts for possible slippage under Coulomb's friction law. The hybrid dynamics of this model is formulated and numerical simulations under representative parameter values reveal several types of stable periodic solutions with stick-slip transitions. Remarkably, it is found that slippage due to low friction can sometimes increase average speed and improve energetic efficiency by significantly reducing the mechanical cost of transport.

Animation of dynamic legged locomotion

This paper is about the use of control algorithms to animate dynamic legged locomotion. Control could free the animator from specifying the details of joint and limb motion while producing both physically realistic and natural looking results. We implemented computer animations of a biped robot, a quadruped robot, and a kangaroo. Each creature was modeled as a linked set of rigid bodies with compliant actuators at its joints. Control algorithms regulated the running speed, organized use of the legs, and maintained balance. All motions were generated by numerically integrating equations of motion derived from the physical models. The resulting behavior included running at various speeds, traveling with several gaits (run, trot, bound, gallop, and hop), jumping, and traversing simple paths. Whereas the use of control permitted a variety of physically realistic animated behavior to be generated with limited human intervention, the process of designing the control algorithms was not automated: the algorithms were "tweaked" and adjusted for each new creature.