Dynamic Locomotion Research Articles

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.

Design and locomotion analysis of two kinds of rolling expandable mobile linkages with a single degree of freedom

This study presents two kinds of rolling robots that are able to roll by deforming their outer shapes with a single degree of freedom. Each robot is an essential multi-loop planar expandable linkage constructed by a concave outer loop and several inner parallelogram loops. In this study, the mechanical design of the robots is introduced. Dynamic rolling process is further analyzed on the basis of zero moment point method, and a morphing strategy is proposed to guarantee a stable dynamic rolling process. A novel passive rolling locomotion is also developed, which enables the robots to roll and stand on a slope. To verify the design, two prototypes are manufactured, wherein the dynamic and passive rolling locomotion are carried out.

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 Novel Superlinearly Convergent Trust Region-Sequential Quadratic Programming Approach for Optimal Gait of Bipedal Robots Via Nonlinear Model Predictive Control

A type of trust region-sequential quadratic programming (TRSQP) approach with superlinearly convergent property is first proposed, investigated, and implemented on bipedal robots based on nonlinear model predictive control (NMPC). NMPC is utilized to predict the system behavior and optimize the control move in a receding horizon way, which will result in recursive efficiency and stability. Considering that the classical line search rules are expensive or hard in particular applications, the attempted trust region search is leveraged to avoid the drawbacks of the classical line search rules. Moreover, the feasible descent direction is contained in the trust region via a novelly truncated technique which avoids to recompute the quadratic programming subproblem (QPS) for the main search direction. Owing to some suitable conditions, the globally/superlinearly convergent performance and well-defined properties are analyzed and verified for the TRSQP. The main result is illustrated on a simple bipedal robot which is called as compass-like bipedal robot (CLBR) through numerical simulations and is used to generate dynamic locomotion via TRSQP and NMPC. Furthermore, to demonstrate the feasibility and superiority of a complex bipedal robot which is called as a high-dimensional bipedal robot, numerical simulations are conducted on the model of a three-link bipedal robot (TLBR) and a five-link robot (RABBIT). Furthermore, simulation results show that the TRSQP approach is effectiveness and superiority through comparing with the classical approaches, which included discrete mechanics and optimal control (DMOC) and hybrid zero dynamic (HZD), and control Lyapunov function-quadratic programming (CLF-QP) for the optimal gait of bipedal robot. In addition, to verify the robustness, the TLBR model with parameter’s disturbance 1.5 times is investigated and analyzed via TRSQP with NMPC technique. Last,this study develops an interesting framework to exploiting control methods on bipedal robots through accurately and effectively solving nonlinear programming problems.

Guided Constrained Policy Optimization for Dynamic Quadrupedal Robot Locomotion

Deep reinforcement learning (RL) uses model-free techniques to optimize task-specific control policies. Despite having emerged as a promising approach for complex problems, RL is still hard to use reliably for real-world applications. Apart from challenges such as precise reward function tuning, inaccurate sensing and actuation, and non-deterministic response, existing RL methods do not guarantee behavior within required safety constraints that are crucial for real robot scenarios. In this regard, we introduce guided constrained policy optimization (GCPO), an RL framework based upon our implementation of constrained proximal policy optimization (CPPO) for tracking base velocity commands while following the defined constraints. We introduce schemes which encourage state recovery into constrained regions in case of constraint violations. We present experimental results of our training method and test it on the real ANYmal quadruped robot. We compare our approach against the unconstrained RL method and show that guided constrained RL offers faster convergence close to the desired optimum resulting in an optimal, yet physically feasible, robotic control behavior without the need for precise reward function tuning.

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.

Variability in locomotor dynamics reveals the critical role of feedback in task control.

Animals vary considerably in size, shape, and physiological features across individuals, but yet achieve remarkably similar behavioral performances. We examined how animals compensate for morphophysiological variation by measuring the system dynamics of individual knifefish (Eigenmannia virescens) in a refuge tracking task. Kinematic measurements of Eigenmannia were used to generate individualized estimates of each fish's locomotor plant and controller, revealing substantial variability between fish. To test the impact of this variability on behavioral performance, these models were used to perform simulated 'brain transplants'-computationally swapping controllers and plants between individuals. We found that simulated closed-loop performance was robust to mismatch between plant and controller. This suggests that animals rely on feedback rather than precisely tuned neural controllers to compensate for morphophysiological variability.

Analysis and Verification on Energy Consumption of the Quadruped Robot with Passive Compliant Hydraulic Servo Actuator

The hydraulic servo actuator with passive compliance (HPCA) is designed for hydraulically-driven quadruped robots. It is characterized by an accumulator that connects to the piston chamber of the hydraulic cylinder to buffer impact forces between a robot’s feet and the ground. This paper studies the energy efficiency of this actuator in the dynamic locomotion of a quadruped robot. Different from the traditional methods of storing potentially recyclable energy using accumulators, the energy-saving principle of HPCA is to utilize the pressure-transition characteristics of the asymmetric hydraulic cylinder with control of the symmetrical valve. The accumulator can store and release oil during the switching of the transition point in each gait cycle of the robot, thereby improving the energy efficiency of the actuator. The influence of the initial inflation pressure and working volume of the accumulator on the energy efficiency of the HPCA is studied by simulation and physical experiments. The results show that the HPCA has a higher energy efficiency that is independent of the physical parameters of the accumulator.

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.

Control of locomotion systems and dynamics in relative periodic orbits

The connection between the dynamics in relative periodic orbits of vector fields with noncompact symmetry groups and periodic control for the class of control systems on Lie groups known as '(robotic) locomotion systems' is well known, and has led to the identification of (geometric) phases. We take an approach which is complementary to the existing ones, advocating the relevance——for trajectory generation in these control systems——of the qualitative properties of the dynamics in relative periodic orbits. There are two particularly important features. One is that motions in relative periodic orbits of noncompact groups can only be of two types: either they are quasi-periodic, or they leave any compact set as \begin{document}$ t\to\pm\infty $\end{document} ('drifting motions'). Moreover, in a given group, one of the two behaviours may be predominant. The second is that motions in a relative periodic orbit exhibit 'spiralling', 'meandering' behaviours, which are routinely detected in numerical integrations. Since a quantitative description of meandering behaviours for drifting motions appears to be missing, we provide it here for a class of Lie groups that includes those of interest in locomotion (semidirect products of a compact group and a normal vector space). We illustrate these ideas on some examples (a kinematic car robot, a planar swimmer).

A Novel Adaptive Zero-Velocity Detector for Inertial Pedestrian Navigation Based on Optimal Interval Estimation

In the foot-mounted inertial pedestrian navigation system, the zero-velocity update (ZUPT) algorithm is an efficient way to bound the inertial error propagation. Therefore, a reliable and accurate zero-velocity detector (ZVD) that adapts to all kinds of locomotion and scenarios plays a vital role in achieving high-precision and long-term pedestrian navigation. The classical threshold-based ZVDs are susceptible to failures during dynamic locomotion due to the fixed threshold. Recent machine-learning-based ZVDs need a huge amount of data to support the model training and their generalization is limited in new testing scenarios. In this paper, we propose a novel adaptive ZVD using the optimal interval estimation. Two filters are used to process the angular rate, aiming at determining a gait cycle. In a gait cycle, the acceleration is mapped to the search space by a special convex function. Based on the features of the data in the search space, a zero-velocity benchmark is calculated for the following interval estimation. The zero-velocity benchmark and the hierarchical iterative search are used to estimate the optimal zero-velocity interval (ZVI). The experiments demonstrate the effectiveness and adaptability of this novel ZVD.

Nonsmooth Optimal Value and Policy Functions in Mechanical Systems Subject to Unilateral Constraints

State-of-the-art approaches to optimal control use smooth approximations of value and policy functions and gradient-based algorithms for improving approximator parameters. Unfortunately, we show that value and policy functions that arise in optimal control of mechanical systems subject to unilateral constraints – i.e., the contact-rich dynamics of robot locomotion and manipulation – are generally nonsmooth due to the underlying dynamics exhibiting discontinuous or piecewise-differentiable trajectory outcomes. Simple mechanical systems are used to illustrate this result and the implications for optimal control of contact-rich robot dynamics.

Neuromotor Dynamics of Human Locomotion in Challenging Settings.

SummaryIs the control of movement less stable when we walk or run in challenging settings? Intuitively, one might answer that it is, given that challenging locomotion externally (e.g., rough terrain) or internally (e.g., age-related impairments) makes our movements more unstable. Here, we investigated how young and old humans synergistically activate muscles during locomotion when different perturbation levels are introduced. Of these control signals, called muscle synergies, we analyzed the local stability and the complexity (or irregularity) over time. Surprisingly, we found that perturbations force the central nervous system to produce muscle activation patterns that are less unstable and less complex. These outcomes show that robust locomotion control in challenging settings is achieved by producing less complex control signals that are more stable over time, whereas easier tasks allow for more unstable and irregular control.

Aggressive dominance can decrease behavioral complexity on subordinates through synchronization of locomotor activities

Social environments are known to influence behavior. Moreover, within small social groups, dominant/subordinate relationships frequently emerge. Dominants can display aggressive behaviors towards subordinates and sustain priority access to resources. Herein, Japanese quail (Coturnix japonica) were used, given that they establish hierarchies through frequent aggressive interactions. We apply a combination of different mathematical tools to provide a precise quantification of the effect of social environments and the consequence of dominance at an individual level on the temporal dynamics of behavior. Main results show that subordinates performed locomotion dynamics with stronger long-range positive correlations in comparison to birds that receive few or no aggressions from conspecifics (more random dynamics). Dominant birds and their subordinates also showed a high level of synchronization in the locomotor pattern, likely emerging from the lack of environmental opportunities to engage in independent behavior. Findings suggest that dominance can potentially modulate behavioral dynamics through synchronization of locomotor activities.

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.

Robust Control of Semi-passive Biped Dynamic Locomotion based on a Discrete Control Lyapunov Function

SUMMARYThis paper focuses on robust control of a simplest passive model, which is established on a DCLF (discrete control Lyapunov function) -based control system, and presents gait transition method based on the study of purely passive walker. Firstly, the DCLF is introduced to stabilize walking process between steps exponentially by modulating the length of next step. Next, the swing leg trajectory from mid-stance position to foot-strike can be planned. Then the control law is calculated to resist external disturbance. Besides, an impulse is added just before foot-strike to realize a periodic walking pattern on flat or uphill ground. With walking terrain varying, the robot can transit to an adaptive walking gait in a few steps. With different push or pull disturbances acting on hip joint and the robot gait transiting on a continuously slope-changing downhill, the effectiveness of the presented DCLF-based method is verified using simulation experiments. The ability to walk on a changing environment is also presented by simulation results. The insights of this paper can help to develop a robust control method and adaptive walking of dynamic passive locomotion robots.

Compact Shape Morphing Tensegrity Robots Capable of Locomotion.

Robustness, compactness, and portability of tensegrity robots make them suitable candidates for locomotion on unknown terrains. Despite these advantages, challenges remain relating to ease of fabrication, shape morphing (packing-unpacking), and locomotion capabilities. The paper introduces a design methodology for fabricating tensegrity robots of varying morphologies with modular components. The design methodology utilizes perforated links, coplanar (2D) alignment of components and individual cable tensioning to achieve a 3D tensegrity structure. These techniques are utilized to fabricate prism (three-link) tensegrity structures, followed by tensegrity robots in icosahedron (six-link), and shpericon (curved two-link) formation. The methodology is used to explore different robot morphologies that attempt to minimize structural complexity (number of elements) while facilitating smooth locomotion (impact between robot and surface). Locomotion strategies for such robots involve altering the position of center-of-mass (referred to as internal mass shifting) to induce “tip-over.” As an example, a sphericon formation comprising of two orthogonally placed circular arcs with conincident center illustrates smooth locomotion along a line (one degree of freedom). The design of curved links of tensegrity mechanisms facilitates continuous change of the point of contact (along the curve) that results from the tip-over. This contrasts to the sudden and piece-wise continuous change for the case of robots with traditional straight links which generate impulse reaction forces during locomotion. The two resulting robots—the Icosahedron and the Sphericon Tensegrity Robots—display shape morphing (packing-unpacking) capabilities and achieve locomotion through internal mass-shifting. The presented static equilibrium analysis of sphericon with mass is the first step in the direction of dynamic locomotion control of these curved link robots.

Dynamic locomotion synchronization of bipedal robot and human operator via bilateral feedback teleoperation.

Despite remarkable progress in artificial intelligence, autonomous humanoid robots are still far from matching human-level manipulation and locomotion proficiency in real applications. Proficient robots would be ideal first responders to dangerous scenarios such as natural or man-made disasters. When handling these situations, robots must be capable of navigating highly unstructured terrain and dexterously interacting with objects designed for human workers. To create humanoid machines with human-level motor skills, in this work, we use whole-body teleoperation to leverage human control intelligence to command the locomotion of a bipedal robot. The challenge of this strategy lies in properly mapping human body motion to the machine while simultaneously informing the operator how closely the robot is reproducing the movement. Therefore, we propose a solution for this bilateral feedback policy to control a bipedal robot to take steps, jump, and walk in synchrony with a human operator. Such dynamic synchronization was achieved by (i) scaling the core components of human locomotion data to robot proportions in real time and (ii) applying feedback forces to the operator that are proportional to the relative velocity between human and robot. Human motion was sped up to match a faster robot, or drag was generated to synchronize the operator with a slower robot. Here, we focused on the frontal plane dynamics and stabilized the robot in the sagittal plane using an external gantry. These results represent a fundamental solution to seamlessly combine human innate motor control proficiency with the physical endurance and strength of humanoid robots.

Visual dynamics cues in learning complex physical interactions

This study investigated the role of visual dynamics cues (VDCs) in learning to interact with a complex physical system. Manual gait training was used as an exemplary case, as it requires therapists to control the non-trivial locomotor dynamics of patients. A virtual analog was developed that allowed naïve subjects to manipulate the leg of a virtual stroke survivor (a virtual patient; VP) walking on a treadmill using a small robotic manipulandum. The task was to make the VP’s leg pass through early, mid, and late swing gait targets. One group of subjects (n = 17) started practice seeing the VP’s affected thigh and shank (i.e., VDCs); a second control group (n = 16) only saw the point-of-contact (VP ankle). It was hypothesized that, if seeing the VP’s leg provides beneficial dynamics information, the VDC group would have better task performance and generalization than controls. Results were not supportive. Both groups had similar task performance, and for the late swing gait target, a decrement in manipulative accuracy was observed when VDCs were removed in a generalization task. This suggests that when learning to manipulate complex dynamics, VDCs can create a dependency that negatively affects generalization if the visual context is changed.

Tuna robotics: A high-frequency experimental platform exploring the performance space of swimming fishes.

Tuna and related scombrid fishes are high-performance swimmers that often operate at high frequencies, especially during behaviors such as escaping from predators or catching prey. This contrasts with most fish-like robotic systems that typically operate at low frequencies (< 2 hertz). To explore the high-frequency fish swimming performance space, we designed and tested a new platform based on yellowfin tuna (Thunnus albacares) and Atlantic mackerel (Scomber scombrus). Body kinematics, speed, and power were measured at increasing tail beat frequencies to quantify swimming performance and to study flow fields generated by the tail. Experimental analyses of freely swimming tuna and mackerel allow comparison with the tuna-like robotic system. The Tunabot (255 millimeters long) can achieve a maximum tail beat frequency of 15 hertz, which corresponds to a swimming speed of 4.0 body lengths per second. Comparison of midline kinematics between scombrid fish and the Tunabot shows good agreement over a wide range of frequencies, with the biggest discrepancy occurring at the caudal fin, primarily due to the rigid propulsor used in the robotic model. As frequency increases, cost of transport (COT) follows a fish-like U-shaped response with a minimum at ~1.6 body lengths per second. The Tunabot has a range of ~9.1 kilometers if it swims at 0.4 meter per second or ~4.2 kilometers at 1.0 meter per second, assuming a 10-watt-hour battery pack. These results highlight the capabilities of high-frequency biological swimming and lay the foundation to explore a fish-like performance space for bio-inspired underwater vehicles.