Compliant Legs Research Articles

Characterization of running with compliant curved legs

Running with compliant curved legs involves the progression of the center of pressure, the changes of both the leg's stiffness and effective rest length, and the shift of the location of the maximum stress point along the leg. These phenomena are product of the geometric and material properties of these legs, and the rolling motion produced during stance. We examine these aspects with several reduced-order dynamical models to relate the leg's design parameters (such as normalized foot radius, leg's effective stiffness, location of the maximum stress point and leg shape) to running performance (such as robustness and efficiency). By using these models, we show that running with compliant curved legs can be more efficient, robust with fast recovery behavior from perturbations than running with compliant straight legs. Moreover, the running performance can be further improved by tuning these design parameters in the context of running with rolling. The results shown in this work may serve as potential guidance for future compliant curved leg designs that may further improve the running performance.

Control oriented model-based simulation and experimental studies on a compliant legged quadruped robot

Quadruped robots offer better maneuverability over wheeled mobile robots. However, a quadruped robot contains many joint actuators which have to operate in a coordinated fashion to achieve the desired locomotion. Joint actuations cause various degrees of disturbance on the robot body and may even destabilize the system. Thus, prior dynamic analysis plays an important role for development of control laws for quadruped locomotion. Here, a three dimensional dynamic model of a quadruped has been developed using the bond graph technique which can be interfaced with various controller models. This model contains a detailed sub-model for telescopic compliant legs. Results from simulations, animations and experiments are discussed. Turning motion at various leg speeds is studied for dynamic stability of the robot. The effect of leg compliance on locomotion parameters is studied which helps in selecting a suitable compliance. Performance measure is carried out using energy efficiency as deciding criteria. Study on energy efficient quadruped structure, energy efficient locomotion gait and foot trajectory have been carried out for designing an efficient quadruped.

Kinetics of bipedal locomotion during load carrying in capuchin monkeys

Facultative bipedalism during load transport in nonhuman primates has been argued to be an important behavior potentially leading to the evolution of obligate, extended limb bipedalism. Understanding the biomechanics of such behavior may lead to insights about associated morphology, which may translate to interpretation of features in the fossil record. Some populations of bearded capuchin monkeys (Sapajus libidinosus) spontaneously carry heavy loads bipedally during foraging activities. This study provides the first data on all three components of ground reaction force for spontaneous bipedalism during load carriage in a nonhuman primate. Five individual S. libidinosus (mean body mass = 2.4 kg ± 0.96) were videorecorded during bipedalism while carrying a stone (0.93 kg) under natural conditions. A force plate was embedded in the path of the monkeys. Spatiotemporal and force data for all three components of the ground reaction force were recorded for 28 steps. Capuchins exhibited a mean vertical peak force per total weight (Vpk) for the hindlimb of 1.19 (sd = 0.13), consistent with those of unloaded capuchins in the laboratory and for other bipedal primates, including humans. Vertical force records suggest that capuchins, along with most nonhuman primates, maintain a relatively compliant leg during both unloaded and loaded locomotion. Like all other primates, loaded capuchins maintained laterally (outward) directed medio-lateral forces, presumably to stabilize side-to-side movements of the center of mass. Medio-lateral forces suggest that at near-running speeds dynamic stability diminishes the need to generate high lateral forces. Vertical force traces exhibited a measurable impact spike at foot contact in 85% of the steps recorded. An impact spike is common in human walking and running but has not been reported in other bipedal primates. This spike in humans is thought to lead to bone and cartilage damage. The earliest biped may have experienced similar impact spikes during bipedal locomotion, requiring compensatory behaviors or anatomical features.

Compliant walking model with a curvy foot reflecting the position of ankle on reproducing the ankle torque profile

Compliant walking model served as a theoretical framework because of the generation of kinetic information of human walking. Moreover, as Whittington applied a curvy foot to the compliant walking model, the exaggerated ratio of the vertical ground reaction force (GRF) to the horizontal GRF and the fixed center of pressure during the single stance phase have been solved. However, giving meaning to the stiffness of the compliant leg became difficult because of the exaggerated height of the foot and the shortened leg length. Moreover, joint torque was not compared with the ankle torque because the joint position is different from the ankle position. Thus, in this study, we modified the compliant walking model with a curvy foot that reflects the configuration of the human leg and examined whether the modified compliant walking model with a curvy foot could represent human walking. We reflected the configuration of the human leg to the model by shortening the height of the curvy foot, lengthening the compliant leg length, and shifting the joint position similar to the ankle position. To confirm the feasibility of the modified model and to examine the effect of shifting the joint position, we compared the simulation data with the experimental data: the GRF obtained with the force plate and the ankle torque obtained with the inverse dynamics through kinematic data. Thus, the modified model can reproduce the symmetric M-shape of the GRF but not the asymmetric shape of the model. Furthermore, the modified model can also produce the joint torque profile similar to the ankle torque profile. In conclusion, considering the human leg configuration can provide us with information about human walking.

Model-Based Development of Leaping in a Hexapod Robot

We report on the model-based development of leaping behavior in a RHex-style hexapod robot. A three-legged model is proposed to analyze the dynamic behavior of leaping, and this serves as a guide for implementing the behavior on the empirical robot. The model has a rigid body and three massless and compliant legs, which have rolling contact with the ground for better modeling the leg behavior of the empirical robot. By investigating the model's behavior, a two-step leaping maneuver is developed. The first step is utilized for adjusting the body pitch, synchronizing the phases of all six legs, and speeding up the body's forward velocity. This provides adequate initial conditions for the second step leaping, which creates a long-distance flight and adequate landing for follow-up running. In addition, we also report on the strategy of stride length regulation. With implementation of the range sensor, the robot can regulate its stride in order to reach a specific and desired position for leaping. The gait transition and initiation of leaping is fully autonomous. The behavioral development is implemented in the RHex-style robot and is evaluated experimentally.

Model-Based Experimental Development of Passive Compliant Robot Legs from Fiberglass Composites.

We report on the methodology of developing compliant, half-circular, and composite robot legs with designable stiffness. First, force-displacement experiments on flat cantilever composites made by one or multifiberglass cloths are executed. By mapping the cantilever mechanics to the virtual spring model, the equivalent elastic moduli of the composites can be derived. Next, by using the model that links the curved beam mechanics back to the virtual spring, the resultant stiffness of the composite in a half-circular shape can be estimated without going through intensive experimental tryouts. The overall methodology has been experimentally validated, and the fabricated composites were used on a hexapod robot to perform walking and leaping behaviors.

Experimental Study on Linear State Feedback Control of Humanoid Robots with Flexible Joints

This paper reports on an experimental study on use of static full state linear feedback for link tracking control of a compliant humanoid leg, cCub. Passive elasticity is used in pitch joints of the cCub legs, offering many benefits in terms of safe interaction as opposed to rigid robots. However, the elasticity makes the link tracking problem more challenging in particular tasks when the robots joints are under large gravitational forces or external disturbances, such as going up the stairs. Therefore, this study explores the experimental results of applying LQR control, formulated using Linear Matrix Inequality in discrete time and implemented on a humanoid robot. A summary of the theory used and a numerical simulation are presented. The proposed LQR controller provides a systematic way of designing the feedback gains with desired closed loop dynamics and it is shown to have good tracking performance in the experiments.

Velocity Control of a Bounding Quadruped via Energy Control and Vestibular Reflexes

In this paper a bio-inspired approach of velocity control for a quadruped robot running with a bounding gait on compliant legs is set up. The dynamic properties of a sagittal plane model of the robot are investigated. By analyzing the stable fixed points based on Poincare map, we find that the energy change of the system is the main source for forward velocity adjustment. Based on the analysis of the dynamics model of the robot, a new simple linear running controller is proposed using the energy control idea, which requires minimal task level feedback and only controls both the leg torque and ending impact angle. On the other hand, the functions of mammalian vestibular reflexes are discussed, and a reflex map between forward velocity and the pitch movement is built through statistical regression analysis. Finally, a velocity controller based on energy control and vestibular reflexes is built, which has the same structure as the mammalian nervous mechanism for body posture control. The new controller allows the robot to run autonomously without any other auxiliary equipment and exhibits good speed adjustment capability. A series simulations and experiments were set to show the good movement agility, and the feasibility and validity of the robot system.

A simple running model with rolling contact and its role as a template for dynamic locomotion on a hexapod robot

We report on the development of a robot’s dynamic locomotion based on a template which fits the robot’s natural dynamics. The developed template is a low degree-of-freedom planar model for running with rolling contact, which we call rolling spring loaded inverted pendulum (R-SLIP). Originating from a reduced-order model of the RHex-style robot with compliant circular legs, the R-SLIP model also acts as the template for general dynamic running. The model has a torsional spring and a large circular arc as the distributed foot, so during locomotion it rolls on the ground with varied equivalent linear stiffness. This differs from the well-known spring loaded inverted pendulum (SLIP) model with fixed stiffness and ground contact points. Through dimensionless steps-to-fall and return map analysis, within a wide range of parameter spaces, the R-SLIP model is revealed to have self-stable gaits and a larger stability region than that of the SLIP model. The R-SLIP model is then embedded as the reduced-order ‘template’ in a more complex ‘anchor’, the RHex-style robot, via various mapping definitions between the template and the anchor. Experimental validation confirms that by merely deploying the stable running gaits of the R-SLIP model on the empirical robot with simple open-loop control strategy, the robot can easily initiate its dynamic running behaviors with a flight phase and can move with similar body state profiles to those of the model, in all five testing speeds. The robot, embedded with the SLIP model but performing walking locomotion, further confirms the importance of finding an adequate template of the robot for dynamic locomotion.

Effect of spine motion on mobility in quadruped running

Most of current running quadruped robots have similar construction: a stiff body and four compliant legs. Many researches have indicated that the stiff body without spine motion is a main factor in limitation of robots’ mobility. Therefore, investigating spine motion is very important to build robots with better mobility. A planar quadruped robot is designed based on cheetahs’ morphology. There is a spinal driving joint in the body of the robot. When the spinal driving joint acts, the robot has spine motion; otherwise, the robot has not spine motion. Six group prototype experiments with the robot are carried out to study the effect of spine motion on mobility. In each group, there are two comparative experiments: the spinal driving joint acts in one experiment but does not in the other experiment. The results of the prototype experiments indicate that the average speeds of the robot with spine motion are 8.7%–15.9% larger than those of the robot without spine motion. Furthermore, a simplified sagittal plane model of quadruped mammals is introduced. The simplified model also has a spinal driving joint. Using a similar process as the prototype experiments, six group simulation experiments with the simplified model are conducted. The results of the simulation experiments show that the maximum rear leg horizontal thrusts of the simplified mode with spine motion are 68.2%–71.3% larger than those of the simplified mode without spine motion. Hence, it is found that spine motion can increase the average running speed and the intrinsic reason of speed increase is the improvement of the maximum rear leg horizontal thrust.

Neuromechanical control for hexapedal robot walking on challenging surfaces and surface classification

The neuromechanical control principles of animal locomotion provide good insights for the development of bio-inspired legged robots for walking on challenging surfaces. Based on such principles, we developed a neuromechanical controller consisting of a modular neural network (MNN) and of virtual agonist–antagonist muscle mechanisms (VAAMs). The controller allows for variable compliant leg motions of a hexapod robot, thereby leading to energy-efficient walking on different surfaces. Without any passive mechanisms or torque and position feedback at each joint, the variable compliant leg motions are achieved by only changing the stiffness parameters of the VAAMs. In addition, six surfaces can be also classified by observing the motor signals generated by the controller. The performance of the controller is tested on a physical hexapod robot. Experimental results show that it can effectively walk on six different surfaces with the specific resistances between 9.1 and 25.0, and also classify them with high accuracy.

Modelling and Design of a Tridimensional Compliant Leg for Bioloid Quadruped

In the growing field of rehabilitation robotics, the modelling of a real robot is a complex and passionate challenge. On the crossing point of mechanics, physics and computer-science, the development of a complete 3D model involves the knowledge of the different physic properties, for an accurate simulation. In this paper, it is proposed the design of an efficient three-dimensional model of the quadruped Bioloid robot setting segmented pantographic legs, in order to actively retract the quadruped legs during locomotion and minimizing large forces due to shocks, such that the robot is able to safely and dynamically interact with the user or the environment.

Design of 3-legged XYZ compliant parallel manipulators with minimised parasitic rotations

SUMMARYThis paper deals with the design of 3-legged distributed-compliance XYZ compliant parallel manipulators (CPMs) with minimised parasitic rotations, based on the kinematically decoupled 3-PPPRR (P: prismatic joint, and R: revolute joint) and 3-PPPR translational parallel mechanisms (TPMs). The designs are firstly proposed using the kinematic substitution approach, with the help of the stiffness center (SC) overlapping based approach. This is done by an appropriate embedded arrangement so that all of the SCs associated with the passive compliant modules overlap at the point where all of the input forces applied at the input stages intersect. Kinematostatic modelling and characteristic analysis are then carried out for the proposed large-range 3-PPPRR XYZ CPM with overlapping SCs. The results from finite element analysis (FEA) are compared to the characteristics found for the developed analytical models, as are experimental testing results (primary motion) from the prototyped 3-PPPRR XYZ CPM with overlapping SCs. Finally, issues on large-range motion and dynamics of such designs are discussed, as are possible improvements of the actuated compliant P joint. It is shown that the potential merits of the designs presented here include a) minimised parasitic rotations by only using three identical compliant legs; b) compact configurations and small size due to the use of embedded designs; c) approximately kinematostatically decoupled designs capable of easy controls; and d) monolithic fabrication for each leg using existing planar manufacturing technologies such as electric discharge machining (EDM).

Kinematic primitives for walking and trotting gaits of a quadruped robot with compliant legs

In this work we research the role of body dynamics in the complexity of kinematic patterns in a quadruped robot with compliant legs. Two gait patterns, lateral sequence walk and trot, along with leg length control patterns of different complexity were implemented in a modular, feed-forward locomotion controller. The controller was tested on a small, quadruped robot with compliant, segmented leg design, and led to self-stable and self-stabilizing robot locomotion. In-air stepping and on-ground locomotion leg kinematics were recorded, and the number and shapes of motion primitives accounting for 95% of the variance of kinematic leg data were extracted. This revealed that kinematic patterns resulting from feed-forward control had a lower complexity (in-air stepping, 2–3 primitives) than kinematic patterns from on-ground locomotion (νm4 primitives), although both experiments applied identical motor patterns. The complexity of on-ground kinematic patterns had increased, through ground contact and mechanical entrainment. The complexity of observed kinematic on-ground data matches those reported from level-ground locomotion data of legged animals. Results indicate that a very low complexity of modular, rhythmic, feed-forward motor control is sufficient for level-ground locomotion in combination with passive compliant legged hardware.

Biomimetic Design and Optimal Swing of a Hexapod Robot Leg

Biological inspiration has spawned a wealth of solutions to both mechanical design and control schemes in the efforts to develop agile legged machines. This paper presents a compliant leg mechanism for a small six-legged robot, HITCR-II, based on abstracted anatomy from insect legs. Kinematic structure, relative proportion of leg segment lengths and actuation system were analyzed in consideration of anatomical structure as well as muscle system of insect legs and desired mobility. A spring based passive compliance mechanism inspired by musculoskeletal structures of biological systems was integrated into distal segment of the leg to soften foot impact on touchdown. In addition, an efficient locomotion planner capable of generating natural movements for the legs during swing phase was proposed. The problem of leg swing was formulated as an optimal control procedure that satisfies a series of locomotion task terms while minimizing a biologically-based objective function, which was solved by a Gauss Pseudospectral Method (GPM) based numerical technique. We applied this swing generation algorithm to both a simulation platform and a robot prototype. Results show that the proposed leg structure and swing planner are able to successfully perform effective swing movements on rugged terrains.

Stable active running of a planar biped robot using Poincare map control

This work formulates the active limit cycles of bipedal running gaits for a compliant leg structure as the fixed point of an active Poincare map. Two types of proposed controllers stabilize the Poincare map around its active fixed point. The first one is a discrete linear state feedback controller designed with appropriate pole placement. The discrete-time control first uses purely constant torques during stance and flight phase, then discretizes each phase into smaller constant-torque intervals. The other controller is an invariant manifold based chaos controller: a generalized Ott, Grebogi and Yorke controller having a linear form and a nonlinear form. Both controllers can stabilize active running gaits on either even or sloped terrains. The efficiency of these controllers for bipedal running applications are compared and discussed.

Quadrupedal bounding with a segmented flexible torso: passive stability and feedback control

In this paper, the implications of torso flexibility on the dynamics of quadrupedal running are examined in a template setting. In the same vein with the spring loaded inverted pendulum, a reductive sagittal-plane model with a segmented flexible torso and compliant legs is introduced to capture the dynamics of bounding in the presence of torso flexibility via a minimum number of variables and parameters. Numerical return map studies of the system in dimensionless setting reveal that a large variety of cyclic bounding motions can be realized passively, through the natural interaction of the model with its environment. Despite the simplicity of the model, the resulting motions correspond to torso bending movements that resemble those in galloping mammals without explicit reliance on the fine structural and morphological details. Furthermore, for certain combinations of the system parameters—in particular the torso and leg relative stiffness—self-stable bounding motions emerge. The implications of the existence of such self-stable bounding orbits to control design are also discussed and a hybrid control law is derived that is capable of stabilizing the system as it encounters significantly large disturbances using only a single actuator located at the torso joint.

Finite Element Analysis of Pedestrian-Bridge Dynamic Interaction

The pedestrian-bridge dynamic interaction problem in the vertical direction based on a bipedal walking model and damped compliant legs is presented in this work. The classical finite element method, combined with a moving finite element, represents the motion of the pedestrian in the dynamic analysis of a footbridge. A control force is provided by the pedestrian to compensate for the energy loss due to the system damping in walking and to regulate the walking performance of the pedestrian. The effects of leg stiffness, the landing angle of attack, the damping ratio of the leg and the mass ratio of the human and structure are investigated in the numerical studies. Simulation results show that the dynamic interaction will increase with a larger vibration level of the structure. More external energy must be input to maintain steady walking and uniform dynamic behavior of the pedestrian in the process. The simple bipedal walking model could well describe the human-structure dynamic interaction.

Compliant Leg Architectures and a Linear Control Strategy for the Stable Running of Planar Biped Robots

This paper investigates two fundamental structures for biped robots and a control strategy to achieve stable biped running. The first biped structure contains straight legs with telescopic springs, and the second one contains knees with compliant elements in parallel with the motors. With both configurations we can use a standard linear discrete-time state-feedback control strategy to achieve an active periodic stable biped running gait, using the Poincare map of one complete step to produce the discrete-time model. In this case, the Poincare map describes an open-loop system with an unstable equilibrium, requiring a closed-loop control for stabilization. The discretization contains a stance phase, a flight phase and a touch-down. In the first approach, the control signals remain constant during each phase, while in the second approach both phases are discretized into a number of constant-torque intervals, so that its formulation can be applied easily to stabilize any active biped running gait. Simulation results with both the straight-legged and the kneed biped model demonstrate stable gaits on both horizontal and inclined surfaces.

Reverse kinetostatic analysis and stiffness synthesis of a spatial tensegrity-based compliant mechanism

This paper presents an analysis of a spatial tensegrity-based mechanism. In this study, a moving body in space is joined to ground by seven compliant leg connectors. Each leg connector is comprised of a spring in series with an adjustable piston. The mechanism is said to be ‘tensegrity-based’ in that it is prestressed when at equilibrium due to compliant linear elements. Two problems are solved in this paper; the reverse kinetostatic analysis and stiffness synthesis problems. In the first problem, the values of the seven spring constants and free lengths are given and the lengths of the seven pistons are determined such that: (1) the top body is given at a desired pose; (2) the top body is at equilibrium while a specified external wrench is applied; and (3) the total potential energy stored in the seven springs equals some desired value. In the second problem, the seven spring constants and the lengths of the seven pistons are determined such that conditions (1) and (2) are met and also the instantaneous stiffness matrix of the top body equals a specified set of matrix values. The paper formulates the solutions of these problems. Numerical examples are also presented.