Spring-loaded Inverted Pendulum Research Articles

Rotary and radial forcing effects on center-of-mass locomotion dynamics

Rotary and radial forcing are two common actuation methods for legged robots. However, these two orthogonal methods of center-of-mass (CoM) forcing have not been compared as potentially alternative strategies of actuation. In this paper, we compare the CoM stability and energetics of running with rotary and radial actuation through the simulation of two models: the rotary-forced spring-loaded inverted pendulum (rotary-forced-SLIP), and the radially-forced-SLIP. We model both radial and rotary actuation in the simplest way, applying them as a constant force during the stance portion of the gait. A simple application of constant rotary forcing throughout stance is capable of producing fully-asymptotically stable motion; however, a similarly constant application of radial forcing throughout the stance is not capable of producing stable solutions. We then allow both the applied rotary and radial forcing functions to turn on or off based on the occurrence of the mid-stance event, which breaks the symmetry of actuation during stance towards a net forward propulsion. We find that both a rotary force applied in the first half of stance and a radial force applied in the second half of stance, are capable of stabilizing running. Interestingly, these two forcing methods improve the motion stability in different ways. Rotary forcing first reduces then greatly increases the size of the stable parameter region when gradually increased. Radial forcing expands the stable parameter region, but only in a moderate way. Also, it is found that parameter region stabilized by rotary and radial forcing are largely complementary. Overall, rotary forcing can better stabilize running for both constant and event-based forcing functions that were attempted. This indicates that rotary forcing has an inherent capability of stabilizing running, even when minimal time-or-event-or-state feedback is present. Radial forcing, however, tends to be more energy efficient when compared to rotary forcing. In addition, a balanced stability and energy efficiency can be achieved by combining both forcing methods. These results may be applied in the future study of how legged animals move.

Compensations during Unsteady Locomotion.

Locomotion in a complex environment is often not steady, but the mechanisms used by animals to power and control unsteady locomotion (stability and maneuverability) are not well understood. We use behavioral, morphological, and impulsive perturbations to determine the compensations used during unsteady locomotion. At the level both of the whole-body and of joints, quasi-stiffness models are useful for describing adjustments to the functioning of legs and joints during maneuvers. However, alterations to the mechanics of legs and joints often are distinct for different phases of the step cycle or for specific joints. For example, negotiating steps involves independent changes of leg stiffness during compression and thrust phases of stance. Unsteady locomotion also involves parameters that are not part of the simplest reduced-parameter models of locomotion (e.g., the spring-loaded inverted pendulum) such as moments of the hip joint. Extensive coupling among translational and rotational parameters must be taken into account to stabilize locomotion or maneuver. For example, maneuvers with morphological perturbations (increased rotational inertial turns) involve changes to several aspects of movement, including the initial conditions of rotation and ground-reaction forces. Coupled changes to several parameters may be employed to control maneuvers on a trial-by-trial basis. Compensating for increased rotational inertia of the body during turns is facilitated by the opposing effects of several mechanical and behavioral parameters. However, the specific rules used by animals to control translation and rotation of the body to maintain stability or maneuver have not been fully characterized. We initiated direct-perturbation experiments to investigate the strategies used by humans to maintain stability following center-of-mass (COM) perturbations. When walking, humans showed more resistance to medio-lateral perturbations (lower COM displacement). However, when running, humans could recover from the point of maximum COM displacement faster than when walking. Consequently, the total time necessary for recovery was not significantly different between walking and running. Future experiments will determine the mechanisms used for compensations during unsteady locomotion at the behavioral, joint, and muscle levels. Using reduced-parameter models will allow common experimental and analytical frameworks for the study of both stability and maneuverability and the determination of general control strategies for unsteady locomotion.

Design and Open-Loop Control of the ParkourBot, a Dynamic Climbing Robot

The ParkourBot climbs in a planar reduced-gravity vertical chute by leaping back and forth between the chute's two parallel walls. The ParkourBot is comprised of a body with two springy legs and its controls consist of leg angles at touchdown and the energy stored in them. During flight, the robot stores elastic potential energy in its springy legs and then converts this potential energy in to kinetic energy at touchdown, when it “kicks off” a wall. This paper describes the ParkourBot's mechanical design, modeling, and open-loop climbing experiments. The mechanical design makes use of the BowLeg, previously used for hopping on a flat ground. We introduce two models of the BowLeg ParkourBot: one is based on a nonzero stance duration using the spring-loaded inverted pendulum model, and the other is a simplified model (the simplest parkour model, or SPM) obtained as the leg stiffness approaches infinity and the stance time approaches zero. The SPM approximation provides the advantage of closed-form calculations. Finally, predictions of the models are validated by experiments in open-loop climbing in a reduced-gravity planar environment provided by an air table.

Control of a Quadruped Robot with Bionic Springy Legs in Trotting Gait

AbstractLegged robots have better performance on discontinuous terrain than that of wheeled robots. However, the dynamic trotting and balance control of a quadruped robot is still a challenging problem, especially when the robot has multi-joint legs. This paper presents a three-dimensional model of a quadruped robot which has 6 Degrees of Freedom (DOF) on torso and 5 DOF on each leg. On the basis of the Spring-Loaded Inverted Pendulum (SLIP) model, body control algorithm is discussed in the first place to figure out how legs work in 3D trotting. Then, motivated by the principle of joint function separation and introducing certain biological characteristics, two joint coordination approaches are developed to produce the trot and provide balance. The robot reaches the highest speed of 2.0 m·s−1, and keeps balance under 250 Kg·m·s−1 lateral disturbance in the simulations. The effectiveness of these approaches is also verified on a prototype robot which runs to 0.83 m·s−1 on the treadmill. The simulations and experiments show that legged robots have good biological properties, such as the ground reaction force, and spring-like leg behavior.

Compliant ankle function results in landing-take off asymmetry in legged locomotion

The spring loaded inverted pendulum (SLIP) model is widely used to predict and explain basic characteristics of human walking and running. Its periodic running solutions can be mirrored at the instant of the vertical orientation of the leg and thus are symmetric between landing and take-off. In contrast, human running shows asymmetries between touchdown and take-off (e.g. shorter brake than push duration, greater mean ground reaction force during braking phase). Yet it is not fully understood whether these asymmetries are caused by asymmetric muscle properties (e.g. velocity-dependent force generation) or the asymmetric lever arm system in the human leg. We extend the SLIP model by a foot segment and a compliant ankle joint. This represents the extended foot contact and the displacement of the center of pressure during contact. With this model we investigate to which extent the landing-take off asymmetry in legged locomotion is caused by this asymmetric lever arm system. We find similar landing-take off asymmetries as in human running suggesting that the asymmetric lever arm system contributes to the asymmetry.

Hybrid control for SLIP-based robots running on unknown rough terrain

SUMMARYRapid and efficient dynamic stability control has been one of the important motivations in legged robot research, especially for legged robots running at high speed and/or on rough terrain. This paper presents a feasible control strategy, named Hybrid Feedback Control (HFC), for running systems based on the spring-loaded inverted pendulum principle (SLIP). The HFC strategy, which comprises two modules (i.e., touchdown angle control and energy compensation), predicts and regulates touchdown angle of the current cycle and need-to-complement energy input of the next cycle through hybrid feedback of flying apex state. This strategy can significantly reduce the computational complexity and enable the system to quickly converge to its control target, meeting the requirements of real-time control. Simulation experiments on various terrains were conducted to verify the adaptability of our HFC strategy. Results of these simulation experiments show that the approach herein can realize the periodical stability control of SLIP systems on different terrain conditions quickly and effectively.

Level locomotion in wood ants: evidence for grounded running

In order to better understand the strategies of locomotion in small insects, we have studied continuous level locomotion of the wood ant species Formica polyctena. We determined the three-dimensional centre of mass kinematics during the gait cycle and recorded the ground reaction forces of single legs utilising a self-developed test site. Our findings show that the animals used the same gait dynamics across a wide speed range without dissolving the tripodal stride pattern. To achieve higher velocities, the ants proportionally increased stride length and stepping frequency. The centre of mass energetics indicated a bouncing gait, in which horizontal kinetic and gravitational potential energy fluctuated in close phase. We determined a high degree of compliance especially in the front legs, as the effective leg length was nearly halved during the contact phase. This leads to only small vertical oscillations of the body, which are important in maintaining ground contact. Bouncing gaits without aerial phases seem to be a common strategy in small runners and can be sufficiently described by the bipedal spring-loaded inverted pendulum model. Thus, with our results, we provide evidence that wood ants perform 'grounded running'.

Performance Analysis and Feedback Control of ATRIAS, A Three-Dimensional Bipedal Robot

This paper develops feedback controllers for walking in 3D, on level ground, with energy efficiency as the performance objective. Assume The Robot Is A Sphere (ATRIAS) 2.1 is a new robot that has been designed for the study of 3D bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having 6 actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits of a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (CMT), a dimensionless measure of energetic efficiency, for walking speeds ranging from 0.5 (m/s) to 1.4 (m/s). A feedback controller is designed that stabilizes the 3D walking gaits, despite the high degree of underactuation of the robot. The 3D results are illustrated in simulation. In experiments on a planarized (2D) version of the robot, the controller yielded stable walking.

Analysis on the performance of the SLIP runner with nonlinear spring leg

The spring-loaded inverted pendulum(SLIP) has been widely studied in both animals and robots. Generally, the majority of the relevant theoretical studies deal with elastic leg, the linear leg length-force relationship of which is obviously conflict with the biological observations. A planar spring-mass model with a nonlinear spring leg is presented to explore the intrinsic mechanism of legged locomotion with elastic component. The leg model is formulated via decoupling the stiffness coefficient and exponent of the leg compression in order that the unified stiffness can be scaled as convex, concave as well as linear profile. The apex return map of the SLIP runner is established to investigate dynamical behavior of the fixed point. The basin of attraction and Floquet Multiplier are introduced to evaluate the self-stability and initial state sensitivity of the SLIP model with different stiffness profiles. The numerical results show that larger stiffness exponent can increase top speed of stable running and also can enlarge the size of attraction domain of the fixed point. In addition, the parameter variation is conducted to detect the effect of parameter dependency, and demonstrates that on the fixed energy level and stiffness profile, the faster running speed with larger convergence rate of the stable fixed point under small local perturbation can be achieved via decreasing the angle of attack and increasing the stiffness coefficient. The perturbation recovery test is implemented to judge the ability of the model resisting large external disturbance. The result shows that the convex stiffness performs best in enhancing the robustness of SLIP runner negotiating irregular terrain. This research sheds light on the running performance of the SLIP runner with nonlinear leg spring from a theoretical perspective, and also guides the design and control of the bio-inspired legged robot.

A novel robotic leg design with hybrid dynamics

A new leg mechanism is designed to achieve optimal behaviour in both the stance and swing phases. The leg is designed such that its dynamics and kinematics optimally mimic those of simpler mechanisms, namely the spring-loaded inverted pendulum (SLIP) and the double pendulum, for the stance and swing phase, respectively. Controllers based on the simpler mechanisms can thus be used, resulting in an effective control strategy that is simple and transparent.

Simulation of Aperiodic Bipedal Sprinting

Synthesis of legged locomotion through dynamic simulation is useful for exploration of the mechanical and control variables that contribute to efficient gait. Most previous simulations have made use of periodicity constraints, a sensible choice for investigations of steady-state walking or running. Sprinting from rest, however, is aperiodic by nature and this aperiodicity is central to the goal of the movement, as performance is determined in large part by a rapid acceleration phase early in the race. The purpose of this study was to create a novel simulation of aperiodic sprinting using a modified spring-loaded inverted pendulum (SLIP) biped model. The optimal control problem was to find the set of controls that minimized the time for the model to run 20 m, and this problem was solved using a direct multiple shooting algorithm that converts the original continuous time problem into piecewise discrete subproblems. The resulting nonlinear programming problem was solved iteratively using a sequential quadratic programming method. The starting point for the optimizer was an initial guess simulation that was a slow alternating-gait "jogging" simulation developed using proportional-derivative feedback to control trunk attitude, swing leg angle, and leg retraction and extension. The optimized aperiodic sprint simulation solution yielded a substantial improvement in locomotion time over the initial guess (2.79 s versus 6.64 s). Following optimization, the model produced forward impulses at the start of the sprint that were four times greater than those of the initial guess simulation, producing more rapid acceleration. Several gait features demonstrated in the optimized sprint simulation correspond to behaviors of human sprinters: forward trunk lean at the start; straightening of the trunk during acceleration; and a dive at the finish. Optimization resulted in reduced foot contact times (0.065 s versus 0.210 s), but contact times early in the optimized simulation were longer to facilitate acceleration. The present study represents the first simulation of multistep aperiodic sprinting with optimal controls. Although the minimized objective function was simple, the model replicated several complex behaviors such as modulation of the foot contact and executing a forward dive at the finish line. None of these observed behaviors were imposed explicitly by constraints but rather were "discovered" by the optimizer. These methods will be extended by addition of musculotendon actuators and joints in order to gain understanding of the influence of musculoskeletal mechanics on gait speed.

Robust hopping based on virtual pendulum posture control

A new control approach to achieve robust hopping against perturbations in the sagittal plane is presented in this paper. In perturbed hopping, vertical body alignment has a significant role for stability. Our approach is based on the virtual pendulum concept, recently proposed, based on experimental findings in human and animal locomotion. In this concept, the ground reaction forces are pointed to a virtual support point, named virtual pivot point (VPP), during motion. This concept is employed in designing the controller to balance the trunk during the stance phase. New strategies for leg angle and length adjustment besides the virtual pendulum posture control are proposed as a unified controller. This method is investigated by applying it on an extension of the spring loaded inverted pendulum (SLIP) model. Trunk, leg mass and damping are added to the SLIP model in order to make the model more realistic. The stability is analyzed by Poincaré map analysis. With fixed VPP position, stability, disturbance rejection and moderate robustness are achieved, but with a low convergence speed. To improve the performance and attain higher robustness, an event-based control of the VPP position is introduced, using feedback of the system states at apexes. Discrete linear quartic regulator is used to design the feedback controller. Considerable enhancements with respect to stability, convergence speed and robustness against perturbations and parameter changes are achieved.

2A1-I04 骨盤運動に着目した2足走行ロボットの開発 (第1報 : 骨盤運動と脚弾性を活用した走行モデルの考案)(移動知と身体性基盤)

A human running motion can be modeled by a spring-loaded inverted pendulum (SLIP). However, the SLIP model does not include human-like pelvic movement which can contribute to increase a jumping force and absorb a landing impact. Based on a running gait analysis, we propose a novel model, named SLIP2 (Spring-Loaded Inverted Pendulum using Pelvis), that is composed of a body mass, a pelvis and leg springs. This model can store elastic energy during running by utilizing pelvic movement in the frontal plane. Additionally, we developed pelvic oscillation control and running velocity control. To evaluate the proposed model and control methods, we performed hopping and running simulations. The simulation results show that the SLIP2 model achieved hopping and running motions. Moreover, it shows that the difference of the pelvic oscillation phase affects the jumping force.

Approximate Perturbation Stance Map of the SLIP Runner and Application to Locomotion Control

This paper presents a novel method of perturbation to obtain the analytic approximate solution to the Spring-Loaded Inverted Pendulum (SLIP) dynamics in stance phase with considering the effect of gravity. This perturbation solution achieves higher accuracy in predicting the apex state variables than the typical existing analytic approximations. Particularly, our solution is validated for non-symmetric trajectory of hopping in a large angle range. Furthermore, the stance controller of the SLIP runner is developed to regulate the apex state based on the approximate apex return map. To compensate the energy variation between the current and desired apex states, a stiffness adjustment of the leg spring in stance phase is presented. The deadbeat controller of the angle of attack is designed to track the regulated apex height and velocity. The simulation demonstrates that the SLIP runner applying the proposed stance controller reveals higher tracking accuracy and more rapidly converges to the regulated apex state.

Multiple-step model-experiment matching allows precise definition of dynamical leg parameters in human running

The spring-loaded inverted pendulum (SLIP) model is a well established model for describing bouncy gaits like human running. The notion of spring-like leg behavior has led many researchers to compute the corresponding parameters, predominantly stiffness, in various experimental setups and in various ways. However, different methods yield different results, making the comparison between studies difficult. Further, a model simulation with experimentally obtained leg parameters typically results in comparatively large differences between model and experimental center of mass trajectories. Here, we pursue the opposite approach which is calculating model parameters that allow reproduction of an experimental sequence of steps. In addition, to capture energy fluctuations, an extension of the SLIP (ESLIP) is required and presented. The excellent match of the models with the experiment validates the description of human running by the SLIP with the obtained parameters which we hence call dynamical leg parameters.

Leg-adjustment strategies for stable running in three dimensions

The dynamics of the center of mass (CoM) in the sagittal plane in humans and animals during running is well described by the spring-loaded inverted pendulum (SLIP). With appropriate parameters, SLIP running patterns are stable, and these models can recover from perturbations without the need for corrective strategies, such as the application of additional forces. Rather, it is sufficient to adjust the leg to a fixed angle relative to the ground. In this work, we consider the extension of the SLIP to three dimensions (3D SLIP) and investigate feed-forward strategies for leg adjustment during the flight phase. As in the SLIP model, the leg is placed at a fixed angle. We extend the scope of possible reference axes from only fixed horizontal and vertical axes to include the CoM velocity vector as a movement-related reference, resulting in six leg-adjustment strategies. Only leg-adjustment strategies that include the CoM velocity vector produced stable running and large parameter domains of stability. The ability of the model to recover from perturbations along the direction of motion (directional stability) depended on the strategy for lateral leg adjustment. Specifically, asymptotic and neutral directional stability was observed for strategies based on the global reference axis and the velocity vector, respectively. Additional features of velocity-based leg adjustment are running at arbitrary low speed (kinetic energy) and the emergence of large domains of stable 3D running that are smoothly transferred to 2D SLIP stability and even to 1D SLIP hopping. One of the additional leg-adjustment strategies represented a large convex region of parameters where stable and robust hopping and running patterns exist. Therefore, this strategy is a promising candidate for implementation into engineering applications, such as robots, for instance. In a preliminary comparison, the model predictions were in good agreement with the experimental data, suggesting that the 3D SLIP is an appropriate model to describe human running in three dimensions. The prediction of stable running based on movement-related leg-adjustment strategies indicates that both humans and robots may not require external targets directing the movement to run in three dimensions based on compliant leg function. This new movement-based reference enables the control of 3D running because leg adjustment is less sensitive and gait stability is separated from directional stability.

Quantifying disturbance rejection of SLIP-like running systems

The speed and maneuverability at which legged animals can travel across rough and cluttered landscapes has provided inspiration for the development of legged robots with similar capabilities. Researchers have developed a number of robots that can run over rough terrain, but there is currently no universally accepted measure for stability in an unstructured environment. This paper considers the effectiveness of a number of stability metrics in predicting the disturbance-rejection behavior of three spring-loaded inverted pendulum (SLIP)-like systems ranging from simple reduced-order models to a one-legged mechanical hopping robot. We show that the gait stability norm utilizing leg-centric indicators and a two-step decay ratio utilizing body-centric indicators provide the best correlation between step response behavior and how the systems perform when running over unknown, uneven terrain. By providing a cross-platform stability comparison, this paper facilitates the development and evaluation of robots and control schemes, such as the active energy removal approach utilized in this paper, that aim to improve the stability of running systems.

Research on the Analyticity of SLIP Model Based on Perturbation Theory and Locomotion Control

An approximate analytical solution of SLIP(spring loaded inverted pendulum) model dynamics in stance phase is proposed based on perturbation theory,aiming at resolving the lack of analytical method to deal with SLIP model dynamics.The Poincare map of apex height in flight phase is established through which the entire dynamics of SLIP model is derived analytically.The error analysis results indicate that the prediction of apex height with approximate solution possesses high approximation precision.Aiming at the locomotion control of SLIP hopper,the deadbeat controller for apex height of SLIP system is designed based on the inverse Poincare map.The tracking simulation of desired step signal is implemented to show that the algorithm proposed in this paper not only can track the desired trajectory but also inhibits overshoot effectively.Furthermore,the equivalence of apex velocity control and height control is verified through analyzing the experiment curve of horizontal velocity of the center of mass,which extends the application range of the algorithm.

Stance leg control: variation of leg parameters supports stable hopping

The spring-loaded inverted pendulum describes the planar center-of-mass dynamics of legged locomotion. This model features linear springs with constant parameters as legs. In biological systems, however, spring-like properties of limbs can change over time. Therefore, in this study, it is asked how variation of spring parameters during ground contact would affect the dynamics of the spring–mass model. Neglecting damping initially, it is found that decreasing stiffness and increasing rest length of the leg during a stance phase are required for orbitally stable hopping. With damping, stable hopping is found for a larger region of rest-length rates and stiffness rates. Here, also increasing stiffness and decreasing rest length can result in stable hopping. Within the predicted range of leg parameter variations for stable hopping, there is no need for precise parameter tuning. Since hopping gaits form a subset of the running gaits (with vanishing horizontal velocity), these results may help to improve leg design in robots and prostheses.

Exploiting the passive dynamics of a compliant leg to develop gait transitions

In the area of bipedal locomotion, the spring-loaded inverted pendulum model has been proposed as a unified framework to explain the dynamics of a wide variety of gaits. In this paper, we present an analysis of the mathematical model and its dynamical properties. We use the perspective of hybrid dynamical systems to study the dynamics and define concepts such as partial stability and viability. With this approach, on the one hand, we identify stable and unstable regions of locomotion. On the other hand, we find ways to exploit the unstable regions of locomotion to induce gait transitions at a constant energy regime. Additionally, we show that simple nonconstant angle of attack control policies can render the system almost always stable.