A Spring-Mass Model of Locomotion With Full Asymptotic Stability

2011 ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Inverted Pendulum ◽  
Center Of Mass ◽  
Legged Locomotion ◽  
Human Locomotion ◽  
Whole Body ◽  
Slip Model ◽  
Mass Model ◽  
Energy Conserving ◽  
Spring Loaded Inverted Pendulum ◽  
Passive Model

The concept of passive dynamic walking and running [5] has demonstrated that a simple passive model can represent the dynamics of whole-body human locomotion. Since then, many passive models were developed and studied: [3,1,2,11]. The later developed Spring-Loaded Inverted Pendulum (SLIP) [1, 4, 11, 2] exhibits stable center of mass (CoM) motions just by resetting the landing angle at each touch down. Also, compared to SLIP, a SLIP-like model with simple flight leg control is better at resisting perturbations of the angle of velocity but not the magnitude [11, 2, 7]. Energy conserving models explain much about whole-body locomotion. Recently, there has been investigations of modified spring-mass models capable of greater stability, like that of animals and robots [9, 10, 8, 12]. Inspired by RHex [6], the Clock-Torqued Spring-Loaded Inverted Pendulum (CT-SLIP) model [9] was developed, and has been used to explain the robust stability of animal locomotion [12]. Here we present a model (mechanism) simpler than CT-SLIP called Forced-Damped SLIP (FD-SLIP) that can attain full asymptotically stability of the CoM during locomotion, and is capable of both walking and running motions. The FD-SLIP model, having fewer parameters, is more accessible and easier to analyze for the exploration and discovery of principles of legged locomotion.

A Spring-Mass Model of Locomotion With Full Asymptotic Stability

2011 ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Mass Velocity ◽  
Inverted Pendulum ◽  
Center Of Mass ◽  
Legged Locomotion ◽  
Reduced Model ◽  
Asymptotically Stable ◽  
Slip Model ◽  
Mass Model ◽  
Improved Stability ◽  
Spring Loaded Inverted Pendulum

A reduced model of legged locomotion, called the Spring Loaded Inverted Pendulum (SLIP) has previously been developed to predict the dynamics of locomotion. However, due to energy conservation, the SLIP model can only be partially asymptotically stable in the center-of-mass velocity. The more recently developed Clock-Torqued Spring Loaded Inverted Pendulum (CT-SLIP) model is fully asymptotically stable, and has a significantly larger stability basin than SLIP, but requires more than twice as many parameters. To more completely explore the parameter space and understand the reason for improved stability, we develop and analyze a further reduced model called the Forced-Damped Spring Loaded Inverted Pendulum (FD-SLIP) model.

A Simple Analytical Tool for Legged Robot Design

2012 ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Analytical Solutions ◽  
Inverted Pendulum ◽  
Legged Locomotion ◽  
Robot Design ◽  
Slip Model ◽  
Legged Robot ◽  
Biologically Relevant ◽  
Analytical Approximations ◽  
Energy Conserving ◽  
Spring Loaded Inverted Pendulum

Although legged locomotion is better at tackling complicated terrains compared with wheeled locomotion, legged robots are rare, in part, because of the lack of simple design tools. The dynamics governing legged locomotion are generally nonlinear and hybrid (piecewise-continuous) and so require numerical simulation for analysis and are not easily applied to robot designs. During the past decade, a few approximated analytical solutions of Spring-Loaded Inverted Pendulum (SLIP), a canonical model in legged locomotion, have been developed. However, SLIP is energy conserving and cannot predict the dynamical stability of real-world legged locomotion. To develop new analytical tools for legged robot designs, we first analytically solved SLIP in a new way. Then based on SLIP solution, we developed an analytical solution of a hip-actuated Spring-Loaded Inverted Pendulum (hip-actuated-SLIP) model, which is more biologically relevant and stable than the canonical energy conserving SLIP model. The analytical approximations offered here for SLIP and the hip actuated-SLIP solutions compare well with the numerical simulations of each. The analytical solutions presented here are simpler in form than those resulting from existing analytical approximations. The analytical solutions of SLIP and the hip actuated-SLIP can be used as tools for robot design or for generating biological hypotheses.

Towards Robustly Stable Musculo-Skeletal Simulation of Human Gait: Merging Lumped and Component-Based Modeling Approaches

2012 ◽  
Author(s):  
Justin Seipel
Keyword(s):  
Inverted Pendulum ◽  
Dynamic Models ◽  
Sagittal Plane ◽  
Center Of Mass ◽  
Human Model ◽  
Whole Body ◽  
Human Gait ◽  
Slip Model ◽  
Spring Loaded Inverted Pendulum ◽  
Low Dimensional

The objective of work presented in this paper is to increase the center-of-mass stability of human walking and running in musculo-skeletal simulation. The approach taken is to approximate the whole-body dynamics of the low-dimensional Spring-Loaded Inverted Pendulum (SLIP) model of locomotion in the OpenSim environment using existing OpenSim tools. To more directly relate low-dimensional dynamic models to human simulation, an existing OpenSim human model is first modified to more closely represent bilateral above-knee amputee locomotion with passive prostheses. To increase stability further beyond the energy-conserving SLIP model, an OpenSim model based upon the Clock-Torqued Spring-Loaded-Inverted-Pendulum (CT-SLIP) model of locomotion is also created. The result of this work is that a multi-body musculo-skeletal simulation in Open-Sim can approximate the whole-body sagittal-plane dynamics of the passive SLIP model. By adding a plugin controller to the OpenSim environment, the Clock-Torqued-SLIP dynamics can be approximated in OpenSim. To change between walking and running, only one parameter representing the preferred period of a stride is changed. The result is a robustly stable simulation of the center-of-mass locomotion for both walking and running that could serve as a first step toward increasingly anatomically accurate and robustly stable musculo-skeletal simulations.

Stabilized Walking of Humanoid NAO using Enhanced Spring-Loaded Inverted Pendulum Model on Uneven Terrain

2022 ◽  
Vol 13 (1) ◽  
pp. 0-0
Keyword(s):  
Path Length ◽  
Inverted Pendulum ◽  
Center Of Mass ◽  
Humanoid Robots ◽  
Slip Model ◽  
Turning Angle ◽  
Uneven Terrain ◽  
Inverted Pendulum Model ◽  
Pendulum Model ◽  
Spring Loaded Inverted Pendulum

In the coming decades, humanoid robots will play a rising role in society. The present article discusses their walking control and obstacle avoidance on uneven terrain using enhanced spring-loaded inverted pendulum model (ESLIP). The SLIP model is enhanced by tuning it with an adaptive particle swarm optimization (APSO) approach. It helps the humanoid robot to reach closer to the obstacles in order to optimize the turning angle to optimize the path length. The desired trajectory, along with the sensory data, is provided to the SLIP model, which creates compatible COM (center of mass) dynamics for stable walking. This output is fed to APSO as input, which adjusts the placement of the foot during interaction with uneven surfaces and obstacles. It provides an optimum turning angle for shunning the obstacles and ensures the shortest path length. Simulation has been carried out in a 3D simulator based on the proposed controller and SLIP controller in uneven terrain.

BIPED HOPPING CONTROL BAzSED ON SPRING LOADED INVERTED PENDULUM MODEL

2010 ◽  
Vol 07 (02) ◽  
pp. 263-280 ◽  
Author(s):  
SEYED HOSSEIN TAMADDONI ◽  
FARID JAFARI ◽  
ALI MEGHDARI ◽  
SAEED SOHRABPOUR
Keyword(s):  
Control Strategy ◽  
Inverted Pendulum ◽  
Hybrid Control ◽  
Sagittal Plane ◽  
Center Of Mass ◽  
Slip Model ◽  
Inverted Pendulum Model ◽  
Wide Range ◽  
Spring Loaded Inverted Pendulum ◽  
Mass Spring

Human running can be stabilized in a wide range of speeds by automatically adjusting muscular properties of leg and torso. It is known that fast locomotion dynamics can be approximated by a spring loaded inverted pendulum (SLIP) system, in which leg is replaced by a single spring connecting body mass to ground. Taking advantage of the inherent stability of SLIP model, a hybrid control strategy is developed that guarantees a stable biped locomotion in sagittal plane. In the presented approach, nonlinear control methods are applied to synchronize the biped dynamics and the spring-mass dynamics. As the biped center of mass follows the mass of the mass-spring model, the whole biped performs a stable locomotion corresponding to SLIP model. Simulations are done to obtain a repeatable hopping for a three-link underactuated biped model. Results show that periodic hopping gaits can be stabilized, and the presented control strategy provides feasible gait trajectories for stance and swing phases.

Foot Contact Interactions of a Clock Torqued Spring-Loaded Inverted Pendulum

2012 ◽  
Author(s):  
Steven Riddle ◽  
Justin Seipel
Keyword(s):  
Disturbance Rejection ◽  
Inverted Pendulum ◽  
Center Of Mass ◽  
Legged Robots ◽  
Contact Friction ◽  
Slip Model ◽  
Specific Interest ◽  
Mass System ◽  
Friction Models ◽  
Spring Loaded Inverted Pendulum

The clock-torqued spring-loaded inverted pendulum (CT-SLIP) model describes the robust dynamic stability properties observed in most animals and some legged robots. However, the model’s behavior is sensitive to changes in liftoff conditions such as those experienced on realistic terrain. Here the incorporation of friction at the foot-ground interface is explored on the CT-SLIP model with specific interest in improving the transient center-of-mass dynamics. Multiple friction models are presented and tuned to reflect a periodic center-of-mass gait. The transient dynamics with friction are analyzed in comparison to the CT-SLIP model and improvements to the settling time and disturbance rejection were found. This addition of foot-ground contact friction may allow for better understanding of center-of-mass system dynamics on realistic terrain.

A Piecewise-Linear Approximation of the Canonical Spring-Loaded Inverted Pendulum Model of Legged Locomotion

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Closed Form ◽  
Inverted Pendulum ◽  
Piecewise Linear ◽  
Closed Form Solution ◽  
Legged Locomotion ◽  
Form Solution ◽  
Piecewise Linear Approximation ◽  
Computational Time ◽  
Slip Model ◽  
Spring Loaded Inverted Pendulum

Here, we introduce and analyze a novel approximation of the well-established and widely used spring-loaded inverted pendulum (SLIP) model of legged locomotion, which has made several validated predictions of the center-of-mass (CoM) or point-mass motions of animal and robot running. Due to nonlinear stance equations in the existing SLIP model, many linear-based systems theories, analytical tools, and corresponding control strategies cannot be readily applied. In order to provide a significant simplification in the use and analysis of the SLIP model of locomotion, here we develop a novel piecewise-linear, time-invariant approximation. We show that a piecewise-linear system, with the only nonlinearity due to the switching event between stance and flight phases, can predict all the bifurcation features of the established nonlinear SLIP model over the entire three-dimensional model parameter space. Rather than precisely fitting only one particular solution, this approximation is made to quantitatively approximate the entire solution space of the SLIP model and capture all key aspects of solution bifurcation behavior and parametric sensitivity of the original SLIP model. Further, we provide an entirely closed-form solution for the stance trajectory as well as the system states at the end of stance, in terms of common functions that are easy to code and compute. Overall, the closed-form solution is found to be significantly faster than numerical integration when implemented using both matlab and c++. We also provide a closed-form analytical stride map, which is a Poincaré return section from touchdown (TD) to next TD event. This is the simplest closed-form approximate stride mapping yet developed for the SLIP model, enabling ease of analysis and numerical coding, and reducing computational time. The approximate piecewise-linear SLIP model presented here is a significant simplification over previous SLIP-based models and could enable more rapid development of legged locomotion theory, numerical simulations, and controllers.

scholarly journals Spring-loaded inverted pendulum goes through two contraction-extension cycles during the single stance phase of walking

2019 ◽  
Author(s):  
Gabriel Antoniak ◽  
Tirthabir Biswas ◽  
Nelson Cortes ◽  
Siddhartha Sikdar ◽  
Chanwoo Chun ◽  
...  
Keyword(s):  
Stance Phase ◽  
Inverted Pendulum ◽  
Center Of Mass ◽  
Legged Locomotion ◽  
Quantitative Model ◽  
Active Elements ◽  
Mechanical Models ◽  
Reaction Forces ◽  
Spring Loaded Inverted Pendulum ◽  
Walking Speeds

AbstractDespite the overall complexity of legged locomotion, the motion of the center of mass (COM) itself is relatively simple, and can be qualitatively described by simple mechanical models. The spring-loaded inverted pendulum (SLIP) is one such model, and describes both the COM motion and the ground reaction forces (GRFs) during running. Similarly, walking can be modeled by two SLIP-like legs (double SLIP or DSLIP). However, DSLIP has many limitations and is unlikely to serve as a quantitative model for walking. As a first step to obtaining a quantitative model for walking, we explored the ability of SLIP to model the single stance phase of walking across the entire range of walking speeds. We show that SLIP can be employed to quantitatively model the single stance phase except for two exceptions: first, it predicts larger horizontal GRFs than empirically observed. A new model - angular and radial spring-loaded inverted pendulum (ARSLIP) can overcome this deficit. Second, even the single stance phase has active elements, and therefore a quantitative model of locomotion would require active elements. Surprisingly, the leg spring undergoes a contraction-extension-contraction-extension (CECE) during walking; this cycling is partly responsible for the M-shaped GRFs produced during walking. The CECE cycle also lengthens the stance duration allowing the COM to travel passively for a longer time, and decreases the velocity redirection between the beginning and end of a step. A combination of ARSLIP along with active mechanisms during transition from one step to the next is necessary to describe walking.

A Spring-Loaded Inverted Pendulum Locomotion Model With Radial Forcing

2012 ◽  
Author(s):  
Peter Larson ◽  
Justin Seipel
Keyword(s):  
Parameter Space ◽  
Inverted Pendulum ◽  
Legged Locomotion ◽  
Radial Force ◽  
Slip Model ◽  
Forcing Function ◽  
Spring Loaded Inverted Pendulum ◽  
The Stability

Recent locomotion models have demonstrated the benefits of hip torques on legged locomotion stability. Here, a simple constant radial forcing function along the leg of the Spring-Loaded Inverted Pendulum (SLIP) model is added. This model is analyzed in order to determine what effect such a radial force might have on the stability of locomotion versus the more commonly used hip-torque forcing. The model is found to be unstable for the vast majority of the parameter space studied, for any amount of added forcing and damping constants. This suggests that simple constant forcing along the leg does not produce stable locomotion, unlike the case where forcing happens via hip torque.

Simple Leg Placement Strategy for a One Legged Running Model

2012 ◽  
Author(s):  
Timothy Sullivan ◽  
Justin Seipel
Keyword(s):  
Stance Phase ◽  
Energy State ◽  
Center Of Mass ◽  
Mass Movement ◽  
Legged Locomotion ◽  
Energy Conserving ◽  
Lower Torque ◽  
The Stability ◽  
Time Required ◽  
Torque Values

The Spring Loaded Inverted Pendulum (SLIP) model was developed to describe center of mass movement patterns observed in animals, using only a springy leg and a point mass. However, SLIP is energy conserving and does not accurately represent any biological or robotic system. Still, this model is often used as a foundation for the investigation of improved legged locomotion models. One such model called Torque Damped SLIP (TD-SLIP) utilizes two additional parameters, a time dependent torque and dampening to drastically increase the stability. Forced Damped SLIP (FD-SLIP), a predecessor of TD-SLIP, has shown that this model can be further simplified by using a constant torque, instead of a time varying torque, while still maintaining stability. Using FD-SLIP as a base, this paper explores a leg placement strategy using a simple PI controller. The controller takes advantage of the fact that the energy state of FD-SLIP is symmetric entering and leaving the stance phase during steady state conditions. During the flight phase, the touch down leg angle is adjusted so that the energy dissipation due to dampening, during the stance phase, compensates for any imbalance of energy. This controller approximately doubles the region of stability when subjected to velocity perturbations at touchdown, enables the model to operate at considerably lower torque values, and drastically reduces the time required to recover from a perturbation, while using less energy. Finally, the leg placement strategy used effectively imitates the natural human response to velocity perturbations while running.

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