Spring-loaded Inverted Pendulum Research Articles

Spring-loaded inverted pendulum goes through two contraction-extension cycles during the single-support phase of walking.

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. In particular, walking can be qualitatively modeled by a simple model in which each leg is described by a spring-loaded inverted pendulum (SLIP). However, SLIP has many limitations and is unlikely to serve as a quantitative model. As a first step to obtaining a quantitative model for walking, we explored the ability of SLIP to model the single-support phase of walking, and found that SLIP has two limitations. First, it predicts larger horizontal ground reaction forces (GRFs) than empirically observed. A new model – angular and radial spring-loaded inverted pendulum (ARSLIP) – can overcome this deficit. Second, although the leg spring (surprisingly) goes through contraction-extension-contraction-extensions (CECEs) during the single-support phase of walking and can produce the characteristic M-shaped vertical GRFs, modeling the single-support phase requires active elements. Despite these limitations, SLIP as a model provides important insights. It shows that the CECE cycling lengthens the stance duration allowing the COM to travel passively for longer, and decreases the velocity redirection between the beginning and end of a step.

Human-like hopping in machines

Template models of legged locomotion are powerful tools for gait analysis, but can also inspire robot design and control. In this paper, a spring-loaded inverted pendulum (SLIP) model is employed to control vertical hopping of a 2-segmented legged robot. Feed-forward and bio-inspired virtual model control using the SLIP model are compared. In the latter approach, the feedback control emulates a virtual spring between hip and foot. The results demonstrate similarity of human and robot hopping. Moreover, the feedback control proves to simplify and improve hopping control. It yields better perturbation recovery and locomotion adaptation and is even easier to tune. Thus, human-like hopping is achievable using a rather simple template-based controller, which ensures the required performance, robustness and versatility.

Dynamic Locomotion Gaits of a Compliantly Actuated Quadruped With SLIP-Like Articulated Legs Embodied in the Mechanical Design

The spring loaded inverted pendulum (SLIP) model has been extensively shown to be fundamental for legged locomotion. However, the way this low-order template model dynamics is anchored in high-dimensional articulated multibody systems describing compliantly actuated robots (and animals) is not obvious and has not been shown so far. In this letter, an articulated leg mechanism and a corresponding quadrupedal robot design are introduced, for which the natural oscillation dynamics is structurally equivalent to the SLIP. On the basis of this property, computationally simple and robust control methods are proposed, which implement the gaits of pronking, trotting, and dynamic walking in the real robotic system. Experiments with a compliantly actuated quadruped featuring only low-performance electrical drives validate the effectiveness of the proposed approach.

Control of Planar Spring–Mass Running Through Virtual Tuning of Radial Leg Damping

Existing research on dynamically capable legged robots, particularly those based on spring–mass models, generally considers improving in isolation either the stability and control accuracy on the rough terrain, or the energetic efficiency in steady state. In this paper, we propose a new method to address both, based on the hierarchical embedding of a simple spring-loaded inverted pendulum (SLIP) template model with a tunable radial damping coefficient into a realistic leg structure with series-elastic actuation. Our approach allows using the entire stance phase to inject/remove energy both for transient steps and in steady state, decreasing the maximum necessary actuator power while eliminating wasteful sources of the negative work. In doing so, we preserve the validity of the existing analytic approximations to the underlying SLIP model, propose improvements to increase the predictive accuracy, and construct accurate, model-based controllers that use the tunable damping coefficient of the template model. We provide extensive comparative simulations to establish the energy and power efficiency advantages of our approach, together with the accuracy of model-based gait control methods.

A simple extension of inverted pendulum template to explain features of slow walking✰

Locomotion involves complex interactions between an organism and its environment. Despite these complex interactions, many characteristics of the motion of an animal's center of mass (COM) can be modeled using simple mechanical models such as inverted pendulum (IP) and spring-loaded inverted pendulum (SLIP) which employ a single effective leg to model an animal's COM. However, because these models are simple, they also have many limitations. We show that one limitation of IP and SLIP and many other simple mechanical models of locomotion is that they cannot model many observed features of locomotion at slow speeds. This limitation is due to the fact that the gravitational force is too strong, and, if unopposed, compels the animal to complete its stance in a relatively short time. We propose a new model, AS-IP (Angular Spring modulated Inverted Pendulum), in which the body is attached to the leg using springs which resist the leg's movement away from the vertical plane, and thus provides a means to model forces that effectively counter gravity. We show that AS-IP provides a mechanism by which an animal can tune its stance duration, and provide evidence that AS-IP is an excellent model for the motion of a fly's COM. More generally, we conclude that combining AS-IP with SLIP will greatly expand our ability to model legged locomotion over a range of speeds.

Can forward dynamics simulation with simple model estimate complex phenomena?: Case study on sprinting using running-specific prosthesis

Surpassing the world record in athletic performance requires extensive use of kinematic and dynamic motion analyses to develop novel body usage skills and training methods. Performance beyond the current world record has not been realized or measured; therefore, we need to generate it with dynamics consistency using forward dynamics simulation, although it is technologically difficult because of the complexity of the human structure and its dynamics. This research develops a multilayered kinodynamics simulation that uses a detailed digital human model and a simple motion-representation model to generate the detailed sprinting performances of individuals with lower extremity amputations (ILEAs) aided by carbon-fiber running-specific prostheses (RSPs), which have complex interactions with humans. First, we developed a digital human model of an ILEA using an RSP. We analyzed ILEA sprinting based on experimental motion measurements and kinematics/dynamics computations. We modeled the RSP-aided ILEA sprinting using a simple spring-loaded inverted pendulum model, comprising a linear massless spring, damper, and mass, and we identified the relevant parameters from experimentally measured motion data. Finally, we modified the sprint motion by varying the parameters corresponding to the RSP characteristics. Here, the forward dynamics have been utilized to simulate detailed whole-body sprinting with different RSP types (including simulated RSPs not worn by the subject). Our simulations show good correspondence with the experimentally measured data and further indicate that the sprint time can be improved by reducing the RSP viscosity and increasing stiffness. These simulation results are validated by the experimentally measured motion modifications obtained with different types of RSPs. These results show that the multilayered kinodynamics simulation using the detailed digital human model and the simple motion-representation model has the capacity to generate complex phenomena such as RSP-aided ILEA sprinting that contains complex interactions between the human and the RSP. This simulation technique can be applied to RSP design optimization for ILEA sprinting.

Unified bipedal gait for autonomous transition between walking and running in pursuit of energy minimization

This paper addresses “unified bipedal gait” control, which autonomously selects the energy-minimized gait from walking and running at any feasible gait speeds. Humans select walking/running at low/high speed in pursuit of energy minimization and transition between them naturally. Despite the quite different behaviors of walking and running, human gaits share an inherent controller. The unified bipedal gait uses the inherent controller, which implements passive dynamic autonomous control (PDAC) based on a damping and spring-loaded inverted pendulum (D-SLIP) model. Although this D-SLIP could cause chaotic motions, compliance in the D-SLIP dynamics switches behaviors between walking and running, that is, low/high compliant legs for walking/running. This property is employed by the virtual holonomic constraint of the PDAC to extract the required characteristics of walking/running from the D-SLIP dynamics while restraining the chaotic motions for asymptotic stability. As a result, the unified bipedal gait bifurcates to walking and running via autonomous transition to minimize energy cost at any feasible gait speeds.

Estimation of unmeasured ground reaction force data based on the oscillatory characteristics of the center of mass during human walking

To enhance the wearability of portable motion-monitoring devices, the size and number of sensors are minimized, but at the expense of quality and quantity of data collected. For example, owing to the size and weight of low-frequency force transducers, most currently available wearable gait measurement systems provide only limited, if any, elements of ground reaction force (GRF) data. To obtain the most GRF information possible with a minimal use of sensors, we propose a GRF estimation method based on biomechanical knowledge of human walking. This includes the dynamics of the center of mass (CoM) during steady human gait resembling the oscillatory behaviors of a mass-spring system. Available measurement data were incorporated into a spring-loaded inverted pendulum with translating pivot. The spring stiffness and simulation parameters were tuned to match, as accurately as possible, the available data and oscillatory characteristics of walking. Our results showed that the model simulation estimated reasonably well the unmeasured GRF. Using the vertical GRF and CoP profile for gait speeds ranging from 0.93 to 1.89 m/s, the anterior-posterior (A-P) GRF was estimated and resulted in an average correlation coefficient of R = 0.982 ± 0.009. Even when the ground contact timing and gait speed information were alone available, our method estimated GRFs resulting in R = 0.969 ± 0.022 for the A-P and R = 0.891 ± 0.101 for the vertical GRFs. This research demonstrates that the biomechanical knowledge of human walking, such as inherited oscillatory characteristics of the CoM, can be used to gain unmeasured information regarding human gait dynamics.

Kinematics of lower limbs during walking are emulated by springy walking model with a compliantly connected, off-centered curvy foot

The dynamics of the center of mass (CoM) during walking and running at various gait conditions are well described by the mechanics of a simple passive spring loaded inverted pendulum (SLIP). Due to its simplicity, however, the current form of the SLIP model is limited at providing any further information about multi-segmental lower limbs that generate oscillatory CoM behaviors and their corresponding ground reaction forces. Considering that the dynamics of the CoM are simply achieved by mass-spring mechanics, we wondered whether any of the multi-joint motions could be demonstrated by simple mechanics. In this study, we expand a SLIP model of human locomotion with an off-centered curvy foot connected to the leg by a springy segment that emulates the asymmetric kinematics and kinetics of the ankle joint. The passive dynamics of the proposed expansion of the SLIP model demonstrated the empirical data of ground reaction forces, center of mass trajectories, ankle joint kinematics and corresponding ankle joint torque at various gait speeds. From the mechanically simulated trajectories of the ankle joint and CoM, the motion of lower-limb segments, such as thigh and shank angles, could be estimated from inverse kinematics. The estimation of lower limb kinematics showed a qualitative match with empirical data of walking at various speeds. The representability of passive compliant mechanics for the kinetics of the CoM and ankle joint and lower limb joint kinematics implies that the coordination of multi-joint lower limbs during gait can be understood with a mechanical framework.

Design and optimization of an elastic linkage quadruped robot based on workspace and tracking error

High acceleration and extreme load are frequently appeared on high-speed locomotion of legged robot’s legs, imposing a challenging trade-off between weight and torque in leg design. This paper proposes a new design paradigm based on cable-drive and elastic linkage to solve the problem. The details of the design procedure are given, including the construction of the single leg. With the optimum design of the linkage mechanism, a combined index of the workspace and tracking error are used as object function, and taking geometrical design parameters of the linkage as optimization parameters. Based on the target workspace and the spring-loaded inverted pendulum model, the best foot trajectory in obstacle climbing and trotting gait are analyzed and illustrated. This paper built linkage cable-drive spring robot based on the legged module integration. Simulations and experiments indicate that linkage cable-drive spring robot performs stable trotting with control of the spring-loaded inverted pendulum model. Linkage cable-drive spring robot prototype experiments results are provided to verify the validity of the new method.

Bionic quadruped robot dynamic gait control strategy based on twenty degrees of freedom

Quadruped robot dynamic gaits have much more advantages than static gaits on speed and efficiency, however high speed and efficiency calls for more complex mechanical structure and complicated control algorithm. It becomes even more challenging when the robot has more degrees of freedom. As a result, most of the present researches focused on simple robot, while the researches on dynamic gaits for complex robot with more degrees of freedom are relatively limited. The paper is focusing on the dynamic gaits control for complex robot with twenty degrees of freedom for the first time. Firstly, we build a relatively complete 3D model for quadruped robot based on spring loaded inverted pendulum (SLIP) model, analyze the inverse kinematics of the model, plan the trajectory of the swing foot and analyze the hydraulic drive. Secondly, we promote the control algorithm of one-legged to the quadruped robot based on the virtual leg and plan the state variables of pace gait and bound gait. Lastly, we realize the above two kinds of dynamic gaits in ADAMS-MATLAB joint simulation platform which testify the validity of above method.

Skipping on uneven ground: trailing leg adjustments simplify control and enhance robustness

It is known that humans intentionally choose skipping in special situations, e.g. when descending stairs or when moving in environments with lower gravity than on Earth. Although those situations involve uneven locomotion, the dynamics of human skipping on uneven ground have not yet been addressed. To find the reasons that may motivate this gait, we combined experimental data on humans with numerical simulations on a bipedal spring-loaded inverted pendulum model (BSLIP). To drive the model, the following parameters were estimated from nine subjects skipping across a single drop in ground level: leg lengths at touchdown, leg stiffness of both legs, aperture angle between legs, trailing leg angle at touchdown (leg landing first after flight phase), and trailing leg retraction speed. We found that leg adjustments in humans occur mostly in the trailing leg (low to moderate leg retraction during swing phase, reduced trailing leg stiffness, and flatter trailing leg angle at lowered touchdown). When transferring these leg adjustments to the BSLIP model, the capacity of the model to cope with sudden-drop perturbations increased.

Morphological and control criteria for self-stable underwater hopping

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

A flight-phase terrain following control strategy for stable and robust hopping of a one-legged robot under large terrain variations

This work demonstrates a simple, once per step, flight-control method for robots running on a planar unknown rough-terrain environment. The robot used to exemplify these control strategies is the ParkourBot, a spring loaded inverted pendulum (SLIP)-based robot. The SLIP model is widely used for the description of humans and animals running motion and has been the basis for many robots. A known control scheme for increasing robustness of the conservative, SLIP model is the swing leg retraction (SLR) method. Despite of the SLR’s popularity, it is not intended to be used on the more realistic, non-conservative damped SLIP model. On the damped SLIP model, the SLR controller failed to provide adequate results, therefore, we have derived a new simple, flight-phase control method called polynomial energy insertion (PEI). The new PEI method is based on the dead-beat solution of the damped simplified instantaneous SLIP (iSLIP) model, which assumes an infinitely stiff spring. Unlike the SLR which, starting from apex, changes the leg angle monotonically during flight, the PEI requires the leg length (hence, energy insertion) to change monotonically throughout the flight phase. Interestingly, the leg angle remains nearly constant. In simulations and experiments, we have compared the newly developed PEI to the previous SLR method. We have found that since the SLR does not control the horizontal velocity, it looses its stability under rough terrain. The PEI method was able to control the horizontal velocity and height from ground and hence showed great improvement in robustness to rough terrain. Moreover, in both simulations and experiments the PEI methods showed an increase in the mean jumps to failure of more than 30% compared to SLR-based controllers.

Simplified Triped Robot for Analysis of Three-Dimensional Gait Generation

[abstFig src='/00290003/08.jpg' width='230' text='The tripedal robot “Martian petit”' ] Significant efforts to simplify the body structure of multi-legged walking robots have been made over the years. Of these, the Spring-Loaded-Inverted-Pendulum (SLIP) model has been very popular, therefore widely employed in the design of walking robots. In this paper, we develop a SLIP-based tripedal walking robot with a focus on the geometric symmetry of the body structure. The proposed robot possesses a compact, light-weight, and compliant leg modules. These modules are controlled by a distributed control law that consists of decoupled oscillators with only local force feedback. As demonstrated through experiments, the simplified design of the robot makes possible the generation of high-speed dynamic locomotion. Despite the structural simplicity of the proposed model, the generation of several gait-patterns is demonstrated. The proposed minimalistic design approach with radial symmetry simplifies the function of each limb in the three-dimensional gait generation of the robot.

A Parallel Actuated Pantograph Leg for High-speed Locomotion

High-speed running is one of the most important topics in the field of legged robots which requires strict constraints on structural design and control. To solve the problems of high acceleration, high energy consumption, high pace frequency and ground impact during high-speed movement, this paper presents a parallel actuated pantograph leg with an approximately decoupled configuration. The articulated leg features in light weight, high load capacity, high mechanical efficiency and structural stability. The similarity features of force and position between the control point and the foot are analyzed. The key design parameters, K1 and K2, which concern the dynamic performances, are carefully optimized by comprehensive evaluation of the leg inertia and mass within the maximum foot trajectory. A control strategy that incorporates virtual Spring Loaded Inverted Pendulum (SLIP) model and active force is also proposed to test the design. The strategy can implement highly flexible impedance without mechanical springs, which substantially simplifies the design and satisfies the variable stiffness requirements during high-speed running. The rationality of the structure and the effectiveness of the control law are validated by simulation and experiments.

SLIP-Model-Based Dynamic Gait Generation in a Leg-Wheel Transformable Robot With Force Control

We report on the model-based development of the pronking and trotting behaviors of a leg-wheel transformable robot. The spring-loaded inverted pendulum (SLIP) model serves as the motion template; its stable running motion is investigated, and the parameters and state conditions suitable for implementation in the robot are selected. Mapping between the SLIP model and the robot is conducted so that the running of the SLIP model can be transformed into the robot's pronking and trotting behaviors. The spring effect of the leg-wheel is achieved using force control with motor current feedback. The robot was operated to initiate four different dynamic behaviors with variations in stiffness and gait based on four different fixed-point trajectories of the SLIP model. The experimental results confirm that the proposed strategy is effective in initiating the dynamic behaviors.

Arbitrary Symmetric Running Gait Generation for an Underactuated Biped Model.

This paper investigates generating symmetric trajectories for an underactuated biped during the stance phase of running. We use a point mass biped (PMB) model for gait analysis that consists of a prismatic force actuator on a massless leg. The significance of this model is its ability to generate more general and versatile running gaits than the spring-loaded inverted pendulum (SLIP) model, making it more suitable as a template for real robots. The algorithm plans the necessary leg actuator force to cause the robot center of mass to undergo arbitrary trajectories in stance with any arbitrary attack angle and velocity angle. The necessary actuator forces follow from the inverse kinematics and dynamics. Then these calculated forces become the control input to the dynamic model. We compare various center-of-mass trajectories, including a circular arc and polynomials of the degrees 2, 4 and 6. The cost of transport and maximum leg force are calculated for various attack angles and velocity angles. The results show that choosing the velocity angle as small as possible is beneficial, but the angle of attack has an optimum value. We also find a new result: there exist biped running gaits with double-hump ground reaction force profiles which result in less maximum leg force than single-hump profiles.

On the periodic gait stability of a multi-actuated spring-mass hopper model via partial feedback linearization

Spring-loaded inverted pendulum (SLIP) template (and its various derivatives) could be considered as the mostly used and widely accepted models for describing legged locomotion. Despite their simple nature, as being a simple spring-mass model in dynamics perspective, the SLIP model and its derivatives are formulated as restricted three-body problem, whose non-integrability has been proved long before. Thus, researchers proceed with approximate analytical solutions or use partial feedback linearization when numerical integration is not preferred in their analysis. The key contributions of this paper can be divided into two parts. First, we propose a dissipative SLIP model, which we call as multi-actuated dissipative SLIP (MD-SLIP), with two extended actuators: one linear actuator attached serially to the leg spring and one rotary actuator attached to hip. The second contribution of this paper is a partial feedback linearization strategy by which we can cancel some nonlinear dynamics of the proposed model and obtain exact analytical solution for the equations of motion. This allows us to investigate stability characteristics of the hopping gait obtained from the MD-SLIP model. We illustrate the applicability of our solutions with open-loop and closed-loop hopping performances on rough terrain simulations.

Suspending loads decreases load stability but may slightly improve body stability

Here, we seek to determine how compliantly suspended loads could affect the dynamic stability of legged locomotion. We theoretically model the dynamic stability of a human carrying a load using a coupled spring-mass-damper model and an actuated spring-loaded inverted pendulum model, as these models have demonstrated the ability to correctly predict other aspects of locomotion with a load in prior work, such as body forces and energetic cost. We report that minimizing the load suspension natural frequency and damping ratio significantly reduces the stability of the load mass but may slightly improve the body stability of locomotion when compared to a rigidly attached load. These results imply that a highly-compliant load suspension could help stabilize body motion during human, animal, or robot load carriage, but at the cost of a more awkward (less stable) load.