Spring-loaded Inverted Pendulum Model Research Articles

On the analysis and control of a bipedal legged locomotion model via partial feedback linearization

In this study, we introduce a new model for bipedal locomotion that enhances the classical spring-loaded inverted pendulum (SLIP) model. Our proposed model incorporates a damping term in the leg spring, a linear actuator serially interconnected to the leg, and a rotary actuator affixed to the hip. The distinct feature of this new model is its ability to overcome the non-integrability challenge inherent in the conventional SLIP models through the application of partial feedback linearization. By leveraging these actuators, our model enhances the stability and robustness of the locomotion mechanism, particularly when navigating across varied terrain profiles. To validate the effectiveness and practicality of this model, we conducted detailed simulation studies, benchmarking its performance against other recent models outlined in the literature. Our findings suggest that the redundancy in actuation introduced by our model significantly facilitates both open-loop and closed-loop walking gait, showcasing promising potential for the future of bipedal locomotion, especially for bio-inspired robotics applications in outdoor and rough terrains.

Intrinsic toe joint stabilization in foot-slip turning motion of musculoskeletal robot with DSP-SLIP model

This paper introduces the foot-slip turning motion by combining a slider-like mechanism with the double-support parallel spring-loaded inverted pendulum model, in which the musculoskeletal robot uses the model to simplify the slip-turning motion with its compliant structure and utilizes the foot muscle to improve its postural stability. The slip-turning motion, characterized by slight movements for swift turning via foot slippage, is advantageous for musculoskeletal robots due to their limited range of movement. The challenge lies in the reduced support area during the motion, which impacts stability. In our previous study, the robot ‘PneuTurn-T’ successfully executed said motion, whereas the details of the turning mechanism were lacking. This study investigated the utilization of leg compliance in the motion and its static postural stability in the landing stance. Experimental results exhibited a leg compression rate derived from the collected data in the early phase of the motion and validated intrinsic toe joint stabilization with foot muscle for passive postural control. The ground reaction force proves the capability to maintain the posture for 130% longer in the foot with plantar intrinsic muscle. Despite structural challenges, this approach shows promise for musculoskeletal robots, highlighting their ability to handle a turning task with simple control.

A three-dimensional spring-loaded inverted pendulum walking model considering human movement speed and frequency

The spring-loaded inverted pendulum (SLIP) model is an effective model to capture the essential dynamics during human walking and/or running. However, most of the existing three-dimensional (3D) SLIP model does not explicitly account for human movement speed and frequency. To address this knowledge gap, this paper develops a new SLIP model, which includes a roller foot, massless spring, and concentrated mass. The governing equations-of-motion for the SLIP model during its double support phase are derived. It is noted that in the current formulation, the motion of the roller foot is prescribed; therefore, only the equations for the concentrated mass need to be solved. To yield model parameters leading to a periodic walking gait, a constrained optimization problem is formulated and solved using a gradient-based approach with a global search strategy. The optimization results show that when the attack angle ranges from 68° to 74°, the 3D SLIP model can yield a periodic walking gait with walking speeds varying from 0.5 to 2.0 m s−1. The predicted human walking data are also compared with published experimental data, showing reasonable accuracy.

The generalized spring-loaded inverted pendulum model for analysis of various planar reduced-order models and for optimal robot leg design

This paper proposes a generalized spring-loaded inverted pendulum (G-SLIP) model to explore various popular reduced-order dynamic models’ characteristics and suggest a better robot leg design under specified performance indices. The G-SLIP model’s composition can be varied by changing the model’s parameters, such as ground contacting type and spring property. It can be transformed into four widely used models: the spring-loaded inverted pendulum (SLIP) model, the two-segment leg model, the SLIP with rolling foot model, and the rolling SLIP model. The effects of rolling contact and spring configuration on the dynamic behavior and fixed-point distribution of the G-SLIP model were analyzed, and the basins of attraction of the four described models were studied. By varying the parameters of the G-SLIP model, the dynamic behavior of the model can be optimized. Optimized for general locomotion running at various speeds, the model provided leg design guidelines. The leg was empirically fabricated and installed on the hexapod for experimental evaluation. The results indicated that the robot with a designed leg runs faster and is more power-efficient.

Design and sequential jumping experimental validation of a musculoskeletal bipedal robot based on the spring-loaded inverted pendulum model.

To effectively control a robot's motion, it is common to employ a simplified model that approximates the robot's dynamics. Nevertheless, discrepancies between the actual mechanical properties of the robot and the simplified model can result in motion failures. To address this issue, this study introduces a pneumatic-driven bipedal musculoskeletal robot designed to closely match the mechanical characteristics of a simplified spring-loaded inverted pendulum (SLIP) model. The SLIP model is widely utilized in robotics due to its passive stability and dynamic properties resembling human walking patterns. A musculoskeletal bipedal robot was designed and manufactured to concentrate its center of mass within a compact body around the hip joint, featuring low leg inertia in accordance with SLIP model principles. Furthermore, we validated that the robot exhibits similar dynamic characteristics to the SLIP model through a sequential jumping experiment and by comparing its performance to SLIP model simulation.

Running Gait and Control of Quadruped Robot Based on SLIP Model.

Legged robots have shown great adaptability to various environments. However, conventional walking gaits are insufficient to meet the motion requirements of robots. Therefore, achieving high-speed running for legged robots has become a significant research topic. In this paper, based on the Spring-Loaded Inverted Pendulum (SLIP) model and the optimized Double leg-Spring-Loaded Inverted Pendulum (D-SLIP) model, the running control strategies for the double flying phase Bound gait and the Rotatory gallop gait of quadruped robots are designed. First, the dynamics of the double flying phase Bound gait and Rotatory gallop gait are analyzed. Then, based on the "three-way" control idea of the SLIP model, the running control strategy for the double flying phase Bound gait is designed. Subsequently, the SLIP model is optimized to derive the D-SLIP model with two touchdown legs, and its dynamic characteristics are analyzed. And the D-SLIP model is applied to the running control strategy of the Rotatory gallop gait. Furthermore, joint simulation verification is conducted using Adams virtual prototyping and MATLAB/Simulink control systems for the designed control strategies. Finally, experimental verification is performed for the double flying phase Bound gait running control strategy. The experimental results demonstrate that the quadruped robot can achieve high-speed and stable running.

Autonomous navigation of underactuated bipedal robots in height-constrained environments

Navigating a large-scaled robot in unknown and cluttered height-constrained environments is challenging. Not only is a fast and reliable planning algorithm required to go around obstacles, the robot should also be able to change its intrinsic dimension by crouching in order to travel underneath height-constrained regions. There are few mobile robots that are capable of handling such a challenge, and bipedal robots provide a solution. However, as bipedal robots have nonlinear and hybrid dynamics, trajectory planning while ensuring dynamic feasibility and safety on these robots is challenging. This paper presents an end-to-end autonomous navigation framework which leverages three layers of planners and a variable walking height controller to enable bipedal robots to safely explore height-constrained environments. A vertically actuated spring-loaded inverted pendulum (vSLIP) model is introduced to capture the robot’s coupled dynamics of planar walking and vertical walking height. This reduced-order model is utilized to optimize for long-term and short-term safe trajectory plans. A variable walking height controller is leveraged to enable the bipedal robot to maintain stable periodic walking gaits while following the planned trajectory. The entire framework is tested and experimentally validated using a bipedal robot Cassie. This demonstrates reliable autonomy to drive the robot to safely avoid obstacles while walking to the goal location in various kinds of height-constrained cluttered environments.

Evaluation of compliant walking locomotion models for civil engineering applications

This study aims to evaluate the suitability of the spring-loaded inverted pendulum (SLIP) model and three of its upgrades for modelling pedestrians in civil engineering applications. Model equations in Cartesian coordinates have been derived for the four SLIP-based models and they have been presented in ready-to-use format. The input parameter space that produces the stable periodic gait has been identified and relevant gait parameters (step frequency, average speed, and dynamic load factor) have then been evaluated against the measured parameter space characterising the pedestrian population. The results indicate that the original SLIP model has multiple drawbacks. Adding roller feet was found to increase the estimates of average walking speed when compared with the SLIP model. Adding damping brings more substantial benefits of improved coverage of the step frequency and dynamic load factor parameter space. Dimensional analysis was found to be a useful means of modelling a new pedestrian having either a new body mass or leg length. In summary, the analysis suggests that inclusion of leg damping in the model is crucial for achieving realistic estimates of the step frequency and dynamic load factor, while modelling foot geometry brings improved estimates of the walking speed.

Emerging Gaits for a Quadrupedal Template Model With Segmented Legs

Abstract Energy-efficient gaits in walking robots can be obtained by designing elastic systems that exhibit naturally emerging locomotion patterns. Biological legged locomotion serves as inspiration, as animals use different gaits to move at certain speeds while minimizing energy consumption. To understand the underlying dynamics of biological locomotion, simplified models have been proposed. The most common one, the SLIP (spring loaded inverted pendulum) model, can explain the effect of the radial elasticity of linear legs and helps to explain locomotion patterns, especially for running behaviors, in different legged systems. However, the SLIP model is inappropriate for the study of stability of limit cycles in systems with articulated legs, which are most commonly used in real robots. This paper introduces a novel quadrupedal template model featuring articulated elastic legs, non-constant leg stiffness, and dynamic leg swing. Numerical simulation with a continuation approach is used to discover the gaits emerging from the natural dynamics of the model, without imposing any contact sequence a priori. The stability of those gaits is also characterized, in order to facilitate the exploitation of the natural model dynamics for generating locomotion patterns for quadrupedal robots.

Dynamic turning and running of a hexapod robot using a separated and laterally arranged two-leg model

We report on the development of separated and laterally arranged two-leg (SLTL) models with/without differentiated leg properties and their use as the dynamic running and turning templates for a hexapod robot. The laterally arranged two-leg morphology enables differential driving for turning. The differentiable leg settings, such as stiffness, enables the model to adopt unbalanced leg arrangements of empirical legged gaits, such as a tripod gait, into consideration. The fixed-point motion of the model was utilized as the main methodology to plan dynamic running and turning, in which the plot of one-step distance versus period was constructed for the legs’ operation point selection and matching. The proposed methodology was experimentally validated using four indices: turning curvature, flight phase, motion stability, and energy efficiency. The experimental results show that the running robot using the SLTL model with differentiated leg stiffness has better energy efficiency than one without by 4%, while the latter model has identical performance to the original spring-loaded inverted pendulum model with rolling contact. As for turning, the robot using the SLTL models with/without differentiated leg stiffness can preserve dynamic turning in all experiments with turning curvatures up to and , respectively, and more than the robot using the original model-less phase-shift turning strategy (). Using the proposed model-based strategy, the flight phase of the robot turning in all curvatures (including straight running) maintains around 20%, the root-mean-squared (RMS) values of pitch and roll remains less than , and the specific resistance () is bounded between . By contrast, the robot using the phase-shifting turning strategy can maintain dynamic motion up to a turning curvature of . A further increase in phase shifting not only does not increase the turning curvature but also changes the robot motion from running to walking. In this case, no flight phase exists, the jumps up significantly, and RMS values of pitch and roll also increase dramatically. In short, the experimental validation confirms the effectiveness of the proposed methodology for initiating the dynamic running and turning of the robot.

Controlling a One-Legged Robot to Clear Obstacles by Combining the SLIP Model with Air Trajectory Planning

Legged animals can adapt to complex terrains because they can step or jump over obstacles. Their application of foot force is determined according to the estimation of the height of an obstacle; then, the trajectory of the legs is controlled to clear the obstacle. In this paper, we designed a three-DoF one-legged robot. A spring-loaded inverted pendulum model was employed to control the jumping. Herein, the jumping height was mapped to the foot force by mimicking the jumping control mechanisms of animals. The foot trajectory in the air was planned using the Bézier curve. Finally, the experiments of the one-legged robot jumping over multiple obstacles of different heights were implemented in the PyBullet simulation environment. The simulation results demonstrate the effectiveness of the method proposed in this paper.

A reduced-order closed-loop hybrid dynamic model for design and development of lower limb prostheses.

This manuscript presents a simplified dynamic human-prosthesis model and simulation framework for the purpose of designing and developing lower limb prosthesis hardware and controllers. The objective was to provide an offline design tool to verify the closed-loop behavior of the prosthesis with the human, in order to avoid relying solely on limiting kinematic and kinetic reference trajectories of (able-bodied) subjects and associated static or inverse dynamic analyses, while not having to resort to complete neuromusculoskeletal models of the human that require extensive optimizations to run. The presented approach employs a reduced-order model that includes only the prosthetic limb and trunk in a multi-body dynamic model. External forces are applied to the trunk during stance phase of the intact leg to represent its presence. Walking is realized by employing the well-known spring-loaded inverted pendulum model, which is shown to generate realistic dynamics on the prosthesis while maintaining a stable and modifiable gait. This simple approach is inspired from the rationale that the human is adaptive, and from the desire to facilitate modifications or inclusions of additional user actions. The presented framework is validated with two use cases, featuring a commercial and research knee prosthesis in combination with a passive ankle prosthesis, performing a continuous sequence of standing still, walking at different velocities and stopping.

Optimizing and designing a leg shape to increase robustness of a running robot on rough terrain

The superior ability of dynamic legged locomotion in traversing rough terrain relative to wheeled or tracked mechanisms comes with the cost of fragile stability. Simple control methods that use only a few basic detection sensors and apply a single controller help robots keep their balance when traversing unforeseen rough terrain. Exploiting multiple controllers simultaneously, such as the free leg length and stiffness in our hopping monopod, can further improve robustness but is often mechanically hard to implement. This work demonstrates that a curved leg shape can improve the robustness of a robot to perturbations in both terrain levels and initial horizontal velocity without complicating the control scheme. Our work develops spring loaded inverted pendulum (SLIP) based models that manifest the coupling of the leg’s parameters and capture the rolling motion. We use these models to find an optimal combination of parameters that maximizes a measure for long-term stability−reaching a desired relative height above terrain. We demonstrate that when traversing unknown rough terrain, such optimal coupling can increase robustness to perturbations in the initial horizontal velocity by 93% relative to the optimal conventional SLIP model. We further demonstrate our results in experiments.

Online whole-stage gait planning method for biped robots based on improved Variable Spring-Loaded Inverted Pendulum with Finite-sized Foot (VSLIP-FF) model

Environmental adaptability and real-time control are significant to the actual application of biped robots. The current Spring-Loaded Inverted Pendulum (SLIP) walking exhibits the compliant interaction with environments. However, the movability and controllability of this model is limited owing to the lack of ankles. Moreover, complicated nonlinear optimization problems in gait generation bring difficulties to real-time control. To overcome these problems, this study proposes an online whole-stage gait planning method to enhance the bipedal walking performance. Firstly, considering the role of ankles, this study applies the proposed template model called Variable Spring-Loaded Inverted Pendulum with Finite-sized Foot (VSLIP-FF) model. Then a Finite State Machine (FSM)-based gait pattern including the corresponding bio-inspired gait strategies is established, which extends the single cyclic gait to the whole-stage gait. Secondly, to realize real-time gait planning, an online gait generator based on a neural network is applied to reduce the calculational burden. Finally, the method is applied on the simulation prototype and real robot platform for verification. Experimental results validate that the proposed method can achieve an autonomous gait with the online planning time of 0.01s, and the step length range is expanded by 37.52% compared with the traditional SLIP model.

Perturbation-based approximate analytic solutions to an articulated SLIP model for legged robots

The classical spring-loaded inverted pendulum (SLIP) model only takes the axial force along the equivalent leg into account without considering the tangential force. This disadvantage limits its potential when it is served as a template for legged locomotion of animals and robots that both have the tangential forces along their legs, even though it is a simple model. To handle the problem, firstly, an articulated/generalized spring-loaded inverted pendulum (ASLIP) model is proposed. Both axial and tangential force along the equivalent leg are considered in the proposed model. Secondly, the analytical approximate solution for the proposed model is addressed through perturbation method. Finally, comparative simulations show that the proposed model can achieve better prediction accuracy than the classical model. It means that the prediction accuracy of approximations depends on not only the methods of analytical approximate solutions, but also the established model. This research provides a more accurate method to predict the movement trajectory of legged locomotion and a more appropriate model representation for the control of legged locomotion.

Stable and Fast Planar Jumping Control Design for a Compliant One-Legged Robot.

Compliant bipedal robots demonstrate a potential for impact resistance and high energy efficiency through the introduction of compliant elements. However, it also adds to the difficulty of stable control of the robot. To motivate the control strategies of compliant bipedal robots, this work presents an improved control strategy for the stable and fast planar jumping of a compliant one-legged robot designed by the authors, which utilizes the concept of the virtual pendulum. The robot was modeled as an extended spring-loaded inverted pendulum (SLIP) model with non-negligible torso inertia, leg inertia, and leg damping. To enable the robot to jump forward stably, a foot placement method was adopted, where due to the asymmetric feature of the extended SLIP model, a variable time coefficient and an integral term with respect to the forward speed tracking error were introduced to the method to accurately track a given forward speed. An energy-based leg rest length regulation method was used to compensate for the energy dissipation due to leg damping, where an integral term, regarding jumping height tracking error, was introduced to accurately track a given jumping height. Numerical simulations were conducted to validate the effectiveness of the proposed control strategy. Results show that stable and fast jumping of compliant one-legged robots could be achieved, and the desired forward speed and jumping height could also be accurately tracked. In addition to that, using the proposed control strategy, the robust jumping performance of the robot could be observed in the presence of disturbances from state variables or uneven terrain.

Analysis and control of a running spring-mass model with a trunk based on virtual pendulum concept

The spring-loaded inverted pendulum model has been one of the most studied conceptual models in the locomotion community. Even though it can adequately explain the center of mass trajectories of numerous legged animals, it remains insufficient in template-based control of complex robot platforms, being unable to capture additional dynamic characteristics of locomotion exhibited in additional degrees of freedom such as trunk pitch oscillations. In fact, analysis of trunk behavior during locomotion has been one of the motivations behind studying the virtual pivot point (VPP) concept, with biological inspiration and basis for both natural and synthetic systems with non-negligible trunk dynamics. This study first presents a comprehensive analysis of the VPP concept for planar running behaviors, followed by a systematic study of the existence and characteristics of periodic solutions. In particular, we investigate how periodic solutions depend on model control parameters and compare them based on stability and energetic cost. We then develop a feedback controller that can stabilize system dynamics around its periodic solutions and evaluate performance as compared to a previously introduced controller from the literature. We demonstrate the effectiveness of both controllers and find that the proposed control scheme creates larger basins of attraction with minor degradation in convergence speed. In conclusion, this study shows that the VPP concept, in conjunction with the proposed controller, could be beneficial in designing and controlling legged robots capable of running with non-trivial upper body dynamics. Our systematic analysis of periodic solutions arising from the use of the VPP concept is also an important step towards a more formal basis for comparisons of the VPP concept with bio-locomotion.

3D-SLIP model based dynamic stability strategy for legged robots with impact disturbance rejection

Inspired by biomechanical studies, the spring-loaded inverted pendulum model is an effective behavior model to describe the running movement of animals and legged robots in the sagittal plane. However, when confronted with external lateral disturbances, the model has to move out of the 2-D plane and be extended to 3-D locomotion. With the degree of freedom increasing, the computational complexity is higher and the real-time control is more and more difficult, especially when considering the complex legged model. Here, we construct a control strategy based on the classical Raibert controller for legged locomotion under lateral impact disturbances. This strategy, named 3D-HFC, is composed of three core modules: touchdown angle control, body attitude angle control and energy compensation. The energy loss in each step is taken into consideration, and the real-time measured energy loss of the current step is adopted to predict that of the next step. We demonstrate the efficiency of the proposed control strategy on a simulated 3D-SLIP lower order model and a simulated running quadruped, which are perturbed by different impact forces. Furthermore, a quadruped bionic prototype named MBBOT was set up, on which lateral impact experiments were designed and implemented. Both simulation and experimental results show that the proposed approach can realize the impact disturbance rejection.

Fast Online Optimization for Terrain-Blind Bipedal Robot Walking With a Decoupled Actuated SLIP Model.

We present an online optimization algorithm which enables bipedal robots to blindly walk over various kinds of uneven terrains while resisting pushes. The proposed optimization algorithm performs high-level motion planning of footstep locations and center-of-mass height variations using the decoupled actuated spring-loaded inverted pendulum (aSLIP) model. The decoupled aSLIP model simplifies the original aSLIP with linear inverted pendulum (LIP) dynamics in horizontal states and spring dynamics in the vertical state. The motion planning can be formulated as a discrete-time model predictive control (MPC) problem and solved at a frequency of 1 kHz. The output of the motion planner is fed into an inverse-dynamics–based whole body controller for execution on the robot. A key result of this controller is that the feet of the robot are compliant, which further extends the robot’s ability to be robust to unobserved terrain variations. We evaluate our method in simulation with the bipedal robot SLIDER. The results show that the robot can blindly walk over various uneven terrains including slopes, wave fields, and stairs. It can also resist pushes of up to 40 N for a duration of 0.1 s while walking on uneven terrains.

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

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.