Bipedal Spring-loaded Inverted Pendulum Research Articles

Optimizing Template Models to Quantifiably Assess Center of Mass Kinematic Reconstruction.

The center of mass (COM) plays a fundamental role in human ambulation, but the redundant nature of the human body adds complexity to mathematically modeling its dynamics. Template models like the Bipedal Spring Loaded Inverted Pendulum (B-SLIP) and the Virtual Pivot Point (VPP) address this complexity by removing the redundancy while retaining desired salient characteristics, such as the COM evolution. However, template models for the COM during human walking have mostly been used for qualitative analysis due to issues such as overestimation of COM vertical displacement. This paper considers a quantifiable template-based analysis of human walking by using an optimization framework to set the model parameter values for matching both explicitly and implicitly considered gait characteristics. Furthermore, it is shown that allowing the leg stiffness of the B-SLIP and VPP model to vary throughout the gait cycle better matches vertical COM trajectories with 54%-63% error reduction. These optimized template models show promise in retaining ground reaction force (GRF) information, which is not explicitly considered during the optimization process. Future work looks to incorporate these optimized trajectories as a reference for control of a lower-limb knee-ankle prosthesis.

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.

Approximate analytical solutions to the double-stance dynamics of the lossy spring-loaded inverted pendulum

This paper introduces approximate time-domain solutions to the otherwise non-integrable double-stance dynamics of the ‘bipedal’ spring-loaded inverted pendulum (B-SLIP) in the presence of non-negligible damping. We first introduce an auxiliary system whose behavior under certain conditions is approximately equivalent to the B-SLIP in double-stance. Then, we derive approximate solutions to the dynamics of the new system following two different methods: (i) updated-momentum approach that can deal with both the lossy and lossless B-SLIP models, and (ii) perturbation-based approach following which we only derive a solution to the lossless case. The prediction performance of each method is characterized via a comprehensive numerical analysis. The derived representations are computationally very efficient compared to numerical integrations, and, hence, are suitable for online planning, increasing the autonomy of walking robots. Two application examples of walking gait control are presented. The proposed solutions can serve as instrumental tools in various fields such as control in legged robotics and human motion understanding in biomechanics.

Unified Modeling and Control of Walking and Running on the Spring-Loaded Inverted Pendulum

This paper addresses the control of steady state and transition behaviors for the bipedal spring-loaded inverted pendulum (SLIP) model. We present an event-driven control approach that enables the realization of active running, walking, and walk–run transitions in a unified framework. The synthesis of the controlled behaviors is illustrated by the notion of hybrid automaton in which different gaits are generated as the sequential composition of SLIP's primary phases of motion. We also propose a novel analytical approximate solution to the otherwise nonintegrable double-stance dynamics of the SLIP model. The analytical simplicity of the solution is utilized in the design and analysis of dynamic walking gaits suitable for online implementation. The accuracy of the approximate solution and its influence on the stability properties of the controlled system are carefully analyzed. Finally, we present two simulation examples. The first demonstrates the practicality of the proposed control strategy in creating human-like gaits and gait transitions. In the second example, we use the controlled SLIP as a planner for the control of a multibody bipedal robot model, and embed SLIP-like behaviors into a physics-based robot simulation model. The results corroborate both the practical utility and effectiveness of the proposed approach.

Walking in circles: a modelling approach.

Blindfolded or disoriented people have the tendency to walk in circles rather than on a straight line even if they wanted to. Here, we use a minimalistic walking model to examine this phenomenon. The bipedal spring-loaded inverted pendulum exhibits asymptotically stable gaits with centre of mass (CoM) dynamics and ground reaction forces similar to human walking in the sagittal plane. We extend this model into three dimensions, and show that stable walking patterns persist if the leg is aligned with respect to the body (here: CoM velocity) instead of a world reference frame. Further, we demonstrate that asymmetric leg configurations, which are common in humans, will typically lead to walking in circles. The diameter of these circles depends strongly on parameter configuration, but is in line with empirical data from human walkers. Simulation results suggest that walking radius and especially direction of rotation are highly dependent on leg configuration and walking velocity, which explains inconsistent veering behaviour in repeated trials in human data. Finally, we discuss the relation between findings in the model and implications for human walking.

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'.

Grounded running in quails: Simulations indicate benefits of observed fixed aperture angle between legs before touch-down

Many birds use grounded running (running without aerial phases) in a wide range of speeds. Contrary to walking and running, numerical investigations of this gait based on the BSLIP (bipedal spring loaded inverted pendulum) template are rare. To obtain template related parameters of quails (e.g. leg stiffness) we used x-ray cinematography combined with ground reaction force measurements of quail grounded running. Interestingly, with speed the quails did not adjust the swing leg's angle of attack with respect to the ground but adapted the angle between legs (which we termed aperture angle), and fixed it about 30ms before touchdown. In simulations with the BSLIP we compared this swing leg alignment policy with the fixed angle of attack with respect to the ground typically used in the literature. We found symmetric periodic grounded running in a simply connected subset comprising one third of the investigated parameter space. The fixed aperture angle strategy revealed improved local stability and surprising tolerance with respect to large perturbations. Starting with the periodic solutions, after step-down step-up or step-up step-down perturbations of 10% leg rest length, in the vast majority of cases the bipedal SLIP could accomplish at least 50 steps to fall. The fixed angle of attack strategy was not feasible. We propose that, in small animals in particular, grounded running may be a common gait that allows highly compliant systems to exploit energy storage without the necessity of quick changes in the locomotor program when facing perturbations.