Pendulum Model Research Articles

Increasing the stability of the spring loaded inverted pendulum model of running with a wobbling mass

Human running can be simulated using a simple model called the spring-loaded inverted pendulum (SLIP). The SLIP model predicts some aspects of running including the self-stabilizing properties of running. In human locomotion energy is dissipated due to the passive motion of the soft tissue. However, little is known about the effects of this energy dissipation on the dynamics of running. This study utilizes a SLIP model with an additional spring-mass-damper system to study the effects of energy dissipation due to an additional wobbling mass on the self-stabilizing properties of human running. It was found that the additional spring-mass-damper system increased the self-stabilizing properties of the SLIP model and increased its robustness to perturbations. This suggests that increasing stability is one of the effects of energy dissipation due to the passive motion of a wobbling mass during human running.

Online Jumping Motion Generation via Model Predictive Control

Legged robots can move on various grounds via walking; however, they have difficulty in moving fast. To overcome this limitation, some legged robots have been studying dynamic motions such as jumping and running. In this article, we present a strategy for generating an optimal jumping motion for legged robots in real time. For convenience, we divided the proposed jumping trajectory into vertical and horizontal directions. The vertical motion trajectory ensures a continuous center of mass position, speed, and acceleration, as well as minimizes the maximum torque and maximum speed of the joints; we generated it via a nonlinear optimal process. Besides, for the horizontal motion, we proposed a novel model predictive control using a height varying inverted pendulum model. In the proposed method, the zero moment point is placed in the support polygon for stable jumping; also, the robot does not slip on the ground and takes off at a desired velocity. We verified the jumping motion using RoK-3, a biped robot developed for the experiment.

Humanoid Gait Pattern Generation with Orbital Energy

A novel approach has been proposed to generate the gait pattern of the humanoid robot with use of orbital energy concept. As the bipedal robots can be modeled as Linear Inverted Pendulum Model (LIPM) and rolling sphere model, the detailed discussion is presented here to generate stable walking pattern with help of this model. The intensive uses and advantages of the orbital energy concept have also been presented compared to different traditional approach of walking pattern generation. In this context, a novel algorithm has been proposed to generate 3D walking pattern of the bipedal robots, and how the 3D walking pattern can be approximated as the 2D walking pattern. All the analytical results are provided and verified with NaO robot (Biped robot).

Biomechanics of Single Stair Climb With Implications for Inverted Pendulum Modeling.

Step-by-step (SBS) stair navigation is used by those with movement limitations or lower-limb prosthetics and by humanoid robots. Knowledge of biomechanical parameters for SBS gait, however, is limited. Inverted pendulum (IP) models used to assess dynamic stability have not been applied to SBS gait. This study examined the ability of the linear inverted pendulum (LIP) model and a closed-form, variable-height inverted pendulum (VHIP) model to predict capture-point (CP) stability in healthy adults executing a single stair climb. A second goal was to provide baseline kinematic and kinetic data for SBS gait. Twenty young adults executed a single step onto stairs of two heights, while attached marker positions and ground reaction forces were recorded. opensim software determined body kinematics and joint kinetics. Trials were analyzed with LIP and VHIP models, and the predicted CP compared to the actual center-of-pressure (CoP) on the stair. Lower-limb joint moments were larger than those reported for step-over-step (SOS) stair gait. Leading knee rather than trailing ankle was dominant. Center-of-mass (CoM) velocity peaked at push-off. The VHIP model accounted for only slightly more than half of the forward progression of the vertical projection of the CoM and was not better than LIP predictions. This suggests that IP models are limited in modeling SBS gait, likely due to large hip and knee moments. The results from this study may also provide target values and strategies to aid design of lower-limb prostheses and powered exoskeletons.

Biomechanical design framework for prosthetic feet: Experimentally validated non-linear finite element procedure

Performance evaluation of prosthetic feet during their design is typically performed experimentally, which may be time and cost intensive. This work presents a first-of-its-kind application of a numerical procedure for the a priori determination of various stance phase biomechanical parameters of a prosthetic foot, such as its roll-over characteristics, centre of pressure trajectory, ankle flexion moment arm and ankle range of motion, to aid in its design. The numerical procedure is based on finite element analysis, which includes geometric, material and contact non-linearity. Boundary conditions emulating the rocker-based inverted pendulum model were employed to evaluate the biomechanical parameters. The finite element model was validated by employing an inverted pendulum-based apparatus using the structurally complex Ottobock Solid Ankle Cushioned Heel (SACH) prosthetic foot as the test device.A comparison of the numerical and experimental results showed low magnitude of errors. For example, the percentage error of the radius of curvature of the roll-over shape was ~0.1%. The differences found appear to be clinically insignificant, which substantiates the reliability of the model. The proposed numerical model can be employed to obtain detailed a priori insights into the biomechanical parameters influencing a prosthetic foot's characteristics during gait, which can better inform the design, analysis and prescription of prosthetic feet.

Legged Robot Running Using a Physics-Data Hybrid Motion Template

We report on the methodology of developing a hybrid model and utilizing it as a template to initiate running behavior in a legged robot. The hybrid model is comprised of a physics-based, rolling, spring-loaded, inverted pendulum (R-SLIP) model, and a data-driven model that compensates for unmodeled dynamics using Gaussian process (GP) regression. The hybrid R-SLIP-GP model retains the R-SLIP's intrinsic running dynamics, as well as improves the template's accuracy without intensive data training and iteration efforts. The proposed hybrid model was evaluated via simulation and in empirical robot running experiments. The results confirm that the added GP-based model greatly improved the model's accuracy, especially discrepancies resulting from the originally difficult-to-model complex leg-ground interactions. In addition, the R-SLIP-GP model was utilized as a complete motion template to design and control the robot's running motion. The experimental results demonstrate that the hybrid template can initiate more stable and power-efficient running motion in the robot than using the R-SLIP template.

Analysis of Autonomous Coordination Between Actuators in the Antagonist Musculoskeletal Model

The McKibben Pneumatic Actuator (MPA) is well-known as a type of soft actuator. As MPA generates tension only in the direction of compression, it is necessary to construct an antagonistic structure to drive a joint by MPAs and to coordinate antagonized MPAs. Similar to MPA, muscles in animals also generate tension only in the direction of contraction. Some studies have reported that animals utilize tension information to coordinate muscles for various autonomous movements. The purpose of this study is to realize autonomous cooperation between antagonized MPAs by applying tension feedback control and analyzing the mechanism of coordination. For this purpose, we verify the effect of tension feedback control on the 1-DOF pendulum model with antagonized MPAs. First, through numerical simulations, it is confirmed that the tension feedback generates various coordinated movements of antagonized MPAs, and the pendulum exhibits a bifurcation phenomenon based on the phase difference of the inputs of MPAs. Thereafter, we develop an actual experimental machine based on the model and confirm the autonomous cooperation between actual MPAs through verification experiments similar to the numerical simulations.

Development of a theoretical model for upright postural control in lower limb prosthesis users

Methods used to assess quiet standing in unilateral prosthesis users often assume validity of an inverted pendulum model despite this being shown as invalid in some instances. The aim of the current study was to evaluate the validity of a proposed unilaterally-constrained pin-controller model in explaining postural control in unilateral prosthesis users. Prosthesis users were contrasted against the theoretical model as were able-bodied controls that stood on a platform which unilaterally constrained movement of the CoP. All participants completed bouts of quiet standing with eyes open, eyes closed and with feedback on inter-limb weight bearing asymmetry. Correlation coefficients were used to infer inverted pendulum behavior in both the anteroposterior and mediolateral directions and were derived from both kinematic (body attached markers) and kinetic (centre of pressure) experimental data. Larger, negative correlation coefficients reflected better model adherence, whilst low or no correlation reflected poorer model adherence. Inverted pendulum behavior derived from kinematic data, indicated coefficients of high magnitude in both mediolateral (all cases range 0.71–0.78) and anteroposterior (0.88–0.91) directions, irrespective of groups. Inverted pendulum behavior derived from kinetic data in the anteroposterior direction indicated validity of the model with large negative coefficients associated with the unconstrained/intact limbs (prosthesis users: − 0.45 to − 0.65, control group: − 0.43 to − 0.72), small coefficients in constrained/prosthetic limbs (prosthesis users: − 0.02 to 0.07, control group: 0.13–0.26) and large negative coefficients in combined conditions (prosthesis users: − 0.36 to − 0.56, control group: − 0.71 to − 0.82). For the mediolateral direction, coefficients were negligible for individual limbs (0.03–0.17) and moderate to large negative correlations, irrespective of group (− 0.31 to − 0.73). Data suggested both prosthesis users’ and able-bodied individuals’ postural control conforms well to that predicted by a unilaterally-constrained pin-controller model, which has implications for the fundamental control of posture in transtibial prosthesis users.

Event-based asynchronous dissipative filtering for fuzzy nonhomogeneous Markov switching systems with variable packet dropouts

This work concerns the event-based asynchronous filtering problem for T-S fuzzy nonhomogeneous Markov switching systems with variable packet dropouts. The discrete-time nonhomogeneous Markov process is adopted to depict mode switching among subsystems, in which the time-varying transition probabilities are characterized by a polytope structure. The variable packet dropouts are developed to describe the randomly occurring packet dropouts, where the arriving rate remains variable and uncertain. Aiming to save the limited network bandwidth, event-triggered strategy and quantization scheme are presented. By establishing the fuzzy-rule-dependent Lyapunov functional and applying a hidden Markov model policy, sufficient criteria are gained and asynchronous filters are designed by solving linear matrix inequalities (LMIs). Finally, the applicability of the proposed filtering strategy is verified by an inverted pendulum model.

Analysis of the energy cost of human walking based on a coupled pendulum model

<p indent="0mm">Based on the phenomenon wherein the muscle groups of the swing leg are nearly quiescent during the swing phase of human walking, a coupled pendulum model is developed to formulate mathematical models of the swing and double-support phases of human walking. The shooting method is used to determine the ankle angle of the stance leg, hip angle, and knee angle of the swing leg. Changes in joint angular velocities during knee-lock and heel-strike are calculated using the principle of momentum conservation, and the rate of energy cost is obtained. The effects of joint frictional torque on the rates of energy cost during knee-lock and heel-strike are also analyzed. Joint elastic components are added to the hip and ankle separately, and their effects on walking gait and energy cost are assessed. Results indicate that the joint angles computed using the mathematical models are very close to the experimental data collected during normal walking. Most of the mechanical energy loss during walking results from the impact at heel strike, which is greater than the energy loss resulting from knee-lock and frictional torque. Elastic components added to the hip can reduce the time of the swing phase, which enables faster walking at the cost of higher energy at heel strike. Elastic components added to the ankle of the stance leg can reduce the energy cost during the heel strike of the swinging leg but does not affect pace. These results provide a reference for the research and design of lower limb exoskeletons and foot prostheses.

Multibody Modeling and Balance Control of a Reaction Wheel Inverted Pendulum Using LQR Controller

In this study, modeling and LQR control of a reaction wheel inverted pendulum system is described. The reaction wheel inverted pendulum model is created by using a 3D CAD platform and exported to Simscape Multibody. The multibody model is linearized to derive a state-space representation. A LQR (Linear-quadratic regulator) controller is designed and applied for balance control of the pendulum. The results show that deriving a state-space representation from multibody is an easy and effective way to model dynamic systems and balance control of the reaction wheel inverted pendulum is successfully achieved by LQR controller. Results are given in the form of graphics.

A new tracking walking adaptive control of uncertain biped robot by state feedback and output feedback

In this paper, a walking algorithm of a biped robot with unknown dynamics has been addressed by proposing a new adaptive control based on the Modified Function Approximation Technique (MFAT) augmented with backstepping control, using state feedback and output feedback. The classical ZMP, the preview controller on the 3- Dimensional Linear Inverted Pendulum Model (3D-LIPM), is a stable walking algorithm but does not allow high accuracy, high speed, and robustness due to the ground environment, and also due to the dynamics uncertainties caused by the complex structure of these kinds of robots. To improve the accuracy of the walking tracking performance of the biped robot and its robustness against dynamics uncertainty and external perturbations, a combination of backstepping control and MFAT strategy have been intended. Recursive Lyapunov function is developed to ensure the closed-loop system stability. Experiment simulation and comparative study were carried out with biped robot to verify the performance and the robustness of the developed control strategy.

SLIP Walking Over Rough Terrain via H-LIP Stepping and Backstepping-Barrier Function Inspired Quadratic Program

We present an advanced and novel control method to enable actuated Spring Loaded Inverted Pendulum model to walk over rough and challenging terrains. The high-level philosophy is the decoupling of the controls of the vertical and horizontal states. The vertical state is controlled via Backstepping-Barrier Function (BBF) based quadratic programs: a combination of control Lyapunov backstepping and control barrier function, both of which provide inequality constraints on the inputs. The horizontal state is stabilized via Hybrid-Linear Inverted Pendulum (H-LIP) based stepping, which has a closed-form formulation. Therefore, the implementation is computationally-efficient. We evaluate our method in simulation, which demonstrates the aSLIP walking over various terrains, including slopes, stairs, and general rough terrains with uncertainties.

An Output Feedback Position/Speed Regulator for a Torque—driven Inertia Wheel Pendulum

This paper presents a position/speed regulator for a torque-driven inertia wheel pendulum where only position measurements are available for feedback. Our contribution is to introduce a controller that does not need joint velocity measurements, and instead it uses the output of a linear filtering of weighted joint position errors to add damping. The proposed controller allows bringing the nonactuated pendulum rod towards its upright position, while the wheel spins asymptotically to a desired constant speed. A complete stability analysis based on Barbashin—Krasovskii’s theorem is presented together with an estimate of domain of attraction. Simulation results upon a torque-driven inertia wheel pendulum model illustrate the performance of the proposed controller.

Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

&lt;p style='text-indent:20px;'&gt;The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.&lt;/p&gt;

Modeling and Simulation of a Human Knee Exoskeleton's Assistive Strategies and Interaction.

Exoskeletons are increasingly used in rehabilitation and daily life in patients with motor disorders after neurological injuries. In this paper, a realistic human knee exoskeleton model based on a physical system was generated, a human–machine system was created in a musculoskeletal modeling software, and human–machine interactions based on different assistive strategies were simulated. The developed human–machine system makes it possible to compute torques, muscle impulse, contact forces, and interactive forces involved in simulated movements. Assistive strategies modeled as a rotational actuator, a simple pendulum model, and a damped pendulum model were applied to the knee exoskeleton during simulated normal and fast gait. We found that the rotational actuator–based assistive controller could reduce the user's required physiological knee extensor torque and muscle impulse by a small amount, which suggests that joint rotational direction should be considered when developing an assistive strategy. Compared to the simple pendulum model, the damped pendulum model based controller made little difference during swing, but further decreased the user's required knee flexor torque during late stance. The trade-off that we identified between interaction forces and physiological torque, of which muscle impulse is the main contributor, should be considered when designing controllers for a physical exoskeleton system. Detailed information at joint and muscle levels provided in this human–machine system can contribute to the controller design optimization of assistive exoskeletons for rehabilitation and movement assistance.

A conical pendulum model of a rotating chain

An experiment is described using double and triple conical pendulums to simulate the behaviour of a rotating chain. The first few rotation modes are calculated theoretically and provide good agreement with experimental results. By contrast, a chain containing many links is difficult to analyse, even though the first few nonlinear modes of a vertically suspended chain are easy to excite and are easily observed using a video camera. The experiment would be suitable as a student project since it is easy to set up and the physics is easily understood, unlike a conventional double pendulum.

Visual Modulation of Human Responses to Support Surface Translation.

Vision is known to improve human postural responses to external perturbations. This study investigates the role of vision for the responses to continuous pseudorandom support surface translations in the body sagittal plane in three visual conditions: with the eyes closed (EC), in stroboscopic illumination (EO/SI; only visual position information) and with eyes open in continuous illumination (EO/CI; position and velocity information) with the room as static visual scene (or the interior of a moving cabin, in some of the trials). In the frequency spectrum of the translation stimulus we distinguished on the basis of the response patterns between a low-frequency, mid-frequency, and high-frequency range (LFR: 0.0165-0.14 Hz; MFR: 0.15–0.57 Hz; HFR: 0.58–2.46 Hz). With EC, subjects’ mean sway response gain was very low in the LFR. On average it increased with EO/SI (although not to a significant degree p = 0.078) and more so with EO/CI (p < 10−6). In contrast, the average gain in the MFR decreased from EC to EO/SI (although not to a significant degree, p = 0.548) and further to EO/CI (p = 0.0002). In the HFR, all three visual conditions produced, similarly, high gain levels. A single inverted pendulum (SIP) model controlling center of mass (COM) balancing about the ankle joints formally described the EC response as being strongly shaped by a resonance phenomenon arising primarily from the control’s proprioceptive feedback loop. The effect of adding visual information in these simulations lies in a reduction of the resonance, similar as in the experiments. Extending the model to a double inverted pendulum (DIP) suggested in addition a biomechanical damping effective from trunk sway in the hip joints on the resonance.

Dynamic walking of humanoid robot on flat surface using amplified LIPM plus flywheel model

PurposeHumanoid robots have complicated dynamics, and they lack dynamic stability. Despite having similarities in kinematic structure, developing a humanoid robot with robust walking is quite difficult. In this paper, an attempt to produce a robust and expected walking gait is made by using an ALO (ant lion optimization) tuned linear inverted pendulum model plus flywheel (LIPM plus flywheel).Design/methodology/approachThe LIPM plus flywheel provides the stabilized dynamic walking, which is further optimized by ALO during interaction with obstacles. It gives an ultimate turning angle, which makes the robot come closer to the obstacle and provide a turning angle that optimizes the travel length. This enhancement releases the constraint on the height of the COM (center of mass) and provides a larger stride. The framework of a sequential locomotion planer has been discussed to get the expected gait. The proposed method has been successfully tested on a simulated model and validated on the real NAO humanoid robot.FindingsThe convergence curve defends the selection of the proposed controller, and the deviation under 5% between simulation and experimental results in regards to travel length and travel time proves its robustness and efficacy. The trajectory of various joints obtained using the proposed controller is compared with the joint trajectory obtained using the default controller. The comparison shows the stable walking behavior generated by the proposed controller.Originality/valueHumanoid robots are preferred over mobile robots because they can easily imitate the behaviors of humans and can result in higher output with higher efficiency for repetitive tasks. A controller has been developed using tuning the parameters of LIPM plus flywheel by the ALO approach and implementing it in a humanoid robot. Simulations and experiments have been performed, and joint angles for various joints are calculated and compared with the default controller. The tuned controller can be implemented in various other humanoid robots

Biped Walking Based on Stiffness Optimization and Hierarchical Quadratic Programming.

The spring-loaded inverted pendulum model is similar to human walking in terms of the center of mass (CoM) trajectory and the ground reaction force. It is thus widely used in humanoid robot motion planning. A method that uses a velocity feedback controller to adjust the landing point of a robot leg is inaccurate in the presence of disturbances and a nonlinear optimization method with multiple variables is complicated and thus unsuitable for real-time control. In this paper, to achieve real-time optimization, a CoM-velocity feedback controller is used to calculate the virtual landing point. We construct a touchdown return map based on a virtual landing point and use nonlinear least squares to optimize spring stiffness. For robot whole-body control, hierarchical quadratic programming optimization is used to achieve strict task priority. The dynamic equation is given the highest priority and inverse dynamics are directly used to solve it, reducing the number of optimizations. Simulation and experimental results show that a force-controlled biped robot with the proposed method can stably walk on unknown uneven ground with a maximum obstacle height of 5 cm. The robot can recover from a 5 Nm disturbance during walking without falling.