Single Inverted Pendulum Model Research Articles

Coordination tending towards an anti-phase relationship determines greater sway reduction during entrainment with a simulated partner

The increased risk of falls in the older aged population demands the development of assistive robotic devices capable of effective balance support. For the development and increased user acceptance of such devices, which provide balance support in a human-like way, it is important to understand the simultaneous occurrence of entrainment and sway reduction in human-human interaction. However, sway reduction has not been observed yet during a human touching an external, continuously moving reference, which rather increased human body sway. Therefore, we investigated in 15 healthy young adults (27.20±3.55 years, 6 females) how different simulated sway-responsive interaction partners with different coupling modes affect sway entrainment, sway reduction and relative interpersonal coordination, as well as how these human behaviours differ depending on the individual body schema accuracy. For this, participants were lightly touching a haptic device that either played back an average pre-recorded sway trajectory (“Playback”) or moved based on the sway trajectory simulated by a single-inverted pendulum model with either a positive (Attractor) or negative (Repulsor) coupling to participant's body sway. We found that body sway reduced not only during the Repulsor-interaction, but also during the Playback-interaction. These interactions also showed a relative interpersonal coordination tending more towards an anti-phase relationship, especially the Repulsor. Moreover, the Repulsor led to the strongest sway entrainment. Finally, a better body schema contributed to a reduced body sway in both the “reliable” Repulsor and the “less reliable” Attractor mode. Consequently, a relative interpersonal coordination tending more towards an anti-phase relationship and an accurate body schema are important to facilitate sway reduction.

Center of gravity estimation of upper body in sitting posture from force plates measurement

Perturbations of the center of gravity (COG) in sitting posture provide significant information for motion sickness and low back pain. Although a motion capture system generally measures the trajectory of the COG, it is difficult to apply the method to measure a vehicle passenger. In a previous study, we proposed a method to estimate the COG in standing posture from a force platform using the single inverted pendulum model. To apply the method to the sitting posture, we adopted a new mechanical model considering that a subject contacts the floor at two points, the seat and under the feet. The purpose of this study is to estimate the upper body COG in sitting posture from force platform measurements and compare the estimation accuracy between the two models, with and without lower limbs. If the method considers the lower limbs, force platforms are needed under the feet in addition to the seat. Since the estimation accuracy may vary depending on the subject’s motion, we implemented a validation test under three conditions: rest sitting test, voluntary motion test, and seat sway test. We checked the estimation accuracy with the correlation coefficient and the root mean square error (RMSE) compared with a motion capture measurement. All experiments demonstrated high estimation accuracy in the lower-limbs-considering model. This method can be used to analyze the behavior of vehicle passengers and may help analyze motion sickness and low back pain.

Quiet standing: The Single Inverted Pendulum model is not so bad after all.

In the study of balance and postural control the (Single) Inverted Pendulum model (SIP) has been taken for a long time as an acceptable paradigm, with the implicit assumption that only ankle rotations are relevant for describing and explaining sway movements. However, more recent kinematic analysis of quiet standing revealed that hip motion cannot be neglected at all and that ankle-hip oscillatory patterns are characterized by complex in-phase and anti-phase interactions, suggesting that the SIP model should be substituted by a DIP (Double Inverted Pendulum) model. It was also suggested that DIP control could be characterized as a kind of optimal bi-axial active controller whose goal is minimizing the acceleration of the global CoM (Center of Mass). We propose here an alternative where active feedback control is applied in an intermittent manner only to the ankle joint, whereas the hip joint is stabilized by a passive stiffness mechanism. The active control impulses are delivered to the ankle joint as a function of the delayed state vector (tilt rotation angle + tilt rotational speed) of a Virtual Inverted Pendulum (VIP), namely a pendulum that links the ankle to the CoM, embedded in the real DIP. Simulations of such DIP/VIP model, with the hybrid control mechanism, show that it can reproduce the in-phase/anti-phase interaction patterns of the two joints described by several experimental studies. Moreover, the simulations demonstrate that the DIP/VIP model can also reproduce the measured minimization of the CoM acceleration, as an indirect biomechanical consequence of the dynamic interaction between the active control of the ankle joint and the passive control of the hip joint. We suggest that although the SIP model is literally false, because it ignores the ankle-hip coordination, it is functionally correct and practically acceptable for experimental studies that focus on the postural oscillations of the CoM.

Dynamic Posture Stabilization of a Biped Robot SUBO-1 on Slope-Changing Grounds

This paper describes a model-based dynamic posture stabilization of a biped robot on slope changing grounds. The ground slope generally is changing when a biped robot is walking outdoors on uneven grounds. Hence, if biped robots are used outside, their robustness against slope-changing grounds is necessary for stable walking. In this paper, a Zero Moment Point (ZMP) controller on level ground was designed as a basic posture stabilizer, and a disturbance observer (DOB) was additionally designed to reject a disturbance due to the ground slope. A simple single inverted pendulum model with a flexible joint is used for the ZMP controller and DOB designs. Finally, the performance of the proposed ZMP-DOB controller was verified through MATLAB simulations and walking in place experiments using a biped robot, SUBO-1 on a slope-changing treadmill.

Reinforcement learning for stabilizing an inverted pendulum naturally leads to intermittent feedback control as in human quiet standing.

Intermittent feedback control for stabilizing human upright stance is a promising strategy, alternative to the standard time-continuous stiffness control. Here we show that such an intermittent controller can be established naturally through reinforcement learning. To this end, we used a single inverted pendulum model of the upright posture and a very simple reward function that gives a certain amount of punishments when the inverted pendulum falls or changes its position in the state space. We found that the acquired feedback controller exhibits hallmarks of the intermittent feedback control strategy, namely the action of the feedback controller is switched-off intermittently when the state of the pendulum is located near the stable manifold of the unstable saddle-type upright equilibrium of the inverted pendulum with no active control: this action provides an opportunity to exploit transiently converging dynamics toward the unstable upright position with no help of the active feedback control. We then speculate about a possible physiological mechanism of such reinforcement learning, and suggest that it may be related to the neural activity in the pedunculopontine tegmental nucleus (PPN) of the brainstem. This hypothesis is supported by recent evidence indicating that PPN might play critical roles for generation and regulation of postural tonus, reward prediction, as well as postural instability in patients with Parkinson's disease.

Coordination of muscle torques stabilizes upright standing posture: an UCM analysis.

The control of upright stance is commonly explained on the basis of the single inverted pendulum model (ankle strategy) or the double inverted pendulum model (combination of ankle and hip strategy). Kinematic analysis using the uncontrolled manifold (UCM) approach suggests, however, that stability in upright standing results from coordinated movement of multiple joints. This is based on evidence that postural sway induces more variance in joint configurations that leave the body position in space invariant than in joint configurations that move the body in space. But does this UCM structure of kinematic variance truly reflect coordination at the level of the neural control strategy or could it result from passive biomechanical factors? To address this question, we applied the UCM approach at the level of muscle torques rather than joint angles. Participants stood on the floor or on a narrow base of support. We estimated torques at the ankle, knee, and hip joints using a model of the body dynamics. We then partitioned the joint torques into contributions from net, motion-dependent, gravitational, and generalized muscle torques. A UCM analysis of the structure of variance of the muscle torque revealed that postural sway induced substantially more variance in directions in muscle torque space that leave the Center of Mass (COM) force invariant than in directions that affect the force acting on the COM. This difference decreased when we decorrelated the muscle torque data by randomizing across time. Our findings show that the UCM structure of variance exists at the level of muscle torques and is thus not merely a by-product of biomechanical coupling. Because muscle torques reflect neural control signals more directly than joint angles do, our results suggest that the control strategy for upright stance involves the task-specific coordination of multiple degrees of freedom.

Dynamics of quiet human stance: computer simulations of a triple inverted pendulum model

For decades, the biomechanical description of quiet human stance has been dominated by the single inverted pendulum (SIP) paradigm. However, in the past few years, the SIP model family has been falsified as an explanatory approach. Double inverted pendulum models have recently proven to be inappropriate. Human topology with three major leg joints suggests in a natural way to examine triple inverted pendulum (TIP) models as an appropriate approach. In this study, we focused on formulating a TIP model that can synthesise stable balancing attractors based on minimalistic sensor information and actuation complexity. The simulated TIP oscillation amplitudes are realistic in vertical direction. Along with the horizontal ankle, knee and hip positions, though, all simulated joint angle amplitudes still exceed the measured ones about threefold. It is likely that they could be eventually brought down to the physiological range by using more sensor information. The TIP systems’ eigenfrequency spectra come out as another major result. The eigenfrequencies spread across about . Our main result is that joint stiffnesses can be reduced even below statically required values by using an active hip torque balancing strategy. When reducing mono- and bi-articular stiffnesses further down to levels threatening dynamic stability, the spectra indicate a change from torus-like (stable) to strange (chaotic) attractors. Spectra of measured ground reaction forces appear to be strange-attractor-like. We would conclude that TIP models are a suitable starting point to examine more deeply the dynamic character of and the essential structural properties behind quiet human stance.Abbreviations and technical termsInverted pendulum body exposed to gravity and pivoting in a joint around position of unstable equilibrium (operating point)SIP single inverted pendulum: one rigid body pivoting around fixation to the ground (external joint)DIP double inverted pendulum: two bodies; external and internal joint operate around instabilityTIP triple inverted pendulum: three bodies; external and both internal joints operate around instabilityQIP quadruple inverted pendulum: four bodies, foot replaces external joint; all three internal joints operate around instabilityEigenfrequency characteristic frequency that a physical system is oscillating at when externally excited at a limited energy levelDOF degree of freedom; in mechanics: linear displacement or angle or combination thereof Mono-articular stiffness: coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in the respective DOFBi-articular stiffness coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in another DOFGRF ground reaction forceHAT segment including head, arms and trunkCOM centre of massCOP centre of pressure in the plane of the force platform surface

Motion planning for a rapid mobile manipulator using model-based ZMP stabilization

SUMMARYThis paper introduces a novel approach to motion planning for a rapid mobile manipulator using inverted pendulum models. Our aim was to realize an actual rapid mobile manipulator with high acceleration and speed performance for an object's delivery. In our research, we developed an actual rapid mobile manipulator called KDMR-1. We proposed simple motion planning methods using a single inverted pendulum model (SIPM) and a double inverted pendulum model (DIPM), which are easily adaptable to a real-time system with only a small computational burden. The SIPM was useful for basic movement but did not provide object carrying capability. For that, a DIPM was proposed. In both models, we designed linear quadratic optimal controllers to stabilize the Zero Moment Point (ZMP). Two kinds of ZMP stabilization strategies were proposed, fixed ZMP and relaxed ZMP. Using these strategies, we realized optimal ZMP stabilizations for a real-time rapid mobile manipulator. For decoupled forward and rotational linear DIPM, we designed a centrifugal acceleration compensation model in the manner of feedback linearization. The experimental results showed high acceleration and speed performances during rapid object delivery.

Intermittent control models of human standing: similarities and differences

Two architectures of intermittent control are compared and contrasted in the context of the single inverted pendulum model often used for describing standing in humans. The architectures are similar insofar as they use periods of open-loop control punctuated by switching events when crossing a switching surface to keep the system state trajectories close to trajectories leading to equilibrium. The architectures differ in two significant ways. Firstly, in one case, the open-loop control trajectory is generated by a system-matched hold, and in the other case, the open-loop control signal is zero. Secondly, prediction is used in one case but not the other. The former difference is examined in this paper. The zero control alternative leads to periodic oscillations associated with limit cycles; whereas the system-matched control alternative gives trajectories (including homoclinic orbits) which contain the equilibrium point and do not have oscillatory behaviour. Despite this difference in behaviour, it is further shown that behaviour can appear similar when either the system is perturbed by additive noise or the system-matched trajectory generation is perturbed. The purpose of the research is to come to a common approach for understanding the theoretical properties of the two alternatives with the twin aims of choosing which provides the best explanation of current experimental data (which may not, by itself, distinguish beween the two alternatives) and suggesting future experiments to distinguish beween the two alternatives.

Application of BP Neural Network Controller on Inverted Pendulum

The traditional PID control, which is based on linearization, is often hard to obtain the optimal control effect on such nonlinear, multiple-output, strongly coupled systems like inverted pendulum. To solve the problem above, the BP neural network controller was developed for inverted pendulum. On the basis of establishing and analyzing the mathematical model of single inverted-pendulum, this paper established the state space expression, and then designed a neural network control system based on BP algorithm. The simulation was researched by Matlab and the running results show that this control has good robustness and can achieve satisfactory control effect.

Theoretical Hill-type muscle and stability: numerical model and application.

The construction of artificial muscles is one of the most challenging developments in today's biomedical science. The application of artificial muscles is focused both on the construction of orthotics and prosthetics for rehabilitation and prevention purposes and on building humanoid walking machines for robotics research. Research in biomechanics tries to explain the functioning and design of real biological muscles and therefore lays the fundament for the development of functional artificial muscles. Recently, the hyperbolic Hill-type force-velocity relation was derived from simple mechanical components. In this contribution, this theoretical yet biomechanical model is transferred to a numerical model and applied for presenting a proof-of-concept of a functional artificial muscle. Additionally, this validated theoretical model is used to determine force-velocity relations of different animal species that are based on the literature data from biological experiments. Moreover, it is shown that an antagonistic muscle actuator can help in stabilising a single inverted pendulum model in favour of a control approach using a linear torque generator.

Intermittent control with ankle, hip, and mixed strategies during quiet standing: A theoretical proposal based on a double inverted pendulum model

Human upright posture, as a mechanical system, is characterized by an instability of saddle type, involving both stable and unstable dynamic modes. The brain stabilizes such system by generating active joint torques, according to a time-delayed neural feedback control. What is still unsolved is a clear understanding of the control strategies and the control mechanisms that are used by the central nervous system in order to stabilize the unstable posture in a robust way while maintaining flexibility. Most studies in this direction have been limited to the single inverted pendulum model, which is useful for formalizing fundamental mechanical aspects but insufficient for addressing more general issues concerning neural control strategies. Here we consider a double inverted pendulum model in the sagittal plane with small passive viscoelasticity at the ankle and hip joints. Despite difficulties in stabilizing the double pendulum model in the presence of the large feedback delay, we show that robust and flexible stabilization of the upright posture can be established by an intermittent control mechanism that achieves the goal of stabilizing the body posture according to a “divide and conquer strategy”, which switches among different controllers in different parts of the state space of the double inverted pendulum. Remarkably, it is shown that a global, robust stability is achieved even if the individual controllers are unstable and the information exploited for switching from one controller to another is severely delayed, as it happens in biological reality. Moreover, the intermittent controller can automatically resolve coordination among multiple active torques associated with the muscle synergy, leading to the emergence of distinct temporally coordinated active torque patterns, referred to as the intermittent ankle, hip, and mixed strategies during quiet standing, depending on the passive elasticity at the hip joint.

A computational human model for exploring the role of the feet in balance

Many studies concerning human balance use computational models that represent the body as a single, double, or triple inverted pendulum while ignoring the feet. Clinical research, however, has begun to more closely examine specific contributions of the feet in balance, leading to a disparity between the state of clinical research and the models used for simulation. Here, we expand the single inverted pendulum model by adding four additional rigid links to represent the feet. Model parameters, equations of motion, actuation based on human musculature, and control based on proprioception are discussed. Computation of ground reaction forces under the heel, forefoot, and toes is also addressed. Simulations focusing on the role of the toes and toe muscles in static balance and forward leaning are presented.

Watching quiet human stance to shake off its straitjacket

The single-inverted pendulum (SIP) model is still the paradigm describing dynamics and control of quiet human stance in the sagittal plane. We used two methods to verify this paradigm. First, in an experimental approach we acquired kinematic data of both legs of ten subjects at high spatial resolution while quietly standing on two force platforms. We calculated all leg joint angles, the belonging joint torques using inverse dynamics and estimates of joint stiffnesses. Some linear correlations and regressions of both local (joint) and global (COM, COP: centre of mass respectively pressure) variables predicted by the SIP model were investigated. All three verification criteria applied to mean values extracted from experimental data revealed that the SIP is not a valid model for quiet human stance. As a second method, we used computer synthesis to demonstrate that a double-inverted pendulum (DIP) model enters a stable attractor when just the “hip” joint torque is regulated, whereas no torque is applied to the “ankle” joint. Here, angle and torque fluctuations are necessary because such a DIP strategy is of inevitable dynamic character. The two predicted eigenfrequencies of this regulated DIP model approximate the upper and lower limits of the main part of measured power spectra of quiet human stance. We suggest this dynamic necessity to be representative of the biological constraints under which a mechanically unstable inverted multi-segment chain must be stabilised.

Effects of joint immobilization on standing balance

We investigated the effect of joint immobilization on the postural sway during quiet standing. We hypothesized that the center of pressure (COP), rambling, and trembling trajectories would be affected by joint immobilization. Ten young adults stood on a force plate during 60 s without and with immobilized joints (only knees constrained, CK; knees and hips, CH; and knees, hips, and trunk, CT), with their eyes open (OE) or closed (CE). The root mean square deviation (RMS, the standard deviation from the mean) and mean speed of COP, rambling, and trembling trajectories in the anterior–posterior and medial–lateral directions were analyzed. Similar effects of vision were observed for both directions: larger amplitudes for all variables were observed in the CE condition. In the anterior–posterior direction, postural sway increased only when the knees, hips, and trunk were immobilized. For the medial–lateral direction, the RMS and the mean speed of the COP, rambling, and trembling displacements decreased after immobilization of knees and hips and knees, hips, and trunk. These findings indicate that the single inverted pendulum model is unable to completely explain the processes involved in the control of the quiet upright stance in the anterior–posterior and medial–lateral directions.