Dynamics Of Inverted Pendulum Research Articles

Modelling strategies supplemental to foot placement for frontal-plane stability in walking.

Walking is unstable and requires active control. Foot placement is the primary strategy to maintain frontal-plane balance with contributions from lateral ankle torques, ankle push-off and trunk postural adjustments. Because these strategies interact, their individual contributions are difficult to study. Here, we used computational modelling to understand these individual contributions to frontal-plane walking balance control. A three-dimensional bipedal model was developed based on linear inverted pendulum dynamics. The model included controllers that implement the stabilization strategies seen in human walking. The control parameters were optimized to mimic human gait biomechanics for typical spatio-temporal parameters during steady-state walking and when perturbed by mediolateral ground shifts. Using the optimized model as a starting point, the contributions of each stabilization strategy were explored by progressively removing strategies. The lateral ankle and trunk strategies were more important than ankle push-off, with their removal causing up to 20% worse balance recovery compared with the full model, while removing ankle push-off led to minimal changes. Our results imply a potential benefit of preferentially training these strategies in populations with poor balance. Moreover, the proposed model could be used in future work to investigate how walking stability may be preserved in conditions reflective of injury or disease.

Ground Reaction Forces and Energy Exchange During Underwater Walking.

Underwater walking was a crucial step in the evolutionary transition from water to land. Underwater walkers use fins and/or limbs to interact with the benthic substrate and produce propulsive forces. The dynamics of underwater walking remain poorly understood due to the lack of a sufficiently sensitive and waterproof system to measure substrate reaction forces (SRFs). Using an underwater force plate (described in our companion paper), we quantify SRFs during underwater walking in axolotls (Ambystoma mexicanum) and Spot prawn (Pandalus platyceros), synchronized with videography. The horizontal propulsive forces were greater than the braking forces in both species to overcome hydrodynamic drag. In axolotls, potential energy (PE) fluctuations were far smaller than kinetic energy (KE) fluctuations due to high buoyant support (97%), whereas the magnitudes were similar in the prawn due to lower buoyant support (93%). However, both species show minimal evidence of exchange between KE and PE, which, along with the effects of hydrodynamic drag, is incompatible with inverted pendulum dynamics. Our results show that, despite their evolutionary links, underwater walking has fundamentally different dynamics compared with terrestrial walking and emphasize the substantial consequences of differences in body plan in underwater walking.

Spherical Inverted Pendulum on a Quadrotor UAV: A Flatness and Discontinuous Extended State Observer Approach

This article addresses the problem of balancing an inverted spherical pendulum on a quadrotor. The full dynamic model is obtained via the Euler-Lagrange formalism, where the dynamics of the pendulum is coupled to the dynamics of the quadrotor, taking as control inputs the torques associated with the yaw, roll, and pitch dynamics, and a control input for the vertical displacement in height. A trajectory tracking control scheme is proposed by means of an active disturbance rejection control based on a discontinuous extended state observer (ADRC-DESO) that allows controlling the system in the translational dynamics of the quadrotor including the rotational dynamics and the inverted pendulum dynamics. To address this problem, the dynamic model is linearized around an equilibrium point, taking into consideration that the system operates in close vicinity of the equilibrium points, thus considerably simplifying the dynamic model. Proving that the linear model is controllable and therefore differentiable flat, flat outputs are proposed around the displacements associated with the three cartesian axes of the Euclidean space, including a dynamic associated with the yaw dynamics of the quadrotor allowing to parameterize the full linear system. Simulation results as well as a convergence analysis validate the performance of the strategy.

Pendsim: Developing, Simulating, and Visualizing Feedback Controlled Inverted Pendulum Dynamics

Sutherland et al., (2023). pendsim: Developing, Simulating, and Visualizing Feedback Controlled Inverted Pendulum Dynamics. Journal of Open Source Education, 6(61), 168, https://doi.org/10.21105/jose.00168

Rethinking margin of stability: Incorporating step-to-step regulation to resolve the paradox

Derived from inverted pendulum dynamics, mediolateral Margin of Stability (MoSML) is a mechanically-grounded measure of instantaneous frontal-plane stability. However, average MoSML measures yield paradoxical results. Gait pathologies or perturbations often induce larger (supposedly “more stable”) average MoSML, despite clearly destabilizing factors. However, people do not walk “on average” – they walk (and sometimes lose balance) one step at a time. We assert the paradox arises because averaging MoSML discards crucial step-to-step dynamics. We present a framework unifying the inverted pendulum with Goal-Equivalent Manifold (GEM) analyses. We identify in the pendulum’s center-of-mass dynamics constant-MoSML manifolds, including one candidate “stability GEM” signifying the goal to maintain some constant MoSML∗. We used this framework to assess step-to-step MoSML dynamics of humans walking in destabilizing environments. While goal-relevant deviations were readily corrected, people did not exploit equifinality by allowing deviations to persist along this GEM. Thus, maintaining a constant MoSML∗ is inconsistent with observed step-to-step fluctuations in center-of-mass states. Conversely, the extent to which participants regulated fluctuations in mediolateral foot placements strongly predicted their regulation of center-of-mass fluctuations. Thus, center-of-mass dynamics may arise indirectly as a consequence of regulating mediolateral foot placements. To help resolve the paradox caused by averaging MoSML, we present a new statistic, Probability of Instability (PoIL), used here to predict lateral instability likelihood. Participants exhibited increased PoIL when destabilized (p = 9.45 × 10−34), despite exhibiting larger (“more stable”) average MoSML (p = 1.70 × 10−15). Thus, PoIL correctly captured people’s increased risk of losing lateral balance, whereas average MoSML did not. PoIL also helps explain why people’s average MoSML increased in destabilizing contexts.

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.

Cooperative Locomotion Via Supervisory Predictive Control and Distributed Nonlinear Controllers

Abstract This paper presents a hierarchical nonlinear control algorithm for the real-time planning and control of cooperative locomotion of legged robots that collaboratively carry objects. An innovative network of reduced-order models subject to holonomic constraints, referred to as interconnected linear inverted pendulum (LIP) dynamics, is presented to study cooperative locomotion. The higher level of the proposed algorithm employs a supervisory controller, based on event-based model predictive control (MPC), to effectively compute the optimal reduced-order trajectories for the interconnected LIP dynamics. The lower level of the proposed algorithm employs distributed nonlinear controllers to reduce the gap between reduced- and full-order complex models of cooperative locomotion. In particular, the distributed controllers are developed based on quadratic programing (QP) and virtual constraints to impose the full-order dynamical models of each agent to asymptotically track the reduced-order trajectories while having feasible contact forces at the leg ends. The paper numerically investigates the effectiveness of the proposed control algorithm via full-order simulations of a team of collaborative quadrupedal robots, each with a total of 22 degrees-of-freedom. The paper finally investigates the robustness of the proposed control algorithm against uncertainties in the payload mass and changes in the ground height profile. Numerical studies show that the cooperative agents can transport unknown payloads whose masses are up to 57%, 97%, and 137% of a single agent's mass with a team of two, three, and four legged robots.

Three degrees of freedom rotary double inverted pendulum stabilization by using robust generalized dynamic inversion control: Design and experiments

The aim of this article was to determine control strategy for balance control of rotary double inverted pendulum system, which is highly nonlinear and unstable under-actuated system. The complexities involved in rotary double inverted pendulum dynamics make this system a useful engineering test bed to test and verify newly designed controllers. In this article, a constraint-based control approach titled robust generalized dynamic inversion is designed and implemented for robust stabilization of rotary double inverted pendulum system. The robust generalized dynamic inversion control is designed in two stages; in the first stage, constraint differential equations of the controlled state variables are prescribed, which encompasses the control objectives. To enforce the constraint dynamics, the equivalent control is realized by means of Moore–Penrose generalized inversion. To enhance robustness, the switching (discontinuous) control is introduced in second stage, whose design principle is based on classical sliding mode control theory. Finally, the controllers obtained in two stages are augmented to form the resultant robust generalized dynamic inversion control law. The proposed controller ensures robustness along with improved time domain performance regardless of system nonlinearities, uncertainties, and unwanted disturbances. The stability analysis is presented for guaranteeing semi-global asymptotically stable closed loop performance via Lyapunov stability criteria. Numerical simulation and experimental investigations are carried out along with comparative analysis, to demonstrate the effectiveness of robust generalized dynamic inversion control algorithm over other conventional control methods.

Active Balancing Control for Unmanned Bicycle Using Scissored-pair Control Moment Gyroscope

A control moment gyroscope (CMG) is capable of generating strong restoration torque with a relatively small-sized flywheel by changing the direction of flywheel momentum. Because the CMG is energy efficient, it has been used for active balancing of a bicycle that has an unstable equilibrium point in its dynamics. A single CMG generates not only the restoration torque component but also an additional unwanted torque component. In contrast, a scissored-pair CMG cancels out the unwanted torque and doubles the restoration torque. This study deals with active balancing to implement an unmanned bicycle by using a scissored-pair CMG. Based on the inverted pendulum dynamics model of a bicycle, a linear quadratic regulation algorithm is presented for balancing control. In order to investigate active balancing, a miniaturized bicycle system is developed, and its 3D solid model is created to obtain the parameter values of the dynamics model. Two kinds of disturbances exist that can cause the instability of a bicycle with an unstable equilibrium point: impulsive external disturbance and static constant disturbance. Experimental results show the performance of active balancing for a bicycle with a scissored-pair CMG against those disturbances.

Learning dynamic control of body yaw orientation.

To investigate the role of gravitational cues in the learning of a dynamic balancing task, we placed blindfolded subjects in a device programmed with inverted pendulum dynamics about the yaw axis. Subjects used a joystick to try and maintain a stable orientation at the direction of balance during 20 100s-long trials. They pressed a trigger button on the joystick to indicate whenever they felt at the direction of balance. Three groups of ten subjects each participated. One group balanced with their body and the yaw axis vertical, and thus did not have gravitational cues to help them to determine their angular position. They showed minimal learning, inaccurate indications of the direction of balance, and a characteristic pattern of positional drifting away from the balance point. A second group balanced with the yaw axis pitched 45° from the gravitational vertical and had gravity relevant position cues. The third group balanced with their yaw axis horizontal where they had gravity-dependent cues about body position in yaw. Groups 2 and 3 showed better initial balancing performance and more learning across trials than Group 1. These results indicate that in the absence of vision, the integration of transient semicircular canal and somatosensory signals about angular acceleration is insufficient for determining angular position during dynamic balancing; direct position-dependent gravity cues are necessary.

On the hybrid inverted pendulum dynamics and its relation to the lateral stability of the walking-like mechanical systems

A simple hybrid mechanical system, tentatively called as the hybrid inverted pendulum, is studied. This study is motivated by the problem of the lateral stability of the planar walking strategies applied to a more realistic settings. Using numerical simulations, the influence of the lateral harmonic perturbations is studied as well. Some complex features indicating the possible chaotic behavior of its forced but bounded dynamics is demonstrated as well.

Learning dynamic balancing in the roll plane with and without gravitational cues.

We determined the relative contributions of gravity-dependent positional cues and motion cues to the learning of roll balance control. We hypothesized that gravity-dependent otolith and somatosensory shear forces related to body orientation would yield better initial performance, more rapid learning, and better retention. Blindfolded subjects rode in a device programmed to roll with inverted pendulum dynamics in a vertical (UPRIGHT) or horizontal plane (SUPINE), and used a joystick to align themselves with the direction of balance. Each subject completed five blocks of four 100s long trials on two consecutive days in one of four groups (n=10 per group): Group 1, UPRIGHT balancing both days; Group 2, SUPINE both days; Group 3, UPRIGHT then SUPINE; and Group 4, SUPINE then UPRIGHT. On Day 1, UPRIGHT subjects showed better initial performance and greater improvement in performance than SUPINE subjects, who showed improvements only in having fewer deviations exceeding ±60deg from the direction of balance. Subjects tested UPRIGHT on both days showed full retention of learning across days and additional Day 2 learning, but subjects tested SUPINE on both days showed partial retention of their marginal learning from Day 1 and little improvement on Day 2. Subjects tested SUPINE on Day 2 after being tested UPRIGHT on Day 1 showed no better performance than subjects tested SUPINE on Day 1. By contrast, there was transfer from SUPINE on Day 1 to UPRIGHT on Day 2. We conclude that absence of gravitationally dependent otolith and somatosensory cues degrades balance performance.

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.

Learning dynamic control of body roll orientation.

Our objective was to examine how the control of orientation is learned in a task involving dynamically balancing about an unstable equilibrium point, the gravitational vertical, in the absence of leg reflexes and muscle stiffness. Subjects (n = 10) used a joystick to set themselves to the gravitational vertical while seated in a multi-axis rotation system (MARS) device programmed with inverted pendulum dynamics. The MARS is driven by powerful servomotors and can faithfully follow joystick commands up to 2.5 Hz with a 30-ms latency. To make the task extremely difficult, the pendulum constant was set to 600°/s(2). Each subject participated in five blocks of four trials, with a trial ending after a cumulative 100 s of balancing, excluding reset times when a subject lost control. To characterize performance and learning, we used metrics derived from joystick movements, phase portraits (joystick deflections vs MARS position and MARS velocity vs angular position), and stabilogram diffusion functions. We found that as subjects improved their balancing performance, they did so by making fewer destabilizing joystick movements and reducing the number and duration of joystick commands. The control strategy they acquired involved making more persistent short-term joystick movements, waiting longer before making changes to ongoing motion, and only intervening intermittently.

Experimental Validation of a Feed-Forward Predictor for the Spring-Loaded Inverted Pendulum Template

Widely accepted utility of simple spring–mass models for running behaviors as descriptive tools, as well as literal control targets, motivates accurate analytical approximations to their dynamics. Despite the availability of a number of such analytical predictors in the literature, their validation has mostly been done in simulation, and it is yet unclear how well they perform when applied to physical platforms. In this paper, we extend on one of the most recent approximations in the literature to ensure its accuracy and applicability to a physical monopedal platform. To this end, we present systematic experiments on a well-instrumented planar monopod robot, first to perform careful identification of system parameters and subsequently to assess predictor performance. Our results show that the approximate solutions to the spring-loaded inverted pendulum dynamics are capable of predicting physical robot position and velocity trajectories with average prediction errors of 2% and 7%, respectively. This predictive performance together with the simple analytic nature of the approximations shows their suitability as a basis for both state estimators and locomotion controllers.

Learning to Walk Fast: Optimized Hip Height Movement for Simulated and Real Humanoid Robots

The linear inverted pendulum model has been used predominantly to generate balanced humanoid walking. This model assumes that the hip height is fixed during the walk. In this paper, generating a fast walk is studied with the main focus on the effect of hip height movement. Our approach is based on modeling the hip height movement and learning its parameters in order to generate a fast walk. The hip height trajectory is generated using Fourier basis functions. The generated trajectory is the input to programmable Central Pattern Generators (CPGs) in order to modulate generated trajectories smoothly. The inverted pendulum model is utilized to model a balanced walking. A numerical approach is presented to control inverted pendulum dynamics. Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is employed to search for appropriate hip height trajectory and walking parameters that optimize walking speed. This approach has been tested not only to obtain fast forward walk but also a fast side walk. Experiments are conducted on both simulated and real NAO robots. The results show that the change from the learned forward walk to learned side walk is performed stably, which confirm the important role of using CPGs. The comparison of the results of the proposed gait model (and development approach) with those obtained using fixed hip height also shows that fixed height walking is slower than variable height walking.

Nonlinear Dynamics of Inverted Pendulum Driven by Airflow

A nonlinear model of inverted pendulum that exhibit unbounded single well φ6 potential is described. The complete equation for one-dimensional wind-induced sway is derived. The harmonic balance method along with Melnikov theory are used to seek the effects of aerodynamic drag forces on the amplitude of vibration, on the structure failure, and on the appearance of horseshoes chaos. Numerical simulations have been performed to confirm analytical investigation.

A robust reinforcement learning using the concept of sliding mode control

In this article, we propose a new control method using reinforcement learning (RL) with the concept of sliding mode control (SMC). Some remarkable characteristics of the SMC method are good robustness and stability for deviations from control conditions. On the other hand, RL may be applicable to complex systems that are difficult to model. However, applying reinforcement learning to a real system has a serious problem, i.e., many trials are required for learning. We intend to develop a new control method with good characteristics for both these methods. To realize it, we employ the actor-critic method, a kind of RL, to unite with the SMC. We are able to verify the effectiveness of the proposed control method through a computer simulation of inverted pendulum control without the use of inverted pendulum dynamics. In particular, it is shown that the proposed method enables the RL to learn in fewer trials than the reinforcement learning method.

Methodology for Zero-moment Point Experimental Modeling in the Frequency Domain

Frequency domain methodology is applied to obtain a nominal model for the Zero-Moment Point (ZMP) stability index of a biped robot in an attempt to establish a relationship between the robot trunk trajectories and the stability margin of the contact surface of the foot (or feet) touching the supporting soil. To this end the biped robot trunk is excited with a variable frequency sinusoidal signal around several operating points. These input oscillations generate other output oscillations that can be analyzed with the help of the ZMP measurement system. The proposed ZMP modeling approach not only considers classical rigid body model uncertainties but also non-modelled robot mechanical structure vibration modes. The non-linear ZMP model is obtained following three consecutive stages: Equivalent inverted pendulum dynamics, where saturation and acceleration upper bounds are taken into account, non-modelled inverted pendulum dynamics, including non-linear effects, and low-pass dynamics defining the system cut-off frequency. The effectiveness of this method is demonstrated in practice with the SILO2 biped robot prototype, and a simple control strategy is implemented in order to validate experimentally the usefulness of the models developed.

The influence of sensory information on two-component coordination during quiet stance

When standing quietly, human upright stance is typically approximated as a single segment inverted pendulum. In contrast, investigations, which perturb upright stance with support, surface translations or visual driving stimuli have shown that the body behaves like a two-segment pendulum, displaying both in-phase and anti-phase patterns between the upper and lower body. We have recently shown that these patterns co-exist during quiet stance; in-phase and anti-phase for frequencies below and above 1 Hz, respectively. Here we investigated whether the characteristics of these basic patterns were influenced by the addition or removal of sensory information. Ten healthy young subjects stood upright on a rigid platform with different combinations of sensory information: eyes were open or closed with or without light touch contact (<1 N) of the right index fingertip with a 5 cm diameter rigid force plate. The in-phase and anti-phase pattern co-exist in both the anterior-posterior (AP) and medial-lateral (ML) directions of sway. The real part of trunk-leg complex coherence decreased with the addition of vision and light touch, corresponding to a transition from the in-phase to anti-phase pattern at a lower frequency. In the AP direction, the decrease was only observed at frequencies below 1 Hz where the in-phase pattern predominates. Additional sensory information had no observable effect at sway frequencies above 1 Hz, where the anti-phase pattern predominates. Both patterns are clearly the result of a double-linked inverted pendulum dynamics, but the coherence of the in-phase pattern is more susceptible to modulation by sensory information than the anti-phase pattern.