Inverted Pendulum Model Research Articles

Nonlinear optimal control for the rotary double inverted pendulum

AbstractThe control problem of the rotary double inverted pendulum (double Furuta pendulum) is nontrivial because of underactuation and strong nonlinearities in the associated state‐space model. The system has three degrees of freedom (one actuated and two unactuated joints) while receiving only one control input. In this article, a novel nonlinear optimal (H‐infinity) control approach is developed for the dynamic model of the rotary double inverted pendulum. First, the dynamic model of the double pendulum undergoes approximate linearization with the use of first‐order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a temporary operating point which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. At a next stage a stabilizing H‐infinity feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation has to be solved at each time‐step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. To implement state estimation‐based control without the need to measure the entire state vector of the rotary double‐pendulum the H‐infinity Kalman filter is used as a robust state observer. The nonlinear optimal control method achieves fast and accurate tracking of setpoints by all state variables of the rotary double inverted pendulum under moderate variations of the control input.

Optimization and evaluation of swing leg retraction for a hydraulic biped robot

AbstractTo improve the locomotion performance of legged robots, the swing leg retraction (SLR) technique is investigated in a hydraulic biped robot. First, the influence of SLR on the locomotion performance of the hydraulic biped robot is analyzed in theory and simulations based on an extended spring load inverted pendulum model. The influence contains three performance indicators: energy loss/effiency, friction/slipping, and impact/compliance. Second, by synthesizing three performance indicators, using unified objective method and particle swarm optimization algorithm, the optimal SLR rate for gait planning based on Bezier curve is addressed. Finally, experiments are implemented to validate the effectiveness and feasibility of proposed method. And, the results show that the SLR technique is useful to reduce the impact force, improve the robot's locomotion stability and make room for impedance performance improvement of compliance controller. This research provides an insight for locomotion control of hydraulic legged robots.

Motor control insights on walking planner and its stability

The application of biomechanics and motor control models in the control of bipedal robots (humanoids and exoskeletons) has revealed limitations of our understanding of human locomotion. A recently proposed model uses the potential energy for bipedal structures to model the bipedal dynamics, allowing us to predict the system dynamics from its kinematics. This work proposes a task-space planner for human-like straight locomotion that targets application in rehabilitation robotics and computational neuroscience. The proposed architecture is based on the potential energy model and employs locomotor strategies from human data. The model generates Centre of Mass (CoM) trajectories, foot swing trajectories and the Base of Support (BoS) over time. The data show that the proposed architecture can generate behaviour in line with human walking strategies for both the CoM and the foot swing. Despite the CoM vertical trajectory not being as smooth as a human trajectory, the proposed model significantly reduces the error in the estimation of the CoM vertical trajectory compared to the inverted pendulum models. The proposed model can also assess stability based on the body kinematics embedding currently used in clinical practice. However, the model also implies a shift in the interpretation of the spatiotemporal parameters of the gait, which are now determined by the conditions for the equilibrium and not vice versa. In other words, locomotion is a dynamic reaching where the motor primitives are also determined by gravity.

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.

Fuzzy-Affine-Model-Based Filtering Design With Memory-Based Dynamic Event-Triggered Protocol

In this article, a novel dynamic event-triggered protocol is constructed to deal with the filtering problem for affine systems presented by Takagi–Sugeno (T–S) fuzzy model. Unlike the traditional fuzzy systems, a unified framework of T–S fuzzy affine systems is formulated, which is more capable of approximation over different operating regions. For achieving desirable performance while saving communication resources, the memory-based dynamic event-triggered protocol is constructed by flexibly exploiting a series of historically triggered packets. The fading channel is time-varying and described as a novel nonhomogeneous Markov process, in which a higher level deterministic switching signal regulates the variation of transition probabilities. By adopting the region-dependent Lyapunov theory, the mean-square exponentially stable, and expected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance for resulting systems is guaranteed. In the end, an inverted pendulum model is applied to verify the results of the theoretical analysis.

Online Running-Gait Generation for Bipedal Robots with Smooth State Switching and Accurate Speed Tracking.

Smooth state switching and accurate speed tracking are important for the stability and reactivity of bipedal robots when running. However, previous studies have rarely been able to synthesize these two capabilities online. In this paper, we present an online running-gait generator for bipedal robots that allows for smooth state switching and accurate speed tracking. Considering a fluctuating height nature and computational expediency, the robot is represented by a simplified variable-height inverted-pendulum (VHIP) model. In order to achieve smooth state switching at the beginning and end of running, a segmented zero moment point (ZMP) trajectory optimization is proposed to automatically provide a feasible and smooth center-of-mass (CoM) trajectory that enables the robot to stably start or stop running at the given speed. To accurately track online the desired speed during running, we propose an iterative algorithm to compute target footholds, which allows for the robot to follow the interactive desired speed after the next two steps. Lastly, a numerical experiment and the simulation of online variable speed running were performed with position-controlled bipedal robot BHR7P, and the results verified the effectiveness of the proposed methods.

Finite-Time Control Synthesis for Discrete Fuzzy Switched Singularly Perturbed Systems

This brief addresses the synthesis for a class of discrete fuzzy switched singularly perturbed systems (SPSs) in finite time. Different from Lyapunov stability, finite-time stability (FTS) is set up to analyze the transient performance, where the state remains within the upper thresholds for a given finite-time interval. In the face of singular perturbation, a singular-perturbation-parameter-dependent Lyapunov function is firstly introduced for fuzzy switched SPSs in the sense of finite time. Inspired by the mode-dependent average dwell time (MDADT) switching rule, a developed FTS is constructed to eliminate an ill-conditioned issue owing to a singularly perturbed parameter. Furthermore, a new fuzzy control strategy is proposed to satisfy the predefined FTS. Finally, the feasibility of the main results is proved using an inverted pendulum model.

A simple model of human walking

Aim. We investigate Alexander’s inverted pendulum model, the simplest mathematical model of human walking. Although it successfully explains some kinematic features of human walking, such as the velocity of the body's centre of mass, it does not account for others, like the vertical reaction force and the maximum walking speed. This paper aims to minimally extend Alexander’s model in such a way as to make it a viable and quantitative model of human walking for clinical biomechanics.Material and methods. In order to compare the predictions of Alexander’s model with experimental data on walking, we incorporate in it a robust phenomenological relation between stride frequency and stride length derived in the literature, and we introduce a step-angle dependent muscle force along the pendulum. We then analytically solve the pendulum's motion equation and find the corresponding analytical expression for the average walking speed.Results. The values of the average walking speed for different heights predicted by our model are in excellent agreement with the ones obtained in treadmill experiments. Moreover, it successfully predicts the observed walking-running transition speed, which occurs when the stride length equals the height of an individual. Finally, our extended model satisfactorily reproduces the experimentally observed ground reaction forces in the midstance and terminal stance phases. Consequently, the predicted value of the (height-dependent) maximum walking speed is in reasonable agreement with the one obtained in more sophisticated models of human walking.Conclusions. Augmented with our minimal extensions, Alexander’s model becomes an effective and realistic model of human walking applicable in clinical investigations of the human gate.

Observer-Based Integral Sliding Mode Controller for a Two-Wheeled Inverted Pendulum

The modeling and control of a two-wheeled inverted pendulum (TWIP) is gaining much interest due to the various advancements in hardware and computing technologies. A TWIP system has many advantages and applications. However, it still faces many challenges such as positioning, disturbance rejection, parameter uncertainties, etc. In this paper, a control scheme based on modern control techniques is presented to overcome these issues. Furthermore, an observer-based robust integral sliding mode controller (ISMC) is proposed for the nonlinear TWIP system. Firstly, a robust integral sliding mode controller is designed to tackle short-term and long-term constant and time-varying disturbances as well as issues relating to parameter variation. Secondly, the reduced-order observer is designed to estimate immeasurable states, enabling simplified implementation. It is then applied to the nonlinear model of the TWIP system, and performance is observed under different transient conditions. A comparison with prevalent controllers in the literature is carried out. This comparison relates to graphical results, time domain specifications, and error performance indices, calculated for states. From this, a significant qualitative improvement can be observed with the proposed controller for all types of transients. In quantitative terms, zero steady-state error with disturbances and a five-fold improvement in settling time with parameter variations are observed. This paper also discusses the software Simulink realization for modeling and control of the TWIP system, which can be valuable for novices working in this area.

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.

Dynamic balance during walking in people with multiple sclerosis: A cross-sectional study.

Dynamic balance disorders are common impairments in People with Multiple Sclerosis (PwMS) leading to gait disorders and a higher risk of falling. However, the assessment of dynamic balance is still challenging and instrumented indexes provide objective and quantitative data of CoM movement and Base of Support, which are considered that are two key factors describing dynamic balance. This study aims at validating recent instrumented indexes based on the inverted pendulum model and characterizing dynamic balance disorders in PwMS. We clinically assessed 20 PwMS and we collected instrumented gait data through an optoelectronic system. Data from 20 Healthy Subjects (HS) were also considered as normative reference. Margin of Stability by HoF (MoS_Hof) and by Terry (MoS_Terry) at midstance, and Foot Placement Estimator (DFPE) at heel strike were calculated in mediolateral (ML) and anteroposterior (AP) directions, for both less affected and most affected sides for PwMS and for dominant and non-dominant side for HS. MoS_HOF well discriminated between PwMS and HS, followed by MoS_TERRY in ML direction (Mos_HOF: PwMS = 130.0 ± 27.2 mm, HS = 106.5 ± 18.6 mm, p < 0.001, MoS_TERRY: PwMS = 75.1 ± 24.3 mm, HS = 56.5 ± 23.4 mm, p < 0.02). MoS_HOF and MoS_TERRY discriminated between sides in both directions in PwMS. DFPE did not discriminate between groups and sides. Moderate correlations were found between all three indexes and clinical balance scales (from r = 0.02 to r = 0.66), energy recovery (from r = -0.77 to r = -0.11), single stance time (from r = -0.11 to r = 0.80) and step length (from r = -0.83 to r = -0.20). MoS_HOF resulted in the best index to describe dynamic balance disorders in PwMS: they keep CoM position far from the lateral and as close as possible to the anterior boundary of the Base of Support as preventive strategies to control balance perturbations. Furthermore, PwMS seem to use different preventive strategies in accordance with the specific lower limb impairments. This alters the physiological gait mechanisms increasing the energy expenditure and decreasing gait quality and dynamic balance.

Online Walking Speed Estimation Based on Gait Phase and Kinematic Model for Intelligent Lower-Limb Prosthesis

Intelligent lower-limb prostheses aims to make amputees walk more comfortably and symmetrically which requires the dynamic altering of gait parameters such as walking speed. Some solutions have been proposed such as direct integration and machine learning methods. The former updates walking speed once after an entire gait cycle and the latter collects large amounts of gait data which are unfriendly to lower-limb amputees. Only by using an inertial measurement unit (IMU) placed on the thigh, this paper proposes a novel online walking speed estimation method to determine the walking speed rapidly and accurately in real-time. A step frequency estimator based on the phase variable and a stride estimator based on the inverted pendulum model is designed to determine the walking speed together. The proposed method is evaluated on a public open-source dataset and the gait data were collected in the lab to verify the effectiveness for able-bodied and prosthetic wearers. The experiment results show that the walking speed estimator offers higher accuracy (RMSE of the able-bodied dataset: 0.051 ± 0.016 m/s, RMSE of the prosthetic wearers‘ dataset: 0.036 ± 0.021 m/s) than the previous works with a real-time frequency of 100 Hz. The results also show that the proposed method has good performances both in static speeds and dynamic speed tracking without much data collection before being applied.

Implementation of damped bipedal inverted pendulum model of pedestrian into FE environment for prediction of vertical structural vibration

Motivated by the limited availability of tools for vibration serviceability assessment of civil engineering structures under pedestrian excitation in finite element (FE) environment, this study proposes a strategy for transferring a damped bipedal inverted pendulum (DBIP) model for a pedestrian from the analytical to the numerical domain. The strategy is implemented in FE-software ANSYS to addresses the key challenge of pedestrian alternating between the left and right stance leg. This is achieved by using COMBIN14 element in conjunction with employing element birth and death techniques to model temporary nature of the foot-structure contact. The implementation in FE environment has been verified against the analytical results of the DBIP model. The proposed method successfully transfers the pedestrian load modelling approach from biomechanics to FE modelling environment typical of civil engineering projects enabling efficient vibration serviceability assessment.

Optimization-Based Reference Generator for Nonlinear Model Predictive Control of Legged Robots

Model predictive control (MPC) approaches are widely used in robotics, because they guarantee feasibility and allow the computation of updated trajectories while the robot is moving. They generally require heuristic references for the tracking terms and proper tuning of the parameters of the cost function in order to obtain good performance. For instance, when a legged robot has to react to disturbances from the environment (e.g., to recover after a push) or track a specific goal with statically unstable gaits, the effectiveness of the algorithm can degrade. In this work, we propose a novel optimization-based reference generator which exploits a linear inverted pendulum (LIP) model to compute reference trajectories for the center of mass while taking into account the possible underactuation of a gait (e.g., in a trot). The obtained trajectories are used as references for the cost function of the nonlinear MPC presented in our previous work. We also present a formulation that ensures guarantees on the response time to reach a goal without the need to tune the weights of the cost terms. In addition, footholds are corrected by using the optimized reference to drive the robot toward the goal. We demonstrate the effectiveness of our approach both in simulations and experiments in different scenarios with the Aliengo robot.

Data-driven nonlinear predictive control for feedback linearizable systems

We present a data-driven nonlinear predictive control approach for the class of discrete-time multi-input multi-output feedback linearizable nonlinear systems. The scheme uses a non-parametric predictive model based only on input and noisy output data along with a set of basis functions that approximate the unknown nonlinearities. Despite the noisy output data as well as the mismatch caused by the use of basis functions, we show that the proposed multi-step robust data-driven nonlinear predictive control scheme is recursively feasible and renders the closed-loop system practically exponentially stable. We illustrate our results on a model of a fully-actuated double inverted pendulum.

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.

Application of Fuzzy Control Algorithm in First-order Inverted Pendulum System

The inverted pendulum system is widely used in the study of control theory. There are many advantages of the inverted pendulum: the equipment is cheap, and it features a simple structure and easy regulation. But since the inverted pendulum system is also a typical nonlinear, multivariable system, it’s a really difficult thing to keep it stable. Because of this, it’s important to figure out a way to stabilize the system. More importantly, new control methods will be discovered during the process, and these new methods can be used in many other different systems. The inverted pendulum system can reflect performances such as robustness, mobility and tracking performance. Its dynamic condition is similar to a human walk, and its steady state is similar to the launch system on a rocket’s motion attitude, so the study of the inverted pendulum system is also important to the robot and rocket area. This paper describes the construction of a mathematical model of a first-order inverted pendulum, and the process of designing the fuzzy controller based on the related fuzzy control rules. It also conducts the optimization of the fuzzy controller using a genetic algorithm. After using MATLAB’s Simulink module to simulate the inverted pendulum system and adjust the parameters of the controller, the system could finally reach a stable status within a short period of time, from all kinds of original status, and have anti-jamming performance.

Reachability Map for Diverse and Energy Efficient Stepping of Humanoids

In legged locomotion, the relationship between different gait behaviors and energy consumption must consider the full-body dynamics and the robot control as a whole, which cannot be captured by simple models. This work studies the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">totality</i> of robot dynamics and whole-body optimal control as a coupled system to investigate energy consumption during balance recovery. We developed a two-phase nonlinear optimization pipeline for dynamic stepping, which generates reachability maps showing complex energy-stepping relations. We optimize gait parameters to search all reachable locations and quantify the energy cost during dynamic transitions, which allows studying the relationship between energy consumption and stepping locations given different initial conditions. We found that to achieve efficient actuation, the stepping location and timing can have simple approximations close to the underlying optimality, resulting in optimal step positions with a 10.9% lower energy cost than those generated by linear inverted pendulum model. Despite the complexity of this nonlinear process, we found that near-minimal effort stepping locations are within a region of attractions, rather than a narrow solution space suggested by a simple model. This provides new insights into the nonuniqueness of near-optimal solutions in robot motion planning and control, and the diversity of stepping behavior in humans.

A self‐stabilised walking gait for humanoid robots based on the essential model with internal states

AbstractWalking stability is one of the key issues for humanoid robots. A self‐stabilised walking gait for a full dynamic model of humanoid robots is proposed. For simplified models, that is, the linear inverted pendulum model and variable‐length inverted pendulum model, self‐stabilisation of walking gait can be obtained if virtual constraints are properly defined. This result is extended to the full dynamic model of humanoid robots by using an essential dynamic model, which is developed based on the zero dynamics concept. With the proposed method, a robust stable walking for a humanoid robot is achieved by adjusting the step timing and landing position of the swing foot automatically, following its intrinsic dynamic characteristics. This exempts the robot from the time‐consuming high‐level control approaches, especially when a full dynamic model is applied. How different walking patterns/features (i.e., the swing foot motion, the vertical centre of mass motion, the switching manifold configuration, etc.) affect the stability of the walking gait is analysed. Simulations are conducted on robots Romeo and TALOS to support the results.

Model Predictive Control of Running Biped Robot

With the feet of a biped robot attached insecurely to a terrain, its stability is strongly affected by the characteristics of the terrain on which it runs. Therefore, for stable bipedal running, online motion control based on the states of the robot and the environment is needed. This paper proposes a method for online motion control of a running biped robot on an uneven terrain based on a dual linear inverted pendulum model (D-LIPM) and hierarchical control which consists of linear model predictive control (MPC) and quadratic-problem (QP) based momentum control. The D-LIPM, which splits the nonlinear dynamics model of the running biped robot into two linear models under some assumptions, is proposed to generate the running motion through linear MPC. The D-LIPM is applied to the proposed hierarchical control for stable bipedal running. In the first stage of hierarchy, linear MPC is employed to generate the trajectory of the center of mass (COM) based on the dynamics of D-LIPM to overcome terrain uncertainties such as elevation levels and surface conditions at foot-landing sites. In the second stage, momentum control based on a QP solver is used to generate the angular motions of the robot while following the COM trajectory. Computer simulations with uncertainties on the running terrain were carried out to measure the performance of the proposed method.