Linear Inverted Pendulum Model Research Articles

Three-dimensional aperiodic biped walking including the double support phase using LIPM and LPM

This paper presents a trajectory generation algorithm with which a three-dimensional (3D) biped robot can perform aperiodic gaits by modifying only a small set of gait parameters. In addition to the double support phase (DSP), the gait can transit smoothly from one single support phase (SSP) to another. We decouple the sagittal and coronal dynamics, firstly. In the sagittal plane, the linear inverted pendulum model (LIPM) is used to generate the walking reference trajectory during the SSP. An extra template model, linear pendulum model (LPM), is added to describe the motion in the DSP. For the coronal plane, we only utilize the LPM for trajectory generation. Thanks to linearity properties, the proposed method can obtain the biped locomotion computationally fast without the need for numerical time-integration. Herein, we also introduce the trajectory generation algorithm for different aperiodic gaits including from standing to walking, stopping walking, as well as speed switch. A full-dynamics 3D humanoid robot is used for the tests of the developed reference trajectory through simulation. The results are promising for implementations.

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.

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.

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

Energy-Efficient Bipedal Walking: From Single-Mass Model to Three-Mass Model

SUMMARYThe work aims to realize energy-efficient bipedal walking by employing the three-mass inverted pendulum model (3MIPM) and compare its energy performance with linear inverted pendulum model (LIPM). To do this, a general optimal index on center of mass (CoM) acceleration is first derived for energetic cost evaluation. After defining the equivalent zero moment point (ZMP) motion, an unconstrained optimization approach for CoM generation is extended for 3MIPM, which can track different ZMP references and address the height variation as well. To make use of the allowable ZMP movement, a constrained optimization method is also employed, contributing to lower energetic cost. Simulation and hardware experiments on a humanoid robot demonstrate that the 3MIPM could achieve higher energy efficiency.

Real-Time Planning and Nonlinear Control for Quadrupedal Locomotion With Articulated Tails

Abstract The primary goal of this paper is to develop a formal foundation to design nonlinear feedback control algorithms that intrinsically couple legged robots with bio-inspired tails for robust locomotion in the presence of external disturbances. We present a hierarchical control scheme in which a high-level and real-time path planner, based on an event-based model predictive control (MPC), computes the optimal motion of the center of mass (COM) and tail trajectories. The MPC framework is developed for an innovative reduced-order linear inverted pendulum (LIP) model that is augmented with the tail dynamics. At the lower level of the control scheme, a nonlinear controller is implemented through the use of quadratic programming (QP) and virtual constraints to force the full-order dynamical model to track the prescribed optimal trajectories of the COM and tail while maintaining feasible ground reaction forces at the leg ends. The potential of the analytical results is numerically verified on a full-order simulation model of a quadrupedal robot augmented with a tail with a total of 20 degrees-of-freedom. The numerical studies demonstrate that the proposed control scheme coupled with the tail dynamics can significantly reduce the effect of external disturbances during quadrupedal locomotion.

Trajectory Planning of Flexible Walking for Biped Robots Using Linear Inverted Pendulum Model and Linear Pendulum Model

Linear inverted pendulum model (LIPM) is an effective and widely used simplified model for biped robots. However, LIPM includes only the single support phase (SSP) and ignores the double support phase (DSP). In this situation, the acceleration of the center of mass (CoM) is discontinuous at the moment of leg exchange, leading to a negative impact on walking stability. If the DSP is added to the walking cycle, the acceleration of the CoM will be smoother and the walking stability of the biped will be improved. In this paper, a linear pendulum model (LPM) for the DSP is proposed, which is similar to LIPM for the SSP. LPM has similar characteristics to LIPM. The dynamic equation of LPM is also linear, and its analytical solution can be obtained. This study also proposes different trajectory-planning methods for different situations, such as periodic walking, adjusting walking speed, disturbed state recovery, and walking terrain-blind. These methods have less computation and can plan trajectory in real time. Simulation results verify the effectiveness of proposed methods and that the biped robot can walk stably and flexibly when combining LIPM and LPM.

Particle Swarm Optimization aided PID gait controller design for a humanoid robot

Gait planning for the humanoid robot is a very essential and basic requirement. The humanoid robot is balanced at two feet; therefore, special attention is required for gait analysis for the execution of assigned tasks. In this paper, the linear inverted pendulum (LIPM) model is considered to simplify the study and to obtain better gait planning of humanoid robot NAO. Center of mass (COM) and zero moment point (ZMP) criterion are applied with the LIPM model for a better understanding of selecting the step length and period. In addition, a PSO (particle swarm optimization) tuned PID (proportional–integral–derivative) controller has been implemented. Sensory data such as the location of obstacles and the target along with the desired trajectory aided inverse kinematics have been embedded to the conventional PID controller, which provides an interim angle to start the navigation. This interim angle has been carried forward to the PSO technique accompanied by the desired trajectory. It tunes the parameters of the conventional PID controller and provides an optimum turning angle, which avoids obstacles and increases the stabilization of the robot while crossing it. It reduces travel time and shortens travel length. PSO technique minimizes the computational complexity and number of iteration because it requires fewer tuning parameters. Simulations are executed on the simulated NAO robot for the conventional PID controller and the proposed controller. To ratify its findings, experiments are carried out on a real NAO robot in laboratory conditions for both the conventional PID controller and the proposed controller. Simulation and experimental results are presenting a good agreement among each other with deviation under 6%. Applying the PSO tuned PID controller provides a predictable gait and reduces the stabilization time and essentially eliminating the overshoot by 25%. A comparative study with various controllers is performed, and the credibility of the evaluated result has been examined using statistical analysis. The proposed controller has been compared with a previously developed technique to ensure its robustness.

Survey on model-based biped motion control for humanoid robots

ABSTRACT Studies of biped control including standing, walking, hopping and running on humanoid robots are reviewed in this survey paper. Model-based approaches standing upon reduced-order dynamics are focused. It begins with revisiting the centroidal dynamics and the zero-moment point, which leads to the linear inverted pendulum (LIP) model and some of its variations. Then, some representative standing and walking control techniques based on the LIP model are reviewed, where the concept of capturability is also discussed. It is followed by consideration of height variation of the center of mass. Enhancement to hopping and running motions, which include aerial phases, is also addressed. Ideas to deal with complex nature as a hybrid system underlying discontinuously varying mechanics mainly due to unilateral constraints on contact forces are concisely explained.

Dynamic and Versatile Humanoid Walking via Embedding 3D Actuated SLIP Model With Hybrid LIP Based Stepping

In this paper, we propose an efficient approach to generate dynamic and versatile humanoid walking with non-constant center of mass (COM) height. We exploit the benefits of using reduced order models (ROMs) and stepping control to generate dynamic and versatile walking motion. Specifically, we apply the stepping controller based on the Hybrid Linear Inverted Pendulum Model (H-LIP) to perturb a periodic walking motion of a 3D actuated Spring Loaded Inverted Pendulum (3D-aSLIP), which yields versatile walking behaviors of the 3D-aSLIP, including various 3D periodic walking, fixed location tracking, and global trajectory tracking. The 3D-aSLIP walking is then embedded on the fully-actuated humanoid via the task space control on the COM dynamics and ground reaction forces. The proposed approach is realized on the robot model of Atlas in simulation, wherein versatile dynamic motions are generated.

Structural Design and Kinematics Simulation of Hydraulic Biped Robot

This paper presents a novel framework of hydraulic-driven biped robot. The target biped robot is a humanoid robot NWPUBR-1 with 12 degrees of freedom (DOF), and its dimensions are close to that of an average male. The joint axis adopts the modular structure design of sensor with angle measurement, and the force sensor is deployed on the side of hydraulic actuator to facilitate force and position control of the robot. Meanwhile, the finite element analysis of critical components is conducted to meet the requirements of mechanical strength in motion. Based on the screw theory, the forward kinematics and Jacobian matrix models are established, and the inverse kinematics of robot is solved by using the analytical geometric method. To achieve real-time control of the robot gait in 3D space, a three-dimensional linear inverted pendulum model (3D-LIPM) is built, and a 3D gait model is generated by combining the zero-moment point (ZMP) theory. In the virtual environment of MATLAB software, the results of programming simulation show that the biped robot can walk stably in the virtual environment, which proves the correctness of 3D-LIPM.

Quadrupedal Locomotion via Event-Based Predictive Control and QP-Based Virtual Constraints

This letter aims to develop a hierarchical nonlinear control algorithm, based on model predictive control (MPC), quadratic programming (QP), and virtual constraints, to generate and stabilize locomotion patterns in a real-time manner for dynamical models of quadrupedal robots. The higher level of the proposed control scheme is developed based on an event-based MPC that computes the optimal center of mass (COM) trajectories for a reduced-order linear inverted pendulum (LIP) model subject to the feasibility of the net ground reaction force (GRF). The asymptotic stability of a desired target point for the reduced-order model under the event-based MPC approach is investigated. It is shown that the event-based nature of the proposed MPC approach can significantly reduce the computational burden associated with the real-time implementation of MPC techniques. To bridge the gap between reduced- and full-order models, QP-based virtual constraint controllers are developed at the lower level of the proposed control scheme to impose the full-order dynamics to track the optimal trajectories while having all individual GRFs in the friction cone. The analytical results of the letter are numerically verified to demonstrate stable and robust locomotion of a 22 degree of freedom quadrupedal robot, in the presence of payloads, external disturbances, ground height variations, and uncertainty in contact models.

Robust Stability Control of Inverted Pendulum Model for Bipedal Walking Robot

This paper proposes robust control for three models of the linear inverted pendulum (one mass linear inverted pendulum model, two masses linear inverted pendulum model and three masses linear inverted pendulum model) which represents the upper, middle and lower body of a bipedal walking robot. The bipedal walking robot is built of light-weight and hard Aluminum sheets with 2 mm thickness. The minimum phase system and non-minimum phase system are studied and investigated for inverted pendulum models. The bipedal walking robot is programmed by Arduino microcontroller UNO. A MATLAB Simulink system is built to embrace the theoretical work. The results showed that one linear inverted pendulum is the worst performance, worst noise rejection and the worst set point tracking to the zero moment point. But two masses linear inverted pendulum models and three masses linear inverted pendulum model have a better performance, a better high-frequency noise rejection characteristic and better set-point tracking to the zero moment point.

A Theoretical Framework for Stability Regions for Standing Balance of Humanoids Based on Their LIPM Treatment

The aim of this paper is to construct a theoretical framework for stability analysis relevant to standing balance of humanoids on top of the linear inverted pendulum model, in which their dynamics between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. Based on the well-known sufficient condition that the contact between the ground and the support leg is stable if the corresponding ZMP is always inside the supporting region, this paper aims at characterizing three types of the associated stability regions. More precisely, assuming no external force disturbances affecting the motion of the humanoids, the stability region of the initial CoM position and velocity values can be explicitly computed by solving a finite number of linear inequalities. The stability regions of time-invariant force disturbances such as impulsive force and constant force disturbances are also dealt with in this paper, where the former is exactly obtained through a finite number of linear inequalities while the latter is approximately derived by using an idea of truncation. Furthermore, time-varying force disturbances of finite energy and finite amplitude are concerned with, and their maximum admissible ${l} _{2}$ and ${l} _{\infty }$ norms are computed in this paper, where the former can be exactly obtained by solving the discrete-time Lyapunov equation while the latter is approximately derived through an idea of truncation. It is further shown for both the truncation ideas that the approximately obtained stability regions converge to the exact stability regions with an exponential order of ${N}$ , where ${N}$ is the truncation parameter. Finally, the effectiveness of the computation methods proposed in this paper is demonstrated through some simulation results.

Controlled Gait Planning of Humanoid Robot NAO Based on 3D-LIPM Model

Gait planning is essential for biped robots because the stability of the two support is very challenging task. A special attention is required to get a stabilized gait for humanoid robots. In this paper, 3D-LIPM (linear inverted pendulum model) is used to solve the gait planning problems of humanoid robots. Humanoid robot NAO is utilized as robotic platform. The structure of the robotic platform decides the period step height and step length. The gait planning consists of model of 3D-LIPM and ZMP (zero moment point). Simulation and experiments are carried out on simulated and real NAO, and showing good agreement with each other. An ideal controlled gait is obtained using the proposed technique.

Oscillation Preventing Closed-Loop Controllers via Genetic Algorithm for Biped Walking on Flat and Inclined Surfaces

In this study, a closed-loop controller is designed to overcome the dynamical insufficiency of the 3D Linear Inverted Pendulum Model (LIPM) via the Genetic Algorithm (GA). The main idea is to still use the 3D LIPM with a closed-loop controller because of its ease at modeling. While suppressing the dynamical flaws only the legs are used, in other words a robot is used which does not have any upper body elements to have a more modular robot. For this purpose, a biped is modeled with the 3D LIPM which is one of the most famous modeling methods of humanoid robots for the ease of modeling and fast calculations during the trajectory planning. After obtaining the simple model, Model Predictive Control (MPC) is applied to the 3D LIPM to find the reference trajectories for the biped while satisfying the Zero Moment Point (ZMP) criteria. The found reference trajectories applied to the full dynamical model on Matlab Simulink and the real biped in the laboratory at Istanbul Technical University. From the simulation results on the flat and inclined surfaces and real-time experiments on a flat surface some dynamical flaws are observed due to the simple modeling. To overcome these flaws a Proportional-Integral (PI) controller is designed, and the optimal value of the controller gains are found by the GA. The results assert that the designed controller can overcome the observed flaws and makes biped move more stable, smoother, and move without steady-state error.

Fuzzy Dynamic Gait Pattern Generation for Real-Time Push Recovery Control of a Teen-Sized Humanoid Robot

A fuzzy dynamic gait pattern generator, which allows a teen-sized humanoid robot to generate, in real-time, a suitable gait pattern when it is hit by an unexpected force, is proposed in this paper. Conventional gait pattern generators usually utilize the ideal Zero Moment Point (ZMP) to plan the trajectory of the Center of Mass (CoM), along with a cycloid to generate steps. However, pre-planned gait patterns cannot deal with unexpected situations, especially instances when the robot experiences an unknown force. Therefore, we propose a dynamic gait pattern generator that leverages the Virtual Force Linear Inverted Pendulum Model (VFLIPM) to adjust the trajectory of the CoM, and which detects balance status by estimating the trajectory of the ZMP using eight high-precision load cell pressure sensors mounted onto the robot's soles. We integrate an accelerometer and the pressure sensors through a fuzzy controller to instantly respond to external forces and generate a suitable gait pattern. When the robot is pushed suddenly, it first adopts a pre-planned gait pattern to replace the current gait. At the same time, the fuzzy controller calculates the recovery gait, with appropriate strides and lean angles to absorb the impact. The proposed method is implemented on the teen-sized humanoid robot, David Junior II, for the Push Recovery event at RoboCup. David Junior II endured a hit with potential energy during walking, which is 1.3 times more robust than when standing.

Підхід до створення системи моделювання гнучкого виробництва

This paper dealing with the problem of generation motion of biped robot, to solve this problem, was introduces a new method of implementation a biped robot walking pattern generation byusing a Model Predictive control (MPC) First, the basic of biped robots’ motion was reviewed and was discussed the elements of motion controller. The dynamics of a biped robot is modeled as a runningcart on a table which gives a convenient representation for generation ZMP trajectory. After reviewing approach of 3D Linear Inverted Pendulum Model, a pattern generator was designed, lastly in MATLAB/Simulink and using Model Predictive Control Toolbox (MPCT) which provides functions, an app, and Simulink blocks was used for designing and simulating a motion generator based on model predictive controllers (MPC).The results of simulation are shown that we can realize such controller for generating trajectory of Center of Mass and Zero Moment point by using the MPC. The smooth trajectory of CoM is generated and the resulted ZMP follows the reference with good accuracy.Ref. 13, pic. 6

An MPC-based two-dimensional push recovery of a quadruped robot in trotting gait using its reduced virtual model

This paper addresses the push recovery of quadruped robots trotting on even terrain. Push recovery is the ability of a legged robot to maintain its balance in the presence of sudden external forces applied to the robot. Due to the nature of this dynamical gait, the quadruped robot can be modeled as a biped robot where each two cross legs of the quadruped are modeled as a virtual leg. Then, the virtual biped model is further reduced to a two-dimensional linear inverted pendulum plus flywheel model (LIPFM). Moreover, using the concept of capture points for the biped model, the desired locations for the center of pressure (COP) of the legs for recovering the robot's balance are calculated by designing a two-dimensional dynamic capture point estimator. A model-based predictive controller (MPC) is then proposed to adjust the footstep locations and generate a new walking-pattern. The resulting redesigned joint angles for the virtual biped model are then transformed back to those of the actual quadruped model. Simulation results show the effectiveness of the proposed method.