Linear Inverted Pendulum Model Research Articles

Stable Rapid Sagittal Walking Control for Bipedal Robot Using Passive Tendon

This paper presents the development, control, and experimental validation of a novel bipedal robot with a passive tendon. The robot, featuring foldable legs, coaxial actuation, and compact folded size, is endowed with a leg configuration with a five-bar mechanism. Based on biological observations of human walking, a passive artificial tendon made of emulsion is fabricated to work in conjunction with a tensioning device, providing adaptive heel touchdown and toe push-off in sync with single-leg movement. The tailored control framework for the bipedal robot is further established with the double-layer architecture. The regulation layer employs the linear inverted pendulum (LIP) model to generate reference trajectory of the center of mass (CoM) with a dead-beat style of parameter adjustment. An inverse-dynamics-based whole-body controller (WBC) is applied to enforce the full-order dynamics of the bipedal robot to reproduce the LIP model’s behavior. We carry out the experiments on the physical prototype to evaluate the walking performance of the developed bipedal robot. The results show that the robot achieves stable walking at the speed of 0.8 m/s (almost twice the leg length/s) and exhibits robustness to external push disturbance.

Reactive bipedal balance: Coordinating compliance and stepping through virtual model imitation for enhanced stability

Compliance and step location adjustments are both important for robots to maintain balance when encountering external disturbances. Compared with controlling separately, unifying compliance with step location adjustments and integrating them into real-time pattern planning gets obvious advantages. However, the controllers’ models differ significantly from each other and from the dynamics of real position-controlled robots, compromising the stabilizing performance. In this paper, we propose a Virtual Model Imitation (VMI) strategy that unifies the controllers’ models and aligns them with actual robot dynamics. Based on this, a reactive pattern planning method is proposed which simultaneously coordinates compliance and step location adjusting strategies. The approach involves emulating the passive dynamic motion of a Linear Inverted Pendulum Model (LIPM) to enable the robot to respond to external disturbances with compliant motions automatically. Additionally, Zero Moment Point Intervened (ZI) estimation is employed to precisely determine the Center of Mass (CoM) state. Real-time optimization of step location is then conducted to stabilize the estimated CoM, actively responding to external disturbances. Importantly, the decision to adjust step location is quantitatively defined. To enhance efficiency, explicit solutions for both ZI Estimation and Step Location Optimization are provided to reduce computational costs. To validate our approach, we conducted simulations and experiments using the biped robot BHR-T, subjecting it to unexpected external pushes and obstacles during walking.

Analytical solution to a time-varying LIP model for quadrupedal walking on a vertically oscillating surface

This paper introduces an analytically tractable and computationally efficient model for legged robot dynamics during locomotion on a dynamic rigid surface (DRS), along with an approximate analytical solution and a real-time walking pattern generator synthesized based on the model and solution. By relaxing the static-surface assumption, we extend the classical, time-invariant linear inverted pendulum (LIP) model for legged locomotion on a static surface to dynamic-surface locomotion, resulting in a time-varying LIP model termed as “DRS-LIP”. Sufficient and necessary stability conditions of the time-varying DRS-LIP model are obtained based on the Floquet theory. This model is also transformed into Mathieu’s equation to derive an approximate analytical solution that provides reasonable accuracy with a relatively low computational cost. Using the extended model and its solution, a walking pattern generator is developed to efficiently plan physically feasible trajectories for quadrupedal walking on a vertically oscillating surface. Finally, simulations and hardware experiments from a Laikago quadrupedal robot walking on a pitching treadmill (with a maximum vertical acceleration of 1 m/s2) confirm the accuracy and efficiency of the proposed analytical solution, as well as the efficiency, feasibility, and robustness of the pattern generator, under various surface motions and gait parameters.

Offline Trajectory Generation for Bipedal Robot Using Linear Inverted Pendulum Model

Reduced order model (ROM)-based controllers have proved to be effective to generate stable bipedal locomotion. However, it is important to understand the limitations and effectiveness of these models without implementing any controllers. This study highlights the versatility of the Linear Inverted Pendulum Model (LIPM) at various walking speeds. Firstly, the Centre of Mass (COM) trajectory has been generated using the LIPM model, and the foot motion trajectory has been created using a sixth-order polynomial function. The trajectory is generated using a predefined step length, speed of locomotion and COM height. Secondly, the task space trajectory has been converted into a joint space trajectory through inverse kinematics for a 6-degree-of-freedom leg. To facilitate the proper walking motion the contact between the foot sole and the ground is implemented. Finally, a simple bipedal robot in MATLAB/Simulink has been modelled and the generated trajectories were implemented.

Design, Control, and Validation of a Symmetrical Hip and Straight-Legged Vertically-Compliant Bipedal Robot.

This paper presents the development, modeling, and control of L03, an underactuated 3D bipedal robot with symmetrical hips and straight legs. This innovative design requires only five actuators, two for the legs and three for the hips. This paper is divided into three parts: (1) mechanism design and kinematic analysis; (2) trajectory planning for the center of mass and foot landing points based on the Divergent Component of Motion (DCM), enabling lateral and forward walking capabilities for the robot; and (3) gait stability analysis through prototype experiments. The primary focus of this study is to explore the application of underactuated symmetrical designs and determine the number of motors required to achieve omnidirectional movement of a bipedal robot. Our simulation and experimental results demonstrate that L03 achieves simple walking with a stable and consistent gait. Due to its lightweight construction, low leg inertia, and straight-legged design, L03 can achieve ground perception and gentle ground contact without the need for force sensors. Compared to existing bipedal robots, L03 closely adheres to the characteristics of the linear inverted pendulum model, making it an invaluable platform for future algorithm research.

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.

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.

Bio-Inspired Model-Based Design and Control of Bipedal Robot

The article aims to describe stages of development of a mobile, anthropomorphic robot. The individual phases of the bio-inspired model-based design of a two-legged robot are described, from the analysis of human walking patterns, through the creation of a simulation model and a control system to the creation and testing the physical model of such a robot. The bio-inspired design is based on an analysis of the movements of the individual parts of a lower human body during walking locomotion. Based on this analysis, the chosen motion model is described as a combination of passive and dynamic walking with a controlled, linear inverted pendulum model. When creating the simulation model, an open kinematic chain consisting of a base (the frame) and two effectors (the legs) is used. The simulation of the virtual model is realized using the software tool Matlab with its toolboxes Simulink and Simscape, and the results confirm the correctness of the design. The feasibility of the design is confirmed by creation of a physical robot skeleton using 3D printing and by a commissioning of the control system based on Atmel ATmega2560 and Raspberry Pi Zero W microcontrollers.

A New Stability Framework for Trajectory Tracking Control of Biped Walking Robots

This article proposes a new stability criterion for biped walking systems on the linear inverted pendulum model, in which the dynamic relationship between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. More precisely, based on the fact that a biped walking robot is stable if its ZMP is always located in the supporting region, we consider whether the ZMP error between its reference and real values stays inside a certain area to guarantee the stability condition. To this end, a norm-based new stability criterion is introduced, in which the temporal supremum of the ZMP error is concerned. Regarding the applicability of the stability criterion, we propose two control approaches to biped walking systems with the consideration of the norm-based stability criterion. In other words, full-state and observer-based feedback control approaches are analyzed in this article, and the initial CoM conditions with respect to the stability criterion are obtained for the two control approaches. We call the sets derived by such conditions the stability regions. Toward a more practical significance, we also deal with the effect of unknown disturbances and sensor noises on the stability of biped walking systems. More importantly, even though computing stability regions intrinsically involves an infinite number of linear inequalities, all the stability regions are shown to be explicitly obtained through only finite numbers of computations in this article. Finally, some simulation results are provided to demonstrate the validity as well as the practical applicability of the developed computation methods.

Dynamic Vertical Climbing Mechanism of Chinese Dragon-Li Cats Based on the Linear Inverted Pendulum Model

Dynamic Vertical Climbing Mechanism of Chinese Dragon-Li Cats Based on the Linear Inverted Pendulum Model

A nonlinear state estimation framework for humanoid robots

This article proposes a novel nonlinear state estimation framework for humanoid robots based on the dynamics of the center of mass (CoM) and dual-loop Kalman filter(DLKF). It effectively fuses the information from inertial measurement unit(IMU), joint encoders, and foot sensitive resistors (FSRs), and provides state estimates for various gait generation algorithms and dynamic balance controllers with CoM or divergence component of motion (DCM) feedback. Compared to the widely used linear models such as the linear inverted pendulum model (LIPM), the nonlinear dynamics of CoM effectively reduce the process error. However, the humanoid robot is a highly complex dynamic system with multiple links and joints, the dynamics of CoM are also a simplification of the whole body dynamics. As a result, it brings non-zero-mean, non-Gaussian, and correlated process error, which the conventional extend Kalman filter (EKF) cannot handle. To this end, the DLKF is adopted to compensate the process error, thus the estimator is robust to the modeling error caused by simplifications. Furthermore, the invariant extended Kalman filter (IEKF) is used for floating base kinematics estimation, and the force/torque (F/T) sensor which is difficult to integrate on smaller or cheaper robots due to the size and cost is not used in our framework. Finally, our nonlinear state estimation framework improves the accuracy of CoM and DCM estimation in a virtual environment simulation using our self-developed Defensor hydraulic humanoid robot.

Balance Recoverability and Control of Bipedal Walkers With Foot Slip.

Low-friction foot/ground contacts present a particular challenge for stable bipedal walkers. The slippage of the stance foot introduces complexity in robot dynamics and the general locomotion stability results cannot be applied directly. We relax the commonly used assumption of nonslip contact between the walker foot and the ground and examine bipedal dynamics under foot slip. Using a two-mass linear inverted pendulum model, we introduce the concept of balance recoverability and use it to quantify the balanced or fall-prone walking gaits. Balance recoverability also serves as the basis for the design of the balance recovery controller. We design the within- or multi-step recovery controller to assist the walker to avoid fall. The controller performance is validated through simulation results and robustness is demonstrated in the presence of measurement noises as well as variations of foot/ground friction conditions. In addition, the proposed methods and models are used to analyze the data from human walking experiments. The multiple subject experiments validate and illustrate the balance recoverability concept and analyses.

Path Tracking Strategy for Quadruped Robots Using a Hierarchical Framework

We present a general path tracking strategy, with continuous gaits, for quadruped robots on flat and rigid terrain. We use a hierarchical framework to decompose the task into unique motion templates (actions) such as translation and turning. A Linear Inverted Pendulum Model (LIPM) based algorithm is applied to create physically correct actions while minimizing the sum of squared accelerations of the CoM. These actions are stored in an online action library and are combined using an autonomous steering system that decides when to use which action. The steering system also computes the input parameters for these actions. We also develop transition maneuvers to execute a smooth switching between these actions. This strategy can help reduce the repeated need to perform comprehensive online computation for motion planning. Moreover, one can enrich the action library to execute more complicated tasks involving different gaits and non-flat terrain. We use an 18 DoF quadruped robot model, and simulations are carried out in MATLAB to demonstrate the capabilities of this approach.

Integrated inverted pendulum and whole-body control design for bipedal robot with foot slip

Low-friction ground surfaces present a particular challenge for control of bipedal walking robots. The onset of foot slip changes the robot dynamics and the commonly designed walking controllers assuming non-slip conditions cannot maintain balance and prevent the walker from falling. We present a two-level robot controller for bipedal walker under foot slip. The high-level control is built on a two-mass linear inverted pendulum (TMLIP) model to generate the appropriate step location such that the robot walker maintains dynamic balance through multiple planned steps. The specified stepping tasks are fed to the low-level control that is based on a whole-body operational space (WBOS) framework. The WBOS controller uses the dynamic model of the multi-link bipedal walker and generates joint torques to maintain balance and follow the given stepping task in the presence of foot slip. Performance of the integrated TMLIP-WBOS control algorithm is demonstrated through extensive simulation studies of a 5-link robotic walker.

Closed‐form solution for real‐time optimal walking pattern generation constrained by the desired amount of capture point

AbstractI present a calculus variations problem for an optimal zero‐moment point (ZMP) pattern, and derive a closed‐form solution satisfying two‐boundary conditions of ZMP and an additional constraint, which is a difference between a current capture point and another. In addition, I present a closed‐form solution for the equation of motion of the linear inverted pendulum model, or a cart‐table model using the optimal ZMP pattern. Through the closed‐form solutions, a current walking pattern is generated with every step period, and is connected with a previous pattern without discontinuity. I demonstrate my optimal walking pattern generator with 13 forward steps, and compare power consumption with walking, using a linearly interpolated ZMP pattern for effectiveness. Moreover, I indicate the walking pattern changes in real‐time, depending on varying step lengths.

Design and Optimization of an Omnidirectional Humanoid Walk: A Winning Approach at the RoboCup 2011 3D Simulation Competition

This paper presents the design and learning architecture for an omnidirectional walk used by a humanoid robot soccer agent acting in the RoboCup 3D simulation environment. The walk, which was originally designed for and tested on an actual Nao robot before being employed in the 2011 RoboCup 3D simulation competition, was the crucial component in the UT Austin Villa team winning the competition in 2011. To the best of our knowledge, this is the first time that robot behavior has been conceived and constructed on a real robot for the end purpose of being used in simulation. The walk is based on a double linear inverted pendulum model, and multiple sets of its parameters are optimized via a novel framework. The framework optimizes parameters for different tasks in conjunction with one another, a little-understood problem with substantial practical significance. Detailed experiments show that the UT Austin Villa agent significantly outperforms all the other agents in the competition with the optimized walk being the key to its success.

Online Foot Location Planning for Gait Transitioning Using Model Predictive Control

This paper proposes an online uniform foot location planning method (UPMPC) based on model predictive control (MPC) for solving the problem of large posture changes during gait transitioning. This method converts the foot location planning into a discrete-time MPC problem. The core part of the method is to complete the planning of the foot location based on the linear inverted pendulum (LIP) model and the simplified robot dynamics model. By unifying the input foot location at each time step, the solution time is shortened. The final simulation experiment compares the results of using the UPMPC and foot location planning method with heuristic function (HF) for gait transitioning, respectively. This result demonstrates that the UPMPC can complete the gait transitioning task and adapt to large changes in posture during gait transitioning. In addition, the results also show the good performance of UPMPC in fixed gait.

A performance comparison between closed form and numerical optimization solutions for humanoid robot walking pattern generation

This article presents the modeling process of the lower part of a humanoid biped robot in terms of kinematic/dynamic states and the creation of a full dynamic simulation environment for a walking robot using MATLAB/Simulink. This article presents two different approaches for offline walking pattern generation: one relying on a closed-form solution of the linear inverted pendulum model (LIPM) mathematical model and another that considers numerical optimization as means of desired output trajectory following for a cart table state-space model. This article then investigates the possibility of introducing solution-dependent modifications to both approaches that could increase the reliability of basic walking pattern generation models in terms of smooth single support–double support phase transitioning and power consumption optimization. The algorithms were coded into offline walking pattern generators for NAO humanoid robot as a valid example and the two approaches were compared against each other in terms of stability, power consumption, and computational effort as well as against their basic unmodified counterparts.