Linear Inverted Pendulum Research Articles

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.

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.

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.

Instantaneous Capture Input for Balancing the Variable Height Inverted Pendulum

Balancing is a fundamental need for legged robots due to their unstable floating-base nature. Balance control has been thoroughly studied for simple models such as the linear inverted pendulum thanks to the concept of the instantaneous capture point (ICP), yet the constant center of mass height assumption limits the application. This letter explores balancing of the variable-height inverted pendulum (VHIP) model by introducing the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">instantaneous capture input</i> (ICI), an extension of the ICP based on its key properties. Namely, the ICI can be computed as a function of the state, and when this function is used as the control policy, the ICI is rendered stationary and the system will eventually come to a stop. This characterization induces an analytical region of capturable states for the VHIP, which can be used to conceptually guide where to step. To further address state and control constraints during recovery, we present and theoretically analyze an explicit ICI-based controller with online optimal feedback gains. Simulations demonstrate the validity of our controller for capturability maintenance compared to an approach based on the divergent component of motion.

Intelligent Locomotion Planning With Enhanced Postural Stability for Lower-Limb Exoskeletons

In this letter, an integrated control strategy is developed for both locomotion trajectory planning and postural stability, enabling shared autonomy between the human and lower-limb exoskeleton. Divergent component of motion (DCM) analysis was employed previously based on the linear inverted pendulum flywheel (LIPF) model to regulate the position of the center of mass (CoM) for humanoid robots. In this study, a new extended model is investigated for the DCM analysis by replacing the previous LIPF model, which is tailored for multi-degree-of-freedom (DOF) exoskeletons. This new model is designed to be personalized for each specific user's body by relaxing the assumption of having the total CoM at the hip joint in the previous LIPF model. Accordingly, the exoskeleton has the authority to ensure the postural stability and viability of locomotion in this human-robot interaction (HRI) by adjusting the upper body position using a DCM-based hip correction strategy. Integrating adaptive central pattern generators (CPGs), the human has enough authority to modify the gait trajectories in real-time, while the amplitude and frequency of walking are constrained to their feasible ranges. The effectiveness of this intelligent controller for safe and stable locomotion is investigated through experimental studies on a lower-limb exoskeleton.

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.

Humanoid Loco-Manipulations Pattern Generation and Stabilization Control

In order for a humanoid robot to perform loco-manipulation such as moving an object while walking, it is necessary to account for sustained or alternating external forces other than ground-feet reaction, resulting from humanoid-object contact interactions. In this letter, we propose a bipedal control strategy for humanoid loco-manipulation that can cope with such external forces. First, the basic formulas of the bipedal dynamics, i.e., linear inverted pendulum mode and divergent component of motion, are derived, taking into account the effects of external manipulation forces. Then, we propose a pattern generator to plan center of mass trajectories consistent with the reference trajectory of the manipulation forces, and a stabilizer to compensate for the error between desired and actual manipulation forces. The effectiveness of our controller is assessed both in simulation and loco-manipulation experiments with real humanoid robots.

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.

Push recovery control for the underactuated lower extremity exoskeleton based on improved capture point concept

PurposeThe purpose of this paper is to use a simple method to enhance the ability of lower limb exoskeletons to restore balance under large interference conditions and to solve the problem that biped robot stability criterion cannot be fully applied to the underactuated lower limb exoskeletons.Design/methodology/approachThe method used in this paper is to construct an underactuated lower extremity exoskeleton ankle joint with a torsion spring. Based on the constructed exoskeleton, the linear inverted torsion spring pendulum model is proposed, and the traditional capture point (CP) concept is optimized.FindingsThe underactuated exoskeleton ankle joint with torsion springs, combined with the improved CP concept, can effectively reduce the forward stepping distance under the same interference condition, which is equivalent to enhancing the balance ability of the lower extremity exoskeleton.Originality/valueThe contribution of this paper is to enhance the balance ability of the exoskeleton of the lower limbs under large interference conditions. The torsion spring is used as the exoskeleton ankle joint, and the traditional CP concept is optimized according to the constructed exoskeleton.

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.

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.

Dynamic Torso Compliance Control for Standing and Walking Balance of Position-Controlled Humanoid Robots

The humanoid robot, one of the ideal unmanned systems, has the potential to perform tasks currently carried out by human workers in industry, manufacturing, service, and disaster response. In such complicated environments with unknown disturbances, compliance is important for a humanoid robot, especially a position-controlled one, to maintain balance while performing a task. This article presents a dynamic torso compliance method for position-controlled humanoid robots to adapt to unknown external disturbances while standing or walking. This method comprises two independent controllers. One controller, which is based on a linear inverted pendulum with a flywheel, is used to realize compliant behavior by controlling the torso postures according to the zero moment point. The other controller, which is based on a double inverted pendulum, is adopted to obtain the dynamic torso movements as well as maintain balance through a quadratic-programming-based controller. With the proposed method, controls for compliance and balance are decoupled, and a position-controlled robot can perform dynamic torso compliance and maintain balance while standing or walking. The performance of the proposed method is validated by simulations and experiments on a position-controlled humanoid robot (BHR-T).

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

State-Space Characterization of Balance Capabilities in Biped Systems with Segmented Feet

The human ability of keeping balance during various locomotion tasks is attributed to our capability of withstanding complex interactions with the environment and coordinating whole-body movements. Despite this, several stability analysis methods are limited by the use of overly simplified biped and foot structures and corresponding contact models. As a result, existing stability criteria tend to be overly restrictive and do not represent the full balance capabilities of complex biped systems. The proposed methodology allows for the characterization of the balance capabilities of general biped models (ranging from reduced-order to whole-body) with segmented feet. Limits of dynamic balance are evaluated by the Boundary of Balance (BoB) and the associated novel balance indicators, both formulated in the Center of Mass (COM) state space. Intermittent heel, flat, and toe contacts are enabled by a contact model that maps discrete contact modes into corresponding center of pressure constraints. For demonstration purposes, the BoB and balance indicators are evaluated for a whole-body biped model with segmented feet representative of the human-like standing posture in the sagittal plane. The BoB is numerically constructed as the set of maximum allowable COM perturbations that the biped can sustain along a prescribed direction. For each point of the BoB, a constrained trajectory optimization algorithm generates the biped’s whole-body trajectory as it recovers from extreme COM velocity perturbations in the anterior–posterior direction. Balance capabilities for the cases of flat and segmented feet are compared, demonstrating the functional role the foot model plays in the limits of postural balance. The state-space evaluation of the BoB and balance indicators allows for a direct comparison between the proposed balance benchmark and existing stability criteria based on reduced-order models [e.g., Linear Inverted Pendulum (LIP)] and their associated stability metrics [e.g., Margin of Stability (MOS)]. The proposed characterization of balance capabilities provides an important benchmarking framework for the stability of general biped/foot systems.

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.

The Spherical Inverted Pendulum: Exact Solutions of Gait and Foot Placement Estimation Based on Symbolic Computation

The gait and the Foot Placement Estimation (FPE) has recently been extended to 3-D spaces by adopting a specific form of a spherical inverted pendulum (SIP). The approach is very attractive, as it does not involve dynamics, but it is based solely on energies and momenta, however the authors (DeHart et al.) introduced several questionable approximations, in order to reach a manageable solution. The scope of the present paper is to revisit this spherical inverted pendulum applied to biped walking, offering an exact solution to the gait and the FPE by using symbolic computation. This is facilitated by exploiting the Kane’s approach to dynamical modelling, and his software environment for symbolic manipulation, called Autolev. It generates explicit formulas describing the energies and angular momenta before/after the impact, along with the mechanics of the impact. As the resulting equations, function of (measurable) angular positions and velocities, are very compact, embedded in a numerical nonlinear solver, are suitable to be implemented in real time and used in practice to control biped robots or lower limb exoskeletons. The two main contributions of the paper are: the recovery of the balance by stepping, in the presence of a push in an arbitrary direction and omnidirectional walking. In this last respect, this specific form of SIP emphasizes the expenditure of energy in the walk. For the first time, at our knowledge, the walk of the SIP, based on energy, has been compared to the simulation of a 12 degrees of freedom biped robot tracking preview signals using the Zero Moment Point (ZMP) of the Linear Inverted Pendulum (LIPM). This quantitatively shows the inefficiency, in terms of energy, of the ZMP-based walk, and the gain due to the recovery of the collision of the flying foot. Similarity in the sagittal plane and differences in the frontal plane of the center of mass trajectories of the two approaches are shown, to open the road to an integration of fully actuated and underactuated controls, for an efficient full-dimensional robot gait to be developed in a future paper.

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.