Zero Moment Point Reference Research Articles

Bio-inspired central pattern generator for development of Cartesian motor skills for a Quadruped Robot

Central Pattern Generator (CPG) plays a significant role in the generation of diverse and stable gaits patterns for animals as well as controlling their locomotion. The main contributions of this paper are the ability to develop the Cartesian motor skills and coordinating legs of the quadruped robot for gait adaption and its nominal characteristics with CPG approach. Primary, a predefined relationship between an excitation signal and essential parameters of the CPG design is programmed. Next, the coordinated oscillator's rhythmic patterns by CPG and accordingly output gait diagrams for each foot of the robot are attained. Then, these desirable features such as predictive modulation and programming the gait event sequences including leg-lifting sequences and step length, duration of the time of each footstep within a gait, coordination of swing and stance phases of all legs are calculated in terms of different spatio_temporal vectors. Furthermore, a novel Cartesian footstep basis function is designed based on the robot characteristics and consequently, the associated spatio-temporal vectors can be inserted to it, which caused to spanning the space of possible gait timing in Cartesian space. Next, Cartesian footstep planner can be computed the swing foot trajectories in workspace along movement axes and then according to these footholds and feet placement, ZMP (Zero Moment Point) reference trajectory will be calculated and obtained. Therefore, COG (Center of Gravity) trajectory can be computed by designing a preview controller on the basis of the desired ZMP trajectory. Finally, to demonstrate the effectiveness of the proposed algorithm, it is implemented on a quadruped robot on both simulation or experimental implementations and the results are compared and discussed with other references.

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.

Two-stage optimization for energy-efficient bipedal walking

This paper proposes a two-stage optimization strategy for energy-efficient gait generation. At the first stage, by tracking the reference zero moment point (ZMP) trajectory, the optimal center of mass (CoM) trajectory, which contributes to the minimal unit energetic cost (UEC) of one step, is solved analytically by using an unconstrained optimization method. At the second stage, to minimize the multi-joint mechanical work, the ZMP reference during the single support phase is optimized by a constrained optimization method. As a result, by considering the feasibility constraints such as the limitation on ZMP movement, the energy-efficient walking patterns can be generated in real-time. Furthermore, the energetic performances under different step parameter configurations, which consist of step length, step duration, and time ratio of double support, are discussed. Simulations and hardware experiments have demonstrated the energetic benefits of the proposed strategy when compared with other state-of-the-art works.

Application of Variable Zero-Moment Point in Walking Control of the Biped Robot

Using the predictive control based on zero-moment point (ZMP), the biped robot can walk comparatively stably. However, the problems such as lack of self-adaptivity are also highlighted mainly on account of modeling errors and environmental perturbations; specifically, the tracking errors of ZMP are generated, leading to a reduced walking stability. To address this problem, in the present work, the expected ZMP was decomposed into the reference ZMP which is pre-planned offline, and the variable ZMP which can be varied in real time. With the addition of the variable ZMP, the outside interferences can be eliminated. By combining the predictive control system and the inverse system of variable ZMP, the walking pattern of the robot with favorable self-adaptivity can be achieved. Finally, the simulation results indicate that the self-adaptivity of the robot can be effectively improved using the proposed control system.

A Method for Bipedal Robot Locomotion Control along an Arbitrary Trajectory

This paper presents an algorithm for stable bipedal walking control along an arbitrary curve. The algorithm starts with a foot planner, which takes a parametrically defined desired path as an input and calculates feet positions and orientations at each step. Zero moment point (ZMP) concept is used for robot stability control. The dynamics of the robot is modeled as a running cart on a table. Given the reference ZMP trajectory, ZMP tracking servo controller based on preview control theory is used to calculate robot’s center of mass (CoM) trajectory. The preview controller is made of three terms: the integral error of ZMP, the state feedback proportional to a current state vector and the preview action which takes into account future values of the desired ZMP position. We propose robot’s state estimator based on linear Kalman filter with measured CoM acceleration and position as system inputs. Swing foot trajectories are calculated using trigonometric functions, since they are simple and can provide zero velocities at contact moments. We put some additional constraints on a system by assuming that there is no forward-backward inclination of the robot trunk and swing foot is always parallel to the ground. To avoid kinematic limits in hip and ankle roll joints we calculate a minimal angle to rotate robot trunk in frontal plane. After position and orientation of robot trunk and swing foot are found we apply inverse kinematics solution for robot legs to get robot joint angles. Finally, joint angles are sent to robot actuators. Software package based on ROS operating system was developed for AR601 robot. Verification tests were executed with robot model in Gazebo simulator. The robot successfully completed series of experiments confirming modelling results.

Quasi-Natural Humanoid Robot Walking Trajectory Generator Based on Five-Mass With Angular Momentum Model

This paper proposes to develop a quasi-natural humanoid robot walking trajectory generator based on five-mass with angular momentum model using feedback–feedforward controller. This approach aims to minimize modeling error and improve the frequency characteristics from nonminimum phase properties so that walking performance and tracking accuracy are enhanced. This proposed model focuses on the angular momentum effects from arm and leg rotation to reduce modeling error to enhance walking performance. Based on pole-zero cancelation using series approximation method, it can overcome the sudden change of the natural zero-moment point reference due to the frequency characteristics in the nonminimum phase control system. The humanoid walking pattern generator is verified and demonstrated using a humanoid robot developed in our laboratory based on the proposed model.

Parameterized gait pattern generator based on linear inverted pendulum model with natural ZMP references

AbstractThis paper presents a parameterized gait generator based on linear inverted pendulum model (LIPM) theory, which allows users to generate a natural gait pattern with desired step sizes. Five types of zero moment point (ZMP) components are proposed for formulating a natural ZMP reference, where ZMP moves continuously during single support phases instead of staying at a fixed point in the sagittal and lateral plane. The corresponding center of mass (CoM) trajectories for these components are derived by LIPM theory. To generate a parameterized gait pattern with user-defined parameters, a gait planning algorithm is proposed, which determines related coefficients and boundary conditions of the CoM trajectory for each step. The proposed parameterized gait generator also provides a concept for users to generate gait patterns with self-defined ZMP references by using different components. Finally, the feasibility of the proposed method is validated by the experimental results with a teen-sized humanoid robot, David, which won first place in the sprint event at the 20th Federation of International Robot-soccer Association (FIRA) RoboWorld Cup.

Learning and Generalization of Compensative Zero-Moment Point Trajectory for Biped Walking

This paper presents an online learning framework for improving the robustness of zero-moment point (ZMP)-based biped walking controllers. The key idea is to learn a feedforward compensative ZMP (CZMP) trajectory from measured ZMP errors during repetitive walking motions by applying iterative learning control theory. The learned CZMP trajectory adjusts the reference ZMP and reduces the effect of unmodeled dynamics at the pattern-generation stage. From individual learned CZMP trajectories of typical walking parameters, we can build up a CZMP database. This database can be used for generating an initial CZMP whenever a new walking pattern is executed. A prediction from the database is done by $k$ -nearest neighbor regression based on the Mahalonobis distance. Compared with state-of-the-art model-based methods, the proposed learning approach is model free and allows online adaptation to constant unknown disturbances. Enhanced walking robustness can be observed from reduced average ZMP error and more robust reaction against external disturbances on the DLR humanoid robot TORO.

Dynamic and Reactive Walking for Humanoid Robots Based on Foot Placement Control

This paper presents a novel online walking control that replans the gait pattern based on our proposed foot placement control using the actual center of mass (COM) state feedback. The analytic solution of foot placement is formulated based on the linear inverted pendulum model (LIPM) to recover the walking velocity and to reject external disturbances. The foot placement control predicts where and when to place the foothold in order to modulate the gait given the desired gait parameters. The zero moment point (ZMP) references and foot trajectories are replanned online according to the updated foothold prediction. Hence, only desired gait parameters are required instead of predefined or fixed gait patterns. Given the new ZMP references, the extended prediction self-adaptive control (EPSAC) approach to model predictive control (MPC) is used to minimize the ZMP response errors considering the acceleration constraints. Furthermore, to ensure smooth gait transitions, the conditions for the gait initiation and termination are also presented. The effectiveness of the presented gait control is validated by extensive disturbance rejection studies ranging from single mass simulation to a full body humanoid robot COMAN in a physics based simulator. The versatility is demonstrated by the control of reactive gaits as well as reactive stepping from standing posture. We present the data of the applied disturbances, the prediction of sagittal/lateral foot placements, the replanning of the foot/ZMP trajectories, and the COM responses.

Biped Walking Pattern Generation Using an Analytic Method for a Unit Step With a Stationary Time Interval Between Steps

In this paper, an analytic method for generating a unit step pattern using a center of mass (COM) constraint is presented. Using a unit step pattern with the COM position constrained in the supporting polygon at the end of each step, a robot can have a stationary time interval between steps to control and stabilize its posture. The walking pattern is based on the linear inverted pendulum model. Assuming radical zero moment point (ZMP) trajectories, a simple solution form of the COM trajectories is formulated. Three types of case studies are presented for an analytic solution, which is based on different constraints on ZMP references, COM, and time differences. The unknown parameters of the COM trajectories in the solution form are formulated for the different cases. One of the cases was tested for long-stride walking with the DRC-HUBO robot developed at the Korea Advanced Institute of Science and Technology for the Defense Advanced Research Project Agency Robotics Challenge. Experiments of long-stride walking on bricks were successfully performed using the unit step pattern with several previous controllers of HUBO.

DEVELOPMENT AND WALKING CONTROL OF EMOTIONAL HUMANOID ROBOT, KIBO

This paper deals with the mechanical design, system integration, and dynamic walking algorithm of KIBO, an emotional biped humanoid robot that has a facial expression mechanism and various human-interactive devices. To emphasize the aesthetic features and marketability of KIBO, the mechanical design was performed after the exterior design stage to conform to all requirements, particularly constraints imposed by the external appearance and human-like link dimensions. For flexible biped walking, a walking pattern generator with variable walking parameters was developed. The walking pattern generator generates both a walking pattern and a corresponding reference zero-moment point (ZMP) pattern simultaneously. For stable biped walking, a walking control strategy using the ZMP and inertial sensor data was developed. In the strategy, we newly proposed a dual ZMP control approach and a posture control approach using an equivalent body inclination, which is calculated from the ZMP and inertial sensor data for robust walking on non-level ground. Finally, the hardware, software architecture, and dynamic walking performance of KIBO were verified through several walking experiments.

목표 ZMP 궤적 기반 휴머노이드 로봇 이족보행의 최적 관절궤적 생성

Humanoid robot is the most intimate robot platform suitable for human interaction and services. Biped walking is its basic locomotion method, which is performed with combination of joint actuator's rotations in the lower extremity. The present work employs humanoid robot simulator and numerical optimization method to generate optimal joint trajectories for biped walking. The simulator is developed with Matlab based on the robot structure constructed with the Denavit-Hartenberg (DH) convention. Particle swarm optimization method minimizes the cost function for biped walking associated with performance index such as altitude trajectory of clearance foot and stability index concerning zero moment point (ZMP) trajectory. In this paper, instead of checking whether ZMP's position is inside the stable region or not, reference ZMP trajectory is approximately configured with feature points by which piece-wise linear trajectory can be drawn, and difference of reference ZMP and actual one at each sampling time is added to the cost function. The optimized joint trajectories realize three phases of stable gait including initial, periodic, and final steps. For validation of the proposed approach, a small-sized humanoid robot named DARwIn-OP is commanded to walk with the optimized joint trajectories, and the walking result is successful.

A Real-Time Center of Gravity Trajectory Generation of a Biped Humanoid under Variable Reference ZMP Trajectory

This paper describes a method for generating a center of gravity trajectory of a biped humanoid robot under variable reference ZMP trajectory. A simple inverted pendulum model (SIPM) is used to calculate a center of gravity (CoG) trajectory from a reference zero moment point (ZMP) trajectory with an analytic form, which is based on the Fourier series. Fundamentally, we used a time segmentation based approach. For each segment, we defined its duration and boundary conditions, which are the key parameters of ZMP trajectory design. After designing the ZMP trajectory in each segment, we can automatically calculate the CoG trajectory by matching the boundary conditions and by calculating the coefficients between the time segments. The reference ZMP trajectory can be changed by updating the boundary conditions during walking. We successfully verified the proposed method through full-body dynamic simulations with variable step length.

Dynamic Balance Control for Biped Robot Walking Using Sensor Fusion, Kalman Filter, and Fuzzy Logic

The walking ability is a fundamental research for biped robots, and there always needs to be a criterion used for judging the walking stability. The zero moment point (ZMP) criterion is one of the useful standards for measuring biped robot walking. Here, a new ZMP trajectory model with adjustable parameters is proposed to modulate the ZMP trajectory both in sagittal and lateral planes and make the ZMP trajectory more flexible. A dynamic balance control (DBC), which includes Kalman filter (KF) and the fuzzy motion controller (FMC), is also designed to keep the body balance and make the biped walking following the desired ZMP reference. In addition, KF is utilized to estimate the system states and reduce the effect caused by noise. Using sensor fusion technique, ZMP error and trunk inclination measured by the force sensor and accelerometer are served as the inputs for FMC, which is presented to correct each joint of the biped robot dynamically. When a biped robot walks under different ground conditions, the coordination of the designed ZMP trajectory and proposed DBC can achieve a successful biped walking. Finally, there are several experiments presented to demonstrate the feasibility and effectiveness of the proposed control scheme.

Motion Planning for Humanoid Robots in Complex Environments Based on Stance Sequence and ZMP Reference

It is a great challenge to plan motion for humanoid robots in complex environments especially when the terrain is cluttered and discrete. To address this problem, a novel method is proposed in this paper by planning the gait according to the stance sequence and ZMP (Zero Moment Point) reference. It consists of two components: an adaptive footstep planner and a walking pattern generator. The adaptive footstep planner can generate the stance path according to the walking rules and adjust the orientation of body relevantly. As the footstep locations are determined, Linear Inverted Pendulum Model (LIPM) is used to generate the walking pattern with a moving ZMP reference. As demonstrated in experiments on the humanoid robot HOAP-2, our method can successfully plan footstep trajectories as well as generate the stable and natural-looking gait in typical cluttered and discrete environments.

Robust Control of Bipedal Humanoid (TPinokio)

A stable walking motion requires effective gait balancing and robust posture correction algorithms. However, to develop and implement such intelligent motion algorithms remain a challenging task for researchers. In order to minimize the modeling errors and disturbances, this paper presents an alternative approach in generating a stable Centre-of-Mass (CoM) trajectory by applying augmented model predictive control. The propose approach is to apply Augmented Model Predictive Control (AMPC) algorithm with on-line time shift and look ahead to process future data to optimize a control signal by minimizing a cost function so that the system is able to track the reference Zero Moment Point (ZMP) as close as possible, and at the same time to limit the motion jerk in order to improve the robot walking stability.

Walking Pattern Generation for a Biped Walking Robot Using Convolution Sum

This paper describes a novel walking pattern generation method for a biped humanoid robot using a convolution sum. For a biped walking model, a single mass inverted pendulum model is generally used and the zero moment point (ZMP) equation is described by a decoupled linear differential equation. As a walking pattern generation method for the robot model, a novel method using a convolution sum is proposed in this paper. From the viewpoint of the linear system response, walking pattern generation can be regarded as a convolution of an arbitrary reference ZMP and the walking pattern for an impulse reference ZMP. For the calculation of convolution, the walking pattern for an impulse reference ZMP is first derived from the analytic walking pattern for a step reference ZMP. The convolution sum is then derived in two recursive forms, which can be applied online and offline, respectively. The proposed algorithm requires low computation power, since the walking pattern equation is composed of a recursive form. As the algorithm is expressed in analytic form, it is not necessary to solve optimization problems or calculate the fast Fourier transform, contrary to previous approaches. A computer simulation of walking demonstrates that the proposed algorithm yields excellent accuracy compared to the preview control method — one of the most highly regarded walking pattern generation methods. In addition, the application on the multi-point mass model is shown with the computer simulation.

Disturbance Compensation Control for a Biped Vehicle

This paper describes an avoidance behavior against unknown forces acting from outside the system on a biped vehicle. External forces that act on a biped vehicle can be divided into two categories: disturbances from the rider of the vehicle and disturbances from outside the system. Disturbances from the rider are measured by a force sensor placed between the rider's seat and the pelvis, while disturbances from outside the system are estimated from zero moment point (ZMP) errors. To guarantee walking stability, the waist position is adjusted to match the measured ZMP to the reference ZMP and the position of the landing foot is adjusted such that the waist trajectory does not diverge. On implementing the developed method on the human-carrying biped robot in an experimental setup, the robot could realize a stable walk even while subjected to unknown external forces in the environment. On applying a pushing force to the walking robot, the robot moved in the pushed direction and away from the source of the externally generated forces. On pushing the robot that was walking forward, the robot stopped moving forward and avoided getting closer to the source of the externally generated forces. We confirmed the effectiveness of the proposed control through these experiments.

From human motion capture to humanoid locomotion imitation Application to the robots HRP-2 and HOAP-3

SUMMARYThis paper presents a method to generate humanoid gaits from a human locomotion pattern recorded by a motion capture system. Thirty seven reflective markers were fixed on the human subject skin in order to get the subject whole body motion. To reproduce the human gait, especially the toes and heel contacts, the front and back edges of the robot's feet are used as support at the start and the end of the double support phase. The balance of the robot is respected using the zero moment point (ZMP) criterion and confirmed by the simulation software OPENHRP (General Robotics, Inc®). First, the feet trajectory as well as the ZMP reference trajectory are defined from the motion of the robot controlled as a marionette with the measured human joint angles. Then a specific inverse kinematic (IK) algorithm is proposed to find the humanoid robot's joint trajectories respecting the constraints of balance, floor contacts, and joint limits. The studied motion presented in this paper is a human walking trajectory containing a start, a movement in a straight line, a stop, and a quarter turn. The method was developed to be easily used for human-like robots of different sizes, masses, and structures and has been tested on the robot HRP-2 (AIST, Kawada Industries, Inc®) and on the small-sized humanoid robot HOAP-3 (Fujitsu Automation Ltd®).

Balancing Control of AIT Leg Exoskeleton Using ZMP based FLC

This paper is focused on the use of Zero Moment Point (ZMP) concept for balancing control of the Asian Institute of Technology Leg EXoskeleton-I (ALEX-I). ALEX-I has been developed to assist patients who suffer from paraplegia or immobility due to the loss of power on lower limbs. The balanced posture set-points (joint trajectories) under ZMP criterion are generated offline. The ZMP based set points are provided as the desired postures to ALEX-I. Fuzzy Logic Controller (FLC) determines the modified set points based on postures balancing sensed by loadcells. Ground Contact Point (GCP) is used to find the “ZMP-like in real time”. GCP data is obtained by placing 4 loadcells forming a force plate on each foot of ALEX-I. This GCP data is then compared with the reference ZMP. Uncertainties of the model parameters, backlash, and joint tolerance are considered as disturbance. The differences of ZMP and GCP on x-z plane are used as the inputs to the FLC. The 4 outputs from FLC are the compensated angles of left and right ankles joints in roll and pitch axes that make the actual ZMP locate in the convex hull of the supporting area.