Zero Moment Point Research Articles

ZMP Tracking Control of an Android Robot Leg on Slope-Changing Ground Using Disturbance Observer and Dual Plant Models

This paper describes a novel zero moment point (ZMP) tracking control strategy using a disturbance observer (DOB) in the presence of ground slope change for balance control of an android robot. With regard to conventional ZMP controls, many researchers have studied ZMP tracking control strategies using an inverted pendulum model on flat level ground, and they have solved a slow response problem of nonminimum phase systems by using suitable feedforward motions called walking patterns. However, the conventional methods lead to ZMP offset errors in the presence of ground slope change; it is hence necessary to quickly eliminate the ZMP offset errors to realize robust balance control. In this paper, we rapidly eliminate the ZMP offset errors through a DOB using a model inversion for robust balance control in the presence of ground slope change. In particular, a dynamic model that uses the projected center of mass (CoM) position on the ground is additionally used as an output to solve a problem that generates an unstable pole during model inversion. Finally, the proposed control strategy is verified through MATLAB simulations and experiments using a real android leg.

Optimal trajectory planning for increased stability of mobile flexible manipulators undergoing large deflection

In this article, a path planning algorithm for nonholonomic mobile flexible manipulators is presented that computes the robot trajectory by maximizing pose stability measures. Zero moment point criterion is used as a performance index for the system stability, which regards the mass moment of inertia of the mobile base. In dynamic analysis, an efficient model is employed to describe the treatment of flexible structure, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. The optimization strategy is based on the indirect solution of the open-loop optimal control problem. Necessary optimality conditions in the form of Pontryagin’s minimum principle are established, which leads to a standard form of a two-point boundary value problem. A new objective function is proposed to improve stability for mobile flexible robot. In order to initially check the validity of the dynamic equations, the proposed model has been implemented and tested on a fixed-base flexible arm undergoing large deflection. A simulation study has been carried out to investigate further the validity and effectiveness of the mobile flexible manipulators on finding the optimal path between two points with stability consideration. The results clearly show the effect of flexibility and tip over stability on the mobile manipulators.

ZMP Based Motion Stability Analysis of a Wheeled Humanoid Robot with Bending Torso

Motion stability is the most important issue to be considered when designing a wheeled humanoid robot with bending torso, as it’s easy to capsize because of its high center of gravity. With ZMP (Zero Moment Point) method the motion stability of a wheeled humanoid robot with bending torso was analyzed. At first, the chain rule was used to model the kinematics of the wheeled humanoid robot, and then the process of calculating the ZMP of the robot was presented. With MATLAB the motion stability of the humanoid robot in three typical conditions is simulated and analyzed, and the simulation results were used to optimize some parameters of a wheeled humanoid robot we are designing.

A novel rhombohedron rolling mechanism

In this paper, a novel multi-loop-coupled mechanism of three degrees of freedom (DOFs) with a rhombohedron shape is presented. The mechanism is composed of six identical mechanisms, which are formed by an octothorp-shaped structure. Based on the corresponding relationship between the mechanism and rhombohedron, the DOF is analyzed by the assembly process of face-units. By kinematic analysis, the center of mass (CM) of the mechanism can be derived. The rolling capability is analyzed by using the Zero-Moment-Point (ZMP) theory. By deforming its shape, this mechanism can realize the gait of going straight, direction adjustment, and climbing slope. Since the structure is symmetrical in nature, its locomotion capability is identical when any of its six faces touches the ground. Two additional postures (surveillance and hiding postures) of the mechanism can increase its surviving ability in the wild. A prototype is manufactured, and rolling experiments are carried out to verify the functions of the mechanism.

Research on Optimization Walking Robot Gait Planning Based on Human Bionics

This paper had analyzed the degree of freedom robot and then simulates the robots and humans similar to the lower limbs, to establish sports model of the robot. The analysis robot has 10 degrees of freedom, in which one leg has five degrees of freedom, namely, a degree of freedom of the knee, hip ankle is assigned two degrees of freedom. Depending on the needs of the kinematic model is divided into forward motion model and lateral movement model. Link by representatives’ form various parts of the robot to simplify the model. Through the establishment of the geometry and the Cartesian coordinate system, analyze the relationship between the joint angle and joint use mathematical functions to represent the correspondence between swing foot and between each joint. By Zero Moment Point (ZMP) stability analysis, to ensure the stability of humanoid robot, obtained center of gravity and zero moment point formula, thus proving humanoid robot walking stability. Using of spline interpolation and polynomial fitting method for solving the hip, the equations of motion of the ankle joint, the knee. By setting the parameters, the use of MATLAB simulation software to obtain motion curve of each joint and humanoid robot designed to simulate obtain humanoid robot gait simulation video.

Stable walking in a biped robot with force-feedback-induced center-of-mass regulation

We report on a novel ZMP feedback control strategy which regulates the center of mass trajectory of a robot based on the measured zero-moment point (ZMP). Based on the trajectory generated by preview control and the linear inverted pendulum model, the controller has an extra model-based control input which is driven by the center-of-mass (CoM) acceleration, which provides sensitive response to disturbance. Thus, the robot motion is balanced between the long-term nominal CoM/ZMP trajectory tracking and the short-term trajectory modification for disturbance rejection. In addition, the controller has two layers of adjustment strategy. When the disturbance is small, the robot is regulated by merely adjusting its CoM trajectory. In contrast, when the disturbance is large, the foot position of the robot is simultaneously changed to provide a more effective response with characteristics of larger intrinsic stability. The performance of the control strategy is also experimentally evaluated using a child-size biped robot.

Force-based stability margin for multi-legged robots

Stability analysis of multi-legged robots is necessary for control especially under dynamic situations. This paper presents the Foot Force Stability Margin, a force-based stability margin that utilizes measured contact normal foot forces as the stability metric, simplifying data and computational requirements. The Foot Force Stability Margin assumes tumbling instability and sufficient friction to prevent slippage. A modified extension of the Foot Force Stability Margin is provided to enhance stability sensitivity to desired robot characteristics. The Foot Force Stability Margin and its modified variant were validated through numerical and physical experiments. The numerical simulations compare the Foot Force Stability Margin with its modified variant, the Force Angle Stability Margin, and the Zero Moment Point. The physical experiments were conducted using a Lynxmotion hexapod robot retrofitted with a Gumstix and force sensitive resistors. The experimental results confirm that the Foot Force Stability Margin and its modified variant are accurate, simple with regard to computational cost, sensitive to robot characteristics, and applicable to flat and uneven terrain, making it practical for use within on-line controllers.

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.

A generic walking pattern generation method for humanoid robot walking on the slopes

Purpose Walking on inclined ground is an important ability for humanoid robots. In general, conventional strategies for walking on slopes lack technical analysis in, first, the waist posture with respect to actual robot and, second, the landing impact, which weakens the walking stability. The purpose of this paper is to propose a generic method for walking pattern generation considering these issues with the aim of enabling humanoid robot to walk dynamically on a slope. Design/methodology/approach First, a virtual ground method (VGM) is proposed to give a continuous and intuitive zero-moment point (ZMP) on slopes. Then, the dynamic motion equations are derived based on 2D and 3D models, respectively, by using VGM. Furthermore, the waist posture with respect to the actual robot is analyzed. Finally, a reformative linear inverted pendulum (LIP) named the asymmetric linear inverted pendulum (ALIP) is proposed to achieve stable and dynamical walking in any direction on a slope with lower landing impact. Findings Simulations and experiments are carried out using the DRC-XT humanoid robot platform with the aim of verifying the validity and feasibility of these new methods. ALIP with consideration of waist posture is practical in extending the ability of walking on slopes for humanoid robots. Originality/value A generic method called ALIP for humanoid robots walking on slopes is proposed. ALIP is based on LIP and several changes, including model analysis, motion equations and ZMP functions, are discussed.

A measure system of zero moment point using wearable inertial sensors

Keeping balance is the premise of human walking. ZMP(zero moment point) is a point where total torque achieves balance. It is an important evaluation parameter of balance ability in walking, since it can be used to better reflect the dynamic balance during walking. ZMP can be used in many applications, such as medical rehabilitation, disease diagnosis, treatment and etc. In this paper, wearable inertial sensors system based on MEMS is used to measure ZMP (zero moment point) during walking, which is cheap, convenient, and free from the restriction of lab. Our wearable ZMP measurement system consists of inertial measurement subsystem and PC real-time monitoring station. Inertial measurement subsystem includes 9-axis inertial sensing nodes, the body communication network and the central node. Inertial sensing nodes are mounted on different parts of the body to collect body posture information in real-time, and then the best estimation of current posture are obtained by Kalman filter. The data from sensors is aggregated to the central node through the CAN bus, and then ZMP is calculated. Finally, it can be showed in the PC monitoring station. Experiments prove the system can achieve real-time ZMP detection during walking.

Dynamic Simulation of Modifiable Bipedal Walking on Uneven Terrain with Unknown Height

To achieve bipedal walking in real human environments, a bipedal robot should be capable of modifiable walking both on uneven terrain with different heights and on flat terrain. In this paper, a novel walking pattern generator based on a 3-D linear inverted pendulum model (LIPM) is proposed to achieve this objective. By adopting a zero moment point (ZMP) variation scheme in real time, it is possible to change the center-of-mass (COM) position and the velocity of the 3-D LIPM throughout the single support phase. Consequently, the proposed method offers the ability to generate a modifiable pattern for walking on uneven terrain without the necessity for any extra footsteps to adjust the COM motion. In addition, a control strategy for bipedal walking on uneven terrain with unknown height is developed. The torques and ground reaction force are measured through force-sensing resisters (FSRs) on each foot and the foot of the robot is modeled as three virtual spring-damper models for the disturbance compensation. The methods for generating the foot and vertical COM of 3-D LIPM trajectories are proposed to achieve modifiable bipedal walking on uneven terrain without any information regarding the height of the terrain. The effectiveness of the proposed method is confirmed through dynamic simulations.

Dynamic Simulation of Modifiable Walking Pattern Generation to Handle Infeasible Navigational Commands for Humanoid Robots

The modifiable walking pattern generation (MWPG) algorithm can handle dynamic walking commands by changing the walking period, step length, and direction independently. When an infeasible command is given, the algorithm changes the command to a feasible one. After the feasibility of the navigational command is checked, it is translated into the desired center of mass (CM) state. To achieve the desired CM state, a reference CM trajectory is generated using predefined zero moment point (ZMP) functions. Based on the proposed algorithm, various complex walking patterns were generated, including backward and sideways walking. The effectiveness of the patterns was verified in dynamic simulations using the Webots simulator.

휴머노이드 로봇의 동보행 안정도에 관한 연구

In this paper, we deal with the dynamic walking of a humanoid robot. In our method, the inverted pendulum model is used as a dynamic model for a humanoid robot in which the Zero Moment Point (ZMP) and COG constraints of the robot are analyzed by considering the motion of the robot as that of an inverted pendulum. The motion of a humanoid robot should be generated by considering the dynamics of the robot, which commonly requires a large amount of computation. If a robot walks from one position to another while keeping the ZMP in the stable region, then the robot remains dynamically stable. The linear inverted pendulum model regards the whole robot as a point mass. It is simple, and relatively less computation is needed; however, it cannot model the whole dynamics of a humanoid robot. We propose a method for modeling a humanoid robot as an inverted pendulum system having 14 point masses. We also show that the dynamic stability of a humanoid robot can be determined more precisely by our method.

Energy-efficient Bio-inspired Gait Planning and Control for Biped Robot Based on Human Locomotion Analysis

In this paper an experiment of human locomotion was carried out using a motion capture system to extract the human gait features. The modifiable key gait parameters affecting the dominant performance of biped robot walking were obtained from the extracted human gait features. Based on the modifiable key gait parameters and the Allowable Zero Moment Point (ZMP) Variation Region (AZR), we proposed an effective Bio-inspired Gait Planning (BGP) and control scheme for biped robot towards a given travel distance D. First, we construct an on-line Bio-inspired Gait Synthesis algorithm (BGSN) to generate a complete walking gait motion using the modifiable key gait parameters. Second, a Bio-inspired Gait Parameters Optimization algorithm (BGPO) is established to minimize the energy consumption of all actuators and guarantee biped robot walking with certain walking stability margin. Third, the necessary controllers for biped robot were introduced in briefly. Simulation and experiment results demonstrated the effectiveness of the proposed method, and the gait control system was implemented on DRC-XT humanoid robot.

Robot's Weight Distribution Influence on Angular Momentum Produced by a Standing Vertical Jump

Vertical jump movement generators usually apply the ZMP (Zero Moment Point) and/or angular momentum equations as restrictions when planning the movements of the robot joints, which can cause undesired oscillations in the torso, compromise accuracy of the jump and/or complicate the search for online solutions. This paper presents a strategy to solve the problem related to angular momentum generated by the movement of the vertical jump, performed by a robot with 3 DoF (Degree of Freedom) and 4-links. Mentioned links correspond to the following parts of the human body: foot, leg, thigh and torso, and its weighted mass distribution and CoM (Center of Mass) position define an angular momentum profile within a specified range of speed values for the jump. In order for that to happen, simulations were performed using MATLAB, considering homogeneous and weighted distributions of total mass between the links of a robot and applying a third-degree polynomial function for the vertical acceleration curve of the CoM. The results suggest that based on the strategy proposed in this paper, it is possible to perform a vertical jump with dynamically stable movements and with angular moment near zero, and, at the same time, minimizing the problems presented in some prior methods.

Modeling, Simulation and Control of the Walking of Biped Robotic Devices, Part II: Rectilinear Walking

This is the second part of a three-part paper. It extends to the free walking results of a previous work on postural equilibrium of a lower limb exoskeleton for rehabilitation exercises. A classical approach has been adopted to design gait (zero moment point (ZMP), linearized inverted pendulum theory, inverse kinematics obtained through the pseudo-inverse of Jacobian matrices). While several ideas exploited here can be found in other papers of the literature, e.g., whole-body coordination, our contribution is the simplicity of the whole control approach that originates logically from a common root. (1) The approximation of the unilateral foot/feet-ground contacts with non-holonomic constraints leads naturally to a modeling and control design that implements a two-phase switching system. The approach is facilitated by Kane’s method and tools as described in Part I. (2) The Jacobian matrix is used to transfer from the Cartesian to the joint space a greater number of variables for redundancy than the degrees of freedom (DOF). We call it the extended Jacobian matrix. Redundancy and the prioritization of postural tasks is approached with weighted least squares. The singularity of the kinematics when knees are fully extended is solved very simply by fake knee joint velocities. (3) Compliance with the contact and accommodation of the swing foot on an uneven ground, when switching from single to double stance, and the transfer of weight from one foot to the other in double stance are approached by exploiting force/torque expressions returned from the constraints. (4) In the center of gravity (COG)/ZMP loop for equilibrium, an extended estimator, based on the linearized inverted pendulum, is adopted to cope with external force disturbances and unmodeled dynamics. Part II treats rectilinear walking, while Part III discusses turning while walking.

Modeling, Simulation and Control of the Walking of Biped Robotic Devices—Part III: Turning while Walking

In part II of this group of papers, the control of the gait of a biped robot during rectilinear walk was considered. The modeling approach and simulation, using Kane’s method with implementation leveraged by Autolev, a symbolic computational environment that is complementary, was discussed in part I. Performing turns during the walk is technically more complex than the rectilinear case and deserves further investigation. The problem is solved in the present part III as an extension of part II. The robot executes a rectilinear walk on a local reference frame whose progression axis is always tangent, and its origin performs the involute of the path curve. The curve is defined by its curvature (osculating circle) and center of curvature (evolute) along the path. Radius of curvature and center can change continuously (in practice at every sampling time). For postural equilibrium, Center of Gravity and Zero Moment Point (COG/ZMP) follow the same preview reference proposed for rectilinear walk (c o g R e f x ( t ) , c o g ˙ R e f x ( t ), c o g R e f y ( t ) , c o g ˙ R e f y ( t )). The effect of the turn on the sagittal plane is negligible and is ignored, while on the frontal plane it is accounted for by an offset on COG reference to compensate for the centrifugal acceleration. The body trunk and local frame rotation, and the generation of the references on this moving frame of the free foot trajectory during the swing deserve attention.

Heel-Contact Toe-Off Walking Pattern Generator Based on the Linear Inverted Pendulum

We propose a novel heel-contact toe-off walking pattern generator for a biped humanoid robot. It is divided in two stages: a simple model stage where a Linear Inverted Pendulum (LIP) based heel-contact toe-off walking model based on the so-called functional rockers of the foot (heel, ankle and forefoot rockers) is used to calculate step positions and timings, and the Center of Mass (CoM) trajectory taking step lengths as inputs, and a multibody dynamics model stage, where the final pattern to implement on the humanoid robot is obtained from the output of the first simple model stage. The final pattern comprises the Zero Moment Point (ZMP) reference, the joint angle references and the end effector references. The generated patterns were implemented on our robotic platform, WABIAN-2R to evaluate the generated walking patterns.

Whole-Body Balancing Walk Controller for Position Controlled Humanoid Robots

Bipedal humanoid robots are intrinsically unstable against unforeseen perturbations. Conventional zero moment point (ZMP)-based locomotion algorithms can reject perturbations by incorporating sensory feedback, but they are less effective than the dynamic full body behaviors humans exhibit when pushed. Recently, a number of biomechanically motivated push recovery behaviors have been proposed that can handle larger perturbations. However, these methods are based upon simplified and transparent dynamics of the robot, which makes it suboptimal to implement on common humanoid robots with local position-based controllers. To address this issue, we propose a hierarchical control architecture. Three low-level push recovery controllers are implemented for position controlled humanoid robots that replicate human recovery behaviors. These low-level controllers are integrated with a ZMP-based walk controller that is capable of generating reactive step motions. The high-level controller constructs empirical decision boundaries to choose the appropriate behavior based upon trajectory information gathered during experimental trials. Our approach is evaluated in physically realistic simulations and on a commercially available small humanoid robot.