Natural ZMP Trajectories for Biped Robot Reference Generation

The hardware compatibility of legged locomotion is often illustrated by Zero Moment Point (ZMP) that has been extensively studied for decades. One of the most popular models for computing the ZMP is the linear inverted pendulum (LIP) model that expresses ZMP as a linear function of the center of mass(COM) and its acceleration. In the real world, however, it may not accurately predict the true ZMP of hardware due to various reasons such as unmodeled dynamics and differences between simulation model and hardware. In this paper, we aim to improve the theoretical ZMP model by learning the real hardware dynamics from experimental data. We first optimize the motion plan using the theoretical ZMP model and collect COP data by executing the motion on a force plate. We then train a new ZMP model that maps the motion plan variable to the actual ZMP and use the learned model for finding a new hardware-compatible motion plan. Through various locomotion tasks of a quadruped, we demonstrate that motions planned for the learned ZMP model are compatible on hardware when those for the theoretical ZMP model are not. Furthermore, experiments using ZMP models with different complexities reveal that overly complex models may suffer from over-fitting even though they can potentially represent more complex, unmodeled dynamics.

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.

Linear Inverted Pendulum model (LIPM) is a classical theory of biped robot gait planning. However, the fixed Zero Moment Point (ZMP) of each step does not conform to the laws of human motion. In this paper, we proposed a VSP-based gait planning algorithm to make the ZMP trajectory comply the law of Heel to Toe and smooth the velocity of the center of mass (CoM). We found it remarkable that combining Virtual Supporting Point (VSP) with classical LIPM increases gait stability. The simulation and hardware experimental results on THU-Walker Platform strongly verified our algorithm.

Most humanoid robots walk in an unhuman-like way with bent knees due to the use of the simplified Linear Inverted Pendulum Model (LIPM) which constrains the Center of Mass (CoM) in a horizontal plane. Therefore it results in high knee joint torque and extra energy consumption. To address this issue, we propose a simple yet efficient control strategy to realize straight leg walking. First, theoretical analyses of simplified models provide insight into Zero Moment Point (ZMP) deviations during straight knee walking. Based on the finding that the deviation is limited comparing to the support polygon, we decide to keep using the LIPM for high-level planning, but let the robot perform straight leg walking automatically via the optimization-based low-level controller. By setting the desired CoM height slightly over the robot's reachable height, the low-level controller will attempt to straighten the robot's leg to reach this vertical reference, in the meanwhile, also satisfy the constraints (i.e. dynamic feasibility, friction cone, torque limits). The simulation results of the humanoid robot WALK-MAN demonstrate the feasibility of proposed control strategy with relatively high energy efficiency. A typical butterfly shape of CoM trajectory was also observed in the frontal plane which is common in human walking.

Human biped locomotion is an ultimate style of biological movement that is a highly evolved function. Biped locomotion by robots is a dream to be attained by the most highly evolved or integrated technology, and research on this has a history of over 30 years. Many methods of generating gaits have been proposed. There has been a tendency to reduce the complicated dynamics of a walking robot to a simple inverted pendulum (Hemami et al., 1973), and to control its motion according to pre-designed time-dependent trajectories while guaranteeing zero moment point (ZMP) conditions (Vukobratovi & Stepanenko, 1972). Although such approaches have successfully been applied to practical applications and nowadays successful biped-himanoids are developed by them, problems on gait performances still remain. Several advanced approaches on the other hand have taken the robot's dynamics into account for generating gaits based on natural dynamics. Miura and Shimoyama studied dynamic bipedal walking without ankle-joint actuation (Miura & Shimoyama, 1984) and they developed robots on stilts whose foot contact occurred at a point. Sano and Furusho accomplished natural dynamic biped walking based on angular momentum using ankle-joint actuation (Sano & Furusho, 1990). Kajita proposed a method of control based on a linear inverted pendulum model with a potential-energy-conserving orbit (Kajita et al., 1992). These approaches utilized the robot’s own dynamics effectively but did not investigate the energy-efficiency by introducing performance indices. It was unclear whether or not efficient gaits were generated. McGeer's passive dynamic walking (PDW) (McGeer, 1990) has provided clues to solve these problems. Passive-dynamic walkers can walk without any actuation on a gentle slope, and they provide an optimal solution to the problem of generating a natural and energy-efficient gait. The objective most expected to be met by PDW is to attain natural, high-speed energyefficient dynamic bipedal walking on level ground like humans do. However, we need to supply power-input to the robot by driving its joint-actuators to continue stable walking on level ground, and certain methods of supplying power must be introduced. Ankle-joint torque is mathematically very important for effectively propelling the robot's center of mass (CoM) in the walking direction, and it is thus required relatively more often than other joint torques. However, to exert ankle-joint torque on a passive-dynamic walker, we need to add feet and this creates the ZMP constraint problem. We clarified that there is a trade-off between optimal gait and ZMP conditions through parametric studies, and O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

This paper focuses on real-time walking pattern generation for humanoid robots with linear inverted pendulum model (LIPM). In general, there are many issues in generating proper walking patterns of center of mass and zero moment point (ZMP) with the LIPM since the LIPM has two drawbacks such as instability and non-minimum phase property. For resolving these difficulties, the paper proposes a new real-time approach by combining a feedback and a feedforward controller. The feedback controller employs a pole placement method which shifts the poles of the LIPM in order to improve system stability. The feedforward controller utilizes advanced pole-zero cancelation by series approximation method (APZCSA) for reducing non-minimum phase property which occurs by an unstable zero and is not able to be dealt with by the feedback controller. In addition, the APZCSA improves the tracking error induced by finite series approximation. Using the two controllers, the proposed method makes the transfer function of overall walking pattern generation system approximately unity and consequently generates a stable walking pattern which follows a desired ZMP according to walking path. The efficiency of the proposed method is verified by walking pattern planning examples and experiments with the humanoid robot MAHRU-R.

This research presents a simple mapping methodology for gait biomechanics of a human being into joint angles of a 10 degrees of freedom (DOF) biped robot. The joint angles are mapped by considering the zero moment point (ZMP) criterion. The walking control of the robot is performed by an optimal state feedback controller. The walking trajectories are planned in the sagittal plane, and they are generated in compliance with the ZMP of the robot – keeping the robot within the support polygon – by dividing the control process in two stages: unique support and double support. A linear inverted pendulum model (LIPM) is used as an approximate single mass model of the robot during gait. Results of this research include simulation-based analysis and real-time implementation results, which show accurate robot movements with limited robustness under slippery platforms.

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.

A novel controller using Center of Gravity (COG) planning based on the Centroidal Moment Pivot (CMP) criterion with Ground Reaction Force (GRF) feedback is presented. High level motion planning of the robot is done by planning a reference CMP trajectory that lies within the support polygon. In order to ensure rotational stability, the controller regulates the distance between the Zero Moment Point (ZMP) and the reference CMP through COG manipulation. A reference COG trajectory is generated from the measured GRF and a simplified model of the rotational dynamics of the robot. The reference COG is then decomposed into reference joint velocities via kinematic resolution of the COG Jacobian. Planar simulations show that modifying the COG trajectory using the CMP criterion with GRF feedback increases the controller's robustness to external disturbances compared to a ZMP based controller. Additionally, the results show that the constant non-zero momentum generated by external disturbances can be regulated by minimizing the difference between the reference CMP and ZMP positions, without explicitly regulating the momentum about the COG.

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.

Walking motion planning based on Divergent Component of Motion (DCM) and Linear Inverted Pendulum Model (LIPM) is one of the alternatives that could be implemented to generate online humanoid robot gait trajectories. This algorithm requires different parameters to be adjusted. Herein, we developed a framework to attain optimal parameters to achieve a stable and energy-efficient trajectory for real robot’s gait. To find the optimal trajectory, four cost functions representing energy consumption, the sum of joints velocity and applied torque at each lower limb joint of the robot, and a cost function based on the Zero Moment Point (ZMP) stability criterion were considered. Genetic algorithm was employed in the framework to optimize each of these cost functions. Although the trajectory planning was done with the help of the simplified model, the values of each cost function were obtained by considering the full dynamics model and foot-ground contact model in Bullet physics engine simulator. The results of this optimization yield that walking with the most stability and walking in the most efficient way are in contrast with each other. Therefore, in another attempt, multi-objective optimization for ZMP and energy cost functions at three different speeds was performed. Finally, we compared the designed trajectory, which was generated using optimal parameters, with the simulation results in Choreonoid simulator.

In this paper, for the first time, experimental tests of complete offline walking gaits generated by the essential model are performed. This model does not make simplifications in the dynamics of the robot, and its main advantage is the definition of desired Zero Moment Point trajectories. The designed gaits are implemented in the NAO robot, where starting and stopping stages are also included. Simulations in MATLAB and Webots, and experiments with the real robot are shown. Also, important remarks about the implementation of walking trajectories in the NAO robot are included, such as dealing with the hip joint shared by both legs. A comparison between the linear inverted pendulum (LIP) model and the essential model is also addressed in the experiments. As expected, the robot fails following the offline gait generated by the LIP model, but it does not with the essential model. Moreover, in order to push the boundaries of the essential model, a complex gait is designed with a vertical motion of the center of mass and an abrupt movement of the arms. As shown in experiments, no external balance controller is required to perform this complex gait. Thus, the efficiency of the essential model to design stable open-loop complex gaits is verified.

In this paper, we solve the gait planning problem by using the Flexible LIP model, which has been shown to be more realistic w.r.t. the LIP for cost-effective or compliant biped robots for gait generation. We extend a stable inversion approach to obtain bounded Center of Mass (CoM) reference trajectories and show several advantages compared to preview control: avoidance of numerical integration, lower computation time, exact tracking of reference Zero Moment Point (ZMP) trajectories, and the ability to come to an immediate stop.

This paper proposes a stabilization method for dynamic walking of a bipedal robot with real-time optimization of capture point trajectories. We used the capture point trajectories to generate the control input, which is the desired zero moment point (ZMP) with a sliding-mode ZMP controller to follow the desired ZMP. This method enables the robot to implement various dynamic walking commands, such as forward stride, lateral stride, walking direction, single support time, and double support time. We also adopted enhanced dynamics with the three mass linear inverted pendulum model (3M-LIPM). First, the compensated ZMP is calculated by both walking commands and kinematic configuration of the robot in closed form. Then, the walking pattern is obtained by using initial and boundary conditions of the 3M-LIPM, which satisfies the walking commands. The capture point (CP) trajectory is optimized in real time to control the walking stability and a capture point tracking controller is used for tracking the optimized CP trajectory, which generates an optimal control input that is near the center of the support polygon. The performance of the proposed stabilization method was verified by a dynamics simulator, Webots, and comparison with the original capture point controller-based walking algorithm is presented.

This paper presents a trajectory generation approach for a 7-DOF biped robot on level ground. Simultaneously rotation of feet in double support phase is considered which leads to high-speed and more similar to human being walking. The zero moment point (ZMP) stability criterion is used to ensure the stability of the bipedal walking robot. Since ZMP trajectory in human walking does not stay fixed, it needs to be a straight line shaped forward ZMP trajectory to have a natural walk. A genetic algorithm based method is proposed to obtain key parameters in trajectory generation such that the ZMP follows a predefined trajectory while minimizing power consumption. Simulation results demonstrate the effectiveness of the proposed method.