Bipedal walking control based on Capture Point dynamics

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.

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.

A robot foot can slip when the horizontal shear force acting on the foot exceeds the frictional force between the foot and the ground. In the linear inverted-pendulum (LIP) model, the vertical height of the center of mass (COM) is kept constant, and the vertical force is always equal to the gravitational force at any walking speed. However, the horizontal force increases upon increasing the walking speed. This restriction on the vertical force in the LIP model can cause the robot foot to slip at fast walking speeds, as the horizontal force can exceed the frictional force, which is proportional to the vertical force. In this study, we present an optimization method to generate vertical COM motion to maintain the utilized coefficient of friction (uCOF) less than the available coefficient of friction between the foot and the ground, and to minimize the mechanical work of the COM. Vertical motions at various speeds are generated using the proposed optimization method. Subsequently, the generated COM motion patterns are used as reference trajectories of the COM in robot simulation. Optimization and simulation results demonstrate that the mechanical work and uCOF decrease because of the vertical motion. This study suggests a way to generate slip-safe and energy-efficient COM patterns, which, in turn, overcome the limitations of the LIP model by adding vertical COM motion.

This paper proposes an analysis of the effect of vertical motion of the center of mass (COM) during humanoid walking. The linear inverted pendulum (LIP) model is classically used to deal with humanoid balance during walking. The effects on energy consumption of the COM height remaining constant for humanoid robots, or varying like human beings are studied here. Two approaches are introduced for the comparison: the LIP which offers the great advantage of analytical solving (i.e., fast and easy calculations), and a numerical solving of the IP dynamics, which allows varying the height of the center of mass during walking. The results are compared using a sthenic criterion in a 3D dynamics simulation of the humanoid robot Romeo (Aldebaran Robotics Company) and show a consequent reduction of the robot torque solicitation when the COM oscillates vertically.

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.

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.

The robust dynamic walking control of humanoid robots is the foundation for their application in complex scenarios. Model Predictive Control (MPC) can predict the robot’s motion state over a future time horizon to obtain optimal control inputs and has achieved significant success in walking control of legged robots. However, the high computational cost makes it challenging to ensure real-time performance when dealing with complex dynamic systems. A novel dynamic walking control framework for humanoid robots is introduced in this paper by integrating an efficient footstep planner and Task Space Inverse Dynamics based Whole Body Controller (TSID-WBC). This framework aims to enhance the dynamic performance and robustness of the humanoid robot’s walking control. Firstly, design a bionic step time generator based on relevant research in biology. Then, the MPC controller, combined with the differential equation solution of the Linear Inverted Pendulum (LIP) dynamics model, is used to predict the optimal footsteps for the humanoid robot. Finally, TSID-WBC is employed to optimize joint torques, joint accelerations, and generalized contact forces based on predefined task priorities. The proposed framework avoids using MPC for predictive control of complex dynamic models. Instead, it innovatively integrates the bionic step time generator proposed in this paper with an MPC based on the differential solution of the LIP dynamics model, resulting in an efficient footstep planner. The simulation and experimental results of the humanoid robot Kuavo demonstrate that the proposed control framework achieves better tracking performance for desired walking speeds and responds quickly to external disturbances.

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.

This paper proposes a capture-point based reactive omnidirectional controller for bipedal locomotion. The proposed scheme, formulated within Model Predictive Control (MPC) framework, exploits concurrently the Center of Mass (CoM) and Capture Point (CP) dynamics. It allows the on-line generation of the CoM reference trajectory and the automatic generation of footstep positions and orientations in response to a given velocity to be tracked, or a disturbance to be rejected by the robot while accounting explicitly for different walking constraints. For instance, in order to cope with disturbance such as a push, the proposed controller not only adjusts the position of the Center of Pressure (CoP) within the support foot, but can also induce at least one step with appropriate length allowing thus to maintain the stability of the robot. Finally, the proposed algorithm is validated through simulations and actual experiments on the humanoid robot iCub.

From which states and with what controls can a biped avoid falling or reach a given target state? What is the most robust way to do these? So as to help with the design of walking robot controllers, and perhaps give insights into human walking, we address these questions using two simple 2-D models: the inverted pendulum (IP) and linear inverted pendulum (LIP). Each has one state variable at mid-stance, i.e., hip velocity, and two state-dependent controls at each step, i.e., push-off magnitude and step length (IP) and step time and length (LIP). Using practical targets and constraints, we compute all combinations of initial states and control actions for the next step, such that the robot can, with the best possible future controls, avoid falling for n steps or reach a target within n steps. All such combinations constitute regions in the combined space of states and controls. Farther from the boundaries of these regions, the robot tolerates larger errors and disturbances. Furthermore, for these models, and thus possibly real bipeds, usually if it is possible to avoid falling, it is possible to reach the target, and if it is possible to reach the target, it is possible to do so in two steps.

In this paper, the concept of divergent component of motion (DCM, also called “Capture Point”) is extended to 3-D. We introduce the “Enhanced Centroidal Moment Pivot point” (eCMP) and the “Virtual Repellent Point” (VRP), which allow for the encoding of both direction and magnitude of the external forces and the total force (i.e., external plus gravitational forces) acting on the robot. Based on eCMP, VRP, and DCM, we present methods for real-time planning and tracking control of DCM trajectories in 3-D. The basic DCM trajectory generator is extended to produce continuous leg force profiles and to facilitate the use of toe-off motion during double support. The robustness of the proposed control framework is thoroughly examined, and its capabilities are verified both in simulations and experiments.

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.

In this paper, we propose a biped walking controller that optimized three push recovery strategies: the ankle, hip, and stepping strategies. We suggested formulations that related the effects of each strategy to the stability of walking based on the linear inverted pendulum with flywheel model. With these relations, we could set up an optimization problem that integrates all the strategies, including step time change. These strategies are not applied hierarchically, but applied according to each weighting factor. Various combinations of weighting factors can be used to determine how the robot should respond to an external push. The optimization problem derived here includes many nonlinear components, but it has been linearized though some assumptions and it can be applied to a robot in real time. Our method is designed to be robust to modeling errors or weak perturbations, by exploiting the advantages of the foot. Hence, it is very practical to apply this algorithm to a real robot. The effectiveness of the walking controller has been verified through abstracted model simulation, full dynamics simulation, and a practical robot experiments.

In order to deal with the problem of imbalance of the elderly in the process of walking on level ground, this paper proposes a hip exoskeleton with series elastic actuator (SEA) to assist the wearer to adjust the step width, step length and step frequency in real time during walking to maintain balance walking. Through the built multi-sensor fusion gait perturbation and perception integrated platform, the experimenter’s autonomous recovery of balance experiment and exoskeleton-assisted human body's recovery of balance experiment under external wrench state were carried out respectively. The perturbations were applied in four directions to the experimenter while walking, which verified the validity of the balance evaluation model and the assisting effect of the exoskeleton in restoring the balance of human walking. Firstly, according to the established kinematics model, the displacement and velocity of the experimenter's center of mass during walking are calculated, and the system is simplified to a linear inverted pendulum model (LIPM) to evaluate the motion state. Secondly, a balance state evaluation model is established based on the instantaneous capture point (ICP) theory, and the balance state evaluation of the human body is realized by judging the positional relationship between the instantaneous capture point and the support range of the feet. Finally, according to the step strategy in the human body balance strategy, the magnitude of the auxiliary torque when the experimenter is out of balance is calculated through the relative position of the center of mass and the instantaneous capture point. Therefore, it can be shown that the balance evaluation model and exoskeleton control strategy we established can provide the balance restoring torque for the experimenter to swing the leg when the system is about to become unbalanced, which can effectively slow down the unbalanced trend.KeywordsHip exoskeletonPerturbation platformSeries elastic actuatorBalance recovery

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.