Inverse kinematics and inverse dynamics for control of a biped walking machine

Analytical techniques are presented for the motion planning and control of a 12 degree-of-freedom biped walking machine. From the Newton-Euler equations, joint torques are obtained in terms of joint trajectories, and the inverse dynamics are developed for both the single-support and double-support cases. Physical admissibility of the biped trajectory is characterized in terms of the equivalent force-moment and zero-moment point. This methodology has been used to obtain reference inputs and implement the feedforward control of walking robots. A simulation example illustrates the application of the techniques to plan the forward-walking trajectory of the biped robot. The implementation of a prototype mechanism and controller is also described.     

正弦驱动与传感反馈结合的双足机器人仿生行走控制

提出了一种正弦驱动与传感反馈结合的双足机器人仿生行走控制方法．所有关节由正弦振荡器驱动, 较之相互耦合的神经元振荡器更加简单;控制参数具有明晰的物理意义,便于对运动模式进行调节．传感反馈表征 了机器人的运动状态,对于保证机器人的稳定行走起着至关重要的作用．将机器人碰地、碰膝等关键运动状态作为 相位反馈,对控制力矩进行相位重置,协调各关节动作,进而实现控制器、机器人、环境的耦合．同时,从节省能量 和仿生的角度,考虑了关节运动的被动特性,确定了各关节力矩的作用区间．仿真结果表明,该控制方法能实现机 器人稳定行走,并具有良好的能效性和自稳定性.     

An approach to analyzing biped locomotion dynamics and designing robot locomotion controls

An approach to analyzing biped locomotion problems is presented. This approach applies the principles of Lagrangian dynamics to derive the equations of motion of locomotion gaits, state-variable techniques to analyze locomotion dynamics, and multivariable feedback to design locomotion controls. A robot model which has no knee joints or feet and is constrained to motion in the sagittal plane is chosen as a sufficiently simple model of a biped to illustrate the approach. A goal of the analysis is the design of a locomotion control for the robot which produces a walking gait having a velocity and stride length similar to those of a human walking gait. The principle feature of the approach is a much deeper understanding of the dynamics of biped locomotion than previous approaches have provided.     

面向离散地形的欠驱动双足机器人平衡控制方法

欠驱动双足机器人在行走中为保持自身的平衡,双脚需要不间断运动.但在仅有特定立足点的离散地形上很难实现调整后的落脚点,从而导致欠驱动双足机器人在复杂环境中的适应能力下降.提出了基于虚拟约束(Virtual constraint,VC)的变步长调节与控制方法,根据欠驱动双足机器人当前状态与参考落脚点设计了非时变尺度缩放因子,能够实时重构适应当前环境的步态轨迹;同时构建了全身动力学模型,采用反馈线性化的模型预测控制(Model predictive control,MPC)滚动优化产生力矩控制量,实现准确的轨迹跟踪控制.最终进行了欠驱动双足机器人的随机离散地形稳定行走的仿真实验,验证了所提方法的有效性与鲁棒性.     

双足机器人自然ZMP轨迹生成方法研究

为了实现双足机器人类人行走,提出了一种基于自然ZMP轨迹的双足机器人步行模式生成方法。在单腿支撑相,根据基于三维线性倒立摆模型,在设定从脚跟到脚趾移动的自然ZMP轨迹后,得到质心轨迹方程;在双腿支撑相采用线性摆模型生成质心轨迹方程。同时给出了在统一坐标系中的多步规划质心轨迹方程。在RoboCup 3D仿真平台实现了采用自然ZMP轨迹的双足机器人类人稳定步行,实验和竞赛结果都验证了该方法的有效性。     

基于深度强化学习的双足机器人斜坡步态控制方法

为提高准被动双足机器人斜坡步行稳定性, 本文提出了一种基于深度强化学习的准被动双足机器人步态控制方法. 通过分析准被动双足机器人的混合动力学模型与稳定行走过程, 建立了状态空间、动作空间、episode过程与奖励函数. 在利用基于DDPG改进的Ape-X DPG算法持续学习后, 准被动双足机器人能在较大斜坡范围内实现稳定行走. 仿真实验表明, Ape-X DPG无论是学习能力还是收敛速度均优于基于PER的DDPG. 同时, 相较于能量成型控制, 使用Ape-X DPG的准被动双足机器人步态收敛更迅速、步态收敛域更大, 证明Ape-X DPG可有效提高准被动双足机器人的步行稳定性.     

Humanoid Robotic System with and without Elasticity Elements Walking on an Immobile/Mobile Platform

This work is concerned with the modeling and analysis of a complex humanoid robotic system walking on an immobile/mobile platform. For this purpose, a software package was synthesized which allows one to select configuration of both the humanoid and the platform. Each joint of the biped and platform can be defined by the user via the motor state (active or locked) and gear type (rigid or elastic). The user can also form very diverse configurations of the humanoid and platform. The software package forms a mathematical model. By selecting system’s parameters the simulation allows user to analyze dynamic behavior of the biped of selected configuration, walking on either an immobile or mobile platform of selected configuration. In the moment when the biped steps on the platform, the latter, by its dynamics, acts on the biped dynamics, and the biped on the other hand, by its characteristics, influences dynamics of the platform motion. These two complex contacting systems form a more complex system, whose mathematical model has to encompass all the elements of coupling between the humanoid joints and platform joints. The phenomenon of coupling is analyzed first on a humanoid robotic system with all rigid elements, which is in contact with the platform mechanism having also all rigid elements. It has been shown that coupling is more influenced when elasticity elements are included into the configuration. Insufficient knowledge of coupling characteristics may present a serious disturbance to the system in the robotic task realization. The deviation of the ZMP (Zero-Moment Point) from the reference trajectory is presented, which implies the need for the synthesis of new control structures for stabilizing biped motion on the immobile/mobile platform. The reference trajectory may be defined in very different ways and from several aspects. Reference trajectory of each joint can be defined so to encompass or not encompass elastic deformations. The control structure for the biped walking on the platform should be defined so that it satisfies the requirement for the ZMP to be within the given boundaries in every sampling instant, which guarantees dynamic balance of the locomotion mechanism in the real regime. The control is defined as CR (Centralized Reference control, calculated from the reference state), plus LO (control via local feedbacks of motor motion with respect to position and velocity). In the case of the biped motion on a mobile platform CR control is defined separately under the real conditions of unknown characteristics of coupling between the two complex systems, as well as unknown elasticity properties. The analysis of simulation results of the humanoid robot motion on a mobile platform gives evidence for all the complexity of this system and shows how much system parameters (choice of trajectory, configuration, geometry, elasticity characteristics, motor, etc.) influence stabilization of its humanoid motion.     

Simultaneous design of optimal gait pattern and controller for a bipedal robot

Nowadays, biped robotics becomes an interesting topic for many control researchers. The biped robot is more adaptable than the other mobile robots in a varied environment and can have more diverse possibilities in planning the motion. However, it falls down easily and its control for stable walking is difficult. Therefore, generation of a desired walking pattern for the biped robot in the presence of some model uncertainties is an important problem. The proposed walking pattern should be also achievable by the designed controller. To achieve this aim and to reach the best control performance, the walking pattern and controller should be designed simultaneously rather than separately. In the present study, an optimal walking pattern is proposed to be tracked by a designed sliding mode controller. In this respect, a genetic algorithm (GA) is utilized to determine the walking pattern parameters and controller coefficients simultaneously. Here, high stability, minimum energy consumption, good mobility properties, and actuator limitations are considered as the important indexes in optimization. Simulation results indicate the efficiency of the proposed scheme in walking the understudy biped robot.     

Development and motion control of a biped walking robot based on passive walking theory

This research aims to develop the biped walking robot that can walk on the horizontal ground and improve walking efficiency by utilizing the theory of the passive walking robot, namely the pendulum principle. For that, two motors were installed on the hip of the robot to generate the control torques to perform a walking motion. The computer simulations with dynamic model were carried out to investigate the walking capability of the system. Experimental robot was developed considering the calculated results. The proportional control law was used in walking experiment. The robot can walk on the horizontal ground with the proposed method.     

3D stable biped walking control and implementation on real robot

A combination of walking control methods was proposed and implemented on a biped robot. The LIPM-based model predictive control (MPC) was adopted to generate a basic stable walking pattern. The stability of pitch and yaw rotation was improved through pitch and yaw momentum control as a supplementation of MPC. It is found that biped robot walking tends to deviate from the planned walking direction if not considering the rotation friction torque in yaw axis under the support foot. There are basically two methods to control yaw momentum, waist and swing arms rotation control. However, the upper body is often needed to accomplish other tasks. Therefore, a yaw momentum control method based on swing leg dynamics was proposed. This idea does not depend on upper body’s motion and is highlighted in this paper. Through experiments, the feasibility of the combination of the control methods proved to be practical in keeping biped robot walking stable both in linear and rotation motion. The pros and cons of the yaw momentum control method were also tested and discussed through comparison experiments, such as walking on flat and uneven terrain, walking with different payloads.     

Fuzzy SVM learning control system considering time properties of biped walking samples

To learn biped walking dynamics accurately, and then compensate time-varying external disturbances timely, a time-sequence-based fuzzy SVM (TSF-SVM) learning control system considering time properties of biped walking samples is proposed. For the first time, time-sequence-based triangular and Gaussian fuzzy membership functions have been proposed for the single support phase (SSP) and the double support phase (DSP), respectively, according to time properties of different biped phases, which provides an effective way to formulate time properties of biped walking samples in the context of time-varying external disturbances. In addition, a time-sequence-based moving learning window (TS-MLW) is proposed for online training of the proposed TSF-SVM. The performance of the proposed TSF-SVM is compared with other typical intelligent methods; simulation results demonstrate that the proposed method is more sensitive to occasional external disturbances, which increases the stability margin and prevents the robot from falling down.     

3D walking biped: optimal swing of the arms

A ballistic walking gait is designed for a 3D biped with two identical two-link legs, a torso, and two identical one-link arms. In the single support phase, the biped moves due to the existence of a momentum, produced mechanically, without applying active torques in the interlink joints. This biped is controlled with impulsive torques at the instantaneous double support to obtain a cyclic gait. The impulsive torques are applied in the seven interlink joints. Then an infinity of solutions exists to find the impulsive torques. An effort cost functional of these impulsive torques is minimized to determine a unique solution. Numerical results show that for a given time period and a given length of the walking gait step, there is an optimal swinging amplitude of the arms. For this optimal motion of the arms, the cost functional is minimum.     

Gait generation and control for biped robots with underactuation degree one

The paper develops a unified feedback control law for n degree-of-freedom biped robots with one degree of underactuation so as to generate periodic orbits on different slopes. The periodic orbits on different slopes are produced from an original periodic orbit, which is either a natural passive limit cycle on a specific slope or a stable periodic walking gait on level ground generated with active control. First, inspired by the controlled symmetries approach, a general result on gait generation on different slopes based on a periodic orbit on a specific slope is obtained. Second, the time-scaling control approach is integrated to reproduce geometrically same periodic orbits for biped robots with one degree of underactuation. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution that scales the original periodic orbit. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed approach is illustrated by two kinds of underactuated biped robots: one has a passive gait on a specific ground slope and the other does not have a natural passive gait.     

面向变高度连续台阶的双足欠驱动步行稳定控制

针对欠驱动双足机器人在已知变高度台阶上的稳定控制,提出了一种基于自适应前馈算法的稳定步行控制策略．首先,考虑地面变形,将地面等效为“弹簧-阻尼”系统,并建立“机器人-台阶”耦合动力学模型．其次,将“机器人-台阶”这一“多输入-多输出”模型简化为由质心位移和速度构成的“单输入-单输出”模型．然后,使用变坡度斜坡等效变高度台阶,根据台阶高度确定等效斜坡倾角和机器人理想步长;同时引入自适应控制系数,并根据等效斜坡倾角调整该控制系数,实现质心对参考速度的跟踪．最后,在台阶高度变化小于0.032 m的环境中进行数值仿真试验,验证控制策略的有效性．仿真结果表明：本文提出的控制策略可以实现已知变高度台阶上的稳定步行．     

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.     

Biped walking pattern generation by a simple three-dimensional inverted pendulum model

For three-dimensional (3D) walking control of a biped robot we analyze the dynamics of a 3D inverted pendulum in which motion is constrained to move along an arbitrarily defined plane. This analysis leads us to a simple linear dynamics, the Three-Dimensional Linear Inverted Pendulum Mode (3D-LIPM). The geometric nature of the trajectories under the 3D-LIPM and a method for walking pattern generation are discussed. A simulation result of a walking control using a 12-d.o.f. biped robot model is also shown.     

Optimization-based analysis of push recovery during walking motions to support the design of rigid and compliant lower limb exoskeletons

Abstract

Lower limb exoskeletons provide a promising approach to allow disabled people to walk again in the future. Designing such exoskeletons and tuning the required actuators is challenging, since the full dynamics of the combined human-exoskeleton system have to be taken into account. In particular, it is important to not only consider nominal walking motions but also extreme situations such as the recovery from large perturbations. In this paper, we present an approach based on push recovery experiments while walking, multibody system models, and least-squares optimal control to analyze the required torques to be generated by the exoskeleton, assuming that the human provides no torque. We consider seven different trials with varying push locations and push magnitudes applied on the back of the subject. In a first study, we investigate the dependency of these total joint torques on the exoskeleton mass – and compare the torques required for a human without exoskeleton to the ones for the human with two different exoskeleton configurations. In a second study, we investigate how optimally chosen passive spring-damper elements can support the required torques in the exoskeleton joints. It can be shown that the active torques can be reduced significantly in the different joints and cases.     

A Novel Method of Gait Synthesis for Bipedal Fast Locomotion

Common methods of gait generation of bipedal locomotion based on experimental results, can successfully synthesize biped joints’ profiles for a simple walking. However, most of these methods lack sufficient physical backgrounds which can cause major problems for bipeds when performing fast locomotion such as running and jumping. In order to develop a more accurate gait generation method, a thorough study of human running and jumping seems to be necessary. Most biomechanics researchers observed that human dynamics, during fast locomotion, can be modeled by a simple spring loaded inverted pendulum system. Considering this observation, a simple approach for bipedal gait generation in fast locomotion is introduced in this paper. This approach applies a nonlinear control method to synchronize the biped link-segmental dynamics with the spring-mass dynamics. This is done such that while the biped center of mass follows the trajectory of the mass-spring model, the whole biped performs the desired running/jumping process. A computer simulation is done on a three-link under-actuated biped model in order to obtain the robot joints’ profiles which ensure repeatable hopping. The initial results are found to be satisfactory, and improvements are currently underway to explore and enhance the capabilities of the proposed method.     

Robust Sliding-mode Control of Nine-link Biped Robot Walking

A nine-link planar biped robot model is considered which, in addition tothe main links (i.e., legs, thighs and trunk), includes a two-segment foot.First, a continuous walking pattern of the biped on a flat terrain issynthesized, and the corresponding desired trajectories of the robot jointsare calculated. Next, the kinematic and dynamic equations that describe itslocomotion during the various walking phases are briefly presented. Finally,a nonlinear robust control approach is followed, motivated by the fact thatthe control which has to guarantee the stability of the biped robot musttake into account its exact nonlinear dynamics. However, an accurate modelof the biped robot is not available in practice, due to the existence ofuncertainties of various kinds such as unmodeled dynamics and parameterinaccuracies. Therefore, under the assumption that the estimation error onthe unknown (probably time-varying) parameters is bounded by a givenfunction, a sliding-mode controller is applied, which provides a successfulway to preserve stability and achieve good performance, despite the presenceof strong modeling imprecisions or uncertainties. The paper includes a setof representative simulation results that demonstrate the very good behaviorof the sliding-mode robust biped controller.     

Model and control of the locomotion of a biomimic musculoskeletal biped

This article presents a biomimic musculoskeletal biped which contains 7 segments and 18 muscles. The muscle model and body dynamics are constructed based on physiological theories. A motor control system is designed to mimic natural human locomotion, which contains a central pattern generator, a regulator, a compensator, and an impedance controller. The recurrent neural oscillator models the central pattern generator, and an artificial neural network is used to design the regulator. From the simulation study, we found that this biped can produce a rhythmic and stable walking movement similar to actual human walking.