Global motion analysis of energy-based control for 3-link planar robot with a single actuator at the first joint

This paper studies nonlinear control of a 3-link planar robot moving in the vertical plane with only the first joint being actuated while the two other revolute joints are passive (called the APP robot below). A nonlinear energy-based controller is proposed, whose objective is to drive the APP robot into an invariant set where the first link is in the upright position and the total mechanical energy converges to its value at the upright equilibrium point (all three links are in the upright position). By presenting and using a new property of the motion of the APP robot, without any condition on its mechanical parameters, this paper proves that if the control gains are larger than specific lower bounds, then only a measure-zero set of initial conditions converges to three strictly unstable equilibrium points instead of converging to the invariant set. This paper presents numerical results for a physical 3-link planar robot to validate the obtained theoretical results and to demonstrate a switch–and–stabilize maneuver in which the energy-based controller is switched to a linear state feedback controller that stabilizes the APP robot at its upright equilibrium point.     

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.     

A simple and quick control strategy for a class of first-order nonholonomic manipulator

This paper presents a simple and quick control strategy for a class of first-order nonholonomic manipulator: planar n-link manipulator with a passive first joint (no gravity). The control target is driving its endpoint from any initial position to any target position. First, we reduce the planar n-link manipulator to a planar three-link one by maintaining the states (angles and angular velocities) of n-3 active links in the initial value all the time. That is, we only adjust the states of the remaining two active links in the whole control process, and these two adjusted active links are chosen to guarantee that the target position is in the reachable region of the planar n-link manipulator by using the enumeration method. Then, we divide the whole control process into two stages for the reduced planar three-link manipulator. In each stage, the manipulator becomes a planar two-link one by maintaining the states of one adjusted active link to be the constant value. The state constraint existing between the passive first link and the adjusted active link is obtained by using the integral characteristics of a planar two-link manipulator. Meantime, the geometric constraint between the position of the endpoint and angles of all joints is obtained based on the homogeneous coordinate transformation method. According to the above two kinds of constraint, the target angles of the two adjusted active links are calculated by employing the particle swarm optimization algorithm. When the two adjusted active links are controlled to their target angles in turn, the control target of the planar n-link manipulator is completed. Finally, simulation results demonstrate that the proposed control strategy is valid and rapid.     

Treadmill walking of the pneumatic biped Lucy: Walking at different speeds and step-lengths

Actuators with adaptable compliance are gaining interest in the field of legged robotics due to their capability to store motion energy and to exploit the natural dynamics of the system to reduce energy consumption while walking and running. To perform research on compliant actuators we have built the planar biped Lucy. The robot has six actuated joints, the ankle, knee and hip of both legs with each joint powered by two pleated pneumatic artificial muscles in an antagonistic setup. This makes it possible to control both the torque and the stiffness of the joint. Such compliant actuators are used in passive walkers to overcome friction when walking over level ground and to improve stability. Typically, this kind of robots is only designed to walk with a constant walking speed and step-length, determined by the mechanical design of the mechanism and the properties of the ground. In this paper, we show that by an appropriate control, the robot Lucy is able to walk at different speeds and step-lengths and that adding and releasing weights does not affect the stability of the robot. To perform these experiments, an automated treadmill was built Published in Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 134–142, July 2008.     

Dynamic delayed feedback control for stabilizing the giant swing motions of an underactuated three-link gymnastic robot

This paper investigates the dynamics of the giant swing motions of an underactuated three-link gymnastic robot moving in a vertical plane by means of dynamic delayed feedback control (DDFC). DDFC, being one of useful methods to overcome the so-called odd number limitation in controlling a chaotic discrete-time system, is extended to control a continuous-time system such as a 3-link gymnastic robot with passive joint. Meanwhile, a way to calculate the error transfer matrix and the input matrix which are necessary for discretization is proposed, based on a Poincaré section which is defined to regard the target system as a discrete-time system. Moreover, the stability of the closed-loop system by the proposed control strategy is discussed. Furthermore, some numerical simulations are presented to show the effectiveness in controlling a chaotic motion of the 3-link gymnastic robot to a periodic giant swing motion.     

基于干扰观测器的柔性关节空间机器人退步自适应控制

研究了载体位置、姿态均不受控的情况下,系统参数不确定的柔性关节空间机器人轨迹跟踪的控制问题.结合系统动量、动量矩守恒关系,利用拉格朗日法推导出系统的动力学模型.为减小系统柔性关节对系统控制精度的影响,采用关节柔性补偿器来等效降低系统关节的柔性.再借助奇异摄动法,针对系统参数不确定的情况,设计了柔性关节空间机器人基于干扰观测器的退步自适应滑模控制方案.该方案不需要对系统惯性参数进行线性化处理,控制器结构简单,且实现了空间机器人期望轨迹的精确跟踪控制.通过平面两杆空间机器人的数值仿真证明了该方法的有效性.     

Stability case study of the ACROBOTER underactuated service robot

The dynamics of classical robotic systems are usually described by ordinary differential equations via selecting a minimum set of independent generalized coordinates. However, different parameterizations and the use of a nonminimum set of (dependent) generalized coordinates can be advantageous in such cases when the modeled device contains closed kinematic loops and/or it has a complex structure. On one hand, the use of dependent coordinates, like natural coordinates, leads to a different mathematical representation where the equations of motion are given in the form of differential algebraic equations. On the other hand, the control design of underactuated robots usually relies on partial feedback linearization based techniques which are exclusively developed for systems modeled by independent coordinates. In this paper, we propose a different control algorithm formulated by using dependent coordinates. The applied computed torque controller is realized via introducing actuator constraints that complement the kinematic constraints which are used to describe the dynamics of the investigated service robotic system in relatively simple and compact form. The proposed controller is applied to the computed torque control of the planar model of the ACROBOTER service robot. The stability analysis of the digitally controlled underactuated service robot is provided as a real parameter case study for selecting the optimal control gains.     

3-PRS并联机器人惯量耦合特性研究

惯量是影响机器人动态性能的主要因素,并联机器人因其多支链耦合的结构特点,关节空间各驱动轴出现惯量耦合的动力学特性,在高速、高加速度运动时易引起控制超调、振动等现象,破坏机器人的动态性能,因此研究并联机器人惯量耦合特性具有重要意义. 以3-PRS 并联机器人为例,通过虚功原理求得惯量矩阵,提出惯量耦合指标,该耦合指标表征了并联机器人在工作空间不同位姿时各驱动轴的耦合惯量大小,并给出了该耦合指标在机器人工作空间内的分布规律. 进一步在一台3-PRS 并联机器人样机上进行了实验验证,结果表明耦合惯量会改变驱动轴负载,负载的改变将最终影响动态性能. 同时各驱动轴的负载变化量随着惯量耦合指标的变大而变大,与理论分析有较好的一致性. 研究成果可帮助评价并联机器人的动力学耦合特性,并可用于并联机器人的结构参数优化及伺服参数调试以提高机器人的动态性能.      

Trajectory planning and tracking control for autonomous bicycle robot

Control of the autonomous bicycle robot offers considerable challenges to the field of robotics due to its nonholonomic, underactuated, and nonminimum-phase properties. Furthermore, instability and complex dynamic coupling make the trajectory planning of the bicycle robot even more challenging. In this paper, we consider both trajectory planning and tracking control of the autonomous bicycle robot. The desired motion trajectory of the contact point of the bicycle’s rear wheel is constructed using the parameterized polynomial curve that can connect two given endpoints with associated tangent angles. The parameters of the polynomial curve are determined by minimizing the maximum of the desired roll angle’s equilibrium of the bicycle, and this optimization problem is solved by the particle swarm optimization algorithm. Then, a control scheme that can achieve full-state trajectory tracking while maintaining the bicycle’s balance is proposed by combining a planar trajectory tracking controller with a roll angle balance controller. Simulation results are presented to demonstrate the effectiveness of the proposed method.     

ZMP based neural network inspired humanoid robot control

This paper concerns ZMP-based control that is inspired by artificial neural networks for humanoid robot walking on varying sloped surfaces. Humanoid robots are currently one of the most exciting research topics in the field of robotics, and maintaining stability while they are standing, walking or moving is a key concern. To ensure a steady and smooth walking gait of such robots, a feedforward type of neural network architecture, trained by the back-propagation algorithm, is employed. The inputs and outputs of the neural network architecture are the ZMPx and ZMPy errors of the robot, and the x, y positions of the robot, respectively. The neural network developed allows the controller to generate the desired balance of the robot positions, resulting in a steady gait for the robot as it moves around on a flat floor, and when it is descending or ascending slopes. In this paper, experiments of humanoid robot walking are carried out, in which the actual position data from a prototype robot are measured in real-time situations, and fed into a neural network inspired controller designed for stable bipedal walking. In addition, natural walking motions on the different surfaces with varying slopes are obtained and the performance of the resulting controller is shown to be satisfactory.     

全柔性空间机器人运动振动一体化输入受限重复学习控制

探究基座、臂、关节全柔性影响下空间机器人动力学模拟、运动控制及基座、臂、关节三重柔性振动主动抑制的问题,设计了不基于系统模型信息的运动振动一体化输入受限重复学习控制算法.将柔性基座与关节等效为线性弹簧与扭转弹簧,柔性臂视为欧拉-伯努利梁模型,利用拉格朗日方程与假设模态法建立动力学模型,然后,用奇异摄动理论将模型分解为包含刚性变量与臂柔性振动的慢变子系统,包含基座、关节柔性振动的快变子系统,并分别设计相应的子控制器,构成了带关节柔性补偿的一体化控制算法.针对慢变子系统,提出输入受限重复学习控制算法,由双曲正切函数,饱和函数与重复学习项构成,双曲正切函数与饱和函数实现输入力矩受限要求,重复学习项补偿周期性系统误差,以完成对基座姿态、关节铰周期轨迹的渐进稳定追踪.然而,为了同时抑制慢变子系统臂的柔性振动,运用虚拟力的概念,构造同时反映臂柔性振动与系统刚性运动的混合轨迹,提出了基于虚拟力概念的输入受限重复学习控制器,保证基座、关节轨迹精确追踪的同时,对臂的柔性振动主动抑制.针对快变子系统,采用线性二次最优控制算法抑制基座与关节的柔性振动.仿真结果表明:控制器适用于一般柔性非线性系统,满足输入力...     

Connectivity-preserving adaptive synchronized tracking of heterogeneous spherical robots with limited communication ranges

A distributed connectivity-preserving problem is addressed for adaptive synchronized tracking of networked heterogeneous nonlinear spherical robots with a dynamic leader. We assume uncertain rolling resistances, limited communication ranges of robots, and that only a small fraction of follower robots have access to the time-varying leader information. Compared with the previous synchronized tracking scheme based on distributed hierarchical sliding surfaces (DHSSs) for multiple underactuated spherical robots, the main contribution of this paper is to present a new error transformation approach of DHSS for preserving the initial connectivity while achieving the synchronized tracking of longitudinal positions of all followers to the dynamic leader and the balance tracking of the swing-up angle of each follower to its swing-up balance angle under the limited communication ranges of robots. A local connectivity-preserving adaptive synchronized tracker is designed using the proposed DHSS with the estimate of the leader velocity and the swing-up balance angular velocity. In addition, we derive a technical lemma to analyze the asymptotic stability and the initial connectivity preservation of the overall closed-loop system in the Lyapunov sense.     

基于虚拟力概念的柔性臂空间机器人模糊退步自适应控制算法设计

讨论了漂浮基柔性臂空间机器人系统的动力学模拟、运动轨迹跟踪控制算法设计及柔性振动主动抑制。采用多体动力学建模方法并结合假设模态法,建立了漂浮基柔性臂空间机器人的系统动力学模型。基于该模型,针对系统惯性参数未知情况,提出了刚性运动基于模糊基函数网络自适应调节的退步控制算法,以完成柔性臂空间机器人载体姿态及机械臂各关节铰的协调运动。然后,为了主动抑制系统柔性振动,运用虚拟力的概念,构造了同时反映柔性模态和刚性运动轨迹的混合期望轨迹,通过改造原有的控制算法,提出了基于虚拟力概念的模糊退步自适应控制算法;这样不但保证了之前刚性运动控制方案对模型不确定的鲁棒性,而且能主动抑制柔性振动,从而提高了轨迹跟踪性能。理论分析及数值仿真算例均表明了控制方法的可行性。     

The Kinematics and the Full Minimal Dynamic Model of a 6-DOF Parallel Robot Manipulator

In this paper we present a particular architecture of parallel robots which has six-degrees-of-freedom (6-DOF) with only three limbs. The particular properties of the geometric and kinematic models with respect to that of a classical parallel robot are presented. We show that inverse problems have an analytical solution. However, to solve the direct problems, an efficient numerical procedure which needs to inverse only a 3 × 3 passive Jacobian matrix is proposed. In a second step, dynamic equations are derived using the Lagrangian formalism where the joint variables are passive and active joint coordinates. Based on the geometrical properties of the robot, the equations of motion are derived in terms of only nine coordinates related by three kinematic constraints instead of 18 joint coordinates. The computational cost of the dynamic model obtained is reduced by using a minimum set of base inertial parameters.     

Fuzzy observer-based command filtered tracking control for uncertain strict-feedback nonlinear systems with sensor faults and event-triggered technology

For the trajectory tracking problem of nth-order uncertain nonlinear systems with sensor faults, a fuzzy controller based on command filtered and event-triggered technology is designed to improve the tracking error of the system. Concurrently, a fault-tolerant control scheme is introduced to effectively solve the problem of sudden output sensor failure. Additionally, the proposed controller can also greatly avoid complexity explosion problem of derivations of virtual control laws, which makes the design of the controller simpler. Furthermore, an effective observer is designed to solve the problem of system state immeasurability. Therefore, the proposed control scheme makes the design of the controller more convenient and flexible. According to Lyapunov stability theory, it is proved that all closed-loop signals are uniformly and ultimately bounded. Finally, two simulation examples of second-order nonlinear system and single-link robot show the effectiveness of the proposed scheme.

     

Adaptive fuzzy formation control for a swarm of nonholonomic differentially driven vehicles

In this paper, an adaptive fuzzy robust H ^∞ controller is proposed for formation control of a swarm of differential driven vehicles with nonholonomic dynamic models. Artificial potential functions are used to design the formation control input for kinematic model of the robots and matrix manipulations are used to transform the nonholonomic model of each differentially driven vehicle into equivalent holonomic one. The main advantage of the proposed controller is the robustness to input nonlinearity, external disturbances, model uncertainties, and measurement noises, in a formation control of a nonholonomic robotic swarm. Moreover, robust stability proof is given using Lyapunov functions. Finally, simulation results are demonstrated for a swarm formation problem of a group of six unicycles, illustrating the effective attenuation of approximation error and external disturbances, even in the case of robot failure.     

Dynamics and trajectory planning of a planar flipping robot

In this paper, a new planar one-legged robot model is firstly proposed, which, unlike previous one-legged robot with springy legs, consists of three revolute joints. Then a novel manner of one-legged locomotion (i.e., ballistic flip) is designed for this robot. A complete flipping gait cycle is composed of four phases: two stance phases and two flight phases. During flight phases, no active control is needed on the knee joint. Rotational motion and translational motion is decoupled from each other in flight phases. Landing of the robot is regarded as an inelastic impulsive impact. During stance phases, the robot model can be simplified as a two-degree-of-freedom rigid manipulator. Based on analysis of kinematics and dynamics of the flip robot, trajectory planning of cyclic flip gait is formulized as a problem of numerical optimization subject to nonlinear constraints such as positive reaction force of ground and finite torque of the joints. One potential application of the flipping robot is space exploration, which urgently requires the legged locomotive robots to be light-weighted and energy efficient.     

Mobile robots with two independent coaxial drive wheels: Dynamics and the schemes of control

A two-wheeled robot and a robot composed of a snakelike chain of two-wheeled objects are considered. The problem of trajectory synthesis and the problem of motion planning and control are studied for these mobile robots. The mobile robots with two independently controlled coaxial wheels and with one or several passive wheels are called robots with differential drive. These problems are formulated in such a way that the construction of dynamically correct and precise control becomes possible for such robots.     

L'espace articulaire de la Robotique Industrielle est un espace vectorielIndustrial Robotics joint space is a vector space

The mathematical modelling of industrial robots is based on the vectorial nature of the n-dimensional joint space of the robot, defined as a kinematic chain with n degrees of freedom. However, in our opinion, the vectorial nature of the joint space has been insufficiently discussed in the literature. We establish the vectorial nature of the joint space of an industrial robot from the fundamental studies of B. Roth on screws. To cite this article: B. Tondu, C. R. Mecanique 331 (2003).     

Par2: a spatial mechanism for fast planar two-degree-of-freedom pick-and-place applications

This paper introduces a new two-degree-of-freedom (dof) parallel manipulator producing two translations in the vertical plane. One drawback of existing robots built to realize these dof is their lack of transversal stiffness, another one being their limited ability to provide very high acceleration. Indeed, these architectures cannot be lightweight and stiff at the same time. The proposed parallel architecture is a spatial mechanism which guarantees a good transversal stiffness. It is composed by two actuated kinematic chains, and two passive chains built in the transversal plane. The key feature of this robot comes from the two passive chains which are coupled to create a kinematic constraint: the platform stays in one plane. A stiffness analysis shows that the robot can be lighter and stiffer than a classical 2-dof mechanism. A prototype of this robot is presented and preliminary tests show that accelerations above 400 ms⁻¹ can be achieved while keeping a low tracking error.