Extended Jacobian inverse kinematics algorithms for mobile manipulators

We consider the inverse kinematic problem for mobile manipulators consisting of a nonholonomic mobile platform and a holonomic manipulator on board the platform. The kinematics of a mobile manipulator are represented by a driftless control system with outputs together with the associated variational control system. The output reachability map of the driftless control system determines the instantaneous kinematics, while the output reachability map of the variational system plays the role of the analytic Jacobian of the mobile manipulator. Relying on a formal analogy between the kinematics of stationary and mobile manipulators we exploit the extended Jacobian construction in order to design a collection of extended Jacobian inverse kinematics algorithms for mobile manipulators. It has been proved mathematically and confirmed in computer simulations that these algorithms are capable of efficiently solving the inverse kinematic problem. Moreover, a choice of the Jacobian extension may lay down some guidelines for the platform‐manipulator motion coordination. © 2002 Wiley Periodicals, Inc.     

A comparison between the Denavit–Hartenberg and the screw-based methods used in kinematic modeling of robot manipulators

This paper aims to integrate didactically some engineering concepts to understand and teach the screw-based methods applied to the kinematic modeling of robot manipulators, including a comparative analysis between these and the Denavit–Hartenberg-based methods. In robot analysis, kinematics is a fundamental concept to understand, since most robotic mechanisms are essentially designed for motion. The kinematic modeling of a robot manipulator describes the relationship between the links and joints that compose its kinematic chain. To do so, the most popular methods use the Denavit–Hartenberg convention or its variations, presented by several author and robot publications. This uses a minimal parameter representation of the kinematic chain, but has some limitations. The successive screw displacements method is an alternative representation to this classic approach. Although it uses a non-minimal parameter representation, this screw-based method has some advantages over Denavit–Hartenberg. Both methods are here presented and compared, concerning direct/inverse kinematics of manipulators. The differential kinematics is also discussed. Examples of kinematic modeling using both methods are presented in order to ease their comparison.     

冗余度机器人操作手的运动灵活性测度

本文在速度水平上讨论了机器人操作手的运动灵活性,根据机器人操作手的运动灵活性完全取决于雅可比矩阵列向量的相关性这一分析,提出用雅可比矩阵的子阵 J_m 的总体相关性指标和相对相关性指标作为机器人操作手的运动灵活性测度,定义了运动灵活性的两个度量指标——绝对灵活度 D_α和相对灵活度 D_r;并论证了绝对灵活度 D_α和可操作度 W,相对灵活度 D_r 和可操作度 W、条件数 K 及最小奇异值σ_m 之间的关系;证明了绝对灵活度 D_α和可操作度 W 的等价性,并给出了可操作度的新的定义.从而全面论证了本文提出的操作手的运动灵活性测度的合理性.     

Approximation of Jacobian Inverse Kinematics Algorithms: Differential Geometric vs. Variational Approach

This paper addresses the approximation problem of Jacobian inverse kinematics algorithms for redundant robotic manipulators. Specifically, we focus on the approximation of the Jacobian pseudo inverse by the extended Jacobian algorithm. The algorithms are defined as certain dynamic systems driven by the task space error, and identified with vector field distributions. The distribution corresponding to the Jacobian pseudo inverse is non-integrable, while that associated with the extended Jacobian is integrable. Two methods of devising the approximating extended Jacobian algorithm are examined. The first method is referred to as differential geometric, and relies on the approximation of a non-integrable distribution (in fact: a codistribution) by an integrable one. As an alternative, the approximation problem has been formulated as the minimization of an approximation error functional, and solved using the methods of the calculus of variations. Performance of the obtained extended Jacobian inverse kinematics algorithms has been compared by means of computer simulations involving the kinematics model of the 7 dof industrial manipulator POLYCRANK. It is concluded that the differential geometric method offers a rapid, while the variational method a systematic tool for solving inverse kinematic problems.     

用于机器人逆运动学分析的几何法及应用

本文采用几何法来分析机器人逆运动学问题,将常见的工业机器人机构分为两级来考虑,避免了常用的雅可比逆阵求解机器人逆运动学问题的复杂性.利用文中所述的方法,通过一个喷漆机器人运动学建模说明了采用此方法的优越性.     

Optimal Extended Jacobian Inverse Kinematics Algorithms for Robotic Manipulators

Extended Jacobian inverse kinematics algorithms for redundant robotic manipulators are defined by combining the manipulator's kinematics with an augmenting kinematics map in such a way that the combination becomes a local diffeomorphism of the augmented taskspace. A specific choice of the augmentation relies on the optimal approximation by the extended Jacobian of the Jacobian pseudoinverse (the Moore--Penrose inverse of the Jacobian). In this paper, we propose a novel formulation of the approximation problem, rooted conceptually in the Riemannian geometry. The resulting optimality conditions assume the form of a Poisson equation involving the Laplace--Beltrami operator. Two computational examples illustrate the theory.      

基于螺旋理论的四面体可展机构运动特性与动力学分析

为了解决星载四面体构架式可展开天线运动学和动力学建模分析困难的问题,建立了一种基于螺旋理论的运动学和动力学特性分析方法．首先分析了3RR-3RRR四面体可展机构的构型机理及几何特征,基于螺旋理论绘制了旋量约束拓扑图,计算了机构的自由度,结果表明其只有1个自由度．然后对其进行了运动学分析,通过运动旋量的组合运算分析了各个构件的速度,并推导得到了机构的雅可比矩阵,采用旋量导数表示构件的6维加速度,推导得到了各个花盘和杆件的角加速度与质心线加速度．最后基于牛顿－欧拉方程和虚功原理建立了动力学方程,并进行了数值计算与仿真验证,数值计算结果与仿真结果完全一致,验证了基于螺旋理论的运动学和动力学特性分析方法的正确性．本文的研究方法在分析过程中各参数物理意义明确,分析过程较为程式化,易于编程,适合应用于此类空间可展开机构的分析中．     

函数矩阵及其微积分的高阶逻辑形式化

函数矩阵广泛应用于动态系统的建模与分析。传统的函数矩阵分析主要采用纸笔演算、数值计算和符号推导的方法,这些方法不能保证提供精确或正确的结果。高阶逻辑定理证明作为一种高可靠的形式化验证方法,可以克服以上不足。在高阶逻辑定理证明器HOL4中对函数向量和函数矩阵相关理论进行形式化,内容包括函数向量和函数矩阵及其连续性、微分、积分的形式化定义和相关性质的逻辑推理证明。为示范函数矩阵形式化的应用,最后给出机器人运动学中旋转矩阵微分公式的形式化验证。     

Endogenous configuration space approach to mobile manipulators: A derivation and performance assessment of Jacobian inverse kinematics algorithms

By a mobile manipulator we mean a robotic system composed of a non-holonomic mobile platform and a holonomic manipulator fixed to the platform. A taskspace of the mobile manipulator includes positions and orientations of its end effector relative to an inertial coordinate frame. The kinematics of a mobile manipulator are represented by a driftless control system with outputs. Admissible control functions of the platform along with joint positions of the manipulator constitute the endogenous configuration space. Endogenous configurations have a meaning of controls. A map from the endogenous configuration space into the taskspace is referred to as the instantaneous kinematics of the mobile manipulator. Within this framework, the inverse kinematic problem for a mobile manipulator amounts to defining an endogenous configuration that drives the end effector to a desirable position and orientation in the taskspace. Exploiting the analogy between stationary and mobile manipulators we present in the paper a collection of regular and singular Jacobian inverse kinematics algorithms. Their performance is evaluated on the basis of intense computer simulations.     

可实现空间取放作业的4自由度并联机器人机构综合

本论文旨在解决非过约束4自由度并联机构综合这一机构学难题。本文在总结已有的4自由度并联机器人综合结果的基础上，结合运动子群的概念和螺旋理论的方法，技术性地解决了约束数目和支链数目之间存在的矛盾。利用解析的手段给出了一套全新的非过约束对称4自由度并联机构设计方法。定义了一系列便于综合该机构的概念，分析了该类机构必须具有的拓扑结构，得出两种类型的该类并联机构。给出了满足设计条件的运动副组成情况及空间装配条件，最后给出了典型机构的设计范例。