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。     

A Dual Neural Network for Kinematic Control of Redundant Manipulators Using Input Pattern Switching

This paper presents a dual neural network for kinematic control of a seven degrees of freedom robot manipulator. The first network is a static multilayer perceptron with two hidden layers which is trained to mimic the Jacobian of a seven DOF manipulator. The second network is a recurrent neural network which is used for determining the inverse kinematics solutions of the manipulator; The redundancy is used to minimize the joint velocities in the least squares sense. Simulation results show relatively good comparison between the outputs of the actual Jacobian matrix and multilayer neural network. The first network maps motions of the seven joints of the manipulator into 42 elements of the Jacobian matrix, with surprisingly smaller computations than the actual trigonometric function evaluations. A new technique, input-pattern-switching, is presented which improves the global training of the static network. The recurrent network was designed to work with the neural network approximation of the Jacobian matrix instead of the actual Jacobian. The combination of these two networks has resulted in a time-efficient procedure for kinematic control of robot manipulators which avoids most of the complexity present in the classical-trigonometric-based methods. Also, by electronic implementation of the networks, kinematic solutions can be obtained in a very timely manner (few nanoseconds).     

Homogeneous weighted motion planning for cooperated manipulator systems under position-torque constrained processing Environment

Real-time motion planning under position and torque constraints is a critical challenge for cooperative manipulator efficiency and safety operation. Real-time motion planning at the velocity level improves computation efficiency, eliminates the complex derivative calculation of the Jacobian matrix, and the velocity planning solution can be used directly for robotic kinematic control. However, little research attention has been attached to handling the position and torque constraints simultaneously at velocity level for cooperative manipulator systems. In this paper, we introduce a novel homogeneous weighted least-norm method (HWLN) for joint velocity redistribution of cooperated manipulators. Within the coupled kinematics-dynamics model of cooperated manipulators, joint position and torque constraints are simultaneously homogenized and taken into account by the constraint performance index. To avoid joint's constraint saturations, two real-time weight updating laws are designed for the joint position and driving torque respectively. The joint velocities of cooperated manipulators are then adaptively redistributed using the pseudo-kinetic-energy minimum optimization criteria. When compared to single manipulator regulation, this strategy takes greater advantage of cooperative redundancy and significantly enhances the position-torque planning performance. Mathematical stability proof is presented. In the meanwhile, numerical experiment results under various joint position and torque constraints demonstrate the effectiveness of the proposed HWLN method. The experimental results for motion planning and control of two 6R ABD-20Kg robotic manipulators are provided.     

Analysis of duality-based interconnected kinematics of planar serial and parallel manipulators using screw theory

In this paper, a method has been proposed to analyze the planar architectures of serial and parallel manipulators, based on the duality associated with their interconnected kinematics. The interconnected kinematics states that model of one architecture can be derived from the kinematic model of the other, using screw theory approach. The performance of the initial and the derived manipulators was evaluated with three criteria: isotropy, maximum force transmission ratio and local transmission index. Without loss of generality, the serial manipulator derived from parallel has better isotropy, while the parallel manipulator derived from serial can be designed to have better force and power transmission.

     

Kinematic analysis and error modeling of TAU parallel robot

The TAU robot presents a new configuration of parallel robots with three degrees of freedom. This robotic configuration is well adapted to perform with a high precision and high stiffness within a large working range compared with a serial robot. It has the advantages of both parallel robots and serial robots. In this paper, the kinematic modeling and error modeling are established with all errors considered using Jacobian matrix method for the robot. Meanwhile, a very effective Jacobian approximation method is introduced to calculate the forward kinematic problem instead of Newton–Raphson method. It denotes that a closed form solution can be obtained instead of a numerical solution. A full size Jacobian matrix is used in carrying out error analysis, error budget, and model parameter estimation and identification. Simulation results indicate that both Jacobian matrix and Jacobian approximation method are correct and with a level of accuracy of micron meters. ADAMS's simulation results are used in verifying the established models.     

On the dynamic model and kinematic analysis of a class of Stewart platforms

In this paper, a dynamic model for a class of Stewart platform (six degrees of freedom parallel link robotic manipulators) is derived by using tensor representation. A set of six Lagrange's equations are obtained. The kinematics analysis for a class of Stewart platform is conducted and a sixteenth order polynomial equation corresponding to the forward kinematic solution of the Stewart platform is obtained, which gives all possible global solutions of a manipulator configuration for a given set of six leg lengths.     

Adaptive synchronised tracking control for multiple robotic manipulators with uncertain kinematics and dynamics

In this study, a new adaptive synchronised tracking control approach is developed for the operation of multiple robotic manipulators in the presence of uncertain kinematics and dynamics. In terms of the system synchronisation and adaptive control, the proposed approach can stabilise position tracking of each robotic manipulator while coordinating its motion with the other robotic manipulators. On the other hand, the developed approach can cope with kinematic and dynamic uncertainties. The corresponding stability analysis is presented to lay a foundation for theoretical understanding of the underlying issues as well as an assurance for safely operating real systems. Illustrative examples are bench tested to validate the effectiveness of the proposed approach. In addition, to face the challenging issues, this study provides an exemplary showcase with effectively to integrate several cross boundary theoretical results to formulate an interdisciplinary solution.     

Internal singularity analysis of a class of lower mobility parallel manipulators with articulated traveling plate

This paper deals with a particular family of lower mobility parallel kinematic manipulators. The four degrees of freedom of the end-effector consist of three translations plus one rotation with a high tilting angle. Robots belonging to this family are first introduced, and a common parameterization is established. Then an extended kinematic model is proposed for this family of robots using a new Jacobian matrix. Relevant information about robot kinematic singularities, internal singularities, and possible end-effector motions can be obtained by resorting to this matrix. The efficiency of this method is proven by applying it to several traveling-plate architectures corresponding to already built robot prototypes. The results of the expected behavior are compared with the prototype's real behavior. The goal of this paper is to show that internal (or constraint) singularities can occur in lower mobility parallel kinematic manipulators, and to underline the influence they have during the design stage.     

Collaborative Multi-MSA Multi-Target Tracking and Surveillance: a Divide &; Conquer Method Using Region Allocation Trees

This work deals with the problem of the accurate task space control subject to finite-time convergence. Kinematic and dynamic equations of a rigid robotic manipulator are assumed to be uncertain. Moreover, unbounded disturbances, i.e., such structures of the modelling functions that are generally not bounded by construction, are allowed to act on the manipulator when tracking the trajectory by the end-effector. Based on suitably defined task space non-singular terminal sliding vector variable and the Lyapunov stability theory, we derive a class of absolutely continuous (chattering-free) robust controllers based on the estimation of a Jacobian transpose matrix, which seem to be effective in counteracting uncertain both kinematics and dynamics, unbounded disturbances and (possible) kinematic and/or algorithmic singularities met on the robot trajectory. The numerical simulations carried out for a 2DOF robotic manipulator with two revolute kinematic pairs and operating in a two-dimensional task space, illustrate performance of the proposed controllers.     

An Adaptive Regulator of Robotic Manipulators in the Task Space

This note addresses the problem of position control of robotic manipulators both nonredundant and redundant in the task space. A computationally simple class of task space regulators consisting of a transpose adaptive Jacobian controller plus an adaptive term estimating generalized gravity forces is proposed. The Lyapunov stability theory is used to derive the control scheme. The conditions on controller gains ensuring asymptotic stability are obtained herein in a form of simple inequalities including some information extracted from both robot kinematic and dynamic equations. The performance of the proposed control strategy is illustrated through computer simulations for a direct-drive arm of a SCARA type redundant manipulator with the three revolute kinematic pairs operating in a two-dimensional task space.     

Approximate Jacobian control with task-space damping for robot manipulators

In this note, we propose two new approximate Jacobian control laws with task-space damping for setpoint control of robot manipulators. The proposed controllers do not require exact knowledge of the Jacobian matrix and dynamics of the robots. We will show that the end-effector's position converges to a desired position in a finite task space even when the kinematics and Jacobian matrix are uncertain. Experimental results are presented to illustrate the performance of the proposed controllers.