Inverse velocity analysis for line guidance five-axis robots

In this paper, inverse velocity problem for five-axis robots is investigated. The conventional method for a five-axis robot is to pseudo-inverse the 6×5 Jacobian matrix. The solution, primarily based on six freedoms inverse velocity analysis, is just an approximation with a least-square error. A five-axis robot can exactly guide an axis-symmetrical tool in 3-D space. Two exact solutions are provided for five-axis robots. One is based on the screw motion of the tool. The other is based on spherical angles of the tool to derive a 5×5 Jacobian matrix. A new type of singular configuration is discovered and is called the task singularity. The moving path of the line shaped tool is constructed as a ruled surface. Analysis of the angular acceleration shows the surface constructed based on the spherical angle representation has better characteristic. It is concluded that for five-axis robots, the tool position is better represented by five parameters rather than six parameters in order to get better solutions for inverse velocity as well as the motion planning.     

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.     

Kinematic modeling of mobile robots by transfer method of augmented generalized coordinates

A kinematic modeling method, which is directly applicable to any type of planar mobile robots, is proposed in this work. Since holonomic constraints have the same differential form as nonholonomic constraints, the instantaneous motion of the mobile robot at current configuration can be modeled as that of a parallel manipulator. A pseudo joint model denoting the interface between the wheel and the ground (i.e., the position of base of the mobile robot) enables the derivation of this equivalent kinematic model. The instantaneous kinematic structures of four different wheels are modeled as multiple pseudo joints. Then, the transfer method of augmented generalized coordinates, which has been popularly employed in modeling of parallel manipulators, is applied to obtain the instantaneous kinematic models of mobile robots. The kinematic models of six different types of planar mobile robots are derived to show the effectiveness of the proposed modeling method. Lastly, for the mobile robot equipped with four conventional wheels, an algorithm estimating a sensed forward solution for the given information of the rotational velocities of the four wheels is discussed. © 2004 Wiley Periodicals, Inc.     

Numerical inverse kinematics for modular reconfigurable robots

The inverse kinematics solutions of a reconfigurable robot system built upon a collection of standardized components is difficult to obtain because of its varying configurations. This article addresses the formulation of a generic numerical inverse kinematics model and automatic generation of the model for arbitrary robot geometry including serial and tree‐typed geometries. Both revolute and prismatic types of joints are considered. The inverse kinematics is obtained through the differential kinematics equations based on the product‐of‐exponential (POE) formulas. The Newton–Raphson iteration method is employed for solution. The automated model generation is accomplished by using the kinematic graph representation of a modular robot assembly configuration and the related accessibility matrix and path matrix. Examples of the inverse kinematics solutions for different types of modular robots are given to demonstrate the applicability and effectiveness of the proposed algorithm. ©1999 John Wiley & Sons, Inc.     

A novel XY-Theta precision table and a geometric procedure for its kinematic calibration

Spatial precision positioning devices are often based on parallel robots, but when it comes to planar positioning, the well-known serial architecture is virtually the only solution available to industry. Problems with parallel robots are that most are coupled, more difficult to control than serial robots, and have a small workspace. In this paper, new parallel robot is proposed, which can deliver accurate movements, is partially decoupled and has a relatively large workspace. The novelty of this parallel robot lies in its ability to achieve the decoupled state by employing legs of a different kinematic structure. The robot repeatability is evaluated using a CMM and so are the actual lead errors of its actuators. A simple geometric method is proposed for directly identifying the actual base and mobile reference frames, two actuator's offsets and one distance parameter, using a measurement arm from FARO Technologies. While this method is certainly not the most efficient one, it yields a satisfactory improvement of the robot accuracy without the need for any background in robot calibration. An experimental validation shows that the position accuracy achieved after calibration is better than 0.339 mm within a workspace of approximately 150 mm×200 mm.     

Position,Jacobian and workspace analysis of a 3-PSP spatial parallel manipulator

This paper investigates the problems of kinematics, Jacobian, singularity and workspace analysis of a spatial type of 3-PSP parallel manipulator. First, structure and motion variables of the robot are addressed. Two operational modes, non-pure translational and coupled mixed-type are considered. Two inverse kinematics solutions, an analytical and a numerical, for the two operational modes are presented. The direct kinematics of the robot is also solved utilizing a new geometrical approach. It is shown, unlike most parallel robots, the direct kinematics problem of this robot has a unique solution. Next, analytical expressions for the velocity and acceleration relations are derived in invariant form. Auxiliary vectors are introduced to eliminate passive velocity and acceleration vectors. The three types of conventional singularities are analyzed. The notion of non-pure rotational and non-pure translational Jacobian matrices is introduced. The non-pure rotational and non-pure translational Jacobian matrices are combined to form the Jacobian of constraint matrix which is then used to obtain the constraint singularity. Finally, two methods, a discretization method and one based on direct kinematics are presented and robot non-pure translation and coupled mixed-type reachable workspaces are obtained. The influence of tool length on workspace is also studied.     

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).     

Kinematic Design of Modular Reconfigurable In-Parallel Robots

This paper describes the kinematic design issues of a modular reconfigurable parallel robot. Two types of robot modules, the fixed-dimension joint modules and the variable dimension link modules that can be custom-designed rapidly, are used to facilitate the complex design effort. Module selection and robot configuration enumeration are discussed. The kinematic analysis of modular parallel robots is based on a local frame representation of the Product-Of-Exponentials (POE) formula. Forward displacement analysis algorithms and a workspace visualization scheme are presented for a class of three-legged modular parallel robots. Two three-legged reconfigurable parallel robot configurations are actually built according to the proposed design procedure.     

Design philosophy of an isotropic six-axis serial manipulator

Presented in this paper is the design philosophy employed for the constructtion of DIESTRO, an isotropic, six-axis, serial manipulator. The kinematic criteria applied so far in manipulator design have been based largely on kinematic solvability, in the sense of allowing for closed-form inverse kinematic solutions. As opposed to this rather limiting criterion, DIESTRO was designed kinematically so as to having a set of configurations in which its Jacobian matrix allows its inversion without roundoff error amplification. Although the basic kinematic chain is of the serial type, this design criterion led to an architecture not admitting closed-form inverse kinematic solutions. The central task was to produce an accurate robot under the prescribed specifications. It is believed that, under similar workspace and load specifications, the particularly challenging design of many other serial manipulators with complex architectures can benefit from the design guidelines given here.     

自由漂浮空间机器人回避动力学奇异的轨迹规划

针对非完整约束导致的自由漂浮空间机器人广义雅可比矩阵动力学奇异性问题,提出一种间接求解机器人逆运动学的方案.该方案通过运动方程分解并构造迭代函数,将动力学奇异转换为运动学奇异问题,避免直接求解逆广义雅可比矩阵,解决广义雅可比矩阵动力学参数相关特性问题,仿真结果表明该方法能够有效地实现自由漂浮空间机器人回避动力学奇异的非完整轨迹规划.     

Path Planning of Free Floating Prismatic-Jointed Manipulators

Reaching a desired position with a specific orientationin space by a robot, mounted on a freely floating base, is an importantpath planning and control problem. Research in this area has mainlyconcentrated on the use of revolute-jointed serial manipulators. It iswell known that the dynamic equations of such manipulators are quitecomplex.In this paper, we propose the use of a 6-link fully prismatic-jointedrobot to achieve a desired position and orientation in space instead of arevolute-jointed robot. The use of pure prismatic-jointed robots forsuch a purpose is counter intuitive. On earth, such a structure is unableto provide a desired orientation to the end-effector. However, it can beshown that in space, arbitrary end-effector orientations are possible.Due to the relative simplicity of kinematics, dynamics and workspace ofprismatic-jointed robots compared to revolute-jointed robots, their useresults in significant computational advantages in path planning andcontrol.Also, in this paper, we adopt an unconventional motion planning methodthat avoids inversion of the Jacobian matrix and results in singularityfree paths for the end-effector. In this method, the joint trajectoriesare considered to be modal sums of basis functions of time. Within this framework, constraints on jointangles and joint rates can be imposed. The results are demonstratedwith an example of a 6-link fully prismatic-jointed robot.     

Analytic singularity analysis of a 4-DOF parallel robot based on Jacobian deficiencies

In this paper, analytic singularity analysis of a 4-DOF parallel robot H4 is addressed. Since a parallel manipulator consisting of several serial chains has complex singularities in the workspace, the determination of singular configurations is very important in design, trajectory planning, and control. The classical method to determine singular configurations is to find the determinant of the Jacobian matrix. However, the Jacobian matrix of a parallel manipulator is complex in general and thus it is not easy to find the determinant of the Jacobian matrix. Therefore, we focus on the analytic singularity analysis of a 4-DOF parallel robot H4 using Jacobian deficiencies. A subset of the whole singularities and the intuitively predictable ones are only derived using Jacobian matrix deficiency. Three type singularities, i.e., overmobility, undermobility and combined singularities, have been presented.     

Neural Network Solution for Forward Kinematics Problem of Cable Robots

Forward kinematics problem of cable robots is very difficult to solve the same as that of parallel robots and in the contrary to the serial manipulators’. This problem is almost impossible to solve analytically because of the nonlinearity and complexity of the robot’s kinematic equations. Numerical methods are the most common solutions for this problem of the parallel and cable robots. But, convergency of these methods is the drawback of using them. In this paper, neural network approach is used to solve the forward kinematics problem of an exemplary 3D cable robot. This problem is solved in the typical workspace of the robot. The neural network used in this paper is of the MLP type and a back propagation procedure is utilized to train the network. A simulation study is performed and the results show the advantages of this method in enhancement of convergency together with very small modeling errors.     

Kinematic modeling of wheeled mobile robots

We formulate the kinematic equations of motion of wheeled mobile robots incorporating conventional, omnidirectional, and ball wheels.¹ We extend the kinematic modeling of stationary manipulators to accommodate such special characteristics of wheeled mobile robots as multiple closed-link chains, higher-pair contact points between a wheel and a surface, and unactuated and unsensed wheel degrees of freedom. We apply the Sheth-Uicker convention to assign coordinate axes and develop a matrix coordinate transformation algebra to derive the equations of motion. We introduce a wheel Jacobian matrix to relate the motions of each wheel to the motions of the robot. We then combine the individual wheel equations to obtain the composite robot equation of motion. We interpret the properties of the composite robot equation to characterize the mobility of a wheeled mobile robot according to a mobility characterization tree. Similarly, we apply actuation and sensing characterization trees to delineate the robot motions producible by the wheel actuators and discernible by the wheel sensors, respectively. We calculate the sensed forward and actuated inverse solutions and interpret the physical conditions which guarantee their existence. To illustrate the development, we formulate and interpret the kinematic equations of motion of Uranus, a wheeled mobile robot being constructed in the CMU Mobile Robot Laboratory.     

全柔性铰链平面并联微动机器人的有限元分析

介绍了并联机器人的特点及其应用,对全柔性铰链平面并联机器人建立了刚性模型,并采用闭环线型原理建立理论运动学线性模型（Jacobian矩阵）,用ANSYS软件对其进行有限元分析,得到有限元运动学模型（Jacobian矩阵值）,讨论两者关系,发现有限元模型比理论模型要精确.     

A generalized kinematic modeling method for modular robots

Modular robots can be defined as reconfigurable mechanical arms which can be automatically controlled using suitable motion control software. In this article, a generalized kinematic modeling method is presented for such modular robots. This method can be used to derive the individual kinematic models of all the mechanical elements that make up the inventory of modular units, independently of their geometry and sequence of assembly into a robot. A general procedure is also presented to derive a global kinematic model of any robot configured using these modular units. The kinematic modeling technique of the units is based on Denavit-Hartenberg (D-H) parameter notation. A provision is also presented for converting “non D-H” parameter transformations, obtained in assembling the kinematic chain, into D-H transformations. This D-H conversion feature allows the modeling technique to preserve its generality when a kinematic model is obtained for the specific robot configuration at hand. The conceptual design of modular robot units that is under development in the Computer Integrated Manufacturing Laboratory (CIML) is also presented to show the feasibility of a modular approach to robot design and to clarify some of the mathematical for mulations developed in the article.     

Forward and inverse kinematics of a 5-DOF hybrid robot for composite material machining

Hybrid robots consist of both serial and parallel mechanisms, which have advantages in stiffness and workspace compared with serial/parallel robots when machining composite material. However, the forward and inverse kinematics of hybrid robots generally do not have analytic solutions. This paper deals with the analytic forward and inverse kinematics solutions of a 5-degree-of-freedom (DOF) hybrid robot which consists with a 3-DOF 2UPU/SP parallel mechanism (PM) and a 2-DOF rotating head. In the forward kinematic problem, a method is proposed to transfer the high order kinematic equation to a 4th-order polynomial based on the Sylvester's dialytic elimination, and the analytic solutions can be further obtained by Ferrari's method. In the inverse problem, the redundant Euler angles expressed by four rotations are firstly proposed for decoupling different motions, then, the closed-form solution of inverse kinematics can be found. Finally, a simulation trajectory is given, and the result shows that the accuracy of the solutions’ calculation reaches femtometer grade and the efficiency reaches microsecond grade; furthermore, an experiment is performed on the prototype to validate the effectiveness of the proposed forward and inverse kinematics.     

Kinematic calibration of a 5-axis parallel machining robot based on dimensionless error mapping matrix

Accuracy problem is one of the most challenging issues for the application of parallel robots in manufacturing industry, and kinematic calibration is a feasible approach to solve it. Although lots of researches have brought up a diversity of calibration methods, there are still rooms for the improvement of the accuracy, efficiency and robustness of these calibration effects. In this paper, an improved method for kinematic calibration of a 5-axis parallel machining robot is proposed, which includes a new forward kinematic solution (FKS) based on dual quaternion and a modified error modeling process leading to dimensionless error mapping matrixes (EMMs). On this basis, an iterative identification procedure is schemed, and the kinematics and identification simulations are carried out. The kinematics simulation results show that the proposed FKS has wider convergence range and faster computation speed than Levenberg-Marquardt algorithm, while the identification simulation results show that the residual pose errors with the proposed dimensionless EMMs are lower than that with the conventional EMM in various units. Additionally, the procedure of the full pose measurement with a laser tracker and an auxiliary tool is introduced, and thereby the contrast experiments of kinematic calibration on the prototype are conducted. The experiment results indicate that the residual position and orientation errors based on the dimensionless EMM decrease by 97.67% and 86.85% of the original values, respectively, at least, and by 76.77% and 38.65% of that based on the conventional EMM, respectively, at most. Consequently, it is further confirmed that the proposed calibration method is effective in enhancing the identification accuracy of the geometric errors and improving the positioning accuracy of the studied parallel robot.     

Formal analysis of the kinematic Jacobian in screw theory

As robotic systems flourish, reliability has become a topic of paramount importance in the human–robot relationship. The Jacobian matrix in screw theory underpins the design and optimization of robotic manipulators. Kernel properties of robotic manipulators, including dexterity and singularity, are characterized with the Jacobian matrix. The accurate specification and the rigorous analysis of the Jacobian matrix are indispensable in guaranteeing correct evaluation of the kinematics performance of manipulators. In this paper, a formal method for analyzing the Jacobian matrix in screw theory is presented using the higher-order logic theorem prover HOL4. Formalizations of twists and the forward kinematics are performed using the product of exponentials formula and the theory of functional matrices. To the best of our knowledge, this work is the first to formally analyze the kinematic Jacobian using theorem proving. The formal modeling and analysis of the Stanford manipulator demonstrate the effectiveness and applicability of the proposed approach to the formal verification of the kinematic properties of robotic manipulators.     

机器人运动分析的一组新等式推证及应用

本文推证了一组刚体旋转矩阵与角速度矢的新等式，以该组等式为基础，文中提出并讨论了一种利用旋转矩阵的导数矩阵分析机器人运动的方法，所提方法在文中的运用表明，它可以取代以往过于繁琐的图解法和递推法，成为一种简明有效的机器人运动分析方法，作为该法应用结果，文中给出了由ｎ个杆件构成的ｎ个自由度机器人雅可比矩阵的一种简明封闭式，具有一定实和价值，这里的方法可作为一般刚体运动分析理论的补充。