Derivation of dual forces in robot manipulators

According to the principle of transference, a compact three-dimensional representation of a rigid body kinematics is obtained by substituting dual for real numbers. This representation has recently been applied to robotics where in addition to its compactness, it allows to constitute the Jacobian matrix explicitly from the product of the dual transformation matrices with no additional computation.This paper introduces the generalized Jacobian matrix. This matrix consists of the complete dual transformation matrices as opposed to the regular Jacobian matrix which consists of specific columns only. The generalized Jacobian matrix relates force and moment at the end-effector to force and moment in all directions, at the joints. It is therefore possible to use the dual transformation matrices to derive, with no additional computation, the full force and moment vector at the robot's joints. Furthermore, the generalized Jacobian matrix also relates motion in all directions at the joints to the motion of the end-effector, an essential relation required at the design stage of robot manipulators (in particular, flexible ones). An extension of these kinematics and statics schemes into dynamics is possible by applying the dual inertia operator as is shown by an example of a three degrees-of-freedom robot.     

含正则支链的并联机器人雅可比简化分析方法

雅可比矩阵对并联机器人的速度、精度、刚度等性能分析有重要意义。通常雅可比分析需要计算与驱动关节运动螺旋不互易而与支链其他运动螺旋均互易的力螺旋,分析过程复杂,计算量大。为此,针对一类常见的特殊支链结构,提出正则支链的概念,研究了正则支链的结构特征。分析了转动副和移动副的做功驱动力螺旋,提出含正则支链的并联机器人雅可比简化分析方法,克服了现有方法中计算驱动关节产生的支链约束螺旋的复杂过程。这种方法不必计算支链约束螺旋就能获得并联机器人的部分雅可比矩阵,进而结合支链约束螺旋获得并联机器人的整体雅可比矩阵,实现驱动关节与动平台之间运动和力的双向映射。方法不仅适用于常见的6自由度并联机器人,也适用于具有正则支链的少自由度并联机器人。     

Properties and structure of dynamic robot models for control engineering applicationsStructure et proprietes des modeles dynamiques robotiques a appliquer au controle automatique

Controller design for robotic manipulators requires a fundamental physical understanding of the properties and structure of dynamic robot models. This paper focuses on the Lagrangian formulation which is attractive from both the dynamic modeling and control engineering points-of-view. Physical and mathematical properties and structural characteristics of the complete dynamic robot model are demonstrated. Implications of the model for control system analysis and design are then indicated. Physical interpretation leads naturally to the decomposition of the model into the positioning arm and end-effector subsystems and motivates the application of decentralized control to robotic manipulators. The authors then propose the application of control the positioning arm and artificial intelligence and intelligent sensors to control the end-effector.     

一种可用于微创手术的并联机构运动学分析与性能优化

与串联机构相比,并联机构具有刚度好,精度高,响应快等优点,适合用于医疗领域。微创穿刺活检等手术要求机器人机构能够输出远中心运动。提出了一种用于微创手术的新型远中心并联机构,其分支内部包含平行四边形闭环子链。采用螺旋理论对机构自由度进行了分析,证明其能输出两个转动运动和一个移动运动,且所有运动均通过远端固定中心。对机构进行位置分析,建立了驱动参数与末端参数之间的映射。通过速度分析建立了雅可比矩阵,并分析得到了机构的奇异位形,包括逆解奇异、正解奇异和混合奇异。应用搜索法分析了机构的工作空间。采用运动/力传递指标对机构进行了性能分析,绘制工作空间内的性能图谱。以优质空间大小为目标对机构进行了尺度优化。     

On the optimum tolerances of structural parameters and kinematic parameters of robit manipulators

The paper presents a matrix form of the positional and orientational erros of the end-effector of manipulators. By using the matrix theory, the paper reaches a conclusion that if the output kinematic errors of a manipulator are minimum, the aberrations of its structural parameters and the kinematic errors in robot joints must be the components of the eigenvector related to the least eigenvalue of the Jacobian. According to this conclusion, the method for choosing the optimum tolerances of structural parameters and kinematic parameters is also given in the paper. A numerical example of a six DOF manipulator is presented.     

Force transmission analyses with dimensionally homogeneous Jacobian matrices for parallel manipulators

To avoid the unit inconsistency problem in the conventional Jacobian matrix, new formulation of a dimensionally homogeneous inverse Jacobian matrix for parallel manipulators with a planar mobile platform by using three end-effector points was presented (Kim and Ryu, 2003). This paper presents force relationships between joint forces and Cartesian forces at the three End-Effector points. The derived force relationships can then be used for analyses of the input/output force transmission. These analyses, forward and inverse force transmission analyses, depend on the singular values of the derived unit consistent Jacobian matrix. Using the proposed force relationship, a numerical example is presented for actuator size design of a 3-RRR planar parallel manipulator.     

A method of verifying force-closure condition for general cable manipulators with seven cables

This paper introduces a systematic method of verifying the force-closure condition for general 6-DOF cable-driven manipulators with seven cables. Force-closure is defined as that the inverse dynamics problem has a feasible solution for any external wrench exerted on the end-effector. For any given configuration of a 7-cable manipulator, the method can easily determine, by examining the Jacobian matrix of the manipulator, whether a solution of all-positive cable forces exists. The necessary and sufficient conditions of the proposed method are mathematically proven. The paper also addresses the application of the proposed method for determination of the force-closure workspace. As a numerical example, the shape, boundary, dimensions, and volume of the force-closure workspace of a 6-DOF, 7-cable manipulator are discussed.     

旋量理论在机械臂末端执行器运动精度分析中的应用研究

虽然D-H参数法已广泛用于机械臂的运动建模和分析,先前文献中建立机械臂手爪运动误差数学模型主要基于D-H参数法或者修正后的D-H参数法,但D-H参数法在技术上存在根本的缺陷,即所有运动都是关于x和z轴的,无法表示关于y轴的运动。因此用D-H参数法所建立的机械臂手爪运动误差数学模型中无法体现关于y轴所产生的误差。为解决此问题,利用旋量理论建立了一模块化机械臂末端执行器运动误差的数学模型。利用算例对这种方法进行了分析,证明利用旋量理论建立误差模型是有效的。     

Dimensional Synthesis of a 3-DOF Parallel Manipulator with Full Circle Rotation

Parallel robots are widely used in the academic and industrial fields. In spite of the numerous achievements in the design and dimensional synthesis of the low-mobility parallel robots, few research ef...     

Reciprocal screw theory based singularity analysis of a novel 3-DOF parallel manipulator

Singularity analysis is an essential issue for the development and application of parallel manipulators.Most of the existing researches focus on the singularity of parallel manipulators are carried out based on the study of Jacobian matrices.A 3-DOF parallel manipulator with symmetrical structure is presented.The novel parallel manipulator employs only revolute joints and consists of four closed-loop subchains connecting to both base and platform via revolute joints.The closed-loop subchain in each chain-leg is a spherical 6R linkage.The motion characteristics of the output link in the spherical 6R linkage with symmetrical structure are analyzed based on the interrelationships between screw systems.The constraints that are exerted on the platform by each chain-leg are investigated applying the concept of generalized kinematic pair in terms of equivalent screw system.Considering the geometric characteristics of the parallel manipulator,the singularity criteria of the parallel manipulator corresponding to different configurations are revealed based on the dependency of screw system and line geometry.The existing conditions of certain configuration that a singularity must occur are determined.This paper presents a new way of singularity analysis based on disposition of constraint forces on the geometrically identified constraint plane and the proposed approach is capable of avoiding the complexity in solving the Jacobian matrices.     

Maximum allowable dynamic load of flexible mobile manipulators using finite element approach

In this paper a general formulation for finding the maximum allowable dynamic load (MADL) of flexible link mobile manipulators is presented. The main constraints used for the proposed algorithm are the actuator torque capacity and the limited error bound for the end-effector during motion on a given trajectory. The accuracy constraint is taken into account with two boundary lines which are equally offset due to the given end-effector trajectory, while a speed-torque characteristics curve of a typical DC motor is used for applying the actuator torque constraint. The finite element method (FEM), which is able to consider the full nonlinear dynamics of mobile manipulators, is applied to derive the kinematic and dynamic equations. In order to verify the effectiveness of the presented algorithm, two simulation studies considering a flexible two-link planar manipulator mounted on a mobile base are presented and the results are discussed.     

Motion Planning Algorithms of Redundant Manipulators Based on Self-motion Manifolds

The current motion planning approaches for redundant manipulators mainly includes two categories: improved gradient-projection method and some other efficiency numerical methods. The former is excessively sensitive to parameters, which makes adjustment difficult; and the latter treats the motion planning as general task by ignoring the particularity, which has good universal property but reduces the solving speed for on-line real-time planning. In this paper, a novel stepwise solution based on self-motion manifold is proposed for motion planning of redundant manipulators, namely, the chief tasks and secondary tasks are implemented step by step. Firstly, the posture tracking of end-effector is achieved accurately by employing the non-redundant joint. Secondly, the end-effector is set to keep stationary. Finally, self-motion of manipulator is realized via additional work on the gradient of redundant joint displacement. To verify this solution, experiments of round obstacle avoiding are carried out via the planar 3 degree-of-freedom manipulator. And the experimental results indicate that this motion planning algorithm can effectively achieve obstacle avoiding and posture tracking of the end-effector. Compared with traditional gradient projection method, this approach can accelerate the problem-solving process, and is more applicable to obstacle avoiding and other additional work in displacement level.     

少自由度并联机器人机构的静力分析

与6自由度并联机器人不同,少自由度并联机器人机构的支链不仅传递驱动力,同时还要向末端提供约束力,然而在对此类机构进行静力分析时,现有的研究都忽略了约束力的传递特性。为此,以微分流形理论为基础,分析指出少自由度并联机器人末端的完整力空间应包括驱动力子空间和约束力子空间,并以此为基础,利用虚功原理,给出一种建立此类机构输入与输出广义力间完整映射的方法。此完整力映射包括驱动力映射矩阵和约束力映射矩阵。通过计算这两个矩阵的条件数,可以分别分析此类机构驱动力和约束力的传递特性。该分析方法的物理意义和数学理论相统一,且概念清晰,便于深入理解少自由度并联机构静力传递的特性。最后,以3-UPU并联机器人机构为例进行实例分析。     

A centro-based characterization of singularities in the workspace of planar closed-loop manipulators

The paper presents a centro-based characterization of singularities in planar multi-DOF mechanisms with arbitrary kinematic structure. A multi-DOF mechanism is treated as an assembly of several single DOF link sets and hence the terms like affected link set, active link set and active centro have been defined. Criteria for end-effector dependent and end-effector independent singularities of a closed-loop manipulator with given output link have been established in the paper in terms of centroes. Based on the relative disposition of the centroes, the singularities have been classified into five types and their physical implications have been explained. End-effector dependent singularities are shown to lie on the coupler curve of a larger single DOF mechanism. A simple linkage transformation has been proposed for the study of manipulators with physical limits on joint motion under the general framework of close-loop manipulators. Several examples have been worked out to demonstrate the centro-based method of singularity analysis.     

两机械臂协调操作的容错运动规划

针对两机械臂协调操作中发生关节坏死这类故障时的容错运动规划问题进行研究。首先,建立了两机械臂协调操作的运动学方程,给出了容错空间的计算步骤。然后,构造了反映机械臂末端运动与容错空间相对位置关系的中心度指标,并基于这一指标提出了机械臂协调操作的容错运动规划算法。利用该算法可以确定被持物体的最优初始位置,从而同时实现故障时刻和故障后的容错操作。最后,利用两面积3R机械臂的仿真实例证明了其有效性。     

A successive approximation algorithm for the inverse position analysis of the general serial manipulators

A new fast numerical algorithm for the solution of the inverse position analysis of the serial manipulator is presented. With the algorithm, the inverse solution of the general serial manipulators can be achieved quickly under the desired precision when we know the position of the three non-collinear end-effector points, which is much easier to be measured than the orientation of the end-effector. The difference between the new numerical algorithm and the general iterative method is that it can search out the solution when the manipulator is at the singular configuration; and the initial configuration used in the successive approximation process may also be the singular configuration. So the convergence domain is bigger than that of the general iterative method. The position analysis of the general 6R serial manipulator is illustrated in the literature as an example and the simulation results to verify the efficiency of the proposed algorithm. Since the three non-collinear end-effector points can be selected at random, the algorithm can be applied to any other serial manipulator.     

基于互易螺旋理论的无奇异完全各向同性移动并联机构型综合

基于互易螺旋理论提出一种无奇异完全各向同性纯移动并联机构型综合的系统方法。首先给出机构末端操作器输出运动的矩阵表达式,然后根据该表达式正雅可比矩阵为非零对角阵的条件确定出机构支路驱动螺旋的表达形式,并依据驱动螺旋与主动螺旋互易积为非零常数的条件确定出主动螺旋的形式,再根据驱动螺旋与同一分支中除主动螺旋外的所有其他运动螺旋互易积都为零的特点,确定出可动非驱动螺旋的类型及其轴线方向,从而按照机构分支连接度的不同进行机构支路的型综合。最后取3条所综合出的机构支路将动平台与静平台联接起来即可得到所期望的机构。最后对一种新型3-CRP无奇异完全各向同性移动并联机构进行运动学分析。结果表明,所提出的型综合方法正确可行。     

Point-based Jacobian formulation for computational kinematics of manipulators

Computational kinematic analysis of mechanisms is a promising tool for the development of new classes of manipulators. In this paper, the authors present a velocity equation to be compiled by general-purpose software and applicable to any mechanism topology. First, the approach to model the mechanism is introduced. The method uses a set of kinematic restrictions applied to characteristic points of the mechanism. The resultant velocity equation is not an input–output equation, but a comprehensive one. In addition, the Jacobian characterizing the equation is dimensionless hence extremely useful for singularity and performance indicators. The motion space of the manipulator is obtained from the velocity equation. Angular velocities are compiled out of three non-collinear point velocities. The procedure is applied to a 3-DoF parallel manipulator to illustrate.     

自由浮动冗余度空间机器人的姿态稳定控制

研究了自由浮动冗余度空间机器人的姿态稳定控制问题。针对自由浮动空间机器人姿态稳定控制的主要特点,通过分析自由浮动冗余度空间机械臂惯性矩阵的伪逆和零空间特性,结合空间机器人的运动学方程推导出了基于本体姿态稳定的广义雅可比矩阵。采用基于该矩阵的分解运动速度控制方法,保证机械臂末端执行器在跟踪期望轨迹的同时不对本体姿态产生干扰。最后,对平面三自由度空间机器人进行了仿真实验,实验结果表明了该方法的有效性。     

A new screw theory method for the estimation of position accuracy in spatial parallel manipulators with revolute joint clearances

This paper presents a novel method based on screw theory for the analysis of position accuracy in spatial parallel manipulators with revolute joint clearances. The method is general, and can tackle with an arbitrary pose error function, expressed as a quadratic function of the end-effector displacement.The method performs a maximization of the pose error function, based on a 2-step computational procedure. The first computational step is analytical and leads to a sub-optimal estimate of the maximum pose error. This analytical solution represents the exact maximum pose error for the calculus of the angular accuracy in the special case of fully translational parallel mechanisms. The second computational step is numerical, and starting from the previous solution, can converge to the exact maximum pose error in a limited number of iterations.The relevance of the method is demonstrated through some application examples, where the worst-case angular and linear position accuracy in translating fully parallel manipulators is determined. As a further contribution, this paper shows how the position accuracy due to joint clearances in parallel manipulators is strictly dependent of the mechanism pose and its association to kinematic isotropy.