Nonholonomic Mobile Manipulators: Kinematics, Velocities and Redundancies

We consider nonholonomic mobile manipulators built from an n _a joint robotic arm and a nonholonomic mobile platform with two independently driven wheels. Actually, there is no efficient kinematic formalism for these systems which are generally characterized by their high number of actuators. So, kinematic modelling is presented with particular emphasis on redundancy. Whereas kinematic redundancy is well known in the holonomic case, it is pointed out that it is necessary to define velocity redundancy in the case of nonholonomic systems. Reduced velocity kinematics based on quasi-velocities are shown to provide an efficient formalism. Two examples of mobile manipulators are presented. Finally, reduced velocity kinematics and velocity redundancy are shown to be adequate tools in order to realize operational task while optimizing criteria such as manipulability.     

Parallel and distributed trajectory generation of redundantmanipulators through cooperation and competition among subsystems

Autonomous distributed control (ADC) is one of the most attractive approaches for more versatile and autonomous robot systems. The paper proposes a parallel and distributed trajectory generation method for redundant manipulators through cooperative and competitive interactions among subsystems composing the ADC that is based on a concept of virtual arms. The virtual arm has the same kinematic structure as the manipulator except that its end point is located on a joint or link of the manipulator. Then the redundant manipulator can be represented by a set of the virtual arms. Trajectory generation and point to point control of the redundant manipulator are discussed, and it is shown that the kinematic redundancy of the manipulator can be utilized positively in the generated trajectories by using the virtual arms.     

An efficient method for inverse dynamics of manipulators based on the virtual work principle

The computational efficiency of inverse dynamics of a manipulator is important to the real-time control of the system. For serial manipulators, the recursive Newton-Euler method has been proven to be the most efficient. However, for more general manipulators, such as serial manipulators with closed kinematic loops or parallel manipulators, it must be modified accordingly and the resultant computational efficiency is degraded. This article presents a computationally efficient scheme based on the virtual work principle for inverse dynamics of general manipulators. The present method uses a forward recursive scheme to compute velocities and accelerations, the Newton-Euler equation to calculate inertia forces/torque, and the virtual work principle to formulate the dynamic equations of motion. This method is equally effective for serial and parallel manipulators. For serial manipulators, its computational efficiency is comparable to the recursive Newton-Euler method. For parallel manipulators or serial manipulators with closed kinematic loops, it is more efficient than the existing methods. As an example, the computations of inverse dynamics (including inverse kinematics) of a general Stewart platform require only 842 multiplications, 511 additions, and 12 square roots.     

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.     

An algorithmic approach to coordinate transformation for robotic manipulators

Coordinate transformation is one of the most important issues in robotic manipulator control. Robot tasks are naturally specified in work space coordinates, usually a Cartesian frame, while control actions are developed on joint coordinates. Effective inverse kinematic solutions are analytical in nature; they exist only for special manipulator geometries and geometric intuition is usually required. Computational inverse kinematic algorithms have recently been proposed; they are based on general closed-loop schemes which perform the mapping of the desired Cartesian trajectory into the corresponding joint trajectory. The aim of this paper is to propose an effective computational scheme to the inverse kinematic problem for manipulators with spherical wrists. First an insight into the formulation of kinematics is given in order to detail the general scheme for this specific class of manipulators. Algorithm convergence is then ensured by means of the Lyapunov direct method. The resulting algorithm is based on the hand position and orientation vectors usually adopted to describe motion in the task space. The analysis of the computational burden is performed by taking the Stanford arm as a reference. Finally a case study is developed via numerical simulations.     

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.     

基于梯度下降和二分法的移动机器人轨迹规划方法

轨迹规划是移动焊接机器人轨迹控制的基础,是该系统中重要的组成部分。为了提高多关节移动焊接机器人轨迹规划的效率和精确性,同时考虑到多关节焊接机器人的运动特性提出了一种梯度下降法和二分法结合的轨迹规划方法。移动焊接机器人由机械杆和机器人移动平台组成,由于移动平台提供移动性使得移动焊接机器人相对固定的机械臂有更大的工作空间。近年来,此类系统的研究已在经学术界和工业界迅猛发展。论文首先建立移动焊接机器人的运动学模型,并且阐述梯度下降法和二分法结合算法的设计步骤。然后,采用典型的正弦波形作为焊缝轨迹,通过仿真验证该方法的应用前景和可行性。     

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.     

Inverse kinematics of planar redundant manipulators via virtual links with configuration index

This article presents a new method for generating inverse kinematic solutions for planar manipulators with large redundancy (hyper-redundant manipulators). The proposed method starts by decomposing a planar redundant manipulator into a series of local planar arms that are either 2-link or 3-link manipulator modules, and connecting the conjunction points between them with virtual links. The manipulator then can be handled by a simple virtual link system, which may be conveniently divided into non-singular and singular cases depending on its configuration. When the virtual link system is no longer effective due to a singular configuration, the displacement of the end-effector is then allocated to virtual links according to a displacement distribution criterion. A dexterity index called the “configuration index” distinguishes the non-singular and singular cases. The concept of virtual link is shown by computer simulations to be simple and effective for the inverse kinematics of a planar hyper-redundant manipulator with a discrete model. In particular, it can be applied to solving the inverse kinematics of a SCARA-type spatial redundant manipulator whose redundancy is included in its planar mechanism. © 1994 John Wiley & Sons, Inc.     

A practical method for the deformation of long-stroke hydraulic manipulators in grasping-handling tasks

Long-stroke hydraulic manipulators are utilized in various grasping-handling tasks, and the flexible deformation of these manipulators is the primary obstacle that affects precise position control of the end-effectors in Cartesian space. This deformation is manifested in the following three aspects: joint deformation, structural deformation and clearance variation. Due to deformation uncertainty, methods that model the hydraulic manipulator as a combination of flexible multibody systems and hydraulic actuators are unsuitable. In this article, we propose an incremental inverse kinematics model (IIKM) as a new approach to solving the above deformation difficulties. The projection method is used to obtain the inverse kinematic analytical solution of long-stroke hydraulic manipulators, which is based on the manipulator deformation in the current configuration (current configuration refers to the arrangement of the manipulator links when the manipulator starts to move to the target position). The proposed method avoids complex flexible multibody modeling and parameter identification, allowing long-stroke hydraulic manipulators to be accurately controlled within a certain neighborhood. An evaluation coefficient is proposed to analyze the calculation accuracy of the IIKM in combination with the success rate obtained from 190 grasping experiments. Through these experiments, we determine the optimal calculation height range of the IIKM in the vertical direction and the optimal calculation position area in the horizontal direction and prove that the IIKM result can guarantee the success of grasping-handling tasks when the end-effector is within the optimal calculation height range.     

Singularities in motion and displacement functions for a 7 degree-of-freedom manipulator

The authors propose a method for the determination of singularities in motion and displacement functions for a seven degree-of-freedom manipulator. The manipulator is considered, hereby, as a set of six degree-of-freedom manipulators. It is proven that two types of singularities in motion can occur at link positions that are independent and dependent with respect to the trajectory to be executed. The relations between the structure of singularity equations and displacement equations are discussed. The derivation of displacement equations for a manipulator with singularities of the second type is based on the idea of modeling of a manipulator by two open kinematic chains and invariants that may be found for these chains. The inverse kinematic equations and the equations of singularities in motion have been derived using a symbolic computer program which can handle manipulators of general structure (with five, six, and seven degrees-of-freedom). This program is written in the Symbolic Computation Language REDUCE.     

A novel formulation for determining joint constraint loads during optimal dynamic motion of redundant manipulators in DH representation

The kinematic representations of general open-loop chains in many robotic applications are based on the Denavit–Hartenberg (DH) notation. However, when the DH representation is used for kinematic modeling, the relative joint constraints cannot be described explicitly using the common formulation methods. In this paper, we propose a new formulation of solving a system of differential-algebraic equations (DAEs) where the method of Lagrange multipliers is incorporated into the optimization problem for optimal motion planning of redundant manipulators. In particular, a set of fictitious joints is modeled to solve for the joint constraint forces and moments, as well as the optimal dynamic motion and the required actuator torques of redundant manipulators described in DH representation. The proposed method is formulated within the framework of our earlier study on the generation of load-effective optimal dynamic motions of redundant manipulators that guarantee successful execution of given tasks in which the Lagrangian dynamics for general external loads are incorporated. Some example tasks of a simple planar manipulator and a high-degree-of-freedom digital human model are illustrated, and the results show accurate calculation of joint constraint loads without altering the original planned motion. The proposed optimization formulation satisfies the equivalent DAEs.     

A general solution for the position,velocity, and acceleration of hyperredundant planar manipulators

A new approach for the solution of the position, velocity, and acceleration of hyperredundant planar manipulators following any twice‐differentiable desired path is presented. The method is singularity free, and provides a robust solution even in the event of mechanical failure of some of the robot actuators. The approach is based on defining virtual layers, and dividing them into virtual/real three‐link or four‐link subrobots. It starts by solving the inverse kinematic problem for the subrobot located in the lowest virtual layer, which is then used to solve the inverse kinematic equations for the subrobots located in the upper virtual layers. An algorithm is developed that provides a singularity‐free solution up to the full extension through a configuration index. The configuration index can be interpreted as the average of the determinants of the Jacobians of the subrobots. The equations for the velocities and accelerations of the manipulator are solved by extending the same approach, and it is shown that the value of the configuration index is critical in maintaining joint velocity continuity. The inverse dynamic problem of the robot is also solved to obtain the torques required for the robot actuators to accomplish their tasks. Computer simulations of several hyperredundant manipulators using the proposed method are presented as numerical examples. © 2002 John Wiley & Sons, Inc.     

Clamping weighted least-norm method for the manipulator kinematic control with constraints

One of the main challenges for kinematic control of manipulators is how to effectively address various constraints, such as obstacle avoidances. Most constraints can be converted into a virtual joint-limit problem by the general weighted least-norm method reported recently. However, a chattering of weighted factors might occur in the presence of disturbances. In this paper, in order to improve the robustness to disturbances, a clamping term forcing joints away from their limits is added to the solution of inverse kinematics, termed as the clamping weighted least-norm method. It is proved that the joint-limit constraint is always guaranteed. The proposed method is applicable to both non-redundant and redundant manipulators; the accuracy of main task is sacrificed only when there is confliction between main task and joint-limit constraint. Experiments on the seven-degree-of-freedom redundant manipulator illustrate the good performance for avoiding joint limits while tracking a given trajectory.     

多移动机械臂系统动力学建模以及稳定性分析

随着社会生产力的发展和发展需求的提高,移动机械臂凭借着自身优势,受到学术界和工业界的广泛关注.但在许多工作场景下,单个移动机械臂有着自由度数以及载荷的限制,无法顺利完成任务.为了更好地满足任务需求,多移动机械臂系统应运而生.在上述工业背景下,本文建立了多移动机械臂系统的动力学模型,并针对该动力学方程进行了稳定性分析.首先通过拉格朗日方程建立单个移动机械臂的动力学方程,将多体动力学软件仿真结果同动力学模型数值计算结果进行对比,验证了模型的正确性.随后联立多个移动机械臂的动力学方程和操作对象的动力学方程,得到封闭形式的多移动机械臂系统的动力学方程.再利用关节位置误差和速度误差设计李雅普诺夫函数,通过反步法获得了关节力矩的控制律.最后在多体动力学软件仿真中,察看轨迹是否能跟踪上期望信号来检验控制律的有效性.     

A new approach to improve absolute positioning accuracy of robot manipulators

This article describes a new calibration system for robot manipulators which improves their absolute positioning accuracy by using parameter-estimation algorithms based on the Newton method. When 3D position data of the specified points on a manipulator and the joint encoder values are input to the calibration system, the system estimates the offset values of joint encoders, link lengths, and position and orientation of the manipulator base coordinate system with respect to the world coordinate system which is difficult to obtain by conventional calibration methods. This calibration system can be applied to various manipulator types by just changing the basic kinematic equations. The system employs an algebraic programming system called REDUCE to automatically reduce the manipulator kinematic equation and partial differential calculus in the Newton method. For efficiency, first only the arm part with three degrees of freedom and then the hand part are calibrated. The experimental results demonstrate the effectiveness of this system by reducing the robot's absolute positioning errors to the order of repeatability errors.     

一种基于指数积的移动机械臂联合标定方法

提出一种基于指数积的移动机械臂联合标定方法,以实现移动平台和机械臂两者间位姿标定与机械臂运动学参数标定模型的统一.机械臂运动学参数标定使用最多的是基于D-H参数法,但D-H参数法无法克服相邻关节平行或接近平行时的奇异性问题,以及建模过程复杂、建模后的模型通用性差等问题.基于指数积的移动机械臂联合标定方法建模时不会因为关节轴平行而出现奇异性问题,建模过程简单.通过对整个系统的运动学方程进行微分运算,获得末端位姿误差和移动机械臂零位状态旋量误差及关节旋量误差的线性化模型.利用伴随矩阵方式建立关节旋量理论值与关节旋量实际值的关系,并通过改变伴随矩阵实现基于最小二乘法的参数辨识计算过程中参数更新.使用高精度激光跟踪仪作为测量工具,通过实验验证所提出方法的有效性.     

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.     

Robot manipulator kinematic compensation using a generalized jacobian formulation

The kinematic error compensation of robot manipulators is at present being attempted by improving the precision of the nominal robot kinematic parameters. This paper addresses the problem of kinematic compensation using a new mathematical joint model proposed to account for shortcomings in existing methods. The corrected manipulator transformation is formulated in terms of “generalized Jacobians”: relating differential errors at the joints to the differential change in the manipulator transformation. The details of application are discussed for a particular industrial manipulator.     

冗余机器人的双向自运动路径规划

冗余机器人的自运动路径规划是在保持手端任务向量不变的情况下,在关节空间内寻找一条连接机器人初始关节构形和期望关节构型的几何路径.本文给出一种双向自运动路径规划算法,其基本思想是使位于初始关节构形的真实机器人和位于期望关节构形的虚拟机器人在自运动流形上运动并收敛到同一关节构形,从而得到一条连接初始和期望关节构形的自运动几何路径.该算法克服了以往算法容易陷入局部极小构形的缺陷.仿真结果证实了算法的有效性.