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.     

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.      

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.     

Iterative learning control and the singularity robust Jacobian inverse for mobile manipulators

The method of iterative learning control, to a large extent, has been inspired by robotics research, focused on the control of stationary manipulators. In this article we deal with the inverse kinematics problem for mobile manipulators, and show that a very basic singularity robust Jacobian inverse can be derived in a natural way within the framework of iterative learning control. To achieve this objective we have exploited the endogenous configuration space approach. The introduced Jacobian inverse defines the singularity robust Jacobian inverse kinematics algorithm for mobile manipulators. A Kantorovich-type estimate of the region of guaranteed convergence of the algorithm is derived. For two example kinematics, this estimate has been computed efficiently.     

A repeatable inverse kinematics algorithm with linear invariant subspaces for mobile manipulators.

On the basis of a geometric characterization of repeatability we present a repeatable extended Jacobian inverse kinematics algorithm for mobile manipulators. The algorithm's dynamics have linear invariant subspaces in the configuration space. A standard Ritz approximation of platform controls results in a band-limited version of this algorithm. Computer simulations involving an RTR manipulator mounted on a kinematic car-type mobile platform are used in order to illustrate repeatability and performance of the algorithm.     

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.     

A Class of Transpose Jacobian‐based NPID Regulators for Robot Manipulators with an Uncertain Kinematics

Transpose Jacobian‐based controllers present an attractive approach to robot set‐point control in Cartesian space that derive the end‐effector posture to a specified desired position and orientation with neither solving the inverse kinematics nor computing the inverse Jacobian. By a Lyapunov function with virtual artificial potential energy, a class of complete transpose Jacobian‐based Nonlinear proportional‐integral‐derivative regulators is proposed in this paper for robot manipulators with uncertain kinematics on the basis of the set of all continuous differentiable increasing functions. It shows globally asymptotic stability for the result closed‐loop system on the condition of suitable feedback gains and suitable parameter selection for the corresponding function set as well as artificial potential function, and only upper bound on Jacobian matrix error and Cartesian dynamics parameters are needed. The existing linear PID (LPID) regulators are the special cases of it. Nevertheless, in the case of LPID regulators, only locally asymptotic stability is guaranteed if the corresponding conditions are satisfied. Simulations demonstrate the result and robustness of transpose Jacobian‐based NPID regulators. © 2002 Wiley Periodicals, Inc.     

Feedback control for robotic manipulator with an uncertain Jacobian matrix

In most applications of robots, a desired path for the end‐effector is usually specified in task space such as Cartesian space. One way to move the robot along this path is to solve the inverse kinematics problem to generate the desired angles in joint space. However, it is a very time consuming task to solve the inverse kinematics problem. Furthermore, in the presence of uncertainty in kinematics, it is impossible to derive the desired joint angle from the desired end‐effector path and the Jacobian matrix of the mapping from joint space to task space. In this article, a feedback control law using an uncertain Jacobian matrix is proposed for setpoint control of robots. Sufficient conditions for the bound of the estimated Jacobian matrix and stability conditions for the feedback gains are presented to guarantee the stability and passivity of the robots. A gravity regressor with an uncertain Jacobian matrix is also proposed for gravitational force compensation when the gravitational force is uncertain. Simulation results are presented to illustrate the performance of the proposed controllers. ©1999 John Wiley & Sons, Inc.     

Dynamically consistent Jacobian inverse for mobile manipulators

By analogy to the definition of the dynamically consistent Jacobian inverse for robotic manipulators, we have designed a dynamically consistent Jacobian inverse for mobile manipulators built of a non-holonomic mobile platform and a holonomic on-board manipulator. The endogenous configuration space approach has been exploited as a source of conceptual guidelines. The new inverse guarantees a decoupling of the motion in the operational space from the forces exerted in the endogenous configuration space and annihilated by the dual Jacobian inverse. A performance study of the new Jacobian inverse as a tool for motion planning is presented.     

Parallel VLSI architectures for real-time kinematics of redundant robots

We describe new architectures for the efficient computation of redundant manipulator kinematics (direct and inverse). By calculating the core of the problem in hardware, we can make full use of the redundancy by implementing more complex self-motion algorithms. A key component of our architecture is the calculation in the VLSI hardware of the Singular Value Decomposition of the manipulator Jacobian. Recent advances in VLSI have allowed the mapping of complex algorithms to hardware using systolic arrays with advanced computer arithmetic algorithms, such as the coordinate rotation (CORDIC) algorithms. We use CORDIC arithmetic in the novel design of our special-purpose VLSI array, which is used in computation of the Direct Kinematics Solution (DKS), the manipulator Jacobian, as well as the Jacobian Pseudoinverse. Application-specific (subtask-dependent) portions of the inverse kinematics are handled in parallel by a DSP processor which interfaces with the custom hardware and the host machine. The architecture and algorithm development is valid for general redundant manipulators and a wide range of processors currently available and under development commercially.     

Gait planning based on kinematics for a quadruped gecko model with redundancy

Recent research on mobile robots has focused on locomotion in various environments. In this paper, a gait-generation algorithm for a mobile robot that can travel from the ground to a wall and climb vertical surfaces is proposed. The algorithm was inspired by a gecko lizard. Our gait planning was based on inverse kinematics using the Jacobian of the whole body, where the redundancy was solved by defining an object function for the gecko posture to avoid collisions with the surface. The optimal scalar factor for these two objects was obtained by defining a superior object function to minimize the angular acceleration of joints. The algorithm was verified through simulation of the gecko model travelling on given task paths and avoiding abnormal joint movements and collisions.     

漂浮基空间机械臂分解运动控制的增广自适应算法

本文讨论了载体位置与姿态均不受控制的漂浮基两杆空间机械臂系统的逆运动学问题 ,推导了系统的运动学、动力学方程．分析表明,结合系统动量守恒及动量矩守恒关系得到 的系统广义Jacobi关系为系统惯性参数的非线性函数．本文证明了,借助于增广变量法可以 将增广广义Jacobi矩阵表示为一组适当选择的惯性参数的线性函数．并在此基础上,给出了 系统参数未知时由空间机械臂末端惯性空间期望轨迹产生机械臂关节铰期望角速度、角加速 度的增广自适应控制算法．仿真运算,证实了方法的有效性．     

Real-time inverse kinematics techniques for anthropomorphic limbs

In this paper we develop a set of inverse kinematics algorithms suitable for an anthropomorphic arm or leg. We use a combination of analytical and numerical methods to solve generalized inverse kinematics problems including position, orientation, and aiming constraints. Our combination of analytical and numerical methods results in faster and more reliable algorithms than conventional inverse Jacobian and optimization-based techniques. Additionally, unlike conventional numerical algorithms, our methods allow the user to interactively explore all possible solutions using an intuitive set of parameters that define the redundancy of the system.     

Kinematic and dynamic models of hybrid robot manipulator for propeller grinding

In this article, we develop a hybrid robot manipulator for propeller grinding and derive its kinematic and dynamic models. The manipulator is constructed by combining a parallel mechanism and a serial one to increase high stiffness as well as workspace. Based on geometric constraints, inverse–direct kinematics and Jacobian are derived to be implemented in real time control. The velocity control is used to measure the surface of a propeller blade and the position control is conducted to grind the removal depth. The dynamic model, which is developed by a motor algebra, can compute the forces and moments acting at a passive joint and an active one. ©1999 John Wiley & Sons, Inc.     

Generating classes of locally orthogonal Gough-Stewart platforms

This paper develops methods for generating classes of orthogonal Gough-Stewart platforms (OGSPs). First, a new, two-parameter class of six-strut OGSPs which leads to isotropic manipulators are found. Next, this class is extended to include redundant Gough-Stewart platforms (GSPs). For an even number of struts, the same algorithm used to generate the six-strut case can be employed. For an odd number of struts, similar essential concepts are used to derive seven-strut and nine-strut OGSPs. Maximization of fault tolerance is implemented for a nine-strut isotropic OGSP. By exploiting invariant properties of the inverse Jacobian, new methods for favorably altering the center of gravity, strut attachment surface, and strut spatial distribution are developed.     

Closed‐loop inverse kinematics algorithm for constrained flexible manipulators under gravity

This work deals with the inverse kinematics problem for a flexible robot manipulator under gravity in contact with a stiff surface. This problem consists of finding the joint and deflection variables for a given tip position and constraint force. The solution algorithm is based on the well‐known closed‐loop inverse kinematics (CLIK) scheme, using the Jacobian transpose, developed for rigid manipulators. The Jacobian employed in the algorithm is obtained by correcting the equivalent rigid manipulator Jacobian with two terms that account for the static deflections due to gravity and contact force, respectively. The algorithm is tested in a number of case studies on a planar two‐link arm. ©1999 John Wiley & Sons, Inc.     

Integer inverse kinematics method using Fuzzy logic

In this paper, we propose an integer inverse kinematics method for multijoint robot control. The method reduces computational overheads and leads to the development of a simple control system as the use of fuzzy logic enables linguistic modeling of the joint angle. A small humanoid robot is used to confirm via experiment that the method produces the same cycling movements in the robot as those in a human. In addition, we achieve fast information sharing by implementing the all-integer control algorithm in a low-cost, low-power microprocessor. Moreover, we evaluate the ability of this method for trajectory generation and confirm that target trajectories are reproduced well. The computational results of the general inverse kinematics model are compared to those of the integer inverse kinematics model and similar outputs are demonstrated. We show that the integer inverse kinematics model simplifies the control process.     

A solution to the inverse kinematics of redundant manipulators

A solution to the inverse kinematics is a set of joint coordinates which correspond to a given set of task space coordinates (position and orientation of end effector). For the class of kinematically redundant robots, the solution is generically nonunique such that special methods are required for obtaining a solution. The method addressed in the paper, introduced earlier and termed “generalized inverse,” is based on a certain partitioning of the Jacobian functional corresponding to a nonlinear relationship of the inverse kinematics type. The article presents a new algorithm for solving the inverse kinematics using the method of generalized inverse based on a modified Newton-Raphson iterative technique. The new algorithm is efficient, converges rapidly, and completely generalizes the solution of the inverse kinematics problem for redundant robots. The method is illustrated by numerical examples.     

Visual servoing of redundant manipulator with Jacobian matrix estimation using self-organizing map

Vision based redundant manipulator control with a neural network based learning strategy is discussed in this paper. The manipulator is visually controlled with stereo vision in an eye-to-hand configuration. A novel Kohonen’s self-organizing map (KSOM) based visual servoing scheme has been proposed for a redundant manipulator with 7 degrees of freedom (DOF). The inverse kinematic relationship of the manipulator is learned using a Kohonen’s self-organizing map. This learned map is shown to be an approximate estimate of the inverse Jacobian, which can then be used in conjunction with the proportional controller to achieve closed loop servoing in real-time. It is shown through Lyapunov stability analysis that the proposed learning based servoing scheme ensures global stability. A generalized weight update law is proposed for KSOM based inverse kinematic control, to resolve the redundancy during the learning phase. Unlike the existing visual servoing schemes, the proposed KSOM based scheme eliminates the computation of the pseudo-inverse of the Jacobian matrix in real-time. This makes the proposed algorithm computationally more efficient. The proposed scheme has been implemented on a 7 DOF PowerCube? robot manipulator with visual feedback from two cameras.     

Repeatability of inverse kinematics algorithms for mobile manipulators

We introduce and examine the property of repeatability of inverse kinematics algorithms for mobile manipulators. Similarly to stationary manipulators, repeatability of mobile manipulators is defined by requiring that a closed path in the task space should be transformed by the inverse kinematics algorithm into a closed path in the configuration space. In a simply connected, singularity-free region of the task space, a necessary and sufficient condition for repeatability is derived as the integrability condition of a distribution associated with the inverse kinematics algorithm.