Efficient closed-form solution of inverse kinematics for a specific six-DOF arm

Inverse kinematics solutions for multi-DOF arms can be classified as analytical or numerical. In general, analytical solutions are preferable to numerical solutions because analytical ones yield complete solutions and are computationally fast and reliable. However, analytical closed-form solutions for inverse kinematics of 6-DOF arms rarely exist for real-time control purposes of fast moving arms. In this paper, we propose a fast inverse kinematics algorithm with a closed-form solution for a specific 6-DOF arm. The proposed algorithm is verified using simulation modules developed by us for demonstrations.     

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.     

A neuro-simulated annealing approach to the inverse kinematics solution of redundant robotic manipulators

The neural-network-based inverse kinematics solution is one of the recent topics in the robotics because of the fact that many traditional inverse kinematics problem solutions such as geometric, iterative and algebraic are inadequate for redundant robots. However, since the neural networks work with an acceptable error, the error at the end of inverse kinematics learning should be minimized. In this study, simulated annealing (SA) algorithm was used together with the neural-network-based inverse kinematics problem solution robots to minimize the error at the end effector. The solution method is applied to Stanford and Puma 560 six-joint robot models to show the efficiency. The proposed algorithm combines the characteristics of neural network and an optimization technique to obtain the best solution for the critical robotic applications. Three Elman neural networks were trained using separate training sets and different parameters, since one of them can give better results than the others can. The best result is selected within three neural network results by computing the end effector error via direct kinematics equation of the robotic manipulator. The decimal part of the neural network result was improved up to 10 digits using simulated annealing algorithm. The obtained best solution is given to the simulated annealing algorithm to find the best-fitting 10 digits for the decimal part of the solution. The end effector error was reduced significantly.     

A neural-network committee machine approach to the inverse kinematics problem solution of robotic manipulators

In robotics, inverse kinematics problem solution is a fundamental problem in robotics. Many traditional inverse kinematics problem solutions, such as the geometric, iterative, and algebraic approaches, are inadequate for redundant robots. Recently, much attention has been focused on a neural-network-based inverse kinematics problem solution in robotics. However, the result obtained from the neural network requires to be improved for some sensitive tasks. In this paper, a neural-network committee machine (NNCM) was designed to solve the inverse kinematics of a 6-DOF redundant robotic manipulator to improve the precision of the solution. Ten neural networks (NN) were designed to obtain a committee machine to solve the inverse kinematics problem using separately prepared data set since a neural network can give better result than other ones. The data sets for the neural-network training were prepared using prepared simulation software including robot kinematics model. The solution of each neural network was evaluated using direct kinematics equation of the robot to select the best one. As a result, the committee machine implementation increased the performance of the learning.     

面向冗余机器人实时控制的逆运动学求解有效方法

针对7自由度冗余机器人实时运动控制,对机器人逆运动学提出了一种新的求解方法.采用位姿分解方式,使7自由度冗余机器人逆运动学简化为4自由度位置逆运动学求解.在梯度投影法得到位置优化解的基础上,利用机器人封闭解公式求得一组优化解.通过对7自由度机器人仿真分析,表明了该方法的有效性.     

Reliability-based approach to the inverse kinematics solution of robots using Elman's networks

The solution of inverse kinematics problem of redundant manipulators is a fundamental problem in robot control. The inverse kinematics problem in robotics is the determination of joint angles for a desired cartesian position of the end effector. For the solution of this problem, many traditional solutions such as geometric, iterative and algebraic are inadequate if the joint structure of the manipulator is more complex. Furthermore, many neural network approaches have been done to this problem. But the neural network-based solutions are not much reliable due to the error at the end of learning. Therefore, a reliability-based neural network inverse kinematics solution approach has been presented, and applied to a six-degrees of freedom (dof) robot manipulator in this paper. The structure of the proposed method is based on using three networks designed parallel to minimize the error of the whole system. Elman network, which has a profound impact on the learning capability and performance of the network, is chosen and designed according to the proposed solution method. At the end of parallel implementation, the results of each network are evaluated using direct kinematics equations to obtain the network with best result.     

A strictly convergent real-time solution for inverse kinematics of robot manipulators

Inverse Kinematics has been recognized as an important problem in robotics applications. A robot independent solution can only be obtained through numerical methods, but most solutions which use this approach have problems with convergence especially near singularity points. This article develops a strictly convergent algorithm and a special-purpose Inverse Kinematics Processor (IKP) to obtain the solution in real time. While the algorithm is based on open-loop integration of rates, the absolute position deviation is used as a criterion to control the iteration, and a feedback mechanism has been especially designed to eliminate problems with long-term drift or with initial errors in the solution. The architecture of the IKP is based on a high-speed floating-point arithmetic processor and is designed to perform the common matrix-vector operations efficiently with a minimum processor cycle time. The algorithm has been simulated on the proposed architecture, and the results show its robustness and real-time capability. For a six degree-of-freedom robot manipulator (for which no closed-form solution exist), the Inverse Kinematics solution may be obtained at an approximate 2 khz rate with an error which is within standard repeatability limits.     

A closed-form solution to natural image matting

Interactive digital matting, the process of extracting a foreground object from an image based on limited user input, is an important task in image and video editing. From a computer vision perspective, this task is extremely challenging because it is massively ill-posed -- at each pixel we must estimate the foreground and the background colors, as well as the foreground opacity ("alpha matte") from a single color measurement. Current approaches either restrict the estimation to a small part of the image, estimating foreground and background colors based on nearby pixels where they are known, or perform iterative nonlinear estimation by alternating foreground and background color estimation with alpha estimation.In this paper we present a closed-form solution to natural image matting. We derive a cost function from local smoothness assumptions on foreground and background colors, and show that in the resulting expression it is possible to analytically eliminate the foreground and background colors to obtain a quadratic cost function in alpha. This allows us to find the globally optimal alpha matte by solving a sparse linear system of equations. Furthermore, the closed-form formula allows us to predict the properties of the solution by analyzing the eigenvectors of a sparse matrix, closely related to matrices used in spectral image segmentation algorithms. We show that high quality mattes for natural images may be obtained from a small amount of user input.     

A study of robot inverse kinematics based upon the solution of differential equations

This article presents an improved numerical algorithm for robot inverse kinematics which is based upon the solution of the first-order differential equations arising from the manipulator's velocity Jacobian relations. The use of the Adams-Moulton predictorcorrector scheme leads to a fourth-order trajectory following in the joint space. The implementation of a strict descent feature for the trajectory error at the end-effector level contributes to the robustness of the algorithm near singularities. The execution of this algorithm is about 2.7 times faster than that of Gupta and Kazerounian,¹⁶ with much of the speed up coming from the use of software optimizations. Several issues related to the accuracy, convergence, speed, real-time computation, and portability of this algorithm are discussed.     

A neuro-genetic-simulated annealing approach to the inverse kinematics solution of robots: a simulation based study

In this study, a hybrid intelligent solution system including neural networks, genetic algorithms and simulated annealing has been proposed for the inverse kinematics solution of robotic manipulators. The main purpose of the proposed system is to decrease the end effector error of a neural network based inverse kinematics solution. In the designed hybrid intelligent system, simulated annealing algorithm has been used as a genetic operator to decrease the process time of the genetic algorithm to find the optimum solution. Obtained best solution from the neural network has been included in the initial solution of genetic algorithm with randomly produced solutions. The end effector error has been reduced micrometer levels after the implementation of the hybrid intelligent solution system.     

A closed-form solution to video matting of natural snow

In this paper, we address the problem on video matting of natural snow in snowing context. By optical features of natural snow and the continuity of a video, we design a suitable temporal filter to recover the background of the video and compute the approximate matte gradient. Using two types of information obtained, we first propose a closed-form solution to video matting of natural snow.     

Exact solution to the inverse kinematics problem of a standard 6-axis robot manipulator

Presented are four sets of exact solutions for the vector of the joint angles {θ_i} pertaining to the inverse kinematics problem of a standard 6-axis robot manipulator with two different kinds of gripper configurations. Here a standard 6-axis robot is meant to be a general computer-controlled revolute robot with base, shoulder, elbow, wrist pitch, wrist yaw, wrist roll, and gripping action. Explicit solutions are obtained using Denavit-Hartenberg homogeneous transformations. Furthermore, the inverse solutions are examined by means of a direct kinematic computer program.     

A study of neural network based inverse kinematics solution for a three-joint robot

A neural network based inverse kinematics solution of a robotic manipulator is presented in this paper. Inverse kinematics problem is generally more complex for robotic manipulators. Many traditional solutions such as geometric, iterative and algebraic are inadequate if the joint structure of the manipulator is more complex. In this study, a three-joint robotic manipulator simulation software, developed in our previous studies, is used. Firstly, we have generated many initial and final points in the work volume of the robotic manipulator by using cubic trajectory planning. Then, all of the angles according to the real-world coordinates (x, y, z) are recorded in a file named as training set of neural network. Lastly, we have used a designed neural network to solve the inverse kinematics problem. The designed neural network has given the correct angles according to the given (x, y, z) cartesian coordinates. The online working feature of neural network makes it very successful and popular in this solution.     

A closed-form solution to tensor voting: theory and applications

We prove a closed-form solution to tensor voting (CFTV): Given a point set in any dimensions, our closed-form solution provides an exact, continuous, and efficient algorithm for computing a structure-aware tensor that simultaneously achieves salient structure detection and outlier attenuation. Using CFTV, we prove the convergence of tensor voting on a Markov random field (MRF), thus termed as MRFTV, where the structure-aware tensor at each input site reaches a stationary state upon convergence in structure propagation. We then embed structure-aware tensor into expectation maximization (EM) for optimizing a single linear structure to achieve efficient and robust parameter estimation. Specifically, our EMTV algorithm optimizes both the tensor and fitting parameters and does not require random sampling consensus typically used in existing robust statistical techniques. We performed quantitative evaluation on its accuracy and robustness, showing that EMTV performs better than the original TV and other state-of-the-art techniques in fundamental matrix estimation for multiview stereo matching. The extensions of CFTV and EMTV for extracting multiple and nonlinear structures are underway.     

基于牛顿-拉夫逊迭代法的6自由度机器人逆解算法

为解决一般6自由度旋转关节机器人逆运动学问题,提出了一种用牛顿一拉夫逊迭代法逐次逼近目标位姿的逆解算法.根据正运动学方程建立雅克比矩阵,采用基于豪斯霍尔德的SVD分解求其伪逆来避免雅克比矩阵的奇异性问题,通过建立迭代规则并逐次迭代找到最优的逆运动学单解,实际应用时无需再建立多解取优策略.本算法具有较好的局部快速收敛性,能够达到较好的精度和速度,并在基于ARM9的嵌人式系统上实现了此算法.相应的测试表明:算法实时性能够满足系统要求,可应用于机器人实时控制系统.     

A method to learn the inverse kinematics of multi-link robots by evolving neuro-controllers

A general method to learn the inverse kinematic of multi-link robots by means of neuro-controllers is presented. We can find analytical solutions for the most used and well-known robots in the literature. However, these solutions are specific to a particular robot configuration and are not generally applicable to other robot morphologies. The proposed method is general in the sense that it is independent of the robot morphology. The method is based on the evolutionary computation paradigm and works obtaining incrementally better neuro-controllers. Furthermore, the proposed method solves some specific issues in robotic neuro-controller learning: it avoids any neural network learning algorithm which relies on the classical supervised input-target learning scheme and hence it lets to obtain neuro-controllers without providing targets. It can converge beyond local optimal solutions, which is one of the main drawbacks of some neural network training algorithms based on gradient descent when applied to highly redundant robot morphologies. Furthermore, using learning algorithms such as the neuro-evolution of augmenting topologies it is also possible to learn the neural network topology which is a common source of empirical testing in neuro-controllers design. Finally, experimental results are provided when applying the method to two multi-link robot learning tasks and a comparison between structural and parametric evolutionary strategies on neuro-controllers is shown.     

A biomimetic approach to inverse kinematics for a redundant robot arm

Redundant robots have received increased attention during the last decades, since they provide solutions to problems investigated for years in the robotic community, e.g. task-space tracking, obstacle avoidance etc. However, robot redundancy may arise problems of kinematic control, since robot joint motion is not uniquely determined. In this paper, a biomimetic approach is proposed for solving the problem of redundancy resolution. First, the kinematics of the human upper limb while performing random arm motion are investigated and modeled. The dependencies among the human joint angles are described using a Bayesian network. Then, an objective function, built using this model, is used in a closed-loop inverse kinematic algorithm for a redundant robot arm. Using this algorithm, the robot arm end-effector can be positioned in the three dimensional (3D) space using human-like joint configurations. Through real experiments using an anthropomorphic robot arm, it is proved that the proposed algorithm is computationally fast, while it results to human-like configurations compared to previously proposed inverse kinematics algorithms. The latter makes the proposed algorithm a strong candidate for applications where anthropomorphism is required, e.g. in humanoids or generally in cases where robotic arms interact with humans.     

Real-time inverse kinematics of the human arm

A simple inverse kinematics procedure is proposed for a seven degree of freedom model of the human arm. Two schemes are used to provide an additional constraint leading to closed-form analytical equations with an upper bound of two or four solutions. Multiple solutions can be evaluated on the basis of their proximity from the rest angles or the previous configuration of the arm. Empirical results demonstrate that the procedure is well suited for real-time applications.     

A fast,robust solution to the Stewart platform forward kinematics

The Stewart platform is a six degree-of-freedom fully-in-parallel linkage well-suited to robotic tasks where structural rigidity and high small motion bandwidth are required. In this article we describe an approach for computing the forward kinematics of this device that is both fast and robust. Our solution is based on the simultaneous solution of three constraint equations using a Newton-Raphson scheme. A well-known property of Newton-Raphson is its tendency to fail when the constraint equations become poorly conditioned, and the main contribution of this article is the development of two algorithms for overcoming this limitation and hence for providing robustness. Certain other matters, such as the singular configurations of the Stewart platform and its assembly modes, are touched upon. © 1996 John Wiley & Sons, Inc.