Synthesis on forward kinematics problem algebraic modeling for the planar parallel manipulator: displacement-based equation systems

Based on a proven exact method which solves the forward kinematics problem (FKP) this article investigates the FKP formulation specifically applied to planar parallel manipulators. It focuses on the displacement-based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator, which is a tripod constituted by a fixed base and a triangular mobile platform attached to three kinematics chains with linear (prismatic) actuators located between two revolute joints. In order to implement the algebraic method, the parallel manipulator kinematics are formulated as polynomial equation systems where the number of equations is equal to or exceeds the number of unknowns. Three geometrical formulations are derived to model the difficult FKP. The selected proven algebraic method uses Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots are isolated using this last system. Each real solution exactly corresponds to one manipulator assembly mode, which is also called a manipulator posture. The FKP resolution of the planar 3-RPR parallel manipulator outputs six complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are given. It is then possible to compare the models and to reject one. Moreover, a number of real solutions are obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.     

A hybrid strategy to solve the forward kinematics problem in parallel manipulators

A parallel manipulator is a closed kinematic structure with the necessary rigidity to provide a high payload to self-weight ratio suitable for many applications in manufacturing, flight simulation systems, and medical robotics. Because of its closed structure, the kinematic control of such a mechanism is difficult. The inverse kinematics problem for such manipulators has a mathematical solution; however, the forward kinematics problem (FKP) is mathematically intractable. This work addresses the FKP and proposes a neural-network-based hybrid strategy that solves the problem to a desired level of accuracy, and can achieve the solution in real time. Two neural-network (NN) concepts using a modified form of multilayered perceptrons with backpropagation learning were implemented. The better performing concept was then combined with a standard Newton-Raphson numerical technique to yield a hybrid solution strategy. Simulation studies were carried out on a flight simulation syystem to check the validity o the approach. Accuracy of close to 0.01 mm and 0.01/spl deg/ in the position and orientation parameters was achieved in less than two iterations and 0.02 s of execution time for the proposed strategy.     

Certified solving of the forward kinematics problem with an exact algebraic method for the general parallel manipulator

This article introduces an exact method to solve the forward kinematics problem (FKP) specifically applied to spatial parallel manipulators. The majority can modeled by the 6-6 parallel manipulator. This manipulator is a hexapod made up of a fixed base and a mobile platform attached to six kinematics chains with linear (prismatic) actuators located between two ball or Cardan joints. In order to implement algebraic methods, the parallel manipulator kinematics will be formulated as polynomial equations systems where the equation number is equal to the unknown numbers. One position-based kinematics model will be identified to solve the difficult FKP. The selected proven algebraic method implements Gröbner bases and constructs an equivalent univariate polynomial system. The exact resolution of this last system determines the real solution which exactly corresponds to the manipulator postures. The FKP resolution of the general 6-6 parallel manipulator outputs 40 complex solutions. We provide several examples of various hexapod types yielding eight real solutions. This algebraic method is exact and computes certified results.     

基于Elman神经网络的Stewart平台位姿正解

Stewart平台广泛应用于运动模拟器、光学、精密定位等领域,然而由于复杂的多元非线性使得位姿正解难以准确得到.针对Stewart平台的位姿正解问题,常规的方法比如迭代法和数值法存在初始值难以选取、计算速度较慢等问题,提出了基于Elman神经网络的位姿正解方法.首先建立Stewart平台支腿长度与平台位姿的运动学模型,然后利用Elman神经网络来实现位姿正解的求解并实验验证.该方法具有良好的动态特性,精度高,能够快速准确的实现Stewart平台位姿正解的求解.实验证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.     

Synthesis of the forward kinematics problem algebraic modeling for the general parallel manipulator: displacement-based equations

This paper closes a triptic to address the issue of the forward kinematics problem (FKP) aimed at certified solving with an exact algebraic method. This solving method was described in the first article published in Advanced Robotics. The second one investigated the formulation specifically applied to the planar parallel manipulators. This third paper is the logical one in the footsteps of the formersones, since it continues the formulation analyses and brings them to the general spatial parallel manipulator. Hence, this paper focuses on the displacement-based equation systems. This paper is the first one to present a synthesis on forward kinematics modeling focusing on finding an optimal mathematical formulation based on the displacement-based equation systems. The majority of parallel manipulators in applications can be modeled by the 6-6 hexapod or so-called Gough platform which is constituted by a fixed base and a mobile platform attached to six kinematics chains with linear (prismatic) actuators located between two revolute or Cardan joints. Again, in order to implement algebraic methods, the parallel manipulator kinematics shall be formulated as polynomial equations systems where the equation number is at least equal to the unknown numbers. Six geometric formulations were derived. The selected algebraic proven method is implementing Gröbner bases from which it constructs an equivalent univariate polynomial system. The resolution of this last system exactly determines the real solutions which correspond to the manipulator postures. The FKP resolution of the general 6-6 parallel manipulator outputs 40 complex solutions. Several instantiations shall be computed in order to select the model which leads to the FKP resolution with the lowest response times and smaller file sizes. It was possible to reject three modelings leading to bad performances or resolution failure. It was possible to determine one formulation where the solving computations were definitely better than the others.     

Comparison of RBF and MLP neural networks to solve inverse kinematic problem for 6R serial robot by a fusion approach

In this paper, a fusion approach to determine inverse kinematics solutions of a six degree of freedom serial robot is proposed. The proposed approach makes use of radial basis function neural network for prediction of incremental joint angles which in turn are transformed into absolute joint angles with the assistance of forward kinematics relations. In this approach, forward kinematics relations of robot are used to obtain the data for training of neural network as well to estimate the deviation of predicted inverse kinematics solution from the desired solution. The effectiveness of the fusion process is shown by comparing the inverse kinematics solutions obtained for an end-effector of industrial robot moving along a specified path with the solutions obtained from conventional neural network approaches as well as iterative technique. The prominent features of the fusion process include the accurate prediction of inverse kinematics solutions with less computational time apart from the generation of training data for neural network with forward kinematics relations of the robot.     

并联3-URS踝关节康复机器人的机构与运动学研究

为了帮助患者进行踝关节康复训练,减轻治疗师工作强度,在分类分析现有的各类型踝关节康复机器人的基础上,设计了一种六自由度并联3-URS踝关节康复机器人。从人体生理结构及康复训练需求出发,设计、优化了康复机器人结构,加工制造了实物样机模型;采用闭环矢量的方法建立了并联机器人运动学模型,结合Rosenbrock-Banana优化函数,将正逆运动学数值求解问题转换为优化问题。以背屈训练轨迹作为数值算例,求解精度可达10-10~10-7mm;结合虚拟样机技术,验证了该并联机器人运动学优化求解方法的可靠性,适用于3-URS并联踝关节康复机器人。     

Multi-fingered robot hand optimal task force distribution: Neural inverse kinematics approach

Grasping and manipulation force distribution optimization of multi-fingered robotic hands can be formulated as a problem for minimizing an objective function subject to form-closure constraints, kinematics, and balance constraints of external force. In this paper we present a novel neural network for dexterous hand-grasping inverse kinematics mapping used in force optimization. The proposed optimization is shown to be globally convergent to the optimal grasping force. The approach followed here is to let an artificial neural network (ANN) learn the nonlinear inverse kinematics functional relating the hand joint positions and displacements to object displacement. This is done by considering the inverse hand Jacobian, in addition to the interaction between hand fingers and the object. The proposed neural-network approach has the advantages that the complexity for implementation is reduced, and the solution accuracy is increased, by avoiding the linearization of quadratic friction constraints. Simulation results show that the proposed neural network can achieve optimal grasping force.     

A length factor artificial neural network method for the numerical solution of the advection dispersion equation characterizing the mass balance of fluid flow in a chemical reactor

Neural Computing and Applications - In this article, a length factor artificial neural network (ANN) method is proposed for the numerical solution of the advection dispersion equation (ADE) in...     

Comparison of Different Methods for Computing the Forward Kinematics of a Redundant Parallel Manipulator

In this paper, three numerical methods are presented to solve the forward kinematics of a three DOF actuator-redundant hydraulic parallel manipulator. It is known, that on the contrary to series manipulators, the forward kinematic map of parallel manipulators involves highly coupled nonlinear equations, whose closed-form solution derivation is a real challenge. This issue is of great importance noting that the forward kinematics solution is a key element in closed loop position control of parallel manipulators. The proposed methods, namely the Neural Network Estimation, the Quasi-closed Solution, and the Taylor series approximation, are using mainly numerical computations, with different ideas to solve the problem in hand. The latter two methods are proposed for the first time in literature to solve the forward kinematics of a parallel manipulator. These methods are compared in detail and the advantages or the disadvantages of each method in computing the forward kinematic map of the given mechanism is discussed. It is shown that a 4th order Taylor series approximation to the problem provides a good compromise for practical applications compared to that of other methods considered in this paper.     

一种基于角度传感器的6-UPS机构正向运动学分析方法

为了提高6自由度并联机构运动学正解的求解效率,增强求解方法的通用性,提出了一种6-UPS（universal-prismatic-spherical）机构的运动学正解算法．首先在6-UPS机构任意分支虎克铰处添加2个角度传感器,测量了虎克铰2个方位的旋转角度．再基于旋转矩阵构建12个方程,通过代数消元对其进行降次处理得到简易的一元二次方程组．最后,求得6-UPS机构的运动学正解．并通过具体数值算例验证所提方法,求出了确定的位置正解．该方法不仅降低了数据处理的难度,且能求得正解的唯一解,避免了并联机构正解存在多解的问题．     

An adaptive-learning algorithm to solve the inverse kinematics problem of a 6 D.O.F serial robot manipulator

An adoptive learning strategy using an artificial neural network ANN has been proposed here to control the motion of a 6 D.O.F manipulator robot and to overcome the inverse kinematics problem, which are mainly singularities and uncertainties in arm configurations. In this approach a network have been trained to learn a desired set of joint angles positions from a given set of end effector positions, experimental results has shown an excellent mapping over the working area of the robot, to validate the ability of the designed network to make prediction and well generalization for any set of data, a new training using different data set has been performed using the same network, experimental results has shown a good generalization for the new data sets.The proposed control technique does not require any prior knowledge of the kinematics model of the system being controlled, the basic idea of this concept is the use of the ANN to learn the characteristics of the robot system rather than to specify explicit robot system model. Any modification in the physical set-up of the robot such as the addition of a new tool would only require training for a new path without the need for any major system software modification, which is a significant advantage of using neural network technology.     

中心引力算法收敛分析及在神经网络中的应用

中心引力优化算法(central force optimization,简称CFO)是一种新型的基于天体动力学的多维搜索优化算法.该算法是一种确定性的优化算法,利用一组质子在万有引力作用下的运动,搜索目标函数在决策空间上的最优值.利用天体力学理论对该算法中质子运动方程进行了深入的研究,并利用天体力学中万有引力定理对质子运动方程进行了推导,建立起天体力学与CFO 算法之间的联系,通过天体力学中数学分析的方法对该算法中质子收敛性能进行了分析,最后,通过严格的数学推导证明出:无论初始时质子是何种分布,CFO 算法中所有的质子始终都会收敛于CFO 空间的确定最优解.作为测试效果,将CFO 算法与常见的BP 训练算法相结合,提出了CFO-BP 训练算法,优化前馈型人工神经网络的权值和结构.实验结果表明,采用CFO-BP 算法优化神经网络比其他常见优化算法有更好的收敛精度和收敛速度.     

3-RRRT并联机器人位置正向求解研究

研究一种3-RRRT型并联机器人机构的运动学正向求解方法。根据3-RRRT型并联机器人机构特点以及关节运动的取值范围,提出了以并联机器人支链中支杆的方向余弦和动平台绝对位置坐标为系统的广义坐标的方法,并详细地推导了3-RRRT型并联机器人运动学模型,通过进一步消除中间变量的方法最终获得了易于正、逆运动学求解的只包含3个驱动关节坐标与动平台3个绝对位置坐标的约束方程组。最后,运用基于Moore—Penwse广义逆的牛顿迭代格式编制了MATLAB运动学正向求解程序,并进行了运动学正向求解数值仿真,结果表明求解程序快速有效。     

Kinematic analysis of a novel 3-DOF actuation redundant parallel manipulator using artificial intelligence approach

Kinematic analysis is one of the key issues in the research domain of parallel kinematic manipulators. It includes inverse kinematics and forward kinematics. Contrary to a serial manipulator, the inverse kinematics of a parallel manipulator is usually simple and straightforward. However, forward kinematic mapping of a parallel manipulator involves highly coupled nonlinear equations. Therefore, it is more difficult to solve the forward kinematics problem of parallel robots. In this paper, a novel three degrees-of-freedom (DOFs) actuation redundant parallel manipulator is introduced. Different intelligent approaches, which include the Multilayer Perceptron (MLP) neural network, Radial Basis Functions (RBF) neural network, and Support Vector Machine (SVM), are applied to investigate the forward kinematic problem of the robot. Simulation is conducted and the accuracy of the models set up by the different methods is compared in detail. The advantages and the disadvantages of each method are analyzed. It is concluded that ν-SVM with a linear kernel function has the best performance to estimate the forward kinematic mapping of a parallel manipulator.     

基于神经网络的并联3自由度机器人位置正解

并联机器人位置正解是机器人运动学中难点问题之一，常规求解方法比较复杂且难度较大，通常需要对大量的非线性方程组进行推导计算且得到的解不唯一。该文提出了一种将人工神经网络用于并联机器人位置正解求解的通用方法，并结合实际机构对并联3自由度机器人进行了具体求解。通过对神经网络拓扑结构的设计以及选取有效的学习算法并用大量的位置反解数据对神经网络进行训练，获得了用于求解位置正解的神经网络模型，该网络可以实现位置正解问题的求解计算，从而避免了复杂的推导和演算。计算机仿真与实验结果表明了该方法的有效性与可行性。     

Artificial neural network for prediction of air flow in a single rock joint

In this paper, an attempt has been made to evaluate and predict the air flow rate in triaxial conditions at various confining pressures incorporating cell pressure, air inlet pressure, and air outlet pressure using artificial neural network (ANN) technique. A three-layer feed forward back propagation neural network having 3-7-1 architecture network was trained using 37 data sets measured from laboratory investigation. Ten new data sets were used for the, validation and comparison of the air flow rate by ANN and multi-variate regression analysis (MVRA) to develop more confidence on the proposed method. Results were compared based on coefficient of determination (CoD) and mean absolute error (MAE) between measured and predicted values of air flow rate. It was found that CoD between measured and predicted air flow rate was 0.995 and 0.758 by ANN and MVRA, respectively, whereas MAE was 0.0413 and 0.1876.     

Robot arm reaching through neural inversions and reinforcement learning

We present a neural method that computes the inverse kinematics of any kind of robot manipulators, both redundant and non-redundant. Inverse kinematics solutions are obtained through the inversion of a neural network that has been previously trained to approximate the manipulator forward kinematics. The inversion provides difference vectors in the joint space from difference vectors in the workspace. Our differential inverse kinematics (DIV) approach can be viewed as a neural network implementation of the Jacobian transpose method for arm kinematic control that does not require previous knowledge of the arm forward kinematics. Redundancy can be exploited to obtain a special inverse kinematic solution that meets a particular constraint (e.g. joint limit avoidance) by inverting an additional neural network The usefulness of our DIV approach is further illustrated with sensor-based multilink manipulators that learn collision-free reaching motions in unknown environments. For this task, the neural controller has two modules: a reinforcement-based action generator (AG) and a DIV module that computes goal vectors in the joint space. The actions given by the AG are interpreted with regard to those goal vectors.     

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.