Dynamic model based nonlinear tracking control of a planar parallel manipulator

Based on the dynamic model, a novel nonlinear tracking controller is developed to overcome the nonlinear dynamics and friction of a planar parallel manipulator. The dynamic model is formulated in the active joint space, and the active joint friction is described with the Coulomb + viscous friction model. A nonlinear tracking controller is designed to eliminate the tracking error by using the power function. The nonlinear tracking controller is proven to guarantee asymptotic convergence to zero of both the tracking error and error rate with the Barbalat’s lemma. The trajectory tracking experiment of the proposed controller is implemented on an actual five-bar planar parallel manipulator both at the low-speed and high-speed motion. Moreover, the control performances of the proposed controller are compared with the results of the augmented PD (APD) controller.     

Calibration of encoder indexing of a redundantly actuated parallel mechanism to eliminate contradicting control forces

This paper presents the kinematic calibration of a two degree-of-freedom redundantly actuated parallel mechanism (RAPM), with the aim of eliminating contradicting control forces (CCF). The kinematic errors in the RAPM induce CCFs, especially in the case of decentralized individual position control, which is the standard control method used in industrial applications. The encoder indexing errors of the actuated joints are known to be of strong influence on the CCFs. Therefore, it is believed that the CCFs will be eliminated if the encoder indexing errors are corrected. We proved this through experiments. We performed the calibration using a projection technique, wherein we projected tracking error terms onto orthogonal complementary terms of the constraint Jacobian between the independent joints and actuated joints. Using this projection technique, the effect of tracking error terms from the joint stiffness and external force is eliminated. During the calibration process, the tracking errors in the actuated joints are measured. Using these errors, we derived the optimal values of the encoder indexing error by minimizing the objective function. We verified the calibration result by comparing the CCFs measured before calibration with those measured after calibration, for the case of individual PID position control. Our results confirmed that the calibration procedure of encoder indexing errors successfully reduces the average norm value of CCFs from 366 N to 13 N.     

On the Characterization of the Dynamic Performances of Planar Manipulators

The characterization of the dynamic performances of a manipulator is important both to compare different manipulators and to improve the dynamic performances of a manipulator during the design stage. In a previous paper the concepts of swiftness and of dynamic isotropy were used to characterize some dynamic performances of 3-dof manipulators. This paper analyzes the usefulness of these concepts for three-dof planar manipulators and shows that the concept of swiftness is still significant, whereas the concept of dynamic isotropy has no practical interest. Moreover, it introduces three new dynamic properties that are useful in the design of a 3-dof planar manipulator. Finally, it proposes some indices that measure both the swiftness and the three new dynamic properties and shows how to use them, both for evaluating the dynamic performances of a given 3-dof planar manipulator and for improving the dynamic performances during its design.     

存在关节力矩输出死区及外部干扰的漂浮基空间机械臂积分滑模神经网络自适应控制

探讨了载体位置和姿态都不受控时,漂浮基空间机械臂在带有关节力矩输出死区及外部干扰情况下轨迹跟踪的控制算法设计问题。死区与外部干扰影响系统的跟踪精度与稳定性。为此引入积分型切换函数,减少外部干扰引起的稳态误差,并利用径向基函数神经网络逼近动力学方程的未知部分,设计了一种积分滑模神经网络控制方案。控制算法的优点是,在死区斜率与边界参数不确定及最优逼近误差上确界未知的条件下,可以利用最优逼近误差、死区及干扰的补偿项来消除影响。李亚普诺夫稳定性分析证明了闭环系统的稳定性,且轨迹跟踪误差将收敛到0的某个小邻域内。仿真算例证实了该控制算法的有效性,实现了空间机械臂的轨迹跟踪控制。     

Velocity and acceleration level inverse kinematic calculation alternatives for redundant manipulators

For both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That is when the desired task possesses lower dimensions than the degrees-of-freedom of a redundant manipulator. The IK calculation can be implemented numerically in several alternative ways not only in case of the redundant but also in the non-redundant case. We study the stability properties and the feasibility of a tracking error feedback and a direct tracking error elimination approach of the numerical implementation of IK calculation both on velocity and acceleration levels. The feedback approach expresses the joint position increment stepwise based on the local velocity or acceleration of the desired TCP trajectory and linear feedback terms. In the direct error elimination concept, the increment of the joint position is directly given by the approximate error between the desired and the realized TCP position, by assuming constant TCP velocity or acceleration. We investigate the possibility of the implementation of the direct method on acceleration level. The investigated IK methods are unified in a framework that utilizes the idea of the auxiliary input. Our closed form results and numerical case study examples show the stability properties, benefits and disadvantages of the assessed IK implementations.

     

Nonlinear adaptive task space control for a 2-DOF redundantly actuated parallel manipulator

A nonlinear adaptive (NA) controller in the task space is developed for the trajectory tracking of a 2-DOF redundantly actuated parallel manipulator. The dynamic model with nonlinear friction is established in the task space for the parallel manipulator, and the linear parameterization expression of the dynamic model is formulated. Based on the dynamic model, a new control law including adaptive dynamics compensation, adaptive friction compensation and error elimination items is designed. After defining a quadratic performance index, the parameter update law is derived with the gradient descent algorithm. The stability of the parallel manipulator system is proved by the Lyapunov theorem, and the convergence of the tracking error and the error rate is proved by the Barbalat’s lemma. The NA controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated parallel manipulator, and the experiment results are compared with the APD controller.     

Tomographic shadowgraphy for three-dimensional reconstruction of instantaneous spray distributions

Tomographic shadowgraphy is an image-based optical technique capable of reconstructing the three dimensional instantaneous spray distributions within a given volume. The method is based on a multiple view imaging setup with inline illumination provided by current-pulsed LEDs, which results in droplet shadows being projected onto multiple sensor planes. Each camera records image pairs with short inter-framing times that allow the trajectories of the individual droplets to be estimated using conventional three-dimensional particle tracking approaches. The observed volume is calibrated with a traversed micro-target. A comparison is made between several photogrammetric and polynomial least-square camera calibration techniques regarding their accuracy in deep volume calibration at magnifications close to unity. A calibration method based on volume calibration from multiple planar homographies at equally spaced z-planes was found to produce the most reliable calibration. The combination of back-projected images at each voxel plane efficiently reproduces the droplet positions in three-dimensional space by line-of-sight (LOS) intensity reconstruction. Further improvement of the reconstruction can be achieved by iterative tomographic reconstruction, namely simultaneous multiplicative algebraic reconstruction technique (SMART). The quality of spray reconstruction is investigated using experimental data from multiple view shadowgraphs of hollow cone and flat fan water sprays. The investigations are further substantiated with simulations using synthetic data.     

A simple and quick control strategy for a class of first-order nonholonomic manipulator

This paper presents a simple and quick control strategy for a class of first-order nonholonomic manipulator: planar n-link manipulator with a passive first joint (no gravity). The control target is driving its endpoint from any initial position to any target position. First, we reduce the planar n-link manipulator to a planar three-link one by maintaining the states (angles and angular velocities) of n-3 active links in the initial value all the time. That is, we only adjust the states of the remaining two active links in the whole control process, and these two adjusted active links are chosen to guarantee that the target position is in the reachable region of the planar n-link manipulator by using the enumeration method. Then, we divide the whole control process into two stages for the reduced planar three-link manipulator. In each stage, the manipulator becomes a planar two-link one by maintaining the states of one adjusted active link to be the constant value. The state constraint existing between the passive first link and the adjusted active link is obtained by using the integral characteristics of a planar two-link manipulator. Meantime, the geometric constraint between the position of the endpoint and angles of all joints is obtained based on the homogeneous coordinate transformation method. According to the above two kinds of constraint, the target angles of the two adjusted active links are calculated by employing the particle swarm optimization algorithm. When the two adjusted active links are controlled to their target angles in turn, the control target of the planar n-link manipulator is completed. Finally, simulation results demonstrate that the proposed control strategy is valid and rapid.     

Terminal sliding mode control for coordinated motion of a space rigid manipulator with external disturbance

The control problem of coordinated motion of a free-floating space rigid manipulator with external disturbance is discussed. By combining linear momentum conversion and the Lagrangian approach, the full-control dynamic equation and the Jacobian relation of a free-floating space rigid manipulator are established and then inverted to the state equation for control design. Based on the terminal sliding mode control （SMC） technique, a mathematical expression of the terminal sliding surface is proposed. The terminal SMC scheme is then developed for coordinated motion between the base＇s attitude and the end-effector of the free-floating space manipulator with external disturbance. This proposed control scheme not only guarantees the existence of the sliding phase of the closed-loop system, but also ensures that the output tracking error converges to zero in finite time. In addition, because the initial system state is always at the terminal sliding surface, the control scheme can eliminate reaching phase of the SMC and guarantee global robustness and stability of the closed-loop system. A planar free-floating space rigid manipulator is simulated to verify the feasibility of the proposed control scheme.     

Development and assessment of a single-image fringe projection method for dynamic applications

A single-image fringe projection profiling method suitable for dynamic applications was developed by combining an accurate camera calibration procedure and improved phase extraction procedures. The improved phase extraction process used a modified Hilbert transform with Laplacian pyramid algorithms to improve measurement accuracy. The camera calibration method used an accurate pinhole camera model and pixel-by-pixel calibration of the phase-height relationship. Numerical simulations and controlled baseline experiments were performed to quantify key error sources in the measurement process and verify the accuracy of the approach. Results from numerical simulations indicate that the resulting phase error can be reduced to less than 0.02 radians provided that parameters such as fringe spacing, random measured intensity noise, fringe contrast and frequency of spatial intensity noise are carefully controlled. Experimental results show that the effects of random temporal and spatial noise in typical CCD cameras for single fringe images limits the accuracy of the method to 0.04 radians in most applications. Quantitative results from application of the fringe projection method are in very good agreement with numerical predictions, demonstrating that it is possible to design both a fringe projection system and a measurement process to achieve a prespecified accuracy and resolution in the point-to-point measurement of the spatial (X, Y, Z) positions.     

An evolutionary approach for the motion planning of redundant and hyper-redundant manipulators

The trajectory planning of redundant robots is an important area of research and efficient optimization algorithms are needed. The pseudoinverse control is not repeatable, causing drift in joint space which is undesirable for physical control. This paper presents a new technique that combines the closed-loop pseudoinverse method with genetic algorithms, leading to an optimization criterion for repeatable control of redundant manipulators, and avoiding the joint angle drift problem. Computer simulations performed based on redundant and hyper-redundant planar manipulators show that, when the end-effector traces a closed path in the workspace, the robot returns to its initial configuration. The solution is repeatable for a workspace with and without obstacles in the sense that, after executing several cycles, the initial and final states of the manipulator are very close.     

Fuzzy adaptive output feedback control for robotic systems based on fuzzy adaptive observer

In this paper, a fuzzy adaptive output feedback control scheme based on fuzzy adaptive observer is proposed to control robotic systems with parameter uncertainties and external disturbances. It is supposed that only the joint positions of the robotic system can be measured, whereas the joint velocities are unknown and unmeasured. First, a fuzzy adaptive nonlinear observer is presented to estimate the joint velocities of robotic systems, and the observation errors are analyzed using strictly positive real approach and Lyapunov stability theory. Next, based on the observed joint velocities, a fuzzy adaptive output feedback controller is developed to guarantee stability of closed-loop system and achieve a certain tracking performance. Based on the Lyapunov stability theorem, it is proved that all the signals in closed-loop system are bounded. Finally, simulation examples on a two-link robotic manipulator are presented to show the efficiency of the proposed method.     

Forward position analysis of 6-3 Linapod parallel manipulators

In this study closed-form solutions to the forward kinematic problems are obtained for a particular type of six degree-of-freedom parallel manipulator called 6-3 Linapod. The 6-3 Linapod parallel manipulators have a 6-3 PSS (or PUS) structure, and forward kinematic solutions are obtained by using the solution procedure for 6-3 SPS (or UPS) manipulators. In this procedure, a 6-3 Linapod is first transformed into its equivalent mechanism, namely an inclined 3RS manipulator, and then the condition that the three spherical joints on the moving platform form an equilateral triangle leads us to obtain three polynomial equations in three unknowns. These equations are solved by using Sylvester dialytic elimination method. Each set of real roots corresponds to a particular configuration of the manipulator. Solutions so obtained are verified by performing inverse position analysis. A method to identify configurations containing crossed links is presented in this study, which is based on the interpretation of link crossing as intersection of a link with a triangle, whose vertices are positions of joints on the corresponding links.     

A particle swarm optimization approach for fuzzy sliding mode control for tracking the robot manipulator

In this paper, an optimal fuzzy sliding mode controller is used for tracking the position of robot manipulator, is presented. In the proposed control, initially by using inverse dynamic method, the known sections of a robot manipulator’s dynamic are eliminated. This elimination is done due to reduction over structured and unstructured uncertainties boundaries. In order to overcome against existing uncertainties for the tracking position of a robot manipulator, a classic sliding mode control is designed. The mathematical proof shows the closed-loop system in the presence of this controller has the global asymptotic stability. Then, by applying the rules that are obtained from the design of classic sliding mode control and TS fuzzy model, a fuzzy sliding mode control is designed that is free of undesirable phenomena of chattering. Eventually, by applying the PSO optimization algorithm, the existing membership functions are adjusted in the way that the error tracking robot manipulator position is converged toward zero. In order to illustrate the performance of the proposed controller, a two degree-of-freedom robot manipulator is used as the case study. The simulation results confirm desirable performance of optimal fuzzy sliding mode control.     

Redundant actuators to achieve minimal vibration trajectory tracking of flexible multibodies: Theory and application

We address the problem of inverse dynamics for flexible multibodies, which arises, in trajectory tracking control of flexible multibodies such as space manipulators and articulated flexible structures. Previous research has resolved this trajectory tracking problem by computing the system inputs for feedforward control of actuators at the joints. Recently, the use of distributed actuators like electro-strictive actuators in flexible structures has introduced a new dimension to this trajectory tracking problem. In this paper we optimally utilize such actuators to aid joint actuators for tracking control, and introduce a new inverse dynamics scheme for simultaneously (1) tracking a prescribed trajectory and (2) minimizing ensuing elastic deflections. We apply this scheme for trajectory tracking of a two-link two-joint planar manipulator with joint motors and distributed electro-strictive actuators. Experimental results are presented to contrast our new scheme with other existing methods.     

Assessing joint angles in human hand via optical tracking device and calibrating instrumented glove

The aim of this paper is to present a method for assessing joint angles in a human hand: a method suitable for the calibration of an instrumented glove. The method is based on an optical tracking device and an inverse-kinematic model of the human hand. It requires only one reflective marker to be attached to each finger and three on the dorsal aspect of the hand in order to assess angles in finger joints. A further three markers are needed to calculate angles in thumb joints. Joint angles assessed through inverse kinematics and with the calibrated glove were validated against reference angles calculated from the centers of rotation of the joints while measuring the finger movements with multiple markers. In fingers, the mean difference between the reference angles and the angles assessed by the glove did not exceed ±7° when the proposed model-based method was used to calibrate the glove. For the thumb the mean error did not exceed ±5° when the reference method was used to calibrate the glove.     

A study on the effect of structure parameters on the dynamic characteristics of a PRRRP parallel manipulator

This paper studies the effect of structure parameters on the dynamic characteristics of a planar PRRRP parallel manipulator. The stiffness model is derived by considering the effect of joint. Based on the stiffness matrix, the vibration equation of this parallel manipulator is investigated to study the dynamic characteristics. The natural frequency is computed, and the effect of Y and Z coordinate on the natural frequency is discussed. Moreover, the sensitivity model of the dynamic characteristic to critical structure parameters is proposed. The thickness of column and leg, the radial stiffness of bearing, and the lumped mass on the end-effector are determined based on the natural frequency and sensitivity index. The results are useful to the structure design of parallel manipulators.     

基于视觉测量与神经网络的工业机器人位姿补偿

为提高六轴工业机器人的绝对定位精度，本文提出了一种利用视觉测量数据通过ELM(Extreme Learning Machine)神经网络实现机器人位姿补偿的新方法．利用固定在机器人末端的手眼相机获取机器人的末端位姿，并借助ELM实现机器人末端执行器从目标位姿到预测指令位姿之间映射，用修正转角代替原转角使机器人末端执行器运行至修正位姿，实现补偿．特别的是，对于使用的ELM，以网络预测均方误差为指标定量选取了网络的最佳参数．相比之前的方法，本文提出的算法具有能够同时高精度补偿姿态角及位置误差的显著优点．为验证该位姿误差补偿方法的有效性，本文进行了实验验证．结果表明，相比较于未补偿前的机器人末端位姿误差，经该方法补偿后的位姿误差被稳定控制在较低水平，平均位置误差降低89.1 %；平均姿态角误差降低96.8 %．除此以外，位置误差与姿态角误差的标准差也分别降低了85.66 %和93.24 %．     

存在关节死区的空间机器人无扰快速终端滑模控制

针对机械臂一般操作过程中运动学的非完整特性进行运动规划时没有考虑机械臂与待抓取目标之间的关系与关节的实际特性, 研究了存在关节死区的漂浮基平面三连杆空间机械臂拦截目标前最后阶段的载体无扰动空间规划与控制. 首先根据拉格朗日第二类方程, 建立存在关节死区的载体位姿均不受控的漂浮基平面三连杆空间机械臂的动力学模型, 推导出三连杆空间机械臂反作用零空间的数学模型, 并对反作用零空间进行向量范数约束算法研究; 进而提出了一种具有抗干扰性与高收敛性的非奇异快速终端滑模控制算法实现系统的姿态无扰控制, 该方法采用变系数双幂次趋近率与非奇异快速终端滑模面相结合的方式, 提高系统状态收敛速度与抗干扰性. 为了消除机械臂关节存在的死区特性, 设计了自适应死区补偿器, 通过自适应控制来逼近死区特性的上界, 以消除关节死区对系统带来的影响, 确保跟踪控制的有效执行. 最后基于Lyapunov函数法证明了系统的稳定性, 并通过系统数值仿真结果验证了存在死区情况下机械臂的各关节角跟踪上无反应空间下的期望轨迹的同时载体的姿态处于稳定状态, 验证了所提方法的有效性.      

State-Space Generalized Predictive Control for Redundant Parallel Robots

Abstract

The article deals with the design and properties of generalized predictive control (GPC) for path control of redundant parallel robots. Redundant parallel classification means redundant number of actuators, i.e., more actuators than degrees of freedom of the robot. Control of such structures suffers from several new control problems like potential inconsistency of steady state positions or nonuniqueness of control actions. The article explains classical direct derivation of GPC and its modification based on square root two-step design of control actions for solving the control problems. As an example for verification of algorithms, a prototype of a planar redundant parallel robot is used. Both design approaches are compared and several possibilities of extensions are presented for taking into consideration additional requirements, like smooth course of actuators or fulfillment of the anti-backlash condition.