Dissociated jerk-limited trajectory applied to time-varying vibration reduction

Jerk-limited trajectories are a widespread solution for the trajectory planning of industrial machines-tools or robots. It is known that jerk limitation can reduce vibrations and in some cases can totally suppress residual vibration induced by a lightly damped stationary mode. However, for systems with time-varying mode, which is classically the case for configuration dependent mode or load mass variations, the previous result vanishes. This paper proposes to extend the jerk-limited profile (JL) properties to time-varying vibration problem. First, a guideline for designing a dissociated jerk-limited profile (DJL) based on simple and pragmatic Finite Impulse Response (FIR) filtering methodology is presented. Following the guideline, the time-varying vibration reduction principle is detailed. Then, experiments conducted on an industrial 3-axes Cartesian manipulator are presented. The experimental results show that the residual vibration magnitude is reduced to less than 23% of the original level obtained with JL profile and the settling time is reduced by 10%, demonstrating the efficiency of the proposed DJL trajectory planning.     

Model-based trajectory planning for flexible-link mechanisms with bounded jerk

This paper deals with the model-based development of optimal jerk-limited point-to-point trajectories for flexible-link robotic manipulators. In the proposed approach, an open-loop optimal control strategy is applied to an accurate dynamic model of flexible multi-body planar mechanisms. The model, which has already been fully validated through experimental tests, is based on finite element discretization and accounts for the main geometric and inertial non-linearities of the linkage. Exploiting an indirect variational solution method, the necessary optimality conditions deriving from Pontryagin's minimum principle are imposed, and lead to a differential Two-Point Boundary Value Problem (TPBVP); numerical solution of the latter is accomplished by means of collocation techniques. The resulting motion and control profiles can be used as feedforward reference signals for a position and vibration control. Considering a lightweight RR robot, simulation results are provided for rest-to-rest, jerk-limited trajectories with minimum actuator jerks and vibrations. However, the strategy under investigation has general validity and can be applied to other types of mechanisms, as well as with different objective functions and boundary conditions. Numerical evidence clearly indicates that the use of a composite cost functional and the imposition of jerk constraints can greatly reduce vibration phenomena during high-speed motion of flexible-link manipulators.     

Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery

Methodologies for planning motion trajectory of parametric interpolation such as non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, most of the algorithms were developed based on the constraints of feedrate, acceleration/deceleration (acc/dec), jerk, and chord errors. The errors caused by servo dynamics were rarely included in the design process. This paper proposes an integrated look-ahead dynamics-based (ILD) algorithm which considers geometric and servo errors simultaneously. The ILD consists of three different modules: a sharp corner detection module, a jerk-limited module, and a dynamics module. The sharp corner detection module identifies sharp corners of a curve and then divides the curve into small segments. The jerk-limited module plans the feedrate profile of each segment according to the constraints of feedrate, acc/dec, jerk, and chord errors. To ensure that the contour errors are bounded within the specified value, the dynamics module further modifies the feedrate profile based on the derived contour error equation. Simulations and experiments are performed to validate the ILD algorithm. It is shown that the ILD approach improves tracking and contour accuracies significantly compared to adaptive-feedrate and curvature-feedrate algorithms.     

带Stewart平台的航天器刚柔耦合动力学建模与仿真分析

本文采用柔性多体系统单向递推组集的建模方法,基于速度变分原理建立了带Stewart平台、柔性帆板和CMG组件的航天器刚柔耦合动力学模型.由于该模型自由度较大,无法满足实时控制的需求,因此建立了简化的Stewart平台等效模型,并通过与柔性Stewart平台完整模型对比,验证了所建立的动力学简化模型的正确性与高效性.分析了星体平台运动及柔性帆板的振动对有效载荷动力学响应的影响,指出了设计Stewart平台的微振动抑制方案时不能忽略下平台的运动及柔性附件的振动.本研究为带Stewart平台的航天器的微振动减振设计与高精度指向提供了有效的技术支撑.     

Feedback Control of Planar Linkages Using a Linearizing Filter: Theory and Experiments

The motion control of mechanical systems with flexible links is investigated. Issues addressed are the modeling and simulations, the design of a feedback control scheme and its implementation on an actual system. The modeling method used is a combination of a spline-based spatial discretization technique and the natural orthogonal complement of the kinematic constraints that eliminates the constraint forces and moments from the mathematical model. The control algorithm consists of two parts, namely, the uncoupling of nonlinear equations of motion and the filtering of consequent nonworking constraint forces. This control scheme is implemented on a prototype four-bar flexible mechanism. Results show that the proposed control scheme provides successful trajectory tracking while suppressing the vibration triggered by a doublet-type of disturbance.     

平面双连杆受限柔性机器人臂的动力学建模*

对一类平面双连杆受限柔性机器人臂的动力学建模问题进行研究，利用Ｄ’Ａｌｅｍｂｅｒｔ－Ｌａｇｒａｎｇｅ原理得到了一组描述该机器人系统运动性态的动力学方程。与已有的动力学模型相比，本文所建立的运动方程和振动方程具有模型准确、结构简单等特点，且具有与传统无约束刚性机器人类似的模型形式，因而有可能直接或间接利用现有的关于刚性机器人运动控制等方面的成果来研究复杂的受限柔性机器人的控制问题。     

柔性机械臂的变结构振动控制研究

当机械臂的质量很轻,尤其是空间应用场合,机器人系统将受到高度柔性限制并且不可避免地产生机械振动.本文为了证实提出的控制不期望残余振动的方法,设计并建立了柔性机器人实验平台.控制方案采用交流伺服电机通过谐波齿轮减速器驱动柔性机械臂,利用粘贴在柔性臂上的压电陶瓷片(PZT)作为传感器来检测柔性臂的振动.对由于环境激励,尤其是在电机转动(机动)时由于电机力矩产生的振动,采用了几种主动振动控制器包括模态PD控制,软变结构控制(VSC)和增益选择变结构方法,进行柔性臂的振动主动控制实验研究.通过实验比较研究,结果表明采用的控制方法可以快速抑制柔性结构的振动,采用的控制方法是有效的.     

Control of a Non-uniform Flexible Beam: Identification of First Two Modes

This paper presents an experimental study implementing the input shaping control of the first two modes of the vibration of a non-uniform flexible cantilever beam having a translating base. Examples of a moving cantilever beam appear in many industrial systems. Vibration suppression of the beam has important implications for improving the effectiveness of such systems. The equations of motion of the cantilever beam, including the moving base, are developed using the extended Hamilton principle. The partial differential equation representing the beam’s dynamics is then transformed into a finite-dimensional model using the Galerkin method. Accordingly, the modal parameter identification procedure is established based on experimental modal analysis. Under the estimated modal parameters, including the natural frequency and damping ratio, single- and two-mode input shaping controllers of three different types (zero vibration, zero vibration derivative, and zero vibration derivative-derivative) are designed for vibration suppression of the beam. Experimental results are discussed, reporting that the two-mode shaper’s vibration suppression was superior to the single-mode shaper. In contrast, the two-mode shaper’s settling time has slightly increased compared to that of the single-mode shaper.

     

Computation of vortex-induced vibrations of long structures using a wake oscillator model: Comparison with DNS and experiments

We consider here the dynamics of flexible slender systems undergoing vortex-induced vibration (VIV). This type of motion results from the coupling between the oscillating wake due to cross-flow and the structure motion. Practical applications are mainly found in the field of ocean engineering, where long flexible structures such as risers or mooring cables are excited by sea currents. The wake dynamics is here represented using a distributed wake oscillator coupled to the dynamics of the slender structure, a cable or a tensioned beam. This results in two coupled partial differential equations with one variable for the solid displacement and one for the wake fluctuating lift. This simplified model of the wake dynamics has been previously validated on simple experiments. Here, comparisons with direct numerical simulation results are done for both uniform and non-uniform flow. Comparison is also performed between the wake oscillator predictions and some experimental results on long cables. The results of those comparisons show that the proposed method can be used as simple computational tool in the prediction of some aspects of vortex induced vibrations of long flexible structures.     

Robust and adaptive boundary control of a stretched string on amoving transporter

Suppression of vibration is an important engineering problem. In this note, control problem of a flexible system that includes a stretched string supported on a transporter is defined and solved. Such a system may be encountered in device manufacturing and process automation. Robust and adaptive control is designed to damp out transverse oscillation of the string via compensating for possible uncertainties in string dynamics and transporter motion. Standard robust control design based on a straightforward Lyapunov argument commonly seen in control design for rigid-body systems is extended to the flexible system. Asymptotically/exponentially and robustly stabilizing controls are found     

Active vibration control of a flexible beam,based on flow source control

This paper presents a new method of actively controlling the vibration of a flexible beam by using a rigid body motion actuator based on flow source control. The proposed flow source controller generates a control input of a rotating angle instead of a torque that acts as an effort source control input. It is shown that the proposed flow source control improves the vibration suppression performance when disturbance forces such as friction forces are involved in the rigid body motion dynamics. The stability and the robustness to disturbance of the flow source controller are compared with an effort source controller. An optimal control theory is used to design the flow source vibration controller and a conventional PD controller is used for the motor position controller. Computer simulations and experimental results on a rotating beam system show that the vibration control performance achieved by the proposed flow source control method is superior to that of an effort source control method.     

Modified PID Control of a Single-Link Flexible Robot

The use of flexible links in robots has become very common in different engineering fields. The issue of position control for flexible link manipulators has gained a lot of attention. Using the vibration signal originating from the motion of the flexible-link robot is one of the important methods used in controlling the tip position of the single-link arms. Compared with the common methods for controlling the base of the flexible arm, vibration feedback can improve the use of the flexible-link robot systems. In this paper a modified PID control (MPID) is proposed which depends only on vibration feedback to improve the response of the flexible arm without the massive need for measurements. The arm moves horizontally by a DC motor on its base while a tip payload is attached to the other end. A simulation for the system with both PD controller and the proposed MPID controller is performed. An experimental validation for the control of the single-link flexible arm is shown. The robustness of the proposed controller is examined by changing the loading condition at the tip of the flexible arm. The response results for the single-link flexible arm are presented with both the PI and MPID controller used. A study of the stability of the proposed MPID is carried out.     

A study on a generalized parametric interpolator with real-time jerk-limited acceleration

The impossibility of exact arc length computation for ‘standard’ parametric curves such as Bézier/B-spline curves makes it difficult to generate a feedrate profile with desired accelerations in real-time. This paper presents a new recursive trajectory generation method that estimates an admissible path increment and determines the initiation of the final deceleration stage according to the distance left to travel estimated at every sampling time, resulting in exact feedrate trajectory generation through jerk-limited acceleration profiles for the parametric curves. The proposed approach allows the feedrate profile to be dynamically adjusted according to geometrical path constraints for the curved path. The simulation result has been also provided to illustrate the generation of smooth feedrate profile encompassing a sequence of mixed NC blocks including traditional linear and circular blocks.     

Vibration predictions and verifications of disk drive spindle system with ball bearings

With areal recording density of hard disk drives (HDD) historically growing at an average of 60% per year, it is becoming increasingly more difficult to maintain the precise positioning required of the ever-smaller GMR heads to read and write data. Any unexpected vibration will cause the data written to a wrong data track, even the vibration amplitude is very small. Consequently, the dynamic behaviors of HDD spindle systems and their potential influence on track misregistration rate must be clearly understood. This paper is to apply an approach based on efficient component mode synthesis (CMS), incorporating multi-body system dynamics technology to predict dynamic characteristics of HDD ball-bearing spindle systems. First, the discrete governing equations of motion for HDD spindle systems, which consist of several flexible and rigid components, are derived through the use of Lagrangian equations. The elastic component modal frequencies and modal shape vectors are then obtained using a finite-element analysis. For ball bearing inherently defects, a mathematical model is used as a time-varying force, resulting in spindle vibrations. The time-varying force and component modal shape vectors are incorporated into the governing equations of the whole spindle systems. An implicit numerical integration method is used to obtain the forced vibration of the HDD spindle system. Finally, the dynamic responses of two typical HDD spindle systems are investigated numerically to predict the significant coupled vibration frequencies, mode shapes and resonance interactions. The results well agree with the solutions predicted by other analytical methods and the experimental results, respectively.     

测定欠阻尼系统阻尼比的一种方法

提出了一种适用于实验测量欠阻尼系统阻尼比的方法，即通过计算相邻两个周期波形的面积之比来测定系统的阻尼比。文中就典型的二阶欠阻尼系统给出了算法 的数学推导，并从数学上证明了该方法比目前常用的用相邻波峰值之比估计阻尼比的方法具有更好的抗干扰能力，同时该方法简单，实时性强。此外，在实际应用中该方法可适用于所有可近似为二阶欠阻尼系统的高阶系统之阻尼测定。文中对一柔性悬垂梁的自由衰减运动进行了实验测定，结果表明用本文提出的方法测定系统的阻尼比比用波峰值之比进行阻尼比估算，其标准差可降低一个数量级。     

欠驱动柔性机器人的振动可控性分析

欠驱动柔性机器人的可控性分析是对其进行有效控制的关键问题. 本文以具有柔性杆的3DOF平面欠驱动机器人为例, 分两步分析系统的可控性. 首先,忽略杆件的弹性变形, 研究欠驱动刚性系统在不同驱动电机位置的状态可控性;然后, 考虑柔性因素, 研究欠驱动柔性系统的结构振动可控性. 结果表明振动可控性是随机器人关节位形和驱动电机位置而变化的, 并且欠驱动刚性机器人的状态可控性对相应的柔性系统的振动可控性有很重要的影响. 最后, 将上述研究方法扩展到具有一个被动关节的N自由度平面欠驱动柔性机器人.     

A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints

In the present study, for the first time, flexible multibody dynamics for a three-link serial robot with two flexible links having active prismatic joints is presented using an approximate analytical method. Transverse vibrations of flexible links/beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the links. In this study, first, vibration analysis of a flexible link sliding through an active prismatic joint having translational motion is considered. A rigid-body coordinate system is used, which aids in obtaining a new and rather simple form of the kinematic differential equation without the loss of generality. Next, the analysis is extended to include dynamic forces for a three-link planar serial robot called PPP (Prismatic, Prismatic, Prismatic), in which all joints are prismatic and active. The robot has a rigid first link but flexible second and third links. To model the prismatic joint, time-variant constraints are written, and a motion equation in a form of virtual displacement and virtual work of forces/moments is obtained. Finally, an approximate analytical method called the “constrained assumed modes method” is presented for solving the motion equations. For a numerical case study, approximate analytical results are compared with finite element results, which show that the two solutions closely follow each other.     

Local asymmetrical corner trajectory smoothing with bidirectional planning and adjusting algorithm for CNC machining

The linear motion command (G01) is a widely used command format in CNC machining. However, the tangential direction discontinuity at the corner junction will cause velocity fluctuations and excite the structural vibration of machine tools. Corner smoothing methods with controlled tolerance are used to obtain continuous smooth motion. Typically, most methods generate a symmetrical cornering trajectory around the angle bisector, and the trajectory generally decelerates first and then accelerates, which limits the velocity increase. In this paper, a novel local asymmetrical corner trajectory smoothing method is proposed, which can realize accelerated/decelerated cornering transition. The proposed method can obtain the analytical solution in one step, which is different from two-step geometric-based corner path smoothing, and can generate the same cornering trajectory in back-and-forth parallel toolpath. In addition, this paper proposes a bidirectional planning and adjusting algorithm for the situation where smoothed cornering paths are close to each other or even overlap. The algorithm can generate a jerk-limited trajectory by coordinating the cornering error of adjacent corners, while respecting the user-specified tolerance. Experimental results demonstrate the effectiveness of the proposed method in contour accuracy and cycle time for CNC machining along short-segmented toolpath.     

宏微复合运动平台的自抗扰控制研究

摩擦是影响机械导轨运动平台精度的主要原因.宏微复合运动平台将无摩擦的柔性铰链与直线平台结合在一起,利用柔性铰链的弹性变形补偿摩擦死区.然而,柔性铰链的固有频率低,其非线性弹性振动严重影响微平台定位精度.为此,本文设计视弹性振动为扰动的自抗扰控制策略,该方法避免了建立非线性弹性振动精准数学模型的困难,利用扩张状态观测器主动估计弹性振动及不确定性,并在微平台位置环补偿之,以保证微平台定位精度.与此同时,在控制律中加入加速度前馈以提高系统响应速度.对于宏平台,采用PID控制作为宏平台位置环的控制策略,并通过宏微双级驱动方式补偿受机械导轨非线性摩擦带来的影响.实验对比结果表明,自抗扰控制在受非线性弹性振动影响时,其抗扰性能、跟踪性能优于传统的PID控制,可保证微平台良好的定位精度.