数控速度规划中的过象限摩擦误差约束

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方晨曦, 张辉, 叶佩青, 梁文勇, 李维谦. 数控速度规划中的过象限摩擦误差约束. 54(6): 822-827[Chenxi FANG, Hui ZHANG, Peiqing YE, et al. High-speed feed-rate planning with friction error constraints[J]. 54(6): 822-827] 复制到剪切板

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数控速度规划中的过象限摩擦误差约束

方晨曦¹, 张辉¹, 叶佩青¹, 梁文勇², 李维谦²

1. 清华大学机械工程系, 精密超精密制造装备及控制北京市重点实验室, 北京 100084

2. 甘肃省数控机床工程技术研究中心, 天水 741024

叶佩青, 研究员, E-mail:yepq@tsinghua.edu.cn

作者简介: 方晨曦(1987#cod#x02014;), 男(满), 江苏, 博士研究生。

基金:国家“十二五”科技重大专项(2011ZX04004-012);

摘要

数控高速加工复杂轨迹时,过象限摩擦误差是轮廓误差的重要来源。将过象限摩擦误差约束加入速度规划模型,能有效弥补伺服系统中的摩擦力反馈补偿缺乏全局信息的缺陷,在保证加工精度的同时实现最优加工效率。首先建立了考虑摩擦影响的机床进给伺服系统动力学模型,结合数学推导与参数正交实验方法,针对光滑复杂轨迹研究了过象限摩擦误差产生机理,提出了系统换向时由摩擦产生的最大跟随误差的预测数学模型,并将其作为约束加入数控速度时间最优规划模型。实验结果表明: 该模型可以较为精确地预测过象限摩擦误差。对比轨迹跟踪结果证明了包含过象限摩擦误差约束的速度规划能有效控制最终轨迹轮廓精度。

关键词: 数控系统; 进给速度规划; 轮廓误差; 摩擦误差

中图分类号:TH273.5 文献标志码:A 文章编号:1000-0054(2014)06-0822-06

High-speed feed-rate planning with friction error constraints

Chenxi FANG¹, Hui ZHANG¹, Peiqing YE¹, Wenyong LIANG², Weiqian LI²

1. Beijing Key Laboratory of Precision/Ultra-Precision Manufacturing Equipments and Control, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

2. Gansu Engineering Research Center for CNC Machine Tool, Tianshui 741024, China

Fund:

Abstract

Feed-rate planning models must be accurate to improve contouring performance and efficiency. Quadrant protrusion errors caused by friction forces are an important source of contour error in high-speed contouring of complex curves. A theoretical analysis is given here to describe the relationships among the curvature, feed rate and quadrant protrusion error. Then, experiments with a wide range of parameters are used to develop a friction error model for feed-rate planning with the friction error as a constraint. Tests show that the model can accurately estimate the friction error and guarantee the contour accuracy requirements.

Keyword: numerical control system; feed-rate planning; contour error; friction error

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数控加工复杂轨迹时,为尽可能缩短加工时间,同时减小因频繁加减速产生的机床震动,数控系统通常采用前瞻技术,在插补运算前,提前读取轨迹段进行速度规划^[1,2],建立符合实际的速度规划约束条件是速度规划研究的核心问题。文[3]提出因机床电机力矩饱和导致的速度与加速度约束,建立了轨迹曲率与电机加速度对速度的限制模型。文[4]将由插补步长引起的弓高误差作为速度规划约束,从而保证轨迹的理论精度满足要求。文[5]为实现速度平滑引入了加加速度约束。但这些研究中通常简化考虑动力学约束,如将电机加速度限制为常量、不考虑伺服进给系统动态过程等。这种对约束的简化处理在可能浪费进给性能的同时^[6],也无法保证加工的轮廓精度^[7,8]。因此,很多学者在规划模型中加入了动态误差约束,根据进给系统跟随输入指令的跟随误差建立轨迹加工轮廓误差模型。

各轴动态误差包括跟随误差和换向时的过象限摩擦误差两部分。其中,基于线性系统模型可以相对精确地估计出各轴沿单向进给时的跟随误差,从而得到速度约束模型。在之前的研究中,过象限摩擦误差则被简化处理成一个常量^[7,8],并且大量实验结果表明,实际进给系统的过象限摩擦误差随着轨迹曲率与进给速度的不断变化而改变,虽然过象限摩擦误差由摩擦现象与反向间隙共同作用形成,但高精度等级丝杠的反向间隙非常小(数量级小于摩擦误差), 因此通常将其忽略。摩擦力本质上是含有非线性特征的动力学现象,如它含有Stick-Slip、 Stribeck Effect等现象^[9]。在摩擦学领域,很多学者提出了各自的摩擦力理论模型^[10,11],这些摩擦力模型在电机伺服控制领域得到了大量研究和应用,许多过象限摩擦误差模型也被提出^[12,13,14],但这些研究偏重于微观机理,在最优控制应用中难以处理电机力矩饱和问题。

速度规划模块位于系统上层,拥有加工轨迹、工艺需求、机床伺服与机械性能等全面的数据信息,该模块可以通过调整进给速度解决数控加工中摩擦误差问题。在数控速度规划研究领域,过象限摩擦误差模型尚未得到大量关注。过象限摩擦误差对于最终加工效果有重要影响,典型例子如圆轨迹加工时,过象限摩擦误差显著超过了单向跟随误差,成为轨迹轮廓误差的主要来源^[15]。因此,研究速度规划模型中的过象限摩擦误差对机床准确执行复杂加工轨迹具有重要意义。

本文首先建立机床进给伺服系统动力学模型并对其参数进行理论分析,然后由参数正交实验得到过象限摩擦误差预测数学模型,并将其加入数控速度时间最优规划模型。实验结果显示,过象限摩擦误差约束能有效控制换向时的轮廓误差。

1 过象限摩擦误差

伺服系统由伺服驱动器、伺服电机、工作台等组成。伺服驱动器一般包括位置环、速度环、电流环等反馈控制模块。忽略电机的机械阻尼与电感,可以得到电机的电流微分方程为

$\begin{matrix} {[(u - y) K_{p} - \overset{\cdot}{θ}] K_{v} - i} K_{c} - \overset{\cdot}{θ} K_{e} = Ri. (1) \end{matrix}$

式中: u为系统输入, y为工作台位置, K_p为位置环增益, θ为电机轴角度, K_v为速度环增益, i为电机电流, K_c为电流环增益, K_e为反电动势系数, R为电机内阻。

电机转动通过联轴器带动滚珠丝杠转动,并通过螺母变为工作台平动。假设滚珠丝杠与工作台之间为弹性连接,则电机与滚珠丝杠部分力矩平衡方程为

$\begin{matrix} i K_{t} = J \overset{\cdot \cdot}{θ} + (y_{m} - y) K K_{bs} . (2) \end{matrix}$

式中: K_t为电机力矩系数, J为电机转动惯量, y_m为丝杠螺母位置, K为连接刚度, K_bs为转动到平动的转换系数。

则工作台部分力平衡微分方程为

$\begin{matrix} (y_{m} - y) K = M \overset{\cdot \cdot}{y} + B \overset{\cdot}{y} + F_{f} sgn (y) . (3) \end{matrix}$

式中: M为工作台质量, B为工作台摩擦阻尼系数, F_f为摩擦力。

系统整体结构如图1所示,其中 s为Laplace算子。

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	图1 进给系统结构示意图

考虑换向时的动力学过程。由于伺服电流环带宽(一般10 kHz左右)远大于位置环带宽(一般1 kHz左右), 可将电流与电机力矩看作线性关系,由式(1)—(3)得到系统微分方程为

$\begin{matrix} \begin{matrix} [(u - y) K_{p} - \overset{\cdot}{θ}] K_{v} K_{c} K_{t} / R = J \overset{\cdot \cdot}{θ} + \\ K_{bs} (M \overset{\cdot \cdot}{y} + B \overset{\cdot}{y} + F_{f} sgn (y)) . (4) \end{matrix} \end{matrix}$

对于参数曲线轨迹 C( u),假设机床在 X- Y平面运动,以角速度 ω通过曲率半径为 r的 X轴换向处,即 x( t) =rsin( ωt), t₀ = 处,则在换向处附近有

$\begin{matrix} \begin{matrix} \overset{\cdot}{θ} (t_{0}) \approx 0, \overset{\cdot}{x} (t_{0}) \approx 0, \\ \overset{\cdot \cdot}{θ} (t_{0}) \approx K_{bs} r ω^{2}, \overset{\cdot \cdot}{x} (t_{0}) \approx r ω^{2} . (5) \end{matrix} \end{matrix}$

此时,摩擦力接近最大静摩擦力 F_m,有 F_f =F_m。带入系统方程,可得换向处位置误差 e_f:

$\begin{matrix} \begin{matrix} u - y \approx \frac{K_{bs} R}{K_{p} K_{v} K_{c} K_{t}} [(M + J) r ω^{2} + F_{m}], \\ e_{f} \approx K [(M + J) v^{2} / r + F_{f} sgn (y)] . (6) \end{matrix} \end{matrix}$

其中:

$\begin{matrix} \begin{matrix} K = K_{bs} R / (K_{p} K_{v} K_{c} K_{t}), \\ r = \frac{{|\frac{dC (u)}{du}|}^{3}}{|\frac{dC (u)}{du} \cdot \frac{d^{2} C (u)}{d u^{2}}|} . \end{matrix} \end{matrix}$

由(6)式可知,换向处位置误差由惯性与摩擦力两部分组成。其中,惯性部分误差大小与速度成正比,与轨迹曲率半径成反比。对于摩擦力部分,由于系统状态在零速区与低速区,即使采用能描述诸多摩擦现象的LuGre动态摩擦模型进行仿真实验,也无法得到实际中摩擦误差高速范围不随速度增加而增加的结果。文[15]对2维混合摩擦模型进行仿真研究也得到了相同的结论。目前,很难得到摩擦力关于曲率半径与速度的显示函数表达。

因此,本文通过正交参数实验建立过象限摩擦误差模型。以第3节所述设备进行实验,使机床在 X- Y 平面做圆轨迹运动,通过改变半径与进给速度,得到多组 X轴过象限摩擦误差。

本文假定曲率半径和进给速度在 r≥10 mm、 v<8 m/min范围内。图2显示了 r=10 mm, v=1 m/min时轨迹轮廓误差实验结果。 X轴过象限摩擦误差明显大于其他位置的轮廓误差,决定了加工最大轮廓误差的大小。在换向处 X轴到达顶点位置后,有一段“平顶”,即对应上述建模分析中,由于摩擦力和惯性力作用,电机驱动力不足造成的进给停滞。

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	图2 r=10 mm v=1 m/min实验轨迹轮廓误差与机床位移图

图3显示了正交参数实验数据拟合曲线。可以看出,当进给速度小于4 m/min时,过象限摩擦误差与轨迹曲率半径成反比,与速度成正比,该趋势符合式(6)的描述和解释; 当进给速度大于4 m/min时,过象限摩擦误差不再随速度增加而增加,可能是因为滑动速度增加导致油膜厚度增加,从而改善了润滑条件。这是LuGre的2维混合等摩擦模型无法描述的。经反复实验可知,图3所得实验结果是稳定的,可以作为拟合数据进行建模。

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	图3 实验数据拟合曲线

2 速度规划模型

设有一段参数路径 P( u) =[ x( u) y( u) z( u)]^T, u∈[0,1] .

目标函数为最短时间完成轨迹,

$\begin{matrix} \min_{\overset{\cdot}{u}} t = \int_{0}^{1} \frac{du}{\overset{\cdot}{u}} . \end{matrix}$

约束条件有速度约束、加速度约束、加加速度约束、跟踪误差约束。

1) 各轴速度约束为

$\begin{matrix} v_{i, \min} \leq v_{i} (t) \leq v_{i, \max}, i = x, y, z. \end{matrix}$

式中: 各轴速度 v_i( t) =v_i( u, $\begin{matrix} \overset{\cdot}{u} \end{matrix}$ ) = $\begin{matrix} \frac{d P_{i} (u)}{du} \end{matrix} \begin{matrix} \overset{\cdot}{u} \end{matrix}$ ; v_i_,max, v_i_,min分别表示速度上、下限,由加工工艺与电机性能共同决定。

2) 加速度约束为

$\begin{matrix} - F_{i, \max} \leq k_{i, a} a_{i} (t) + k_{i, v} v_{i} (t) \leq F_{i, \max} . \end{matrix}$

式中: a_i( t) =a_i( u, $\begin{matrix} \overset{\cdot}{u} \end{matrix}$ , $\begin{matrix} \overset{\cdot \cdot}{u} \end{matrix}$ ) = $\begin{matrix} \frac{d v_{i} (t)}{dt} \end{matrix}$ = $\begin{matrix} \frac{d^{2} P_{i} (u)}{d u^{2}} \end{matrix} \begin{matrix} {\overset{\cdot}{u}}^{2} \end{matrix}$ + $\begin{matrix} \frac{d P_{i} (u)}{du} \end{matrix} \begin{matrix} \overset{\cdot \cdot}{u} \end{matrix}$ 为各轴加速度; F_i_,max, F_i_,max分别表示电机驱动力上、下限。

3) 加加速度约束为

$\begin{matrix} J_{i, \min} \leq J_{i} (t) \leq J_{i, \max} . \end{matrix}$

J_i( t) =J_i( u, $\begin{matrix} \overset{\cdot}{u} \end{matrix}$ , $\begin{matrix} \overset{\cdot \cdot}{u} \end{matrix}$ , $\begin{matrix} \overset{\cdot \cdot \cdot}{u} \end{matrix}$ ) = $\begin{matrix} \frac{d a_{i} (t)}{dt} \end{matrix}$ = $\begin{matrix} \frac{d^{3} P_{i} (u)}{d u^{3}} \end{matrix} \begin{matrix} {\overset{\cdot}{u}}^{3} \end{matrix}$ +3 $\begin{matrix} \frac{d^{2} P_{i} (u)}{d u^{2}} \end{matrix} \begin{matrix} \overset{\cdot}{u} \end{matrix} \begin{matrix} \overset{\cdot \cdot}{u} \end{matrix}$ + $\begin{matrix} \frac{d P_{i} (u)}{du} \end{matrix} \begin{matrix} \overset{\cdot \cdot \cdot}{u} \end{matrix}$ 为各轴加加速度; J_i_,max, J_i_,min分别表示加加速度上、下限。

4) 跟踪误差约束在各轴单方向运动时为

$\begin{matrix} e_{i} (t) \leq E_{\max} . \end{matrix}$

式中: 各轴跟踪误差为 e_i( t) =K_v v_i( t) +K_c a_i( t), E_max表示加工允许最大轮廓误差。

在换向时,对于任意平滑曲线,其在某轴换向处可以被看作以某速度通过某半径圆弧。根据理论分析与实验结果,将摩擦误差拟合为关于速度的分段函数。设过象限摩擦误差约束为

$\begin{matrix} e_{f} \leq E_{\max} . \end{matrix}$

式中: $\begin{matrix} \{\begin{matrix} e_{f} = αv / r, v \leq e_{fmax} r / α \\ e_{f} = e_{fmax}, v > e_{fmax} r / α \end{matrix} \end{matrix}$

v代表合成速度。当 v>e_fmax r/α时,过象限摩擦误差即为常值 e_fmax; 反之,则将 v与 e_f拟合为一次函数, α为拟合系数。

综上所述,数控微段连续速度规划问题可表述为如下静态最优问题,可直接使用Matlab非线性规划工具箱求解。

$\begin{matrix} \min_{\overset{\cdot}{u}} t = \int_{0}^{1} \frac{du}{\overset{\cdot}{u}} \end{matrix}$

s.t. v_i_,min≤ v_i( t)≤ v_i_,max, i=x, y, z,

$\begin{matrix} \begin{matrix} - F_{i, \max} \leq k_{i, a} a_{i} (t) + k_{i, v} v_{i} (t) \leq F_{i, \max}, \\ J_{i, \min} \leq J_{i} (t) \leq J_{i, \max}, \\ e_{i} (t) \leq E_{\max}, \\ e_{f} \leq E_{\max} . \end{matrix} \end{matrix}$

3 实验验证

实验平台为自主研发的高精度铣床与GT100型数控系统(见图4), 其中 X轴为龙门结构,负载相对较重,被选为实验对象。 X、 Y轴均配有光栅尺,分辨率为0 .5 μm,光栅尺信号采集设备为NI公司DAQ系统,采集频率为100 MHz。轨迹运动参数设定如下: 指令速度为3 m/min, 插补周期为4 ms, 各轴最大加速度为0 .2 g, 最大允许轮廓误差为20 μm。

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	图4 实验设备

测试路径为平面参数样条曲线,见图5。整条轨迹包含3个 X轴换向区(A、 B、 C)。将轨迹输入数控系统经过速度规划并插补输出至机床进给系统,由采集系统同步采集光栅信号得到实际轨迹位置。实验结果见图6—8。

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	图5 实验测试路径

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	图6 无过象限摩擦约束速度规划结果图

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	图7 包含过象限摩擦约束速度规划结果

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	图8 X轴换向点处实测轨迹

图6与7分别为没有过象限摩擦约束与包含过象限摩擦约束的整条轨迹速度规划结果。可以看出,包含过象限摩擦约束后, X轴换向时 Y轴由3 m/min的指令速度降为1.2 m/min左右,在换向处降低了合成矢量速度,轨迹其余部分通过速度没有受到影响。由于换向处降速,轨迹整体加工时间由3.88 s略微增加到3.92 s, 没有对加工效率产生显著影响。

图8分别显示了测试轨迹3个 X轴换向点处的轮廓跟随情况。图8a、 8c、 8e为无过象限摩擦约束模型规划实验结果, 3个 X轴换向点最大轮廓误差分别为33 .5 μm, 32.7 μm, 34.9 μm, 均超出最大允许轮廓误差;图8b、 8d、 8f为包含过象限摩擦误差约束模型规划实验结果, 3个 X轴换向点最大轮廓误差分别为21 .0 μm, 18.4 μm, 19.7 μm,基本满足最大允许轮廓误差要求。

测试实验结果表明,通过正交参数实验得到的过象限摩擦误差约束模型,可以在轮廓误差允许范围内对换向时的通过速度提供有效指导,在不降低加工效率的情况下,避免换向处轮廓误差超过设定值,是保证加工轨迹轮廓精度的必要条件。

4 结论

本文从理论上分析了过象限摩擦误差的产生机理,将复杂轨迹换向处局部抽象为圆轨迹,由实验数据拟合得到过象限摩擦约束模型,完善了数控速度规划模型,保证了最终轨迹的轮廓精度满足要求。实验结果表明: 本文所提出的应用于数控速度规划的过象限摩擦误差模型具有良好的可行性,易于应用于实际。

下一步的工作将探索更加精确的摩擦误差模型,考虑负载与轴换向位置等其他影响因素,并探索更精确且实用的实验数据拟合方法; 同时,将研究建立适于表达圆轨迹换向过程的摩擦模型,将系统动力学与摩擦模型结合在一起,为摩擦误差建模提供参考。

The authors have declared that no competing interests exist.

参考文献

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[2]	ZHANG Lixian, SUN Ruiyong, GAO Xiaoshan, et al. High speed interpolation for micro-line trajectory and adaptive real-time look-ahead scheme in CNC machining[J]. Science China: Technological Sciences, 2011, 54(6): 1481-1495. [本文引用:1]
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2011

1.205

0.0

SHI

Chuan

, YE

Peiqing.

The look-ahead function-based interpolation algorithm for continuous micro-line trajectories[J]. International Journal of Advanced Manufacturing Technology, 2011, 54(5-8): 649-668.

Continuous micro-line segments having micron magnitude are also used to approach complicated trajectories in computer numerical control (CNC) machining. Traditional linear interpolation processes micro-line segments separately. The rather small lengths of the segments result in low machining efficiency, and the frequent acceleration and deceleration also reduce the lifetime of the motor. To solve the problems, in this paper, we present an interpolation algorithm that is based on the look-ahead function. The look-ahead-based feedrate planning is elaborated from the aspects of trajectory feedrate description, feedrate preplanning, and feedrate planning. With the function of feedrate planning, CNC systems can predict the variation of the curvature of the coming trajectory so that adaptively determining whether accelerates interpolation feedrate to enhance machining efficiency or decelerates the feedrate to guarantee the machining precision. In addition, the trajectory interpolation for micro-line segment trajectories is analyzed as well, which realizes the generalization of the coordinates for each interpolation step. The implementation of the proposed algorithm in our CNC system is also studied. Finally, two machining experiments are provided to verify the accuracy and efficiency of the proposed interpolation algorithm.

1.Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA 2.Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, 100084 People’s Republic of China

2011

0.0

ZHANG

Lixian

, SUN

Ruiyong

, GAO

Xiaoshan

, et al. High speed interpolation for micro-line trajectory and adaptive real-time look-ahead scheme in CNC machining[J]. Science China: Technological Sciences, 2011, 54(6): 1481-1495.

CNC machining plays an important role in mechanical manufacturing. A key issue is to improve the machining feedrate while keeping the machining precision and satisfying the acceleration constraints of the CNC machine. For the consecutive micro-line segments interpolation, the velocities at the junction of two segments are the bottlenecks for the machining efficiency. This paper proposes a multi-period turning method to improve the feedrate at the junctions using the linear acceleration and deceleration mode, which utilizes the maximal acceleration capabilities of the NC machine while satisfying the machining precision. A new and more efficient look-ahead method and a feedrate override method are also proposed to boast the global machining speed. The proposed algorithm has been implemented on Blue Sky NC System, and experimented in real material manufacturing. Compared with several existing algorithms, the current algorithm can improve the manufacturing time ranging from 50% to 180%, depending on the machining parameters, and also results in better machining quality. In addition, the algorithm also satisfies the need of real-time interpolation.

1. Key Laboratory of Mathematics Mechanization, AMSS, Chinese Academy of Sciences, Beijing 100190, China; 2. Graduate University of Chinese Academy of Sciences, Beijing 100049, China;

2008

0.0

2012

1.205

0.0

FAN

Wei

, GAO

Xiaoshan

, YAN

Wei

, et al. Interpolation of parametric CNC machining path under confined jounce[J]. International Journal of Advanced Manufacturing Technology, 2012, 62(5-8): 719-739.

Abstract
Aiming at reducing the computer numerically controlled (CNC) machining vibration and increasing machining quality, an interpolation method for parametric tool paths with confined jounce, jerk, acceleration, and speed is proposed. An acceleration/deceleration profile with confined jounce, jerk, acceleration, and speed is proposed, and it is shown that this profile is time optimal to change the speed from one value to another under the given constraints. For a given parametric tool path, the velocity function is obtained by first computing the critical points of the tool path where the radius of curvature reaches extremal values, then determining the feasible maximal speeds at the critical points, and finally using the jounce confined acceleration/deceleration profile to connect the speeds at the adjacent critical points. A vibration experiment is conducted, which shows that vibration of the CNC machine decreases significantly for motions under confined jounce than that under confined acceleration and jerk. Simulation for two real CNC models are given to show the feasibility of the method.

<li>1.KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China</li>

2012

1.07

0.0

2000

0.0

... 这种对约束的简化处理在可能浪费进给性能的同时^[6],也无法保证加工的轮廓精度^[7-8] ...

2007

0.786

0.0

2013

1.205

0.0

2004

1.669

0.0

Guesalaga

Modelling end-of-roll dynamics in positioning servos[J]. Control Engineering Practice, 2004, 12(2): 217-224.

Abstract

End-of-roll dynamics is a characteristic of pointing systems and positioning servos when change in movement direction occurs. Factors such as backlash and friction cannot be easily controlled nor predicted under these circumstances, and although an accurate model for friction exists, the literature is still lacking of a representation for the integrated effect backlash and friction dynamics in these loops. This paper proposes a new mathematical model to represent backlash effects in positioning servos demonstrating its accuracy to predict tracking errors of a naval gun mount. The model introduces a continuous hysteresis representation, different from classical discontinuous models, showing a behaviour closer to what has been observed empirically. The new model can be useful for high performance positioning servo-control applications.

... 摩擦力本质上是含有非线性特征的动力学现象,如它含有Stick-Slip、 Stribeck Effect等现象^[9] ...

1985

0.758

0.0

1996

0.0

1994

2.919

0.0

1999

1.3

0.0

2004

0.0

2001

0.0

... 过象限摩擦误差对于最终加工效果有重要影响,典型例子如圆轨迹加工时,过象限摩擦误差显著超过了单向跟随误差,成为轨迹轮廓误差的主要来源^[15] ...