【机械类毕业论文中英文对照文献翻译】基于机床混合模型的参数曲线高速插补速度极值分析
【机械类毕业论文中英文对照文献翻译】基于机床混合模型的参数曲线高速插补速度极值分析,机械类毕业论文中英文对照文献翻译,机械类,毕业论文,中英文,对照,对比,比照,文献,翻译,基于,机床,混合,模型,参数,曲线,高速,速度,极值,分析
毕业设计文献翻译 院(系)名称工学院机械系 专业名称机械设计制造及其自动化 学生姓名贾玉芳 指导教师杨汉嵩2012年 03月 27日 毕业设计 文献综述 院(系)名称工学院机械系 专业名称机械设计制造及其自动化 学生姓名贾玉芳 指导教师杨汉嵩2010年 03月 27日 黄河科技学院毕业设计(文献翻译) 第 7 页基于机床混合模型的参数曲线高速插补速度极值分析 塞巴斯蒂安四蒂马尔,日达吨法鲁克美国加州大学戴维斯分校,机械系和航空工程系,美国 加州956162005年7月7日收稿, 2006年3月23修订 ,2006年4月10日发表摘 要算法是随着估算进给速度的曲率的变化而发展的,这确保了一个3轴的最低运动时间,数控机床受固定轴加速度范围和驱动电机输出扭矩特性的轴速度约束。对于由一个多项式参数曲线指定一个路径,最优时间的进给速度确定一个分段曲线函数的参数解析与细分,对应限制一个轴的加速度饱和常数。进给速度之间的始发点段,可通过数值计算的解决方法。对于细分固定加速度的(平方)的最佳进给速度是合理的曲线参数。对于速度依赖加速度范围,最佳进给速度在一种新的超越函数,其值及封闭的形式表达可有效地计算使用,实时控制一个特殊的算法。最佳进给速度推导出一个实时插补算法,可以直接从驱动器的解析路径描述机器。从实验结果的执行情况看,时间最优的3轴数控由于采用开放式构架的软件驱动进给速度控制器给出。该算法是一种显著的改善蒂马尔支持SD,法鲁克逆转录,史密斯给付,博亚杰夫建议。算法的时间最优控制沿着弯曲的数控机床刀具路径。机器人集成制造2005; 21:37-53,因为除了电压限制运动排除了沿直线或接近直线路径段任意高速的可能性。2006爱思唯尔版权所有。关键词:3轴加工,进给速度的函数,加速度的极值,时间最优路径遍历,噪音控制1 简介时间最优控制在以往的研究领域,机器人技术1-7和数控加工8-10关注与一个指定的路径最短时间穿越了一系统具有已知的动态和在指定的范围运动的执行机构。该方案解决这些问题的一个典型招致控制“噪音”战略,其中至少有一个输出系统饱和执行器在每个瞬间整个路径遍历。这些研究通常假定驱动器常与对称力极限(独立驱动的速度和方向)而且一般不解决问题的速度,超过该范围执行器可以发挥最大的力量。固定场直流电动机是最常见的定位在机器人及数控加工轮廓的应用11。由于他们的扭矩输出是成正比对电枢电流,恒转矩对称限制反映了马达的最大电流容量电枢绕组。保持恒转矩输出不断变化的有关电枢电压反电动势的(正比于电机转速)否则控制电枢电流供应10。除了电枢,电流限制应用电枢电压可能会受到限制的问题引起的电机特性或电枢电源。 这样电压限制限制了生产的运动能力最大输出扭矩,速度有限的范围内。超出此范围,最大适用电枢电压不电枢电流是限制因子电机扭矩输出,速度依赖造成最大力矩电机的增加呈线性下降速度10。 在3轴加工中,最大电流容量一轴驱动电机施加一个恒定的加速度限制在轴速度降低,最大电压容量规定在较高轴速度依赖加速度极限速度。从目前有限的过渡到电机轴的操作发生在过渡速度。在下面的速度过渡的速度,最高轴加速度保持不变。在速度大于过渡的速度,最大轴加速度线性轴的速度下降,在下降到零轴空载速度。为了保证时间的最优路径遍历符合这两个驱动器电流和电压的限制,算法必须考虑到这两个常数和在每台机器轴加速度限制。这本文推广了以前的研究结果9用人唯一不变的加速度式(1假设高速任意核算结果,如果路径中包含扩展线性段),并介绍了新算法现实的时间来计算最优进给速度为笛卡尔与驱动电机轴数控机床同时受电压和电流限制。列入的加速式招致重大,定性以较早的算法在许多方面的变化9,其中包括一套可行的进给速度和加速组合的速度限制曲线(可变编码);可能的切换不同类型点;以及进给速度的极值函数的形式相平面轨迹。然而,对于笛卡尔数控与轴独立驱动的机器,它仍然是可能的以获取基本上封闭形式解的进给速度,由于计算能力的根源某些多项式方程。我们首先回顾了第2个DC电机运行并在第3轴加速度范围。我们介绍了最低时的遍历问题常和速度依赖轴弯曲的路径加速度限制在第4节,我们得出进给速度恒和速度的表达式依赖极值加速度轨迹。饲料加速度限制,可变长编码,和进给速度破发点,然后对第5-7分别进行讨论。经过讨论的进给速度计算在第8和实时数控插补算法在第9,我们目前的细节进给速度计算和机实施效果。在第10条的几个例子。最后,第11节总结我们的结果并提出了一些结论说这番话的。2 直流电动机转矩限制为加深对轴的性质背景,适当的直角式加速度数控机床,我们开始与一固定场区的简要概述了通常用于驱动小型至中型电机。铣床(见其更完整的细节操作 10)。该方程管运作电机是也就是说,电机的输出转矩T是成正比的,电枢电流I,反电动势是成正比。电机角速度,电枢和应用电压V等于反电动势和总结的压降电枢电阻R的KT和柯相称因素,所谓的扭矩常数和反电动势常数,是内在的物理一个给定的电机性能的影响。从这些表现形式,你可以很容易地推导出电动机转矩转速的关系在给付是失速扭矩,和无负载速度。所以,电机转矩降低,线性电动机的速度增加,从时到时。 参见12更完整的细节。在发动机启动和低速时,反电动势E是小相比,施加电压V,以及限流设备是用来限制电流I为(大约)常数的最大值,以防止伊利姆电枢绕组的损坏。因此,电机转矩输出保持恒定在整个低转速范围的操作。随着马达的加快,电枢电压应用最终达到最大电机或电源供应器额定电压。这发生在过渡的速度,定义对于速度高于催产素大,电枢电压(而不是比目前的)是在电机转矩限制因素输出。在电压限制,扭矩T线性下降随着电机转速澳,下降至零,空载转速的实现。图(1)描述了电机的制约电流和电压范围,和在为积极和消极的马达速度。该约束定义两个平行带,其交集形成定义可行的制度直流电动机运行。所有受理的组合电动机的扭矩和速度,按照给定的电枢电流和电压范围,在这个谎言。 对超出的部分延伸无负载在每个方向符合再生电机,制动其中意味着外部扭矩申请。由于没有这样的扭矩可在驱动器中的数控机床马达,可行的扭矩范围/速度降低状态来表示空载速度最高电机转速,高产的六面平行四边形,如图1所示。 这六个面平行四边形定义了三个不同的直流马达转速范围,具有鲜明的最低和每最大扭矩限制,即:3 轴加速度限制在高速加工8,13,14惯性力可能称霸切削力,摩擦等,尤其是工具路径的高曲率。会计轴惯性,轴的速度和加速度是成比例的力矩电机和电机速度分别。考虑,也就是说,x轴。如果它是有效质量的Mx和驱动,由驱动电机通过弹性模量Kx(即滚珠丝杆,线性轴速度是关系到汽车的角相应的轴加速度以电动机转矩T是ax=KxT/Mx。注意到进给速度可被视为一个数量级v和载体由单位路径切线的特定方向,我们有和电机转速为因此,上面导出的转矩限制相当于X轴加速度限制其中VT是轴过渡的速度,V0的是轴空载速度,我们定义通过对速度的依赖加速度限制,轴速度VX始终保持在区间轴转速范围内 ,最低轴加速度和最高限额都是固定的,因此,这被称为制度的不断限制在X轴。轴速度范围,为其中一个加速度是固定的二是依靠速度,被称为混合为X轴的限制制度。在制度不变的限制,加速范围可写为。对于混合限制制度,加速范围可能表现在表格在路径遍历,每个轴在一个月内运作,其加速度限制制度独立于其他轴,每一个都可能加速极限之间切换,按照制度与工具的变化路径几何形状和进给速度。因此,有四个加速度限制制度的可能组合,其中的x,y轴,Z轴(见表1)。对于一个平面曲线,涉及的仅有的两个机轴运动,有三个可能的组合:常量/恒,恒/混合,和混合/混合。每个组合的加速度极限,除了要具体分析计算的时间最优进给速度。4 时间最优的进给速度考虑到学位曲线描述的路径与对照点。如果指弧长沿曲线测量,我们定义参数速度切线的单位和(主轴)和正常向量曲率(4)定义与此相反,与我们可以写现在假设我们遍历与进给速度(速度的曲线)指定由该函数。由于衍生金融工具方面时间t和参数x,我们以点表示和素数,分别为,由有关速度和加速度向量由每个点给出由切向分量的消失如果V 是常数,而正常(向心力)组件的如果消失K=0。其时的进给速度(衍生的加速度)给出的角度来看,我们希望尽量减少沿线rx遍历时间,开始和结束休息时,受限制的加速度表格(3)和其他类似用语机轴。这些要求可以在以下方面措辞以下优化问题使得其中指的是笛卡尔每一个组成部分,正如在第3节,轴加速度的形式是4.1 恒定加速度轨迹从关系和我们可以写对于给定的曲线的X轴组件(说)一个定义为加速的在我们写,因为它是方便工作对进给速度平方(见9详情)。在一个不断加速阶段极值加速度限制,其中一个组成部分,是加速等于加上或减去相应的约束,一条件是产生一个为q的线性微分方程如果x是加快轴,这个方程承认为(平方)进给速度,即封闭形式解。定义为其中积分常数C是取决于指定的一个已知点,对轨迹:关于进一步解决(10)的方法详情中可以 在9 找到。4.2 加速度极值轨迹考虑到当x轴(假定)执行一个加速极值,通过定义加速度极值约束决定进给速度v形式。通过以上描述,就是在这种情况下推导出进给速度的微分方程在我们不断介绍方程(11)是一阶变系数非线性微分方程。这对来说,可以专门写作Robotics and Computer-Integrated Manufac paths form Previous studies of time-optimal control in the fields of the speed and direction of actuation), and generally do not actuators can exert their maximum force. Fixed-field DC motors are common to most positioning armature voltage may be subject to limits arising from the motor characteristics or armature power supply. Such voltage limits confine the ability of the motor to produce ARTICLE IN PRESS the maximum output torque to a finite range of speeds. Beyond this range, maximum applied armature voltage not armature currentis the factor limiting the motor 0736-5845/$-see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2006.07.002 C3 Corresponding author. E-mail addresses: sdtimarucdavis.edu (S.D. Timar), faroukialgol.engr.ucdavis.edu (R.T. Farouki). robotics 17 and CNC machining 810 were concerned with the minimum-time traversal of a prescribed path by a system with known dynamics and specified bounds on the motive-force capacity of its actuators. The solutions to such problems characteristically incur a bang-bang control strategy, in which the output of at least one system actuator is saturated at each instant throughout the path traversal. These studies typically assume actuators with constant and symmetric force limits (independent of and contouring applications in robotics and CNC machin- ing 11. Since their torque output is directly proportional to the armature current, the constant symmetric torque limits reflect the maximum current capacity of the motor armature windings. Constant torque output is maintained by continuously varying the armature voltage in relation to the back EMF (proportional to the motor speed) or otherwise controlling the armature current supply 10. In addition to the armature current limits, the applied CNC machine subject to both fixed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of the axis drive motors. For a path specified by a polynomial parametric curve, the time-optimal feedrate is determined as a piecewise-analytic function of the curve parameter, with segments that correspond to saturation of the acceleration along one axis under constant or speed- dependent limits. Break points between the feedrate segments may be computed by numerical root-solving methods. For segments that correspond to fixed acceleration bounds, the (squared) optimal feedrate is rational in the curve parameter. For speed-dependent acceleration bounds, the optimal feedrate admits a closed-form expression in terms of a novel transcendental function whose values may be efficiently computed, for use in real-time control, by a special algorithm. The optimal feedrate admits a real-time interpolator algorithm, that can drive the machine directly from the analytic path description. Experimental results from an implementation of the time-optimal feedrate on a 3-axis CNC mill driven by an open-architecture software controller are presented. The algorithm is a significant improvement over that proposed in Timar SD, Farouki RT, Smith TS, Boyadjieff CL. Algorithms for time-optimal control of CNC machines along curved tool paths. Robotics Comput Integrated Manufacturing 2005;21:3753, since the addition of motor voltage constraints precludes the possibility of arbitrarily high speeds along linear or near-linear path segments. r 2006 Elsevier Ltd. All rights reserved. Keywords: 3-Axis machining; Feedrate functions; Acceleration constraints; Time-optimal path traversal; Bang-bang control; Real-time interpolators 1. Introduction address the question of the range of speeds over which the Algorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axis Time-optimal traversal of curved under both constant and speed-dependent Sebastian D. Timar, Department of Mechanical and Aeronautical Engineerin Received 7 July 2005; received in revised Abstract turing 23 (2007) 563579 by Cartesian CNC machines axis acceleration bounds Rida T. Farouki C3 g, University of California, Davis, CA 95616, USA 23 March 2006; accepted 10 April 2006 ARTICLE IN PRESS torque output, resulting in a speed-dependent maximum torque that decreases linearly with increasing motor speed 10. In 3-axis machining, the maximum current capacity of an axis drive motor imposes a constant acceleration limit at lower axis speeds, and the maximum voltage capacity imposes a speed-dependent acceleration limit at higher axis speeds. The transition from current-limited to voltage- limited operation of the motor occurs at the axis transition speed. At speeds below the transition speed, the maximum axis acceleration remains constant. At speeds greater than the transition speed, the maximum axis acceleration decreases linearly with the axis speed, dropping to zero at the axis no-load speed. To guarantee that time-optimal path traversals conform to both actuator current and voltage limits, algorithms must account for the regimes of both constant and speed- dependent acceleration limits on each machine axis. This paper generalizes the results of a previous study 9 employing only constant acceleration bounds (an assump- tion that incurs arbitrarily high speeds if the path contains extended linear segments), and introduces new algorithms to compute realistic time-optimal feedrates for Cartesian CNC machines with axis drive motors subject to both current and voltage limits. The inclusion of speed- dependent acceleration bounds incurs significant, qualita- tive changes to many aspects of the earlier algorithm in 9including the set of feasible feedrate and feed acceleration combinationsv; a; the nature of the velocity limit curve (VLC); the different types of possible switching points; and the form of the feedrate function for extremal phase-plane trajectories. Nevertheless, for Cartesian CNC machines with independently driven axes, it is still possible to obtain an essentially closed-form solution for the time- optimal feedrate, given the ability to compute the roots of certain polynomial equations. We begin by reviewing DC motor operation in Section 2 and the axis acceleration bounds in Section 3. We introduce the problem of minimum-time traversal of curved paths with constant and speed-dependent axis acceleration limits in Section 4, and we derive feedrate expressions for constant and speed-dependent extremal acceleration trajectories. Feed acceleration limits, the VLC, and feedrate break points are then addressed in Sections 57, respectively. Following a discussion of the feedrate computation in Section 8, and the real-time CNC inter- polator algorithm in Section 9, we present details of feedrate computation and machine implementation results for several examples in Section 10. Finally, Section 11 summarizes our results and makes some concluding remarks. 2. DC motor torque limits As background for understanding the nature of the axis S.D. Timar, R.T. Farouki / Robotics and Computer-In564 acceleration bounds appropriate to Cartesian CNC ma- chines, we begin with a brief overview of the fixed-field DC motors that are commonly used to drive small-to-medium milling machines (see 10 for more complete details of their operation). The equations governing the operation of fixed- field motors are T K T I; EK E o; V EIR, i.e., the motor output torque T is proportional to the armature current I, the back EMF E is proportional to the motor angular speed o, and the applied armature voltage V is equal to the sum of the back EMF and the voltage drop across the armature resistance R. The proportionality factors K T and K E , called the torque constant and back EMF constant, are intrinsic physical properties of a given motor. From these expressions, one can easily derive the motor torquespeed relation T T s 1C0 o o 0 C18C19 , (1) where T s K T V=R is the stall torque, and o 0 V=K E is the no-load speed. Hence, the motor torque decreases linearly with increasing motor speed, from T T s at o 0toT 0atoo 0 .See12 for more complete details. At motor start-up and low speeds, the back EMF E is small compared to the applied voltage V, and a current- limiting device is used to constrain the current I to an (approximately) constant maximum value I lim to prevent damage to the armature windings. Hence, the motor torque output remains constant at T lim K T I lim throughout the low-speed range of operation. As the motor speeds up, the applied armature voltage eventually reaches the maximum motor or power supply voltage rating, V lim . This occurs at the transition speed, defined by o t V lim C0I lim R K E . (2) For speeds greater than o t , the armature voltage (rather than the current) is the limiting factor on the motor torque output. At the voltage limit, the torque T decreases linearly with increasing motor speed o, dropping to zero when the no-load speed o 0 is attained. Fig. 1 depicts the motor constraints imposed by the current and voltage limits, I lim and V lim ,intheo; Tplane for both positive and negative motor speeds. The constraints define two parallel strips, whose intersection forms a paralellogram that defines the feasible regime of DC motor operation. All admissible combinations of motor torque and speed, consistent with the given armature current and voltage limits, lie within this paralellogram. The portions of the paralellogram extending beyond the no-load speed in each direction (ooC0o 0 and o4o 0 ) correspond to regenerative braking of the motor, which implies application of an external torque. Since no such tegrated Manufacturing 23 (2007) 563579 torque is available in the context of CNC machine drive motors, the range of feasible torque/speed states is reduced speed-dependent acceleration limits, the axis speed v x always remains in the intervalC0v 0 ;v 0 C138. Within the axis speed range v x 2C0v t ;v t , the mini- mum and maximum axis acceleration limits are both fixed, and hence this is referred to as the constant limits regime for ARTICLE IN PRESS drive abc constant constant constant mixed constant constant mixed mixed constant mixed mixed mixed to indicate the no-load speed as the maximum motor speed, yielding the six-sided parallelogram shown in Fig. 1. The six-sided parallelogram defines three distinct DC motor speed ranges, each with distinct minimum and maximum torque limits, namely: C0T lim o 0 o o 0 C0o t pTp T lim for C0o 0 popC0o t , C0T lim pTp T lim for C0o t popo t , C0T lim pTp T lim o 0 C0o o 0 C0o t for o t popo 0 . 3. Axis acceleration limits In high-speed machining 8,13,14 inertial forces may dominate cutting forces, friction, etc., especially for tool T Fig. 1. Left: the maximum current and voltage limits impose constant and speed-de (shaded) of feasible motor torque/speed values. Right: since the motors that of feasible torque/speed values is truncated to form a six-sided parallelogram. S.D. Timar, R.T. Farouki / Robotics and Computer-In paths of high curvature. Accounting for the axis inertias, the axis speeds and accelerations are proportional to the motor speeds and motor torques, respectively. Consider, say, the x-axis. If it has effective mass M x and is actuated by a drive motor through a ball screw of modulus K x (i.e., the linear axis velocity v x is related to the motor angular speed o by v x o=K x ), the axis acceleration correspond- ing to motor torque T is a x K x T=M x . Noting that the feedrate may be regarded as a vector of magnitude v and direction given by the unit path tangent tt x ; t y ; t z ,we have v x t x v and the motor rotational speed is oK x t x v. Hence, the torque limits derived above are equivalent to the x-axis acceleration limits C0 A x v 0 v x v 0 C0v t pa x pA x for C0v 0 pv x pC0v t , C0 A x pa x pA x for C0v t pv x pv t , C0 A x pa x pA x v 0 C0v x v 0 C0v t for v t pv x pv 0 , 3 where v t is the axis transition speed, v 0 is the axis no-load speed, and we define A x K x T lim =M x . By virtue of the T pendent torque limits, respectively, forming a four-sided parallelogram CNC machine axes will not exceed the no-load motor speed, the region Table 1 The four possible combinations of acceleration-limited regimes for a 3-axis CNC machine (here a; b; c denotes any permutation of the axes x; y; z) Axis tegrated Manufacturing 23 (2007) 563579 565 the x-axis. The axis speed ranges v x 2C0v 0 ;C0v t and v x 2v t ;v 0 , for which one acceleration limit is fixed and the other is speed dependent, are called the mixed limits regimes for the x-axis. In the constant limits regime, the acceleration bounds may be written as a x A x , with a x C61. For the mixed limits regime, the acceleration bounds may be expressed in the form A x g x v 0 C0v x v 0 C0v t and C0g x A x , where g x C01 for v x 2C0v 0 ;C0v t i.e., t x o0, and g x 1 for v x 2v t ;v 0 i.e., t x 40. Similar considerations apply to the y- and z-axis. During a path traversal, each axis operates within one of its acceleration limit regimes independently of the other axis, and each may switch between the acceleration limit regimes in accordance with variations in the tool path geometry and feedrate. Consequently, there are four possible combinations of acceleration-limited regimes among the x-, y-, z-axis (see Table 1). For a planar curve, ARTICLE IN involving motion of only two machine axes, there are three possible combinations: constant/constant, constant/mixed, and mixed/mixed. Each combination of acceleration limits incurs a specific analysis to compute the time-optimal feedrate. The case in which all axes are in the constant regime is covered by our earlier study 9, but cases involving one or more of the axes in the mixed regime have not been previously addressed. 4. Time-optimal feedrates Consider a path described by a degree-n Bezier curve rx X n k0 p k n k C18C19 1C0 x nC0k x k ; x20;1C138 (4) with control points p k x k ; y k ; z k , k0; .; n 15.Ifs denotes arc length measured along the curve, we define the parametric speed by sxjr 0 xj ds dx . The unit tangent and (principal) normal vectors and the curvature of (4) are defined by t r 0 s ; n r 0 C2r 00 jr 0 C2r 00 j C2t; k jr 0 C2r 00 j s 3 (5) and, conversely, with s 0 r 0 C1r 00 =s we may write r 0 st; r 00 s 0 ts 2 kn. (6) Now suppose we traverse the curve with feedrate (speed) specified by the function vx. Since derivatives with respect to time t and the parameter xwhich we denote by dots and primes, respectivelyare related by d dt ds dt dx ds d dx v s d dx , the velocity and acceleration vectors at each point are given by v_rvt; ar _vtkv 2 n. (7) The tangential component _vt of a vanishes if vconstant, while the normal (centripetal) component kv 2 n vanishes if k0. The time derivative of the feedrate (the feed acceleration) is given in terms of x as _vvv 0 =s. We wish to minimize the traversal time along rx, starting and ending at rest, subject to acceleration limits of the form (3) and analogous expressions for the other machine axes. These requirements can be phrased in terms of the following optimization problem: min vx T Z 1 0 s v dx (8) such that S.D. Timar, R.T. Farouki / Robotics and Computer-In566 A i;min pa i xpA i;max for x20;1C138, Z1C0 v t v 0 . Eq. (11) is a first-order, non-linear differential equation with variable coefficients. It may be written exclusively in terms of x as 00 0 C18C19 2 where ix; y; z refers to each of the Cartesian components a x ; a y ; a z of a. As noted in Section 3, the axis acceleration bounds A i;min , A i;max are of the form C0A i ;A i or A i g i v 0 C0v i v 0 C0v t ;C0g i A i . 4.1. Constant acceleration trajectories From the relations (5), (7), ss 0 r 0 C1r 00 , and _vvv 0 =s, we may write a vv 0 s 2 r 0 v 2 s 3 sr 00 C0s 0 r 0 . For a given curve rxxx; yx; zx the x-axis component (say) of the acceleration a is defined by a x q 0 2s 2 x 0 q s 3 sx 00 C0s 0 x 0 , (9) where we write qv 2 , since it is convenient to work with the square of the feedrate (see 9 for further details). During an extremal acceleration phase under constant acceleration limits, one component of the acceleration is equal to plus or minus the corresponding bound, a condition that yields a linear differential equation for q. If x is the extremally accelerating axis, this equation admits a closed-form solution for the (squared) feedrate, namely q s x 0 C16C17 2 C2a x A x x, (10) where the integration constant C is determined by specifying a known point x C3 ; qx C3 on the trajectory: Cx 0 x C3 =sx C3 2 qx C3 C02a x A x xx C3 . Further details of the solution method for (10) may be found in 9. 4.2. Speed-dependent acceleration trajectories Consider the determination of the feedrate v when the x- axis (say) executes an extremal acceleration defined by a speed-dependent acceleration bound, of the form described above. The differential equation governing the feedrate under such circumstances is t x _vkn x v 2 A x Zv 0 t x vC0 g x A x Z 0, (11) where we introduce the constant PRESS tegrated Manufacturing 23 (2007) 563579 vv 0 x x 0 C0 s s v 2 A x Zv 0 svC0 g x A x Z s x 0 0. feedrate consistent with the axis constraints, and the range a min x; vpapa max x; v of possible feed accelerations at each feedrate v less than v lim x. In the case of constant acceleration bounds on all axes, the acceleration constraints at each curve point x describe strips in the v 2 ; a plane, bounded by parallel line pairs. The intersection of these strips defines a parallelogram, whose interior constitutes the set of feasiblev 2 ; avalues, and whose right-most vertex defines v lim x. For each feedrate v less than v lim x, the upper parallelogram boundary defines the maximum feed acceleration a max x; v, and the lower parallelogram boundary defines the minimum feed acceleration a min x; v. We refer the reader to 9 for complete details. In the case of mixed acceleration bounds, either the lower or the upper constraint involves both v and v 2 ,as well as a, and is thus not describable by a linear relation in ARTICLE IN PRESS To obtain a closed-form integration of this equation, we note that vv 0 x 00 x 0 C0 s 0 s C18C19 v 2 1 2 s x 0 C16C17 2 d dx x 0 s v C18C19 2 . Hence, since g 2 x 1, we obtain d dx x 0 s v v 0 C18C19 2 2 g x A x Zv 2 0 x 0 1C0g x x 0 s v v 0 C18C19 . Writing ux 0 =sv=v 0 , this gives u du dx g x A x Zv 2 0 x 0 1C0g x u, which is amenable to separation of variables, giving Z udu 1C0g x u g x A x Zv 2 0 Z x 0 dx. Noting again that g 2 x 1, this can be integrated to obtain 1C0g x uC0ln1C0g x u g x A x Zv 2 0 xc, the integration constant c being determined from a known initial condition. We note that g x ug x x 0 =sv=v 0 satisfies 0pg x up1, since 0pv=v 0 p1, C01px 0 =sp1, and g x has the same sign as x 0 =s. Hence, the argument of the logarithm occurring above is between 0 and 1. Now let ck be the transcendental function that is defined implicitly as the solution of the equation ckC0lnckk. (12) By differentiating, we see that dc dk C0 ck 1C0ck , and hence the function ckis monotone decreasing if its range is confined to 0pckp1. The corresponding domain is 1pkp1. Using the function c, we can write the feedrate explicitly in terms of the curve parameter x as vxg x v 0 sx x 0 x 1C0c g x A x Zv 2 0 xxc C18C19C20C21 . We regard ck as a basic transcendental function, of similar stature to the trigonometric or logarithmic func- tions. To use it in the context of real-time motion control, an efficient means to evaluate this function is required. Re-writing (12) in the form ckexpC0kexpck (13) yields the iteration sequence for ckdefined by c r expC0kexpc rC01 ; r1;2; . (14) with a suitable starting approx
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