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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 current—is 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: sdtimar@ucdavis.edu (S.D. Timar), farouki@algol.engr.ucdavis.edu (R.T. Farouki). robotics [1–7] and CNC machining [8–10] 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;21:37–53], 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) 563–579 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 [9]—including the set of feasible feedrate and feed acceleration combinationsev; aT; 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 5–7, 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; E?K E o; V ?EtIR, 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 torque–speed 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 ?0ato?o 0 .See[12] 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 ,intheeo; TTplane 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 o4to 0 ) correspond to regenerative braking of the motor, which implies application of an external torque. Since no such tegrated Manufacturing 23 (2007) 563–579 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 interval?C0v 0 ;tv 0 C138. Within the axis speed range v x 2eC0v t ;tv t 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 to o 0 C0o t pTpt T lim for C0o 0 popC0o t , C0T lim pTpt T lim for C0o t popto t , C0T lim pTpt T lim o 0 C0o o 0 C0o t for to t popto 0 . 3. Axis acceleration limits In high-speed machining [8,13,14] inertial forces may dominate cutting forces, friction, etc., especially for tool ω 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 t?et x ; t y ; t z T,we have v x ?t x v and the motor rotational speed is o?K x t x v. Hence, the torque limits derived above are equivalent to the x-axis acceleration limits C0 A x v 0 tv x v 0 C0v t pa x ptA x for C0v 0 pv x pC0v t , C0 A x pa x ptA x for C0v t pv x ptv t , C0 A x pa x ptA x v 0 C0v x v 0 C0v t for tv t pv x ptv 0 , e3T 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 ω 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) 563–579 565 the x-axis. The axis speed ranges v x 2eC0v 0 ;C0v t T and v x 2etv t ;tv 0 T, 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 2eC0v 0 ;C0v t T—i.e., t x o0, and g x ?t1 for v x 2etv t ;tv 0 T—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 Be′zier curve rexT? X n k?0 p k n k C18C19 e1C0 xT nC0k x k ; x2?0;1C138 (4) with control points p k ?ex k ; y k ; z k T, k?0; ...; n [15].Ifs denotes arc length measured along the curve, we define the parametric speed by sexT?jr 0 exTj? 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 ?er 0 C1r 00 T=s we may write r 0 ?st; r 00 ?s 0 tts 2 kn. (6) Now suppose we traverse the curve with feedrate (speed) specified by the function vexT. Since derivatives with respect to time t and the parameter x—which we denote by dots and primes, respectively—are 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?_r?vt; a?€r? _vttkv 2 n. (7) The tangential component _vt of a vanishes if v?constant, while the normal (centripetal) component kv 2 n vanishes if k?0. The time derivative of the feedrate (the feed acceleration) is given in terms of x as _v?vv 0 =s. We wish to minimize the traversal time along rexT, 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 vexT 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 exTpA i;max for x2?0;1C138, Z?1C0 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 i?x; 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 ;tA 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 _v?vv 0 =s, we may write a? vv 0 s 2 r 0 t v 2 s 3 esr 00 C0s 0 r 0 T. For a given curve rexT?exexT; yexT; zexTT the x-axis component (say) of the acceleration a is defined by a x ? q 0 2s 2 x 0 t q s 3 esx 00 C0s 0 x 0 T, (9) where we write q?v 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 eCt2a x A x xT, (10) where the integration constant C is determined by specifying a known point ex C3 ; qex C3 TT on the trajectory: C?ex 0 ex C3 T=sex C3 TT 2 qex C3 TC02a x A x xex C3 T. 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 _vtkn x v 2 t A x Zv 0 t x vC0 g x A x Z ?0, (11) where we introduce the constant PRESS tegrated Manufacturing 23 (2007) 563–579 vv 0 t x x 0 C0 s s v 2 t 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 ex; vTpapa max ex; vT of possible feed accelerations at each feedrate v less than v lim exT. In the case of constant acceleration bounds on all axes, the acceleration constraints at each curve point x describe strips in the ev 2 ; aT plane, bounded by parallel line pairs. The intersection of these strips defines a parallelogram, whose interior constitutes the set of feasibleev 2 ; aTvalues, and whose right-most vertex defines v lim exT. For each feedrate v less than v lim exT, the upper parallelogram boundary defines the maximum feed acceleration a max ex; vT, and the lower parallelogram boundary defines the minimum feed acceleration a min ex; vT. 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 t 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 u?ex 0 =sTev=v 0 T, this gives u du dx ? g x A x Zv 2 0 x 0 e1C0g x uT, 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 e1C0g x uTC0lne1C0g x uT? g x A x Zv 2 0 extcT, the integration constant c being determined from a known initial condition. We note that g x u?g x ex 0 =sTev=v 0 Tsatisfies 0pg x up1, since 0pv=v 0 p1, C01px 0 =spt1, 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 cekT be the transcendental function that is defined implicitly as the solution of the equation cekTC0lncekT?k. (12) By differentiating, we see that dc dk ?C0 cekT 1C0cekT , and hence the function cekTis monotone decreasing if its range is confined to 0pcekTp1. The corresponding domain is 1pkp1. Using the function c, we can write the feedrate explicitly in terms of the curve parameter x as vexT?g x v 0 sexT x 0 exT 1C0c g x A x Zv 2 0 exexTtcT C18C19C20C21 . We regard cekT 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 cekT?expeC0kTexpecekTT (13) yields the iteration sequence for cekTdefined by c r ?expeC0kTexpec rC01 T; r?1;2; ... (14) with a suitable starting approx