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Robotics and Computer-Integrated Manufacturing 23 (2007) 563579 Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds Sebastian D. Timar, Rida T. Farouki? Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA Received 7 July 2005; received in revised form 23 March 2006; accepted 10 April 2006 Abstract Algorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axis CNC machine subject to both fi xed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of the axis drive motors. For a path specifi ed 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 fi xed 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 effi ciently 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 signifi cant 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 Previous studies of time-optimal control in the fi elds of robotics 17 and CNC machining 810 were concerned with the minimum-time traversal of a prescribed path by a system with known dynamics and specifi ed bounds on the motive-force capacity of its actuators. The solutions to suchproblemscharacteristicallyincurabang-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 the speed and direction of actuation), and generally do not address the question of the range of speeds over which the actuators can exert their maximum force. Fixed-fi eld DC motors are common to most positioning 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 refl ect 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 armature voltage may be subject to limits arising from the motor characteristics or armature power supply. Such voltage limits confi ne the ability of the motor to produce the maximum output torque to a fi nite range of speeds. Beyond this range, maximum applied armature voltage not armature currentis the factor limiting the motor ARTICLE IN PRESS 0736-5845/$-see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2006.07.002 ?Corresponding author. E-mail addresses: (S.D. Timar), (R.T. Farouki). 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 currentandvoltagelimits.Theinclusionofspeed- dependent acceleration bounds incurs signifi cant, qualita- tive changes to many aspects of the earlier algorithm in 9includingthesetoffeasiblefeedrateandfeed acceleration combinations v;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 acceleration bounds appropriate to Cartesian CNC ma- chines, we begin with a brief overview of the fi xed-fi eld 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 fi xed- fi eld motors are T KTI;E KEo;V E IR, 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 KTand KE, 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 Ts1 ? o o0 ? ,(1) where Ts KTV=R is the stall torque, and o0 V=KEis the no-load speed. Hence, the motor torque decreases linearly with increasing motor speed, from T Tsat o 0 to T 0 at o o0. 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 Ilimto prevent damage to the armature windings. Hence, the motor torque output remains constant at Tlim KTIlimthroughout 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, Vlim. This occurs at the transition speed, defi ned by ot Vlim? IlimR KE .(2) For speeds greater than ot, 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 o0is attained. Fig. 1 depicts the motor constraints imposed by the current and voltage limits, Ilimand Vlim, in the o;T plane forbothpositiveandnegativemotorspeeds.The constraints defi ne two parallel strips, whose intersection forms a paralellogram that defi nes the feasible regime of DC motor operation. All admissible combinations of motortorqueandspeed,consistentwiththegiven armaturecurrentandvoltagelimits,liewithinthis paralellogram. The portions of the paralellogram extending beyond the no-load speed in each direction (oo ? o0and o4 o0) correspond to regenerative braking of the motor, which implies application of an external torque. Since no such torque is available in the context of CNC machine drive motors, the range of feasible torque/speed states is reduced ARTICLE IN PRESS S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579564 to indicate the no-load speed as the maximum motor speed, yielding the six-sided parallelogram shown in Fig. 1. The six-sided parallelogram defi nes three distinct DC motor speed ranges, each with distinct minimum and maximum torque limits, namely: ?Tlim o0 o o0? ot pTp Tlimfor ? o0pop ? ot, ?TlimpTp Tlimfor ? otpop ot, ?TlimpTp Tlim o0? o o0? ot for otpop o0. 3. Axis acceleration limits In high-speed machining 8,13,14 inertial forces may dominate cutting forces, friction, etc., especially for tool 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 Mxand is actuated by a drive motor through a ball screw of modulus Kx(i.e., the linear axis velocity vxis related to the motor angular speed o by vx o=Kx), the axis acceleration correspond- ing to motor torque T is ax KxT=Mx. Noting that the feedrate may be regarded as a vector of magnitude v and direction given by the unit path tangent t tx;ty;tz, we have vx txv and the motor rotational speed is o Kxtxv. Hence, the torque limits derived above are equivalent to the x-axis acceleration limits ? Ax v0 vx v0? vt paxp Axfor ? v0pvxp ? vt, ? Axpaxp Axfor ? vtpvxp vt, ? Axpaxp Ax v0? vx v0? vt for vtpvxp v0,3 where vtis the axis transition speed, v0is the axis no-load speed, and we defi ne Ax KxTlim=Mx. By virtue of the speed-dependent acceleration limits, the axis speed vx always remains in the interval ?v0;v0?. Within the axis speed range vx2 ?vt;vt, the mini- mum and maximum axis acceleration limits are both fi xed, and hence this is referred to as the constant limits regime for the x-axis. The axis speed ranges vx2 ?v0;?vt and vx2 vt;v0 , for which one acceleration limit is fi xed 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 axAx, with ax ?1. For the mixed limits regime, the acceleration bounds may be expressed in the form Ax gxv0? vx v0? vt and? gxAx, where gx ?1 for vx2 ?v0;?vti.e., txo0, and gx 1 forvx2 vt;v0i.e.,tx40.Similarconsiderations 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 possiblecombinationsofacceleration-limitedregimes among the x-, y-, z-axis (see Table 1). For a planar curve, ARTICLE IN PRESS T T Fig. 1. Left: the maximum current and voltage limits impose constant and speed-dependent torque limits, respectively, forming a four-sided parallelogram (shaded) of feasible motor torque/speed values. Right: since the motors that drive CNC machine axes will not exceed the no-load motor speed, the region of feasible torque/speed values is truncated to form a six-sided parallelogram. 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 abc constantconstantconstant mixedconstantconstant mixedmixedconstant mixedmixedmixed S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579565 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 specifi c 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 rx X n k0 pk n k ? 1 ? xn?kxk;x 2 0;1?(4) with control points pk xk;yk;zk, k 0;.;n 15. If s denotes arc length measured along the curve, we defi ne the parametric speed by sx jr0xj ds dx . The unit tangent and (principal) normal vectors and the curvature of (4) are defi ned by t r0 s ;n r0? r00 jr0? r00j ? t;k jr0? r00j s3 (5) and, conversely, with s0 r0? r00=s we may write r0 st;r00 s0t s2kn.(6) Now suppose we traverse the curve with feedrate (speed) specifi ed 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 _ r vt;a r _ vt kv2n.(7) The tangential component _ vt of a vanishes if v constant, while the normal (centripetal) component kv2n vanishes if k 0. The time derivative of the feedrate (the feed acceleration) is given in terms of x as _ v vv0=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 Ai;minpaixpAi;maxfor x 2 0;1?, where i x;y;z refers to each of the Cartesian components ax;ay;azof a. As noted in Section 3, the axis acceleration bounds Ai;min, Ai;maxare of the form ?Ai;AiorAi giv0? vi v0? vt ;?giAi. 4.1. Constant acceleration trajectories From the relations (5), (7), ss0 r0? r00, and _ v vv0=s, we may write a vv0 s2 r0 v2 s3 sr00? s0r0. Foragivencurverx xx;yx;zxthex-axis component (say) of the acceleration a is defi ned by ax q0 2s2 x0 q s3 sx00? s0 x0,(9) where we write q v2, 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 x0 ? ?2 C 2axAxx,(10) wheretheintegrationconstantCisdeterminedby specifying a known point x?;qx? on the trajectory: C x0x?=sx?2qx? ? 2axAxxx?. 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 defi ned by a speed-dependent acceleration bound, of the form described above. The differential equation governing the feedrate under such circumstances is tx_ v knxv2 Ax Zv0 txv ? gxAx Z 0,(11) where we introduce the constant Z 1 ? vt v0 . Eq. (11) is a fi rst-order, non-linear differential equation with variable coeffi cients. It may be written exclusively in terms of x as vv0 x00 x0 ? s0 s ? v2 Ax Zv0 sv ? gxAx Z s2 x0 0. ARTICLE IN PRESS S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579566 To obtain a closed-form integration of this equation, we note that vv0 x00 x0 ? s0 s ? v2 1 2 s x0 ? ?2 d dx x0 s v ?2 . Hence, since g2 x 1, we obtain d dx x0 s v v0 ?2 2 gxAx Zv2 0 x01 ? gx x0 s v v0 ? . Writing u x0=sv=v0, this gives u du dx gxAx Zv2 0 x01 ? gxu, which is amenable to separation of variables, giving Z udu 1 ? gxu gxAx Zv2 0 Z x0dx. Noting again that g2 x 1, this can be integrated to obtain 1 ? gxu ? ln1 ? gxu gxAx Zv2 0 x c, the integration constant c being determined from a known initial condition. We note that gxu gxx0=sv=v0 satisfi es 0pgxup1, since 0pv=v0p1, ?1px0=sp 1, and gxhas the same sign as x0=s. Hence, the argument of the logarithm occurring above is between 0 and 1. Now let ck be the transcendental function that is defi ned implicitly as the solution of the equation ck ? lnck k.(12) By differentiating, we see that dc dk ? ck 1 ? ck , and hence the function ck is monotone decreasing if its range is confi ned to 0pckp1. The c