PARAMETER IDENTIFICATION OF A FEED DRIVE FOR HIGH SPEED MACHINE TOOLS

Transcription

1 PARAMEER IDENIFICAION OF A FEED DRIVE FOR HIGH SPEED MACHINE OOLS Agustín Casquero, agustincasquero@gail.co Facultad de Ingeniería, Universidad Nacional de La Papa. Calle esq. 9, General Pico (636), La Papa, Argentina Rogelio Hecker, hecker@ing.unlpa.edu.ar Facultad de Ingeniería, Universidad Nacional de La Papa - CONICE. Calle esq. 9, General Pico (636), La Papa, Argentina. Diego Vicente, vicente@ing.unlpa.edu.ar Facultad de Ingeniería, Universidad Nacional de La Papa. Calle esq. 9, General Pico (636), La Papa, Argentina Marcelo Flores, gflores@ing.unlpa.edu.ar Facultad de Ingeniería, Universidad Nacional de La Papa. Calle esq. 9, General Pico (636), La Papa, Argentina Abstract. Appropriate control techniques can be tuned to reduce the vibrations induced in High Speed Machining (HSM) due to the excitation of the vibration odes. herefore, the present work proposes an estiation procedure to obtain the paraeters for the low frequency odel and the frequency of the first resonant ode. A feed drive with ball screw, lineal ball guides, brushless otor and a rotational encoder was used to run the experients. First, the otor was excited with two different input signals: a train of frequency variable steps and a train of steps with odulated aplitude and constant frequency. he easured rotational position was then used to estiate the paraeters for the low frequency odel and a coulob friction value using the least square technique. A set of estiates was perfored with different torque aplitude for both signals, showing the best results for the odulated aplitude signal at the iu torque. Afterwards, the otor was excited with a chirp signal running fro 5 Hz to 28 Hz. he Fast Fourier ransfor fro the easured position shows a resonant frequency at 55 Hz, identified as the first vibration ode of this syste. herefore, the procedure was successful for the estiation of the low frequency odel copleented with the coulob friction and the location of the first resonant ode. his odel can now be used to tune controllers with an adequate closed-loop bandwidth for HSM. Keywords: feed drive, paraeter estiation, vibration odes. INRODUCION Servo-control loops anage the axes of achine tools based on requireents of velocity, position, and acceleration. he success of odeling and control techniques applied to this level sets the basis for production of parts with high quality as well as cycle tie reduction, Hecker et al. (28). oday, High Speed Machining (HSM) is possible due to the advances in cutting tools and spindle speed. Particularly, the axes of a achine for HSM require tracking of high acceleration and high jerk trajectories, which in turn deands high bandwidth closed-loops. However, this solution can excite the vibration odes of the syste, inducing vibrations degrag the real trajectory and causing direct consequences on parts precision. herefore, it is convenient to have a representative odel in order to apply adequate control schees to reduce these negative effects. his odel ust represent the syste with a certain degree of coplexity, which could vary depeng on the particular application. here are several ethods to obtain a odel, inclug paraetric or non paraetric odels. In the case of paraetric odels, soe paraeters can be easured directly, but soe others are difficult to easure or can vary with tie, such as Coulob or viscous friction, therefore ust be identified. In this case, the syste is a linear positioning axis, which is coposed by a brushless otor connected to a ball screw that drives a carriage, guided by lineal ball bearings. First, a rigid odel plus a coulob friction odel are proposed and identified. wo type of signal were used, both at different levels of otor current. he estiated paraeters where analyzed for each set of experients and the optial values were proposed, based on soe priori paraeter knowledge. he first vibration ode can be identified to augent the rigid odel and then to tune an adequate controller, like notch filter, to avoid vibrations in high bandwidth closed-loops. Previous research shows that the first vibration ode of this kind of feed-drive systes can go fro 65 Hz (Sith, 999) to 349 Hz (Varanasi, 999), where a bandwidth of at least Hz is recoended for HSM. his work proposed to excite the syste with a chirp signal and analyze the response with a Fast Fourier transfor to identify the frequency of this resonant ode.

2 2. SYSEM DESCRIPION he syste is a linear positioning axis, as it is shown in Fig.. It is basically coposed by a brushless otor with a rotational encoder, a ball screw, a oving carriage and lineal ball bearings. he brushless otor has a iu speed of 36 rp, a noinal torque of 3.53 N and a iu torque of 2,2 N. his otor is attached to a lead-screw drive, with a /rev lead, which is fixed to the base by eans of two bearings, a rigid one in one side and a floating one in the other side. he nut of the screw-drive is preloaded, in order to iniize backlash, and it is connected to the oving carriage by eans of the nut holg. he oving carriage is guided by two linear bearings, which supports the top and side loads, and allows the carriage to ove in a straight line. otor coupling carriage guideways rigid bearing ball-nut screw Figure : Syste outline he otor has a rotational encoder that provides, in cobination with the ball screw, 2.5 µ of resolution in the linear position. he echanical syste is connected to a real tie control syste that is ounted in an industrial PC, and serves as the connection between the user and the syste. he control syste consists of a PCI expansion board with an onboard processor that runs a real tie operating syste. he control syste is connected to the hardware using an analog output to coand the otor torque (by eans of its power aplifier) and a quadrature encoder input in order to read the angular position of the otor. he analog output has a 2 bits resolution and the quadrature decoder has a 6 bits counter, expandable to 32 bits by software. 3. LOW FREQUENCY MODEL he otor controller tries to set, in a closed-loop configuration, a otor current proportional to the input signal. In general, the bandwidth of this current closed-loop is uch higher than the frequency range of interest of the echanical syste. herefore, this dynaics is discarded and the input to the syste is directly the otor current, or in this case, the otor torque. Now, a low frequency odel that includes the viscous friction is considered. First, the rotational behavior of the syste can be represented by J && θ = B & θ () c θ r where J c is the cobined inertia of the otor, the screw and the coupling, θ is the angular coorate, B θ is the rotational viscous coefficient, is the otor torque, and r is the reaction torque applied by the carriage. Siilarly, the linear behavior of the syste can be represented by M && x = F B x& F (2) r x p where M is the carriage ass, B x is the linear viscous coefficient, F r is the reaction to the resistant torque applied by the carriage to the screw, F p is a disturbing force, and x is the linear coorate. he rotational and the linear equations are coupled by the screw-nut interface considering that r = Frr, where r is the screw lead, and also tacking into account that r = x θ. Using these relations, Eq. () and Eq. (2) can be cobined to obtain && x 2 x& 2 = ( J + r M) + ( Bθ + r B ) + rf r r c x p (3) he transfer function fro the otor torque to the linear position is found by applying the Laplace transfor to Eq. (3) and considering that no disturbing forces are present (F p =) X() s r () = ( ) s s sj + B (4)

3 where J is the syste cobined inertia and B represents the syste cobined viscous coefficient, which can be written in ters of the previously defined paraeters as follows 2 2 J Jc r M and B Bθ r Bx = + = + (5) he continuous odel represented by Eq. (4) is transfored to discrete variables using a Zero Order Hold to obtain X( z) k z+ k2 ( ) = ( )( ) z z z p (6) where B r r J k = ( p+ ln p), k2 = ( + p pln p),and p = e (7) B ln p B ln p and where is the sapling tie. Now, the odel can be used to know the order of the syste in a paraetric identification process. 3.. Paraeters estiation ethod In order to obtain the paraeters of the low frequency odel, a siilar procedure as the one proposed by Erkorkaz and Altintas (2) is followed. his approach uses the Least Squares Estiation (LES) ethod, but taking into account the Coulob friction of the syste. herefore, the results fro the estiation process are the estiated values for the cobined inertia, cobined viscous friction of the syste, and dynaic Coulob friction. Fro Eq. (6), adg the Coulob friction ( f ), the following equation in differences can be forulated ( f ) ( f ) x ( + p) x + p x = k + k (8) ( k) ( k ) ( k 2) ( k ) ( k ) 2 ( k 2) ( k 2) and writing the equation in ters of x(k) x = ( + p) x p x + k + k k k (9) ( k) ( k ) ( k 2) ( k ) 2 ( k 2) f ( k ) 2 f ( k 2) Now, this equation can be siplified defining the following variable Δ x( k) = x( k) x( k ) () Additionally, the friction ter is Eq (9) is replaced by a siplified odel of the Coulob friction, described by the following equation + = VP ( wk ( )) + VN ( wk ( )) () f where + and represent the positive and negative dynaic torque, respectively, whereas VP and VN are two functions that depend on the velocity as follows if ω ( k) > ωin if ω( k) < ω VP ( ω( k) ) = VN ( ω( k) ) = if ω( k) ωin if ω( k) ωin in (2) where the paraeter ω in is a iniu non-cero velocity value, used to iniize the effects of error and noise in the velocity calculation, specially around zero velocity. Using the friction odel, Eq. (9) can be rewritten to obtain + Δ x( k) = p Δ x( k ) + k ( k ) + k ( k 2) k VP ( w( k ) ) 2 + k VN ( wk ( ) ) k VP ( wk ( 2) ) k VN ( wk ( 2) ) 2 2 (3) Using Eq. (3) with N saples, the atrixes for the LSE are obtained, and these can be written as

4 Y = Φθ (4) where Y, Φ y θ are defined as p Δx Δx L Δx k L Y = θ= Φ= (2) (3) ( N ) (2) (3) ( N ) Δx(3) k2 () (2) ( N 2) x L Δ (4) + k VP(2) VP(3) VP( N ) L L k VN (2) VN (3) LVN ( N ) Δx( N ) + k 2 VP() VP(2) LVP( N 2) k 2 VN () VN (2) LVN ( N 2) (5) And the vector that iniizes the estiation error is found by (Franklin et al, 998) ( ) θ= Φ Φ Φ Y (6) Once that θ (p k k 2 ) is calculated, the transfer function in Eq. (6) can be obtained by replacing the identified coefficients. With this equation the continuous representation represented by Eq. (4) can be found, by using the Matlab function D2C, fro where the values of J and B can be calculated, where the screw lead (r) is a known paraeter. he values of the dynaic Coulob friction (both positive and negative) can also be obtained; they can be calculated by the following equations, based on the identified data k k k k = = k+ k2 k+ k2 (7) 3.2. Experients and results A set of experients were conducted to get the needed data for the identification process. wo types of exciting signals were used: One consisting of a series of steps of varying sign and height (as proposed by Erkorkaz and Altintas, 2); and the other consisting on a series of steps of varying sing and frequency, but with constant height. Both types of signals can be seen in Fig. 2. Motor orque [N] ie [s] Motor orque [N] ie [s] Figure 2: Varying Height and Varying Frequency Exciting Signal hese signals were applied to the syste, with different values of iu current, i.e.: A,.5 A, 2 A, 2.5 A, 3 A, 3.5 A, 4 A and 4.5 A. For each signal the continuous paraeters of the syste were estiated (cobined inertia J and cobined viscous friction B). he coparison for the paraeters obtained with each exciting signal is shown in Fig. 3.

5 J [kg 2 ].6 x Variable Freq. Variable Ap. Calculated B [kg 2 / s] Motor Current [A] x -3 5 Variable Freq. Variable Ap Motor Current [A] Figure 3: J and B odel s paraeters obtained with the exciting signasl, varying the iu otor torque (or current) It can be seen in Fig. 3 that, for both signals, the paraeters identified (J and B) see to stabilize at a given value as the iu torque is increased. It can be also seen that for currents over 2.5 A (for the frequency varying signal) and over 3.5 A (for height varying signal) the identified values tend to increase, but this is due to a saturation of the otor speed, which introduce a non-linearity into the syste. Fro the graphic of the inertia, it can be seen that the signal of steps of varying height is the one that best approaches the theoretical inertia for this syste (doted line), so it can be concluded that this type of signal is the best aong the ones used. Finally, the values of the estiated paraeters were chosen fro the experient which has the best approxiation to the theoretical value of the cobined inertia, this is for the varying height signal with a iu current of 3.5 A. he identified values are resued here as ˆ 4 2 ˆ 4 2 ˆ+ ˆ J = 8,885 kg B = 6,6 kg =,65 N =,62 N (8). FIRS VIBRAION MODE IDENIFICAION he syste was excited with a high frequency signal to identify the first vibration ode. he signals were applied during a short period of tie, while the position and velocity were easured. After that, a Fourier ransfor analysis was applied to the position and velocity data, to deterine the frequency of the vibration ode. As an excitation signal, a chirp type signal was used, coanded directly to the otor current aplifier. his kind of chirp signal was suggested by various authors, such as Ljung (999) and Franklin et al. (998), and can be generated using the following equation t a t at ( ωe ω ) s ref = sin ( ωi ) ωi = ωs + = < < ( ) t t t ( a) t < t < t at K t K a t t a t t (9) where ref is the reference otor torque function, is the iu otor torque, ω i is the instant frequency, which stars at ω e and ends at ω e, K is the odulation function that is defined by the total signal tie (t ) and the percent of the total tie that the signal is odulated (a). he instant frequency (ω i ) of the signal ( ref ) varies linearly with tie, and the aplitude is odulated at the beginning and the end by the saturation function K. his kind of signal generates a nice window of excitation frequencies in the range of interest, as Fig. 4 shows.

6 4.. Experients and results For the identification of the first resonant ode of the syste, chirp type signals were used, as described before. o do this, it is necessary to define the chirp paraeters like: Starting and eng frequency, iu torque, total tie, and odulation. In order to identify the first vibration ode, the window of excitation frequency should be in the range of this expected resonant ode. o have soe rough idea about the frequency of the vibration ode a detailed odel of the syste was siulated. his odel, developed by Vicente et al. (27), takes into account the first four axial vibration odes and the first four angular vibration odes of the syste, and the coupling between the. Using this odel it was found that the frequency of the first resonant ode was about 67 Hz, for the carriage positioned at the center of the axis. With the approxiated inforation of the vibration frequency, the starting and eng frequency were chosen to be 5 Hz and 28 Hz respectively. he iu torque of the signal was fixed at 3.55 N (5 A). he total length of the experient was liited to 2.4 s, with a sapling tie of.3 s, and the initial and final odulation was about 2 % of the experient tie. Once the position and velocity data were obtained using the defined exciting function, the FF was applied to these signals, which are shown in Fig. 4. he presence of a peak at 55 Hz can be seen in the figure, both in position and in velocity. he FF of the exciting signal is also plotted..2 FF of the Exciting Signal Frequency [Hz] FF of the Angular Positon Frequency [Hz] FF of the Angular Speed Frequency [Hz] Figure 4: FF of the excitation signal, the angular position and the angular speed, respectively. he experient was repeated varying soe paraeters, like starting and eng frequency and sapling tie, and in all cases a peak at 55 Hz was observed. It should be note that all the experients were ade with the carriage in the center of the positioning syste, so the resonant frequency identified will be for that position. 5. CONCLUSIONS A ethod to identify a low frequency odel, inclug the Coulob friction effect, of a linear feed drive syste was presented. his ethod was tested experientally using different excitation signal, where the best set of identified paraeters was proposed. Alternatively, the frequency of the first vibration ode was obtained to augent the rigid odel. he first resonant ode was clearly observed in the FF of the velocity and the position signals for a convenient excitation signal. he augented odel of the syste in this work can be used to select and tune control techniques for feed-drives requiring high bandwidth closed-loops.

7 6. REFERENCES Erkorkaz, K and Altintas, Y., 2: High speed CNC syste design. Part II: odeling and identification of feed drives, International Journal of Machine ools & Manufacture, 4, Franklin, G. F., Powell, J. D. and Workan, M., 998: Digital Control of Dynaic Systes, 3 rd edition, Addison Wesley. Ljung, Lennart (999): Syste Identification heory for the User, 2 nd edition, Prentice Hall. Vicente, Diego A., Hecker, Rogelio L. y Flores, Gustavo M, 27: Dynaic Modeling of Lead Screw Drives using Ritz Series, XII Reunión de rabajo en Procesaiento de la Inforación y Control, Río Gallegos Santa Cruz - Argentina. Hecker, Rogelio L., Flores, Gustavo M., Xie, Quin and Haran, Roberto, 28: A Review of Machine-ools Servocontrol Level, Latin Aerican Applied Research An International Journal, Vol 38, No., 85-94,. Sith, D. A., 999, Wide bandwidth control of high-speed illing achine of feed drives, PhD dissertation, University of Florida. Varanasi, K. K., 22: On the Design of a Precision Machine for Closed-Loop Perforance, MS thesis, Massachusetts Institute of echnology, Cabridge, Massachusetts. 7. RESPONSIBILIY NOICE he authors are the only responsible for the printed aterial included in this paper.