NON-LINEAR ANALYSIS OF THE SMOOTH PURSUIT EYE MOVEMENT SYSTEM OF THE MACAQUE, by Lei and S.^Stone | Submitted as partial requirement for the degree of Master's of Science at the University of California, Berkeley, California. Acknowledgements . The author would like to thank professor Shankar Sastry for his Insight and patience both of which were essential to the completion of this project. He is also indebted to professor Steve Llsberger, Lauren Westbrook, and Ed Morris whose experimental results were graciously made available to him and used both in the development and testing of the model presented here. THE FAR SIDE/GARY LARSON ABSTRACT Using the time domain data of Llsberger and others, a nonlinear model of the Smooth Pursuit system of the macaque was constructed. The sinusoidal response of the model was determined using a modified form of the describing function method. The results were compared to actual frequency domain data in one monkey. The discrepancy between the model's response and the actual data is Interpreted as being the result of a prediction process which is assisting the Smooth Pursuit during predictable tracking. In this way, a quantitative estimate of the role of the predictor is made at least for frequencies between .7 and 2 Hz and for amplitudes between .3 and 2 deg. I. Introduction and Background. In the macaque and all foveate animals, the -function o-f the oculomotor system is to put the image o-f an object o-f interest on the -fovea, the region o-f the retina where the highest resolution of visual processing is passible, and maintain that image on the fovea by compensating -for any movements o-f the objects or the head with the appropriate eye movements. This task is achieved by several subsystems each producing a speci-fic category o-f eye movement: the saccadi_c_s^stem which produces rapid reor i entat i on o-f gaze is used to scan the visual environment and orrect -for di -f -f erences between the object (desired) position and eye position, the vesti.bul^o-ocuLa.C reflex produces smooth eye movements equal and opposite to that o-f the head thereby stabilizing images during head turns, and the smgpth_gursuit (SP) system produces smooth eye movements appropriate -for stabilizing the image o-f a moving target on the -fovea. To characterize what aspects of the visual experience are used by the SP system, Rashbass Cl] per-formed experiments in which position and velocity error were in con-flict and determined that target motion not target position is the relevant input to the system. It was concluded that the SP system can be modelled as simple velocity controller. (Recent evidence has shown that Rashbass' results are not universally applicable and that in certain situations position errors can also produce smooth eye movements [23 C3D but these e-f-fects are small and not relevant in the paradigms used below so will be ignored here). On purely theoretical and logical grounds an eye velocity positive -feedback loop was postulated C4D to account -for the accurate steady-state tracking when the -feedback velocity error signal is small. This hypothesis was con-firmed by two experimental results: 1) in both monkey and man, retinal error velocity is linearly correlated with eye acceleration during steady-state sinusoidal tracking C5D, and 2) in open loop conditions, constant eye velocity is maintained in the absence o-f visual input C63. The SP system there-fore can be considered as a -feedback system driven by image motion velocity on the retina and whose output is eye acceleration used to modulate ongoing eye velocity. Despite the 1OO msec delay in the system response C73, this model predicts relatively accurate steady-state tracking of approximately constant velocity targets: the SP system will accelerate the eye up to the appropriate velocity (the saccadic system will then correct any remaining position error) and hold it there by modulating eye velocity only when image slippage occurs. It does not however account -for the accurate tracking o-f sinusoidal motion where target velocity is constantly changing (predictably) and the delay should but does not cause significant phase lag between target and eye velocity C53. A predictor (target trajectory estimator) is there-fore postulated to account for the data. Many models described how the SP system could predict C81 , but as yet no study has attempted to quantify the actual prediction process going on during steady-state tracking o-f sinusoids. The goal o-f the present study is to use in-formation about the open loop time domain response of the SP system to contruct a model that will be used to estimate the theoretical response to sinusoidal inputs without the benefit of prediction. The model's response can then be compared with the actual open loop sinusoidal response of the system. In this way a quantitative measure of the predictor's contribution to the frequency response can made (at least within the frequency and amplitude range examined). It is hoped that the use of modelling to determine the overall response of the predictor will, in combination with future experimental results, help elucidate how the SP system overcomes its delay and dynamics and is able to track predictable targets accurately. I I. The Model . The construction of the model was based on the results o-f Lisberger and Westbrook on the initiation o-f Smooth Pursuit eye movements to steps in target velocity C63. They found that the SP system responds to steps in target velocity by producing an eye acceleration with a latency of about 1OO msec. whether the eye was initially at rest < i ni ti ati on) or v*s tracking a constant velocity target (maintenance). They concluded 1) that the 1OO msec. delay in the tracking response represents a true input time delay rather than a "start-up" delay and 2) that the initial 1<~>0 msec after the onset of eye movement can be considered open loop because visual feedback is also delayed by 1OO msec. . Schematic illustrating the open loop interval ? B. True open loop response to a velocity step (stimulus sho with the closed loop response superimposed. They first studied the effect of initial target position on the response to velocity steps and found that the initial 10O msec, could be subdivided into an "early" (~O-3O msec., largely independent of initial position) and a "late" (~3O-1OO msec., strongly spatially dependent) response. They therefore divided the 1OO msec. open loop interval into 2O msec. bins and the 0-2O msec. bin was taken as a measure of the early response while the 6O-8O msec. bin was assumed to be characteristic of the late response. The late response is strongest -for targets moving near and towards the fovea (fig. 2A) and proportional to the velocity step size (-fig. 2B) up to around 45 deg/sec. I- 400 200 100 A. M ItO 1M wtootty Ute/**c) Fig. 2 f) . Spatial weighting of the B. Magnitude of the late response size. late response (60- SO msec bin); as a function of velocity step The response peaks for a position between 0-3 deg. off the fovea and is about zero for positions >1O deg if the target is moving away from the fovea and for positions >2O deg if approaching the fovea. This spatial weighting of the late response (the output of element A) is represented in the model by a non-linear function with hysteresis (fig. 3). Fig. 3 Non-linear element f) illustrating the hysteretic nature of the spatial weighting of the late response. B is the peak angle parameter C*0-3 deg). The model velocity gain (element B) is constructed from the information in -figure 2B and -from the results o-f Morris and Lisberger C3DC93 that show that the low velocity (less than ~O-5 deg/sec) slope is much higher than the asymptotic slope (-fig. 4A) . (The low velocity slope m varied between about 26.5 sec-1 during initiation with dim background illumination and 33.33 sec-1 during maintenence in the dark). Figure 4B illustrates the non-linear idealized gain element used in the model -for the late or velocity pathway. t %* 100 2.10 MS - (T) 8 Fig. 4 ft. Lou velocity responses; B. Non-linear element B nith ION velocity slope parameter m C*26-33 deg/sec/sec> and velocity elboN parameter fr C"mrS deg). The early response is only weakly spatially weighted (not shown) but is strongly related to the acceleration o-f the target at the onset o-f the velocity "steps" (actually velocity traperoids). It is approximately proportional to the acceleration used to achieve final velocity up to around 125 deg/sec2 (-fig. 6A) and is weighted by the -final velocity up to 15 deg/sec (-fig. 5A) . The model idealizations o-f the gain and velocity weighting o-f the early or acceleration pathway are shown in -figures 6B and 5B respectively. l H 55- i.,tk '* >00 IfcO Ta-**r A cede <.*<- 5 ft. Velocity Weighting of the early response (0-20 msec bin); B. Non-linear element C. Fig. 6 ft. Early response (JO msec after onset) as a function of the acceleration used to achieve 5 deg/sec velocity; B. Non-linear element D. In -figure 7, the O deg/sec curve represents the pursuit response to a target which is already moving at constant velocity when pursuit is initiated (pure velocity step) while the in-f deg/sec curve represents the response to a target which is initially stationary and "instantaneously" accelerates up to speed at time zero (velocity step plus acceleration impulse at time zero). Assuming dynamic linearity and independence (see below), the di-f-ference between the two curves can there-fore be considered as the acceleration impulse response (fig. 7B) and the O deg/sec curve can be considered as the velocity step response (fig. 7A) . The dynamics of the late or velocity pathway was determined to have the transfer function l/(.O4s + 1) and that of the early or acceleration the transfer function l/(.Ols + 1)". (The responses of the idealized model transfer functions are graphed along with the data in fig. 7). 200 100 lOO HC to Fig. 7 ft. Velocity step rsponse with < inf deg/sec/sec> and Mithout (O deg/sec/sec) acceleration impulse? B. Acceleration impulse response. The complete model is given in -figure 8 with parameters 8, m,2f that vary within the constraints given above. It is constructed such that the input, target motion as a -function o-f time (where the position and -first two derivatives o-f motion are the only relevant aspects o-f the stimuli), produces an eye acceleration through two parallel pathways. The model assumes that the time delay is at the input end, that the static non-linearities discussed above -follow with the weighting parameters (value O-l) being multiplicative at the gain output, and that the dynamics o-f both pathways are linear (i.e. independent o-f amplitude), independent (i.e. that the dynamics are independent o-f the value o-f the weighting parameter), and that they act as low pass filters at the output end such that higher harmonics are removed. (The -first two assumptions are critical to the interpretation o-f the data in -figure 7). I < (- lu T :iu| I c CP - 9 - III. Analysis. Open loop steady-state sinusoidal data [10] show that despite the SP system's non-linearities that the sinusoidal response for amplitudes less than 2 deg and -frequencies between .7 and 2 Hz. is quasi linear i.e. that the system phase shifts and scales the sinusoidal input but does not produce much harmonic distortion. In the case o-f a linear system the gain and phase are -functions o-f -frequency only whereas -for a quasi linear system they may be functions o-f amplitude as well. This -fact suggested that the method o-f describing -functions (DF's) might be appropriate -for the sinusoidal analysis o-f the SP model. This method merely assumes that the steady-state periodic response o-f any non-linear element written as a Fourier series can be truncated after the first term without significant loss of signal. The element therefore can be characterized by the amplitude and phase of the first harmonic of the output as a function of input amplitude and frequency. The DF itself is a complex gain calculated as the ratio of the first complex coefficient in the Fourier expansion of the response to Asinwt divided by A. In the case of static nonl i near i ti es, it is independent of frequency and given by the following integral CUD: V' ' (1) The sinusoidal analysis of the SP model must deviate from classical DF theory because the weighting factor non-linearities are even functions and therefore have no first harmonic component in their output (see Appendix A). In addition their effect is multiplicative so that not only the D.C. term but also higher harmonics must be considered because the product will contain first harmonic terms produced by the multiplication of higher order harmonics. Specifically, the weighting factor elements produce a D.C. term and even harmonics (aO, a2, a4, a6. . . ) whereas the gain elements (odd functions) produce no D.C. term and odd harmonics (bl , b3, b5. . . > so the product will contain first harmonic terms from the factors aO*bl, a2*bl, a2*b3, a4*b3, a4*b5, etc. In this case, only the first two terms of this expansion are kept because the higher order terms are considerably smaller and because in this way the gain elements could be analyzed in the usual DF manner. For the weighting factor non-linearities however both the D.C. gain NO and second harmonic complex gain N2 were calculated using the following equations (derived by dividing the appropriate Fourier coefficient by the input amplitude): IV. Computation. (Details of the computation of the model output can be -found in Appendices A, B and C) . The classical "first order" DF's Mere calculated -for the odd non-linear elements B and D using the integral -formula given above (EQ. 1) yielding! < II < tf <4> < I 1 < 45 < 12 < 125 (5) 125 < 12 with II the amplitude o-f the target velocity sinusoid and 12 that o-f the acceleration (see -fig. 8). I-f A and -f are the amplitude and frequency o-f the actual input target position sinusoid then Il=2*PI*f*A and 12 = 2 * PI * -f * 1 1 . and m are de-fined in -fig. 4. The -first order DF's -for the even nonl i near i ti es A and C are zero. The zeroth order (D.C. ) and second order terms were calculated using equations (2) and (3) respectively (see Appendix A) . < A < A < 10 defined in The Bode Plot of interest is that of eye velocity E with respect to target velocity T because of the available data for verification of the model (see next section) therefore the phase of t will be defined as zero and all phases will be calculated with respect to this. The n-th order (n > O) output of the non-linear elements were calculated using the -following formula where np and nq are the real and imaginary parts o-f the n-th order DF respectively: Input Amp. *1r\6 f + A< X #sin(nwt - arctan (np/nq) + (PI/2) + input phase) (B) which when the DF is real valued simplifies to: Input Amp. * np * sin(nwt + input phase) and the D.C. term (zeroth order output) is given by: Input Amp. * Zeroth Order DF The above formulae were used to calculate the outputs of the various non-linear elements: YO, Y2 *> , Xl(f), WO, W2 (/3 > , X2(^>. (The notation >: (y) meaning ampl itude (phase) and the variables defined in fig. 8). The outputs of the multiplier stages were calculated using standard trigonometric identities (see Appendix C). The D.C. term of the weighting factor multiplies the output of the gain element yielding a first order term and the second order term of the weighting factor also produces a first order term. The sum of these two first order output terms is the output of the multiplier: Viith 'o^ ^x(^) N* OC 4 X vi ttr*V c.s . * . (9) (10) The two multiplier output terms, Z1(.) and Z2(ty are then filtered linearly yielding 01 (J,) and O2 <) , the outputs of the velocity and acceleration pathways respectively. These two are then added yielding the output eye acceleration which is integrated to get eye velocity. (11) The calculation of the model output was done on using the program Ismodel written in programming copy of the program is in Appendix B. a PDP-11/23 language C. A V. Results and Discussion. The results of the simulation are shown in Table 1. The parameters 8, m, and f , were optimired by eye within the constraints set -forth in section II. The third and -fourth columns are the output of the model in -fig. 8, the -fifth and sixth columns the output o-f the model i -f the early or acceleration pathway is ignored, and the seventh and eighth columns the experimental results o-f Lisberger (1O) o-f the actual open loop -frequency domain response o-f one monkey. The optimization procedure was not simply a curve -fitting procedure because 1) the allowed values were severely constrained -from the experimental data, 2) the number o-f parameters (3) did not offer enough degrees of freedom for any real "manipulation" of the output, 3) the simulation was insensitive to any change in the value of within its constrained value interval, and 4) the optimal values for m and within the constrained bounds were very close to the best a Br.i9r.i estimates. It was performed more to determine the effect of the parameter modulation on the performance than to match the data. As mentioned above, the model performance is insensitive to changes in so it is difficult to choose a value with any conviction but the data of Morris (9) suggests that it may be as low as .5 deg or lower so the value of .5 was used in the simulation in Table 1. The low velocity slope m of the velocity gain element has a value that varies between 26.5 /sec for initiation of pursuit with dim background and 33.3 /sec for maintenance (modulation of an ongoing eye velocity) of pursuit in the dark C33C9D. The a priori best estimate is therefore 33.3 given that the conditions under which it was derived most closely match those under which the frequency domain data was found. The optimal value was found to be 32 /sec. Increases in m had the effect of almost uniformly raising the gains for all amplitudes and frequencies. The velocity gain curve bends starting at around 1.5 deg/sec and achieves a new constant slope at around 5 deg/sec. Most of the gain attenuation occurs at the low end of the interval so the best a priori guess would be somewhere between 1.5 and 3 deg/sec. The optimal value of "o was determined to be 2.75 deg/sec. Increasing If had the effect of decreasing the high frequency and amplitude attenuation in gain. One encouraging result of this is that the optima found within the constrained intervals seemed also to be the global optima in that values outside the intervals may the simulation less accurate. As can be seen from the results in Table 1 and fig. 9, the model relatively accurately predicts the gain data exept that the high frequency attenuation in the model occurs at somewhat too low a frequency (at <1 rather than >1 Hz) and is somewhat too broad in nature. No manipulation of the parameters could improve the overall shape of the gain curve. The phase data, as bid. . odel Output -for Peak Angle Parameter elocity Elbow Parameter "K = 2.75 Value Theta 0. 50 ow Velocity Slope |ys* 32.000 flip. Preq. Gain Phase Gain Phase Gain Phase Model Model No Ace. No Ace. Data Data .32 0.7 6.75 -125. 1 6.74 -125.5 6. 13 -7. 1 1.0 4.67 -139.6 4.65 -140.6 5.32 -47.7 .32 2.0 1.89 -183.7 1.81 -189.4 0.96 -38.5 .42 0.7 6.73 -125.0 6.71 -125.6 .42 1.0 4.66 -139.4 4.64 -140.7 4.90 -36.7 .42 2.0 1.63 -180.9 1.52 -189.7 0.59 -45.6 .62 O.7 6.64 -123.6 6.61 -124.5 .62 1.0 3.93 -137.5 3.88 -139.8 6.21 -19.3 .62 2.0 1.34 -173.8 1. 15 -189.6 1.29 -42.2 .83 0.7 5.90 -123.0 5.87 -124.3 5.34 22.4 .83 1.0 3.29 -136. 1 3.23 -139.7 5.35 6.0 .83 2.0 1.25 -167.0 0.97 -189.8 1.40 -55.2 .25 0.7 4.64 -121.7 4.59 -124.3 .25 1.0 2.62 -133.0 2.52 -139.7 4.61 -16.4 .25 2.0 0.62 -180.4 0.80 -190.0 1.36 -40.5 .66 0.7 3.95 -120.2 3.88 -124.2 3.87 35. 1 ,66 1.0 2.31 -129.5 2. 15 -139.7 3.71 14.5 .66 2.0 0.58 -183.2 0.71 -190.2 1. 17 -49.3 tie a. sdel Output -for Peak Angle Parameter elocity Elbow Parameter t = 2.75 Value Theta =0. 50 DW Velocity SI ope *n 32.000 ccel erati on Slope Factor = 10 .OOO reel eration Cuto-f-f = 25O.OOO mp. Preq. Gain Phase Gain Phase Gain Phase Model Model No Ace. No Ace. Data Data .32 0.7 6.87 -121. 1 6.74 1 25. 5 6. 13 -7. 1 .32 1.0 4.93 -131.3 4.65 -14O.6 5.32 -47.7 .32 2.0 3. 14 -152.6 1.81 -189.4 0.96 -38.5 .42 0.7 6.89 -119.8 6.71 -125.6 .42 1.0 5.02 -128.8 4.64 -140.7 4.90 -36.7 .42 2.0 3.48 -144.2 1.52 -189.7 0.59 -45.6 .62 0.7 6.91 -115.9 6.61 -124.5 .62 1.0 4.57 -120.2 3.88 -139.8 6.21 -19.3 .62 2.0 4.44 -134.2 1. 15 -189.6 1.29 -42.2 .83 0.7 6.32 -111.7 5.87 -124.3 5.34 22.4 .83 1.0 4.32 -111.4 3.23 -139.7 5.35 6.0 .83 2.0 6.07 -291.5 0.97 -189.8 1.40 -55.2 .25 0.7 5.46 -102.0 4.59 -124.3 .25 1.0 4.57 -97.4 2.52 -139.7 4.61 -16.4 .25 2.0 3.05 -211.4 0.80 -19O.O 1.36 -40.5 .66 0.7 5.26 -92.7 3.88 -124.2 3.87 35. 1 .66 1.0 5.26 -88.6 2. 15 -139.7 3.71 14.5 .66 2.0 1.97 -47.9 0.71 -190.2 1. 17 -49.3 * ' 5 3 O M O D- --O-- O- -o --O-* --D- --o-- a I.CC C Os. TV, ->>. I*C O >D -" expected, is not at all predicted by the model. One very interesting result however is that the slope of the phase curve is approximately correct especially -for the higher amplitude curves and that there is a constant phase discrepancy o-f about 140 deg -for all frequencies and amplitudes examined. It should also be noted that the acceleration pathway contributes very little to the response as can be seen -from a comparison o-f columns -four and six, and -five and seven. In an e-f-fort to examine the potential role of the acceleration pathway (assuming the gain estimate used in the model construction was possibly inaccurate and too low), the acceleration gain and saturation point were manipulated arbitrarily while attempting to improve the fitting of the data. c 0, *jt / A * ^r tf ^l,f f . A,\ t-i 222 O r, ~ c . re 2, y__ 7 ; ielcU ^ < us- - > us A-3- D r. H* , r\ 1 A n ~ I *>. d J^ A 7 i ^ A b - = I7 -crt ^ / Jc in/ J i ,^ rect-,f,e*A .v, S . t V. 7 / /-.A.-. fl I O (\ c -f )A L . ^^L > T" T" -. fc 4 ** > fa*.-& In J-frk.-Ofi *c*^,)i.iL/^-vHa)A * nA/_ v y (^.^^ (c -- &/A r L n A ^ 10, &) A - e / tb 1Cut DC .o or A- 7 ^ 2. - cos n /l H Sn is M c - iclude static double aC63 = { . 32, . 42, . 62, . 83, 1 . 25, 1 . 66> ; static double fC33 ={.7,1.,2.>; static double gdataC63C33 = C6. 13, 5. 32, . 96, 0. , 4. 9, . 59, 0. , 6. 21 , 1 . 29, 5.34,5.35, 1.4,O. ,4.61, 1.36,3.87,3.71, 1. 17> ; static double pdata[63C33 O7. 1 , -47. 7, -38. 5, 0. , -36. 7, -45. 6, 0. , -19. 3, -42.2,22.4,6. ,-55.2,O. , -16. 4, -4O. 5, 35. 1, 14.5, -49. 33- ; double il 0, i2 0, wO 0, w2 O, xl 0; double >:2 0, yO O, y2 0, zl O, z2 0, ol 0, o2 0; double alpha 0, beta 0, theta 0, eta , slope 0; double phil O, phi2 O, psi 1 O, psi2 0, ki 1 O, ki2 Oj double nap 0, naq 0; int i O, j 0; double gainC63C33 C0> , phaseC63C33 CO, gC63C33 <03 , pC63C33 <0>; double sqrtO, cos ( ) , sin(), asinO, atanO, atan2(), power () ; Lain ( ) put-fmt ("Peak Angle Parameter (theta) ? "); putch(-O12) ; get-fmt ("V.d", ? ; put-fmt ("Velocity Elbow Parameter (eta) 7 "); putch(-012) ; getfmt ("*/.d", ? for ( j = ; j < 3 ; j++) < /* Velocity Pathway */ il = 2 * 3.1415926 * fCj] * aCiD; phil = - .2 * 3.1415926 * fCjD; /* Velocity Gain Element */ i-f (i 1 > eta) xl = (.6366 * (slope - 6.2222) * (asin(eta/il) + (eta/il) * sqrt ( 1 -power (eta/i 1,2.))) +6. 2222) *i 1 1 el se x 1 = si ope * i 1 ; /* Spatial Weighting */ i-f ( aCi 3 > theta) { yO = ( (2OO - 25 * theta) * 3.1415926 + 3O * theta * asin(theta / aCi:> + 30*aCi 3*sqrt (1 -power (theta/aCi 3,2. ) ) ) / (3.1415926 * (10 + theta) * (20 - theta) ) : nap = (2 * (-30 * theta * (1 - power (theta/aCi 3,2. )) - .66667 * aCi3 * (10 - 2 * theta + 3O * power (theta/aCi3,3. ) ) ) )/(3. 1415926*aCi3 * (10 + theta) * (2O theta)); naq = (2O * sqrtd - power ( theta/aC i 3 , 2. ) ) * (1 + 5 * power (theta/aCi 3,2. ))) / if 1 /n =:O->A* r 1 o - thDta) t (?ft - theta)): y^ = sqrt (power (nap, ^'. )--power (naq,^. )> * aiij; alpha = phil - atan2 (nap, naq) ; 3- else < yO = 10 / (10 + theta) ; y2 = (8 t aCiD) / (3 t (10 + theta) t 3. 1415926) ; alpha = phil + 1.5708; /* Crass-Product Calculation */ zl = xl * sqrt (power (yO, 2. ) * . 25 * power (y2, 2. ) + yO * y2 * cos(2tphil - alpha 1.5708)); psil = 1.57O8 - atan2(2 * yO * cos (phil) y2 * cos(alpha - phil + 1.5708), 2 * yO * sin(phil) + y2 * sin(alpha - phil + 1.5708)); /t Linear Filtering #/ ol = zl / sqrt(power(2 * 3. 1415926** C j 3 , 2. ) *. 0016 +1 ) ; kil = psil - atan(.08 * 3.1415926 * -f C j 3 ) ; /t Acceleration Pathway t/ i2 = 2 * 3.1415926 * i C j 3 * il; phi 2 = phil + 1.57O8; /* Acceleration Gain Element #/ i-f (12 > 125) x2 = 1.28 * .6366 * (asi n ( 125/i 2) + (125/i2) * sqrtd. - power (125./i2,2. ) ) ) * i2; else x2 = 1.28 * i2; /* Velocity Weighting */ if (il > 15) { wO = (3.1415926 - 2*asin ( 15/i 1 ) + .1333 * il * (1 - sqrtd - power (15/i 1,2. ))))/ (3. 1415926 til); w2 = 1.2732 * ((.O222 - 5 / power (i 1 , 2. ) ) * sqrtd - power (15/i 1,2. ))- .O222) til; i-f (w2 > 0) beta = phi 1 + 1.5708; else beta = phil - 1.5708; > else { wO = . O4244 til; w2 = .02829 til; beta = phil - 1.5708; /t Cross-Product Calculation t/ z2 = x2 t sqrt (power (wO, 2. ) + . 25 t power (w2, 2.) + wO t w2 t cos(2 t phi2 beta - 1.5708)); psi2 = 1.5708 - atan2(2 t wO t cos (phi 2) -- w2 t cos (beta phi 2 * 1.5708) ,2 t wO t sin(phi2) + w2 t sin(beta - phi2 * 1.57OB)); /t Linear Filtering t/ o2 = z2 / power ( . OO3948 t -fCj] t -FCj3 * 1,2.); /* uutput calculation */ j gCi 3C j3 = ol/i2; / "* * pCiDCjD = 57.29578 * (kil 1.5708)} gainCi3Cj3 = sqrt (power (ol , 2. ) +power (o2, 2. ) + 2*ol*o2 * cos(ki 1 - ki2) ) / 12; phaseCiDCjD = 57.29578 * ( atan2(ol * cos(kil) + o2 * cos(ki2),ol * sin(kil) + o2 *sin(ki2)))| /* Printout */ eptext ("Model Output -for Peak Angle Parameter Value Theta = "); epdoub (t beta, 3, 2) ; epchar ( ' \n' ) ; eptext ("Velocity Elbow Parameter Eta = "); epdoub (eta, 3, 2) ; epchar C\n' ) ; eptext ("Low Velocity Slope = "); epdoub (slope, 5,3); epchar ( ' \n' ) ; epchar ( ' \n' ) ; eptext ("Amp. ") ; epchar C\t' ) ; eptext ("Freq. ") ; epchar C\t' ) ; eptext ("Gain") ; epchar (' \t' ) ; eptext ("Phase") ; epchar C\t' ) ; eptext ("Gain") ; epchar <'\t' ) ; eptext ("Phase") ; epchar C\t' ) ; eptext ("Gain") ; epchar C\t' ) ; eptext ("Phase") ; epchar ( ' \n' ) ; epchar (' \t' ) ; epchar C\t' ) ; eptext ("Model ") ; epchar C\t' ) ; eptext ("Model ") ; epchar C\t' ) ; eptext ("No Ace. ") ; epchar C\t' ) ; eptext ("No Ace. ") ; epchar C\t' ) ; eptext ("Data") ; epchar C\t' ) ; eptext ("Data") ; epchar ( * Xn' ) ; epchar (' \n' ) ; for (i = ; i < 6 ; i++) for (j = ; j < 3 ; j++) < epdoub (aCi 3,5,2) ; epchar C\t' ) ; epdoub (f [ j 3 , 3. 1 ) ; epchar (' \t" ) ; epdoub (gainCi 3 L j 3 , 5, 2) ; epchar (' \t 7 ) ; pnrlnuh (ohaspC i3Ci3.4.1) epdouib (gLiJLjJib,^); epchar <'\t' )| epdoub (pC i ] L j 3 , 4 , 1) ; epchar C\t' ) ; i-f (gdatati DC jD '= 0. ) { epdoub (gdataCi DC jD, 5,2) ; epchar C\t' ) 5 epdoub (pdataCi DC j 3,4, 1) } epchar C\n' ) ; else epchar C\n' ) ; epchar ( ' \-f ' ) ; .c II e> ^0 ~A t"rvi\ j 3 L E> s. f A n P *.l -a. - 4 c- C- * "A. \ o( ), 6 oF A- t Ue o^t r oV fc F l* * y n/\ J_ v Kt c 7 e v. x, T\ F.Uccxw c> a f M Jt c< C\ o^ v\ v\K$e * vvo, 1 V (C - ' V ^ vv (ft - oP c\ Vo t We ^Ko ,c^a i i rr GENERAL LIBRARY U.C. BERKELEY III I II II III I! I II 111