Reduction of Encoder Measurement Errors in UKIRT Telescope Control System Using a Kalman Filter

Transcription

1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY Reduction of Encoder Measurement Errors in UKIRT Telescope Control System Using a Kalman Filter Yaguang Yang, Nick Rees, and Tim Chuter Abstract The tracking performance of the United Kingdom Infra-Red Telescope (UKIRT) is directly related to the accuracy of the position measurement which is given by an encoder. It is found that the encoder has some deterministic and random measurement errors which cause consistent wobbling of the telescope. Therefore, a Kalman filter with deterministic error input model is adopted in this embedded real-time control system to reduce both deterministic and random measurement error effects. Our extensive test shows that the system tracking performance is improved. Astronomers report confirms our result. Index Terms Computer control, Kalman filter, modeling, real-time system, telescopes. I. INTRODUCTION THE United Kingdom Infra-Red Telescope (UKIRT) is designed to explore invisible scientific objects. Therefore, the performance of tracking a given coordinate frame of some invisible scientific target is very important. The control system designed to achieve this goal is depicted in a simplified diagram, Fig. 1. The desired star position (understandable by the telescope) is calculated based on several mathematical transformations over a sphere (this program is called the Kernel). The real telescope pointing is determined by a high-resolution encoder to guarantee the desired high quality of the tracking performance. The control strategy is straightforward as described in the upper part of Fig. 1, where controller 1 gets the error signal from the kernel and the encoder outputs, and controller 2 is not used. This error signal is used to control the telescope to the desired position. Our experience in UKIRT, however, indicated that the telescope experienced consistent wobbling in right ascension (RA) direction when this scheme is used. Though this wobbling was suspected to be a problem caused by measurement errors of the fine encoder, there has been little research on the problem. Very little literature on encoder measurement errors is available except some internal materials from manufacturers, for example [1]. Similarly there is scant information on how these errors could possibly affect the telescope performance. To the best of our knowledge, there is no research on improving these measurement errors by using algorithms such as a Kalman filter. Instead, many telescopes (including UKIRT) turn to use another very efficient scheme, Manuscript received July 12, 2000; revised August 25, Manuscript received in final form March 12, Recommended by Associate Editor R. Middleton. Y. Yang is now with the CIENA Corporation, Linthicum, MD USA. N. Rees and T. Chuter are with the Joint Astronomy Center, University Park, Hilo, HI USA. This work was done when Y. Yang was with the Joint Astronomy Center, University Park, Hilo, HI USA. Publisher Item Identifier S (02) the so-called auto-guiding method, which utilizes a visible guiding star nearby the invisible scientific target. The idea of this auto-guiding method is to use two instruments on the same telescope for these two (visible and invisible) targets. Scientists input the coordinates of the two astronomical objects to the computer, and the kernel program calculates where the light beams from the two objects will be projected on the focal plane at run time. The positions of the two instruments on the focal plane are therefore determined. These positions can be adjusted automatically by a cross head control system (cf. Fig. 1) such that the scientific instrument receives photons from the invisible scientific target and the other instrument (CCD) 1 receives photons from the visible guiding star. Once the two instruments point correctly to the two objects, the cross head system will be locked, i.e., the two instruments are fixed on the focal plane. The guiding process starts at this point. During the guiding process, if the visible star is not in the center of the CCD, i.e., if the CCD is not pointing correctly to the visible star, then the scientific instrument is not pointing correctly to the invisible scientific target, because the distance of the light beams from these two stars on the focal plan is fixed. A correction scheme, therefore, uses the position information of the visible star on the CCD to control the telescope such that the CCD is always pointing to the visible guiding star, thereby the scientific instrument is always pointing to the scientific invisible target. This strategy (turning on controller 2 and turning off controller 1) is described in the lower part of Fig. 1. Unfortunately, not all invisible scientific targets have some nearby visible stars. In such a case, we have no choice but to use the encoder. This means, we need to switch back and forth between two controllers depending on the condition whether there is a bright star near the scientific target. Another way to get better pointing information is to use better encoder with a higher cost. The Gemini telescope, one of the most advanced telescopes in the world, uses 30-bit encoder in its specification [2]. However, UKIRT has no plan to use a better encoder. Therefore, we carefully studied the wobbling problem and linked the periodical oscillation of the telescope to some special errors of the encoder. This leads us to use a Kalman filter to provide better feedback information. The remainder of this paper is organized as follows. In Section II, we explain how the encoder measurement error is related to the known wobbling problem of the telescope. In Section III, we give details on how the measurement error is reduced by the Kalman filter. We present our test results in Section IV, followed by our conclusion. 1 CCD stands for charge couple device. It receives visible star signal and transforms the signal to electronic information in a very fast speed. Many telescopes use it to determine visible stars position /02$ IEEE

2 150 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002 Fig. 1. Simplified diagram of UKIRT telescope control system. Fig. 2. Power spectrum density of UKIRT mount control error signal. II. DETERMINISTIC AND RANDOM ENCODER ERRORS As discussed above, if there is no visible star near the target, i.e., the loop in the lower part of Fig. 1 is open, the control performance relies directly on two important facts: 1) How accurate can the kernel software provide demanded position given all the complex effects. The current algorithm takes account of very small effects, much smaller than one arc-second (an arc-second degree). Therefore, the calculated desired position is reasonably accurate. 2) How accurate can the encoder measure the telescope position so that we will have correct error information to control the telescope. For the encoder with 24 bits, the resolution is roughly arc-second. If the measurement is accurate and if the control system is well designed, the error of the closed-loop system may not be smaller than arc-second, but should be reasonably close to the resolution limit of the encoder. It was observed that the telescope did not work well when a guiding star was not available, in particular, the telescope had a significant oscillation (up to more than one arc-second) at about 0.76 Hz. This frequency dependent error can easily be seen from the power spectrum density analysis for the control error signal, see Fig. 2. The power spectrum density is generated by using MATLAB function psd for the error signal obtained in the following conditions: 1) the fast guider loop is open and 2) the telescope moves at a speed of 15 arc-seconds/second (the nominal speed of the telescope when observing). Since the encoder has physical slots to represent radians, one slot represents arc-seconds. Note that the speed of the telescope (and the speed of the encoder) is 15 arc-seconds per second, therefore it takes seconds for the encoder to move from one slot to the next one, or the frequency from one slot to the next one is about Hz. Hence, we believe that the

3 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY Fig. 3. Fine track read out system. frequency dependent control error at 0.76 Hz is at least partially related to the periodic error of the encoder as it moves from one slot to the next slot. We begin our investigation on how the encoder measures the telescope position, which is illustrated in Fig. 3 [1]. Fig. 3 shows some important components of the encoder: an illuminator, a code disk with slots on the edge, 2 a set of four slit plates, and detector arrays. When the telescope rotates, the code disk of the encoder moves with the telescope while all other parts of the encoder described in Fig. 3 are static. By counting the number of the slots on the code disk which move over the slit 2 In the diagram, only part of the disk is illustrated with black and white colors. The black part represents the solid disk where the light from the illuminator cannot go through, and the white part represents the slot where the light from the illuminator can go through. The same convention is used for slit plates. plates, one knows the position of the telescope at any time with the resolution of 16 bits. It is worthwhile to note that the four slit plates are designed in such a way that they have certain phase shift from the code disk. For example, when the code disk is in the position described in Fig. 3, the first plate is aligned perfectly with the code disk, so all light going through the slot of the code disk will go through the slot of the slit plate; the second plate is not aligned at all with the code disk at that position, no light will arrive at the detector; the third and the fourth plates also have some phase shift with respect to the code disk, and some but not all light will arrive at these two detectors. As code disk moves along the telescope, each detector will receive different magnitude of light at different time depending on the relative position of code disk and slit plates. The received signals in all these 4 detectors are sinusoidal functions with some fixed phase shift

4 152 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002 Fig. 4. The effect of displacement of detect arrays. relations. This will generate four sinusoidal currents and with the phase shift described as in Fig. 3. Therefore, by combining the current and, then and, we can obtain a sine function and a cosine function. Using these six sinusoidal functions and the phase relationship of these functions, one can get extra eight bits of information by using the so-called arc tangent multiplier, thereby the resolution of the encoder can reach 24 bits. This estimation will be error free only if everything is accurate. But this is not always the case. For instance, if the phase shift on the slit plate is not in the position as designed (the slit array displacement error), we will have an approximately sinusoidal measurement error which repeats at every slot (a code cycle) of the code disk; in particular, when the encoder spins at a speed of 15 arc-seconds/second (the nominal speed when the telescope tracks a star), the repetition of this measurement error is roughly at 0.76 Hz. This argument is illustrated in Fig. 4. Here the samples are taken at every radian and 256 samples are a cycle of radian. Fig. 4(a) is the plot of, where and are each 1 unit magnitude with nominal phase difference, Fig. 4(b) is the plot of with the same magnitude but the phase has a small shift from nominal position. Fig. 4(c) shows that the error [the difference of Figs. 4(a) and (b)] is approximately a sinusoidal function. Similarly, if the the magnitudes of and/or is not exactly equal to the design value, there will also have some approximately sinusoidal errors. Finally, by applying arc tangent multiplier, the sine error will be passed over to the output [1]. Assume that the above described scenario indeed happens, then the encoder will have a nearly sinusoidal measurement error, moreover, this error will be time invariant, and it depends only on the relative position of the code disk and the slit plates. Since the slit plates are fixed, it actually only depends on the code disk position or the telescope position. If we run the telescope at a different speed, it will take a different time for the encoder to move from one slot to the next slot because the arc distance between any two physical slots is fixed. If the largest periodic control error is related to this periodic measurement error, we will see that the frequency, at which the largest control error occurs, will change to a frequency that is equal to the frequency of the movement of the encoder from one slot to the next one. The test that moves the telescope at different speeds verifies our expectation. Using a plot similar to Fig. 2, we see that the error peak at 0.76 Hz on the spectrum plot shifts to a different frequency if we run the telescope at a different speed. We then run the telescope at a much faster speed (90 arc-seconds per second) than its designed speed (15 arc-seconds per second). In this case, the heavy telescope, like a low-pass filter, should have little response to the high-frequency error 3 correction. Therefore, the telescope should move in a roughly constant speed, and the encoder measurement should provide similar information. But, by careful examination, we find from the test data that the real encoder signal can be fitted very well by a constant speed plus position variations described by a sine function. This means that the sinusoidal position variation could be a measurement error when slots are used to give 24-bit resolution. Moreover, the phase of the sine function is time invariant, it is only related to the encoder position. We also note that the frequency of the sine function is exactly the same as we calculated based on the assumption that it is equal to the encoder speed 3 The frequency of the periodic measurement error, related to the distribution of the 2 slots, is six times higher now ((90=19:7754) 4:55 Hz) because the speed of the telescope is six times faster than nominal speed.

5 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY (arc-seconds per second) divided by arc-seconds, the distance between two slots. This means that our assumption is correct. There are also some random errors in encoder measurement signal. The most obvious one is from the resolution limit of the encoder. Since the encoder has only 24 bits resolution, it cannot distinguish position difference smaller than radian, or arc-second. Assume that the encoder is perfectly manufactured without any error except the resolution limit, then, the difference between the real position and the measurement output will be smaller than arc-second. Since this error depends on where the real (but not measurable) position is, it can be viewed as a random variable because the system is constantly adjusted. We will use a Kalman filter to improve the encode measurement and therefore to reduce the wobbling of the system. III. POSITION ESTIMATION BY A KALMAN FILTER We use radians as the measurement unit in all formulas in this section. First, we only consider the deterministic measurement error. Let be the real telescope position at sampling time, which is to be estimated. Let be the deterministic measurement error as shown in Fig. 4(c), and let be the observed position from the measurement. Then we have From this relation, we have Let us define then, we have Since the deterministic measurement error is approximately a periodic function repeated in every radians with fixed magnitude and fixed phase, these parameters are time invariant. Also Fig. 2 shows that it may have a second harmonic effect, therefore, is modeled by where and are obtained off-line by best curve fitting for a very large measurement data set. This system [described by (1) (5)] may be solved without difficulty by an extended Kalman filter if, for the estimated and one-step predicted telescope positions and obtained by the extended Kalman filter, and are small (cf. [3]). But it is difficult to have some rigorous analysis on the approximation quality. We choose to use another method in our practice. Notice that if can be obtained based on, then is available when is observed, hence the system can be solved by a linear Kalman filter. An easy way to approximate is to replace in (5) by the observation (since the actual (1) (2) (3) (4) (5) may never be available and its value is to be estimated). This gives But this simplification introduces some error. A better way is to utilize some approximation method. The idea is described below. Since where and are all constants, and is given by the encoder, the solution for this equation is simply a fixed point problem described by. The problem can be solved by the so called fixed point method. There are plenty of books, e.g., [4] and [5], that discuss the method and its convergence properties. Applied to our problem, one can readily show that the fixed point algorithm converges to a unique fixed point from any initial guess, because it is straightforward, by using the mean value theorem, to show that is a contract map. The claim follows directly from invoking Banach fixed point theorem (cf. [5]). Therefore we get the first approximation Repeat this procedure, the second approximation is Fig. 5 shows, and the error is reduced significantly after two repetitions. This suggests that we use (6) (7) (8) (9) (10) Now we incorporate random effects into the above model. Let be the random measurement error including the random cutoff effect for the measurement signal smaller than radian. Let be the random effects associated with the state equation. The model is then given by and (11) (12) and are known when is observed. Let be the mean of its argument, be the variance of its argument. Assume that, where if and otherwise,, and. These assumptions are proved to be practically reasonable and enable us to use Kalman filter formulas directly to estimate the real encoder position. If the statistical properties of and are also time invariant, then, and therefore

6 154 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002 Fig. 5. Approximation errors.. All these parameters will be obtained by analytic/experimental methods (the details are explained below). Let be the estimated telescope position at sampling time (assume, the mean of at the initial state), be the one-step predicted telescope position at sampling time be the filter variance matrix at sampling time (assume, the variance of the initial state), be the one-step prediction of the variance matrix at sampling time be Kalman filter gain, then the Kalman filter iteration formulas 4 are given as follows: (13) (14) (15) (16) (17) Since contains an error caused by the cutoff of the measurement signal smaller than radians, a reasonable choice about is radians. Based on ex- 4 An early effort to include deterministic input in Kalman filter problem can be found in [6]. For the system described by (11) and (12) with Y = 0, one can find the Kalman filter formulas in a widely referenced book [3]; for the random system described by (11) and (12) with Y = U, one can find the Kalman filter formulas in [7]. For the general case described by (11) and (12), the Kalman filter formulas is given in [8] which is in Chinese (We cannot find any English reference addressing the general case described by (11) and (12)). But it is not so difficult to get the Kalman formulas because the idea used in the proof in [8] is very similar to the one used in [3]. tensive experiments, 5 is selected to be radians, is selected to be, and is selected to be zero. IV. TEST RESULTS The proposed Kalman filter has been implemented in the UKIRT mount control system. The computer source code is written in C/EPICS and is running in VxWorks real-time system. The EPICS GUI window running on Sun/Unix machine can communicate with the computer control system running on VxWorks real time machine. One of the main GUI windows for mount control system is shown in Fig. 6. The Kalman filter related widgets are located in the lower part of the window (where kk1 is, kk2 is, phase1 is, and phase2 is ). We can turn on and off the Kalman filter by clicking on and off buttons next to the widget kalman- Filter. We can also choose what kind of filter we want to use by clicking 0, 1, or 2 after the widget filter, where 0 uses a simple correction without considering random effects, 1, uses the Kalman filter with the first approximation (6) for, and 2 uses the Kalman filter with the second approximation (10). We can monitor the error outputs based on the difference between kernel and the signal from the encoder, or based on the difference between kernel and the signal from the Kalman filter, or from the CCD. In the first two cases, we can use an instrument hard wired to the control system. This instrument can draw real-time response curves for any selected signals. If the tracking star is visible, we can see 5 We can achieve this because we have designed this system to be able to change many parameters at run time. We can also monitor the telescope performance following every change in a very efficient way, by using a powerful control software environment EPICS, cf. [9] and [10]. We will discuss this in detail in Section IV.

7 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY Fig. 6. The GUI interactive interface of the mount system. Fig. 7. Simplified diagram of UKIRT telescope control system. from another GUI window the (real-time) standard deviation of the tracking star from its desired position by using the signal from the CCD camera (cf. Fig. 7). We can also choose if we want to record all above mentioned data by clicking a button after the widget logging. The recorded data can be easily analyzed by different tools, for instance, by MATLAB. Moreover, we can select, and from the GUI window at run time by selecting and entering data from the fields of the corresponding widgets. 6 This makes it possible for us to choose, and by trial and error, because we can set these parameters from the GUI window at run time and we can monitor 6 Q and J widgets are removed from final GUI window because their effects on the tracking performance are less significant than R and S, and they seem to be related to R. the control performance from the outputs immediately. This also makes our other test tasks much easier and more efficient. The modified system diagram used in our test is slightly different from the original one. Fig. 7 shows the simplified control system. All configuration switches are implemented by software, and reconfiguration can easily be done via the GUI window (cf. Fig. 6). Our first goal is to decide if the Kalman filter really improves the control system. This can easily be done by selecting the button on or off, if the button is on, we can choose button 0, or 1, or 2, then examine the response from the instrument which monitors various error outputs. The conclusion to our first question is clear: using the Kalman filter is better than not using the Kalman filter, using the Kalman filter with (10)

8 156 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002 Fig. 8. Error power spectrums of different control schemes. to estimate is better than using the Kalman filter with (6) to estimate, and using the Kalman filter with (6) to estimate is better than the simple correction. Therefore, in the test described below, we will discuss the improvement by the Kalman filter only for the case when (10) is used. Since the encoder output is not accurate, to verify if the Kalman filter really improves the measurement, we use bright stars instead of faint ones in our test so that we can always use the CCD to get the accurate star position no matter what control scheme is used. The main advantages of using the CCD to determine the telescope position are 1) the optical measurement has significantly better resolution and 2) the CCD does not have an error similar to and is therefore much reliable. This is particularly important in tracking control scheme when the encoder measurement or the Kalman estimation is used in feedback, and when we are not sure about the reliability of the measurement from the encoder and the Kalman filter. By using the position information from the CCD, we can reliably evaluate the effect of the Kalman filter and see if it really improves the measurement or not. We run the telescope in three modes in our test (cf. Fig. 7): 1) the tracking control scheme using the output of the Kalman filter and controller 1 to control the telescope, while turning on the CCD camera to record star position (but controller 2 is not used), so that we can analyze the star position from different measurements, the CCD camera, the encoder measurement, and the estimated star position from the Kalman filter; 2) the tracking control scheme using encoder measurement and controller 1 to control the telescope with the Kalman filter off, also the CCD camera is on to record star position from CCD point of view (but controller 2 is not used); and 3) the guiding control scheme using the position information from the CCD and controller 2 to control the telescope, but also record position signals obtained from the encoder and the Kalman filter (controller 1 is not used in this case). A typical comparison result of the different control schemes is presented in Fig. 8. Fig. 8 shows the error power spectrum plots based on the signal observed from the CCD measurement, with different control schemes, i.e., with the encoder feedback, or with the Kalman filter feedback, or with the CCD feedback. We see that the fast guider is the best, because it has used better measurement information (which is not available if our scientific target is an invisible star, and if there is no bright star nearby). The Kalman filter reduces the oscillation significantly at 0.76 Hz comparing to the tracking scheme without the filter, this is our main purpose. A summary of the improvement for all test stars is listed in Table I. V. CONCLUSION The oscillation problem in UKIRT telescope control system has been investigated. The problem is pinned down to the encoder measurement error. The Kalman filter is introduced to reduce the deterministic and random effects. Since encoders are

9 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY TABLE I THE PERFORMANCE COMPARISON WITH/WITHOUT THE KALMAN FILTER widely used in many tracking problems, this idea could be extended to all of the similar applications. REFERENCES [1] E. Pearson, M. Kansky, and N. Tobey, Presentation of 24 bit optical encoder system, unpublished, [2] J. Wikes and C. Carter, Mount Control System-Control System Design Description. Cambridge, U.K.: Royal Greenwich Observatory, internal publication. [3] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, [4] C. Woodford, Solving Linear and Non-Linear Equations, West Sussex, U.K.: Ellis Horwood, [5] V. I. Istratescu, Fixed Point Theory. D., Dordrecht, The Netherlands: Reidel, [6] B. Friedland, Treatment of bias in recursive filtering, IEEE Trans. Automat. Contr., vol. AC-14, pp , [7] G. Chen, G. Chen, and S.-H. Hsu, Linear Stochastic Control Systems. Boca Raton, FL: CRC, [8] Z. Wang, Basic Modern Control Theory, Beijiang, PRC: National Defense Industry Press, [9] M. R. Kraimer, Epics Release Beta 11, Input/Output Controller Application Develop s Guide: Web publication, Argonne National Laboratory, [10] J. O. Hill, Epics r3.12 Channel Access Reference Manual: Web publication, Los Alamos National Laboratory, ACKNOWLEDGMENT The authors would like to thank the Associate Editor and anonymous reviewers for their useful suggestions which helped them to improve this paper significantly.