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Tracking and Kalman Filtering Made Easy Eli Brookner Copyright # 1998 John Wiley & Sons, Inc ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic) g–h AND g–h–k FILTERS 1.1 WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR Let us first start by indicating why tracking and prediction are needed in a radar Assume a fan-beam surveillance radar such as shown in Figure 1.1-1 For such a radar the fan beam rotates continually through 360
, typically with a period of 10 sec Such a radar provides two-dimensional information about a target The first dimension is the target range (i.e., the time it takes for a transmitted pulse to go from the transmitter to the target and back); the second dimension is the azimuth of the target, which is determined from the azimuth angle (see Figure 1.1-1) the fan beam is pointing at when the target is detected [1] Figures 1.1-2 through 1.1-6 show examples of fan-beam radars Assume that at time t ¼ t the radar is pointing at scan angle
and two targets are detected at ranges R and R ; see Figure 1.1-7 Assume that on the next scan at time t ¼ t ỵ T (i.e., t ỵ 10 see), again two targets are detected; see Figure 1.1-7 The question arises as to whether these two targets detected on the second scan are the same two targets or two new targets The answer to this question is important for civilian air traffic control radars and for military radars In the case of the air traffic control radar, correct knowledge of the number of targets present is important in preventing target collisions In the case of the military radar it is important for properly assessing the number of targets in a threat and for target interception Assume two echoes are detected on the second scan Let us assume we correctly determine these two echoes are from the same two targets as observed on the first scan The question then arises as to how to achieve the proper association of the echo from target on the second scan with the echo from g–h AND g–h–k FILTERS Figure 1.1-1 Example of fan-beam surveillance radar Figure 1.1-2 New combined Department of Defense (DOD) and Federal Aviation Administration (FAA) S Band fan-beam track-while-scan Digital Airport Surveillance Radar (DASR) ASR-11 This primary system uses a 17-kW peak-power solid-state ‘‘bottle’’ transmitter Mounted on top of ASR-11 primary radar antenna is L-band openarray rectangular antenna of colocated Monopulse Secondary Surveillance Radar (MSSR) Up to 200 of these systems to be emplaced around the United States (Photo courtesy of Raytheon Company.) WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR Figure 1.1-3 Fan-beam track-while-scan S-band and X-band radar antennas emplaced on tower at Prince William Sound Alaska (S-band antenna on left) These radars are part of the Valdez shore-based Vessel Traffic System (VTS) (Photo courtesy of Raytheon Company.) target on the first scan and correspondingly the echo of target on the second scan with that of target on the first scan If an incorrect association is made, then an incorrect velocity is attached to a given target For example, if the echo from target on the second scan is associated with the echo from target of the first scan, then target is concluded to have a much faster velocity than it actually has For the air traffic control radar this error in the target’s speed could possibly lead to an aircraft collision; for a military radar, a missed target interception could occur The chances of incorrect association could be greatly reduced if we could accurately predict ahead of time where the echoes of targets and are to be expected on the second scan Such a prediction is easily made if we had an estimate of the velocity and position of targets and at the time of the first scan Then we could predict the distance target would move during the scanto-scan period and as a result have an estimate of the target’s future position Assume this prediction was done for target and the position at which target is expected at scan is indicated by the vertical dashed line in Figure 1.1-7 Because the exact velocity and position of the target are not known at the time of the first scan, this prediction is not exact If the inaccuracy of this prediction is known, we can set up a  3 (or  2) window about the expected value, where  is the root-mean-square (rms), or equivalently, the standard deviation of the sum of the prediction plus the rms of the range measurement This window is defined by the pair of vertical solid lines straddling the expected position If an echo is detected in this window for target on the second scan, g–h AND g–h–k FILTERS Figure 1.1-4 Fan-beam track-while-scan shipboard AN=SPS-49 radar [3] Two hundred ten radars have been manufactured (Photo courtesy of Raytheon Company.) then with high probability it will be the echo from target Similarly, a  3 window is set for target at the time of the second scan; see Figure 1.1-7 For simplicity assume we have a one-dimensional world In contrast to a term you may have already heard, ‘‘flatland’’, this is called ‘‘linland’’ We assume a target moving radially away or toward the radar, with x n representing the slant range to the target at time n In addition, for further simplicity we assume the target’s velocity is constant; then the prediction of the target position (range) and velocity at the second scan can be made using the following simple target equations of motion: x nỵ1 ẳ x n ỵ T x_ n x_ nỵ1 ẳ x_ n 1:1-1aị 1:1-1bị where x n is the target range at scan n; x_ n is the target velocity at scan n, and T the scan-to-scan period These equations of motion are called the system dynamic model We shall see later, once we understand the above simple case, WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR Figure 1.1-5 L-band fan-beam track-while-scan Pulse Acquisition Radar of HAWK system, which is used by 17 U.S allied countries and was successfully used during Desert Storm Over 300 Hawk systems have been manufactured (Photo courtesy of Raytheon Company.) Figure 1.1-6 New fan-beam track-while-scan L-band airport surveillance radar ASR23SS consisting of dual-beam cosecant squared antenna shown being enclosed inside 50-ft radome in Salahah, Oman This primary radar uses a 25-kW peak-power solidstate ‘‘bottle’’ transmitter Mounted on top of primary radar antenna is open-array rectangular antenna of colocated MSSR This system is also being deployed in Hong Kong, India, The People’s Republic of China, Brazil, Taiwan, and Australia g–h AND g–h–k FILTERS Figure 1.1-7 Tracking problem that we can easily extend our results to the real, multidimensional world where we have changing velocity targets The –, ––, and Kalman tracking algorithms described in this book are used to obtain running estimates of x n and x_ n , which in turn allows us to the association described above In addition, the prediction capabilities of these filters are used to prevent collisions in commercial and military air traffic control applications Such filter predictions also aid in intercepting targets in defensive military situations The fan-beam ASR-11 Airport Surveillance Radar (ASR) in Figure 1.1-2 is an example of a commercial air traffic control radar The fan-beam marine radar of Figure 1.1-3 is used for tracking ships and for collision avoidance These two fan-beam radars and those of the AN/SPS-49, HAWK Pulse Acquisition Radar (PAR), and ASR-23SS radars of Figures 1.1-4 to 1.1-6 are all examples of radars that target tracking while the radar antenna rotates at a constant rate doing target search [1] These are called track-while-scan (TWS) radars The tracking algorithms are also used for precision guidance of aircraft onto the runway during final approach (such guidance especially needed during bad weather) An example of such a radar is the GPS-22 High Performance Precision Approach Radar (HiPAR) of Figure 1.1-8 [1–4] This radar uses electronic scanning of the radar beam over a limited angle (20
in azimuth, WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR Figure 1.1-8 Limited-scan, electronically scanned phased-array AN/GPS-22 HiPAR Used for guiding aircraft during landing under conditions of poor visibility [1–3] Sixty systems deployed around the world [137] (Photo courtesy of Raytheon Company.) Figure 1.1-9 Multifunction PATRIOT electronically scanned phased-array radar used to dedicated track on many targets while doing search on time-shared basis [1–3] One hundred seventy-three systems built each with about 5000 radiating elements for front and back faces for a total of about 1.7 million elements [137] (Photo courtesy of Raytheon Company.) 10 g–h AND g–h–k FILTERS Figure 1.1-10 Multifunction shipboard AEGIS electronically scanned phased-array radar used to track many targets while also doing search on a time-shared basis [1, 3] Two hundred thirty-four array faces built each with about 4000 radiating elements and phase shifters [137] (Photo courtesy of Raytheon Company.)
in elevation) instead of mechanical scanning [1–4] An example of a wideangle electronically scanned beam radar used for air defense and enemy target intercept is the PATRIOT radar of Figure 1.1-9 used successfully during Desert Storm for the intercept of SCUD missiles Another example of such a radar is the AEGIS wide-angle electronically scanned radar of Figure 1.1-10 The Kalman tracking algorithms discussed in this book are used to accurately predict where ballistic targets such as intercontinental ballistic missiles (ICBMs) will impact and also for determining their launch sites (what country and silo field) Examples of such radars are the upgraded wide-angle electronically steered Ballistic Missile Early Warning System (BMEWS) and the Cobra Dane radars of Figures 1.1-11 and 1.1-12 [1–3] Another such wideangle electronically steered radar is the tactical ground based 25, 000-element X-band solid state active array radar system called Theater High Altitude Area WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR 11 Figure 1.1-11 Upgrade electronically steered phased-array BMEWS in Thule, Greenland [1] (Photo courtesy of Raytheon Company.) Figure 1.1-12 Multifunction electronically steered Cobra Dane phased-array radar (in Shemya, Alaska) Used to track many targets while doing search on a time-shared basis [1, 3] (Photo by Eli Brookner.) 12 g–h AND g–h–k FILTERS Figure 1.1-13 A 25,000-element X-band MMIC (monolithic microwave integrated circuit) array for Theater High Altitude Area Defense (THAAD; formerly GBR) [136, 137] (Photo courtesy of Raytheon Company.) Figure 1.1-14 Multifunction electronically steered two-faced Pave Paws solid-state, phase-steered, phased-array radar [1–3] (Photo by Eli Brookner.) Defense (THAAD; formerly called GBR) system used to detect, track, and intercept, at longer ranges than the PATRIOT, missiles like the SCUD; see Figure 1.1-13 [136, 137] Still another is the Pave Paws radar used to track satellites and to warn of an attack by submarine-launched ballistic missiles; see Figure 1.1-14 [1–3] ... Company.) WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR Figure 1.1-3 Fan-beam track-while-scan S-band and X-band radar antennas emplaced on tower at Prince William Sound Alaska (S-band antenna... where the echoes of targets and are to be expected on the second scan Such a prediction is easily made if we had an estimate of the velocity and position of targets and at the time of the first... velocity at scan n, and T the scan-to-scan period These equations of motion are called the system dynamic model We shall see later, once we understand the above simple case, WHY TRACKING AND PREDICTION