Unified Tracking and Fusion for Airborne Collision Avoidance using Log-Polar Coordinates

Transcription

1 Unified Tracking Fusion for Airborne Collision Avoidance using Log-Polar Coordinates Dietrich Fränken Data Fusion Algorithms & Software Cassidian D-8977 Ulm, Germany Andreas Hüpper Data Fusion Algorithms & Software Cassidian D-8977 Ulm, Germany Abstract Collision avoidance applications require state estimators that are able to deliver estimates of relevant quantities sufficient quality under hard real-time constraints. In this paper, we will present a unified approach to tracking data fusion in this context. The proposed approach is easy to implement allows for a kinematics integration of data stemming from a variety of sensor types. Initialization, prediction, update in a common state space extended Kalman filter are elaborated. Simulation results show strengths weaknesses of the proposed approach. Keywords: Collision avoidance, polar tracking, log-polar coordinates, unobservable states. I. INTRODUCTION Collision avoidance separation of air traffic participants are important safety tasks, both for manned unmanned platforms, see, e. g., [] [3]. Air vehicles may be equipped different sensor suites that can be used for estimating relative position speed of intruders respect to ownship where a single sensor type will most likely not suffice to provide the required estimation accuracy [4], so fusion is a must. Measurement spaces may differ partly or completely between sensors, but a common representation of the estimated state is desirable for ease of implementation data association between contributions stemming from different sensors. And, the state should directly contain all information relevant for benchmarking the criticality of an intruder. In this paper, we consider air vehicles equipped one or more of the following sensors: 3D radar, passive optical, identification friend or foe (IFF) interrogator omnidirectional antenna, airborne collision avoidance system (ACAS). Here the particularity lies in the fact that only the radar delivers information that is complete respect to range, bearing, elevation, i. e., to polar position. In contrast to that, output of passive optical sensors lacks range information. IFF interrogator omnidirectional antenna does not provide bearing measurements the elevation may not be determinable if the intruder does not transmit its altitude (e. g., via mode C). Finally, bearing information delivered by ACAS often is barely reliable. For tracking purposes in case of angular-only measurements, the use of modified polar coordinates for the state space has been proposed [5]. When doing so, the estimation of all other components in the state vector is not influenced by the range that remains unobservable as long as the ownship does not out-maneuver the intruder. In a later publication, it has been shown that, for the D bearings-only tracking case, the replacement of the inverse of the range in that original proposal by the logarithm of the range in an extended Kalman filter in those coordinates leads to even better more robust estimation results [6]. In this paper, we will recall the fundamental properties of those log-polar coordinates argue why they are especially suitable for airborne collision avoidance applications. We will elaborate an extended Kalman filter that can be used for unified tracking fusion in airborne collision avoidance. We investigate which of the state variables are observable can thus be determined from multiple incomplete polar measurements of the different interesting types. Filter initiation methods are presented. We also talk about suitable Kalman filter updates discuss how tracks stemming from different sources can be fused. Thereafter, we derive the (somewhat tedious) Jacobians of the non-linear state propagation equations that are needed for covariance propagation. Simulation examples conclude our presentation. II. LOG-POLAR COORDINATES For the sake of conciseness, we assume throughout this paper a constant velocity motion model for the platform a (possibly noisy) motion model of the same type for the intruder. In the following presentations, positions as well as their polar representations (ranges angles) are considered in a Cartesian coordinate system constant orientation, e. g., an east-north-up (ENU) system centered in moving the ownship position. Velocities are referenced in the same coordinate system. We use the conventional (relative) elevation angle ε a (relative) bearing angle β measured in mathematical positive direction from the x axis, i. e., x r cos β cos ε y r sin β cos ε z r sin ε () 46

2 With the unitary rotation matrices cos β sin β B(β) sin β cos β () cos ε sin ε E(ε) (3) sin ε cosε the position p [x, y, z] T following eq. () can be written as p rb(β)e(ε)u x (4) the unit vector u x [,, ] T. Differentiation of () respect to time ordering of terms yields the velocity v v x v y x y rb(β)e(ε) r/r ω (5) v z z ε the projected bearing rate ω β cos ε (6) Now, log-polar coordinates use as state variables, in addition to the angles β ε, the logarithmic range ρ log(r/r) ρ r/r (7) some normalizing range R as well as the three rate components ρ, ω, ε. Herein, eq. (5) documents that the three rate components form a basis in a scale-free (normalized) velocity space the ρ axis being co-aligned the line of sight (LOS) from platform to intruder the ω ε axes being orthogonal both to LOS as well as to each other. Further benefits of using this particular set of state variables become apparent when a constant velocity motion model is considered. For now, assume that the intruder moves a constant (relative) velocity out additive noise. Denote subscript all quantities at some given time t subscript all at some time t t + T. For brevity, we will further write q [ ρ, β, ε ] T q [ ρ, ω, ε ] T (8) respect to its second component, q is not really the time derivative of q, so we abused notation here a bit for reasons of convenience in our derivations. Starting from the equations p p + T v v v (9) in view of β β + β B(β )B(β )B(β ) () β β β one obtains in combination eqs. (4) (5) the propagation correspondence r B E u x r γ γ E (u x + T q ) () when using the short-h notation E E(ε ), E E(ε ), B B(β ). From that, there follows both r γ r ρ ρ +log( γ ) () as well as γ ( + ρ T ) +(ω T ) +( ε T ) (3) cos(β )X / σ sin(β )Y / σ (4) γ [ X,Y,Z ] T σ [ X,Y, ] T (5) plus finally sin(ε )Z / γ cos(ε ) σ / γ (6) There is some singularity in these equations for γ for σ but we will except that from our further presentations. The state propagation equations are completed by looking at the rates. Due to eqs. (5) (), the identity v v yields q γ ET B T E q (7) At some points in the following derivations, it turns out to be helpful to rewrite this, using eq. (), as T q u x ET B T γ E u x (8) where the elements of E T B T are given according to eqs. (3) to (6) there thus holds c X +s Z E T B T γ γ E u x c Y σ γ (9) n σ γ n s σ c Z X () the abbreviations c cos(ε ) s sin(ε ). The use of log-polar coordinates hence produces additive increments for both log-range bearing. It is a known decisive feature that these increments as well as all other propagation equations depend neither on the range nor on the bearing of the intruder, but only on the last four components ε, ρ, ω, ε of its state. Because there holds c X + s Z + ρ T γ is independent of ε, eq. (8) in combination eq. (9) shows that, in addition to the aforementioned propagation independence on range bearing, the propagation equation of the normalized range rate ρ r/r does not depend on elevation: T ρ + ρ T γ () We note that the undisturbed straight line relative movement of the intruder uniquely determines a plane through the origin of the platform-oriented coordinate system (unless the intruder is on a collision course or on an escape course, i. e., moving 47

3 just into the direction opposite to collision). The normal vector of that plane is given by n p v () p v Now, the movement can also be described in a coordinate system whose z axis is co-aligned n. If ρ, β, ε, ρ, ω, ε denote the quantities of the state vector respect to that rotated coordinate system, then (relative) elevation ε (relative) elevation rate ε will be zero, while (logarithmic) range normalized range rate are not affected by the rotation of the coordinate system. A formal replacement of each variable by in eqs. () to (8) while honoring ε ε leads to the two-dimensional log-polar propagation equations in the variables ρ ρ, β, ρ ρ, ω. Aswehave ε, there always holds ω ω cos(θ) ε ω sin(θ) (3) ω ω + ε (4) some angle θ. An inspection of eqs. (8) (9) now not only determines θ, but also yields ω γ ω r r ω (5) This equation comes to no surprise, as it is just the conservation law of the angular momentum. So, we see that also the propagation equation of the effective bearing rate ω does not depend on elevation. Log-polar coordinates contain states that are directly relevant for collision avoidance applications. By evaluating! v T p v T (p + T c v ) (6) one finds that the time to the closest approach T c vt p v T v ρ ρ + ω + ε ρ ρ + ω (7) can be determined from the rates alone. The corresponding closest distance r c is proportional to the initial range r factor r c r γ c ω + ε ω ρ + ω + ε ρ + ω III. EXTENDED KALMAN FILTER (8) In order to implement a recursive state estimator in logpolar coordinates, an extended Kalman filter is set up. We start the discussion of the details initialization update before coming to the prediction equation. The former is more interesting due to the different variants of polar measurements we encounter, most of them being incomplete respect to position. Prediction will be common to (almost) all cases. Clearly, more advanced non-linear estimation techniques like the unscented Kalman filter [7] or the particle filter [8] can be used to obtain better estimation results, but that is beyond the scope of this paper. A. Initialization Update In the following, we identify for each measurement type the minimum number of observations required for determining that subset of initial states q q that can be determined from perfect measurements in an analytical form. Although this inherently answers the question of observability approaches to deal this question in a quantitative way by, e. g., evaluating Fisher infomation matrices as in [9] are not discussed here we will see that these equations are in general not always suitable to initialize a Kalman filter. Hence, alternatives are then pursued. As usual, there exists a variety of way for updating a filter in case of a non-linear measurement equation. We choose to convert such measurements into state space do a linear Kalman filter update those converted measurements. This method proves to be robust precise enough for our applications in most of the cases. ) 3D Radar: Consider a radar that measures both bearing elevation as well as range plus possibly range rate. Two such observations at different times can be used to determine γ r /r, given bearing elevation values, T q γ E T B E u x u x (9) then immediately delivers the missing rate quantities. We initialize the extended Kalman filter by applying this relationship to the noisy polar position measurements use possibly measured range rates r m r m to refine the initial estimate so obtained. An update bearing elevation is trivial as these are direct measurements of state variables. Range r range rate r are the remaining measurement variables a common error covariance matrix R that in general is not diagonal due to inner correlations in the plot extraction. The Jacobian for transforming the measurement sub-space r, r into the state sub-space ρ, ρ is easily computed we use J ρr RJ T ρr [ ] ρ / [ ] r J ρr [ ] ρ r r ρ (3) as equivalent error covariance in state sub-space. ) Passive Optical Sensor: Consider a sensor that delivers bearing elevation measurements out any further output. In that case, one has to take into account three measurement pairs bearing elevation obtained at different time instances t, t, t. Obviously, the range is never observable out range measurement due to the scaleinvariance of the problem, but it is possible to determine all remaining state variables, i. e., ρ, ω, ε as follows: An evaluation of eq. () for two different durations T t t T t t in combination eq. () leads to the identity T γ B E u x T E u x T γ B E u x T E u x (3) 48

4 that can be rewritten to (T T )E u x [ B E u x (B E u x ) ] [ T γ T γ ] (3) From there, γ then again eq. (9) all rates can be uniquely determined if only if the vectors B E u x B E u x are not collinear, i. e., if there are differences in bearing /or elevation. Seeing no such differences implies that the intruder is either on a collision course or on an escape course ( thus there holds w ). There is no way to distinguish between those two cases, the estimation of the time to collision based on angular measurements alone is impossible in either one. For noisy measurements, it is by no means guaranteed that eq. (3) delivers positive solutions γ γ. And, the estimates obtained by evaluating this equation are often not of sufficient quality. Here, it is better to initialize the Kalman filter by finding those initial values ε, ρ, ω, ε that maximize the (log-)likelihood of the observed measurements ε m, βi m βm i β m, ε m i i,...,n for some number n of observations pairs (minimum three). Thus, χ (εm ε ) σ ε n + i [ (ε m i ε i) σ ε + (βm i β i) σ β ] (33) is (numerically) minimized, where the ε i βi are computed by the analytic propagation equations of section II. 3) Passive Optical Sensor Size Measurements: The weakness of not being able to observe rates in case of collision/escape can be overcome by further exploitation of the images produced by the optical sensors. The image processing therein used for plot track detection can also deliver angular extent of the object. Figure. Size of object angular extent α. Consider a reference plane distance d from the origin perpendicular to the line of sight (LOS) towards the center of the intruder. Determine the one-dimensional extent e of the intruder in that plane by intersecting it the LOSs towards the borders of the intruder. It is related to the (unknown) onedimensional width w of the object in the plane parallel to the previous one (unknown) range r via e d w (34) r I. e., the range is inverse proportional to the measured extent e, but the value w d r e is unknown. But, if w d can safely be assumed to be constant over some relevant time interval for a particular intruder, then two measurements e e(t ) e e(t ) can be used to compute log(e /e )log(r /r )log(r /R) log(r /R) (35) So, in view of eq (7), log(e /e ) t t ρ ρ t t ρ(t ) (36) provides a suitable pseudo-measurement of ρ that can directly be used in a linear Kalman filter update step. We thus note that the log-range comes into play quite naturally when incorporating size information into the passive optical tracking (in fact, two-dimensional log-polar coordinates are quite popular in video tracking applications). Regression models in log(r/r) taking into account more than two size measurements can be used to obtain improved pseudo-measurements of ρ for sensors high update rate. These pseudo-measurements have uncorrelated errors if the sets of size measurements used are disjoint. Finally, if the measurement report by the sensor is a (one-dimensional) angular extent α asshowninfig., eq. (36) is to be replaced by log ( tan(α/) tan(α /) ) t t ρ(t ) (37) With the additional information of a normalized range rate measurement ρ, all rates are always observable upon the second detection. We note that, from eqs. () (), the norm γ ut x E T B E u x ρ T c cos(β )c + s s ρ T (38) is immediately given, the rest then follows as before. Again, the γ resulting from this equation is not guaranteed to be positive for noisy measurements. But, adjusting χ in eq. (33) to incorporate also the differences between observations ρ m i predictions ρ i is easy, so we can again initialize based on a maximized (log-)likelihood. The described approach to make use of size information is sensitive to changes in aspect angle of the intruder. Sensitivity can be reduced (although not completely eliminated) by using, instead of two pseudo-measurements based on onedimensional extents in orthogonal directions, a single pseudo measurement that computes e as the square root of the number of pixels in the reference plane (or α as the square root of some measured solid angle). 4) IFF Omnidirectional Antenna or ACAS: The systems considered in this subsection provide at least range information plus possibly altitude difference ( thus something equivalent to logarithmic range elevation), but absolutely no or only barely reliable bearing information. We start the case of range-only measurements a large number of publications treats this challenging tracking problem, see, e. g., [9] the references cited therein 49

5 note that two successive range measurements r r provide a value γ r /r. Now, examine From that, there follows γ i (+ ρ T i ) +( ω T i ) (39) T γ T γ (T T )+T T (T T ) ρ (4) which gives a unique solution for ρ. With this, eq. (39) also delivers ω ω + ε, thus, time to closest approach closest distance can be completely be determined from three range values alone. However, in analogy to what has been observed in the previous sections, the value ω in eq. (39) can become negative for noisy measurements. Consequently, we minimize respect to initial values r ω a χ as in eq. (33), but now using differences between measured ranges ri m predicted ranges r i obtained from eqs. () (5). Without elevation information, an extended Kalman filter running in a two-dimensional log-polar coordinate system as described in section II can be used. The tilt of the rotated coordinate system respect the platform-oriented ENU system is completely unknown but constant. This fact is easily honored in association data fusion. In case elevation or altitude difference measurements are available in addition, a distinction between elevation rate (magnitude of) projected bearing rate is possible. The analytic equations resulting in this case are omitted here as is a detailed discussion of the corresponding maximum-likelihood initialization. The discussion of the range/elevation case is concluded by noting that, for transforming from measurement sub-space range altitude difference to state sub-space logarithmic range elevation, the Jacobian [ ρ ε is to be used. ] / [ r z ] [ r tan(ε) / cos(ε) ] (4) B. Some Remarks on Tracking Fusion Before completing the picture the covariance prediction equations of the filter, a few words regarding a unified tracking data fusion approach are at h. In the above cases of optical sensors ( or out size measurements) IFF/ACAS, we have identified some states that are not observable at all (range for optical, bearing for IFF altitude ACAS, bearing elevation for IFF out altitude). Those unobservable states do not impact the derived prediction equations, so arbitrary default values can be assumed in state prediction out corrupting any of the results for the observable states as well as for the increments of the remaing ones. But, we will see that the range (being unobservable for optical sensors) does have an influence on the covariance prediction in presence of process noise. Then, the initially assumed range must be carefully chosen (e. g., honoring a maximum detection range of the sensor) in order to get coherent (although not correct) range estimates in the further process. So, we suggest to maintain the range in the state throughout. In particular, this makes it easier to react upon ownship maneuvers when range does become observable (by kinematic ranging). Like in all fusion systems, several general architecture approaches exist for systems used in airborne collision avoidance. Due to the different nature of the measurements their different update rate, it seems a good idea to first do plot-based data-association tracking individually for each sensor to perform afterwards a track-to-track data association where all observable states can be taken into account. E. g., for association of passive optical including size measurements IFF based on range altitude difference, the quantities ε, ρ, ω, ε can be evaluated. Passive optical IFF based out elevation information can be compared by using ρ ω vs. ω + ε. Table I MEASURED AND OBSERVABLE STATES Source Measured states Observable states Radar r, β, ε, r ρ, β, ε, ρ, ω, ε Optical, no size β, ε β, ε, ρ ( ), ω, ε Optical size β, ε, ρ β, ε, ρ, ω, ε IFF omni/acas, no altitude r ρ, ρ, ω IFF omni/acas altitude r, z ρ, ε, ρ, ω, ε ( ) observable only if not on collision/escape course Fused kinematics estimates can be obtained either by running a common extended Kalman filter associated plots or, ignoring the correlation between estimation errors in case of process noise, by convex combination of the sensor level tracks. In the latter case, the fact that certain data is not observable therein has to be taken into account just like the fact that the sign of ω or ω is unknown for the IFF/ACAS track. Table I summarizes measured observable states for the different types of sources considered in this paper. Of course, in such a hierachical structure, there is the possibility to feed back, e. g., existence information from the fusion center to the sensor level tracking systems in order to help track initiation or to avoid premature track drops in case of low detection probabilities. C. Prediction For the extended Kalman filter, we use the Jacobians of the non-linear prediction equation to approximate the variance after prediction for a target model out noise. Additive noise on the constant velocity motion in Cartesian space is incorporated by suitable Jacobians, too. The following equations represent merely one of many possible ways of writing these Jacobians. ) Noise-free Motion: Our derivation of the Jacobians starts noting the auxiliary relationships γ γ T γ γt γ γ γ σ σ T σ σt γ γ σ (4) (43) 5

6 as well as Furthermore, there holds γ ε (44) σ X Z ε σ γ ε Z X (45) (46) finally, in view of eq. (), γ T E (47) q Now, we can compute the Jacobian J for the prediction equation from log-polar to log-polar space described in the previous section. It is a two by two block matrix, its top left block is given by J qq q Y Z σ (48) q X σ The only non-zero entries in the bottom left block J qq q / q are again, derivatives respect to ρ β are zero we have discussed that ρ does not depend on ε, either ω n ε T γ σ 3 ε Y ε T σ 3 (49) The top right block can be evaluated by noting that eq. (47) there holds computing plus as well as ε γ One obtains J q q T J q q q q q γ T E (5) ρ γ / γ γt γ γ γ (5) β γ sin(β ) [ γ cos(β ) Y,X, ] σ (5) sin(ε ) [ γ cos(ε ) X Z, Y Z, σ ] σ γ (53) + ρ T γ ω T γ ε T γ Y c X Y s σ σ σ n σ γ Z ω T σ γ m σ γ (54) m c σ + s Z X (55) In combination eqs. (8). (9), J q q q q q γ T E (56) can be used to compute the rows of J q q as [ ] T ρ (+ ρ T )ω T q γ 4 [ ] T ω q [ ] T ε q (+ ρ T ) (ω T ) ( ε T ) γ 4 (+ ρ T ) ε T γ 4 c Y ( σ (+ ρ T )+c X γ ) σ 3 γ 3 σ γ Y ( σ + γ ) σ 3 γ 3 c Y ( σ ε T s X γ ) σ 3 γ 3 c ZY γ +(+ ρ T )n σ σ 3 γ 4 ω T (n γ (+ ρ T )Z σ ) σ 3 γ 4 (X σ c s Z Y ) γ + ε Tn σ σ 3 γ 4 (57) (58) (59) The Jacobians are used in the extended Kalman filter to propagate estimation error covariance. Starting from some P, the propagated covariance is approximated as P J P J T (6) The equations for the Jacobians as derived above are somewhat tedious. Alternatively, they can be expressed by honoring transformations from to Cartesian space. The Jacobian for getting to Cartesian from log-polar is a lower block-triagonal matrix. Its blocks can be computed by defining the matrix T T(β,ε) B(β)E(ε) (6) verifying the identities rt rt ρ (6) cos ε rt β rt cos ε sin ε sinε (63) rt ε rt (64) From there, one gets J pq p q rt cosε (65) as well as J v q v rt (66) q 5

7 plus J vq v ρ ω cos ε ε q rt ω ρ cos ε ε sin ε ε ωsin ε ρ (67) The elements of the lower triangular Jacobian for transformation to log-polar from Cartesian space are thus given by J qp q p / cos ε T T (68) r as well as plus J qv q v r TT (69) J qp q p ρ ω ε ω ε tan ε ρ T T (7) r ε ω tan ε ρ In view of the propagation equation (9), the Jacobian J is equivalently expressed by [ ][ ][ ] Jqp J I T I Jpq (7) J qp J qv I J v q J v q where I is the identity matrix of dimension three. It should be noted that again, for non-observable states, default values can safely be used to perform the transformations when computing the propagation of the covariance in case of zero process noise. ) Motion Process Noise: The Jacobian of the transformation from Cartesian to log-polar space also is to be used if the motion of the intruder is assumed to be not strictly constant velocity but if some additional process noise is honored. Typically, a white acceleration process noise covariance [ ] DT Q 3 /3 DT / DT (7) / DT the matrix of noise levels q xy D q xy (73) q z in Cartesian space is assumed. In order to incorporate it, eq. (6) is replaced by P J P J T [ Jqp + J qp J qv ] [ J T Q qp J T q ] p J T q v (74) Due to the assumed rotational symmetry of the process noise covariance, bearing again has no influence on the transformed covariance values in log-polar space, but the range does. IV. SIMULATION RESULTS In order to validate the presentations above, we have simulated two different scenarios intruders performing noise-free constant velocity motions. Initial position in both scenarios was p T [3, 3, 75] m, but the scenarios differed in initial speed, which was v T [, 6, 5] m/s for the first one v T [ 48, 4, ] m/s for the second. Here, the intruder was basically on a passing course in the first scenario, while its course was close to collision ( a closest distance of 95 m) in the second. We ran the described extended Kalman filters (assuming some small process noise) for various cases incorporating either single or multiple sensor source inputs. We used a common sample time T s, the following accuracies have been chosen: Radar σ r m, σ β σ ε.5deg, σ r 5m/s, optical σ β σ ε.deg, σ ρ.5deg/s, IFF/ACAS σ r 3m, σ z 4m. Figs. 3 show the simulation results for some typical single runs as well as root mean square (RMS) errors resulting from Monte Carlo runs each. We note that most of the outcomes are as could have been expected. In general, fused results are better than that obtained using only one of the contribution sensors. We confirm that observability of ρ is weak for the optical sensor out size information as long as the intruder virtually remains on a collision course (for larger ranges). Once significant values of bearing rate /or elevation rates have occurred (for smaller ranges), ρ is estimated high quality. We note that, not surprisingly, the gain in performance using additional size measurements is driven by the quality of those measurements where pixel effects play a major role for intruders being far away from the sensor. And, we see that poor quality size measurements can even lead to some degradation of the estimates in cases where the rates are strongly observable by means of angular values only (for the passing intruder as well as after the pass-by for the close-tocollision intruder). For the passing intruder, we see the usual convergence behavior of a Kalman filter for the observable rates. In contrast to that, observability becomes increasingly weaker as the almost-colliding intruder comes closer (until it has passed by). Here, the filter does not yield satisfactory results for the effective bearing rate in the range-only case, future investigations have to show whether estimation accuracy can be improved by more advanced (but still not computationally too deming) non-linear filtering techniques. V. CONCLUSION We have investigated a unified approach to tracking data fusion for airborne collision avoidance applications that provides a simple way of incorporating measurements from a large variety of sensors into the estimation process. The presented approach is based on a consequent use of a state representation in log-polar coordinates that are known to directly contain values relevant for collision avoidance. We have elaborated initialization, prediction, update of an 5

8 5 8 Range.6 Bearing. Elevation Time to closest approach r / m 4 β/π ε/π. T c / s Normalized range rate.4 5 Effective bearing rate Elevation rate. 5 Closest distance 4 (dr/dt)/r / (/s).... (dβ/dt)cos(ε) / (/s).... (dε/dt) / (/s).. r c / m Range error.5 x 3 Bearing error.5 x 3 Elevation error Time to closest approach error 5 8 RMS / m 5 RMS / π.5 RMS / π.5 RMS / s x 3 Norm. range rate error 5 4 x 3 Eff. bearing rate error 5 4 x 3 Elevation rate error 5 Closest distance error 5 RMS / (/s) 4 3 RMS / (/s) 3 RMS / (/s) 3 RMS / m Figure. Single run results (two top rows) measurements (circles) estimates (dots) for single-sensor runs as well as root mean square errors (solid lines) from Monte Carlo runs (two bottom rows) for an intruder on a passing course. Used sensors/sensor combinations are radar, optical w/o size, optical size, IFF w/o altitude, IFF w/o altitude plus optical w/o size, radar plus optical w/o size, radar plus optical size plus IFF w/o altitude, truth. 8 Range. Bearing.5 Elevation Time to closest approach r / m 4 β/π.6.4 ε/π.5 T c / s 5. 5 (dr/dt)/r / (/s) 5 Normalized range rate (dβ/dt)cos(ε) / (/s). 5 Effective bearing rate.... (dε/dt) / (/s).5 5 Elevation rate.5 r c / m 5 5 Closest distance Range error.5 x 3 Bearing error.5 x 3 Elevation error Time to closest approach error 5 8 RMS / m 5 RMS / π.5 RMS / π.5 RMS / s Norm. range rate error.5 5 Eff. bearing rate error.3 5 Elevation rate error. 5 Closest distance error RMS / (/s)..5 RMS / (/s)..5. RMS / (/s).6.4 RMS / m Figure 3. Single run results (two top rows) root mean square errors from Monte Carlo runs (two bottom rows) for an almost-collision course. extended Kalman filter running in these coordinates given according simulation results. Further studies will concentrate on the use of an unscented Kalman filter the question about how that may further improve estimation results. REFERENCES [] Airborne collision avoidance system (ACAS). Eurocontrol. [Online]. Available: page/acas Startpage.html [] Unmanned aircraft systems ATM collision avoidance requirements. Eurocontrol. [Online]. Available: content/public/documents/cause report v3.pdf [3] Mid air collision avoidance system (MIDCAS). [Online]. Available: [4] A. R. Lacher, D. R. Maroney, A. D. Zeitlin. Unmanned aircraft collision avoidance Technology assessment evaluation methods. The Mitre Corporation. [Online]. Available: tech papers/tech papers 8/7 95/7 95.pdf [5] V. Aidala S. Hammel, Utilization of modified polar coordinates for bearings-only tracking, IEEE Trans. Automatic Control, vol. 8, no. 3, pp , 983. [6] B. L. Scala M. Morele, An analysis of the single sensor bearingsonly tracking problem, in Proc. ISIF Intern. Conf. on Information Fusion (FUSION), 8, pp [7] S. J. Julier J. K. Uhlmann, Unscented filtering nonlinear estimation, in Proceedings of the IEEE, vol. 9, no. 3, 4, pp [8] B. Ristic, S. Arulampalam, N. Gordon, Beyond the Kalman Filter. Artech House, 4. [9] B. Ristic, S. Arulampalam, J. McCarthy. Target motion analysis using range-only measurements: Algorithms, performance application to Ingara ISAR data. DSTO TR 95. [Online]. Available: /DSTO-TR-95%PR.pdf 53