Spatial Alignment of Passive Radar and Infrared Image Data on Seeker

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1 Available online at Procedia Engineering 4 (011) International Conference on Advances in Engineering Spatial Alignment of Passive Radar and Infrared Image Data on Seeker Jin ang a, Rui Ding b, a* a Department of Automation, eijing University of Posts and elecommunications, eijing, China b Department of Electronic, Capital Normal University, eijing, China Abstract he line-of-sight (LOS) and LOS rate is essential for proportional navigation. However, on the hybrid seeker with passive radar sensor and infrared image sensor, they can not be obtained directly for they are observed in different observing space. hen, observations obtained from passive radar sensor and infrared image sensor need to be share a consistent index of space, that is spatial alignment. he exact LOS and LOS rate dynamic models of passive radar sensor and infrared image sensor are constructed respectively. 011 Published Elsevier Ltd. Open access under CC Y-NC-ND license. Selection and/or peer-review under responsibility of ICAE011. Keywords: LOS; LOS rate; Spatial Alignment 1. Introduction Most modern homing missiles make use of PN (proportional navigation) guidance law based on LOS (line of sight) vector and its rate [1][]. he hybrid seeker system utilizes passive radar sensor and infrared image sensor to get the information of LOS and LOS rate. More precise result can be obtained fusing data from the two kinds of sensor [3]. However, the observed signals from passive radar and for the infrared image sensor have different form and norm. o accomplish sensor fusion, it is necessa for different sensors to agree on a common notion of space. hat is, observations obtained from different sensors need to be share a consistent index of space.. LOS and LOS rate from passive radar he principle of angle measurement from passive radar is that, two or more independent antennae accept the signal from target in the same time, and then compare the measurements to get the relative * Corresponding author.el.: address: tangjin@bupt.edu.cn Published Elsevier Ltd. doi: /j.proeng Open access under CC Y-NC-ND license.

2 650 Jin ang and Rui Ding / Procedia Engineering 4 (011) angle information of the target. his angle measurement value is the vector relative the place where the antennae are located, while the LOS is relative to the missile body coordinate. herefore, the angle measurement need transformed to LOS, and then LOS rate is acquired for the proportional navigation [4]. First of all, the body coordinate frame OXYZ is built, which is glued on the missile body. Define a right-handed coordinate system as showed in the figure 1, where OX always points to the missile nose direction along the roll axes, OY is the axis of pitch, and OZ is the axis of row which completes the mutually orthogonal right-handed coordinate system. Providing that there is an inherent angleθ betwe en the signal reference plan of passive radar and the body coordinate frame, which is usually decided the mechanical position of antennae, then the radar signal coordinate frame OXYZ R R R can be built. One way to describe the rotation (the change in orientation) of a rigid body is means of Euler angles. hen, the radar signal coordinate frame can be described with that OXR has same direct ion with OX, while OYR and OZR can be regard as the rotated OY and OZ angle - θ (the rotation directio ns are represented the right hand rule). hen, the coordinate-transformation matrix from the radar signal coordinate frame OXYZ R R R to the body coordinate frame OXYZ is the following: LR = 0 cosθ sinθ 0 sinθ cosθ (1) It is quit convenient to get the position vector ( x,, ) b yb zb in the body coordinate frame from the target position vector ( x, y, z ) in the radar signal coordinate frame through the formula: r r r ( x, y, z ) = L R( x, y, z ) () b b b r r r However, through passive radar, the relative distance between the missile and its target is unobservable, and the only the angle bias vector (, ) is measurable. the angle bias vector (, ) means that the target direction can be reached twice successive rotation from the radar coordinate frame OXYZ R R R: first, rotate OXYZ R R R around the axes OZR with angle, and then rotate the new "OXYZ R R R" around it new axes "OY R" with angle to acquire the target coordinate frame OXYZ P P P (the rotation directions are represented the right hand rule), and the angle has relation to location vector ( x, y, z ), and comply to: r r r tan = y / x = z / x + y (3) r r, tan r r r Substitution of (3) in () lead to: x 1 b x r 1 yb 0 cos sin tan xr cos tan sin tan 1 tan = θ θ = θ + θ + xr = A xr z 0 sin cos b θ θ tan 1 tan x r sinθtan cosθtan 1 tan + + (4) he line of sight (LOS) between missile and target in the body coordinate frame is defined the same way as that in the radar signal coordinate frame, which is: tan = y / x = z / x + y (5) b b, tan b b b hen from formula (4) (5), the line of sight between missile and target is: arctan( ) A LOSradar = = arctan( / 1 + A ) (6)

3 Jin ang and Rui Ding / Procedia Engineering 4 (011) Since the coordinate frame OXYZ P P P is successively rotated from the radar coordinate frame OXYZ R R Rthrough the Euler angle (, ), then the coordinate-transformation matrix LPR is: cos 0 sin cos sin 0 LPR = R( ) R( ) = sin cos 0 (7) sin 0 cos hen with corresponding reverse rotate, the radar coordinate frame OXYZ R R R can also be 1 1 transformed from OXYZ P P P with matrix LRP : LRP = R ( ) R ( ).y the basic Euler angle equation [5, the relation between the Euler angle velocity and of the angular velocity vector is : ω x x ω n ) ω bp 0 (si rp 0 0 (si 0 n ) R y 1 y 1 ω ( ) ω + 0 = (cos RP = ωrp = R + 0 = (cos ), ωp = bp = R ( ) ) z z ω 0 ω rp bp 0 R RP,Where represents the angular velocity vector of the target coordinate relative to the radar coordinate, and ωp represents the angular velocity vector of the target coordinate relative to the missile body coordinate. From (), the equation between them is: ωp = LRωRP. Finally, the LOS rate can be got: (sin / sin ) LOSRradar = = (sinθcos ) (cos ) θ (8) + Figure 1. Geomet of passive radar Figure. Geomet of infrared image sensor 3. LOS and LOS rate from infrared image As the infrared image sensor is mounted on the two-degree-of-freedom (-DOF) gimbals, more parameters are need for the LOS and LOS rate [6][7]. he body coordinate frame OXYZ is built same as that in section ; the gimbals coordinate frame OXGYGZG, whose axes OXG points to the direction same with gimbals, can be reached twice successive rotation from the body coordinate frame OXYZ : first, rotate OXYZ around the axes OZ with angle η, and then rotate the new "OXYZ" around it new axes "OY" with angle λ (the rotation directions are represented the right hand rule).he infrared image sensor is fixed with the gimbals and the optical axis is regarded the same as the gimbals axis OXG for convenience. y using sensor information of the target position, the gimbals are controlled to point to the target. hat is the gimbals axis OXG always approaches the target direction OXP. However, Even if the target is tracked sensor pointing control, it is not necessarily on the optical axis [8]. he target and the seeker vehicle platform (i.e., the body) may be maneuvering and the sensor stabilization and pointing cannot be made perfect. Consequently, the target position may be somewhere off the optical axis, as illustrated in figure.

4 65 Jin ang and Rui Ding / Procedia Engineering 4 (011) A new coordinate system OXYZ P P P is introduced in such a way that the target is on the OXP axis. his system is further defined two Euler rotations of the gimbals system to the target: first (yaw) about the axis OZG, the (pitch) about the subsequent axis "OY G".he target is then on the axis "OX G", a condition that determines the size of the rotations. Since the OXp axis is pointing to the target permanently, we call this system the target coordinate frame. he target angular position (, ) is measured the sensor. For an IR sensor this is accomplished image processing in combination with a detection process. he angular position is then determined from the position of a certain detector element in the focal plane. With the Euler rotate angle( λη),, the coordinate-transformation matrix from the gimbals coordinate frame OXYZ to the body coordinate frame OXGYGZG is the following: cosλ 0 sin λcosη sin η 0 G ()( λ η) sin η cosη 0 (9) L = R R = sin λ 0 cosλ he coordinate-transformation matrix from the body coordinate frame OXGYGZG to the gimbals 1 1 coordinate frame OXYZ is reverse of (9): LG = R () η R () λ.if the target location can be defined as r G = ( xg, yg, zg) in the gimbals coordinate frame, while r (, b, b) = xb y z in the body coordinate frame, then we can get: r =L G r G.And the target angular (, ) observed can be deduced from the target position vector: tan = yg / xg, tan = - zg / xg + y,and then lead to: g (cos ηcos λ sin ηtan (cos ηsin λtan ) 1 tan ) xb cos ηcos λ sin η cos ηsin λ x g C y b sin ηcos λ cos η sin ηsin λ (tan ) x ( sin ηcos λ+ cos ηtan (sin ηsin g = = λtan ) 1+ tan ) x g= D x z g sin 0 cos b λ λ (tan ) 1+ tan x E g sin λ (cosλtan ) 1+ tan (10) he equation between the line of sight (LOS) between missile and target in the body coordinate frame and the position vector is presented in (5). From (5) and (10), we get : arctan( / ) arctan( / ) yb xb D C (11) LOSimage = = = arctan( zb / xb yb ) + arctan( E/ C + D ) With the Euler rotate angle (, OX Y ), the coordinate-transformation matrix between the gimbals OXYZ coordinate frame G GZG and the body coordinate frame P P P is the following: LPG = R ( ) R( ), 1 LGP = R ( ) 1 R ( ).From basic kinematics, the following equation holds: Where ω P G ωp = ωg + ωgp = ωg +LGωG P is the angle velocity of target relative to missile body in the body coordinate frame, is the angle velocity of gimbals relative to missile body in the body coordinate frame, ω GP velocity of target relative to gimbals in the body coordinate frame. On the other hand, the relation between the Euler angle velocity and of the angular velocity vector can be got from basic Euler angle (sin (sin ) ) ηλ (sin ) equation: G ω (13) P = (cos ), G (cos ηλ ), GP (cos ) ω = ω =. η Substitution of (13) in (1) lead (sin ) (sin (sin ) (cos cos sin sin cos ) (cos sin ) ) ηλ (sin ) ηλ+ η λ + η η λ F to: (cos ) (cos ) (cos ) (cos ) ηλ G ηλ+ ( sin ηcos λsin + cos ηcos = + L = = ) + (sinηsin λ) G (14) η η+ (sin λsin ) + (cos λ) H (1) ω G is the angle

5 Jin ang and Rui Ding / Procedia Engineering 4 (011) he LOS rate in the missile body coordinate frame is deduced from (14), and that is: G / cos LOSRimage = = H (15) 4. Conclusion Since hybrid seeker has passive radar and IR image sensor, the target s angle position could be estimated more precisely with data fusion. However, the observations from two kinds of sensor are different in space and time. hen the exact LOS and LOS rate dynamic models of passive radar sensor and infrared image sensor are constructed respectively. Form (6) and (11), the angle information of target from radar and IR image is transformed to LOS in the missile body coordinate; and form (8) and (15), the angle velocity information of target from radar and IR image is transformed to LOS rate in the missile body coordinate. For application, the simplifying of calculation need more work. References [1] R.R.Allen, and S.S.lackman, Angle-only racking with a MSC Filter, IEEE/AIAA 10th Digital Avionics Systems Conference, Los Angeles, pp , Oct [] aek lyul Song, and ae Yoon Um, Practical Guidance for Homing Missiles with earings-only Measurements, IEEE ransactions on Aerospace and Electronic Systems, Vol 3, No. 1, pp , [3] David L. Hall, An Introduction to Multi-sensor Data Fusion, Proceedings of the IEEE, VOL. 85, NO. 1, pp. 6-3, Janua 1997 [4] Nesline. F W, and Zarchan. P, Line-of-sight Reconstruction for Faster Homing Guidance, Journal of Guidance, Control, and Dynamics,Vol 8, pp. 3-8, [5] A. L. Schwab,J. P. Meijaard, How to draw Euler angles and utilize Euler parameters, Proc. IDEC/ CIE 006, ASME 006 International Design Engineering echnical Conferences &Computers and Information in Engineering Conference September 10-13, 006. [6] Jacques Waldmann, Line-of-sight Rate Estimation and linearizing Control of an Imaging Seeker in a actical Missile Guided Proportional Navigation, IEEE ransactions on control systems technolo, Vol.10, No. 4, pp , July 00. [7] ertil Ekstrand, racking Filters and Models for Seeker Applications, IEEE ransactions on aerospace and electronic systems Vol. 37, No. 3 July 001. [8] Zhe Lin, Yu Yao, Ke-mao Ma, he Design of LOS Reconstruction Filter for Strap-down Imaging Seeker, Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, Guanhou, 18-1 August 005.

3D Transformations. CS 4620 Lecture Kavita Bala w/ prior instructor Steve Marschner. Cornell CS4620 Fall 2015 Lecture 11

3D Transformations. CS 4620 Lecture Kavita Bala w/ prior instructor Steve Marschner. Cornell CS4620 Fall 2015 Lecture 11 3D Transformations CS 4620 Lecture 11 1 Announcements A2 due tomorrow Demos on Monday Please sign up for a slot Post on piazza 2 Translation 3 Scaling 4 Rotation about z axis 5 Rotation about x axis 6