Methods and apparatus to determine a target location in body coordinates

A processor for determining a position of a doppler radar target in aircraft body coordinates, said processor configured to:

• calculate values for doppler circle equations, in doppler coordinates, based upon a range to the target, a vehicle velocity, and a center frequency and bandwidth of a doppler swath filter;
• calculating an interferometric circle equations in body coordinates based upon a range to the target, and an interferometric angle;
• transform the doppler circle equations into body coordinates utilizing received pitch, roll and yaw information;
• and
• calculate an intersection of the interferometric circle equations with the transformed doppler circle equations, the intersection being the position of the target in aircraft body coordinates.

A body coordinate processor comprising

• a doppler circle equation processor configured to calculate values for doppler circle equations, in doppler coordinates, based upon a range to the target, a vehicle velocity, and a center frequency and bandwidth of a doppler swath filter;
• an interferometric circle equation processor configured to calculate an interferometric circle equations in body coordinates based upon a range to the target, and an interferometric angle;
• a doppler to body coordinate transformation processor configured to transform the doppler circle equations into body coordinates utilizing received pitch, roll and yaw information;
• and
• an intersection processor configured to calculate an intersection of the interferometric circle equations with the transformed doppler circle equations, the intersection being the position of the target in aircraft body coordinates.

A method for determining a position of a doppler radar target in body coordinates using a body coordinate processor, the body coordinate processor including a doppler circle equation processor, an interferometric circle equation processor, a doppler to body coordinate transformation processor, and an intersection processor said method comprising:

• calculating a radius of a doppler circle, Rd, according to Rd=target range×sin(β), and calculating a distance of the doppler circle, Xd, from the aircraft according to Xd=target range×cos(β), where β=cos−1((Fc×L)/(2×V)), and Fc is a swath filter center frequency, V is velocity, L is wavelength, and β is an angle with respect to a line of flight with the doppler circle equation processor;
• calculating an interferometric circle radius, Ri, according to Ri=target range×cos(a), and calculating a location of the interferometric circle on a Ym axis according to Ym=target range×sin(a) with the interferometric circle equation processor, the processor configured to receive a target range and an interferometric angle, a;
• determining a velocity vector in body coordinates, from navigation data, N, (in pitch, roll, and yaw according to [LeftBracketingBar] V X N V Y N V Z N [RightBracketingBar] ⁢ [LeftBracketingBar] TRANSPOSE ⁢ ⁢ MATRIX [RightBracketingBar] = [LeftBracketingBar] V X BODY V Y BODY V Z BODY [RightBracketingBar] ,
• where the transpose matrix is given by [LeftBracketingBar] cos ⁡ ( ψ ) ⁢ cos ⁡ ( θ ) - sin ⁡ ( ψ ) ⁢ cos ⁡ ( θ ) sin ⁡ ( θ ) cos ⁡ ( ψ ) ⁢ sin ⁡ ( θ ) ⁢ sin ⁡ ( φ ) - sin ⁡ ( ψ ) ⁢ cos ⁡ ( φ ) - sin ⁡ ( ψ ) ⁢ sin ⁡ ( θ ) ⁢ sin ⁡ ( φ ) - cos ⁡ ( ψ ) ⁢ cos ⁡ ( φ ) - cos ⁡ ( θ ) ⁢ sin ⁡ ( φ ) cos ⁡ ( ψ ) ⁢ sin ⁡ ( θ ) ⁢ sin ⁡ ( φ ) + sin ⁡ ( ψ ) ⁢ sin ⁡ ( φ ) cos ⁡ ( ψ ) ⁢ sin ⁡ ( φ ) - sin ⁡ ( ψ ) ⁢ sin ⁡ ( θ ) ⁢ sin ⁡ ( φ ) - cos ⁡ ( θ ) ⁢ cos ⁡ ( φ ) [RightBracketingBar] ,
• and ψ is azimuth, θ is pitch and Φ is roll, and determining velocity unit vectors, which are direction cosines, in body coordinates according to ax=Vx/(Vx2+Vy2+Vz2)½, ay=Vy/(Vx2+Vy2+Vz2)½, and az=Vz/(Vx2+Vy2+Vz2)½, with the doppler to body coordinate transformation processor;
• and
• calculating body coordinates according to X1=D×ax, Y1=D×ay, Z1=D×az, where a velocity vector D, is calculated as R×cos(β), and β=cos−1(Fcx×/2×V), where β is the doppler angle, Fc is the swath filter center frequency, R is the range to the target, V is (Vx2+Vy2+Vz2)½, and L is the wavelength of the radar, and calculating target coordinates according to y=R×sin(A), where A is a measured phase angle in body coordinates, z=(−b±(b2−4ac)½)/(2×a), where a=1+(Z1K1)2, b=(−4Z1×KT/(2X1)2), and c=(KT/2X1)2−KA, KA is calculated as (R×cos(A))2, KB is calculated as (R×sin(B))2, KY=(y−Y1)2, KT is calculated as KT=KA+KY−KB+X12+Z12, and x=(KA−z2)½, with the intersection processor.