decomp - = a 2 x vθ cos θ a x a y vθ-a z sin θ a x a z...

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ME/ECE 739 Rotation Matrix Decompositions - Professor Nicola J. Ferrier 1 Roll,Pitch Yaw The row-pitch-yaw rotations are represented as: R = Rot [ z, φ ] Rot [ y, θ ] Rot [ x, ψ ] R = cos φ cos θ cos φ sin θ sin ψ - sin φ cos ψ cos φ sin θ cos ψ + sin φ sin ψ sin φ cos θ sin φ sin θ sin ψ + cos φ cos ψ sin φ sin θ cos ψ - cos φ sin ψ - sin θ cos θ sin ψ cos θ cos ψ The solution 1 for θ in ( - π/ 2 , π/ 2): φ = atan2( r 21 , r 11 ) (1) θ = atan2( - r 31 , q r 2 32 + r 2 33 ) (2) ψ = atan2( r 32 , r 33 ) (3) and the solution for θ in ( π/ 2 , 3 π/ 2): φ = atan2( - r 21 , - r 11 ) (4) θ = atan2( - r 31 , - q r 2 32 + r 2 33 ) (5) ψ = atan2( - r 32 , - r 33 ) (6) (7) Euler Parameterization R = Rot [ z, φ ] Rot [ y, θ ] Rot [ z, ψ ] R = cos φ cos θ cos φ - sin φ sin ψ - cos φ cos θ sin ψ - sin φ cos ψ cos φ sin θ sin φ cos θ cos ψ + cos φ sin ψ - sin φ cos θ sin ψ + cos φ cos ψ sin φ sin θ - sin θ cos ψ sin θ sin ψ cos θ Decomposition solution for r 13 6 = 0 and r 23 6 = 0 (singularity when sin θ = 0): φ = atan2( r 23 , r 13 ) (8) θ = atan2( q r 2 13 + r 2 23 , r 33 ) for θ (0 , π ) (9) ψ = atan2( r 32 , - r 31 ) (10) OR φ = atan2( - r 23 , - r 13 ) (11) θ = atan2( - q r 2 13 + r 2 23 , r 33 ) for θ ( - π, 0) (12) ψ = atan2( - r 32 , r 31 ) (13) Angle-Axis Decomposition R = Rot [ k = ( a x , a y , a z ) , θ
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Unformatted text preview: ] = a 2 x vθ + cos θ a x a y vθ-a z sin θ a x a z vθ + a y sin θ a x a y vθ + a z sin θ a 2 y vθ + cos θ a y a z vθ-a x sin θ a x a z vθ-a y sin θ a y a z vθ + a x sin θ a 2 z vθ + cos θ where vθ = versine( θ ) = 1-cos θ Decomposition (singularity at θ = 0 , π, . .. and there are two solutions for ± θ ): θ = arccos (( r 11 + r 22 + r 33-1) / 2) (14) a T = 1 2 sin θ [ r 32-r 23 , r 13-r 31 , r 21-r 12 ] T for θ 6 = π, (15) 1 We use the quadrant specific arctan function atan2( y, x ) = tan-1 ( y/x ). Caution! atan2 is defined differently in various software packages - check specific definition, i.e. atan2( y, x ) = tan-1 ( y/x ) vs. tan-1 ( x/y )...
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This note was uploaded on 08/08/2008 for the course ME 739 taught by Professor Ferrier during the Spring '06 term at University of Wisconsin.

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