{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

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 , 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 , - 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 = 0 and r 23 = 0 (singularity when sin θ = 0): φ = atan2( r 23 , r 13 ) (8) θ = atan2( 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( - 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
This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern