16.333: Lecture #3 Frame Rotations Euler Angles Quaternions Fall 2004 16.333 3–1 Euler Angles ? For general applications in 3D, often need to perform 3 separate rotations to relate our “inertial frame” to our “body frame” – Especially true for aircraft problems ? There are many ways to do this set of rotations - with the variations be based on the order of the rotations – All would be acceptable – Some are more commonly used than others ? Standard: start with the body frame (x,y,z) aligned with the inertial (X,Y,Z), and then perform 3 rotations to re-orient the body frame. ? Rotate by ψ about Z x ? ,y ? ,z ? ? ? Rotate by θ about y ? x ?? ,y ?? ,z ?? ? fi Rotate by φ about x ?? ? x,y,z Euler angles: – ψ ～ Heading/yaw – θ ～ Pitch – φ ～ Roll Fall 2004 16.333 3–2 Can write these rotations in a convenient form: ? ? ? ? ?? ? ? ? x ? cψ sψ 0 X X ?? y ? ? = ? ?sψ cψ 0 ?? Y ? = T 3 (ψ) ? Y z ? 0 0 1 Z Z ? ? ? ?? ? ? ? x ?? cθ 0 ?sθ x ? x ? ? y ?? ? = ? 0 1 0 ?? y ? ? = T 2 (θ) ? y ? ? z ?? sθ 0 cθ z ? z ? ? ? ? ?? ? ? ? x 1 0 0 x ?? x ?? ? y ? = ? 0 cφ sφ ?? y ?? ? = T 1 (φ) ? y ?? ? z 0 ?sφ cφ z ?? z ?? which combines to give: ? ? ? ? x X ? y ? = T 1 (φ)T 2 (θ)T 3 (ψ) ? Y ? z Z ? ?? ? cθcψ cθsψ ?sθ X ? = ? ?cφsψ + sφsθcψ cφcψ + sφsθsψ sφcθ ?? Y sφsψ + cφsθcψ ?sφcψ + cφsθsψ cφcθ Z ? Note that the order that these rotations are applied matters and will greatly change the answer – matrix multiplies of T i must be done consistently. Fall 2004 16.333 3–3 ? To get the angular velocity in this case, we have to include three terms: ? ψ ˙ about Z ? θ ˙ about y ? fi φ ˙ about x ?? which we combine to get ?ω ? ω in terms of its components in ?nal frame (body) Want to write ? – Use the rotation matrices Example: rotate ψ ˙ about Z ≡ z ? ? ? ? 0 – In terms of X, Y, Z, frame rotation rate has components ? 0 ? , ψ ˙ which is the same as in frame x ? , y ? , z ? – To transform a vector from x ? , y ? , z ? to x, y, z, need to use T 1 (φ)T 2 (θ) ? ? 0 – Similar operation for θ ˙ about y ? ≡ y ?? use T 1 (φ) on ? φ ˙ ? ? 0 Final result: ? ? ? ? ? ? ? ? ? ω x 0 0 φ ˙ ω b = ? ω y ? = T 1 (φ)T 2 (θ) ? 0 ? + T 1 (φ) ? θ ˙ ? + ? 0 ? ω z ψ ˙ 0 0 ly rojec- ,z ? Fall 2004 16.333 3–4 Visualization: Can write ? ? ˙ ? ˙ ω = φ + ? θ ˙ + ψ ? ˙ ? ˙ ? ˙ But φ,θ,ψ do not form a mutual orthogonal triad Need to form the orthogonal p tions onto the body frame x, y ? ? ? ? ? ? ? ? ω x 0 0 φ ˙ ω b = ? ω y ? = T 1 (φ)T 2 (θ) ? 0 ? + T 1 (φ) ? θ ˙ ? + ? 0 ? ω z ψ ˙ 0 0 Final form ? ˙ ω x = φ ? ψ ˙ sin θ ˙ ω y = θ ˙ cos φ + ψ cos θ sin φ ˙ ω z = ?θ ˙ sin φ + ψ cos θ cos φ With inverse: ? φ ˙ = ω x + [ω y sin φ + ω z cos φ] tan θ θ ˙ = ω y cos φ ? ω z sin φ ψ ˙ = [ω y sin φ + ω z cos φ] sec θ Fall 2004 16.333 3–5 ? Need to watch for singularities at θ = ±90 ? | | If we limit ? ψ ≤ 2π0 ≤ π π θ? 2 ≤ ≤ 2 φ < 2π0 ≤ then any possible orientation of the body can be obtained by per- forming the appropriate rotations in the order given. ? These are a pretty standard set of Euler angles ? ? ? ? ? ? Fall 2004 16.333 3–6 Quaternions ? Theorem by Euler states that any given sequence of rotations can be represented as a single rotation about a single ?xed axis ? Quaterions provide a convenient parameterization of this e?ective axis and the rotation angle ? ? ? ? ? ? b 1 b 2 b 3 ? ? ? ? E sin ζ/2 cos ζ/2 b = = b 4 where E is a unit vector and ζ is a positive rotation about E Notes:? – ?b? = 1 and thus there are only 3 degrees of freedom in this formulation as well – If b represents the rotational transformation from the reference frame a to reference frame b, the frame a is aligned with frame b when frame a is rotated by ζ radians about E ? In terms of the Euler Angles: sin θ ?2(b 2 b 4 + b 1 b 3 )= φ = arctan 2 1 + b 2 2(b 2 b 3 ? b 1 b 4 ), 1 ? 2(b 2 2 ) 2 + b 2 ψ = arctan 2 2(b 1 b 2 ? b 3 b 4 ), 1 ? 2(b 2 3 ) Pros: ? – Singularity free; Computationally e?cient to do state propagation in time compared to Euler Angles Cons: ? – Far less intuitive - less appealing ? Refs: Kuipers, Quaternions and rotation sequences, 1999 Princeton University Press.