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Euler angle sequences

(Definition)

An Euler angle sequence is a rotation matrix that is completely determined by three parameters, called Euler Angles. These Euler angles are represented by the ϕ , 𝜃 and ψ variables with each corresponding to a rotation about an axis. There are several different conventions. Only one will be shown here, since it is more important to understand the underlying math thoroughly.

A list of the Euler Angle rotation matrices for different sequences

Euler 123 sequence

R3(ψ)R2(𝜃)R1(ϕ) = ⌊ ⌋ cψc𝜃 cψs𝜃sϕ + sψcϕ − cψs𝜃cϕ + sψsϕ ⌈ − s c − s s s + c c s s c + c s ⌉ ψ 𝜃 ψ 𝜃 ϕ ψ ϕ ψ 𝜃 ϕ ψ ϕ s𝜃 − c𝜃sϕ c𝜃cϕ

Euler 132 sequence

R2(ψ)R3(𝜃)R1(ϕ) = ⌊ ⌋ cψc𝜃 cψs𝜃cϕ + sψsϕ cψs 𝜃sϕ − s ψcϕ ⌈ − s𝜃 c𝜃cϕ c𝜃sϕ ⌉ sψc𝜃 sψs 𝜃cϕ − cψsϕ s ψs𝜃sϕ + cψc ϕ

Euler 121 sequence

R1(ψ)R2(𝜃)R1(ϕ) = ⌊ ⌋ c𝜃 − s𝜃sϕ s 𝜃cϕ ⌈ − s s c c − s c s c s + s c c ⌉ ψ 𝜃 ψ ϕ ψ 𝜃 ϕ ψ ϕ ψ 𝜃 ϕ − s𝜃cψ − sψcϕ − cψc𝜃sϕ − sψsϕ + cψc𝜃cϕ

Euler 131 sequence

R1(ψ)R3(𝜃)R1(ϕ) = ⌊ ⌋ c𝜃 s𝜃cϕ s𝜃sϕ ⌈ − cψs𝜃 cψc𝜃cϕ − sψsϕ cψc𝜃sϕ + sψcϕ ⌉ sψs𝜃 − sψc𝜃cϕ − cψsϕ − sψc𝜃sϕ + cψcϕ

Euler 213 sequence

R3(ψ)R1(𝜃)R2(ϕ) = ⌊ ⌋ cψcϕ + sψs𝜃sϕ sψc𝜃 − cψsϕ + sψs𝜃cϕ ⌈ − s c + c s s c c s s + c s c ⌉ ψ ϕ ψ 𝜃 ϕ ψ 𝜃 ψ ϕ ψ 𝜃 ϕ c𝜃sϕ − s𝜃 c𝜃cϕ

Euler 231 sequence

R1(ψ)R3(𝜃)R2(ϕ) = ⌊ ⌋ c𝜃cϕ s𝜃 − c𝜃sϕ ⌈ − cψs𝜃cϕ + sψsϕ cψc𝜃 cψs𝜃sϕ + sψs𝜃sϕ ⌉ sψs𝜃cϕ + cψsϕ − sψc𝜃 − sψs𝜃sϕ + cψcϕ

Euler 212 sequence

R2(ψ)R1(𝜃)R2(ϕ) = ⌊ cψcϕ − sψc𝜃sϕ sψs𝜃 − cψsϕ − sψc𝜃cϕ ⌋ ⌈ ⌉ s𝜃sϕ c𝜃 s𝜃cϕ sψcϕ + cψc𝜃sϕ − cψs𝜃 − sψsϕ + cψc𝜃cϕ

Euler 232 sequence

R2(ψ)R3(𝜃)R2(ϕ) = ⌊ ⌋ cψc𝜃cϕ − sψsϕ cψs 𝜃 − cψc𝜃sϕ − sψcϕ ⌈ − s𝜃cϕ c𝜃 s𝜃sϕ ⌉ sψc𝜃cϕ + cψsϕ sψs 𝜃 − sψc𝜃sϕ + cψcϕ

Euler 312 Sequence

R2(ψ)R1(𝜃)R3(ϕ) = ⌊ cψcϕ − sψs𝜃sϕ cψs ϕ + s ψs𝜃cϕ − sψc 𝜃⌋ ⌈ ⌉ − sϕc𝜃 c𝜃cϕ s𝜃 sψcϕ + cψs𝜃sϕ sψs ϕ − cψs𝜃cϕ cψc𝜃

Euler 321 sequence

R1(ψ)R2(𝜃)R3(ϕ) = ⌊ ⌋ c𝜃cϕ c𝜃sϕ − s𝜃 ⌈ − cψsϕ + sψs𝜃cϕ cψcϕ + sψs𝜃sϕ sψc𝜃 ⌉ sψsϕ − cψs𝜃cϕ − sψcϕ + cψs𝜃sϕ cψc𝜃

Euler 313 sequence

R3(ψ)R1(𝜃)R3(ϕ) = ⌊ c c − s s c c s + s c c s s ⌋ ⌈ ψ ϕ ψ ϕ 𝜃 ψ ϕ ψ 𝜃 ϕ ψ 𝜃 ⌉ − sψcϕ − cψsϕc𝜃 − sψsϕ + cψc𝜃cϕ cψs𝜃 s𝜃sϕ − s 𝜃cϕ c𝜃

Euler 323 sequence

R3(ψ)R2(𝜃)R3(ϕ) = ⌊ ⌋ cψc𝜃cϕ − sψsϕ cψc𝜃sϕ + sψcϕ − cψs𝜃 ⌈ − sψc𝜃cϕ − cψsϕ − sψc𝜃sϕ + cψcϕ sψs𝜃 ⌉ s𝜃cϕ s𝜃s ϕ c𝜃


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Other names:  "Euler angle rotation matrix", "Euler Angle Sequence"

Cross-references: Euler 323 sequence, Euler 313 sequence, Euler 321 sequence, Euler 312 Sequence, Euler 232 sequence, Euler 212 sequence, Euler 231 sequence, Euler 213 sequence, Euler 131 sequence, Euler 121 sequence, Euler 132 sequence, Euler 123 sequence, Euler Angle, Euler Angles, parameters, matrix
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This is version 6 of Euler angle sequences, born on 2005-08-19, modified 2026-02-16.
Object id is 71, canonical name is EulerAngles.
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Classification:
Physics Classification: 45.40.-f (Dynamics and kinematics of rigid bodies)

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