Davenport chained rotations

[2] The general problem consists of obtaining the matrix decomposition of a rotation given the three known axes.

[3] Davenport proved that any orientation can be achieved by composing three elemental rotations using non-orthogonal axes.

For example, when used for vehicles, which have a special axis pointing to the "forward" direction, only one of the six possible combinations of rotations is useful.

The interesting composition is the one able to control the heading and the elevation of the aircraft with one independent rotation each.

In the adjacent drawing, the yaw, pitch and roll (YPR) composition allows adjustment of the direction of an aircraft with the two first angles.

As before, describing the attitude of a vehicle, there is an axis considered pointing forward, and therefore only one out of the possible combinations of rotations will be useful.

For instance, the intrinsic rotations x-y’-z″ by angles α, β, γ are equivalent to the extrinsic rotations z-y-x by angles γ, β, α.

The Euler or Tait-Bryan angles (α, β, γ) are the amplitudes of these elemental rotations.

For instance, the target orientation can be reached as follows: The above-mentioned notation allows us to summarize this as follows: the three elemental rotations of the XYZ-system occur about z, x’ and z″.

For instance, represents a composition of intrinsic rotations about axes x-y’-z″, if used to pre-multiply column vectors.

This is standard practice, but take note of the ambiguities in the definition of rotation matrices.

The Euler or Tait-Bryan angles (α, β, γ) are the amplitudes of these elemental rotations.

For instance, the target orientation can be reached as follows: In sum, the three elemental rotations occur about z, x and z.

For instance, represents a composition of extrinsic rotations about axes x-y-z, if used to pre-multiply column vectors.

This is standard practice, but take note of the ambiguities in the definition of rotation matrices.

For instance, the intrinsic rotations x-y’-z″ by angles α, β, γ are equivalent to the extrinsic rotations z-y-x by angles γ, β, α.

This is standard practice, but take note of the ambiguities in the definition of rotation matrices.