Davenport chained rotations

In physics and engineering, Davenport chained rotations are three chained intrinsic rotations about body-fixed specific axes. Euler rotations and Tait–Bryan rotations are particular cases of the Davenport general rotation decomposition. The angles of rotation are called Davenport angles because the general problem of decomposing a rotation in a sequence of three was studied first by Paul B. Davenport.^[1]

The non-orthogonal rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Being rotation axes are solidary with the moving body, the generalized rotations can be divided in two groups (here x, y and z refer to the non-ortogonal moving frame):

• generalized Euler rotations (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)
• generalized Tait–Bryan rotations (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z).

Most of the cases belong to the second group, being the generalized Euler rotations a degenerated case in which first and third axes are overlapping.

Davenport rotation theorem

Davenport possible axes for steps 1 and 3 given Z as the step 2

The general problem of decomposing a rotation in three composed movements about intrinsic axes was studied by P. Davenport, under the name "generalized Euler angles", but later these angles were named "Davenport angles" by M. Shuster and L. Markley.^[2]

The general problem consists in obtaining the matrix decomposition of a rotation given the three known axes. In some cases one of them can be repeated. This problem is equivalent to a decomposition problem of matrices^[3]

Davenport proved that any orientation can be achieved by composing three elemental rotations using non-orthogonal axes. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations).

According to the Davenport theorem, a unique decomposition is possible if and only if the second axis is perpendicular to the other two axes. Therefore axes 1 and 3 must be in the plane orthogonal to axis 2.^[4]

Therefore decompositions in Euler chained rotations and Tait–Bryan chained rotations are particular cases of this. The Tait–Byran case appears when axes 1 and 3 are perpendicular, and the Euler case appears when they are overlapping.

Complete system of rotations

Image 2: Airplane resting on a plane

A set of Davenport rotations is said to be complete if it is enough to generate any rotation of the space by composition. Speaking in matrix terms, it is complete if it can generate any orthonormal matrix of the space, whose determinant is +1. Due to the non-commutativity of the matrix product, the rotation system must be ordered.

Sometimes the order is imposed by the geometry of the underlying problem. For example, when used for vehicles, which have an 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 to adjust the direction of an aircraft with the two first angles. A different composition like YRP would allow to establish the direction of the wings axis, which is obviously not useful in most cases.

Tait–Bryan chained rotations

Image 1: The principal axes of an aircraft

Tait–Bryan rotations are a special case in which the first and third axes are perpendicular among them. Assuming a reference frame <x,y,z> with a convention as in the image 2, and a plane with <yaw, pitch, roll> axes like in the image 1, lying horizontal on the plane <x,y> in the beginning, after performing intrinsic rotations Y, P and R in the yaw, pitch and roll axes (in this order) we obtain something similar to image 3.

Image 3: Heading, elevation and bank angles after yaw, pitch and roll rotations (Z-Y’-X’’)

In the beginning :

• the plane roll axis is on axis x of the reference frame
• the plane pitch axis is on axis y of the reference frame
• the plane yaw axis is on axis z of the reference frame

The rotations are applied in order yaw, pitch and roll. In these conditions, the Heading (angle on the horizontal plane) will be equal to the yaw applied, and the Elevation will be equal to the pitch.

The matrix expression for the three Tait–Bryan rotations in dimension 3 is:

$R_x(\phi) = \mathrm{Roll}(\phi) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \phi & -\sin \phi \\ 0 & \sin \phi & \cos \phi \end{bmatrix}$

$\begin{align} \\ R_y(\theta) = \mathrm{Pitch}(\theta) = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix} \end{align}$

$\begin{align} \\ R_z(\psi) = \mathrm{Yaw}(\psi) = \begin{bmatrix} \cos \psi & -\sin \psi & 0 \\ \sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{align}$

The matrix of the composed rotations is M = Yaw·Pitch·Roll. Of the six possible combinations of yaw, pitch and roll, this combination is the only one in which the heading (direction of the roll axis) is equal to one of the rotations (the yaw) and the elevation (angle of the roll axis with the horizontal plane) is equal to other of the rotations (to the pitch).

Euler chained rotations

Starting position of an aircraft to apply proper Euler angles

Euler rotations appear as the special case in which the first and third axes of rotations are overlapping. These Euler rotations are related to the proper Euler angles, which were thought to study the movement of a rigid body such or a planet. The angle to define the direction of the roll axis is normally named "longitude of the revolution axis" or "longitude of the line of nodes" instead of "heading", which makes no sense for a planet.

Anyway, Euler rotations can still be used when speaking about a vehicle, though they will have a weird behavior. As the vertical axis is the origin for the angles, it is named "inclination" instead of "elevation". 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.

The combination depends on how the axis are taken and what the initial position of the plane is. Using the one in the drawing, and combining rotations in such a way that an axis is repeated, only roll–pitch–roll will allow to control the longitude and the inclination with one rotation each.

The three matrices to multiply are:

$R_Z(\phi) = \mathrm{Roll}_1(\phi) = \begin{bmatrix} \cos \phi & -\sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$R_y(\theta) = \mathrm{Pitch}(\theta) = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$

$R_Z(\psi) = \mathrm{Roll}_2(\psi) = \begin{bmatrix} \cos \psi & -\sin \psi & 0 \\ \sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{bmatrix}$

In this convention Roll₁ imposes the "heading", Pitch is the "inclination" (complementary of the elevation), and Roll₂ imposes the "tilt".

Conversion to extrinsic rotations

A rotation represented by Euler angles (α, β, γ) = (−60°, 30°, 45°), using z-x’-z″ intrinsic rotations

The same rotation represented by (γ, β, α) = (45°, 30°, −60°), using z-x-z extrinsic rotations

Davenport rotations are usually studied as an intrinsic rotation composition, because of the importance of the axes fixed to a moving body, but they can be converted to an extrinsic rotation composition, in case it could be more intuitive.

Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. 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. Both are represented by a matrix

$R = X(\alpha) Y(\beta) Z(\gamma)$

if R is used to pre-multiply column vectors, and by a matrix

$R = Z(\gamma) Y(\beta) X(\alpha)$

if R is used to post-multiply row vectors. See Ambiguities in the definition of rotation matrices for more details.

See also

• Matrix decomposition
• Euler angles
• Givens rotation

References

1. P. B. Davenport, Rotations about nonorthogonal axes
2. M. Shuster and L. Markley, Generalization of Euler angles, Journal of the Astronautical Sciences, Vol. 51, No. 2, April–June 2003, pp. 123–123
3. J. Wittenburg, L. Lilov, Decomposition of a finite rotation in three rotations about given axes
4. M. Shuster and L. Markley, Generalization of Euler angles, Journal of the Astronautical Sciences, Vol. 51, No. 2, April–June 2003, pp. 123–123