Euler Angles

1. NASA Standard

1.1 Rotation Matrix

Euler angles are three successive rotation angles that determine the orientation of a reference frame (x, y, z) with respect to another reference frame (x_o, y_o, z_o).  They can be defined in various ways, and the notation has not been standardized.  See for example http://mathworld.wolfram.com/EulerAngles.html. The choice of Euler angles to be made in what follows is referred to in http://www.martinb.com/  as the NASA Standard Airplane convention.  The orientation matrix obtained in this last reference has a sign error.

Let   and  be sets of orthonormal right-handed base vectors corresponding to axes (x, y, z) and (x_o, y_o, z_o), respectively.  The process of rotating   into   uses two intermediate sets of axes, having base vectors  and, respectively.  Let

b₃ = rotation angle about  bringing  into

b₂ = rotation angle about  bringing  into

b₁ = rotation angle about  bringing  into

The way in which these three angles are defined from the geometry of given frames and  will be dealt with after obtaining the rotation matrix resulting from the successive application of the three rotations.  We have

           (1)                

           (2)           

               (3)

Substituting from (1) into (2), and from (2) into (3) yields

                                                      (4)

where

                                                                    (5)

Performing the matrix product, and adopting the abbreviating notation

(c₁,  c₂,  c₃) = (cos b₁, cos b₂, cos b₃)                                 (6)

(s₁,  s₂,  s₃) = (sin b₁, sin b₂, sin b₃)                                      (7)

yields

              (8)

The figure shows how angles b₃, b₂, and b₁ are defined from given axes (x, y, z) and (x_o, y_o, z_o).  Plane (y, z) intersects plane (x_o, y_o) along a line oriented and labeled as axes y₁ and y₂.  Axis z₁ coincides with axis z_o, and axis x₁ is on the perpendicular to axes y₁ and z₁, and is oriented such that the system (x₁, y₁, z₁) is right-handed.  Being perpendicular to axis z_o, axis x₁ is in the (x_o, y_o) plane.  b₃ is the angle of rotation about axis z_o that rotates axes (x_o, y_o, z_o) into axes (x₁, y₁, z₁).  Axes (x₂, z₂) are defined in the plane perpendicular to axis y₂, with axis x₂ coinciding with the x axis.  Axes (x₂, z₂) and (x₁, z₁) are coplanar since they are all perpendicular to the same axis

y₂ or y₁.  b₂ is the angle of rotation about axis y₁ that rotates axes (x₁, y₁, z₁) into axes (x₂, y₂, z₂).  Finally, axes (y₂, z₂) and (y, z) are coplanar since they are all perpendicular to the same axis x₂ or x.  b₁ is the angle of rotation about axis x that rotates axes (x₂, y₂, z₂) into axes (x, y, z).  All rotation angles are defined according to the right-hand rule.

The preceding geometric constructions fails if plane (y, z) coincides with plane (x_o, y_o).  In that case, axes x and z_o are collinear.  Two cases need to be considered

Case

If , axes x and z_o coincide, and axes x_o, y_o, y, and z are coplanar.  Let

b₃ = rotation angle about the z_o axis that brings axis y_o into axis y.

Axis x_o is thereby rotated into axis x₁, which is collinear with axis z, but oriented in the opposite direction. Thus, noting that, frame  is rotated into.  A rotation of  about the y axis by angle

b₂ =  -p /2                                        (9)

brings axis x₁ into axis x, and axis z₁ into axis z, thus frame  into frame.  We can apply the general results to the present case by letting R₁ be the identity matrix, and substituting b₂ =  -p /2  into the expression of R₂ in Eq. (2).  This yields

                               (10)

and

                         (11)

We thus obtain

                     (12)

Case

If , axis x is collinear with z_o and opposite in direction, and axes x_o, y_o, y, and z are coplanar.  Let

b₃ = rotation angle about the z_o axis that brings axis y_o into axis y.

Axis x_o is thereby rotated into axis x₁, which coincides with axis z. Thus, noting that, frame  is rotated into frame.  A rotation of  about the y axis by angle

b₂ =  p /2                                             (13)

brings axis x₁ into axis x, and axis z₁ into axis z, thus frame  into frame.  Substituting b₂ =  p /2  into the expression of R₂ in Eq. (2), then applying Eq.(11) yields

                                   (14)

                 (15)         

1.2 Angular Velocity

Let (b₁, b₂, b₃) be functions of time, and  be the corresponding angular velocity of  relative to.  The motion of  relative to, or its absolute motion, may be considered as composed of three relative motions:

a) the motion of  relative to, whose angular velocity is

b) the motion of  relative to, whose angular velocity is

c) the motion of  relative to, whose angular velocity is

By a theorem of rigid body kinematics, the absolute angular velocity is the sum of the relative angular velocities.  Thus

                                         (16)

To obtain the components, of on, we have

                 (17)

From (3) and (2) we have  and.  Eq. (17) turns into

                  (18)

or, with reference to Eqs. (3) and (4),

 {1 0 0}  + {second column of R₁} + {third column of R}

From Eqs. (3) and (8),

 {1 0 0} + {0  cos b₁ -sin b₁} + {-sin b₂    sin b₁cos b₂   cos b₁cos b₂}    (19)

    = {-sin b₂     cos b₁+sin b₁cos b₂     -sin b₁+cos b₁cos b₂}            (19)

For the components, of  on, we have

                  (20)

or, with reference to Eqs. (4) and (1),

 [first row of R] ^T + [second row of R₃] ^T + {0 0 1}

From Eqs. (8) and (1),

 {cosb₂cosb₃    cos b₂sin b₃  -sin b₂}  + {-sinb₃  cosb₃   0} + {0 0 1}  (21)

       =  {cosb₂cos b₃ -sin b₃     cosb₂sin b₃+cos b₃     -sin b₂+}       (21)

Case : b₁ = 0,  b₂ = -p/2

                                        (22)

Case : b₁ = 0,  b₂ = p/2

                                      (23)

2. Nutation, Precession, Spin

A traditional way of defining Euler Angles is associated with the analysis of a body rotating about a body-fixed axis, chosen as the z axis, which is itself rotating about a fixed axis, chosen as the z_o axis.  The problem is to describe the orientation of body-fixed axes (x, y, z) with respect to fixed axes (x_o, y_o, z_o).   An intermediate axis x' is defined along the perpendicular to axes z and z_o.  Axis x' thus lies in the (x_o, y_o) plane, and is also along the intersection of the (x, y) plane with the (x_o, y_o) plane.  The Euler angles are:

q  = angle between the z and z_o axes

y  = angle between the x_o and x' axes

j  = angle between the x' and x axes

Two intermediate sets of axes are defined by successive rotations as follows:

a) rotation about axis z_o by angle y brings axes (x_o, y_o, z_o) into axes (x₁=x', y₁, z₁=z_o).  

b) rotation about axis x' by angle q brings axes (x₁=x', y₁, z₁=z_o) into axes (x', y', z'). 

c) rotation about axis z by angle j brings axes (x', y', z') into axes (x, y, z).

In the context of a body rotating about a body-fixed z axis, such as a top, y, q, and j are called, precession, nutation, and spin, respectively. For a fixed q, the precession describes the rotation of the axis of the top about the vertical z_o axis.  The nutation represents the inclination of the axis of the top with respect to the vertical, and the spin represents the rotation angle of the top about its axis.

2.1 Rotation Matrix

As before, let   and  be sets of orthonormal right-handed base vectors corresponding to axes (x, y, z) and (x_o, y_o, z_o), respectively, and let  and correspond to axes (x₁=x', y₁, z₁=z_o) and (x', y', z'), respectively.  We have

and

We obtain

2.2 Angular Velocity

The angular velocity of the (x, y, z) frame is the sum of the intermediate relative angular velocities, i.e.

The components of the angular velocity on the fixed axes are

Thus

or

The components on the body-fixed axes are

Thus

or

The components on the (x', y', z') axes are

Thus

or,