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3.2. TRANSFORMATIONS 13
the popular Aerospace Sequence, which is the zyx sequence, it can be expressed as a matrix product
as
R = Rφ x Ry θ Rz ψ
⎡
⎤ ⎡
⎤ ⎡
⎤
1 0 0 cos θ 0 −sin θ cos ψ sin ψ 0
= ⎣0 cos φ sin φ⎦
⎣ 0 1 0 ⎦ ⎣−sin ψ cos ψ 0⎦
0 −sin φ cos φ sin θ 0 cos θ 0 0 1
This equation states, that the first rotation is through the Heading angle ψ about the z-axis, followed
by a rotation through the Elevation angle θ about the new y-axis. The final rotation is through the
Bank angle φ about the newest x-axis.
Euler Angles are a simple way to represent rotation sequences. They allow to describe the rotation
of an arbitrary frame into any other frame. Nevertheless they have two main disadvantages [8, p. 156].
First the representation for a given orientation is not unique and second, the interpolation between two
angles is difficult. The phenomenon that comes from this disadvantages is known as Gimbal lock.
3.2.3 Quaternions
Quaternions were invented by Hamilton in 1843 as a so-called hyper-complex number of rank 4. They
are another method to represent rotations, which solve the problems of the Euler Angles. The most
important rule for quaternions is
i 2 = j 2 = k 2 = i j k = −1
A quaternion is defined as the sum
q = q 0 + q = q 0 + iq 1 + jq 2 + kq 3
where q 0 is called the scalar part, and q is called the vector part of the quaternion. The scalars
q 0 , q 1 , q 2 , q 3 are called the components of the quaternion. Further i, j and k are the standard orthonormal
basis in R 3 and defined as
The quaternion rotation operator L q is defined as
i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)
w = L q (v) = qvq ∗ (3.1)
where w, v are vectors of R 3 , q is a quaternion and q ∗ is the complex conjugate of this quaternion. For
the equation to be valid, the vector v has to be converted into a pure quaternion which is v = 0 + v.
The algebraic action of Equation 3.1 is illustrated in Figure 3.4.
In this section only a brief overview of quaternions with the basic definitions were given. A
complete explanation of quaternions can be found in [17].