Modelling of Mechanical Systems I
Lecture 2
Subject: Euler angles, Quaternions and Kinematics
Objectives:
1. To provide understanding of different rotation parameter alternatives and their
application areas
2. To provide the students knowledge about rigid body kinematics for the
direction cosine matrix, Euler angles and the quaternion
Literature: Thomas Bak, Modeling of Mechanical Systems, Aalborg University 2002.
page 15-26.
Appendix C in “Autonomous Aircraft”, 2005, Finn Jensen, Daniel Pedersen
Quaternions
Exercises:
1. Let the B frame orientation relative the A be given by the (3-2-1) Euler angles
(10, 25, -15) degrees.(θ1 = 10, θ2 = 25, θ3 = -15). Find the corresponding
principal axis and angle.
2. What is the geometric condition for which the (3-2-1) Euler angles to
encounter a mathematical singularity.
3. Show that the two rotation matrices C(n, θ1) = C1 and C(m, θ2) = C2 commute
in multiplication if and only if:
nxm=0
4. Show that C.7 in appendix C is equal to 3.51 in Thomas Bak notes