1. Chris Hall Aerospace and Ocean Engineering cdhall@vt.edu
Attitude Kinematics
Chris Hall Aerospace and Ocean Engineering

2. Dynamics = Kinematics + Kinetics
Translational dynamics (Newton’s 2nd Law)
2nd order includes kinematics and kinetics
1st order separates the two
kinematics
kinetics

3. Dynamics = Kinematics + Kinetics
Rotational dynamics (Euler’s Law)
Implies both kinematics and kinetics
h is the angular momentum, g is the torque
1st order separates the two ?
kinematics
kinetics

4. Translational vs Rotational
Linear momentum = mass  velocity
d/dt (linear momentum) = applied forces
d/dt (position) = linear momentum/mass
Angular momentum = inertia  angular velocity
d/dt (angular momentum) = applied torques
d/dt (attitude) = “angular momentum/inertia”

5. Back to Reference Frames
Denote reference frames as triads of mutually orthogonal unit vectors

6. Right-Handedness

7. Vectors
A vector is an abstract mathematical object with two properties: length or magnitude, and direction

8. Vector Notation Summary
is a vector, is its magnitude
is the “1” component
is a 31 matrix of components in an unspecified reference frame
is a 31 matrix of components in
The vector is an abstract mathematical object with direction and magnitude; its matrix representation is a 31 matrix of scalars

9. Rotations Suppose we know components in body frame
And we want to know components in inertial frame
Frames are related by a 33 Rotation Matrix So

10. Rotations Continued
The equation is a linear system of the form A x = b
Thus, to determine the components in the inertial frame, we need to determine R and solve the linear system
Can write components of R as
Thus (direction cosines)
Rotation matrix aka Direction Cosine Matrix

11. Rotations Continued
Can also write the rotation matrix as the dot product of two “vectrices”:
Can show that the inverse of a rotation matrix is simply its transpose
Thus a rotation matrix is an “orthonormal” matrix: its rows and columns are components of mutually orthogonal unit vectors

12. Rotation Notation As dot product of “vectrices”:
As matrix transforming vectors from one frame to another
As matrix with rows and columns being unit vectors of one frame expressed in the other
The matrix is the rotation from to

13. Attitude Kinematics Representations
The rotation matrix represents the attitude
A rotation matrix has 9 numbers, but they are not independent
There are 6 constraints on the 9 elements of a rotation matrix (what are they?)
Thus rotation has 3 degrees of freedom
There are many different sets of parameters that can be used to represent or parameterize rotations
Euler angles, Euler parameters (aka quaternions), Rodrigues parameters (aka Gibbs vectors), Modified Rodrigues parameters, …

14. Euler Angles
Leonhard Euler ( ) reasoned that the rotation from one frame to another can be visualized as a sequence of three simple rotations about base vectors
Each rotation is through an angle (Euler angle) about a specified axis
Let’s consider the rotation from to using three Euler angles
The first rotation is about the axis, through angle
The resulting frame is denoted or

15. Euler Angles (second rotation)
The second rotation is about the axis, through angle
The resulting frame is denoted or
The rotation matrix notation for the “simple”rotations is
Ri(j) denotes a rotation about the I axis. The subscript on R defines which simple rotation axis is used, and the subscript on  defines which of the three angles in the Euler sequence it is

16. Euler Angles (third rotation)
The third rotation is a “1” rotation, through angle
The resulting frame is the desired body frame, denoted or

17. An Illustrative Example

18. Illustrative Example (continued)

19. Illustrative Example (concluded)

20. Rotation Matrix  Euler Angles
Knowing the 9 numbers in the rotation matrix, we can compute the 3 Euler angles
Quadrant checks are imperative.
Use atan2(y,x)

21. What you need to know about Euler Angles
Given a sequence, say “3-1-2” for example, derive the Euler angle representation of R
Be sure to get the order correct
Given a sequence and some values for the angles, compute the numerical values of R
Be sure to know the difference between degrees and radians
Given the numerical values of R, extract numerical values of the Euler angles associated with a specified sequence
Be sure to make appropriate quadrant checks, and to check your answer

22. Euler’s Theorem
The most general motion of a rigid body with a fixed point is a rotation about a fixed axis.
The axis, denoted a, is called the eigenaxis or Euler axis
The angle of rotation, is called the Euler angle or the principal Euler angle

23. Observations Regarding and
Since the Euler axis is the eigenvector of R associated with the eigenvalue 1
Thus every rotation matrix has an eigenvalue that is equal to +1
This fact justifies the term eigenaxis for the Euler axis
This parameterization requires four parameters

24. Extracting and from R
Just as we need to be able to compute Euler angles from a given rotation matrix, we need to be able to compute the Euler axis and Euler angle:
What do you do about the = 0 case?

25. Another Four-Parameter Set
The Euler parameter set, also known as a quaternion, is a four-parameter set with some advantages over the Euler axis/angle set:
The vector component, q, is a 3 1, whereas the scalar component, q4, is, well, a scalar
The quaternion is denoted by , a 4 1 matrix

26. and To compute the rotation matrix using the quaternion:
To compute the quaternion using the rotation matrix:

27. Summary of Kinematics Notation
Several equivalent methods of describing attitude or orientation:
Rotation matrix = DCM = vectors of one frame expressed in the other = dot products of vectors of one frame with those of the other
Euler angles: 3 2  2 = 12 different sets
Euler axis/angle: unit vector and angle
Euler parameters = quaternions: unit 4 1
You must be able to compute one from the other for any given representation
Next: How does attitude vary with time?

28. Differential Equations of Kinematics
Given the velocity of a point and initial conditions for its position, we can compute its position as a function of time by integrating the differential equation
We now need to develop the equivalent differential equations for the attitude when the angular velocity is known

29. Euler Angles and Angular Velocity
One frame at a time, just like we developed rotation matrices in terms of Euler angles

30. Adding the Angular Velocities
The three angular velocities are expressed in different frames. To add them,we need to rotate them all into the same frame. Typically, we use the body frame, but this is not always the case.
We already have in body frame.

31. Complete the Operation
Carry out the matrix multiplications and add the three results: Or

32. Kinematic Singularity in the Differential Equation for Euler Angles
Note that for this Euler angle set, the Euler rates go to infinity when cos q2  0
The reason is that near q2 = p/2 the first and second rotations are indistinguishable
For the “symmetric” Euler angle sequences (3-1-3, 2-1-2, 1-3-1, etc) the singularity occurs when q2 = 0 or p
For the “asymmetric” Euler angle sequences (3-2-1, 2-3-1, 1-3-2, etc) the singularity occurs when q2 = p/2 or 3p/2
This kinematic singularity is a major disadvantage of using Euler angles for large-angle motion

33. Euler Axis/Angle and Euler Parameters
Euler axis and angle differential equations
Singularity when   0 or 2p
Euler parameter differential equations
No singularity!

34. Typical Problem Involving Angular Velocity and Attitude
Given initial conditions for the attitude (in any form), and a time history of angular velocity, compute R or any other attitude representation as a function of time
Requires integration of one of the sets of differential equations involving angular velocity