7 VectorsA vector is an abstract mathematical object with two properties: length or magnitude, and direction
8 Vector Notation Summary is a vector, is its magnitudeis the “1” componentis a 31 matrix of components in an unspecified reference frameis a 31 matrix of components inThe vector is an abstract mathematical object with direction and magnitude; its matrix representation is a 31 matrix of scalars
9 Rotations Suppose we know components in body frame And we want to know components in inertial frameFrames are related by a 33 Rotation MatrixSo
10 Rotations ContinuedThe equation is a linear system of the form A x = bThus, to determine the components in the inertial frame, we need to determine R and solve the linear systemCan write components of R asThus (direction cosines)Rotation matrix aka Direction Cosine Matrix
11 Rotations ContinuedCan also write the rotation matrix as the dot product of two “vectrices”:Can show that the inverse of a rotation matrix is simply its transposeThus 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 anotherAs matrix with rows and columns being unit vectors of one frame expressed in the otherThe matrix is the rotation from to
13 Attitude Kinematics Representations The rotation matrix represents the attitudeA rotation matrix has 9 numbers, but they are not independentThere are 6 constraints on the 9 elements of a rotation matrix (what are they?)Thus rotation has 3 degrees of freedomThere are many different sets of parameters that can be used to represent or parameterize rotationsEuler angles, Euler parameters (aka quaternions), Rodrigues parameters (aka Gibbs vectors), Modified Rodrigues parameters, …
14 Euler AnglesLeonhard Euler ( ) reasoned that the rotation from one frame to another can be visualized as a sequence of three simple rotations about base vectorsEach rotation is through an angle (Euler angle) about a specified axisLet’s consider the rotation from to using three Euler anglesThe first rotation is about the axis, through angleThe resulting frame is denoted or
15 Euler Angles (second rotation) The second rotation is about the axis, through angleThe resulting frame is denoted orThe rotation matrix notation for the “simple”rotations isRi(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 angleThe resulting frame is the desired body frame, denoted or
20 Rotation Matrix Euler Angles Knowing the 9 numbers in the rotation matrix, we can compute the 3 Euler anglesQuadrant 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 RBe sure to get the order correctGiven a sequence and some values for the angles, compute the numerical values of RBe sure to know the difference between degrees and radiansGiven the numerical values of R, extract numerical values of the Euler angles associated with a specified sequenceBe sure to make appropriate quadrant checks, and to check your answer
22 Euler’s TheoremThe 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 axisThe 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 1Thus every rotation matrix has an eigenvalue that is equal to +1This fact justifies the term eigenaxis for the Euler axisThis parameterization requires four parameters
24 Extracting and from RJust 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 scalarThe 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 otherEuler angles: 3 2 2 = 12 different setsEuler axis/angle: unit vector and angleEuler parameters = quaternions: unit 4 1You must be able to compute one from the other for any given representationNext: 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 equationWe 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 0The reason is that near q2 = p/2 the first and second rotations are indistinguishableFor the “symmetric” Euler angle sequences (3-1-3, 2-1-2, 1-3-1, etc) the singularity occurs when q2 = 0 or pFor the “asymmetric” Euler angle sequences (3-2-1, 2-3-1, 1-3-2, etc) the singularity occurs when q2 = p/2 or 3p/2This 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 equationsSingularity when 0 or 2pEuler parameter differential equationsNo 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 timeRequires integration of one of the sets of differential equations involving angular velocity