POSITION & ORIENTATION ANALYSIS

This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the finite motion of a rigid body in 3-D space After this lecture, the student should be able to: Appreciate the distinction between direction of rotation and axis of rotation Express the motion of a rigid body in terms of the motion of a point of the body and the rotation of the body Position & Orientation Analysis

General Motion += TranslationRotation Summary of Previous Lecture Translation: defined by the position between the origins of frame w.r.t. frame {X,Y,Z}. Translation will not change the orientation between the frames: Orientation: The changes in the orientation of the can be viewed as a result of rotations of frame w.r.t. frame {X, Y, Z}. R(t) times vector before rotation = Vector after rotation

Rotation between two configurations In general, given any arbitrary orthonormal set attached to the rigid body The above shows that if two configurations F(t 2 ) and F(t 1 ) are known relative to F(t 0 ), then we can find the configuration of F(t 2 ) relative to F(t 1 ) using R, where Therefore:

Properties of Rotational tensor R is proper orthogonal: For an orthonormal set F: Where S is a skew-symmetric matrix:

Pure Rotation So far, we use R (a matrix) to represent the rotation. Is there another way we can represent the rotation? Consider a clock. To define the rotation of the hand represented by vector “AB”, we need at least 2 things: the axis of rotation (unit vector) and the angle of rotation  about axis A B “AB(t 0 )” before rotation B* “AB(t 1 )” after rotation P Axis of rotation  positive Axis of rotation  (angle of rotation)

Axis of Rotation for Pure Rotation Notice that regardless of how the clock is orientated, once we know the axis of rotation and the angle of rotation, we can define the orientation of final vector “AB(t 1 )” if given the initial vector “AB(t 0 )”. To find the axis of rotation, we need a point P so that A P Axis of rotation Note: if you rotate AP any angle about AP, you will still end up with AP

Let A be an n  n matrix. If there exists a and a nonzero n  1 vector such that then is called an eigenvalue of A and is called an eigenvector of A corresponding to the eigenvalue Refresh: Now: R is a 3  3 matrix and is a 3  1 vector. Relate =1, A  R, Therefore, is the eigenvector of R corresponding to =1 Unit vector for axis of rotation is Axis of Rotation for Pure Rotation

Example: Axis of Rotation for Pure Rotation Determine the direction of rotation given by The direction of rotation is the same as the axis of rotation. Given R, we can solve for the eigenvector of R corresponding to =1 to get Solution:

Example: Axis of Rotation for Pure Rotation

Multiply 3 rd eqn by -5 and add it to 1 st eqn to eliminate

Example: Axis of Rotation for Pure Rotation Divide 2 nd eqn by and simplify using the known result:

Example: Axis of Rotation for Pure Rotation (Remember this axis of rotation. We need it later in another e.g.)

Angle of Rotation for Pure Rotation For the clock, the hand is always perpendicular to, the axis of rotation. Generally, given a point C on the rigid body, the line “AC” may not be perpendicular to axis. Nevertheless, once the axis of rotation is obtained from R, we can always resolve “AC” into 2 components: the first component AC n is normal and the second component AC p is parallel to the axis : A P Axis of rotation C AC p AC n Time t 0

At time t 1, AC has rotated an angle of  about axis to AC* : AC p A P Axis of rotation C AC n C*  But Therefore Angle of Rotation for Pure Rotation

In other word, their cross product will be a vector in the direction of axis. We can find once we determine axis. AC p A P* Axis of rotation C AC n C*  Notice that axis is normal to both and Therefore Where  is the angle of rotation Angle of Rotation for Pure Rotation But

Angle of Rotation for Pure Rotation Therefore, given R and, we can find the axis of rotation

Example: Angle of Rotation for Pure Rotation Determine the angle of rotation of a point C of the rigid body if Solution: The axis of rotation for R has been found from the previous example to be

Example: Angle of Rotation for Pure Rotation With

Example: Angle of Rotation for Pure Rotation With

Example: Angle of Rotation for Pure Rotation

Story so far …. Therefore

Explicit Expression of R given axis and angle of Rotation Given R we can obtain the axis of rotation and the angle of rotation. If we are given the axis of rotation and the angle of rotation, then we should be able to derive the rotational matrix R as Given  and where

Explicit Expression of R given axis and angle of Rotation Example 1: Find R for a rotation of  radians about [1, 0, 0] T Solution: where This is a rotation about the X-axis

Explicit Expression of R given axis and angle of Rotation Example 2: Find R for a rotation of  radians about [0, 1, 0] T Solution: where This is a rotation about the Y-axis

Explicit Expression of R given axis and angle of Rotation Example 3: Find R for a rotation of  radians about [0, 0, 1] T Solution: where This is a rotation about the Z-axis

This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the finite motion of a rigid body in 3-D space The following were covered: The distinction between direction of rotation and axis of rotation The motion of a rigid body in terms of the motion of a point of the body and the rotation of the body Summary