Rotations and Translations Cherevatsky Boris

Mathematical terms The inner product of 2 vectors a,b is defined as: The cross product of 2 vectors is defined as: A unit vector will be marked as:

Representing a Point 3D A three-dimensional point A is a reference coordinate system here

Representing a Point 3D (cont.) Once a coordinate system is fixed, we can locate any point in the universe with a 3x1 position vector. The components of P in {A} have numerical values which indicate distances along the axes of {A}. To describe the orientation of a body we will attach a coordinate system to the body and then give a description of this coordinate system relative to the reference system.

Example

Description of Orientation is a unit vector in B is a coordinate of a unit vector of B in coordinates system A (i.e. the projection of onto the unit direction of its reference)

Example Rotating B relative to A around Z by

Example In general:

Using Rotation Matrices

Translation

Combining Rotation and Translation

What is a Frame ? A set of four vectors giving position and orientation information. The description of the frame can be thought as a position vector and a rotation matrix. Frame is a coordinate system, where in addition to the orientation we give a position vector which locates its origin relative to some other embedding frame.

Arrows Convention An Arrow - represents a vector drawn from one origin to another which shows the position of the origin at the head of the arrow in terms of the frame at the tail of the arrow. The direction of this locating arrow tells us that {B} is known relative to {A} and not vice versa. Mapping a vector from one frame to another – the quantity itself is not changed, only its description is changed.

Example Rotating a frame B relative to a frame A about Z axis by degrees and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0]T?

Extension to 4x4 We can define a 4x4 matrix operator and use a 4x1 position vector

Example If we use the above example we can see that:

P in the coordinate system A

Formula

Compound Transformation

Several Combinations

Example

Example

Notes Homogeneous transforms are useful in writing compact equations; a computer program would not use them because of the time wasted multiplying ones and zeros. This representation is mainly for our convenience. For the details turn to chapter 2.

Euler Theorem In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. How can we compute the axis of rotation? (Eigenvector corresponding to eigenvalue 1).

Quaternions The quaternion group has 8 members: Their product is defined by the equation:

Quaternions Algebra We will call the following linear combination a quaternion. It can be written also as: All the combinations of Q where a,b,c,s are real numbers is called the quaternion algebra.

Quaternion Algebra By Euler’s theorem every rotation can be represented as a rotation around some axis with angle . In quaternion terms: Composition of rotations is equivalent to quaternion multiplication.

Example We want to represent a rotation around x-axis by 90 , and then around z-axis by 90 :

Rotating with quaternions We can describe a rotation of a given vector v around a unit vector u by angle : this action is called conjugation. * Pay attention to the inverse of q (like in complex numbers) !

Rotating with quaternions The rotation matrix corresponding to a rotation by the unit quaternion z = a + bi + cj + dk (with |z| = 1) is given by: Its also possible to calculate the quaternion from rotation matrix: Look at Craig (chapter 2 p.50 )

Rodrigues formula We would like to rotate a vector v around a unit vector u with angle the rotated vector will be: