I. Introduction

1. Configuration

a. DoF

  1. Planar

    2 translational degrees of freedom and one rotational degree of freedom

  2. Spacial

    3 translational degrees of freedom and 3 rotational degree of freedom

b. Status representation

  1. Displacement & Orientation

    Use the body frame ( xyz & its orirtentaion), include:

    1. position in space
    2. vector (principle axes)

    image.png

  2. Velocity & Acceleration

    Use the differential of the body frame.

2. Displacement

image.png

image.png

3. Orientation

image.png

a. Relationship

The upper left corner represents the reference coordinate system for observation. As shown in the figure above, it represents the expression of B in A space.

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Use column vector to represent “B relative to A”.

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b. Direct cosines

The expression of B in A space can be viewed as the projection of B frame vectors onto A.

II. Rotation

1. Rotation Matrix