Introduction to Rigid Body Motions

In robotics, describing where a robot is and how its links are oriented is everything. Some of the building blocks include coordinate frames, position vectors, rotation matrices. Let's take a quick peek at them!

Coordinate Frames A coordinate frame is simply a set of reference axes ($x, y, z$) attached to a specific point, acting like a mini 3D graph paper. In robotics, we constantly juggle multiple frames: a fixed "world" frame anchored to the room, and moving frames attached to each joint of the robot.

Position Vectors To locate a physical point $p$ in space relative to a reference frame $\{A\}$, we use a position vector:

Think of these coordinates as walking instructions: take $p_x$ steps along the $x$-axis, $p_y$ along the $y$-axis, and $p_z$ along the $z$-axis. Crucially, position is relative. If you change your viewpoint to a new coordinate frame, the numerical values of the vector change, even though the physical point hasn't moved.

Rotation Matrices While a position vector handles translation, orientation requires a matrix. We represent the orientation of a frame $\{B\}$ relative to frame $\{A\}$ using a rotation matrix $R_{AB}$. This is built by placing the unit axes of $\{B\}$ expressed in $\{A\}$'s coordinates side by side:

Rotation matrices belong to a group called $SO(3)$ and must satisfy two strict rules:

These rules ensure that a rotation matrix only rotates space without stretching, shearing, or reflecting it like a mirror. A powerful property of these matrices is that their inverse is simply their transpose ($R^{-1} = R^T$), making it effortless to reverse a perspective.

Changing Perspectives To convert a point measured in frame $\{B\}$ into the coordinates of frame $\{A\}$, we multiply it by the rotation matrix:

By multiplying these matrices together ($R_{AC} = R_{AB} R_{BC}$), we can chain transformations across multiple robot links.

In the future we'll put these blocks together and see them in motion!