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Position

Position of point $B$ relative to point $A$:

In 3D space positions are represented by vectors $\mathbf{r} \in \mathbb{R}^3$

It is necessary to define a reference frame $A$ and to express the vector in this frame:

With:

• $\mathcal{A}$: Frame $\mathcal{A}$ is used to express position vector $\mathbf{r}_{AB}$
• $\mathbf{r}_{AB}$: Position vector of point $B$ with with respect to the origin of frame $\mathcal{A}$

The unit vectors $\left ( \mathbf{e}^\mathcal{A}_x, \mathbf{e}^\mathcal{A}_y, \mathbf{e}^\mathcal{A}_z \right )$ of frame $\mathcal{A}$ form an ortho-normal basis of $\mathbb{R}^3$.

An alternative notation:

Meaning: Vector $\mathbf{p}$ to $B$ with respect to coordinate frame ${A}$. (The bound vector $\mathbf{p}$ pointing from the origin of ${A}$ to the point $B$, Where everything is in the coordnate frame ${A}$).

Representation of Positions

Cartesian Coordinates

Where $\mathbf{\chi}_P$ are the stacked parameters of the position representation.

The position vector is given by

Where:

• $\mathbf{e}^\mathcal{A}_i$: Unit vector of dimension $i$ in frame $\mathcal{A}$

Cylindrical coordinates

A position vector is given by

Spherical coordinates

A position vector is given by

Where:

• $\theta$: azimuthal angle
• $\phi$: polar angle (sometimes $\varphi$)

Pose

To fully describe the configuration of a rigid body (pose) a position and a rotation is needed.

A pose $\xi$ in 2D has 3 parameters $(x, y, \theta)$. It can be considered as the motion (translation and rotation) of a coordinate frame.

The pose of a coordinate frame can be described with respect to another coordinate frame:

Meaning: Pose $\xi$ of coordinate frame ${B}$ with respect to frame ${A}$.

Sources

Notes taken from:

Robot Academy