Position and coordinate systems

The simplest object we can study in kinematics is the point, also known as particle in classical mechanics. We can describe a point \(A\) in the Euclidean space by providing a vector \({}^\calB \bfp_A \in \mathbb{R}^3\) of coordinates in the reference frame \(\calB = (B, {}^\calB \bfe_x, {}^\calB \bfe_y, {}^\calB \bfe_z)\).

Cartesian coordinates

The three vectors \(({}^\calB \bfe_x, {}^\calB \bfe_y, {}^\calB \bfe_z)\) form an orthonormal basis of \(\mathbb{R}^3\). We can use it to express the Cartesian coordinates of \(A\), which consist of:

  • The signed distance \(x\) along the x-axis.
  • The signed distance \(y\) along the y-axis.
  • The signed distance \(z\) along the z-axis.

Mathematically:

\begin{equation*} {}^\calB \bfp_A = \begin{bmatrix} x \\ y \\ z \end{bmatrix} = x\,{}^\calB \bfe_x + y\,{}^\calB \bfe_y + z\,{}^\calB \bfe_z \end{equation*}

Graphically:

Cartesian coordinates in the Euclidean space

On a side note, the figure above introduces a color convention from computer graphics where the x-axis (sagittal) is red, the y-axis (lateral) green and the z-axis (vertical) blue. This convention is common in off-the-shelf robotics software such as RViz and OpenRAVE.

Cylindrical coordinates

Cylindrical coordinates represent the position of a point relative to an axis of interest, typically the vertical z-axis, and a direction of interest, typically the sagittal x-axis. They consist of three numbers:

  • The distance \(\rho\) from \(A\) to the z-axis.
  • The angle \(\varphi\), called azimuthal angle (or just the azimuth, which sounds way cooler), between the vector from \(B\) to \(A\) and \(\bfe_x\).
  • The axis coordinate \(z\).

Mathematically, we can map the vector \({}^\cal B\bfchi_A\) of cylindrical coordinates to Cartesian ones by:

\begin{equation*} {}^\calB \bfchi_A = \begin{bmatrix} \rho \\ \varphi \\ z \end{bmatrix} \ \Longrightarrow \ {}^\calB \bfp_A({}^\calB \bfchi_A) = \begin{bmatrix} \rho \cos \varphi \\ \rho \sin \varphi \\ z \end{bmatrix} \end{equation*}

Graphically:

Cylindrical coordinates in the Euclidean space

These coordinates are sometimes useful to analyze quantities that act on a body frame of a car or mobile robot. For example, when an external disturbance is applied to a humanoid robot, looking at the direction and amplitude of the disturbance in the horizontal plane can be more insightful than looking at individual x- and y-coordinates.

Spherical coordinates

Spherical coordinates represent the position of a point by:

  • Its distance \(r\) to the origin \(B\) of the reference frame.
  • The angle \(\varphi\), the azimuth™, identical to the one used in cylindrical coordinates.
  • The angle \(\theta\), called polar, corresponding to the amount of rotation to apply in the resulting plane to go from the "pole" to the desired point.

Mathematically, we can map the vector \({}^\cal B\bfsigma_A\) of spherical coordinates to Cartesian ones by:

\begin{equation*} {}^\calB \bfsigma_A = \begin{bmatrix} r \\ \varphi \\ \theta \end{bmatrix} \ \Longrightarrow \ {}^\calB \bfp_A({}^\calB \bfsigma_A) = \begin{bmatrix} r \cos \varphi \sin \theta \\ r \sin \varphi \sin \theta \\ r \cos \theta \end{bmatrix} \end{equation*}

Graphically:

Spherical coordinates in the Euclidean space

Observations

Difference between the object and its coordinates

These three coordinate systems can be used equivalently to represent the position of our point \(A\) in the reference frame \(\calB\). Note how we have made, and will consistently make, the distinction between the point itself \(A\) and the vector \({}^\calB \bfp_A\) (of coordinates (with respect to a given reference frame)) used to represent this point mathematically. The same point can for instance be written \({}^\calC p_A\) in another reference frame \(\calC\) translated with respect to \(\calB\). This mental gymnastics of keeping track of both the object and its representation will become all the more important as we move on to linear velocities, angular velocities, twists, etc.

Omitting the reference frame

When there is no ambiguity, the reference frame exponent is often dropped and the position of point \(A\) is abbreviated as \(\bfp_A\). Yet, physical quantities are always expressed with respect to a reference frame, so implicitly there is always an underlying frame to any vector \(\bfp_A\) of position coordinates.

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