# Configuration space (Robotics)

The configuration space (also called C-space) of a robot is a specification of the position of all the points of a robot. The dimension of the C-space is called the degrees of freedom (DOF), which is the minimum number needed to represent the robot. For e.g. if the robot has 2 angles, the C-space is the 2D torus. A robot is defined to consits of rigid bodies (called links) connected together by joints.

The C-space can be parametrized both explicitly or implicitly.

Explicit parametrization uses the minimum number of coordinates, in a sphere for example, $$(\phi,\lambda) \in [-90,90]\times[-180,180]$$ for the latitude and longitude. While simple, not necessarily being topologically equivalent to the Euclidean space means representation would have poor behaviour at some parts of the space in the form of singularities and discontinuities.

Implicit parametrization makes use of embedding the surface in a higher dimensional space that is easier to work with, typically the Euclidean space. Here, a sphere can be represented as $$\{(x,y,z)\in \mathbb{R}^3 x^2+y^2+z^2=1\}$$. This results in no discontinuities or singularities but at the expense of greater complexity in representing the coordinates.

Instead of representing the C-space of a robot in terms of the exact amount of its DOF, we can make use of holonomic constraints (also sometimes called integral constraints).

The task space is the space where the robot's task can be naturally expressed. E.g. when the controlling position of a position of a pen on a flat piece of paper, the task space is $$\mathbb{R}^2$$, and when controlling the position and orientation of a rigid body, the task space is the 6D space of rigid body configurations.

The work space is a specification of the reachable configurations of the end-effector (end of the robot), usually defined in Cartesian coordinates.

The set of positions reachable from all possible orientations is called the dextrous workspace.

## Thoughts

• Much if not all of the ideas come from Northwestern robotics course lectures
• Look further into the difference between points and joints in the first sentence.
• Nash embedding theorem argument for why we are allowed to use implicit parametrizations.

Created: 2022-03-13 Sun 21:45

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