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How to make a configuration space of a robot manipulator from a work space ? it's robot manipulator is 3 degree of freedom. Is a robot manipulator with more degrees of freedom the same process ?. Do more links affect the config space?

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I am not exactly sure what the question is, (maybe you are confused on terminology), but I will take a shot.

Forward Kinematics (FK) defines the transform between a robot's joint space (aka configuration space) and its end-effector's task space (aka operational space). While Inverse Kinematics (IK) defines the opposite; the transform from task space to joint space. A robot's workspace is simply the volume of space that the end-effector can reach.

For serial-link arms (aka open kinematic chain), "regular" robot arms, computing the FK gets more complicated as you add degrees of freedom, but is still relatively straight-forward. There are closed-form solutions for IK for certain arm styles and DoF, but is more complicated in general. Typically this is done numerically for larger DoF.

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Let's start from the definition. (Well, there are actually a few possible definitions depending on the system but let's talk about a basic one first.)

A configuration $q$ is a vector of robot joint values. So as in your case, a configuration is a 3D vector. The configuration space $\mathcal{C}$ is then the space of those configurations, i.e., $\mathcal{C} = \mathbf{R}^{3}$.

There are two types of configurations in $\mathcal{C}$, those that are in collision and those that are collision-free (i.e., not colliding with itself or any environment). To construct a configuration space is then equivalent to creating a map of $\mathcal{C}$ which shows collision region (obstacles). To explicitly do that, you need to discretize $\mathcal{C}$ then test at each discretized configuration if it is collision-free. For example, suppose all of your three joints can move from $-180^\circ$ to $180^\circ$ with discretization resolution of $1^\circ$, you then have a grid of the size $361^3$. This means you need to test all of $361^3$ (almost 50 million) points to explicitly construct the collision regions and hence the map of $\mathcal{C}$.

This process is the same for more DOFs as well. And having more links means that your configuration space will be of higher dimension. We cannot tell anything much about the obstacle regions in the configuration space since their geometries will generally be very complex.

You can see from above that even constructing the map of $\mathcal{C}$ of dimension $3$ is a very computationally expensive process. Therefore, people generally do not construct explicitly the configuration space. For example, in motion planning, using a sampling-based planner together with a collision detection module can efficiently substitute the construction of the full map of $\mathcal{C}$.

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