3.3 The Dimension of the Configuration Space

In 3.2 Obstacles and the Configuration Space, we stuck to 2 dimensional robots. Now we’re going to up the ante to more complex systems.

Suppose the robot is a point that can move in the plane. So, $q=(x,y)\in Q={\mathbb{R}}^2$. Once we have chosen a reference coordinate frame fixed somewhere in sapce. Thus, the robot has two degrees of freedom, and the configuration space is two-dimensional.

Now consider a system consisting of 3 point robots, $A$, $B$, $C$, that are free to move independently in the plane. We need the coordinates of all robots to specify the full configuration, 2 degrees of freedom for each, 6 degrees of freedom in total. so $Q={\mathbb{R}}^6$.

Real robots are sets of rigid bodies connected by joints, not points that move independently. So, suppose the robot is a planar rigid body that can translate and rotate in the plane. $A$, $B$, $C$ are 3 points fixed to the body. We decide to choose the position of $A$, but have now specified the relative distance to $A$ of $B$ and $C$. We can only determine $B$ as lying on a circle around $A$, so our only freedom is $\theta$. The position of $A$ and $\theta$ entirely determines $C$s position.

These constraints are defined as holonomic constraints, because they can be expressed purely as a function of the configuration variables (and possibly time). Each linearly independent holonomic constraint reduces the configuration space by 1.

$g(q,t)=0$ Applying the same counting method (and imagining that all points are in ${\mathbb{R}}^3$, we arrive at 3 degrees of freedom for $A,(x_a,y_a,z_a)$ , and 2 degrees of freedom for B (placed on a sphere a fixed distance from A), leaving a circle for which C can be placed (just 1 degree of freedom), meaning this system has 6 degrees of freedom!

Kinematically redundent robots have multiple solutions for a given end effector pose. A robot is kinematically redundent when the degrees of freedom in the workspace is less than the degrees of freedom in configuration space

// this chapter also delves into weird robots and how to compute DoF for linked robots and such, not going to bother with notes for that