The 3-Arms Case
With all basic knowledge in hand, we move to the next member consisting of 3 arms.
Let's consider first the 2-arms machine and its associated configuration space represented by a sphere. By adding an additional arm to this machine, we restrict the possible states of this machine (2-arms machine). States in which the distance between the central joint and the newly placed anchor is larger then the arm's length are no longer reachable. Therefore, the constraint given by the third leg restricts the configuration space to a certain region of the sphere. Now consider the 2-arms machine with the additional arm as a 3-arms machine, is it possible to represent its configuration space by the restricted region on the sphere? By all means no, since to each state in the restricted area the third joint could take two positions (excluding those where the third arm is stretched). This means that the configuration space consists of two copies of the restricted region on the sphere, these regions are connected along their borders. Because each border consists of all points where the third arm is stretched, we must identify these two borders.
|States with 2 possible bending of the restricting arm -States inside the restricted region.||States in which the restricting arm is stretched - States on the border of the restricted region.||Unreachable states - States out of the restricted region.|
|How can we find the nature of the restricted region on sphere? If we know how its border looks like then we might know how this area looks like. For Example if the border turns out to be a circle on the sphere then it would divide the sphere into two regions, one of them is the restricted area and must look like the left image. This object is homeomorphic to a simple disk. With the help of the next Java Applet you can examine how the border looks like:|
Restricted region of a sphere with circle shaped border
|Before jumping to conclusions about the restricted region, we must remember that our analysis is of a topological nature. That is we have found the border of this region only to be homeomorphic to a circle. The border might turn up to be a very complicated curve on the sphere. Can we still say how the restricted area looks like? To our aid comes a mathematical theorem called "The Jordan curve Theorem" which states that every curve homeomorphic to a circle on a sphere divides the sphere into two regions, each region homeomorphic to a disk. (This statement seems rather trivial, but it is not!). Therefore, each of our two copies of restricted area is homeomorphic to a disk:||
A Sphere dissected into 2 regions by a circle.
If we finally identify the regions' borders we end up with a sphere.
|1. Connect the regions along the borders||2. The unified region is Homeomorphic to a Sphere|
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