Basic Machines

We are now ready to investigate the configuration space of a special kind of machines. The machines we will investigate in this lesson are built in the following way: We take N copies of 2-leg machines, all legs with equal length. Next we create a new central joint by combining all tips of the machines. Finally we limit the movement of the machine by taking the anchors apart, so no leg could rotate in a full circle. We will call the distinguished member with N arms: N-arms machine.

We will find out the configuration spaces by investigating machines, starting with simple ones and continuing to more complicated machines. Before examining machines from our family, let's examine a somewhat simpler machine shown in the applet below:

By adding one more bar to the above machine we meet the first member of our family consisting of just two anchors. As in the previous case the left joint can move in an arc. If we fix the left joint in place and let the other two joints move freely, the resulting motion is the movement of the previous machine, which is therefore represented by a circle. Thus to each fixed position of the left joint the restricted movement corresponds to a circle, with two exceptions: if the left joint is placed at the one of the endpoints of its arc of movement, the other two joints can not move and the corresponding configuration space is a point. Similar to the former machine the corresponding circles get smaller as the left joint approaches the end of the arc. Finally we can describe the configuration space of the 2-arms Machine by a sphere as shown in the applet below: For every fixed position of the left joint the possible motion is represented by the red circle on the configuration space. |

In the next pages we will examine more complicated members of our family, but for now let us try to analyze some properties which are common to all members. In the previous lesson we have introduced the local property of geometrical objects and defined surfaces as objects that locally behave like the 2-dimensional plane. Since we saw that the configuration space of the 2-arms machine is a sphere its local property is 2-dimensional, what can we say about the local properties of other members? All our machines have a distinguished joint: the central joint, which is connected to each arm. Let’s ask two questions regarding this joint:

If this joint is in a position where no arm is stretched, in which directions can it move? |

Since the whole machine is placed in a two dimensional plane, the central joint is free to move in all directions of this plane. None the less, it can not move endlessly in one direction since eventually one of the arms will stretch, but we can draw a small enough disk containing the joint's original position. The central joint's motion is bounded by this disk. The points on the border of this disk represent states in which there is at least one arm stretched.

If we fix the central joint in what positions can the other joints be in? |

Because the other joints are locked between the anchors and the central joint, if we fix the central joint, the other joints can not move without breaking the bars, which is also a breaking of the rules of movement. Also if the arm is not stretched then it can take two positions:

What can we make out of these two observations? If the central joint
position is not on the border of its movement disk (no arm is stretched) this point
corresponds to 2 |