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. Lets 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 2N states in the machine's configuration space each state represents the exact positions of the other joints (each joint can take only two positions). So the movement disk corresponds to 2N copies of disks in the configuration space, the interiors of each two distinct disks are distinct and only points on the borders can be common to two disks. In other words, the configuration space is constructed by connecting the 2N disks along the borders. Each state in the configuration space which is an interior state of one the disks has a 2-dimensional local property. We can not say yet what is the local property of points on the borders of the disks, but it can't be more then 2-dimensional. It appears that the configuration spaces are some kind of surfaces (if we will be lucky and points on the border will also have 2-dimensional property). In order to find out exactly what kind of surface it represents we will need to study how are the disks borders connected. This will be our motivation when we examine more complicated machines in the next pages.
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