Movement Of Machines.

One way to study the laws of movement for machines is by examining the various states the machine could take. A machine's state is the exact position of all its joints, bars and anchors. We call the space of all the machine's states, the Configuration Space. You may find this notion very confusing at first, therefore as a first example let us examine a very simple machine and try to study its configuration space:

 The position of the single joint in this machine describes completely the state of the machine, because if we know the joint's position we know the exact location of the single bar and anchor. Thus, the circle this joint  follows while the machine rotates would represent the machine's configuration space. Can  we describe the configuration space as two separate lines as in the following example? Something odd is happening  while the machine is moving away from the state represented by the end of the upper line. No matter how small we rotate the machine in the clockwise direction there is a jump on our 'configuration space'. We do not regard this example as a valid representation for the one-leg machine's configuration space. We require that "close" enough states of the machine would be represented by  "close" enough points on the configuration space. In our next trial we try to represent the configuration space as a single point. Indeed, "close" representations represent "close" states, but by looking at the single point of the configuration space, we cannot differentiate between the various states of the machine. Therefore, this is not a valid representation of the configuration space; the single point represents more then one state of machine.

A valid representation of the configuration space must follow the above two rules that we now summarize:

1. Representation is continuous: "Close" states must be represented by "close" points.
2. Representation is in One-To-One correspondence: Each point in the representation represents exactly one state of the machine.

The first representation for configuration space of the one-leg machine as a circle follows these two rules and is therefore a valid representation. In General, Representations which follow these rules are called Maps.

 Notice that the representation depends not only on the geometrical object, but also on the exact mapping between the states of the configuration space and points on this object. As an example here is another representation of the configuration space of  the 1-leg machine as a circle. While the leg completes a 360o turn, the green point on the circle has completed two 2 rounds on the circle. Therefore, the mapping does not follow our second rule: representation is not in one-to-one correspondence. Each point on the circle represents 2 possible states of the machine. positions of the joint.

Before studying maps of machines we will look on some more familiar maps on the next page.

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