I should demure to Peter Nach. as he is the motion guru in here, but cos I can't resist getting in first
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1. Single axis is fairly self expalantory. One motion planner, one encoder feedback, and one motor being controlled to postion. These three things combined are called an "axis".
2. Axis and half. Same as above but now there are two encoder feedback channels. The extra one is used solely for either display purposes, or more commonly as a "master axis" that is then ratio'ed or geared, or cammed to the other "slave axis".
3. Multi-axis. Simply extends the single axis concept to as many "axis'" as the system supports....typically 8,16 or 32. In this case the system usually supports complex gearing motions between mulitple axis, allowing interpolated, circular, and helical motions to be generated. Usually any axis may be configured a Master or Slave.
Some systems support an axis type called "Imaginary" that have a motion planner only, ie no feedback or motor output. These are useful for complex gearing systems.