@George:
I must admit I smiled when I read your post since somewhat similar thoughts have passed through my mind in the past couple of years.
However, words enter the English language through repeated usage which facilitates communication, and in the field of robotics the adjective "holonomic" when applied to a robot has come to mean that the robot can translate (X,Y) and rotate (theta) simultaneously and independently.
This usage of the word is a good thing, in the sense that it is widely understood and there is no other word available that captures quite the same meaning.
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many people in the FRC community (and many people working in the field of robotics in general) use the word "holonomic" interchangeably with the word "omnidirectional"
an omnidirectional robot does not have velocity constraints, holonomic or otherwise, that is the point of building them!
Omnidirectional robots remove the nonholonomic constraints,
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A unicycle has nonholonomic constraints (see Mason, pages 31 & 32), so is it not omnidirectional?
What is your definition of omnidirectional? Do you consider a “crab” drive to be omnidirectional? If not, what technical term do you suggest be used? But if so, what word should then be used for, say, a mecanum, to distinguish its freedom of motion from a crab’s?
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Linguistically, at some point in time, people started calling robots with nonholonomic constraints "nonholonomic robots". This usage is technically incorrect.
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Mason (e.g. c.f. p31) uses the phrase nonholonomic to refer to systems with nonholonomic constraints.
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Nonholonomic constraints refer to velocity constraints that do not result in an algebraic constraint on the position of the robot.
Holonomic constraints are those that result in an algebraic constraint on the position of the robot.
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The above is a good non-technical way to explain it. Some experts, when arguing the finer points, would say however that there are nonholonomic constraints which do result in an algebraic constraint on the position of the robot (Mason, p27)... although Mason goes on to say "the robotics literature often neglects this point, and in fact this book will henceforth also neglect it".