What is the idea behind calling configuration space a metric space?

Question

In my robotics textbook ¹ , the configuration space has been defined configuration space the following way:

A configuration space is a metric space compromising of all given configuration of a system.

Then he goes one about how we can define a metric on a configuration space of a system and some other ideas on based on it... I can't understand this at all.

What exactly is the concept of configuration space and secondly what is the idea behind defining distances behind it?

After some quick googling, I understood that a metric space is just a set endowed with a notion of distance but I can't understand how it connects to robotics. What exactly is the idea behind what the author is trying to say here?

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¹ Mechanics of Robot Manipualtion, Matthew T. Mason,page-11

Answer 1

I did not read the book. But one of possible thinks that came to my mind is this:

There is big field in mathematica dedicated to preciselly define some terms (set, metric, metric space, ...) and prove a lot of thinks about them. Once you can prove, that something is "metric space" then you can use any result from this part about "metric space" and safely apply it to your problem, knowing, that it fits.

So many authors use this "reduction on known problem" to be able use all knowledge collected about the "known problem" and apply it on the problem they have (and maybe then add something new).

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Say I want win a contest on running thru maze in less time (and I have 2 tries).

Then I would just prove, that the maze can be described as graph where crossing are nodes and ways between them are weighted edges, where the weight is travel time.

Then I can first run just "map" all the maze and in second run use https://en.wikipedia.org/wiki/Shortest_path_problem known graph theory to find "shortest path", in my case fastest (as I maesure the time, not distance).

And then I would folow the fastest way and the rest just dependes on how good motor and navigatition I have, knowing I cannot get better result any other way (I can get the same result at the best).

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So I think, that the author of the book took the same approach - he "proved", that configurations can be described as some already known mathematic constructs and then he use proven knowledges about those mathematical constructs to apply them on configurations and get desired results "fast, easy and proven way".

Answer 2

In regard to your question about what a configuration space is:

Lets pretend we have a car (which is kinda like a mobile robot).

The C-space can be though of as the locations on the map where there is no collision. But what does that really mean for a mobile vehicle?

For example if we are traveling 50 mph towards a brick wall and our breaks are out, you can imagine that at some point it will be too late to turn away from the wall and we will crash into it at some point in the future. So even though the space BEFORE we collided (but after it was too late to turn away) was technically free open space, it was not a free C-space because there was no action we could take at that point which would have avoided a collision.

So what the C-space models is the entire set of valid states of the car, which therefore considers its position, kinematic motion, and steering constraints. These should satisfy some property, like our ability to not crash into the wall. I think the understanding that the C-space is a function of more than just position is an important thing to understand. Because it can be a function of so many things it can be challenging to visualize it in our dimension limited 3D world.

I bet there is some robotics guru who will disagree with my characterization of C-space, and I look forward to learning more from them, but I hope this helps answer the question or spawns at least one productive thought.