Why is the C- Space topology for a 2R robot a torus?

Question

Currently doing a course on robotics - skip to 1:12 in the video below where Kevin Lynch describes the C-Space topology of a 2R robot to be a torus. Why did he rotate the circle for joint 1 to be perpendicular to that of joint 2?

https://youtu.be/z29hYlagOYM?list=PLggLP4f-rq01z8VLqhDC94W2nWpWpZoMj&t=72

Also shouldn't the C-Space for such a contraption be a annular disc? notice the difference between the fig1 and fig2 in the illustration I drew.

I get that the range being [0,2π] wrap around to form a torus, but I am confused about how Kevin Lynch approached this problem, aren't both joints operating on the xy plane?

Answer 1

The torus doesn't represent the actual motion of joints in space it is used to represent the C-space only.

You need a C-space that can represent all the combinations of two different angles i.e., all pairs ($\theta_1, \theta_2$) so that by picking a point on the C-space you can uniquely determine the location of both the joints.

If you use the annular ring and you pick any point on the annular ring so you can find out the angle of the first joint but how do you represent the angle of second joint.

But if you take a torus the point from the centre will represent the angle of the first joint and the angle around the peripheral ring of the torus will represent angle of the second joint as i tried to show in the image.

I tried to draw so that it is easier to visualise(pretty bad actually but hope it helps)

So if you take a disc it wouldn't be possible to uniquely describe the location of $\theta_2$ and if you try you will get overlapping circles.

Again, don't confuse C-space with the actual motion of the joints. it is a space which is used to map each possible position of the 2 joints to a unique point on some region(here the torus).