Geometry & Trig Quiz

[Ether]

Originally Posted by Aaron.Graeve

I disregarded the width and length of the wheelbase because the initial and final coordinates were already based on the center of geometry, meaning I could treat this as an omnidirectional unicycle bot

Good move. I put that in there as a distraction.

Actually, since the bot as described does not rotate, any point on the bot could have been used (as long as it was the same point for both the initial and final coords).

Phi = arctan((x1-x0)/(y1-y0))
Phi = Alpha + Sigma
Alpha + Sigma = arctan((x1-x0)/(y1-y0))
Sigma = arctan((x1-x0)/(y1-y0)) - Alpha

This is my solution for the posted problem; I trust it has errors, but I am not certain where they are, so any critiques or suggestions are welcome.

Nice work. Reps to you. You're 95% there...

The formula you gave gives the "correct" angle if y1>y0, but gives the "reverse" angle (180 degrees off) if y1<y0. If y1=y0, it crashes.

For example:

1) if x0=0, y0=0, x1=10, y1=-1, and alpha=90 degrees, the formula gives sigma= -174.3 degrees instead of +5.7 degrees.

2) if x0=0, y0=0, x1=10, y1=0, and alpha=60 degrees, the formula crashes instead of sigma=30 degrees.

There's a simple way fix the formula that does not involve any conditional logic in your code.

Additionally, the wheels must be turned Sigma - Beta from their initial configuration

Let's call that angle "delta". Is there a simple formula (not involving conditional logic) that will compute the shortest value for delta? (For example if delta is 270 degrees the formula should give -90 degrees).

[Aaron.Graeve]

From my reasoning, the code only crashes due to a division by zero in the arctangent function. This could be fixed by using arcsin ((x1-x0)/sqrt ((x1-x0)^2+(y1-y0)^2)). This crashes if the inital and final positions are the same, but the problem is meaningless if you are already at yor final position.

Regarding the smallest Delta, either a conditional check on the absolute value of Delta being larger than 180 degrees or running the delta through arcsin(sin(Delta)) could do the trick. I know the second suggestion is implemetation dependent and not the ideal method but it is a method that I think will work in most cases.

[Ether]

Originally Posted by Aaron.Graeve

This could be fixed by using arcsin ((x1-x0)/sqrt ((x1-x0)^2+(y1-y0)^2)).

That actually made the problem worse. It gives the wrong answer if y1<y0.

For example: If x0=0, y0=0, x1=10, y1=-1, and alpha=90 degrees, that formula gives sigma= -5.7 degrees instead of the correct +5.7 degrees.

There's a simple alternative to arctan(dx/dy) that gives the correct angle for all values of dx and dy except when they are both zero.

Regarding the smallest Delta, either a conditional check on the absolute value of Delta being larger than 180 degrees or running the delta through arcsin(sin(Delta)) could do the trick. I know the second suggestion is implemetation dependent and not the ideal method but it is a method that I think will work in most cases.

There's a calculation that produces the smallest angle without conditional logic or trig functions.

[Aaron.Graeve]

The simple alternative to atan(dy/dx) wouldn't happen to be atan2(dy, dx), would it? I am unsure if the problem permits us to use computer functions that happen to wrap up the conditonal logic nicely.

Edit: I just saw the thread with StangPS and the video. I see you did live up to your comments about disecting the math. Well played. I feel foolish for reinventing the wheel now.

[Ether]

Originally Posted by Aaron.Graeve

The simple alternative to atan(dy/dx) wouldn't happen to be atan2(dy, dx), would it?

Yes it would. atan and atan2 have different behaviors.

atan's range is limited to -90 < atan < +90. So if the problem involves an angle like 95.7 degrees (as in the previously posted example), atan cannot return that value.

atan2's range is -180 <= atan2 <= +180 and it handles the quadrantal angles without crashing. Also, in many implementations (Octave is an example) atan2 handles the arguments (0,0) and returns 0.

You can wrap conditional logic around atan in your own code to produce your own version of atan2, but it's better to use the library version if it's available. Pages 152 thru 158 of P J Plauger's classic 1992 book The Standard C Library shows a reference implementation of atan and atan2.

I don't think you re-invented the wheel. Your diagram and accompanying explanation stated the simplifying assumptions explicitly, used a consistent reference coordinate system, and were very clear and easy to understand.