calculating position using follower wheels

[Ether]

Originally Posted by flameout

this is way too easy when all I need to do is change a script I already wrote.

Yeah; I had already prepared Question 6 back when everyone was using a closed-form analytical solution for Q(t). You beat me to the punch by setting up your numerical integration script.


Question 6 solution:

Spoiler for solution:

Position: ( -3.7271, -4.0749)
Distance: 60.8613

I think you are showing more decimal places than are warranted for the accuracy of your solution method. I'm differing from you in the third decimal place for X and Y. Compiled C on 8-year-old 32-bit Windows XP machine takes roughly 40ms to compute.

Can someone else weigh in with their numbers for this?

[flameout]

Originally Posted by Ether

Yeah; I had already prepared Question 6 back when everyone was using a closed-form analytical solution for Q(t). You beat me to the punch by setting up your numerical integration script.

Makes sense.

Originally Posted by Ether

Question 6 solution:

Spoiler for solution:

Position: ( -3.7271, -4.0749)
Distance: 60.8613

I think you are showing more decimal places than are warranted for the accuracy of your solution method.

I never bothered to do any accuracy analysis, so that's probably the case.

After lowering the tolerances and calculating bounds for the error (total error should be less than 1e-7), I got the following revised figures:

Spoiler for solution:

Position: (-3.7350, -4.0685)
Distance: 60.8668

New parameters and error analysis:
Relative and absolute tolerances set to 10^-12 (previously 10^-6): The error at each step should not exceed max(10^-12, |x * 10^-12|), where x is the current value of the ODE solution
Number of ODE solver steps: 1022
Maximum element of the state: 60.8668
Upper bound for error: max(10^-12, |60.8668 * 10^-12|) * 1022 = 6.2267e-08

[Ether]

Originally Posted by flameout

After lowering the tolerances and calculating bounds for the error (total error should be less than 1e-7), I got the following revised figures...

Spoiler for solution:

Position: (-3.7350, -4.0685)
Distance: 60.8668

That matches my results out to the number of decimal places shown, except I'm getting a 6 instead of a 5 in the fourth decimal place of the Y value.

[flameout]

Originally Posted by Ether

That matches my results out to the number of decimal places shown, except I'm getting a 6 instead of a 5 in the fourth decimal place of the Y value

I re-ran my script once for each of MATLAB's ODE solvers, adjusting the tolerances as necessary to get a reasonable solve time.

Every solver agreed through the fifth decimal place (for the Y value). The stiff solvers needed significantly larger error tolerances and took more steps; their predicted error was much higher.

The nonstiff solvers all agreed through the seventh decimal place.

The lowest predicted error was obtained through the ode113 solver, using a relative tolerance of 100 * epsilon and an absolute tolerance of 10^-12. According to my error estimate, all of the following digits are correct:

Spoiler for solution:

Position: (-3.73497544, -4.06852591)
Distance: 60.86682701

It was interesting going through all the solvers -- many of the stiff solvers had error estimates in excess of 10^-4. I guess this shows that the choice of solver really can have an effect on the error, and not just the solution time.

[Ether]

Originally Posted by flameout

Every solver agreed through the fifth decimal place (for the Y value).

The rotate_CW function has some nasty behavior near zero. So I tried replacing the numerical integration of that function in the region T<0.01 with an analytic solution of a 3rd order Taylor expansion. I'm now matching your results with a time step of 1 microsecond.

Bleary-eyed, time to call it quits for the night.

[Ether]

The large black rectangle in the attached sketch represents a top view of a robot.

The forward direction of the robot is upwards in the sketch.

The 4 red arrows represent 4 unpowered instrumented (with encoders) omni follower wheels. For each of the wheels, the arrow points in the + direction for that wheel.


Question7

Find the formula for the FWD (forward), STR (strafe right), and RCW (rotate clockwise) robot motions in terms of the X₁, Y₁, X₂, and Y₂ wheel speeds.

[RyanCahoon]

Spoiler for Question 7:

FWD=(Y1+Y2)/2
STR=(X1+X2)/2
RCW=(X1-X2)/L=(Y2-Y1)/W

[Ether]

Originally Posted by RyanCahoon

Spoiler for Question 7:

FWD=(Y1+Y2)/2
STR=(X1+X2)/2
RCW=(X1-X2)/L=(Y2-Y1)/W

Looks good to me. Anybody disagree or have questions?