Curve Driving Generator

[Ether]

Originally Posted by theNerd

what if the current heading angle was a slope of infinity?

Pick a coordinate system whose origin is the starting point and whose positive X axis aligns with the starting direction of motion.

Compute the ending point (X₂,Y₂) and slope m₂ in that coordinate system.

Then the cubic becomes simply Y = aX³ + bX²

and the formulas for a and b are simply

a = (m₂X₂ - 2Y₂)/X₂³

b = (3Y₂ - m₂X₂)/X₂²

[Whippet]

Why not use tank control mode? That way, you can drive curves, but it takes a little more practice to use. It's just a matter of willingness to learn, but will save a lot of time programming. So it's kind of a double-edged sword.

[Ether]

Originally Posted by Whippet

Why not use tank control mode? That way, you can drive curves, but it takes a little more practice to use. It's just a matter of willingness to learn, but will save a lot of time programming.

That's not what the OP wants to discuss:

Originally Posted by theNerd

Lets not discuss why we would or would not do this but instead, focus on how. Thanks!

[MichaelBick]

Originally Posted by Whippet

Why not use tank control mode? That way, you can drive curves, but it takes a little more practice to use. It's just a matter of willingness to learn, but tank mode can do curves, and will save a lot of time programming. So it's kind of a double-edged sword.

I agree. One of the reasons that a lowered center drive(I'm assuming not 4 wheel or treads, as I would not try these maneuvers with those drivetrains) is so competitive, is because it needs very little programming. Sure, some teams have more complex code(I'm thinking 254 with their physics engine for turn radius) that is really nice, it is not worth the effort for the negligible gain, unless you have extra time(like maybe getting a camera working and shooting accurately).

[dcherba]

The real problem you have is one of reference frame. The curve it self is not important as it represents some path on the playing field frame of reference. The incremental control of the robot is based on its frame of reference to the center of the robot for example.
This is a common problem for aircraft, satellites and just about any navigation problem.
The solution is to use a series of translations and rotations. The best way to do that is with quaternions as they do the rotation without ambiquity.

http://en.wikipedia.org/wiki/Quatern...atial_rotation
has a good primer on the subject. This is commonly used for computer graphics and simulations.

Have fun

[Ether]

This is not a 3D problem. It's a problem in a 2D flat plane, and a skid-steer vehicle has only 2 degrees of freedom (rotation and fwd/reverse translation) in that plane.

To execute a smooth* path from an arbitrary point A (starting with heading angle Alpha) to point B (ending with heading angle Beta) the vehicle's instantaneous translation and/or rotation rates must continuously change over time in order to follow that path and arrive at the destination with the proper heading.

So the problem is:

1) given the starting and ending points and headings, what criteria are to be used to define the desired path? smoothest* curve? Shortest distance* ? Shortest time? radius of curvature* never less than some specified minimum value?

2) how to compute the curve defined by those specifications

3) how to use the curve to determine the required instantaneous vehicle rotation and translation at any point along the curve

4) how to use the answer to #3 to determine the left and right wheel speeds (for skid steer) at each point along the curve

5) is the path to be computed once (at the start) and then executed to completion without feedback, or is the path periodically re-computed to minimize accumulated drift errors

For a subset of the possible starting and ending points and headings, it is possible, as explained in earlier posts in this thread, to compute a cubic polynomial which defines a smooth path. For a given vehicle forward speed, this polynomial can be used to determine the required instantaneous vehicle rotation rate at each point along the path.

In another earlier post, dbeckwith suggested a Bezier curve. I believe the instantaneous vehicle rotation rate for a given vehicle forward speed can also be computed from the parametric Bezier equations, but I haven't actually done it yet.

The bottom line here is that the OP stated a problem without much context and asked that the discussion be limited to "how" and not "why". It makes for an interesting academic discussion, but I suspect that if we had greater insight into what was trying to be accomplished the solution might be different.



*notes:

smooth path: perhaps for esthetic reasons? the OP did not specify why

shortest distance: rotate-in-place at point A, drive straight to point B, rotate-in-place. probably not what the OP would consider a "smooth" curve

limited minimum radius of curvature: many skid-steer vehicles don't like a small turning radius. they hop and jerk.

[theNerd]

Originally Posted by Ether

[smooth path: perhaps for esthetic reasons? the OP did not specify why

Sorry for the confusion guys,

The reason for this problem is in aiding the drive in autonomous and teleop - more so autonomous. Also, this is just an exercise for the future programmers of our team. I had noticed that although the rotate in place is the "shortest distance" method it is not always the fastest - especially for a very smooth control - as that the robot must slow down, rotate, and then speed back up. While if the robot drove in a curve it would never need to stop, only change wheel speeds. So then, I'm looking for the fastest route possible. (The reason I did not want to discuss why is because most times I post a thread there always seems to be a discussion as to why not or why and completely turn away from the answer I wanted. I apologize for not being more specific).

I am planning on integrating this with distance PID loops to control the drive. Essentially this is an add-on to a drive system I am creating that pulls from a list of coordinates that I want to go to, sets the left and right wheel distances that the robot must travel to get to that point, and then feeds those distances into a PID loop. Once the error in the pid loop reaches around 0-5% it then pulls the next coordinate to go to. I also, figured that a coordinate system relative to the robots current position would be more beneficial than a coordinate system that is fixed for what I am trying to do.

The equations that the robot uses to determine left and right distances use the arc length of the path, and the angle through which the robot will turn while it is driving along that path. If the angle is 0 then it drives in a straight line, if the distance is 0 then it rotates in place. The equations are thus:

double rightDistance = ((wheelWidth*angle) + (2*distance))/2;
double leftDistance = ((2*distance) - (wheelWidth*angle))/2;

The advantage of this is that I only need one set of equations to drive curves, or perform the traditional rotate and drive.

I am also planning on working with trapezoidal motion profiling in this. I am using these equations:

x = x0 +dx(P(u));
P(u) = -2u^3+3u^2;
u = currentTime - startTime/delta_time;

Where x is the distance travelled in 1 dimension and dx is delta_x and x0 is the startingX --> which is normally 0. P(u) is a function with these properties: f(0)=0, f'(0) =0; f(1)=1, f'(1)=0 and P(u)=[0,1] and u=[0,1].
There is a second set of equations I plan on replacing the above with that include max acceleration and max velocity, I just haven't yet because I'm not sure how to go about testing for those due to a very very small workspace - and running the robot on blocks is inaccurate for our heavy bot.

[MichaelBick]

254 has something like this build into their drive code, though as I said before, I believe it also integrates the current speed of the robot(faster speed, bigger arc, slower speed, smaller arc). I would check their code, as they have made it public, and it is very refined.

[Tom Line]

Is this for teleop or autonmous?

In autonmous, getting a robot to drive an accurate curve is fairly early.

Have a PID loop for distance connected to each side of your drive train, and another PID loop based on a desired angle and the gyro that is added into to your final drivetrain output values. The robot will then drive curves to reach the endpoints, assuming gains are set correctly.

[theNerd]

Originally Posted by MICHAELABICK

254 has something like this build into their drive code, though as I said before, I believe it also integrates the current speed of the robot(faster speed, bigger arc, slower speed, smaller arc). I would check their code, as they have made it public, and it is very refined.

I have checked their code and the sections that I have seen are not well documented and I am confused on the logic behind their code.

[Ether]

Food for thought

See attached graph.

There's a Segway with a 2-foot wheelbase sitting at the origin.

Its heading is aligned with the +X axis.

I drive the Segway so that the center of its axle follows the black Bezier curve.

At the end of the path, the Segway's heading is parallel to the X axis and its coordinates are x=10 y=3.

The red curve shows the track of its left wheel, and the green curve shows the track of its right wheel.

The red, black, and green curves all have the SAME LENGTH (~10.6807 feet), but they are three DIFFERENT SHAPES.

If I were to command the left and right sides of a skid-steer vehicle to each travel exactly 10.6807 feet, the chances that the vehicle would follow the black curve are pretty slim. In fact, if the left and right sides were controlled identically and perfectly, it would travel in a straight line and end up on the X axis with coordinates x=10 y=0.

Wow. All I have to say is Ether = Genius.

I'll email my programmers this thread, since it will put them to good use for the next 9 months or so.

[theNerd]

Originally Posted by Ether

Food for thought

See attached graph.

There's a Segway with a 2-foot wheelbase sitting at the origin.

Its heading is aligned with the +X axis.

I drive the Segway so that the center of its axle follows the black Bezier curve.

At the end of the path, the Segway's heading is parallel to the X axis and its coordinates are x=10 y=3.

The red curve shows the track of its left wheel, and the green curve shows the track of its right wheel.

The red, black, and green curves all have the SAME LENGTH (~10.6807 feet), but they are three DIFFERENT SHAPES.

If I were to command the left and right sides of a skid-steer vehicle to each travel exactly 10.6807 feet, the chances that the vehicle would follow the black curve are pretty slim. In fact, if the left and right sides were controlled identically and perfectly, it would travel in a straight line and end up on the X axis with coordinates x=10 y=0.

Ok, so arc length would not bee what I am looking for but left and right wheel speeds? (Since the integrals of the three curves are different?)
Also, I noticed with the 3rd degree polynomial solution I can not compute a curve for a pi/2 rotation for the robot at the destination point.
How would I go about working with bezier curves as I have never worked with them in my life .

[Ether]

Originally Posted by theNerd

How would I go about working with bezier curves as I have never worked with them in my life .

First play around with these Java applets to get a feel for what a Bezier curve is:

http://www.math.ucla.edu/~baker/java/hoefer/Bezier.htm

http://www.math.ubc.ca/~cass/gfx/bezier.html

... then I will try to answer your questions

[theNerd]

Originally Posted by Ether

... then I will try to answer your questions
]

Ok, I am ready. I have quite a good feel for Bezier curves now....which is great timing since my Calc class just hit parametric equations.