The polynomial idea works… theoretically… However, from experience does this solution fit if we used 2nd degree polynomials instead of cubic? Also, for this type of problem is it easier to do matrix math instead of algebra?
You cannot fit a 2nd degree polynomial to 2 arbitrary points in the XY plane if the tangents at both points are specified.
A 2nd degree polynomial has only 3 adjustable parameters, and there are 4 equations to be satisfied:
Y[sub]1[/sub] = f(X[sub]1[/sub])
Y[sub]2[/sub] = f(X[sub]2[/sub])
Tangent1 = f**’**(X[sub]1[/sub])
Tangent2 = f**’**(X[sub]2[/sub])
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.
has a good primer on the subject. This is commonly used for computer graphics and simulations.
Have fun
*This is not a 3D problem. It’s a problem in a 2D flat plane, and a skidsteer 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:

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?

how to compute the curve defined by those specifications

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

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

is the path to be computed once (at the start) and then executed to completion without feedback, or is the path periodically recomputed 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: rotateinplace at point A, drive straight to point B, rotateinplace. probably not what the OP would consider a “smooth” curve
limited minimum radius of curvature: many skidsteer vehicles don’t like a small turning radius. they hop and jerk.
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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 addon 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 05% 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.[/quote]
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.
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.
*Food for thought
See attached graph.
There’s a Segway with a 2foot 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 skidsteer 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.
*
tracks equations.pdf (16.1 KB)
tracks equations.pdf (16.1 KB)
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.
The reason I suggested them is because they are parametric, so I thought they might be easier to deal with, and also you can control the shape of the curve in without changing the tangents at the endpoints easily, which I’m not sure you can do with cubic curves.
This is all really cool by the way, I’d love to try to implement this in our robot, even if it isn’t entirely practical or needed, just because it’s such an interesting problem. Another layer of complexity to add with our robot though is that we use Mecanum drive, so basically we could have the robot drive along the curve with the X and Y coordinates being affected by forward and strafing motion, and the heading just constantly changing by itself from the first tangent to the last.
I think that actually brings up another problem though, which I’ve been wondering about for a while: how do you get a robot with Mecanum drive to drive completely straight while turning at a constant rate (as in, the overall motion of the robot is in a straight line, but it is constantly rotating)? This might be going way beyond what the OP was looking for, but it was just something I was thinking about since we have Mecanums.
If you have a gyro, so that you know the heading of the robot, you can transform the driver’s commands into “fieldcentric” commands1. So, for example, pushing forward on the joystick would mean “translate straight downfield, regardless of your current heading”. So the robot could be rotating continuously as it travels a straight line.
In actual practice, it doesn’t work perfectly, because of sensor accuracy and mechanical imperfections (pesky friction and free play etc), but with a welldesigned robot built with fine craftsmanship, you can get pretty close.
1The equations to do this can be found here; or you can use the gyro input to the mecanum drive provided in WPILib
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*
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 .
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.
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
Bezier curves are defined by a formula that can be generalized for any degree, but for the most common, degree 3, the equation for any point along the curve is: B(t) = (1  t)^3 * P0 + 3 * (1  t)^2 * t * P1 + 3 * (1  t) * t^2 * P2 + t^3 * P3 for any t between 0 and 1, inclusive. So you have four control points in a degree 3 curve which you can use to define the shape of the curve, its endpoints, and its tangents at the endpoints (the applet Ether linked to are great for seeing how these affect the curve). I’m not a math genius like Ether so I’m not really sure exactly how to implement them into your question, but in my experience with computer graphics I’ve found that Bezier splines are easier to control in a way that makes sense than cubic splines.
The Wikipedia article on Bezier curves has a lot of cool information on them: http://en.wikipedia.org/wiki/Bezier_curves
Also if you needed to get the slope of a Bezier curve at some point for your pathing algorithm, I found this: http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/bezierder.html
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.
I have been conducting Mathcad webinars for FIRST teams. A search turned up this conversation and I have implemented the Matrix Solution as a Mathcad worksheet. The attached .zip folder contains both a Mathcad Prime 2.0 and a .pdf version of the worksheet.
PTC is once again offering Mathcad Prime for free to participants in FIRST Robotics programs. Contact FIRSTsupport@ptc.com if you are interested in obtaining a copy of Mathcad Prime.
You can learn how to create Mathcad worksheets like this on our Wednesday Night Robotalk Webinars (7 pm EST). This Wednesday night (Sept 19 2012) we will focus on the tools used in the attached worksheet. To register for the webinar click here.
Chris
Curve Driving Generator.zip (700 KB)
Curve Driving Generator.zip (700 KB)
Thanks for posting this Chris. Unfortunately, the example you picked is a case where a cubic polynomial is surely not what you’d want. A Bezier curve would give a much more desirable curve for the robot to follow. Does Mathcad support Bezier curves?
Attached is a Bezier curve for the example you gave.
Also, there’s an Excel spreadsheet here for those who don’t have Maxima* installed:
http://www.chiefdelphi.com/media/papers/2713
 Maxima is a free opensource Computer Algebra System, available for download at the link below. Highly recommended:
http://maxima.sourceforge.net/
*
Bezier Maxima macro.txt (834 Bytes)
Bezier Maxima macro.txt (834 Bytes)
@Ether: The example in the .pdf was more deliberate than unfortunate. I chose a tangent line slope that would exagerate the effect and help high school students to make connections to their work in schools. (I am an educator, not an engineer.)
I included the .pdf more to show what a Mathcad worksheet looks like. I think it is more transparent than either Excel or Maxima because Mathcad uses standard math notation. The .pdf was not meant to be the primary contribution.
The Mathcad worksheet is a computational document, so it is possible to change the parameters. The worksheet is therefore a general method for calculating a cubic polynomial given the endpoints and tangent slopes. As you observed in your post, there are some examples where the solution is efficient, others where it is not. With the Mathcad worksheet, it is possible to quickly derive and plot a path and determine whether or not it is a solution worth implementing.
It is certainly possible to build a Mathcad worksheet to generate the Bezier Curve. I will add this to my “Want to Do” list and post it to the Thread when I cross it off. Thanks for the suggestion and the Excel worksheet. I’ll probably convert your Excel spreadsheet to Mathcad.
Ah. Thanks for that clarification. In that context, I think your example is excellent.
I’m both an engineer (now retired after 33 years) and an educator. I share your passion about teaching.