Calculate Speed of a Non Standard X Drive

[Xenon27]

I’m well aware that for a standard X drive with 45 degree angled wheels, the forward speed of the robot is theoretically 1.41 times faster than the speed of a tank drive with the same wheel size and rpm.

But what about with an X drive that does not have its wheels angled at 45 degrees? I’ve been experimenting with an angle shifting X drive in CAD, and I was curious, how would one calculate the theoretical forward speed of an X drive with a variable wheel angle? I can’t seem to find any resources on this, since everyone just seems to know that a standard 45 degree x drive moves at 1.41 times as a 0 degree x drive (aka tank drive)

For discussion I’m considering an x drive where the wheels are facing perfectly forward to be 0 degrees and when the wheels are facing perfectly inwards (perpendicular to the forward direction of the robot) to be 90 degrees.

From experience I know that as the angle of the wheels nears 90 degrees, the theoretical speed of the robot increases. (I once made a little rig with angled omni wheels and found that when angled shallower than 45 degrees, the rig moved faster forwards than when the wheels were angled to be 45 degrees)

So basically my question is, how does one figure out the theoretical forward speed of an x drive just from a wheel angle?

this is an example of an x drive with an angle greater than 45 degrees.

[jpearman]

It doesn’t answer your question, but if you haven’t seen it, this was sort of fun.

Shifting X-Drive

[image] Yet another transmission system, this one allows you to shift from normal 4-wheel tank drive to 45 degree x-drive with the power of pneumatics! Video: http://www.youtube.com/watch?v=KtsbIVxHNb8 Read the full write-up (including close-up pictures) of the drivetrain here: http://www.aura.org.nz/archives/1141 So what’s the point of having an x-drive? Apart from being able to strafe, it goes faster than a tank drive too! Don’t believe us? Read this arti…

[Xenon27]

yeah I’ve seen that. pretty cool, I’m attempting something similar except with a variable angle instead of just 0 or 45 degrees, and powered by a fifth motor instead of pneumatics.

[mvas]

@Xenon27

So basically my question is, how does one figure out the theoretical forward speed of an x drive just from a wheel angle?

You break up the total velocity vector of each wheel into component parts and sum the parts. The direction and magnitudes of the resultant vector can be derived from trigonometric functions

[Ethan5956F]

@Kyle1 it is your time to shine

[2775Josh]

The multiplier will be secant(wheel offset) where 0 is straight forward wheels. Secant(90degrees) is undefined, which makes sense.

[9MotorGang]

I would calculate it using force vectors.

[Xenon27]

@2775Josh

The multiplier will be secant(wheel offset) where 0 is straight forward wheels. Secant(90degrees) is undefined, which makes sense.

you sure about that?

because secant(45) is 1.9, but we all know that 45 degree X drives have a speed multiplier of 1.41

[Xenon27]

@mvas

You break up the total velocity vector of each wheel into component parts and sum the parts. The direction and magnitudes of the resultant vector can be derived trigonometric functions (these will probably be for non-right triangles )

not entirely sure how to do that tbh.
are the velocity vectors the two sides of an imaginary right triangle with the wheel as the hypotenuse?

[2775Josh]

You’re on radians lmao. No worries, just type degrees after or switch your mode.

[Xenon27]

@2775Josh

You’re on radians lmao. No worries, just type degrees after or switch your mode.

ahh lol google calculator defaults to radians for some reason.

[holbrook]

Here’s a graph showing how the multipliers for speed (red) and torque (blue) would change depending on wheel angle, with 0 degrees being parallel to the direction of motion.

A cool demo with a robot like this would be to simulate a continuously variable transmission, with the wheels starting a 0 degrees when the robot begins moving and gradually angling towards 90 as it picks up speed. How high could you get that speed multiplier before there isn’t enough torque to sustain motion? Would be fun to find out.

[Xenon27]

I get the feeling that as the angle gets closer to 90 degrees, the actual tolerances in the system will cause issues before the power losses from friction. Because if one wheel is angled at say 70 degrees, and another wheel is angled at 72 degrees (just due to slop in the angling system) then the two wheels would be trying to pull the robot forward at significantly different speeds.

a way you could fix this is have a really precisely placed hard stop at the greatest angle you would ever go, that way when the wheels go that aggressively angled they all have a very close angle.

[technik3k]

@Xenon27, I think you will find these two discussions informative and interesting:

@technik3k

There was recently a very interesting thread on Chief Delphi where they discussed pros and cons of the Mecanum vs X-Drive (which they call Poor Man’s Mecanum or PMM).

This post by GeeTwo describes an asymmetric X-Drive:

Chief Delphi – 20 Jun 19

Mecanum vs Poor Man’s Mecanum

Another difference which has not been noted here between Mecanum and what OP refers to as PMM: With the same motors, gearing, and wheel size, Mecanum accelerates 41% better than PMM, while PMM has a 41% higher free speed than Mecanum. The big…

A wrinkle on PMM I experimented with a couple of years ago is one I called Pakuni Drive†, named after the simian aliens from Land of the Lost and their attribute of walking forwards like humans, but running sideways like my little brother Vernon when he was 18 months old. The idea is that I toed in the wheels ~22.5° rather than 45°. (It turns out that the acute angle on a 5-12-13 right triangle is closer to 22.62°, so that’s what I really did.) Anyway, this gives a forward acceleration which is 2.4x better in the forward direction than the lateral (strafe) direction, but a free speed which is 2.4x better in the lateral. My intention was to start the robot accelerating in the “forward” direction, then rotate the robot 90 degrees to “shift” to a speed 2.4x as fast for longer (cross-field) runs. My attempts with an arduino and no gyro or other feedback nor radio input were unsuccessful, so I’m putting the concept out there in case anyone wants to try this with better control systems.

† My earlier name for this before inspiration hit was roto-shifter, as will be explained in later sentences.

@technik3k

In addition to video link in OP, anybody interested in this topic may want to check AURA’s experiment video as well as their tutorials.

To answer the question about torques, here is the graph of the force and velocity as the function of x = angle between proposed wheel orientation and the default orientation as in the tank drive (straight forward):

The unit of force is a single motor max (stalling) output and the unit of velocity is similarly its max (idle) velocity. In case when x=0 (tank drive) the force is 2 (two motors) and velocity is 1. Then, in the ideal case, force will drop as the function of cos(x) and velocity will grow reciprocally (dashed lines).

However, in the real world, friction, especially in the omni rollers, will make things less than ideal. As the angle (x) increases so will the friction loses and both force and velocity will start underperforming their ideal curves more and more (solid lines on the graph). Real formulas would be more complicated and I made friction coefficient high for the illustration purposes, but at some angle (x) the wheels will either slip or the friction will be consuming almost all available motor power, PTCs will be tripping, and both velocity and torque will effectively go to zero.

Note that power curves are not the real power output of the motors, since you cannot have both max torque and max velocity at the same time. Still they are useful for comparison of what you could expect from your setup.

So to answer original question I would agree with @Cody and recommend against 30 / 60 degree drive, because at higher angles it will be more sensitive to extra friction and you will be losing power fast.

But it would make a great entry for the eng. notebook to build a test chassis and measure actual forces you could get out of it. And, if you like what you get, it may work great for you.

[mvas]

@Xenon27

are the velocity vectors the two sides of an imaginary right triangle with the wheel as the hypotenuse?

Exactly right…

Here’s a better visual –– just imagine that arrow is the output velocity of one of the wheels and the dashed lines are it’s component parts. The angle can be arbitrarily picked.

I wonder if you could use Calculus and do an optimization problem based on the constraints of the motor outputs. You would solve for maximum speed based on the wheel angle when the motors are operating at maximum efficiency. Maybe there are too many variables for one problem, but it can actually be quite simple when the other parameters are defined.

[9935E]

@2775Josh

The multiplier will be secant(wheel offset) where 0 is straight forward wheels. Secant(90degrees) is undefined, which makes sense.

This is how you do it.
secant is 1 divided by the cosine of x