Like many rovers, Square One relies on skid steering. I knew that a skid steer vehicle can turn corners but I had no idea whether it would be precise enough for autonomous driving. I did some experiments to find out. The results were pleasantly surprising.

If you’re not sure what skid steering is then have a look at this article for a recap.

TLDR: Just skip to the end of the article for my conclusions if you’re in a hurry.

Cornering Requirements

I’d been thinking of a corner as a section of a circle. If the rover is driving down the middle of a 60 cm corridor then a left hand turn is just an anticlockwise turn of radius 30 cm which goes on for 90 degrees.

To find out if this would be feasible for PiWars I needed to answer the following questions:

• can the rover drive round a corner smoothly?
• would I be able to control the corner radius?
• would I be able to stop at the correct angle?

Basic Theory

If you think of the rover driving all the way round a perfect circle then the inside wheels must follow a smaller circle than the outside wheels.

If the wheel speeds are proportional to the size of these circles then the centre of the rover should follow the mid-point which is the yellow line.

Given:

• vo - speed of outside wheels
• vi - speed of inside wheels
• r - radius of turn
• w - width of rover (wheel separation)
• v - speed of the rover

then:

• vo = v(r + w/2)
• vi = v(r - w/2)
  

A Bit More Theory

In reality, the basic theory only applies for vehicles with true steering under ideal circumstances. That’s definitely not true for skid steering.

I thought I’d find out if the academics have some better maths. Here’s what they said:

• the basic equations definitely don’t apply to skid steering
• you need to account for the tyres skidding and slipping
• skid steering involves a lot of skidding and slipping
• we’ve no idea how much
• if we just wave our arms a lot whilst muttering about torque and instantaneous centres of rotation then maybe you’ll stop bugging us
• meanwhile here’s some really complex maths that won’t help

… so that was dead end.

Experiments

Square One can control the speed of each wheel independently using a PID controller. I used the equations above to set the target speeds for each wheel and started driving in circles.

Q1 - Driving Smoothly

It was immediately clear that the rover can drive in a nice smooth arc.

Q2 - Controlling the Radius

I now spent several hours driving the rover in circles and measuring the actual radii that it followed. I checked out three factors that I thought might influence the result:

• desired radius (varied from 100 mm to 500 mm)
• the level of grip (smooth wooden floor vs carpet)
• the speed of the rover (varied from 200 mm/s to 1250 mm/s)

From previous measurements I knew that the wooden floor and the carpet had coefficients of friction of around 0.7 and 1.4 respectively. The PiWars challenges should be somewhere in that range.

I found that the actual radius of a turn is always much bigger than the desired radius. The first big surprise is that the ratio between the two doesn’t change much. On the wooden floor it’s around 1.8 times irrespective of the desired radius.

Even more surprisingly, it’s not very different on carpet either. I could only do limited runs on carpet because of problems with the motors hitting either maximum or minimum power. The readings I did get suggest a ratio of about 1.7.

The effect of speed was also a little unexpected. Speed does affect the turning circle. Faster runs create slightly tighter turns. The difference isn’t huge though. The error varied between 1.9 and 1.6.

I decided that I can just divide the target radius by a factor of 1.75 and I’ll get a turn that’s good enough.

Q3 - Stopping at the Correct Point

I have Square One set up to keep turning until the rover has travelled a certain distance. It calculates the distance by averaging the distance that each wheel has travelled. For this experiment, I compared the distance given by the rover to the circumference of the actual circle that it turned. i.e. I’m was discounting the difference between the desired and actual turn radius.

As you can see from this graph, the estimated distance varies by about +-10% of the expected distance. Once again, the errors are slightly smaller on carpet but are otherwise very similar.

Other Observations

In most of my tests, the rover started out running wide before settling down to a steady state. This leaves room to improve my control code which may modify the results a little.

I can log the amount of throttle applied to each wheel. It’s clear that the amount of power fed to the different wheels varies considerably. In a tight turn on a high grip surface, the rear outside wheel will reach full power. Meanwhile, the inside wheels will be braking. In some tests the turn radius was worse than expected because an individual wheel either couldn’t go any faster or couldn’t go any slower.

Conclusions

If the rover can control the speed and power of each wheel independently then skid steering works very well. This assumes that the rover is driven on a smooth consistent surface such as a road or floor.

• the rover turns in nice smooth arcs
• the radius of the turn is about twice as large as expected
• speed has a larger effect on the actual radius than anything else
• the level of grip doesn’t matter much
• the average distance travelled by the four wheels is a very close to the distance travelled by the rover. i.e. odometry works quite well
• tight turns require very high power levels from the outside rear wheel