How to Make a Robot Car Go in a Straight Line#
After messing around with my micropython-based esp8266 chip, I’ve finally got to the point where I can make the robot go in a straight line.
The Hardware#
My robot car is a dual-motored car. This means that unavoidably I’ll need to correct the tiny differences between the two motors so as to make it go in a straight line. To achieve this goal, I bought an MPU9250 motion sensor which can detect its acceleration \(\mathbf{a}\), angular velocity \(\mathbf{\omega}\) and the earth’s magnetic field \(\mathbf{B}\).
The Original Algorithm#
My original algorithm is to increaase the speed of one motor and decrease the speed of the other when I detect the robot is turning to one direction. As the motors are controlled by the voltages supplied, when I apply a voltage difference between the two motors, the motors will operate at different speeds.
The following is a theoretical analysis of my original algorithm.
The scalar angular velocity \(\omega\) can be determined by projecting the vector onto the gravitational unit vector:
\[ \omega = \vec{\omega}\cdot\frac{\vec{g}}{|\vec{g}|} \]The angular velocity can be approximated as a linear function of \(V\), the voltage difference between the two motors I apply:
\[ \omega(V) = a + bV \]The voltage difference is assumed to have a linear relationship with respect to he angle \(\theta\) and the angular velocity \(\omega\):
\[ V(\theta, \omega) = p\theta + q\omega \]In step 3, \(\theta\) is determined by numerically integrating \(\omega\):
\[ \theta = \int_0^t\omega(t)\,dt \]Then a differential equation can be set up and solved with separation of variables:
\[\begin{split} \begin{align*} \dot{\theta} &= a + b(p\theta + q\dot{\theta}) \\ (1-bq)\frac{d\theta}{dt} &= a + bp\theta \\ \int_0^\theta\frac{d\theta}{a + bp\theta} &= \int_0^t\frac{1}{1-bq}dt \\ \frac{1}{bp}\ln\left|\frac{a + bp\theta}{a}\right| &= \frac{1}{1-bq}t \\ 1 + \frac{bq}{a}\theta &= \exp\left(\frac{bp}{1-bq}t\right) \\ \theta &= \frac{a}{bq}\left(\exp\left(\frac{bp}{1-bq}t\right)-1\right) \\ \end{align*} \end{split}\]
This implies that my original algorithm should never be able to achieve a perfect straight line.
Nevertheless, I tried it out.
Note
The units of \(p\) and \(q\) were all messed up because I was using degrees and the voltages I used were not in volts either. I omitted them because they are irrevelant here.
\(p\) |
\(q\) |
\(\Delta t\) |
Result |
|---|---|---|---|
10 |
20 |
0.2 |
oscillates |
10 |
10 |
0.2 |
tilts about 25 degrees |
50 |
10 |
0.2 |
oscillates |
20 |
10 |
0.2 |
tilts about 25 degrees |
20 |
12 |
0.2 |
tilts about 15 degrees |
20 |
10 |
0.6 |
tilts about 5 degrees |
(\(\Delta t\) is the period with which I update the voltages)
As you can see, the results are far from satisfactory. Oscillation is not shown in the theoretical derivation but it was probably a result of latency. Another significant problem was that none of them works on relatively smooth surfaces. (It oscillates violently)
The Solution#
I eventually solved this problem by using accumulative voltage difference with weight decay.
Instead of using a fixed formula for \(V\), I update the value of \(V\) with this recursive formula:
where \(\alpha\) is a constant in the interval \((0, 1)\). This way, more recent measurements of \(\theta\) and \(\omega\) constitute a larger portion of \(V\) while older ones constitute a smaller portion.
This not only smooths out the curve for \(V\) but also implies that I will be able to reduce \(\Delta t\) to a smaller value and thus gain more accurate control.