Week 1: Meet your car

What you will be able to do

Define the lateral error e and read its sign (left of lane centre is positive)
Proof Quiz questions on lateral error and the tempting shortcut
Summarise a whole drive as mean |e|, max |e| and RMS, and say what each one hides
Proof Your lane_error_stats passes the checker for Exercise A in the lab
Count the steps and seconds a car spends off its lane with one line of NumPy
Proof Your steps_off_lane passes the checker for Exercise B in the lab

Warm-up review

Nothing is due today. Pass a week's quiz and its questions come back here at growing intervals.

Lecture

Is this car any good?

You have just been handed a car that steers itself. It rolls along a two-lane road at 8 m/s, about 29 km/h, and nobody is touching the wheel. Before you change a single line of its code, you need an answer to a plain question: is it any good?

"Good" needs a number. By the end of this lesson you will take a whole drive, nearly two thousand moments long, and boil it down to three numbers, and one count, that say how well the car kept its lane. You will also know which of the three to trust for which question, and which tempting number to never trust at all.

Here is where you are heading. This is the report card for the car you were handed, after one lap of a test road:

Figure 1. The provided car's lap of the gentle loop. The blue line is how far the car sat from the middle of its lane at each moment of the lap. Three numbers sum it up, each drawn as a dashed line: 0.38 m off centre typically, 0.44 m when big errors count extra, and 0.53 m at the worst moment. The painted lines are 1.8 m from the lane centre, far above all three.

Every number on that card comes from a recording of the drive, and in the lab you compute each one yourself. The rest of this lesson rebuilds the card, one piece at a time.

Optional: how we got 29 km/h

A speed in metres per second says how many metres you cover in one second. To say it in kilometres per hour, count kilometres instead of metres (1 km is 1000 m, so divide by 1000) and count in hours instead of seconds (1 h is 3600 s, so multiply by 3600):

\begin{aligned}8\ \text{m/s} &= 8 \times \frac{3600}{1000}\ \text{km/h} \\ &= 8 \times 3.6\ \text{km/h} \\ &= 28.8\ \text{km/h}\end{aligned}

That is about 29 km/h.

The car you were handed

Twenty times a second, once every 0.05 seconds, the car runs the same loop:

The camera takes a picture of the road ahead.
The perception box looks at the picture and reports where the lane is.
The controller box turns that report into a steering angle.
The car moves a little, and the camera takes the next picture.

Figure 2. The loop the car runs every 0.05 seconds: camera, then perception box, then controller box, then the car moves, and round again. In this lesson both boxes stay closed. You stand outside the loop and measure where the car went.

In this lesson you open neither box. You instrument the car instead: you record what it did and judge the result, the way a driving examiner judges a driver without reading their mind.

Approximation. The car lives in a flat-world teaching simulator. The road is paint on a perfectly flat plane, and the car moves like a bicycle (a simple model that treats each pair of wheels as one wheel) with a 2.7 m wheelbase, the distance from its front axle to its rear axle. Real cars roll, slip and bounce. Here none of that happens, on purpose, so that every wobble you see has one cause.

Placeholder. Both boxes inside the provided car are stand-ins. You build your own perception box in Your first perception box and your own controller in Closing the loop.

Here is what the camera sees, and the path the car drove:

Figure 3. Top: the 320 by 180 pixel camera picture, 80 m into the gentle loop, with the first bend coming up ahead. Bottom: a top-down view of the stretch from 70 m to 110 m, with the car as a dot at the 80 m mark and the path the provided car drove through it, measured from the lane centre.

The car is tested on three roads. straight is 200 m of straight road. gentle is a stadium-shaped loop: two 100 m straights joined by two wide bends of radius 80 m. curvy is a 30 m straight followed by six tight quarter-circle bends of radius 20 m, alternating left and right.

Guess first: which road do you expect the provided car to find hardest, and why?

Figure 4. The three test roads from above, at the same scale. Each is labelled with its start and the end of its route, and an arrow gives the direction of travel. straight is a 200 m line, gentle is a 702.7 m loop of two long straights and two wide bends, and curvy is 218.5 m of tight bends that flip between left and right.

Answer: curvy. Its bends are a quarter of the radius of gentle's, so the car must turn four times as sharply, and each bend turns the other way from the last. The provided car does not finish it. You will see exactly where it goes wrong near the end of this lesson.

Optional: how we got 702.7 m and 218.5 m

Each length is the distance along the road, measured along the yellow line, from the start of the route to its end, added up piece by piece. A circle of radius $r$ has circumference $2\pi r$ , where $\pi \approx 3.1416$ is the fixed ratio of any circle's circumference to its diameter. So a half circle is $\pi r$ long and a quarter circle is $\frac{\pi}{2} r$ .

gentle is two straights of 100 m and two half-circle bends of radius $r = 80$ m:

\begin{aligned}&2 \times 100 + 2 \times \pi \times 80 \\ &= 200 + 502.65 \\ &= 702.65 \\ &\approx 702.7 \text{ m}\end{aligned}

curvy is a 30 m straight and six quarter-circle bends of radius 20 m:

\begin{aligned}&30 + 6 \times \frac{\pi}{2} \times 20 \\ &= 30 + 188.50 \\ &= 218.50 \\ &\approx 218.5 \text{ m}\end{aligned}

The paint of an open road runs 40 m past the end of the route, so the camera still has lines to look at as the car finishes. Those 40 m are not part of the route, which is why the figure marks the route's end with a square.

Lateral error

Watch the car from above. The lane is 3.6 m wide, with a dashed yellow line on its left and a solid white line on its right. At some moments the car sits dead centre; at others it has drifted left or right.

At each moment you want to know one thing: how far sideways is the car from the middle of its lane? Measure it at the front axle, the line between the two front wheels, because that is where the simulator mounts the camera.

Try a number. If the front axle sits half a metre to the left of the lane centre, the answer is "0.5 m, to the left". Saying "to the left" every time gets tedious, so give left a plus sign and right a minus sign: +0.5 m. A car 0.3 m to the right of centre scores −0.3 m. A car exactly on the lane centre scores 0.

Call this number $e$ , read simply "e", for error. Its name is the lateral error; lateral means sideways.

The lane is 3.6 m wide, so each painted line is 1.8 m from the middle: the yellow line at $e = +1.8$ m and the white line at $e = -1.8$ m. When the front axle reaches a line, the simulator ends the run.

Figure 5. A lane seen from above. A car whose front axle is 0.5 m left of the lane centre has e = +0.5 m; a car 0.3 m right of it has e = −0.3 m. The yellow line on the left is at e = +1.8 m and the white line on the right is at e = −1.8 m.

Left is positive everywhere in this course: for offsets, for headings (which way the car points), for steering angles. Almost every sign bug you will write comes from forgetting it.

Sometimes you care how far off the car is, not which side. Then you drop the sign and write $|e|$ , read "the size of e". A car at −0.3 m and a car at +0.3 m both have $|e| = 0.3$ m, and the run ends when $|e|$ reaches 1.8 m.

Optional: who else measures this

You are not the first to pick this number. Stanford's robot car Stanley, which won the 2005 DARPA Grand Challenge desert race, steered by exactly this measurement:

"The key error metric is the cross-track error, x(t), […] which measures the lateral distance of the center of the vehicle's front wheels from the nearest point on the trajectory." (Thrun et al., 2006, p. 684)

They write it x(t), read "x of t". Cross-track error is Stanley's name for what this course calls $e$ .

A drive is a list of numbers

The simulator measures $e$ at every step, 20 times a second. One lap of gentle takes the provided car 88.85 s, so the whole lap is a list of 1,778 numbers, written $e_1, e_2, \ldots, e_N$ , where $N = 1778$ is how many there are and the small number under each $e$ says which step it came from.

A list of measurements taken one after another in time is called a time series. In the lab the recording arrives as an object called tel, short for telemetry, the record of everything the car measured on its drive. Its lateral errors are tel.lat, one number per step.

Guess first: gentle turns left twice. While the car is going round a left-hand bend, do you expect $e$ to be positive (car left of centre) or negative (car right of centre)?

Figure 6. The provided car's lateral error over one lap of gentle, step by step, plotted against the distance along the road. The two grey bands mark the two bends. The blue line is e.

Answer: positive. In each bend the car drifts toward the inside of the turn, and the inside of a left-hand bend is on the left. On the straights $e$ stays within a few millimetres of 0: its lowest value on the whole lap is one millimetre right of centre, too small to see in the figure. Through both bends the car is left of centre the whole way.

Optional: how we got N = 1778

The simulator records one value at the start of every step, beginning with step $k = 0$ at time $t = 0$ . Step $k$ is recorded at time $t = 0.05,k$ seconds, and the lap ends at the step where the car has driven the whole route, at $t = 88.85$ s. That last step is

k_{\text{last}} = \frac{88.85}{0.05} = 1777

Counting the steps $k = 0, 1, 2, \ldots, 1777$ gives $1777 + 1 = 1778$ values. The first value, $e_1$ , is the one recorded at step $k = 0$ . There is one more value than there are 0.05 s gaps between them, just as a fence with 1777 gaps has 1778 posts.

You may notice that $88.85 \times 8 = 710.8$ m is more than the 702.7 m of the road. That is right: the car drives in the right-hand lane, which is the outside lane of gentle's left-hand bends, so its own path is longer than the road. Measured on the recording, the path is 8.1 m longer: about 11 m extra from driving on the outside, less about 3 m given back when the car drifts toward the inside of each bend.

The tempting shortcut

You have 1,778 numbers and you want one. The obvious move is to average them. A small average sounds like a car that stays near the middle.

Try it on a short drive first. Here are five lateral errors, in metres:

e = (0.1,\ -0.3,\ 0.2,\ -0.4,\ 0.4)

Guess first: this car was off centre at every one of its five steps. What do you think its average is?

Figure 7. The five errors as dots across the lane, drawn the same way as the lane figure above: left of the lane centre (positive) on the left, right of it (negative) on the right. The three positive dots add to +0.7 m and the two negative dots add to −0.7 m.

Answer: zero. Add them up and divide by how many there are:

\begin{aligned}&0.1 + (-0.3) + 0.2 + (-0.4) + 0.4 \\ &= 0.0 \\[4pt] &\frac{0.0}{5} = 0.0 \text{ m}\end{aligned}

A perfect score, for a car that was never once on the lane centre. The errors to the left cancelled the errors to the right.

This happens on real drives too, not only on made-up ones. Take two cars that differ from the provided one in a single way: when they are off centre, they push back toward the lane centre harder. The 2× car pushes back twice as hard as the provided car, and the 4× car four times as hard. (How a steering setting like that works is the subject of Closing the loop; here they are simply two more cars.) Both drive the whole of curvy without leaving the lane. The provided car cannot join this comparison: it runs out of lane on curvy, as you will see below.

Their plain averages are +0.023 m for the 2× car and +0.024 m for the 4× car. Rounded to two decimal places, both are +0.02 m.

Guess first: the plain average cannot tell these two cars apart. Did they drive about equally well?

Figure 8. Both cars on curvy, plotted against the distance along the road, on the same scale. Top: the 2× car swings left of the lane centre and then right of it, by up to 0.82 m. Bottom: the 4× car stays within 0.35 m of the lane centre the whole way. The dashed line on each panel is that car's average, +0.02 m, and it sits in the same place on both panels.

Answer: no. The 2× car swings from 0.72 m left of the lane centre to 0.82 m right of it, while the 4× car never goes more than 0.35 m from it. The 2× car's swings to the left and to the right cancel almost exactly, so its average lands right next to the 4× car's. On a fairer measure, which you meet next, the 2× car is 0.50 m from the lane centre on average and the 4× car 0.09 m. One car is nearly six times further off than the other, and the plain average shows no difference at all.

The plain average does answer a question, just a different one: on average, which side does the car favour? That is called the car's bias. It is worth knowing, but it is not a score, and it must never be used as one.

Three honest numbers

The fix is to stop errors cancelling. Count every error as a cost, whichever side of the line it fell on. There are three good ways to do that, and each answers its own question.

Mean absolute error: how far off, typically?

Drop the signs first, then average. On the five errors:

|e| = (0.1,\ 0.3,\ 0.2,\ 0.4,\ 0.4)

They add up to $1.4$ , and $1.4 / 5 = 0.28$ . So this car sat 0.28 m from centre, typically.

Figure 9. The same five dots, with the two right-of-centre dots flipped across the lane centre to the positive side, their old places shown faded. With every error counted as a size, nothing cancels, and the average of the five sizes is 0.28 m.

Written out in full for these five steps, the calculation is

\begin{aligned}&|e_1| + |e_2| + |e_3| + |e_4| + |e_5| \\ &= 0.1 + 0.3 + 0.2 + 0.4 + 0.4 \\ &= 1.4 \\[4pt] &\frac{1.4}{5} = 0.28 \text{ m}\end{aligned}

For a lap of 1,778 steps you would never write out every term, so mathematicians use a shorthand:

\text{mean}|e| = \frac{1}{N}\sum_{k=1}^{N} |\color{#58C4DD}{e_k}|

Read it aloud as "one over N, times the sum, for k from 1 to N, of the size of e-k". The tall symbol $\sum$ is the Greek capital letter sigma, and it means "add up what follows", here for every step from $k = 1$ to $k = N$ .

Confusion is normal here. Sigma is only shorthand for a long sum. If it slows you down, read it as the written-out sum above; it means exactly that.

This number is the mean absolute error, written mean $|e|$ . Absolute is the mathematician's word for "with the sign dropped".

Maximum absolute error: how bad was the worst moment?

Sometimes the typical moment does not matter. What matters is the single worst one, because that is when a wheel might touch the line. Take the largest of the sizes: for the five errors, $0.4$ m. The shorthand is

\max|e| = \max_{k} |\color{#58C4DD}{e_k}|

read "the largest, over all steps k, of the size of e-k". Its name is the maximum absolute error.

Root mean square: how far off, counting big errors extra?

The third number squares each error before averaging, which turns every error into a positive cost and makes the big ones count for more. On the five errors:

Square each one: $(0.01,\ 0.09,\ 0.04,\ 0.16,\ 0.16)$ .
Take the mean of the squares: they add to $0.46$ , and $0.46 / 5 = 0.092$ .
Take the square root: $\sqrt{0.092} = 0.303$ m.

There is a picture behind those three steps. Squaring a length gives the area of a square with that side.

Figure 10. Each of the five errors drawn as a square whose side is the error's size, so its area is the error squared. The two 0.4 m errors make the two biggest squares. Averaging the five areas gives 0.092 square metres, and a square of that area has a side of 0.303 m: that side is the RMS.

Squares of metres are square metres, and the square root brings you back to metres. That is why this number can sit beside mean $|e|$ on the same scale.

The shorthand reads from the inside out, the same order you just worked in: square, take the mean, take the root.

\text{RMS}(e) = \sqrt{\frac{1}{N}\sum_{k=1}^{N} \color{#58C4DD}{e_k}^{2}}

Its name is the steps read backwards: root mean square, or RMS.

Now look back at the report card at the top of this lesson. For the provided car's lap of gentle, mean $|e|$ is 0.38 m, RMS is 0.44 m and max $|e|$ is 0.53 m, and you can say what each of them means.

Why RMS punishes big errors

Here are two five-step drives:

Drive A: $(0.2,\ 0.2,\ 0.2,\ 0.2,\ 0.2)$ . Always a little off.
Drive B: $(0,\ 0,\ 0,\ 0,\ 1.0)$ . Perfect, then one big swerve.

Both have the same mean $|e|$ : $1.0 / 5 = 0.2$ m.

Guess first: do they have the same RMS? If not, which is bigger, and by roughly how much?

Figure 11. Drive A and Drive B as squares, one above the other. Drive A has five small squares of side 0.2 m. Drive B has nothing for four steps, then one big square of side 1.0 m, with as much area as twenty-five of Drive A's squares. Drive A's RMS is 0.2 m; Drive B's is 0.447 m.

Answer: Drive B's RMS is more than twice Drive A's.

A: every square is $0.04$ , their mean is $0.04$ , and $\sqrt{0.04} = 0.2$ m.
B: the squares are $(0, 0, 0, 0, 1.0)$ , their mean is $0.2$ , and $\sqrt{0.2} = 0.447$ m.

The swerve of 1.0 m is 5 times the steady 0.2 m, but its square is 25 times as big. Squaring grows big numbers much faster than small ones, so one large error dominates the average of the squares.

That matches driving. Being 0.2 m off centre all the time never brings a wheel near a line. Being 1.0 m off for a moment might. When the worst moments matter most, look at RMS and max $|e|$ , not only at mean $|e|$ .

One more fact you can check on every example here: RMS is never smaller than mean $|e|$ (0.303 ≥ 0.28, and 0.447 ≥ 0.2). The two are equal only when every $|e_k|$ is the same, as in Drive A. The more uneven the errors, the wider the gap.

Counting steps off the lane

Some questions have a yes-or-no answer at each step. "Was the car more than 0.9 m from centre just now?" The course uses 0.9 m, half the 1.8 m from the middle to a line, as its off-lane limit. Why 0.9 m and not 1.8 m? $e$ is measured at the front axle, a single point, and a typical real car is about 1.8 m wide (an approximation), so with its axle 0.9 m out a wheel is already near the paint. The simulator's car is a point with no width, but the limit is set as if it had one. A step is off the lane when

|\color{#58C4DD}{e_k}| > 0.9 \text{ m}

The comparison is strict: a step at exactly 0.9 m is on the limit, not over it.

Figure 12. The lane from above, with two shaded strips along its edges: the off-lane zone, from 0.9 m to 1.8 m either side of the lane centre. A car whose front axle is anywhere in a strip is off the lane, though its front axle is still inside the painted lines; at the painted line, 1.8 m out, the run ends.

To count off-lane steps, ask the question at every step and count the yeses. With the five errors and a stricter limit of 0.35 m, the sizes $(0.1, 0.3, 0.2, 0.4, 0.4)$ give no, no, no, yes, yes: 2 steps. To turn a count of steps into time, divide by 20 steps per second.

Guess first: the provided car leaves curvy 48 m into the 218.5 m route. Before it went off, how long had it spent more than 0.9 m from centre: a blink, or whole seconds?

Figure 13. The provided car's distance from centre on curvy, plotted against the distance along the road. The dashed grey line is the 0.9 m off-lane limit and the yellow line at 1.8 m is the painted yellow line. The steps above 0.9 m are shaded. The run ends at 48 m, where the car reaches the line.

Answer: whole seconds. The car spends 40 steps more than 0.9 m from centre, which is $40 / 20 = 2.0$ seconds, before it reaches the line at 48 m. All 40 are in the first bend, over the last 15 m before the car leaves, from 33 m to 48 m.

What does doing nothing score?

Is 0.38 m good? A number means little until you have something to set it against. So score the laziest car imaginable: one that holds the wheel dead straight and never steers at all. Call it the do-nothing car.

Guess first: on which of the three roads, if any, does the do-nothing car get a perfect score?

Figure 14. The do-nothing car's path, in grey, against the provided car's, in blue. On straight the grey path runs exactly down the middle of the lane. On gentle it runs straight on at the first bend and its run ends 117 m into the route; a magnified inset shows the two paths splitting.

Answer: straight. A car that starts in the middle of a straight lane and never turns stays in the middle, so all three of its numbers are 0.00 m, the same as a perfect driver's. That is why one road is never enough: on straight, a car with no steering at all is indistinguishable from a flawless one. On gentle the do-nothing car is gone 14.7 s into the lap, 117 m along the road, at the first bend.

Its mean $|e|$ on gentle comes out at only 0.10 m, lower than the provided car's 0.38 m. That is not a better drive. It is an average over the 117 m the car managed before its run ended, and most of that was straight road. A number from an unfinished run cannot be set against a full lap. Compare two cars' numbers only when both drove the same road to the end.

Now 0.38 m has its place. A perfect car scores 0; a car that never steers crosses the painted line at the first bend. The provided car finishes the lap, but it uses up to 0.53 m of its 1.8 m of room to do it. That is the bar your own controller has to beat in Closing the loop.

Arrays, not loops

In the lab, tel.lat is a NumPy array (NumPy is Python's standard library for arrays, imported as np): a list of numbers that you can do arithmetic on all at once. You could compute mean $|e|$ with a Python loop:

total = 0.0
for x in e:
    total += abs(x)
mean_abs = total / len(e)

NumPy lets you write the formula instead:

mean_abs = np.abs(e).mean()

Figure 15. The five errors as a row of boxes. np.abs(e) makes a new row with every sign dropped; .mean() adds that row up and divides by five, giving 0.28. Below, np.abs(e) > 0.35 makes a row of True and False, and .sum() counts the Trues: 2.

np.abs(e) makes a new array holding the size of every element, and .mean() adds them up and divides by how many there are. In the same way, e ** 2 squares every element, np.sqrt takes a square root, .max() finds the largest element, and a comparison such as np.abs(e) > 0.9 asks its question of every element and returns an array of True and False. Python counts True as 1 and False as 0, so .sum() of that array is the number of off-lane steps. Applying one operation to a whole array at once is called vectorisation.

Vectorised code is shorter, it reads like the maths, and it runs much faster, because the loop happens inside fast code written in C instead of in Python, one element at a time. You will time the difference in the lab. The grader checks your answers against the plain-loop calculation, so the two styles must agree.

Recap

A drive is a time series of lateral errors $e$ , one every 0.05 s, with left of centre positive.
The plain average of $e$ lets left and right cancel. It measures the car's bias, never its quality.
mean $|e|$ says how far off the car typically was, RMS counts big errors extra, max $|e|$ is the worst moment, and counting steps with $|e| > 0.9$ m says how long the car was off the lane.

Next question: twenty times a second, the camera hands the perception box a picture. What is a picture to a computer, and how could you find a yellow line in one? That is the next lesson, Images are arrays.

Figure 16. The camera's picture of the gentle loop, with a small patch of the yellow line magnified until its pixels show as squares. Three squares are labelled with their three numbers, their amounts of red, green and blue. To the computer, the picture is nothing but numbers like these.

Go deeper

NumPy: the absolute basics for beginners (sections "Array fundamentals" and "Basic array operations")The official guide to everything this lesson used: making arrays, abs, mean, max and comparisons.
CS231n Python Numpy Tutorial (sections "Arrays" and "Array math")A compact tour from Stanford's computer-vision course. Its "Boolean array indexing" part prepares you for Images are arrays.
Thrun et al., Stanley: The Robot that Won the DARPA Grand Challenge (2006), §9.2 "Steering Control", p. 684How the winning car of the 2005 desert race defined its cross-track error and steered to shrink it. Read the definition now, and come back for the steering law after Closing the loop.

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Checkpoint quiz

Teach it back

Explain this week as if to a friend who has never seen it. Then paste it, with the prompt below, into any AI tutor and ask it to find the gaps.

Show the tutor prompt

You are a Socratic tutor for the lesson "Meet your car" of a self-driving car course. I am a programmer with rusty maths and no machine-learning background. I have just read a lecture and done a lab on measuring how well a car keeps its lane.

What the unit taught (use these definitions exactly):
- The lateral error e is the sideways distance in metres from the lane centre to the car's front axle. Left of centre is positive, right is negative.
- A drive is a time series e_1 ... e_N, one value every 0.05 s (20 per second).
- mean |e| is the typical distance from centre; max |e| is the worst moment; RMS = sqrt(mean of e squared) is the typical distance with big errors counted extra. RMS is never smaller than mean |e|.
- A step is off the lane when |e| > 0.9 m, strictly.
- The misconception to probe: "the average of the signed errors measures how well the car keeps its lane". It does not, because left and right errors cancel. The signed mean measures bias (which side the car favours).

How to run the session:
1. Ask me ONE question at a time, then wait for my answer. Start by asking me to explain, in my own words, what the lateral error is and what its sign means.
2. Never give me the answer. If I am wrong or vague, ask a smaller question, or give me a tiny set of numbers to work through by hand, so I find the mistake myself.
3. Within the first three questions, test the misconception above: for example, give me a drive whose errors swing left and right, and ask whether its signed mean is a fair score.
4. Ask at least one question where I must compute a number (a mean |e|, an RMS or a count off the lane) from three to five values you choose, and one where I must predict what a line of NumPy prints.
5. Keep your turns short: two or three sentences, then the question.
6. After about five exchanges, stop. List what I explained well, and what I should revisit, naming these lecture sections: "Lateral error", "A drive is a list of numbers", "The tempting shortcut", "Three honest numbers", "Why RMS punishes big errors", "Counting steps off the lane", "What does doing nothing score?", "Arrays, not loops".

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