Steering swings a camera too little to see ahead

A turn moves the picture, not the blind point turned the middle picture of a camera driving forward and found the turn useless to the blind disc ahead: a turn changes where a picture points, not where it was taken, and the depth of the mark at the direction of travel comes only from where pictures were taken. A sway gives the blind centre a depth, not a good one had already found what does help — a sideways step — and how little a small one buys: a centimetre of sway gave the centre a depth good only to 144 per cent at eight metres.

That essay’s first assumption was a camera turning about its own centre. A camera on a vehicle does not. A vehicle yaws about its rear axle, and a camera mounted a distance ahead of the axle is swung sideways by that distance times the sine of every yaw. Every small steering correction then buys a sway the camera would otherwise lack. The question was whether that lever arm, with the steering an ordinary drive has, closes the blind disc at the ranges a vehicle cares about.

It does not, and the numbers say by how much.

A turn that swings what is ahead of the axle #

The frames below are a vehicle driving straight, half a metre between frames, its rear axle held to the lane’s centre line and its heading wandering by small yaws: none at the first frame, 0.6 degrees at the second, −0.5 at the third.

A camera over the rear axle turns with each yaw and goes nowhere: its three positions lie on the lane’s line, and its depth at the direction of travel is as blind as the earlier essays found. A camera behind the windscreen, a metre and a half ahead of the axle, is swung 1.6 centimetres one way and 1.3 the other. One on the front bumper, 3.6 metres ahead, is swung 3.8 and 3.1. Each camera is also turned by the same yaws, which moves its picture and adds nothing; the swing is the part that counts, because it is a baseline across the direction of travel. Drag the third frame’s heading and the swing follows it in proportion: a degree and a half puts the bumper camera 9.4 centimetres off the line and the windscreen camera 3.9 — at the most, a baseline about the width of a hand.

What the swing buys straight ahead #

The depth a pixel of reading error costs the mark at the direction of travel, eight metres ahead, is the measure the earlier essays used for the blind disc’s centre. With random yaws at the second and third frames it varies from drive to drive, so the figure takes the median of eighty.

With half a degree of steering noise at each frame, a camera half a metre ahead of the axle reads the depth straight ahead to 289 per cent; behind the windscreen, a metre and a half ahead, to 96 per cent; on the bumper, 3.6 metres ahead, to 40 per cent. A quarter of a degree doubles every number and a whole degree halves it. The law is as simple as the geometry: the cost is about 0.72 metre-degrees divided by the product of the arm and the yaw, because the sway it buys is their product, and whole pixels cut space into shells is the reminder that a depth’s precision is set by its baseline.

No curve in the figure reaches the 10 per cent line, the precision at which a depth starts to be usable. The lever arm is real — a bumper camera sees the road ahead in depth two and a half times better than a windscreen camera for the same steering — and it is small.

Why only the product counts #

The law in the figure is exact enough to be worth deriving, because it says which of the two quantities a designer can trade for the other. A yaw $\psi$ about the rear axle carries a camera $a$ metres ahead of it sideways by $a\sin\psi$ and backwards by $a(1-\cos\psi)$, and turns it by $\psi$. For the yaws a vehicle makes, a degree or less, the backward part is a few millionths of the arm and the sideways part is $a\psi$ to better than one part in ten thousand.

The turn contributes nothing to the depth at the direction of travel. A turning frame can be straightened found a pure turn removable without knowing the scene, because it moves every point’s image by an amount that does not depend on depth; a camera that knows its own turn — and the triangulation here does, since the frames’ orientations are given — takes it out exactly and is left with where the pictures were taken. What is left is three camera centres: one on the lane’s line, two swung sideways by $a\psi_2$ and $a\psi_3$. Two rays that do not meet is the reminder that a triangulated depth’s precision is set by the angle between the rays, and at the direction of travel that angle comes entirely from the sideways swings.

So the depth error straight ahead is inversely proportional to the swing, and the swing is the arm times the yaw. Doubling the arm and halving the steering leaves the depth exactly where it was. That is what makes the requirement a single number in metre-degrees, and it is why the figures’ curves are parallel lines on logarithmic axes: every one of them is the same curve, shifted by the product.

It worsens with range #

The eight-metre mark is a near one for a vehicle. The same sway gives a worse depth further away, and not slowly.

The bumper camera’s forty per cent at eight metres becomes seventeen at four metres and 180 at thirty-two; the windscreen camera’s goes from forty at four metres to 433 at thirty-two. A sideways baseline $b$ gives a depth error of about $Z\sigma/(fb)$ as a share of the depth, which grows in proportion to the range — depth is a reciprocal made the same point about any pair. The range at which a vehicle most needs to know what is ahead of it, a braking distance or more, is where the lever arm’s sway is least use. A car at fifty kilometres an hour needs something like twenty-five metres to stop; at twenty-four metres the bumper camera’s depth straight ahead is good to 134 per cent and the windscreen camera’s to 321 — not a depth at all, only a statement that something is there. What a vehicle at speed can read straight ahead from one camera is the thing’s size in the picture and how fast that size grows, which is a time to contact and not a distance.

At four metres the forward steps start to help as well. Half a metre closer to a surface four metres away is a large fraction of the distance, the direction of travel moves off the epipole of the nearer pairs, and the blind disc — which is centred on the direction of travel only for a surface far away — no longer covers the mark exactly. An epipole in the picture leaves a blind disc described the disc as the region a pure forward step leaves poorly ranged; very close in, that region shrinks without any sway at all.

More frames collect more swing #

Three frames is the minimum the earlier essays used. A camera keeps taking pictures, and each one is swung by its own yaw.

For a surface sixteen metres ahead, three frames give the windscreen camera 208 per cent and the bumper camera 87; five frames, 107 and 45; nine, 59 and 24; seventeen, 24 and 10. More frames help in two ways at once: each adds its own sway, and the later frames are nearer the surface. By seventeen frames the vehicle has driven eight of the sixteen metres, and the bumper camera finally reads the surface to ten per cent — as much because it has halved the distance as because it has collected more swing.

That is the honest reading of the lever arm’s value. Over a second or so of driving, a bumper camera’s steering-noise baseline and its approach together bring a surface ahead into depth. Over three frames, which is the window a vehicle has to react to something appearing ahead of it, they do not.

What a usable depth would need #

The law makes the requirement a single number: the arm times the yaw, in metre-degrees, that brings the depth error straight ahead to ten per cent.

At eight metres a ten per cent depth needs 7.2 metre-degrees. The requirement grows almost in proportion to the range — 3.0 metre-degrees at four metres, 15.6 at sixteen, 24.1 at twenty-four — because the depth error a fixed sway leaves grows in proportion to the range, and the sway needed to hold it at ten per cent grows with it; at four metres it falls a little below proportion, because there the forward steps themselves begin to range the surface. A bumper camera, 3.6 metres ahead of the axle, would need two degrees of yaw at every frame; a windscreen camera nearly five. At sixteen metres the numbers double, at twenty-four they triple. Ordinary lane-keeping corrects by fractions of a degree over a second or more, not by degrees every half-metre; a vehicle yawing two degrees every half-metre of travel is weaving across its lane.

So the answer to the earlier essay’s question is geometric and negative. A metre of lever arm and ordinary steering noise do not close the disc at the ranges a vehicle cares about, and no mounting on an ordinary vehicle gives the direction of travel a usable depth from steering alone. A forward camera placed at the front of a vehicle rather than behind the windscreen sees the road ahead in depth materially better, by the ratio of the arms — two and a half times on a car — but “better” here means a depth error of forty per cent rather than a hundred.

How long an arm, and how much steering #

The two numbers in the product are both set by things other than a camera designer’s wishes, and it is worth putting ordinary values on them.

The arm is the distance from the rear axle forward to the camera. On a car with a wheelbase of about 2.7 metres, a camera behind the windscreen sits roughly a metre and a half ahead of the rear axle and one on the front bumper roughly three and a half — the two mountings drawn here. A long-wheelbase van or a bus puts its front several metres further forward, and a camera at the front of a bus has perhaps twice the bumper arm of a car. Twice the arm halves the depth error: a bus’s front camera at half a degree of steering would read eight metres ahead to about twenty per cent, still twice the usable ten.

The steering is the harder number to raise. The frames here are half a metre apart, which at ten metres a second — a town speed — is a twentieth of a second. Two degrees of yaw every frame, the bumper camera’s requirement at eight metres, is a yaw rate of forty degrees a second: the rate of a car taking a tight corner, not of one holding a lane. Lane-keeping corrections are small and slow, a fraction of a degree over a second or more, and at each twentieth of a second they amount to a few hundredths of a degree. The half-degree yaws in the figures are, if anything, generous for a vehicle driving straight.

Slower travel does not help the product, though it changes the frames. At walking pace the same half-metre spacing takes longer, a driver’s corrections accumulate more yaw between frames, and the swing per frame grows; but the depth that matters to a slow vehicle is also nearer, and the forward steps themselves start to range it, which is the regime the range figure’s four-metre point showed.

What the geometry offers instead #

The blind point ahead is a fact about a camera that moves along its own line of sight, and the ways round it are the ways the earlier essays found: a baseline that does not depend on the vehicle’s motion. A second camera beside the first gives the direction of travel a fixed baseline at every frame, the image of the other eye being the whole theory of it. A lever arm is a way of borrowing a baseline from the vehicle’s own wandering, and the vehicle does not wander enough.

It is still worth knowing what the arm gives, because it gives it for nothing. A bumper camera on a vehicle whose driver corrects by half a degree a frame has a baseline of three or four centimetres at every frame without any second camera, and at close range — the four metres of a parking manoeuvre, say — that is a seventeen per cent depth straight ahead from one camera. A sway gives the blind centre a depth, not a good one found a centimetre of sway worth 144 per cent; the lever arm is the way a vehicle turns its steering into those centimetres, and the numbers above are what they come to.

What was assumed #

The vehicle yaws about its rear axle. A car’s instantaneous centre of rotation in a gentle steering correction is close to the line of its rear axle, which is what makes the arm the distance to that axle. A vehicle that yaws about some other point — a truck with a trailer, a vehicle skidding — swings its camera by a different arm.

The yaws are independent from frame to frame. Real steering corrections are slow and correlated, so the heading drifts smoothly rather than jumping; a smooth drift of the same size swings the camera by the same amount over several frames but gives the frames’ sways a common trend, which is less useful than independent ones — the numbers here are, if anything, generous.

The frames’ turns are known exactly. The triangulation is given each frame’s orientation, as a vehicle with a good gyroscope would supply it. A turn misread by a small angle is indistinguishable, at the direction of travel, from a sideways shift of the ray, and with sways of a few centimetres a turn error of a few hundredths of a degree is already comparable to what the sway buys; a camera that estimates its own yaw from the pictures spends part of its sway doing so, and the depths here are the best case.

The rear axle keeps to the lane’s centre. A vehicle that also drifts sideways as a whole adds a sway of its own, from the whole vehicle rather than from the arm, and every camera, whatever its mounting, gains it.

Still open: whether a slow drift across the lane is the sway that matters #

The third assumption is where a vehicle’s real sideways motion lies. Drivers do not hold the lane’s centre to a centimetre; a car wanders across its lane by tens of centimetres over a few seconds, slowly, and that wander moves every camera on it sideways by the same amount, far more than any lever arm’s swing.

The measurement that settles what that is worth drives the camera down a lane with a slow sideways wander of a stated size and period, over a stated number of frames, and asks how the depth at the direction of travel and the blind disc depend on the wander — whether a car’s ordinary drift across its lane over a second or two gives the road ahead a usable depth where its steering corrections do not, and so whether the blind point is closed not by where the camera is mounted but by how long it is allowed to look.