mobile robots from scratch, part 8: lidar, how it measures

The direct way to measure a distance with light is to send a short pulse and time how long it takes to come back. Light covers the distance twice, there and back, so

Light is fast: a wall 4 m away sends the pulse back after 26.7 ns. To measure to 1 cm, the lidar has to time the echo to 67 ps, which is why time-of-flight lidars have special timing electronics. This is direct time of flight (dToF).

The echo also gets weaker with distance, roughly as \(1/d^2\) for a wall: at 4 m it’s 16 times weaker than at 1 m. The range a lidar can reach is set by when the echo sinks into the noise, and depends on how much light the surface reflects.

Cheap lidars often don’t time anything. They shine a laser dot, and a small camera a few centimetres to the side sees where the dot lands. A near wall puts the dot at one edge of the camera’s image, a far one near the centre. The geometry is a triangle, like the stereo cameras of part 11:

with \(b\) the baseline between laser and camera, \(f\) the camera’s focal length in pixels and \(x\) where the dot lands.

The catch is that the error grows with the square of the distance. With a 5 cm baseline and the dot located to a tenth of a pixel, the error is 0.3 cm at 1 m, 2.6 cm at 3 m and 18.3 cm at 8 m. Time of flight’s error stays about the same (here 1.5 cm, for 100 ps of timing jitter). Move the baseline: a wider one helps, but makes the sensor bigger.

Another way avoids timing a short pulse. Make the light’s brightness rise and fall smoothly, at a frequency \(f\), and compare the wave that comes back with the one sent out. The echo is delayed, so its wave is shifted, by a phase

Measuring a phase is easier than timing picoseconds. This is amplitude-modulated continuous wave (AMCW), used in many depth cameras.

But a phase repeats every full turn. At 10 MHz the wave repeats every 15 m, so a wall at 16 m reads as 1.01 m. A higher frequency measures more finely but repeats sooner: at 100 MHz, every 1.5 m. Real sensors combine two or more frequencies to tell the cases apart.

One distance isn’t a picture. So a 2-D lidar puts its range finder on a turning head, and takes ranges as it goes round. This one takes 4,000 a second and turns 10 times a second: 400 points a turn, 0.9° apart.

That spacing is in angle, so the points spread out with distance: 7.9 cm apart on a wall 5 m away. Spin faster and the scan is fresher but sparser. At 20 Hz, 200 points, 15.7 cm apart at 5 m; at 5 Hz, 800 points, but each scan is 0.2 s old by the time it’s finished (part 9 deals with that).

All the points lie in one flat slice of the room, at the height of the lidar. A table top above it, or a chair leg below it, is invisible.

Not every beam comes back. Black surfaces absorb most of the light. Glass lets it through, and a mirror sends it elsewhere. A wall hit at a very glancing angle scatters too little light back. And past its maximum range, the lidar hears nothing.

So a real scan has gaps and noisy points. Drivers report a missing return as “no reading” (in ROS, inf or a range outside the valid limits), and every algorithm downstream has to cope with that. A missing reading isn’t the same as free space: the glass door is still there.

A 2-D spinning lidar is what most indoor robots use: a few tens to a few hundred dollars, one flat slice, good enough for mapping and localisation on a flat floor.

3-D lidars stack many beams, 16 to 128, on one spinning head, each at a different tilt, giving a 3-D point cloud. They’re what self-driving cars and outdoor robots use. Others point the beam with no spinning head: a tiny tilting MEMS mirror, a flash of light over the whole scene read by a sensor of pixels, or an optical phased array that steers the beam electronically. These are called solid-state.

All of them measure range in the ways above.