M5-T1 — LiDAR Point Clouds for SLAM

Learning objectives

After this lesson, you should be able to:

• Describe how a spinning or solid-state LiDAR produces a point cloud and what each point contains.
• Convert range, azimuth and elevation to Cartesian coordinates and say what a ring is.
• Estimate the distortion of a sweep from the robot’s speed and angular rate.
• List the data a LiDAR SLAM front end needs besides the points: timestamps, extrinsics and sensor model.

The intuition

Imagine photographing a spinning fan with a slit camera that scans across the image over a tenth of a second. If you walk while you press the button, the picture is smeared: the left side of the image was taken a little earlier than the right. A spinning LiDAR is that slit camera. One revolution takes 0.1 s at 10 Hz. A robot driving at 2 m/s moves 20 cm during that time; turning at 100°/s it rotates 10°. The scan is a sequence of measurements taken from slightly different places, filed together as if from one.

A LiDAR scan has a duration. Every point has its own time, and therefore its own robot pose.

The mathematics

What a point is

A mechanical LiDAR (Velodyne, Ouster, Hesai) has a column of laser emitters at fixed elevation angles ω_k (the rings: 16, 32, 64 or 128) rotating around a vertical axis. At azimuth α_j each emitter reports a range r (from the time of flight), an intensity, and a timestamp. In the sensor frame the point is

r

range in metres, typically 0.5–100 m for automotive and 0.1–30 m for indoor sensors, with noise of a few centimetres.

ω_k

elevation (vertical) angle of ring k. A 16-ring sensor spans, for example, ±15° in 2° steps.

α_j

azimuth of the beam at the time of measurement.

t_i

time of the point within the sweep, from 0 to the sweep period T.

A frame (a “scan”) is one full revolution, 10 or 20 Hz, and contains 10⁴–2·10⁵ points. Solid-state and non-repetitive-scan sensors (Livox, some automotive units) cover a field of view of 70–120° with different patterns; they need different feature extraction and have fewer rings to rely on (M5-T3).

Motion distortion

If the sensor moves with constant linear velocity v and angular velocity ω during a sweep of period T, a point measured at time t = sT (s ∈ [0, 1]) was seen from the pose

Treating all points as if taken from the pose at s = 0 puts the point at s in the wrong place by roughly the displacement of the sensor since then, rotated into the measurement: a point at range r suffers a lateral error of about r · ω T s for a rotation and v T s for a translation.

Sources of the pose during the sweep

Other data the front end needs

• Per-point timestamps (or enough to reconstruct them from azimuth and the rotation rate). Without them, deskewing is guessing.
• The extrinsic transform from the LiDAR to the robot body or IMU (M2-T2, M6-T8) and its time offset.
• A sensor model: minimum and maximum range, noise, ring angles, blind regions (the robot’s own body).
• Returns that are not points: no-return pixels, saturated intensity, multi-path ghosts near glass and retro-reflectors.

A worked example: how bad is the distortion?

A 10 Hz LiDAR (T = 0.1 s) on a robot looking at a wall 5 m to its left.

1. Straight driving. The robot drives at 2 m/s, parallel to the wall. During the sweep it moves 2 × 0.1 = 0.20 m. The points measured at the end of the sweep are, in a frame fixed at the start, 0.2 m ahead of where the sensor thinks. Along a wall parallel to the motion this stretches the wall by 20 cm but does not bend it. A wall perpendicular to the motion, however, is shifted: the front-facing points taken late in the sweep are 20 cm closer than those at the start: a flat wall appears tilted by about 20 cm across the sweep.
2. Turning. The robot rotates at 100°/s. During the sweep the heading changes by 10°. A wall point at 5 m has a tangential displacement of 5 × 0.1745 = 0.87 m between the first and last point of the sweep, if undone. A straight wall of 5 m width becomes visibly curved.
3. Mid-sweep point. A point at s = 0.5 was measured from the pose halfway: 0.10 m ahead and 5° turned. Deskewing transforms it back into the frame at s = 0 by applying the inverse of T(0.5): a rotation of −5° and a translation of −0.10 m.
4. Compare with noise. The sensor noise is about 2 cm and the typical registration accuracy 1–5 cm. A 20 cm to 87 cm distortion is 10–40 times larger. Ignoring it would make scan matching converge to a compromise that is biased in the direction of motion.
5. It scales with speed. At walking speed (1.2 m/s, 30°/s) the distortion is 12 cm and 0.26 m: smaller, but still above the registration accuracy. This is why even slow handheld mapping needs deskewing.

The lab’s first stage gives you a smeared map and asks you to repair it.

Engineering implications and limitations

Engineering implications

• Check timestamps first. Drivers differ: some give per-point times (time field), some only the frame time and the azimuth. Plot the point time against azimuth to verify.
• Know the ring layout. Many algorithms (LOAM, LeGO-LOAM) use the ring index to find a point’s neighbours along the scan line. A reordered point cloud (voxel-filtered, merged) loses it.
• Mind the blind spots and the robot’s own body: mask them with a range or angular filter, or the matcher will lock onto the robot.
• Understand the sensor modes. Dual return, intensity calibration, and rotation rate change the data. A mode change that is not reflected in the configuration has caused many a failed run.
• Log raw packets or point clouds with timestamps. Debugging a failed map later without the raw data is guesswork.

Limits

Constant-velocity deskewing fails when the robot accelerates or turns sharply during a sweep. LiDAR also sees poorly through rain, fog and dust, on glass, black surfaces and at grazing angles, and a spinning sensor is low-resolution in elevation. The best deskewing is only as good as the motion estimate used for it: it is the first place where the SLAM pipeline depends on its own output.

Common misconceptions

Engineering takeaways

1. A point is a range, an azimuth, an elevation (ring), an intensity and a time; Cartesian coordinates follow from spherical geometry.
2. A scan is a sweep: its duration (0.05–0.1 s) means that points come from different robot poses.
3. The distortion is v·T for translation and r·ω·T for rotation: tens of centimetres, bigger than registration accuracy.
4. Per-point timestamps and an estimate of the motion during the sweep are required for deskewing.
5. Check the data (timestamps, rings, blind spots) before blaming the algorithm.

References and further reading

Primary papers and authoritative documentation.

• Zhang, J. and Singh, S. (2014). LOAM: Lidar odometry and mapping in real-time. Robotics: Science and Systems: motion distortion handled by linear interpolation of the pose over a sweep.
• Velodyne Lidar, VLP-16 User Manual and Programming Guide; Ouster, Sensor Documentation (docs.ouster.dev): data formats, per-point timestamps and ring layouts.
• Dellenbach, P., Deschaud, J.-E., Jacquet, B. and Goulette, F. (2022). CT-ICP: Real-time elastic LiDAR odometry with loop closure. IEEE International Conference on Robotics and Automation.
• Barfoot, T. D. (2017). State Estimation for Robotics. Cambridge University Press, chapter 6: interpolating poses.

Knowledge check

Three conceptual questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

What’s next

With the data understood, the next lesson covers the preprocessing that every LiDAR pipeline applies before it registers a scan.

Related content

Nothing below is required to finish this lesson.

• Before
  M4-T7: Bundle Adjustment

  Define reprojection-error optimisation over cameras and 3D points and its Schur-complement structure.

• Lab
  M5-L1: Fix Point-Cloud Preprocessing

  A map is smeared and slow: repair deskewing, ground removal and downsampling.