M1-T5 — Odometry and Motion Models

Learning objectives

After this lesson, you should be able to:

• Derive the forward speed and turn rate of a differential-drive robot from wheel speeds.
• Integrate encoder increments into a 2D pose and explain the midpoint approximation.
• Identify the main systematic and random sources of odometry error.
• Explain why even a calibrated odometry error grows without bound, and why that makes it an input rather than a solution.

The intuition

A differential-drive robot has two driven wheels on one axle. If both turn at the same speed it goes straight. If one is faster it curves toward the slower one. If they turn in opposite directions it spins on the spot. By counting how far each wheel turned, the robot can compute how it moved, then add that movement to its previous pose. This is called dead reckoning, the same method sailors used with a log line and a compass.

The weakness is clear from the idea: every step builds on the last. There is nothing in dead reckoning that checks the answer against the world, so a mistake in step 1 is carried forever.

The mathematics

A wheel encoder counts ticks. With N ticks per revolution and wheel radius r, a count of Δn ticks means the wheel travelled

For the left and right wheels this gives Δs_L and Δs_R. With wheelbase b, the robot’s forward motion and turn in that step are

r

wheel radius in metres. A 1% error in r is a 1% error in all distances.

b

wheelbase: the distance between the contact points of the two wheels. Errors here corrupt the turn angle.

Δs

distance moved by the robot centre during the step.

Δθ

heading change during the step, positive counter-clockwise.

N

encoder ticks per wheel revolution.

The pose is updated by moving along the arc. The standard midpoint model uses the heading at the middle of the step:

Using θ at the start of the step would be less accurate, and using the end would be biased the other way: the midpoint is exact for straight lines and a good approximation for small arcs.

Error sources

The systematic errors are the large ones, and they are repeatable. That is good news: a repeatable error can be measured and removed.

A worked example: one step and one square

A robot has wheel radius r = 0.05 m, wheelbase b = 0.30 m and an encoder with N = 1000 ticks per revolution. Pose is (0, 0, 0).

1. Ticks to distance. In one step the left wheel counts 80 ticks and the right 100. Δs_L = 2π × 0.05 × 80 / 1000 = 0.02513 m. Δs_R = 2π × 0.05 × 100 / 1000 = 0.03142 m.
2. Robot motion. Δs = (0.03142 + 0.02513) / 2 = 0.02827 m. Δθ = (0.03142 − 0.02513) / 0.30 = 0.02094 rad (1.2°). The right wheel went farther, so the robot turned left (positive).
3. Pose update. The midpoint heading is 0 + 0.02094 / 2 = 0.01047 rad. x = 0.02827 × cos(0.01047) = 0.02827 m, y = 0.02827 × sin(0.01047) = 0.000296 m. New pose (0.0283, 0.0003, 0.0209).
4. Now a square, with a 1% wheelbase error. The robot drives four 2 m sides, turning 90° at each corner. If the true wheelbase is 0.30 m but the model uses 0.303 m (1% too large), the computed turn is smaller than the real one: each corner is under-estimated by about 1% of 90°, or 0.9°. After four corners the heading estimate is off by about 3.6°, yet the robot truly turned 360°.
5. Result. Integrating the four sides with the mis-estimated headings (0°, 89.1°, 178.2°, 267.3°) ends about 0.09 m from the start: the believed path does not close. One percent in one parameter turned into a closure error of 9 cm after 8 m of driving, and the error is systematic, so a longer run makes it larger.

This is exactly what the lab does with a tool that lets you switch each error source on and off.

Engineering implications and limitations

Using odometry well

• Calibrate first. Drive a known square and a known straight line; solve for r_L, r_R and b. This often removes most of the error for free (see the next lab).
• Treat it as short-term information. Over a metre it is excellent; over a kilometre it is useless. Fuse it with measurements that do not drift.
• Model the noise honestly. A motion model gives the probability of a pose after a command. Heading noise should depend on how much the robot turned and how far it travelled.
• Detect slip. If the wheels spin but an IMU or scan matching says the robot did not move, odometry must be down-weighted. Climbing a ramp or driving on a wet floor are typical.
• Publish the covariance. Downstream SLAM needs to know how much to trust odometry; a fixed constant is a common source of bad maps.

Limits

No calibration removes slip, floor irregularities or the wheel’s deformation under load. A calibrated robot still drifts; it drifts more slowly. And the error is not just growing, it is unbounded: after a long drive the uncertainty of the position is larger than the building. That is why mapping and place recognition are needed: they supply measurements that are tied to the world rather than to the path.

Common misconceptions

Engineering takeaways

1. Differential drive: v = (v_R + v_L) / 2 and ω = (v_R − v_L) / b.
2. Odometry integrates increments into a pose; the midpoint model is the standard update.
3. Systematic errors in r and b dominate and can be calibrated; slip and noise cannot.
4. Heading error compounds into position error, so turning is where odometry hurts most.
5. Odometry is unbounded in error and must be corrected by measurements of the world.

References and further reading

Primary papers and authoritative documentation.

• Borenstein, J. and Feng, L. (1996). Measurement and correction of systematic odometry errors in mobile robots. IEEE Transactions on Robotics and Automation 12(6). UMBmark: the standard calibration method.
• Siegwart, R., Nourbakhsh, I. and Scaramuzza, D. (2011). Introduction to Autonomous Mobile Robots, 2nd ed., MIT Press, chapters 3 and 5: kinematics and error models.
• Thrun, S., Burgard, W. and Fox, D. (2005). Probabilistic Robotics, chapter 5: velocity and odometry motion models.
• Kelly, A. (2013). Mobile Robotics: Mathematics, Models and Methods. Cambridge University Press: error propagation for dead reckoning.

Knowledge check

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

What’s next

Time to try it. The second lab gives you a robot that returns to its start but whose odometry says it is metres away, and asks you to find out why.

Related content

Nothing below is required to finish this lesson.

• Before
  M1-T4: Robot Poses and Coordinate Frames

  Describe a 2D pose and the world, odom, base and sensor frames, and read a frame-chain statement correctly.

• Lab
  M1-L2: Diagnose Drifting Odometry

  A robot returns to its start but odometry says it is metres away: find the dominant error source and calibrate it.