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Robots Under Uncertainty: Estimating Position from Noisy Sensors
Robot Learning

Foundations

  1. 01The Position Update: Gain from Prediction and Measurement
  2. 02Robotics Foundations: Sensors, Actions, and Motion
  1. 03Week 1: Robot Modeling and Kinematics

Foundations

  1. 04Robots Under Uncertainty: Estimating Position from Noisy Sensors
  1. 05Week 2: Dynamics and State Estimation
  2. 06Week 3: Control Fundamentals
  3. 07Week 4: Teleoperation and Data Collection
  4. 08Week 5: Imitation Learning
  5. 09Week 6: Reinforcement Learning for Robotics
  6. 10Week 7: Sim2Real Pipelines and IsaacLab
  7. 11Week 8: Foundation Models for Manipulation — ACT and Action Chunking
  8. 12Week 9: Flow Matching and Diffusion for Robot Policies
  9. 13Week 10: Vision–Language–Action Models
  10. 14Week 11: Fine-Tuning and Adaptation
  11. 15Week 12: Safety, Constraints, and Reliability
  12. 16Week 13: Multi-Robot and Multi-Task Learning
  13. 17Week 14: Sim2Real Capstone
Week 00· Robot Learning· Foundations5 min read

Robots Under Uncertainty: Estimating Position from Noisy Sensors

A rover is in the aisle. It has to stop at a shelf, and the only position it can hold is a blend of a prediction and a noisy measurement.

Sensor trust 0.50. Sensor noise 0.50. Estimate 3.35 m.

The amber square is the prediction at 3.50 m, and it does not move. The teal dot is the measurement at 3.21 m, and the cloud around it widens as sensor noise rises. The ink line is the true position at 3.00 m. The rover never gets that number. The open circle is the estimate. It slides from the prediction toward the measurement as sensor trust rises.

NumberWhat it isCan the rover compute it?
True positionWhere the rover physically is, the ink lineNo
PredictionWhere the motion model says it is, the amber squareYes
MeasurementWhat the sensor reports, the teal dotYes, with noise
EstimateThe blend the open circle showsYes, and later decisions use this

The two sliders are one weight. Sensor trust is the share given to the measurement. Sensor noise is one minus that share, so dragging either slider moves the other. The estimate is the prediction plus that weight times the gap:

estimate=prediction+weight×(measurement−prediction)\text{estimate} = \text{prediction} + \text{weight} \times (\text{measurement} - \text{prediction})estimate=prediction+weight×(measurement−prediction)

Set sensor trust to 0.80. The circle sits at 3.27 m, because 3.50+0.80×(3.21−3.50)=3.273.50 + 0.80 \times (3.21 - 3.50) = 3.273.50+0.80×(3.21−3.50)=3.27. Trust 0 leaves the circle on the prediction. Trust 1 puts it on the measurement. Why 0.80 is the right share, and how a shove the motion model never saw shows up in a longer trace, is the next page.

Learning Outcomes
  • Separate true position, prediction, measurement, and estimate on the picture
  • Move one weight and predict which way the estimate circle goes
  • Compute the blend for a stated weight
Prerequisites

Robotics Foundations: Sensors, Actions, and Motion is the sense, decide, act loop, and why every sensor is imperfect. From the probability track, Probability Spaces and Events through Bayes' Rule and Generative Classification are the background for the claim that a reading is evidence about the state, not the state itself.

Browser lab: one blend#

The cell computes the same blend the circle shows. Change weight, press Run, and check the printed estimate against the picture.

python · runs in browser
prediction = 3.50
measurement = 3.21
weight = 0.80
estimate = prediction + weight * (measurement - prediction)
print(f"estimate {estimate:.2f} m")

Knowledge Check#

Five checks against the picture. Drag the sliders, then answer.

Exercise · Multiple choice

Set sensor noise to the high end. Where does the estimate circle sit, compared with a low noise setting?

Closer to the amber prediction
Closer to the teal measurement
On the ink truth line
Off the aisle
Exercise · Multiple choice

Move the sensor trust slider toward the sensor. Where does the estimate circle go?

Toward the amber prediction
Toward the teal measurement
Onto the ink truth line
It stays put
Exercise · Multiple choice

You raise sensor trust. What does the noise slider do?

It stays where it was
It falls, because both sliders set the same weight
It rises with trust
It jumps to the true position
Exercise · Multiple choice

What happens to the ink line when you drag either slider?

Nothing. It is the true position, and the rover cannot compute it
It follows the estimate circle
It jumps to the measurement
It widens into the cloud
Exercise · Multiple choice

The weight is 1. The estimate equals which number?

The prediction, 3.50 m
The measurement, 3.21 m
The true position, 3.00 m
The average of the truth and the prediction

Key takeaways#

  • The ink line is the truth. The rover does not have it.
  • The amber square is the prediction and the teal dot is the measurement.
  • One weight blends them. Noise is the complement of trust, so the sliders agree.
  • The open circle is the estimate, and it is what later decisions should use.

Further reading#

The derivation of that weight, the gain formula, and the longer tracking lab are in The Position Update.

  • Sebastian Thrun, Wolfram Burgard, and Dieter Fox, Probabilistic Robotics (MIT Press, 2005). Chapter 3 is the Kalman filter this picture is a one-number case of.
  • Greg Welch and Gary Bishop, An Introduction to the Kalman Filter (University of North Carolina, TR 95-041). The same gain, with the covariance algebra written out.

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← Previous
Week 1: Robot Modeling and Kinematics
Next →
Week 2: Dynamics and State Estimation
On this page
  • Browser lab: one blend
  • Knowledge Check
  • Key takeaways
  • Further reading