Robots rely on sensors to understand the world around them:

• Sensors measure distance, light, pressure, and more
• But sensor readings are often noisy or imperfect
• The raw reading rarely equals the true physical value
• Linear regression helps model the relationship between a sensor reading and the true measurement

Linear regression fits a line to data to model the relationship between two variables.

\[y = \beta_0 + \beta_1 x + \varepsilon\]

• \(x\) = sensor reading
• \(y\) = actual distance
• \(\beta_0\) = intercept
• \(\beta_1\) = slope
• \(\varepsilon\) = error

A robot’s ultrasonic distance sensor was tested against a laser rangefinder (ground truth).

The shaded band shows the 95% confidence interval around the fitted line.

Running lm() on this data gives the fitted model:

\[\hat{y} = 0.42 + 1.03x\]

Example prediction: If the robot’s sensor reads 10.0, the estimated actual distance is:

\[\hat{y} = 0.42 + 1.03(10) = 10.72 \text{ m}\]

This fitted line can be used in real-time to correct raw sensor output before feeding it into navigation algorithms.

sensor <- c(5.1, 6.3, 7.0, 8.4, 9.2, 10.5, 11.0, 12.1, 13.3, 14.7)
actual <- c(5.5, 6.6, 7.4, 8.7, 9.8, 10.9, 11.4, 12.5, 13.8, 15.2)
df <- data.frame(sensor = sensor, actual = actual)

model <- lm(actual ~ sensor, data = df)
summary(model)

From the regression output:

• The slope \(\hat{\beta}_1 \approx 1.03\) means for every 1-unit increase in sensor reading, actual distance increases by about 1.03 meters
• The intercept \(\hat{\beta}_0 \approx 0.42\) is a small calibration offset
• A high \(R^2\) value (close to 1) confirms the sensor readings are strongly linearly related to actual distance
• This model can be deployed on the robot to convert raw sensor output into calibrated distance estimates in real time