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What is robot mapping?

Robot mapping means the robot builds a model of the environment as it moves.

State estimation
Finding the robot’s best guess of its condition or position.

Mapping
Building a map of the environment.

Localization
Finding the robot’s location inside the map.

SLAM
Doing mapping and localization at the same time.

Navigation
Moving the robot from one place to another safely.

Motion planning
Planning the best path for the robot to move.

What is SLAM?

SLAM means the robot finds its own pose and builds a map of the environment at the same time.

Localization → location
Finding where the robot is.

Mapping → environment
Building the shape of the environment.

SLAM → both at same time
Finding location + building map together.

Definition of the SLAM Problem

Given:

  • Robot controls
  • Robot observations Wanted:
  • Map of the environment
  • Path of the robot

Bayes Filter explanation

Goal:
Technically, the Bayes Filter estimates the probability of the robot’s state (x_t) given all past measurements (z_{1:t}) and controls (u_{1:t}):

$$ p(x_t \mid z_{1:t}, u_{1:t})
$$ Intuition: It answers the question, “Where am I right now?” using everything the robot has sensed and done.

Prediction Step:
Technically: The filter uses the motion model to predict the new state from the previous state:

$$ \bar{p}(x_t) = \int p(x_t \mid u_t, x_{t-1}) , p(x_{t-1} \mid z_{1:t-1}, u_{1:t-1}) , dx_{t-1}
$$

Intuition: “If I moved like this, where could I possibly be?” The robot accounts for uncertainty in motion (wheels slip, sensors drift).

Update Step:
Technically: The predicted state is corrected using the sensor model:

$$ p(x_t \mid z_{1:t}, u_{1:t}) = \eta , p(z_t \mid x_t) , \bar{p}(x_t)
$$

Intuition: “Do my eyes, lidar, or cameras agree with my guess?” The robot compares its prediction with what it observes and adjusts its position estimate accordingly.

Uncertainty Handling:
Technically: Motion and sensor noise are explicitly modeled using probability distributions, often Gaussian.

Intuition: “I think I moved 1 meter, but maybe it was slightly more or less.” The robot knows its movements and sensors are imperfect.

Recursive Nature:
Technically: The posterior at time (t-1) becomes the prior for time (t), so the robot only needs the previous estimate to update its belief.

Intuition: “I only need to remember where I was last step and what I just sensed.” No need to store the entire history.

Special Cases:

  • Kalman Filter: linear system with Gaussian noise
  • Extended Kalman Filter (EKF): non-linear system, approximated as linear
  • Particle Filter: represents any probability shape using samples

Intuition: These are different ways for the robot to guess and correct depending on how messy the environment is.

Core Intuition:
Bayes Filter = Guess + Check repeatedly.

  • Prediction = guess from motion
  • Update = check with sensors
  • Probabilities = robot’s confidence

It keeps repeating this cycle, continuously refining the robot’s belief of where it is.