Probabilistic Robotics and Probability Theory

Probabilistic Robotics

• Probabilistic robotics is a new approach to robotics that addresses uncertainty in robot perception and action.
• The core concept is to explicitly represent uncertainty using probability theory.
• Uncertainty arises from several factors: unpredictable environments, limited sensors, unpredictable robot actuation (e.g., control noise), and computational limitations in real-time systems.
• State estimation, typically combining position, velocity, orientation, and angular velocity, is crucial for robust robot control. Probability theory is the key, enabling robust estimation.

Axioms of Probability Theory

• Probability (P(A)) denotes the likelihood of a proposition (statement) A being true.
• P(True) = 1, P(False) = 0.
• 0 ≤ P(A) ≤ 1.
• P(A or B) = P(A) + P(B) - P(A and B)

Random Variables

• Random variables can take on different values, each with a specific associated probability.
• Discrete random variables have a countable number of values; the probabilities sum to 1, and lie between 0 and 1.
• The probability mass function (PMF) expresses the probability of each value for a discrete variable.
• Continuous random variables can take on infinitely many values within a given range.
• Probability density functions (PDFs) describe the probability of a continuous variable falling within a range—the area under a PDF curve.

Joint and Conditional Probability

• Joint probability measures the likelihood of two events occurring together.
• Conditional probability is the likelihood of an event given another event has occurred.
• P(x,y) is the joint probability of x and y co-occurring.
• If X and Y are independent, P(x,y) = P(x)*P(y); they are uncorrelated.
• P(x|y) is the conditional probability of x given y, which can be calculated as P(x,y)/P(y)

Bayes' Formula

• Bayes' Theorem provides a method for calculating a conditional probability without the joint probability.
• It's used for calculating probability of an event based on prior knowledge of conditions
• P(x|y) = [P(y|x) * P(x)] / P(y)

State Estimation

• State estimation aims to infer the state of a robot from sensor data.
• Robot state typically includes position, velocity, orientation.
• The real-world process of estimating these variables is often complex since they are not directly observed. Sensors provide partial information and are subject to noise.

Actions and Modeling Actions

• Actions carried out by a robot affect the world; they are not always deterministic.
• Models must consider the outcome of actions using conditional probability.
• P(x/u,x') models how an action u affects the state from x' to x.

Combining Evidence

• Bayesian filters incorporate new measurements into our belief about the robot state.
• The probability of a robot being at a particular state considering past observations and actions is updated recursively. 

Hidden Markov Models (HMM)

• HMMs are models that show how an observable event (Y) is influenced by another unobserved process (X).
• In the context of robotics, HMMs relate sensor readings to robot state.

Bayes Filters in Robotic Localization

• Bayes filters are algorithms for continuously updating the most likely position of a robot in a coordinate system using sensor data.
• It’s a recursive process.

Bayes Filter Framework

• The framework consists of:
  • A sequence of observations (Z) and action data (U).
  • A sensor model (P(z|x)) and an action model (P(x/u,x')).
  • A prior probability of the system state (P(x)).
  • The estimate of the state (X).
  • The posterior belief (Bel(x)), which is the probability of a given state given the sequence of observations and actions.

Explore the essentials of probabilistic robotics, which tackles uncertainty in robot perception and action through probability theory. This quiz covers key concepts such as random variables, state estimation, and the axioms of probability to enhance your understanding of robot control in unpredictable environments.