Process Model

The state vector is a mathematical representation of everything the system needs to know about itself at a given time. It is the variable being estimated and predicted.

• Typical elements: Position, orientation (attitude), linear velocity, angular velocity, sensor biases.
• Dimensionality: Defines the complexity of the estimation problem. A 3D drone might have a 16-state vector (position, velocity, quaternion orientation, gyro biases).
• Example: For a ground robot, the state could be x = [x_position, y_position, heading_angle, forward_velocity]^T.

The dynamics function, x_{k+1} = f(x_k, u_k, w_k), is the heart of the process model. It mathematically describes how the state evolves.

• Control Input (u): Known commands to the system (e.g., wheel velocities, propeller thrusts, steering angle).
• Function (f): Encodes the physics. For a simple robot, this might be kinematics. For a drone, it involves rigid-body dynamics.
• Process Noise (w): Models unaccounted-for disturbances (e.g., wind, wheel slip, modeling inaccuracies). Its covariance (Q) defines the prediction's uncertainty growth.

The form of the dynamics function f determines the required estimation algorithms.

• Linear Models: x_{k+1} = F_k * x_k + B_k * u_k + w_k. Here, F is the state transition matrix. The Kalman Filter provides an optimal solution.
• Nonlinear Models: The vast majority of robotic systems. Examples include:
  • Unicycle Model: [x; y; θ] updated with velocity and turn rate.
  • IMU Kinematics: Orientation integration using quaternions or rotation matrices.
• Linearization: Nonlinear models are often approximated (e.g., using a Jacobian) for use with the Extended Kalman Filter (EKF).

The process noise covariance matrix, Q, quantifies the uncertainty introduced by the prediction step. It is a critical tuning parameter.

• Represents: Uncertainty from the w_k term in the dynamics. This includes unmodeled dynamics, external disturbances, and discretization errors.
• Effect: A larger Q tells the estimator to "trust the new measurements more" because the model is uncertain. A smaller Q weights the prior prediction more heavily.
• Tuning: Incorrect Q leads to poor estimator performance—overly large Q causes jittery estimates; overly small Q causes the filter to be sluggish and ignore sensors.

Process models are implemented in discrete time on digital computers, requiring the conversion of continuous-time dynamics.

• Time Step (Δt): The interval between prediction steps. Must be synchronized with sensor data rates.
• Discretization: The continuous differential equation dx/dt = g(x,u) must be integrated over Δt to form the discrete function f(x_k, u_k). For linear systems, this involves computing the matrix exponential.
• Example: The simple kinematic model x_{k+1} = x_k + (v * cos(θ_k)) * Δt is a discretized version of continuous motion.

The process model implements the prediction step of the recursive Bayesian estimation framework.

• Bayesian Prediction: bel(x_t) = ∫ p(x_t | x_{t-1}, u_t) * bel(x_{t-1}) dx_{t-1}.
• The Model's Role: The term p(x_t | x_{t-1}, u_t) is the state transition probability, which is defined by the process model f and the process noise distribution w.
• Algorithm Specifics: The Kalman Filter assumes this distribution is Gaussian. The Particle Filter uses the process model to propagate each particle forward, sampling from this distribution.

The overarching goal of using a process model. State estimation is the process of inferring the internal, often unmeasured, variables (e.g., position, velocity, orientation) of a dynamic system from a sequence of noisy sensor measurements. It combines the prediction from the process model with the correction from a measurement model.

• Core Problem: Determining x_k (state at time k) given z_1:k (all measurements up to time k).
• Frameworks: Includes Bayesian filtering, smoothing, and batch optimization.
• Applications: Essential for robot localization, target tracking, and inertial navigation.

The mathematical counterpart to the process model. A measurement model (or observation model) describes how sensor data is generated from the true state of the system. It is formally expressed as z_k = h(x_k, v_k), where h() is the function, x_k is the state, and v_k is measurement noise.

• Role in Filtering: Provides the expected sensor reading for a given state, enabling the comparison with actual data to correct the state estimate.
• Examples: A camera's perspective projection model, a LiDAR's 3D point generation model, or a GPS receiver's pseudorange model.

The foundational probabilistic framework that defines how process and measurement models are used sequentially. Bayesian filtering recursively applies Bayes' theorem to estimate the posterior probability distribution of the state p(x_k | z_1:k).

The recursive cycle consists of two steps:

• Prediction Step: Uses the process model to propagate the state belief forward: p(x_k | z_1:k-1) = ∫ p(x_k | x_k-1) * p(x_k-1 | z_1:k-1) dx_k-1.
• Update Step: Uses the measurement model to incorporate the latest observation: p(x_k | z_1:k) ∝ p(z_k | x_k) * p(x_k | z_1:k-1).

Algorithms like the Kalman Filter and Particle Filter are specific implementations of this framework.

A specific, common type of process model. A motion model explicitly describes the kinematic or dynamic evolution of a robot's pose (position and orientation). It is central to odometry and localization.

• Kinematic Models: Describe motion geometry without considering forces (e.g., differential drive, ackermann steering).
• Dynamic Models: Incorporate forces, mass, and inertia (e.g., a quadrotor's dynamics derived from Newton-Euler equations).
• Example (Differential Drive): Predicts new robot pose [x', y', θ'] based on wheel encoder ticks, wheel radius, and baseline. Often used as the process model in robot EKF-SLAM.

The key mathematical object representing uncertainty in probabilistic process models. In algorithms like the Kalman Filter, the process model predicts both the mean state and its covariance matrix P. This matrix quantifies the estimated uncertainty of the state and the correlations between state variables (e.g., how error in x-position correlates with error in velocity).

• Process Noise Covariance (Q): A design parameter added during the prediction step. It models the uncertainty in the process model itself (e.g., unmodeled dynamics, slippage). Tuning Q is critical for filter performance.
• Propagation: The process model's Jacobian F_k is used to propagate the covariance: P_k|k-1 = F_k P_k-1|k-1 F_k^T + Q_k.

A navigation technique that is the direct application of a process model without subsequent sensor correction. Dead reckoning estimates a vehicle's current position by using a previously determined position and advancing it based on known or estimated speeds, heading, and elapsed time.

• Pure Integration: It directly integrates velocity or acceleration measurements from proprioceptive sensors (wheel encoders, IMUs).
• Fundamental Limitation: Errors in the process model (e.g., wheel slip, IMU bias) accumulate without bound, leading to drift. This is why dead reckoning is almost always fused with absolute sensing (e.g., GPS, landmarks) in a state estimation framework.
• Example: Calculating a robot's position by counting wheel revolutions, assuming no wheel slip.