motionmodel

Motion model

A motion model describes how the state of a system evolves from one time step to the next, given the current state and, optionally, control inputs. It reflects the dynamics of the system and is used in simulation, planning, and state estimation (for example in robotics and autonomous vehicles). Uncertainty is represented separately as process noise.

In discrete time, a common formulation is x_k = f(x_{k-1}, u_k) + w_k, where x_k is the state at time k, u_k is the control input, f is the state transition function, and w_k is process noise (often modeled as Gaussian). Sensor measurements relate to the state through an observation model z_k = h(x_k) + v_k, with measurement noise v_k.

Common motion models include:

- Constant velocity: for a planar robot, x_k and y_k are updated by the previous velocity, while velocities may be held constant or damped.

- Constant turn rate and velocity (CTRV): suitable for turning motion, updating position using current speed and turn rate.

- Bicycle model: used for wheeled vehicles, with state typically including position, orientation, and velocity; kinematics account for steering angle and wheelbase.

- Random walk or white-noise acceleration: simple models where acceleration is treated as random input.

Model choice affects estimation and planning. Linear models align with standard Kalman filters, while nonlinear models require extended or unscented Kalman filters or particle filters. Time discretization, typically via a time step Δt, influences stability and accuracy. The motion model is often complemented by a measurement model, forming a complete state-space representation used in localization, tracking, and SLAM.