systems.builtin.deterministic.continuous.DubinsVehicle

systems.builtin.deterministic.continuous.DubinsVehicle(*args, **kwargs)

Dubins vehicle - kinematic car model with unicycle dynamics.

Physical System:

A simplified model of a car or mobile robot that can move forward and rotate, but cannot move sideways (nonholonomic constraint).

The vehicle is modeled as a point with: - Position (x, y) in the plane - Heading angle θ - Forward velocity v (control input) - Angular velocity ω (control input)

Key constraint: The vehicle must move in the direction it’s pointing (no lateral sliding). This is called a nonholonomic constraint.

Coordinate Frame:

  • Inertial frame: Fixed (x, y) coordinates
  • Body frame: Moves and rotates with vehicle
  • Heading θ: Angle from x-axis to vehicle’s forward direction

State Space:

State: x = [x, y, θ] - x: Horizontal position [m] - y: Vertical position [m] - θ (theta): Heading angle [rad] * θ = 0: pointing right (along +x axis) * θ = π/2: pointing up (along +y axis) * θ = π: pointing left * θ = 3π/2 or -π/2: pointing down

Control: u = [v, ω] - v: Forward velocity [m/s] * v > 0: move forward * v < 0: move backward * v = 0: stopped - ω (omega): Angular velocity [rad/s] * ω > 0: turn left (counterclockwise) * ω < 0: turn right (clockwise) * ω = 0: straight motion

Output: y = [x, y, θ] - Full state observation (position and heading)

Dynamics:

The kinematic equations (Dubins car model):

ẋ = v·cos(θ)
ẏ = v·sin(θ)
θ̇ = ω

Position dynamics: - Vehicle moves in direction θ at speed v - cos(θ) and sin(θ) project velocity onto x and y axes - No motion perpendicular to heading (nonholonomic constraint)

Heading dynamics: - Directly controlled by angular velocity ω - Independent of forward velocity (can rotate in place if v=0)

Physical interpretation: - The vehicle is like a bicycle: must point where it’s going - Cannot slide sideways (like a car on dry pavement) - Minimum turning radius determined by maximum ω/v ratio

Turning Radius:

When moving in a circle (v constant, ω constant): R = v/ω (radius of circular path)

  • Tighter turn: increase ω or decrease v
  • Larger turn: decrease ω or increase v
  • Straight line: ω = 0

Parameters:

This implementation has no physical parameters - it’s a pure kinematic model. Further modifications to this model may include: - Maximum speed v_max - Maximum angular velocity ω_max - Minimum turning radius R_min = v_max/ω_max

Equilibria:

Stationary at origin: x_eq = [0, 0, θ*] (any heading θ*) u_eq = [0, 0] (no velocity)

Note: Equilibria form a manifold - any (x, y, θ*) with u = [0, 0]. The system is marginally stable (doesn’t return to equilibrium on its own).

See Also:

PathTracking : Error dynamics for path following PVTOL : Flying vehicle with similar kinematics CartPole : Another nonholonomic system