systems.builtin.deterministic.continuous.SymbolicQuadrotor2D

systems.builtin.deterministic.continuous.SymbolicQuadrotor2D(*args, **kwargs)

Planar quadrotor (quadcopter) - second-order formulation.

Physical System:

A quadrotor constrained to move in a 2D vertical plane with two rotors providing thrust. The system has: - 3 degrees of freedom: (x, y) position and pitch angle θ - 2 control inputs: thrust forces from left and right rotors - Underactuated: 3 DOF controlled by 2 inputs - Nonlinear coupling between rotation and translation

State Space:

State: x = [x, y, θ, ẋ, ẏ, θ̇] Position coordinates: - x: Horizontal position [m] (positive right) - y: Vertical position [m] (positive up) - θ (theta): Pitch angle [rad] (positive counterclockwise) * θ = 0: level orientation

Velocity coordinates:
- ẋ (x_dot): Horizontal velocity [m/s]
- ẏ (y_dot): Vertical velocity [m/s]
- θ̇ (theta_dot): Angular velocity [rad/s]

Control: u = [u₁, u₂] - u₁: Left rotor thrust [N] - u₂: Right rotor thrust [N] Both must be non-negative in physical systems (thrust-only)

Output: y = [x, y, θ] - Measures position and orientation

Dynamics:

The equations of motion are: ẍ = -(u₁ + u₂)/m · sin(θ) ÿ = (u₁ + u₂)/m · cos(θ) - g θ̈ = L/I · (u₁ - u₂)

Physical interpretation: - Total thrust (u₁ + u₂) provides vertical lift and horizontal acceleration - Differential thrust (u₁ - u₂) creates torque for rotation - Gravity acts downward with acceleration g - Thrust direction rotates with pitch angle θ

Parameters:

length : float, default=0.25 Half-distance between rotors [m]. Larger L → more control authority for rotation (more torque from differential thrust). mass : float, default=0.486 Total mass of quadrotor [kg]. Based on Crazyflie 2.0 specs. inertia : float, default=0.00383 Moment of inertia about center of mass [kg⋅m²]. gravity : float, default=9.81 Gravitational acceleration [m/s²].

Equilibrium:

Hovering equilibrium (level flight): x_eq = [x, y, 0, 0, 0, 0] (any (x, y), level, stationary) u_eq = [mg/2, mg/2] (each rotor supports half the weight)

Default Physical Parameters:

• Mass: 0.025 kg (25 grams)
• Length: 0.046 m (rotor arm)
• Inertia: 0.00383 kg⋅m²
• Gravity: 9.81 m/s²

See Also:

SymbolicQuadrotor2DState : Full-state observation variant PVTOL : Similar dynamics but different parameterization CartPole : Another underactuated 2D system