systems.builtin.deterministic.continuous.SymbolicPendulum2ndOrder

systems.builtin.deterministic.continuous.SymbolicPendulum2ndOrder(
    *args,
    **kwargs,
)

Inverted pendulum - second-order formulation (returns ONLY acceleration).

CRITICAL BEHAVIOR: The forward() method returns ONLY θ̈, not [θ̇, θ̈]. This is the correct formulation for second-order systems.

Physical System:

Identical physics to SymbolicPendulum, but formulated as a second-order differential equation rather than a first-order state-space system.

State Space:

State: x = [θ, θ̇] (same as first-order variant)

Dynamics Representation:

Second-order form: θ̈ = -(β/I)θ̇ + (g/l)sin(θ) + τ/I

The forward() method computes and returns ONLY the acceleration θ̈.

State-space conversion (handled automatically): dx/dt = [θ̇ ] = [ θ̇ ] [θ̈ ] [f(θ, θ̇, τ)]

The GenericDiscreteTimeSystem integrator handles the conversion: 1. Calls forward(x, u) to get θ̈ 2. Integrates θ̈ to get θ̇{k+1} 3. Integrates θ̇ to get θ{k+1} 4. Returns x_{k+1} = [θ_{k+1}, θ̇_{k+1}]

Parameters:

m, l, beta, g : Same as SymbolicPendulum

Properties:

order : int = 2 Marks system as second-order (changes integration behavior) nq : int = 1 Number of generalized coordinates (just θ)

Notes:

• forward() output shape is (1,) for scalar acceleration, NOT (2,)
• The state x is still (2,) = [θ, θ̇]
• Integrators automatically handle the state-space conversion
• Can use different integration methods for position vs velocity
• Linearization returns full 2×2 state-space matrices

See Also:

SymbolicPendulum : First-order state-space formulation SymbolicQuadrotor2D : Another second-order system (3 accelerations)