Inverted Pendulum Cart-Pole Virtual Lab

2D DYNAMICS WORKBENCH | Free Open-Loop Dynamics

Bob Trace Forces

PASSIVE RUN

ANGLE θ 0.00°

ANG VEL θ̇ 0.00 rad/s

CART POS x 0.00 m

CART VEL ẋ 0.00 m/s

TOTAL ENERGY 0.00 J

CONTROL FORCE u 0.00 N

Real-Time Strip Chart (θ(t) & u(t))

θ (rad) u (N/10)

Phase Plane (θ vs θ̇)

Stage 1: Physical Parameters

RK4 INTEGRATION

Experiment with nonlinear free dynamics derived from Lagrangian mechanics. Displace the pendulum or apply impulsive external forces to observe energy dissipation.

Cart Mass (M) 1.20 kg

Pendulum Bob Mass (m) 0.20 kg

Rod Length (l = L/2 to COM) 0.50 m

Cart Friction (b) 0.10 N·s/m

Pivot Damping (b_θ) 0.01 N·m·s

Initial Drop Angle θ₀ 170°

Mathematical Derivation & Equations

Applying Euler-Lagrange equations with generalized coordinates q = [x, θ]ᵀ:

(M + m)ẍ - mlθ̈ cos θ + mlθ̇² sin θ + bẋ = u

mlẍ cos θ + I_rod θ̈ - mgl sin θ + b_θ θ̇ = 0

Linearization around the unstable equilibrium (θ = 0, x = 0) yields the state-space form ẋ = Ax + Bu. The Riccati equation minimizes quadratic cost:

J = ∫ [xᵀ Q x + uᵀ R u] dt ⟹ u = -K x