🎯 Inverted Pendulum: The Classic Control Problem

A free pendulum has a stable equilibrium hanging straight down (θ = 180°): nudge it and gravity pulls it back. The cart-pole's upright point (θ = 0°) is the opposite kind of fixed point — the real nonlinear equations balance there in theory (net torque is zero), but any tiny nudge grows instead of shrinking, because gravity's restoring torque near θ = 0 acts on the pole in the direction that increases the tilt, not the direction that corrects it. That is the textbook definition of an unstable equilibrium, and it is why this simulation needs an active controller running every frame — turn it off (set all gains to 0) and the pole falls immediately, with no push needed.

This simulation numerically integrates the standard coupled cart-pole ODEs (after Sutton, Barto & Anderson, 1983, with added friction): θ̈ = [g·sinθ − cosθ·((F + m·l·θ̇²·sinθ − b·ẋ)/(M+m)) − c·θ̇/(m·l)] / [l·(4/3 − m·cos²θ/(M+m))], and ẍ = [F + m·l·(θ̇²·sinθ − θ̈·cosθ) − b·ẋ] / (M+m), where M is cart mass, m is pole mass, l is the half-length of the pole, F is the applied cart force, and b, c are small viscous friction terms. These are solved with a fixed small time step every frame — nothing about the fall or the recovery is pre-animated.

Linearizing the equations above near θ = 0 shows the closed loop is only stable once Kp exceeds g·(M + cart mass), roughly 19–20 N/rad for this rig's masses — below that threshold gravity's destabilizing torque outweighs the proportional correction and the pole falls, slowly at first and then rapidly, no matter how it started. That threshold is a real, derivable property of this exact system, not a tuned-in constant, which is why the Weak gains preset (Kp well under it) reliably demonstrates a slow, genuine loss of balance.

Excessive proportional gain overreacts to small angle errors, overshooting the correction and exciting oscillation; excessive derivative gain amplifies the angular-rate term so aggressively that the actuator repeatedly saturates at ±22 N and the control force starts chattering between its limits. Both failure modes are visible directly in this simulation: watch the force arrow flicker and the cart shudder back and forth as the oscillation grows until the pole tips past recovery — a real instance of the classic PID-tuning tradeoff between responsiveness and stability.

Balancing and repositioning are coupled in a real cart-pole: to move the cart, you first have to lean the pole slightly in the direction of travel and let gravity accelerate it, exactly like a person balancing a broomstick who leans it to walk it across a room. This simulation implements that as cascade control — a fixed outer loop turns the cart-position error into a small commanded tilt (clamped to about 8°), and the angle PID loop you tune then tracks that tilt instead of a fixed zero. That is why pushing the setpoint slider far while the gains are weak can itself cause a fall — it is the same, harder dual-objective control problem faced by real Segways and rocket landers.

Push → and Push ← apply an instantaneous kick to the pole's angular velocity θ̇, the same effect a real shove near the top of the pole would have. Whether the controller rejects it depends entirely on the current gains and the size of the kick relative to the actuator's 22 N force limit — a well-tuned loop absorbs it and returns to vertical, while a weak or badly-tuned one cannot generate enough corrective force in time and the pole falls, exactly as the real, unmodified physics dictates.

It is one of the simplest systems that is simultaneously nonlinear, underactuated (one motor, two degrees of freedom) and genuinely unstable, which makes it a standard stress test for both classical control (PID, state-space/LQR, pole placement) and modern reinforcement learning, where "CartPole" is one of the first environments almost every RL algorithm is benchmarked against. Its rail end-stops and angle-failure bound in this simulation echo the original 1983 Barto–Sutton–Anderson formulation (±2.4 m, ±12°), though the visual fail threshold here is set a little wider so you can actually watch the fall happen instead of the trial ending instantly.