Control & mechanics · Force limits

Cart-Pole Balance

Tilt a real three-dimensional cart-pole and release it. Watch feedback catch the pole, return the cart, and lose control when the motor cannot deliver the requested push.

Tilt from upright $0$

Rod angular speed $0$

Delivered cart force $0$

Requested cart force $0$

Cart position $0$

System horizontal momentum $0$

Signed motor work $0$

Maximum sampled work-balance residual $0$

Release & actuator

Catch it. Then find the limit.

Initial tilt

\theta_0

8.00\,{}^\circ

Initial rod angular speed

\omega_0

0.00\,\mathrm{rad/s}

Motor off or feedback on

b_{\rm on}

1.00\,

Nominal response rate

\lambda

2.00\,\mathrm{s^{-1}}

Force limit

F_{\max}

12.00\,\mathrm N

Pivot-to-bob length

l

1.00\,\mathrm m

Integration: standard or fine

b_{\rm fine}

0.00\,

Positive tilt leans right from vertical. The gold handle below edits the initial tilt; sliders also work with the keyboard. Playback is one quarter of physical speed so the response is easy to follow.

Cart mass is one kilogram; the point bob is one quarter kilogram. The visible cart, bob and rod thickness are schematic. The rail is an unlimited ideal guide; displayed rail ends are not stops.

Full-state linear feedback places four local poles at the negative nominal response rate. Force is clipped to the selected limit. Control acts only within sixty degrees of upright; local stability is not a global recovery guarantee.

Eight physical seconds, fourth-order Runge–Kutta, 8000 or 16000 steps and cubic Hermite interpolation. Signed work is integrated with the same dynamics; the displayed residual checks stored step states. Saturation and controller switching reduce smoothness. Compare fine integration for trajectory accuracy.

Ideal frictionless pivots, prescribed planar constraint and no collision surfaces. A failed rod may rotate through the schematic stand. No sensor delay, structural flex, electrical losses, regenerative battery model, hardware safety guarantee or swing-up controller is included. Fast rotor markings can alias in real-time playback; angle and exported data retain physical time.

Ideal point bob, massless rigid rod and frictionless unlimited rail. Local state feedback with force saturation; no swing-up, impacts, sensor noise or motor electrical model.

Start with the essentials

Focus question: Why does a rightward lean first require a rightward push?
One-sentence intuition: Horizontal base acceleration turns the pole through the contact constraint. The controller also tracks cart position and velocity so the base returns.

Core mathematical model

Nonlinear cart-pole dynamics

(M+m)\ddot x+ml\cos\theta\,\ddot\theta-ml\sin\theta\,\dot\theta^2=F,\qquad l\ddot\theta+\cos\theta\,\ddot x=g\sin\theta

Tilt is measured toward the right from upright. The rod is massless and the bob is a point mass.

Local full-state control

F_d=k_xx+k_v\dot x+k_\theta\theta+k_\omega\dot\theta,\qquad F=\operatorname{clip}(F_d,-F_{\max},F_{\max})

Gains are recomputed from the chosen length and response rate, placing four unsaturated local poles at the negative response rate. All gain signs follow this upright angle convention.

Mechanical work audit

E=\tfrac12(M+m)\dot x^2+ml\dot x\dot\theta\cos\theta+\tfrac12ml^2\dot\theta^2+mgl(\cos\theta-1),\qquad\dot E=F\dot x

The coupling term is essential. Potential is zero at upright; energy change, not its arbitrary offset, is compared with signed motor work.

Common difficulties

Feedback is not unlimited

Typical misconceptionTurning feedback on guarantees recovery from every release.

Better mental modelCompare the limit presets. These are local balancing controllers with finite drive capacity, not swing-up algorithms.

A stable pole still needs base control

Typical misconceptionAngle feedback alone returns the cart to its start.

Better mental modelThe force law uses cart position and velocity as well as pole tilt and angular velocity.

Run the experiment

01
Release
Drag the gold initial-tilt handle, then play.
What to observe: Follow the rod and signed motor demand.
02
Find the limit
Choose Too little force, then raise the force limit.
What to observe: Look for separation between requested and delivered actuation.
03
Audit the energy
Go to the end and export the record. Compare standard and fine integration.
What to observe: Mechanical energy change follows signed motor work.

Primary theory: MIT · Cart-Pole · ETH Zürich · 1D reaction-wheel model, Section II