Control & mechanics · Momentum storage

Reaction-Wheel Balance

Balance a pivoted rod by accelerating its flywheel. Change wheel radius, available torque or initial spin, and see why a stable rod can leave a fast-spinning rotor behind.

Tilt from upright $0$

Rod angular speed $0$

Delivered rotor torque $0$

Requested rotor torque $0$

Motor-relative wheel speed $0$

Absolute rotor angular 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}}

Stall torque limit

\tau_{\max}

1.20\,\mathrm{N\,m}

Uniform wheel radius

R

0.18\,\mathrm m

Initial absolute wheel speed

\nu_0

0.00\,\mathrm{rad/s}

Motor-relative no-load speed

\Omega_{\rm nl}

180.00\,\mathrm{rad/s}

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.

Uniform rod: 0.30 kilograms, length 0.80 metres. Uniform wheel: 0.35 kilograms; its center sits at the rod tip. Radius changes rotor inertia, not the rod length or center-of-mass position. The hub and stand are schematic and massless.

Gravity-compensated, critically damped angle control acts only within sixty degrees of upright. It does not unload wheel momentum. The current limit and linear motor-relative torque-speed envelope are ideal design assumptions, not a calibrated motor.

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.

One-plane rigid rod and centered uniform disk. Ideal torque-speed envelope and gravity-compensated angle feedback. No swing-up, momentum unloading, impacts or out-of-plane gyroscopic motion.

Start with the essentials

Focus question: Why can the pole balance while the rotor keeps its angular momentum?
One-sentence intuition: A motor applies equal and opposite torques. The rod can settle while angular momentum remains stored in the wheel; gravity supplies external torque about the pivot.

Core mathematical model

Body and rotor dynamics

I_b\ddot\theta=G\sin\theta-\tau,\qquad J\dot\nu=\tau,\qquad\dot\phi=\nu

The rotor angle and speed are absolute; the shaft sees the difference between rotor and rod speeds. Body inertia includes rod and wheel-center translation, excluding rotor axial spin.

Unsaturated angle feedback

\tau_d=G\sin\theta+I_b(\lambda^2\theta+2\lambda\dot\theta),\qquad\ddot\theta+2\lambda\dot\theta+\lambda^2\theta=0

The second relation holds only while the motor delivers the requested torque in the active upright neighborhood.

Power at the relative shaft

E=\tfrac12 I_b\dot\theta^2+\tfrac12J\nu^2+G(\cos\theta-1),\qquad\dot E=\tau(\nu-\dot\theta)

Motor work can be positive or negative. This is mechanical shaft work; electrical consumption and losses are omitted.

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.

Absolute and relative rotation

Typical misconceptionA steady rod means the flywheel has stopped.

Better mental modelRead the absolute rotor angular momentum and motor-relative speed separately.

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 Low speed ceiling, then compare a larger wheel.
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