Control & mechanics · Momentum storage

Reaction-Wheel Balance

How can spinning up a wheel keep a rod upright? Every time the motor speeds the wheel up one way, it twists the rod the other way with an equal torque. The controller times that twist to cancel gravity's tipping and settle the rod upright. Once the rod is still, the wheel can keep spinning, because nothing has taken its spin away. The motor's torque fades as it nears top speed, so a small wheel or a low speed limit can let the rod fall.

Keep a pivoted rod upright by speeding up or slowing down its flywheel. Change the wheel radius, the available torque or the initial spin, and see why the rod can come to rest while the wheel is still spinning fast.

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 the fall, then find where control fails

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\,

A positive tilt leans right of vertical. Drag the gold handle below to set the starting tilt, or use the sliders with the keyboard. Playback runs at one quarter of physical speed so the response is easy to follow.

The uniform rod has a mass of 0.30 kilograms and a length of 0.80 metres. The uniform wheel has a mass of 0.35 kilograms, with its centre at the rod tip. Changing the wheel radius changes the rotor's inertia but not the rod length or the centre-of-mass position. The hub and stand are schematic and massless.

The angle controller cancels gravity, is critically damped and acts only within sixty degrees of upright. It does not bleed off the wheel's momentum. The current limit and the torque limit, which falls linearly with the rotor's speed relative to the motor, are ideal design assumptions, not a calibrated motor.

Each run covers eight physical seconds with fourth-order Runge–Kutta in 8000 or 16000 steps, plus cubic Hermite interpolation. The signed work is integrated with the same dynamics, and the residual shown checks the stored step states. Saturation and controller switching make the motion less smooth. To judge trajectory accuracy, compare with the fine integration.

The pivots are ideal and frictionless, the motion is held to a plane, and there are no collision surfaces. A rod that falls may swing through the schematic stand. Sensor delay, structural flex, electrical losses, a regenerative battery model, hardware safety guarantees and a swing-up controller are not included. Fast rotor markings can alias during real-time playback, but the angle and exported data keep physical time.

A rigid rod moves in one plane and carries a uniform disk that spins about its center. The motor follows an ideal torque-speed limit, and the angle feedback compensates for gravity. Swing-up, momentum unloading, impacts and out-of-plane gyroscopic motion are not modeled.

Start with the essentials

Focus question: Why can the pole balance while the rotor keeps its angular momentum?
One-sentence intuition: The motor twists the wheel and the rod with equal and opposite torques. The rod can come to rest while angular momentum stays stored in the wheel. Gravity is the external torque about the pivot.

Core mathematical model

Equations of motion for rod and rotor

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

The rotor's angle and speed are measured in the lab frame; the motor shaft sees the difference between the rotor and rod speeds. The body's moment of inertia includes the rod and the wheel's mass carried at its center, but not the rotor's spin about its own axle.

Angle feedback below the motor limit

\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 equation holds only near upright, where the controller is active, and only while the motor can deliver the requested torque.

Motor power and the change in energy

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. It is the mechanical work done at the shaft; electrical consumption and losses are left out.

Common difficulties

Feedback has limits

Typical misconceptionOnce feedback is on, the system will recover from any release.

Better mental modelCompare the limit presets. These controllers balance only near upright and have limited drive strength. They are not swing-up controllers.

Absolute and relative rotation

Typical misconceptionIf the rod is still, the flywheel must have stopped.

Better mental modelLook separately at the rotor's absolute angular momentum and at its speed relative to the rod, which is what the motor sees.

Run the experiment

01
Tilt and release
Drag the gold handle to set the initial tilt, then press play.
What to observe: Watch the rod, and watch how hard and in which direction the controller drives the motor.
02
Find the motor's limit
Choose Low speed ceiling, then try a larger wheel.
What to observe: Look for the moment when the motor stops delivering what the controller asks for.
03
Compare energy with motor work
Go to the end of the playback and export the record. Then compare standard and fine integration.
What to observe: The change in mechanical energy matches the work done by the motor, which is negative whenever the motor brakes.

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