An Inertia Wheel Pendulum is an underactuated mechanical system made of a free-swinging pendulum arm with a motor-driven flywheel mounted at its tip. The Spong reaction-wheel pendulum at the University of Illinois popularised it as a control benchmark. By accelerating the wheel, the motor generates a reaction torque that swings the passive pendulum joint up and holds it inverted. The purpose is to test nonlinear and energy-based controllers on a system with one actuator and two degrees of freedom — a clean, repeatable rig for proving swing-up and balance algorithms before they go onto a robot.
Inertia Wheel Pendulum Interactive Calculator
Vary wheel inertia, wheel acceleration, arm inertia, and bearing friction to see the reaction torque available to swing the passive pendulum arm.
Equation Used
The calculator applies the article relationship tau = I x alpha: accelerating the flywheel creates an equal and opposite reaction torque on the passive pendulum arm. Pivot friction is subtracted before estimating arm angular acceleration.
- Reaction torque is equal and opposite to the wheel acceleration torque.
- Bearing friction is treated as a constant opposing torque.
- The result is an instantaneous torque and acceleration estimate, not a full swing-up simulation.
- Gravity, motor saturation, and controller dynamics are not included.
The Inertia Wheel Pendulum in Action
The mechanism has two bodies and one motor. The pendulum arm pivots freely on a passive bearing — no actuator there. The wheel sits at the far end and spins about its own axis, driven by a small DC or BLDC motor. When you command the motor to accelerate the wheel, Newton's third law dumps an equal and opposite torque straight into the pendulum arm. That reaction torque is the only thing you can use to move the arm, which is why this is called an underactuated system — fewer actuators than degrees of freedom.
Why build it this way? Because it forces the controller to be smart. You cannot just push the arm where you want it. You have to inject energy in pulses, timed with the swing, until the pendulum has enough kinetic energy to reach the top — classic energy shaping control. Once near the inverted position, the controller switches to a linear balancing law (usually LQR) that uses small wheel accelerations to correct tilt. The transition between swing-up and balance is the hard part, and it is exactly why the rig exists as a nonlinear control benchmark.
If the tolerances are wrong, the whole thing falls apart. Pivot bearing friction above roughly 0.5 mNm static breakdown torque eats your swing-up energy budget — the pendulum stalls below vertical. Wheel imbalance over about 1 g·mm injects a sinusoidal disturbance at wheel speed that the LQR cannot reject cleanly, and you see a limit-cycle wobble at the top. Encoder resolution below 2000 counts/rev on the pendulum joint kills your velocity estimate at low speeds — the controller chases noise and saturates the motor. Most failed builds we see are one of those three.
Key Components
- Passive pendulum arm: The unactuated link, typically 200–400 mm long with a moment of inertia of 0.001–0.01 kg·m² about its pivot. It carries the wheel and motor at its tip and swings on a low-friction bearing — a pair of preloaded R4 deep-groove ball bearings is standard, with breakaway torque under 0.3 mNm.
- Reaction wheel (flywheel): A symmetric disc, usually brass or steel, sized so its rotational inertia is 5–15% of the pendulum's pivot inertia. Too light and you cannot inject enough torque impulse to swing up; too heavy and the wheel saturates speed-wise before the pendulum reaches the top.
- DC or BLDC motor: Bolted rigidly to the arm, rotor coupled directly to the wheel — no gearbox, because backlash kills the controller. Typical units are 20–50 W flat brushless motors like a Maxon EC-flat 45 or a T-Motor MN-series, capable of 5000+ RPM and 0.05–0.2 Nm continuous torque.
- Pendulum joint encoder: Reads the arm angle at 2000 counts/rev minimum, ideally 4000+. Optical encoders like the US Digital E5 or magnetic units like the AMS AS5048A are common. Resolution drives velocity-estimate noise, which directly limits LQR gain.
- Motor encoder or hall sensors: Measures wheel position and speed for current commutation and outer-loop control. Resolution requirements are looser than the pendulum encoder — 1000 counts/rev is usually enough because the wheel runs fast and you average over many counts.
- Real-time controller: Runs the swing-up energy-shaping law and the LQR balance law at 1 kHz minimum. Quanser's QUBE-Servo, dSPACE MicroLabBox, and Speedgoat boxes are typical lab choices; a Teensy 4.1 with carefully written code handles it for hobby rigs.
Industries That Rely on the Inertia Wheel Pendulum
You will not find an Inertia Wheel Pendulum doing useful work in a factory. It exists almost entirely as a research and teaching rig — but the underlying principle, using a reaction wheel to control attitude through a passive joint, shows up in serious hardware. Spacecraft attitude control is the obvious cousin. Self-balancing one-wheeled robots, gyroscopically stabilised motorcycles, and reaction-wheel-stabilised cubes like Cubli at ETH Zürich all use the same physics. Anywhere you cannot push directly against the world but you can spin a mass, this mechanism's lessons apply.
- Control engineering education: The Quanser QUBE-Servo 2 reaction wheel pendulum used in undergraduate control labs at hundreds of universities, including MIT 6.302 and ETH Zürich's Recursive Estimation course.
- Robotics research: Mark Spong's original reaction wheel pendulum at the University of Illinois Urbana-Champaign, the canonical rig that introduced energy-shaping swing-up control in the 1995 Spong paper.
- Aerospace attitude control: Reaction wheel assemblies on the Hubble Space Telescope and the Kepler observatory — same torque-coupling principle, scaled up and triplicated for redundancy.
- Self-balancing robotics: ETH Zürich's Cubli, a 15 cm cube that jumps up onto an edge and balances using three orthogonal reaction wheels — direct descendant of the inertia wheel pendulum.
- Two-wheeled vehicle stabilisation: Lit Motors C-1 prototype electric motorcycle and the Honda Riding Assist concept, both using control-moment gyros and reaction wheels to stay upright at zero speed.
- Nonlinear control benchmarking: Standard test rig in the IFAC and IEEE control conference community for evaluating new swing-up algorithms — the Furuta pendulum and acrobot share the same benchmark role.
The Formula Behind the Inertia Wheel Pendulum
The most useful number for sizing this rig is the maximum wheel acceleration you can extract from the motor, because that sets the peak reaction torque available to the pendulum. At the low end of the typical operating range — small bench rigs with a 50 g wheel and a 20 W motor — you get a few hundredths of a Nm of reaction torque, which is just enough to swing up a 250 mm arm in 4–6 pumps. At the high end, with a 300 g wheel and a 100 W motor, you push past 0.3 Nm and swing up in one or two pumps but risk wheel speed saturation before vertical. The sweet spot for a teaching rig sits around 0.05–0.1 Nm peak reaction torque, where the swing-up takes 3–4 visible pumps and the balance dynamics stay slow enough for a student to see what the controller is doing.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| τreact | Reaction torque applied to the pendulum arm by accelerating the wheel | N·m | lbf·ft |
| Iwheel | Mass moment of inertia of the reaction wheel about its spin axis | kg·m² | slug·ft² |
| αwheel | Angular acceleration of the wheel commanded by the motor | rad/s² | rad/s² |
| mw | Mass of the reaction wheel disc | kg | lb |
| rw | Radius of the reaction wheel disc (solid disc, I = ½ m r²) | m | ft |
Worked Example: Inertia Wheel Pendulum in a university control-systems teaching rig
A mechatronics lab in Delft is building an inertia wheel pendulum for a graduate nonlinear-control course. The pendulum arm is 300 mm long with a pivot inertia of 0.0040 kg·m². The reaction wheel is a brass disc, 80 mm diameter, 8 mm thick, mass 0.32 kg, giving I<sub>wheel</sub> = ½ × 0.32 × 0.040² = 2.56 × 10<sup>-4</sup> kg·m². The motor is a Maxon EC-flat 45 brushless rated 0.255 Nm peak. The team needs to know the reaction torque available across the realistic command range so they can size the swing-up controller and predict how many energy pumps the rig will need.
Given
- Iwheel = 2.56 × 10⁻⁴ kg·m²
- αnominal = 300 rad/s²
- αlow = 100 rad/s²
- αhigh = 1000 rad/s²
- Motor peak torque = 0.255 N·m
Solution
Step 1 — at nominal command, the controller asks for 300 rad/s² of wheel acceleration. Compute the reaction torque dumped into the pendulum:
That is roughly 30% of the motor's peak torque, well within continuous rating, and produces a pendulum angular acceleration of τ / Iarm = 0.0768 / 0.0040 = 19.2 rad/s² at the arm. At this level the swing-up takes about 3 visible energy pumps over 2–3 seconds — slow enough that a student watching can see each pump build amplitude.
Step 2 — at the low end of the typical operating range, 100 rad/s² wheel acceleration:
That is barely above the pivot bearing's static friction band on a typical R4 bearing pair (around 0.005–0.01 Nm including seal drag). The pendulum can still swing up, but it needs 6–8 pumps and the controller has almost no margin against bearing stiction — if the bearings are not run-in, the rig will refuse to leave the bottom.
Step 3 — at the high end, 1000 rad/s² wheel acceleration, demanded by an aggressive energy-shaping gain:
That equals the motor's peak torque exactly — the controller saturates the current loop. In theory the pendulum could swing up in a single pump. In practice the wheel hits its speed limit (around 7000 RPM on this Maxon unit) before the pendulum reaches vertical, the wheel cannot accelerate any further, the reaction torque collapses to zero, and the arm falls back. The sweet spot for this rig sits between 200 and 500 rad/s² — fast swing-up, no saturation, clean handoff to the LQR balance law.
Result
At the nominal 300 rad/s² command, the rig delivers 0. 0768 N·m of reaction torque into the pendulum — enough to swing up a 0.0040 kg·m² arm in 3 pumps over about 2.5 seconds. At the low end (100 rad/s²) the torque drops to 0.0256 N·m and the swing-up stretches to 6–8 pumps with no margin against bearing friction; at the high end (1000 rad/s²) the 0.256 N·m demand saturates the motor and the wheel runs out of speed before the pendulum reaches vertical. If your measured swing-up time is longer than predicted, the most common causes are: (1) wheel-shaft coupling slop above 0.1° backlash, which lets the motor accelerate without dragging the wheel for the first millisecond of each pump, (2) motor current loop bandwidth below 1 kHz, which delays torque delivery and de-tunes the energy-shaping controller, or (3) pendulum encoder mounting eccentricity above 0.05 mm, which feeds a once-per-revolution error into the velocity estimate and bleeds energy out of the controller's pump timing.
Inertia Wheel Pendulum vs Alternatives
The Inertia Wheel Pendulum is one of three classic underactuated benchmarks. Picking between them depends on what you want to teach or test — they share the family resemblance but each stresses a different controller weakness.
| Property | Inertia Wheel Pendulum | Furuta Pendulum | Acrobot |
|---|---|---|---|
| Actuated joint location | Wheel at tip, pendulum joint passive | Base joint actuated, pendulum passive | Elbow joint actuated, shoulder passive |
| Typical swing-up time (teaching rig) | 1–3 s, 2–4 pumps | 0.5–2 s, 1–3 pumps | 2–5 s, 3–6 pumps |
| Pendulum encoder resolution required | ≥2000 counts/rev | ≥2000 counts/rev | ≥4000 counts/rev (two passive states) |
| Mechanical complexity | Low — one motor, no slip ring | Medium — slip ring or bus on rotating arm | Medium — two arms, cable management |
| Cost (lab-grade rig) | $2,500–$6,000 (Quanser QUBE) | $5,000–$12,000 | $8,000–$18,000 |
| Saturation failure mode | Wheel speed saturation before vertical | Base motor torque saturation | Elbow torque saturation, shoulder runaway |
| Best fit application | Energy-shaping + LQR teaching, reaction-wheel attitude analogue | Classic inverted pendulum control, RL benchmarks | Gymnast-style multi-link control, hybrid systems |
| Lifespan of a typical lab build | 10+ years (no wear path except bearings) | 5–10 years (slip ring wear) | 5–10 years (cable flex fatigue) |
Frequently Asked Questions About Inertia Wheel Pendulum
That limit-cycle wobble at the top almost always traces to wheel imbalance, not controller tuning. A static imbalance of even 0.5–1 g·mm injects a sinusoidal disturbance torque at wheel rotation frequency. While the wheel spins down through your LQR's natural frequency band — typically 1–5 Hz — the disturbance crosses the controller bandwidth and the LQR cannot reject it.
Quick diagnostic: spin the wheel up to a known speed with the pendulum clamped vertical, then release the clamp and log the arm angle. If the wobble frequency tracks wheel speed, it is imbalance. Fix it on a static balancer — get the residual under 0.2 g·mm and the wobble disappears.
Rule of thumb: Iwheel between 5% and 15% of Ipendulum measured about the pivot. Below 5% and the motor cannot inject enough energy per pump — you spend the whole swing-up budget fighting bearing friction. Above 15% and the wheel mass at the tip dominates the pendulum's own inertia, the natural frequency drops, and the swing-up gets sluggish because each pump moves the wheel a long way before transferring useful energy.
For a 250–350 mm arm with a small brushless motor in the 30–50 W class, a brass or steel disc 70–90 mm diameter and 6–10 mm thick lands you in the sweet spot every time.
Pick the Inertia Wheel Pendulum if you want to teach energy-shaping and reaction-torque thinking — the physics maps directly onto spacecraft attitude control and self-balancing robots, which students find motivating. Pick the Furuta if your course focus is classic inverted-pendulum LQR and you want a faster, more visually dramatic swing-up.
Mechanically the inertia wheel rig is simpler — no slip ring on a rotating arm, no cable management around a base joint. That matters if you have 12 rigs in a teaching lab and want them all running 5 years from now without service calls.
Almost always the motor and wheel inertia at the tip. CAD captures the geometry but most students forget that the motor rotor inertia adds to the pendulum's effective tip mass when the wheel is held fixed (as it is during free-swing identification). On a flat brushless motor the rotor inertia is 50–150 g·cm² — not huge, but it sits at the longest moment arm.
Re-identify the natural frequency experimentally with a logarithmic decrement test, then back-calculate Ipendulum. Use that number for controller design, not the CAD value. Your LQR gains will land within 5% on the first try instead of needing field tuning.
Coulomb friction at the pivot bearing. Simulation models almost always use viscous damping only, because Coulomb friction is a pain to identify and makes the equations non-smooth. On the real rig, the breakaway torque at low arm velocity bleeds energy out of every pump, and your simulated energy budget overestimates how high the pendulum will get.
Two fixes: identify the Coulomb term with a slow rotation test and add it to the controller's energy estimate, or just bump up the energy-shaping gain by 20–30% as a margin. The second works fine for teaching rigs. For research papers, identify it properly.
Direct-drive only. A gearbox kills the rig in two ways. Backlash — even 0.1° on a planetary set — destroys the controller because the reaction torque transmission becomes discontinuous; the wheel accelerates freely through the backlash before the pendulum sees any torque, and your energy pumps lose timing. Gearbox friction also adds load-dependent loss that breaks the clean I × α relationship the controller relies on.
If you need more torque, go bigger on the motor or the wheel inertia, not the gear ratio. A 100 W flat brushless direct-drive will outperform a 30 W motor with a 5:1 planetary every time on this rig.
1 kHz minimum, 2 kHz preferred. The dominant pole of the inverted pendulum sits around 5–8 rad/s for a typical 300 mm arm, and the unstable mode at the top has a similar time constant. To control an unstable system cleanly you want 50–100× the unstable pole frequency in your sample rate — that puts you at 250–800 Hz absolute floor. Add velocity-estimate filtering and quantisation effects from the encoder and the practical floor lifts to 1 kHz.
If you are seeing a high-frequency chatter at the top that sounds like the motor is buzzing, your loop rate is probably the limiter, not the controller gains. An Arduino Uno at 100 Hz will not balance this rig. A Teensy 4.1 at 2 kHz will.
References & Further Reading
- Wikipedia contributors. Inertia wheel pendulum. Wikipedia
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