The Reaction-Wheel Inverted Pendulum, or RWIP, is a legend in control engineering circles. Conceptually simple, it hides a minefield of difficulties: the upright position is an unstable equilibrium, meaning the slightest deviation grows exponentially until the rod crashes down. The system is also underactuated, a technical term with a blunt meaning. There are two mechanical degrees of freedom, the rod angle and the wheel angle, but only one control input, the voltage applied to the direct-current motor that spins the reaction wheel. The wheel is a momentum-exchange device: spin it one way and the reaction torque nudges the pendulum the other way. This is the same physical principle that keeps spacecraft attitude control systems honest, and it has spawned celebrated variants such as the One-Wheel Cubli, a three-dimensional pendulum balanced by a single flywheel. The RWIP captures all the essential drama of stabilizing an unstable equilibrium while remaining tractable enough for a wide range of synthesis methods, which is why it has anchored decades of teaching and research.

The IFPR prototype is a study in low-cost ingenuity. A rigid metal rod rotates about a horizontal pivot on a metal frame. At one end sits a reaction wheel rigidly fixed to the shaft of a brushed DC motor; at the opposite end, a counterweight reduces the gravitational torque about the pivot and, with it, the control effort the motor must deliver. A slip ring with a central bore routes the motor power leads through the pivot itself, allowing the rod to rotate freely without twisting its own cables. Incremental encoders measure the rod angle at 600 pulses per revolution and the motor shaft angle at 350 pulses per revolution; derivative filters applied to these readings produce the angular velocities that full-state feedback requires. An ESP32 microcontroller acquires the encoder data and drives an H-bridge with pulse-width-modulation signals, while the control algorithms run in MATLAB/Simulink on a desktop computer linked over serial, sampling every millisecond. That 1 kHz rate is no accident: the fastest closed-loop eigenvalue of the designs sits below 10 rad/s, so the sampling rate exceeds the closed-loop bandwidth by more than two orders of magnitude, letting the engineers treat the discrete implementation as quasi-continuous without a discrete-time redesign.

The mathematical backbone of the work is a model derived from Lagrange’s equations, a choice made deliberately. The Lagrangian route yields the equations of motion of the two-degree-of-freedom underactuated system directly in terms of the generalized coordinates, without the tedium of computing and eliminating internal reaction forces at the pivot and motor mount. Better still, the kinetic and potential energy functions constructed in the derivation are precisely the quantities manipulated by the energy-based swing-up law, giving methodological consistency between modeling and control design. The pendulum dynamics were then coupled with the DC-motor dynamics, including the brush-friction term and the back-electromotive-force effect, and linearized about the upright equilibrium. Because the wheel angle is a cyclic coordinate that does not enter the equations of motion, the state vector shrinks from four variables to three, rod angle, rod angular velocity, and wheel angular velocity, without losing any information relevant for stabilization. The resulting linearized model was verified to be controllable, the precondition for both synthesis methods.

Here the study confronts a practical paradox. The reaction-wheel motor simply lacks the torque to lift the pendulum from its stable downward equilibrium to the upright position by direct action. The pendulum must first be swung up by a maneuver that injects energy through the system’s natural oscillations. The team adopted the energy-control approach of Åström and Furuta in its simplified bang-bang form: the actuator is driven to its saturation limits according to the sign of the pendulum velocity, pumping energy in at each oscillation cycle and progressively increasing the amplitude. Because the bang-bang law is undefined at rest, an initialization pulse during the first second dislodges the pendulum and produces the non-zero velocity the law needs. Once the pendulum nears the vertical, an angle-based switching law takes over, partitioning the angular range into three regions: a swing-up region where energy is injected, a coasting region where the input is set to zero and the pendulum glides on residual motion, and a stabilization region within a narrow cone of the vertical where the state-feedback controller engages. The threshold was chosen with care: at the switching angle, the position term of the feedback law alone already demands about 10.3 volts of the available 12, so substantially wider cones would saturate the motor.

Two full-state feedback controllers were designed on this common model, matched in closed-loop bandwidth so the comparison isolates the synthesis criterion rather than arbitrary tuning differences. The first, pole placement, assigns the closed-loop eigenvalues directly, offering a transparent design parameter at the cost of no explicit handle on control effort. The dominant pair was placed at about 1 rad/s, giving a settling time of roughly 4 seconds after capture, while a third eigenvalue, mainly tied to wheel velocity, sits a decade faster and below the 20 rad/s bandwidth of the derivative filter. The second, the Linear Quadratic Regulator, returns the gain that minimizes a quadratic performance index penalizing both state deviation and control effort, but only indirectly shapes the closed-loop dynamics. Its weighting matrices were refined iteratively from a Bryson-type starting point, scaled by the inverses of the squared maximum admissible deviations, until capture occurred without saturation. Solving the continuous-time algebraic Riccati equation yielded the optimal gain, and the resulting closed-loop eigenvalues landed at approximately the same locations as the pole-placement design, about 1 rad/s, ensuring a fair fight.

Both controllers were evaluated in simulation and implemented experimentally under the same disturbance scenario, with performance quantified through four integral indices: the integral of the absolute error (IAE), the integral of the squared error (ISE), and their time-weighted counterparts (ITAE and ITSE). Each index tells a different story. ISE punishes large errors harshly and is sensitive to oscillations; IAE accumulates error with less emphasis on peaks; ITAE weights errors more heavily as time progresses, exposing long-lasting steady-state error; ITSE combines both traits, amplifying the penalty for large late errors. Over the full response, including the swing-up maneuver, the two controllers performed comparably, with experimental index differences below 5 percent. Over the stabilization phase alone, the LQR posted marginally lower IAE, ISE, and ITSE while pole placement claimed the lower ITAE, but neither design was uniformly superior. The simulation told a nearly identical tale, with the two controllers practically indistinguishable, differences below 0.1 percent over the full response.

The discrepancies between simulation and experiment proved as instructive as the controller comparison. In simulation, the swing-up takes longer because, in the absence of friction and mechanical backlash, the bang-bang law accumulates more energy per cycle, overshooting the stabilization cone with higher velocity and stretching the transient. In the experiment, the relation inverts: the real system never rests exactly at equilibrium. Bearing and brush friction, mechanical backlash, the finite encoder resolution, derivative-filter noise, and the H-bridge dead zone together sustain a small-amplitude oscillation around the upright position, inflating the experimental stabilization indices by factors of two to three over simulation. Under the disturbance pulses, however, the root-mean-square response differed by only a factor of 1.4 to 1.5, indicating that the closed-loop dynamics is well reproduced by the model and that the index gap originates almost entirely from the unmodeled steady-state residual.

To settle whether the modest controller differences reflected the synthesis method or mere run-to-run variability, the team ran a repeatability campaign of fifteen complete swing-up, stabilization, and disturbance runs per controller, thirty runs in total. The outcome was conclusive: the mean stabilization-phase indices of the two designs differed by only 0.6 to 1.4 percent, while the coefficient of variation within each controller ranged from 5.6 to 13.4 percent. The difference between synthesis methods was about one order of magnitude smaller than the variation between repetitions of the same method. The two designs are therefore equivalent in performance on this prototype, and the choice between them should rest on design considerations, the systematic nature of the LQR weighting procedure or the transparency of direct eigenvalue assignment, rather than on measured results. The same campaign doubled as a robustness gauntlet: across sixty independent disturbance events, every one was rejected without loss of stabilization, with a largest angular excursion of 9.90 degrees, using at most 5.78 volts of the available 12, a margin of more than half, and a peak wheel speed of 400 rpm against the 880 rpm motor rating.

A simulation-based sensitivity sweep of the switching threshold, varied across twenty values, confirmed low sensitivity of the closed loop within its operating envelope: capture occurred in every case, stabilization held for every threshold from 0.2 to 0.5 rad, and the stabilization-phase indices varied by less than 1 percent. Below 0.2 rad, the pendulum was still captured, but the stabilization region became narrower than the angular excursion produced by the disturbance, so control was handed back to the swing-up branch and the pendulum fell, an undesired behavior the adopted threshold of 0.3 rad comfortably avoids, providing a margin of 1.5 times the largest excursion actually measured on hardware. The work stands as a practical validation and a didactic comparison of state-feedback synthesis methods applied to the RWIP, a platform for teaching and applied research built from accessible components. Future directions include extending the model with friction and backlash, sweeping disturbance amplitudes to find the largest rejectable one, testing robustness under payload variation, and using fuzzy Takagi-Sugeno models to address the upright equilibrium without relying on a single linearization. For now, the pendulum in Jacarezinho keeps balancing, indifferent to which controller holds it upright.

Subject of Research: State-feedback control of a reaction-wheel inverted pendulum prototype

Article Title: State-feedback control of a reaction-wheel inverted pendulum prototype

Article References: Souza, R. H. S. D., Santos, J. B. D., Silva, L. D. D., Breganon, R., & Alves, U. N. L. T. (2026). State-feedback control of a reaction-wheel inverted pendulum prototype. Frontiers in Control Engineering, 7, Article 1905949. https://doi.org/10.3389/fcteg.2026.1905949

Image Credits: AI Generated

DOI: 10.3389/fcteg.2026.1905949

Keywords: State-feedback, control, reaction-wheel, inverted, pendulum, prototype, scientific research