The core insight of the new work is that no single controller suits every operating regime of a self-balancing wheelchair. Near the upright equilibrium, the system behaves almost like a linear inverted pendulum, and a linear quadratic regulator, or LQR, is an ideal tool: it is computationally cheap, mathematically well understood, and delivers smooth, accurate regulation when deviations are small. But when the chair is tilted far from vertical, say 35 to 45 degrees, the linear approximation breaks down badly. The nonlinear terms in the equations of motion, particularly the sine of the tilt angle and the coupling between translation and rotation, dominate the behavior, and an LQR designed around the upright point loses authority rapidly. Nonlinear model predictive control, or NMPC, can handle these large deviations because it re-solves an optimization problem at every sampling instant using the full nonlinear dynamics, but that repeated online optimization carries a heavy computational price and is not always superior to simpler methods once the system is already close to equilibrium.

The hybrid NMPC–LQR scheme resolves this tension with an elegant switching law. Whenever the magnitude of the pitch angle exceeds a predefined switching threshold, the NMPC controller takes command, exploiting its nonlinear recovery capability to drive the chair back toward upright under explicit constraints on the available wheel torque. Once the tilt angle falls below the threshold, control authority is handed to the LQR, which stabilizes the system efficiently in the near-equilibrium region. The result, according to the team’s simulations, is a controller that recovers faster than either standalone method while cutting the average computation time from 48.04 minutes for standalone NMPC to 35.00 minutes for the hybrid scheme across repeated simulation runs, a meaningful reduction in the online optimization burden that matters for real-time embedded implementation.

Choosing the switching threshold is not a trivial matter, and the researchers devoted an entire methodological framework to it. If the threshold is set too small, NMPC remains active until the chair is nearly upright, wasting computational resources and delaying the efficient local regulation that LQR provides. If it is set too large, the controller hands over to LQR while the system is still in a strongly nonlinear region, where the linear controller’s assumptions are invalid and recovery performance degrades. The team first performed a linearization-error analysis, comparing the nonlinear model against its linearized counterpart across angular ranges of 0.015 to 0.150 radians, and found that the mismatch in pitch angular acceleration grows gradually within that interval but rises sharply beyond it, establishing 0.015 to 0.150 radians as a defensible search range.

Within that range, the researchers evaluated six candidate thresholds using a normalized five-index criterion that combines settling time, steady-state error, steady-state standard deviation, drift rate, and maximum steady-state deviation into a single composite cost. The weights were deliberately chosen to prioritize steady-state accuracy, since precise balance near upright is the primary requirement for occupant safety and comfort. Among the candidates tested under nominal model parameters, a threshold of 0.100 radians yielded the lowest composite cost, representing the best trade-off between transient recovery and steady-state regulation. The team went further with a sensitivity analysis, varying the equivalent upper-body mass and pendulum length independently, and discovered a clear empirical relationship: heavier occupants require smaller switching thresholds, so that NMPC stays in charge over a wider angular range. The fitted relation, with a coefficient of determination of 0.945, offers designers a practical rule for adapting the threshold to different users.

The modeling foundation beneath all of this is a reduced-order description of the wheelchair–occupant system derived through the Lagrange method. The chair and its occupant are treated as a rigidly connected inverted pendulum, with the two driving wheels assumed identical and in pure rolling contact with the ground. Nonholonomic rolling constraints are eliminated using a null-space projection, which removes the constraint reaction forces from the equations and yields a compact two-degree-of-freedom model in longitudinal displacement and pitch angle. The authors are candid about the limits of these assumptions: yaw, roll, tire deformation, wheel slip, and active occupant motion are not modeled, and the rigid-body assumption holds only when the occupant sits relatively still. Rather than pretending these effects do not exist, the team treated them as unmodeled disturbances and stress-tested the controller against them.

Those stress tests form the empirical heart of the study. In comparative simulations with initial tilt angles of 35, 40, and 45 degrees, the hybrid controller achieved the shortest settling time under all three conditions, and at 45 degrees it also delivered the smallest steady-state error, standard deviation, and maximum deviation. Against a sliding-mode control benchmark, a popular robust nonlinear method, the contrast was stark: the sliding-mode controller failed to recover from 40 and 45 degree tilts, while the hybrid controller remained stable across every tested initial angle. Disturbance-rejection trials told a similar story. Under pulse disturbances of 50 to 80 newtons, the chair exhibited a secondary deflection but never diverged, and under persistent step disturbances of 10, 20, and 30 newtons the peak deviations grew to 3.62, 5.89, and 8.11 degrees respectively, yet always remained bounded and returned toward equilibrium.

Robustness to real-world imperfections was examined through flat-ground friction tests in which the static friction coefficient was varied from 0.03 to 0.30, spanning low-, medium-, and high-friction contact conditions. The wheelchair did not overturn under any of them, although the high-friction case produced larger residual pitch oscillations, suggesting that stronger wheel–ground interaction can amplify small fluctuations near equilibrium. The authors interpret these results as bounded recovery rather than exact convergence, an honest framing that reflects the difficulty of the problem. Finally, the team validated the strategy on a desktop-scale physical prototype built on the WHEELTEC B570 architecture, using an STM32F103C8T6 microcontroller and an MPU6050 inertial sensor. In balance-recovery experiments starting from 25, 30, and 35 degree tilts, the prototype returned to a small neighborhood of upright in every case, with transient oscillations that grew as the initial angle increased but never threatened stability.

The significance of this work extends beyond wheelchairs. Two-wheeled inverted-pendulum systems appear in segways, delivery robots, and personal mobility devices, and the challenge of balancing fast nonlinear recovery against efficient local stabilization is universal. What distinguishes this study is its methodological rigor in quantifying the handover point between controllers: rather than picking a switching threshold by intuition, the team grounded it in linearization-error analysis, optimized it with a multi-index criterion, and characterized how it should shift with user mass. That kind of disciplined threshold selection addresses a recognized gap in the hybrid control literature, where switching boundaries are often chosen heuristically and their parameter dependence is rarely measured.

Limitations remain, and the authors enumerate them plainly. The model is planar and does not capture yaw, roll, lateral instability, or the coupled dynamics of a moving human body. The stability argument rests on local LQR analysis and extensive closed-loop validation rather than a rigorous global Lyapunov proof for the switched nonlinear system, which remains an open theoretical problem. The prototype is desktop-scale, and full-scale trials on slopes and uneven terrain are still to come. Future work, the team says, will pursue human–wheelchair coupled modeling, adaptive threshold calculation, explicit friction compensation, and robust NMPC with terminal constraints. For now, the study offers a concrete, tested blueprint for making self-balancing wheelchairs safer under exactly the conditions that frighten their users most: a sudden jolt, a heavy passenger, or a chair tipped dangerously far from vertical.

Subject of Research: Hybrid nonlinear model predictive and linear quadratic balance control for two-wheeled self-balancing wheelchairs

Article Title: Hybrid nonlinear model predictive and linear quadratic balance control for a two-wheeled self-balancing wheelchair

Article References: Sun, H., Zhang, X., Gu, Y., Wang, S., Chen, W., Zhang, S., Xu, T., Yu, H., & Meng, Q. (2026). Hybrid nonlinear model predictive and linear quadratic balance control for a two-wheeled self-balancing wheelchair. Mechanical Sciences, 17(2), 713-729. https://doi.org/10.5194/ms-17-713-2026

Image Credits: AI Generated

DOI: 10.5194/ms-17-713-2026

Keywords: self-balancing wheelchair, model predictive control, LQR, nonlinear control, inverted pendulum, switching threshold, disturbance rejection, assistive robotics, robustness, prototype experiments, rehabilitation engineering, underactuated systems

The central challenge the researchers tackled is a fundamental tension in flight control. On one hand, engineers want to guarantee that the aircraft’s tracking errors stay within strict bounds, a property known as prescribed performance, so that the jet never strays outside a safe corridor during its descent. On the other hand, the aircraft’s control surfaces and engine thrust are physically limited. When a controller demands more deflection or thrust than the actuators can deliver, a condition called actuator saturation, the aircraft temporarily stops obeying commands and the error can breach the very boundaries the performance function was designed to enforce. Once those boundaries are violated, many prescribed performance controllers fail catastrophically, because their mathematics assumes the error always remains inside a constrained domain.

The Harbin team’s solution is elegantly pragmatic: rather than pretending saturation will never happen, they built a controller that bends without breaking. At the heart of the method lies a predefined-time prescribed performance function, a mathematical curve that dictates how quickly and how tightly the landing error must shrink. Unlike conventional exponential convergence guarantees, which depend on initial conditions, a predefined-time guarantee ensures the error converges within a time chosen by the designer, no matter where the aircraft starts. This is a significant advantage in carrier operations, where the approach window is short and unforgiving and where an autopilot that converges ‘eventually’ is not good enough.

To work with these constraints mathematically, the researchers employed an error transformation technique that converts the constrained error domain into an equivalent unconstrained one. In essence, the transformation warps the coordinate system so that as long as the transformed error remains bounded, the physical landing error is guaranteed to stay inside its prescribed funnel. This allows standard control design tools to be applied to a problem that would otherwise involve hard, nonlinear boundaries. The transformed error then feeds into a nonsingular fast terminal integral sliding mode controller, a robust control strategy renowned for its ability to reject external disturbances. The ‘terminal’ component drives errors to zero in finite time rather than merely asymptotically, while the ‘nonsingular’ design avoids the mathematical pitfalls, such as division by zero, that plague naive terminal sliding mode formulations.

The truly novel ingredient, however, is the flexible adjustment mechanism embedded within the performance function itself. When the controller detects that actuators are saturating, meaning the aircraft cannot fully execute the commanded deflections, the mechanism adaptively relaxes the local performance boundaries, giving the aircraft room to recover rather than demanding the impossible. Once the saturation episode passes, the boundaries tighten again, and the landing proceeds with its original accuracy targets intact. This adaptive flexibility prevents the boundary violations that would otherwise cause the error transformation to blow up and the controller to fail. Complementing this mechanism, the team designed an auxiliary system that maintains stability during saturation events, a well-known strategy in the control literature that the authors integrated carefully with their performance framework.

Disturbance rejection receives equally thorough treatment. The turbulent airwake behind a carrier’s island superstructure is modeled in the study as a set of external disturbances acting on the aircraft’s translational and rotational dynamics, with components that vary sinusoidally over time to mimic the chaotic flow field. To counter these, the researchers developed a predefined-time disturbance observer, an algorithm that estimates the unknown disturbance signals and feeds the estimates directly into the control law as compensation. Because the observer converges in predefined time, the compensation kicks in quickly enough to matter during the brief final approach. The team also introduced a predefined-time reference model to generate the command signals that the inner control loops track. This choice eliminates a chronic computational problem in conventional backstepping control designs, where repeated differentiation of command signals leads to ‘explosion of complexity,’ an exponential growth in the analytic terms that must be computed at each step.

The aircraft model used in the study is a full six-degree-of-freedom representation of a carrier-based aircraft, complete with the coupling between roll, pitch, and yaw dynamics that arises from the aircraft’s inertia properties, aerodynamic coefficients for lift, drag, and side force, and the contributions of the elevator, aileron, rudder, and direct lift flap. The guidance problem is framed as following a moving deck landing path, with the controller tasked with tracking the ship’s motion while respecting output constraints on altitude and attitude errors. This comprehensive modeling matters, because simplified models often hide exactly the coupling effects that make real carrier approaches so treacherous, particularly the way lateral gusts induce rolling moments that must be countered within fractions of a second.

To benchmark their approach, the authors compared FPTPPC against three competing strategies in simulation: dynamic inverse control, a classical technique that cancels the aircraft’s nonlinearities through model inversion; predefined-time control without prescribed performance; and standard prescribed performance control without the flexible saturation mechanism. The results showed that the proposed method reduced path following errors and improved overall landing accuracy relative to all three baselines. The simulations also demonstrated that the flexible adjustment mechanism kept the controller stable and accurate through saturation events that would have destabilized a rigid prescribed performance design. While the study is computational rather than flight-tested, the consistency of the improvements across the comparison set lends weight to the practical promise of the approach.

The implications extend beyond naval aviation. The core ideas, predefined-time convergence guarantees, flexible performance boundaries that tolerate actuator limits, and disturbance observers that compensate within a designer-chosen horizon, are directly transferable to other safety-critical control problems, including spacecraft attitude control, unmanned aerial vehicle operations in gusty conditions, and robotic systems with limited actuator authority. Related work in the same journal and elsewhere has begun applying predefined-time and anti-saturation techniques to fault-tolerant flight control, suggesting a broader research movement toward controllers that are honest about physical limits rather than assuming them away. For the moment, the Harbin team’s contribution stands as a careful piece of control engineering: a mathematically rigorous answer to the question of what an autopilot should do when the ideal command and the possible command diverge, at the worst possible moment, a few hundred feet above a moving flight deck.

Subject of Research: Automatic carrier landing control using flexible predefined-time prescribed performance control under output constraints, actuator saturation, and airwake disturbances

Article Title: Flexible Predefined-Time Prescribed Performance Control for Automatic Carrier Landing with Output Constraints and Actuator Saturation

Article References: Wang, Z., Zhu, Q., Kong, L., & Wang, L. (2026). Flexible Predefined-Time Prescribed Performance Control for Automatic Carrier Landing with Output Constraints and Actuator Saturation. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01239-y

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01239-y

Keywords: automatic carrier landing, prescribed performance control, predefined-time control, actuator saturation, sliding mode control, airwake disturbance, flight control systems, disturbance observer, carrier-based aircraft, nonlinear control, landing accuracy, Harbin Engineering University