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