Comparative analysis of LQR and PID control for a reaction-wheel stabilized cube under actuator saturation constraints

1. Introduction

Reaction wheels are the dominant actuator for fine attitude control in small spacecraft and CubeSats, where the absence of external reaction forces and the strict power and mass budgets make momentum-exchange devices the natural choice [9]. Ground-based testbeds emulating these dynamics are essential for hardware-in-the-loop verification of attitude-control algorithms before flight. Among such testbeds, the Cubli – a cube that balances on its edge using internal flywheels [1], [2] – has become a de-facto benchmark because its inverted-pendulum dynamics combine instability, underactuation, and saturation-limited actuation in a compact platform [4], [6]. The same mathematical structure appears in CubeSat momentum-bias control and in small inspector-satellite design, which makes the Cubli a representative case study for reaction-wheel attitude-control systems in general.

The control of reaction-wheel inverted pendulums has been studied extensively. Energy-based swing-up [3] and passivity-based laws [5] established the foundations of jump-up control. Gajamohan et al. [1], [2] demonstrated edge and corner balancing of the Cubli, Muehlebach and D’Andrea [6] developed a full nonlinear analysis of corner balancing, Trimpe and D’Andrea [4] addressed sensor estimation and self-tuning LQR, and Fankhauser et al. [7] reported an integrated modelling and control study on a related balancing cube. Block et al. [8] gave a comprehensive treatment of the reaction-wheel pendulum, while Prasad et al. [10] and Trentin et al. [11] reported comparative PID-vs-LQR studies on related platforms. Markley and Crassidis [9] supply the spacecraft attitude-control background relevant here.

Three engineering-oriented gaps persist. (G1) Actuator saturation is typically introduced as a post-design Saturation block placed downstream of an LQR or PID law synthesized for an unconstrained linear plant; the resulting domain of attraction is not mapped against actuator capability, so designers lack quantitative actuator-sizing criteria [1], [6], [7]. (G2) Comparative studies of LQR and PID for the Cubli are sparse and typically use different plants, cost functions, and saturation assumptions, which makes their conclusions difficult to transfer to a new design [10], [11]. (G3) The jump-up maneuver is usually presented as a qualitative demonstration; a quantitative torque-margin requirement linking motor stall torque to body inertia and tilt amplitude has not been reported, although this is the single most important sizing constraint for any practical implementation [1].

The contribution of this paper is threefold and addresses G1-G3 directly. First, a unified CAD-driven Simscape Multibody pipeline is described in which the SolidWorks assembly is exported with preserved inertial properties, linearized about the unstable equilibrium with the Model Linearizer, and used to synthesize both LQR and PID controllers that are then validated on the same nonlinear plant with identical saturation limits. Second, the relationship between ${\tau }_{max}$ and the achievable domain of attraction is mapped, showing a saturation-dominated regime at low torque and diminishing returns above an identifiable threshold. Third, a quantitative torque-margin criterion for the jump-up maneuver is established and used to show that the hobby-grade Nidec 24H BLDC motor (${\tau }_{max} =$ 0.04 N·m) falls short of the requirement by a factor of 12.5.

2. System model

3. Results

Both controllers were tested on the identical Simscape Multibody model with ${\tau }_{max}=$0.04 N⋅m (Nidec 24H specifications) and initial tilt angle ${\theta }_0=$3°. Table 2 summarizes the performance metrics.

To quantify the relationship between actuator capacity and stabilization performance, the LQR controller was tested across a range of ${\tau }_{max}$ values. Table 3 presents the domain of attraction (maximum initial tilt angle for successful stabilization) and the feasibility of the jump-up maneuver (raising the cube from a flat face to edge balancing).

The results reveal a strongly nonlinear relationship (Fig. 2) between ${\tau }_{max}$ and the domain of attraction. A 5× increase in torque (0.04 → 0.20 N·m) expands the domain by a factor of 6.7 (3° → 20°), while a subsequent 2.5× increase (0.20 → 0.50 N·m) yields only a 2.25× expansion (20° → 45°). This diminishing-returns characteristic indicates that the actuator-constrained system operates in a saturation-dominated regime at low torque levels, where the controller spends most of its time at the torque limit.

Fig. 1Settling time comparison: LQR achieves stabilization in 0.5 s vs. 3 s for PID (θ0= 3°, τmax= 0.04 N⋅m)

Fig. 2Domain of attraction as a function of maximum actuator torque. The jump-up threshold (45°) requires τmax≥ 0.5 N·m

The jump-up was executed from ${\theta }_0$ $=$ –45° to $\theta =$ 0° using an impulse of 0.5 N·m over 0.5 s superposed on the LQR command (Fig. 3); the peak flywheel velocity reached ≈ 470 rad/s and post-impulse settling required ≈ 0.4 s with ≈ 3° residual overshoot. A conventional Stateflow SpinUp → Brake → Balance machine was also evaluated but stalled in Brake under saturation, because the wheel could not decelerate fast enough; the continuously active impulse-plus-LQR architecture avoided this failure mode.

Fig. 3Jump-up maneuver from θ0= –45°: tilt angle θ(t) and flywheel velocity φ˙t. The orange region marks the impulse phase

4. Discussion

The performance gap between LQR and PID follows from the feedback structure: LQR uses the wheel state $\phi ̇$, which carries the system’s angular momentum, while PID reconstructs it implicitly through I/D action, an information loss that anti-windup cannot recover. The nonlinear ${\tau }_{max}$-to-domain map (Fig. 2) shows two regimes: below 0.20 N·m the actuator limit dominates and extra torque buys area in the stability region roughly linearly; above 0.20 N·m the linear regime takes over and returns diminish; crossing 0.50 N·m enables the jump-up by delivering $mgl∆\theta /J_w$ within the 0.5 s window, justifying the observed 12.5× torque margin. The conventional Stateflow SpinUp → Brake → Balance jump-up [1], [2] stalls under saturation because the brake phase cannot decelerate the wheel fast enough; the impulse-plus-LQR architecture used here removes this failure mode and is, to the authors’ knowledge, the first saturation-aware Cubli jump-up reported with quantitative validation. The same reaction-wheel dynamics govern fine attitude control of small spacecraft [9], so the sizing rule ${\tau }_{max}\ge mgl \mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{max}\right)$transfers to reaction-wheel selection on CubeSat air-bearing testbeds. The study is restricted to single-axis edge balancing without sensor-noise modelling; LQG re-design with explicit margin re-evaluation and hardware-in-the-loop tests are planned next.

5. Conclusions

A unified CAD-driven Simscape Multibody pipeline was used to compare LQR and PID controllers for a 130 mm reaction-wheel Cubli-type platform under realistic actuator saturation, addressing gaps G1-G3 of the introduction. Three results emerge. Under ${\tau }_{max}$= 0.04 N·m, LQR settles in 0.5 s at 85 rad/s peak wheel velocity against 3 s and 120 rad/s for PID, a 6× speed-up and 30 % energy reduction; PID retains a marginally larger initial-tilt envelope (4° vs. 3°). The domain of attraction grows nonlinearly with ${\tau }_{max}$ (3°, 8°, 20°, 45° at 0.04, 0.10, 0.20, 0.50 N·m), with diminishing returns above 0.20 N·m. The jump-up requires ${\tau }_{max}$ ≥ 0.50 N·m (12.5× Nidec 24H), and the conventional Stateflow approach stalled under saturation while the impulse-plus-LQR architecture succeeded. The pipeline and sizing rule transfer directly to CubeSat ground testbeds; future work will extend to three-axis balancing, Kalman estimation, and hardware-in-the-loop validation.