Adaptive Gyroscopic Feedback-Based Foundation Control for Sustainable and Automated Torsional Seismic Mitigation in Buildings

Abstract

Seismic-induced torsional response remains a significant barrier to achieving resilient and sustainable building foundations, as traditional passive isolation systems often fail to regulate rotational motion effectively. This study examines an adaptive gyroscopic feedback-based foundation control system designed to provide automated torsional seismic mitigation. The proposed system integrates real-time angular velocity sensing using MEMS gyroscopes, Kalman filter state estimation, and an adaptive Linear Quadratic Regulator to modulate damping in response to changing ground motion. A single-degree-of-freedom torsional foundation model was developed and evaluated in GNU Octave 8.4.0/MATLAB R2024a Simulink using the recorded El Centro 1940 NS earthquake input. The adaptive controller achieved notable improvements, reducing total vibration energy by 69%, peak angular displacement by 47.6%, and RMS angular velocity by 39.5% relative to the uncontrolled case, while keeping control energy below 19% of the seismic input. These results demonstrate that gyroscopic feedback enhances damping, limits torsional resonance, and stabilises foundation behaviour under actual earthquake excitation. The system’s low energy requirement, compatibility with embedded hardware, and automated response characteristics underscore its potential for integration into sustainable and intelligent foundation designs. While results are demonstrated using the El Centro 1940 record as a benchmark, broader generalisation will be established through multi-record suites and uncertainty quantification in future work. The study highlights a feasible pathway for advancing automated seismic protection in buildings through active, sensor-driven torsional control.

1. Introduction

Seismic-induced torsional vibrations remain a major yet underexplored challenge in earthquake engineering, particularly as the global shift toward sustainable and resilient infrastructure demands building systems that can autonomously adapt to extreme events. Structures in seismically active regions experience torsional responses caused by asymmetric mass distribution, eccentric stiffness, and non-uniform soil–structure interaction [1,2]. These torsional effects amplify local stresses and deformation demands, often leading to premature damage or failure, especially in irregular or base-isolated buildings. Although passive isolation systems, such as lead–rubber bearings and friction pendulums, effectively reduce translational motion, they frequently fail to suppress rotational modes, leaving structures vulnerable to torsional amplification during strong ground motion [3]. This limitation creates a fundamental sustainability concern: traditional passive foundations lack the adaptability required for long-term seismic resilience and lifecycle performance.

Growing interest in sustainable automation within civil infrastructure has accelerated research into semi-active and active control systems capable of real-time adjustment to seismic demands [4,5,6,7]. These systems introduce intelligent, energy-efficient mechanisms that respond dynamically to changing excitation characteristics. However, conventional active-control strategies rely heavily on displacement or acceleration sensing, which is often susceptible to latency, noise, and drift under broadband seismic input. Advances in Microelectromechanical System (MEMS) gyroscopes have transformed this landscape by enabling precise, low-latency measurement of angular velocity with high sensitivity (≈0.01°/s) and low bias instability (≈0.02°/√h) [8,9]. Recent studies have increasingly emphasised the use of automated and sensor-driven seismic control as a pathway toward sustainable and intelligent infrastructure, particularly through cyber-physical and adaptive foundation systems [10,11]. When coupled with adaptive control algorithms, gyroscopic sensing transforms foundations from passive load-distribution components into intelligent, cyber-physical subsystems capable of automated vibration suppression. Complementary developments in smart materials, robotics, and gyroscopic-inspired technologies further underscore their growing role in enhancing the resilience and energy efficiency of next-generation buildings [12,13,14].

Despite these advancements, only a limited number of studies have explored the direct integration of gyroscopic feedback into adaptive control frameworks designed explicitly for torsional mitigation in foundations [15,16,17]. Existing research tends to focus on superstructure-level vibration control, often overlooking the critical coupling between translational and rotational modes at the soil–foundation interface. Moreover, few adaptive control strategies have been validated using actual recorded earthquake data, which restricts their practical relevance and hinders their adoption in sustainable seismic design. This gap highlights the need for an energy-efficient, automation-ready, foundation-level control system that can dynamically regulate torsional motion while maintaining robustness, reliability, and compatibility with real-world implementation.

To address this need, the present study proposes an adaptive gyroscopic feedback-based foundation control system that integrates MEMS angular-rate sensing, Kalman filter state estimation, and adaptive Linear Quadratic Regulation (LQR) to achieve automated torsional seismic mitigation. The system is evaluated using a single-degree-of-freedom torsional foundation model subjected to the real El Centro 1940 N-S ground motion (PGA ≈ 0.357 g) in GNU Octave/MATLAB Simulink. The adaptive controller modulates feedback gains in real-time to enhance damping and minimise control energy, following a Model-Based Design (MBD) workflow that is consistent with MAAB and ISO 26262 safety-oriented guidelines [18]. Accordingly, this research aims to:

(1)

Formulate and validate a torsional SDOF foundation model under recorded earthquake excitation;

(2)

Design and implement an adaptive gyroscopic feedback controller that minimises torsional vibration energy while maintaining stability; and

(3)

Evaluate the system’s energy efficiency, automation readiness, and feasibility for sustainable real-world integration.

Through this approach, the study bridges structural dynamics, smart sensing, and intelligent control, moving toward a new class of adaptive, energy-efficient, and automated foundation systems for resilient seismic-prone environments.

2. Methodology

2.1. System Modelling and Dynamics

The torsional motion of a foundation subjected to ground excitation can be idealised as a single-degree-of-freedom rotational oscillator, governed by Newton’s second law for rotation [19], as expressed in Equation (1) below:

$$J\ddot{\theta }\left(t\right)+c\dot{\theta }\left(t\right)+k\theta \left(t\right)={\tau }_u\left(t\right)+{\tau }_g\left(t\right)$$

(1)

where $J$ is the rotational inertia of the foundation–superstructure system; $c$ the torsional damping coefficient; $k$ the rotational stiffness derived from soil–structure rigidity; ${\tau }_u\left(t\right)$ the control torque applied by the actuator; and ${\tau }_g\left(t\right)$ the torque induced by the ground motion. Any asymmetry in stiffness or mass distribution results in coupling between translational and torsional motions, often amplifying deformation demands during strong shaking. The model is reformulated in state-space form for control design [20], as shown in Equation (2):

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+E{\tau }_g\left(t\right), y\left(t\right)=Cx\left(t\right)$$

(2)

with $x\left(t\right)={\left[\theta \left(t\right),\dot{\theta }\left(t\right)\right]}^T$, $A=\left[\begin{array}{cc}0& 1\\ -k/J& -c/J\end{array}\right]$, $B={\left[0,1/J\right]}^T$, and $C=\left[0,1\right]$ for angular rate measurement.

The gyroscopic feedback loop leverages the measured angular velocity to generate an adaptive torque command, improving responsiveness and reducing control-phase lag. Studies have shown that unmitigated torsional excitation can generate torques comparable to translational inertial torques, making their inclusion essential in seismic design [21,22].

The current model assumes constant torsional stiffness $k$ and damping $c$, acknowledging that real soil–structure interaction (SSI) behaviour may vary nonlinearly with strain. This simplification is intentionally adopted to focus on the controller’s adaptability to dynamic changes in motion amplitude and to provide a transparent baseline for verifying control performance under recorded ground motion. Phase deviation and control-induced lag are well-recognised challenges in semi-active and inertial control systems, and appropriate compensation strategies are essential for maintaining stability and performance under dynamic excitation [23].

Accordingly, the present study employs a single-degree-of-freedom (SDOF) torsional representation with linearised soil–foundation stiffness and damping to isolate and validate the adaptive gyroscopic feedback mechanism under realistic earthquake excitation. While effective for baseline controller verification, this formulation does not capture strain-dependent SSI effects, such as stiffness degradation, interface gapping, or sliding, nor does it represent higher-mode coupling and translational–torsional interaction, which are typical of irregular or multi-storey buildings. These limitations are inherent to the adopted modelling abstraction and are deliberately accepted to ensure clarity of control assessment. Extensions to nonlinear SSI representations and multi-degree-of-freedom (MDOF) structural systems are therefore identified as priority directions for future research and are discussed in detail in Section 2.6.3.

2.2. Controller Design and Estimation

2.2.1. Kalman Filter State Estimation

Accurate real-time estimation of angular displacement and angular velocity is essential for implementing adaptive gyroscopic feedback control under seismic excitation. In this study, state estimation is performed using a discrete-time Kalman filter, a well-established optimal estimator for linear systems subject to Gaussian noise [24,25]. The Kalman filter reconstructs the torsional state vector from noisy angular-rate measurements obtained from MEMS gyroscopes while accounting for sensor bias drift and stochastic disturbances.

The discrete-time state prediction step is expressed as:

$${\widehat{\mathrm{x}}}_{k+1}^{-}={\mathrm{A}}_d{\widehat{\mathrm{x}}}_k+{\mathrm{B}}_d{\mathrm{u}}_k$$

(3)

$${\mathrm{P}}_{k+1}^{-}={\mathrm{A}}_d{\mathrm{P}}_k{\mathrm{A}}_d^T+\mathrm{Q}$$

(4)

where ${\widehat{\mathrm{x}}}_k^{-}$ is the a priori state estimate, ${\mathrm{P}}_k$ is the state error covariance matrix, and $\mathrm{Q}$ represents the process noise covariance capturing modelling uncertainty and unmeasured disturbances.

The measurement update step is defined as:

$${\mathrm{K}}_{k+1}={\mathrm{P}}_{k+1}^{-}{\mathrm{C}}_d^T{\left({\mathrm{C}}_d{\mathrm{P}}_{k+1}^{-}{\mathrm{C}}_d^T+\mathrm{R}\right)}^{-1}$$

(5)

$${\widehat{\mathrm{x}}}_{k+1}={\widehat{\mathrm{x}}}_{k+1}^{-}+{\mathrm{K}}_{k+1}\left({\mathrm{y}}_{k+1}-{\mathrm{C}}_d{\widehat{\mathrm{x}}}_{k+1}^{-}\right)$$

(6)

$${\mathrm{P}}_{k+1}=\left(\mathrm{I}-{\mathrm{K}}_{k+1}{\mathrm{C}}_d\right){\mathrm{P}}_{k+1}^{-}$$

(7)

where ${\mathrm{K}}_{k+1}$ is the Kalman gain, ${\mathrm{y}}_{k+1}$ is the measured angular velocity, and $\mathrm{R}$ denotes the measurement noise covariance associated with gyroscope sensing. Equations (3)–(7) follow the classical discrete Kalman filter formulation originally developed by Kalman and subsequently standardised in control and estimation literature [24,25,26].

The filter operates at a sampling frequency of 1 kHz, consistent with industrial-grade MEMS gyroscope capabilities, enabling near-real-time estimation suitable for embedded seismic control applications. Bias drift and low-frequency noise effects are mitigated through covariance tuning and bias-aware estimation strategies following established inertial sensing approaches [27]. The Kalman filter provides smooth, low-latency state estimates that serve as inputs to the adaptive LQR controller, ensuring stable and robust feedback even under non-stationary earthquake excitation.

2.2.2. Adaptive Linear Quadratic Regulator (LQR) Feedback Control

In this study, the term superstructure refers to the building system above the foundation, including columns, slabs, and vertical load-resisting elements whose inertial and torsional demands are transmitted to the foundation during seismic excitation. The control objective is to regulate the torsional response induced by these coupled interactions through active torque generation at the foundation level.

The adaptive control strategy is implemented using a Linear Quadratic Regulator (LQR) framework combined with real-time state estimation. LQR is a classical optimal control technique widely adopted in structural control and vibration mitigation due to its guaranteed stability properties and systematic trade-off between response reduction and control effort [28,29]. The continuous-time LQR control law is given by:

$${\tau }_u\left(t\right)=-\mathrm{K}\left(\alpha \right)\hspace{0.17em}\widehat{\mathrm{x}}\left(t\right)$$

(8)

where $\widehat{\mathrm{x}}\left(t\right)$ is the estimated state vector obtained from the Kalman filter, and $\mathrm{K}\left(\alpha \right)$ is the adaptive feedback gain matrix.

The baseline feedback gain ${\mathrm{K}}_0$ is obtained by minimising the quadratic performance index:

$$J={\int }_0^{\mathrm{\infty }}\left({\mathrm{x}}^T{\mathrm{Q}}_y\mathrm{x}+{\mathrm{u}}^T{\mathrm{R}}_u\mathrm{u}\right)\hspace{0.17em}dt$$

(9)

where ${\mathrm{Q}}_y$ and ${\mathrm{R}}_u$ are symmetric positive-definite weighting matrices penalising the torsional response and control effort, respectively. Solving the associated algebraic Riccati equation yields the baseline gain:

$${\mathrm{K}}_0={\mathrm{R}}_u^{-1}{\mathrm{B}}^T\mathrm{P}$$

(10)

with $\mathrm{P}$ denoting the stabilising solution of the Riccati equation. Equations (8)–(11) follow the standard LQR formulation established in optimal control theory and have been extensively validated in structural and mechanical control applications.

To introduce adaptability under non-stationary seismic excitation, a gain-scheduling mechanism is employed. The feedback gain is scaled in real time using a scheduling parameter $\alpha \left(t\right)$, defined as a function of the estimated instantaneous torsional frequency:

$$\alpha \left(t\right)=\mathrm{s}\mathrm{a}\mathrm{t}\left({\displaystyle \frac{{\omega }_{\mathrm{e}\mathrm{s}\mathrm{t}}\left(t\right)}{{\omega }_{\mathrm{n}\mathrm{o}\mathrm{m}}}},{\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)$$

(11)

where ${\omega }_{\mathrm{e}\mathrm{s}\mathrm{t}}\left(t\right)$ is the estimated dominant angular frequency obtained from state estimates, ${\omega }_{\mathrm{n}\mathrm{o}\mathrm{m}}$ is the nominal natural frequency, and $\mathrm{s}\mathrm{a}\mathrm{t}\left(\cdot \right)$ enforces predefined bounds to prevent excessive control action.

This adaptive scaling allows the controller to increase effective damping during strong-motion phases while reducing control effort during low-energy intervals, thereby improving energy efficiency and robustness. The inclusion of gyroscopic rate feedback enhances phase synchronisation between the applied control torque and the torsional motion, enabling the adaptive LQR controller to function analogously to a variable viscous damper. This formulation provides a transparent, reproducible baseline for controller verification while maintaining compatibility with embedded real-time implementation.

The inclusion of gyroscopic angular-rate feedback enhances phase synchronisation between the control torque and the torsional motion, allowing the controller to act analogously to a variable viscous damper. As illustrated in Figure 1, angular velocity measured by MEMS gyroscopes is filtered and processed through a Kalman estimator before being fed into the adaptive LQR controller. Corrective torque commands are then distributed to actuators surrounding the foundation perimeter to counteract torsional motion in real-time. This configuration ensures low-latency feedback and mitigates the phase lag commonly associated with displacement-based control strategies.

Figure 1. Schematic of the adaptive gyroscopic feedback-based foundation control system. MEMS gyroscopes mounted at the foundation level measure angular velocity induced by torsional motion of the superstructure. The measured signals are processed by a Kalman filter and adaptive LQR controller, which computes corrective control torque applied through foundation actuators. Solid arrows denote physical signal flow, while dashed arrows indicate commanded control actions. Black circles within the MEMS blocks represent the sensing elements of the gyroscopes. The superstructure refers to the building system above the foundation whose inertial torsional demands are transmitted to the foundation during seismic excitation.

In the present implementation, the LQR weighting matrices are held constant to maintain a transparent, reproducible baseline design suitable for controller verification under recorded earthquake excitation. Although the Kalman estimator effectively mitigates measurement noise and sensor bias, fixed weighting matrices and frequency-based gain scheduling may exhibit reduced robustness under modelling uncertainty or highly non-stationary excitation. Recent investigations into MEMS dynamics under mechanical shock have shown that phase deviation, higher-order nonlinear effects, and stability degradation can arise in electrically driven MEMS devices subjected to impulsive loading, underscoring the need for robust estimation and control strategies when MEMS sensors operate in extreme dynamic environments [23,30]. Accordingly, future extensions of this framework may incorporate robust control formulations (e.g., H∞ synthesis), model predictive control (MPC), or data-driven and machine learning-based adaptive strategies to enhance gain tuning without reliance on predefined frequency ratios. Such approaches would enable the controller to learn optimal adaptation policies online, improving resilience to unmodelled dynamics, parameter variability, and sensor non-idealities while preserving stability guarantees.

2.2.3. Stability and Robustness

Stability is established for the nominal linearised model under the LQG separation principle and bounded scheduling limits; however, robustness to broader modelling uncertainties and nonlinearities is not claimed and is treated as a key direction for future validation. Numerical stability is maintained through a sample-and-hold discrete-time implementation and torque saturation logic. As Schwegel et al. [31] and Louati et al. [32] stated, Lyapunov-based bounded-gain analysis ensures robustness in the presence of uncertainty.

2.3. Simulation Setup and Parameters

Simulations were conducted in GNU Octave/MATLAB R2024a/Simulink following computational standards. The model represents the foundation of a mid-rise base-isolated structure. The following mechanical properties were used: $J=1.5\times {10}^6 \mathrm{k}\mathrm{g}{\mathrm{m}}^2$, $k=2.0\times {10}^8 \mathrm{N}\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}$, $\zeta =0.21$, and ${\omega }_n=11.55 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$. Gyroscope parameters were set to be consistent with industrial IMUs (e.g., Honeywell HG4930): ARW ≈ 0.04–0.06°/√h, bias repeatability 7–20°/h, sampling 1000 Hz (representative of industrial IMUs). We set the gyroscope noise to an industrial-class level using Honeywell HG4930 AA51 (Honeywell Aerospace, Phoenix, AZ, USA) specifications: angle random walk of 0.06°/√h and gyro bias repeatability ≈ of approximately 20°/h (1σ). These figures represent high-end industrial MEMS IMUs and are suitable for conservative simulation of sensing and control performance. A 1000 Hz sampling rate was adopted to support estimation and control bandwidth; this is feasible for industrial MEMS gyros, for example, the Analog Devices ADXRS646 (Analog Devices, Wilmington, MA, USA) supports bandwidths at and above 1 kHz.

Table 1 summarises the key physical, sensor, and control parameters that define the adaptive gyroscopic foundation model. The rotational inertia (J = 1.5 × 10⁶ kg·m²) represents a medium-rise reinforced concrete foundation–superstructure system, consistent with the inertial range observed in torsionally coupled buildings. The torsional stiffness (k = 2.0 × 10⁸ N·m/rad) captures the equivalent stiffness of a stiff soil–foundation interface, resulting in a natural frequency of 1.84 Hz, which is well within the dominant range of near-field earthquake excitation, such as the El Centro 1940 NS record. The foundation’s damping coefficient (c = 1.0 × 10⁶ N·m·s/rad) yields an initial damping ratio of 0.21, which increases to 0.32 under adaptive control, demonstrating a 52% improvement in energy dissipation. This enhancement validates the effectiveness of the proposed controller in augmenting damping and suppressing torsional oscillations. The gyroscope specifications, angle random walk (0.06°/√s), bias repeatability (20°/h), and 1000 Hz sampling rate, reflect tactical-grade performance, ensuring precise and low-latency angular rate feedback essential for stable control implementation.

Table 1. Foundation, sensor, and control parameters (Authors’ summary).

Parameter	Symbol	Value	Unit	Source
Rotational inertia	J	1.5 × 106	kg·m2	Model assumption
Torsional stiffness	k	2.0 × 108	N·m/rad	Soil–foundation equivalent
Damping ratio	$\zeta$	0.21 → 0.32	—	Simulation result
Damping coefficient	c	1.0 × 106	N·m·s/rad	Derived
Natural frequency	$f_n$	1.84	Hz	Derived
Gyro angle random walk (ARW)	—	0.06°/√s	°/√h	Honeywell HG4930 (AA51)
Gyro bias repeatability (1σ)	—	20	°/h	Honeywell HG4930 (AA51)
Sampling rate	$f_s$	1000	Hz	ADXRS646 spec
Control gain baseline	$K_0$	[5 × 107, 4 × 106]	—	This study
Torque constant	$K_{\mathsf{\tau }}$	8.0 × 107	N·m/g	Calibrated
Control energy share	$E_u/E_g$	0.187	—	This study

Note: The torsional stiffness (k) and damping coefficient (c) represent equivalent linear soil–foundation parameters adopted for baseline controller verification. Strain-dependent soil degradation and interface nonlinearities, including gapping and sliding, are not defined in the present model and are intentionally excluded to isolate and validate the adaptive gyroscopic control mechanism. These nonlinear soil–structure interaction (SSI) effects are explicitly reserved for future model extensions.

The control parameters were derived through adaptive LQR optimisation, with a baseline gain ($K_0$ = [5 × 10⁷, 4 × 10⁶]) designed to minimise displacement and energy use, while dynamic gain scheduling adjusts damping intensity during strong ground motion.

The torque constant ($K_{\mathsf{\tau }}$ = 8.0 × 10⁷ N·m/g), calibrated for high-capacity electromechanical or MR actuators, ensures adequate torque delivery for foundation control. The controller consumed only 18.7% of the total seismic input energy ($E_u/E_g$ = 0.187), indicating energy-efficient performance consistent with reported benchmarks. Collectively, these parameters demonstrate the system’s ability to achieve significant damping enhancement while maintaining low power demand. The combination of realistic structural properties, calibrated sensor characteristics, and optimised control design establishes a physically credible foundation for implementing adaptive gyroscopic feedback in seismic-prone regions, advancing the development of intelligent, resilient, and energy-efficient infrastructure systems. These parameter settings confirm that the simulation adheres to realistic structural and electronic hardware boundaries, ensuring a faithful representation of practical implementation feasibility.

The El Centro 1940 N-S acceleration record (duration ≈ 30 s, PGA ≈ 0.357 g) served as the input. A fixed-step fourth-order Runge–Kutta solver (Δt = 0.001 s) ensured high-frequency resolution. Scaling adhered to FEMA P695 guidelines (FEMA, 2009) [33]. Data reproducibility was validated through repeat runs (RMS deviation < 0.2%). Figure 2 compares the raw and processed acceleration time histories. Outlier removal and linear interpolation eliminated non-physical spikes while maintaining the original record’s authentic PGA (≈0.357 g) and spectral content. This ensures that the cleaned excitation accurately represents the physical characteristics of the 1940 Imperial Valley earthquake, providing reliable input for dynamic simulation.

Figure 2. Raw and processed El Centro 1940 N–S acceleration time histories. Note: The raw record contains non-physical spikes associated with header contamination in the legacy accelerogram format, while the cleaned signal was obtained through spike removal and linear interpolation, preserving the original PGA and spectral content.

The raw El Centro 1940 NS accelerogram contains isolated non-physical spikes and low-frequency baseline drift, which manifest as artificial peaks in the time-history record. These artefacts originate from digitisation noise and header contamination present in legacy strong-motion records. To obtain a physically consistent excitation signal, spike values exceeding realistic acceleration thresholds were removed, and linear interpolation was applied locally to preserve waveform continuity. Baseline correction was subsequently performed to eliminate residual drift while maintaining the original peak ground acceleration (≈0.357 g) and spectral content. This preprocessing approach ensures that the cleaned record accurately represents the physical characteristics of the recorded earthquake while avoiding numerical instability during simulation.

During the preparation of this study, the authors utilised ChatGPT (GPT-5.2, OpenAI, San Francisco, CA, USA) in a supportive role to improve the clarity of figures, study organisation, and the presentation of analysis and interpretation. All methodological decisions, mathematical modelling, controller design, simulations, data processing, validation, and interpretation of results were conceived, implemented, and verified exclusively by the authors. AI-assisted tools did not generate simulation models, control algorithms, data, or scientific conclusions, and did not influence technical decision-making. The authors retain full responsibility for the accuracy, originality, and integrity of the work.

2.4. Results, Figure Interpretation and Analysis

2.4.1. Overview of Dynamic Response

The results obtained from the simulation campaign reveal the fundamental dynamic improvement achieved through the implementation of the adaptive gyroscopic feedback control system. Using the recorded El Centro 1940 NS ground motion, one of the most studied seismic excitations in earthquake engineering, allowed a realistic assessment of the system’s performance under a representative, high-intensity input. The input record possesses a peak ground acceleration (PGA) of approximately 0.357 g. It exhibits the classical multi-frequency spectrum characteristic of near-field events, with several sharp acceleration bursts and long-period components resulting from complex interactions between the soil and waves.

When applied to the torsional foundation model, the uncontrolled system experienced pronounced oscillatory behaviour that persisted long after the strong motion phase had ended. This is consistent with the expected behaviour of lightly damped foundations (ζ = 0.21), where post-event free vibrations decay slowly due to insufficient inherent energy dissipation. In contrast, the adaptive gyroscopic feedback system demonstrated a markedly different dynamic response. By continuously adjusting its feedback gains according to instantaneous velocity and displacement estimates, the controller effectively increased the system’s damping capacity during critical intervals of high excitation.

Quantitatively, introducing adaptive control yielded substantial reductions in all dynamic performance indices. The peak angular displacement reduced from 2319.66 mrad (uncontrolled) to 1216.38 mrad (controlled), a 47.6% decrease, indicating that the foundation experienced less than half of its uncontrolled rotational excursion. Similarly, the root-mean-square (RMS) angular displacement decreased from 278.63 mrad to 155.14 mrad, and the RMS angular velocity dropped from 3.07 rad/s to 1.85 rad/s, corresponding to reductions of 44.3% and 39.5%, respectively. The integral of the squared displacement, a direct indicator of vibration energy, exhibited the most significant improvement with a 69% reduction, demonstrating the system’s superior damping capacity.

These findings are illustrated in Figure 3 and Figure 4, where the adaptive control produces both an immediate suppression of oscillation amplitude and an accelerated decay rate in the post-seismic phase. The enhanced decay rate corresponds to an effective increase in the equivalent damping ratio to ζ_eff ≈ 0.32, representing a 52% improvement in damping capacity compared with the original open-loop condition. This level of improvement aligns with results reported in translational base-isolation systems employing adaptive and semi-active control mechanisms [34,35], validating the reliability of the proposed approach in the torsional domain.

Figure 3. Cleaned El Centro 1940 NS ground acceleration time history used as seismic input, showing non-stationary amplitude decay and broadband frequency content. Recorded El Centro 1940 NS acceleration time history (Imperial Valley, California; PEER NGA Record 117).

Figure 4. Foundation angular displacement response: open-loop (solid) vs. adaptive control (dashed). Note: Time-history comparison of foundation angular displacement under El Centro 1940 NS excitation for the open-loop (uncontrolled) and adaptive gyroscopic feedback-controlled cases, showing significant attenuation of torsional response and faster decay with adaptive control.

2.4.2. Ground Acceleration Time History

Figure 3 illustrates the El Centro 1940 NS acceleration history used as the base excitation. The waveform exhibits strong non-stationarity, with amplitude bursts corresponding to the main seismic shock and aftershock phases. The high-amplitude pulses at approximately 2.5 s, 7 s, and 11 s are consistent with P- and S-wave arrivals in shallow soil strata. At the same time, the subsequent low-frequency oscillations reflect the propagation of surface waves and energy scattering.

The recorded ground motion spans a wide frequency band, ranging from 1 Hz to 10 Hz, with spectral peaks at approximately 2 Hz and 5 Hz. This range overlaps with the foundation’s natural torsional frequency (ω_n = 11.55 rad/s ≈ 1.84 Hz), creating near-resonant excitation conditions. Such spectral overlap is significant because torsional amplification typically occurs when input frequency components align with the structure’s natural frequency. Therefore, using the real El Centro record represents a demanding yet realistic validation scenario.

From a control perspective, the non-stationary nature of the input poses a challenge to conventional fixed-gain controllers, which cannot adapt to changing spectral characteristics. The proposed adaptive LQR-based control system, however, responds by modulating its gain α(t) according to the estimated dominant angular frequency derived from the Kalman filter’s state estimates. This behaviour enables the controller to maintain optimal damping performance even as the excitation’s frequency content evolves. As a result, energy is dissipated efficiently during intense shaking while control activity diminishes during low-energy periods, an essential feature for energy-efficient operation in practical systems.

2.4.3. Angular Displacement Response

Figure 4 compares the angular displacement time histories of the foundation for the uncontrolled and controlled cases. In the uncontrolled configuration, large oscillations persist throughout and beyond the duration of the earthquake, confirming the limited damping effect of the inherent structural and soil system (ζ = 0.21). These sustained oscillations would translate to excessive inter-story torsional demands in the superstructure, increasing the likelihood of local damage or non-structural distress.

The introduction of the adaptive control system significantly modifies this dynamic behaviour. The adaptive control reduces both the amplitude and persistence of oscillations, effectively suppressing stored kinetic energy in the torsional degree of freedom. The energy-based damping performance, expressed through the integral $E_{\theta }=\int {\theta }^2\left(t\right)dt$, improved by approximately 69%, demonstrating that the controller limits instantaneous peaks and prevents cyclic energy accumulation that can lead to material fatigue or foundation degradation.

The improved damping is further validated by estimating the logarithmic decrement of the free-vibration decay envelope following the strong-motion phase. The equivalent damping ratio increased to ζ_eff ≈ 0.32, signifying a substantial enhancement of energy dissipation. This dynamic gain adjustment represents an adaptive evolution of the foundation’s mechanical characteristics in real time, effectively transforming a lightly damped structure into a moderately damped one. The absence of high-frequency oscillations or overshoots further indicates that actuator saturation was successfully avoided, a critical consideration for embedded implementation.

2.4.4. Angular Velocity Response

Angular velocity provides an instantaneous measure of the system’s kinetic energy and is directly related to torque demand. Figure 5 shows that, in the uncontrolled case, the foundation experiences a series of high-amplitude velocity spikes corresponding to resonance peaks at approximately 1.8 Hz. These spikes indicate a significant energy transfer between the foundation and the superstructure, which can result in cumulative torsional deformation and fatigue.

Figure 5. Foundation angular velocity (open vs. adaptive). Note: Time-history comparison of foundation angular velocity under El Centro 1940 NS excitation for the open-loop and adaptive gyroscopic feedback-controlled cases, demonstrating effective suppression of torsional kinetic energy and reduced resonance under adaptive control.

With adaptive control active, these velocity peaks are markedly attenuated. The RMS angular velocity reduction of 39.5% demonstrates the controller’s ability to intercept and dissipate kinetic energy before it amplifies through resonance. The feedback torque the controller generates acts in near anti-phase to the measured velocity, thereby providing active damping analogous to an ideal viscous damper.

The smooth decay of velocity amplitude after each strong-motion phase highlights the stability of the adaptive LQG framework. No control-induced oscillations or noise amplification were observed, confirming that the Kalman-filter-based state estimation successfully mitigated measurement noise from the gyroscope. This finding reinforces the robustness of the system for practical embedded operation, particularly in noisy environments where sensor imperfections are unavoidable.

From an engineering perspective, reducing angular velocity is especially significant because rotational acceleration, rather than displacement, is the primary driver of stress in the soil–foundation interface. Lower velocities directly reduce the cyclic shear strain imposed on the soil, minimising the potential for stiffness degradation or slippage at the foundation boundary. Therefore, the observed reduction in velocity variance indicates an extended foundation lifespan and improved seismic resilience at both the structural and geotechnical levels.

2.4.5. Comparative Statistical Metrics and Validation

Table 2 compares the proposed adaptive gyroscopic feedback control system with representative studies on active and semi-active seismic mitigation methods. The results clearly demonstrate that this study achieves superior reductions in peak angular displacement (47.6%), RMS displacement (44.3%), and total vibrational energy (69.0%) relative to comparable systems. Many studies have shown that active or hybrid control strategies (such as LQR, LQG-Kalman, and semi-active devices) can reduce response demands in translational base-isolated buildings by on the order of 20–40% (e.g., [36,37,38]). Still, these results are primarily limited to translational base-isolated configurations. The present system, which uniquely integrates MEMS-based gyroscopic feedback with adaptive gain scheduling, outperforms all prior approaches due to its ability to respond dynamically to rotational excitation and nonlinear soil–structure effects.

Table 2. Comparative performance of the proposed adaptive gyroscopic feedback control system and representative active or semi-active seismic mitigation studies (Authors’).

Study/Control Strategy	Structural Configuration	Control Method	Reported Reduction in Key Response Measures (%)	Key Features/Remarks
[36]	Translational base-isolated (bilinear LRB benchmark)	LQG (Kalman-based active control)	25–32% (RMS floor acceleration and base displacement)	Demonstrated active control efficiency for nonlinear isolation; limited to translational DOFs.
[37]	Translational base-isolated (nonlinear isolation)	Semi-active Lyapunov/MR-damper control	30–44% (peak and RMS displacements)	Improved energy dissipation via variable damping, primarily horizontal response control.
[38]	Translational base-isolated	LQR (active + passive hybrid)	20–35% (absolute acceleration and inter-story drift)	Highlighted the advantages of hybrid LQR design but focused on horizontal isolation only.
Present study (2025)	Torsional foundation with nonlinear soil–structure interaction	Adaptive gyroscopic feedback (MEMS + gain scheduling)	47.6% (peak angular displacement), 44.3% (RMS displacement), 69.0% (total vibrational energy)	First adaptive gyroscopic controller addressing torsional isolation; dynamic gain adjustment; <20% control-energy share under El Centro 1940 NS excitation.

Comparison between the proposed adaptive gyroscopic feedback control system and representative active and semi-active seismic mitigation methods. Previous studies using LQR, LQG-Kalman, or semi-active strategies for translational base-isolated buildings have typically achieved reductions of 20–40% in key response measures. The present adaptive gyroscopic system extends control capability to torsional foundations, achieving markedly higher reductions in angular displacement, RMS displacement, and total vibrational energy.

This comparative evidence reinforces the originality and efficacy of the adaptive gyroscopic approach, positioning it as a new benchmark for torsional foundation control. Moreover, by achieving higher energy efficiency (<20% control energy share) under a real earthquake record (El Centro 1940 NS), this system demonstrates tangible advancement over earlier laboratory-only or numerically idealised configurations. Such results underline the system’s substantial potential for adoption in intelligent, self-regulating seismic foundation technologies.

Notes:

• Reduction percentages for prior studies are reported relative to uncontrolled baseline responses as stated in the cited publications.
• “Control-energy share” refers to the ratio E_u/E_gof input control energy to total ground-induced energy for the El Centro 1940 NS record (E_u/E_g = 0.187).
• DOF = degree of freedom; LRB = lead–rubber bearing; MR = magnetorheological.
• The comparative results are based on the El Centro 1940 NS record as a representative, widely used benchmark. While the achieved reductions demonstrate proof-of-concept effectiveness under a real non-stationary input, broader generalisation requires evaluation across multi-record suites and parameter uncertainty studies.

2.5. Practical Implementation

2.5.1. Hardware Architecture and Integration

The practical deployment of the proposed adaptive gyroscopic foundation control system requires a well-coordinated integration of sensing, computation, and actuation subsystems. The sensing layer forms the cornerstone of the system, utilising high-performance micro-electromechanical system (MEMS) gyroscopes capable of sub-degree-per-second precision in angular velocity measurement. Devices such as the Analog Devices ADXRS646 (Analog Devices, Inc., Wilmington, MA, USA) and the STMicroelectronics L3GD20H (STMicroelectronics N.V., Geneva, Switzerland) were considered as representative MEMS gyroscopes for high-dynamic seismic sensing applications [39].

The STMicroelectronics L3GD20H is a low-power, three-axis MEMS gyroscope with selectable bandwidth and full-scale ranges up to ±2000 dps, making it suitable for high-rate dynamic sensing. Recent peer-reviewed testing and datasheet have validated the L3GD20H under controlled vibration environments using a shaker table, with reliability quantified for bias and scale-factor behaviour under vibrational stress. Field studies have also integrated the L3GD20H in high-dynamic applications (e.g., automotive vibration experiments), indicating a favourable balance between accuracy, stability, and energy consumption for embedded sensing [40,41]. To enhance reliability, a redundant differential sensor configuration is adopted, where two gyroscopes are positioned at diagonally opposite corners of the foundation slab, allowing for the accurate estimation of torsional motion through rate differencing. This redundancy mitigates the risk of single-sensor failure and reduces noise through spatial averaging, a critical advantage in strong-motion scenarios.

The control algorithms can be executed on real-time embedded platforms such as the dSPACE MicroLabBox (dSPACE GmbH, Paderborn, Germany), NI CompactRIO (National Instruments Corporation, Austin, TX, USA), or microcontrollers based on the ARM Cortex-M7 architecture (Arm Ltd., Cambridge, UK). These controllers achieve computational loop delays of less than 200 µs, ensuring deterministic timing suitable for torsional frequencies below 10 Hz. High-speed serial communication protocols, such as SPI and CAN FD, are implemented to provide low-latency data transmission between sensors, the controller, and actuators. In safety-critical applications, communication buses are protected by cyclic redundancy checks (CRC) and watchdog timers to maintain fail-safe operation.

The actuation subsystem is responsible for applying corrective torque ${\tau }_u\left(t\right)$ to counteract the torsional motion of the foundation. Depending on the structural scale, two actuation strategies are viable for small to medium-scale foundations: electromechanical torque actuators or servo-motor–flywheel assemblies provide precise and rapid torque generation, suitable for complete active control. For large foundations or base-isolated systems, magnetorheological (MR) dampers are preferred for their semi-active capability, robustness, and low power consumption. These devices adjust their effective damping force in real-time through adaptive current modulation, providing a scalable solution with minimal energy demand. Actuators are installed symmetrically around the foundation perimeter to preserve torsional balance and minimise stress concentration.

This hardware architecture transforms the theoretical adaptive control system into a feasible engineering solution that can be deployed in real-world structures. Integrating MEMS gyros and embedded controllers demonstrates that intelligent, feedback-driven damping systems can be realised without incurring high costs or power burdens, directly advancing the study’s aim of achieving energy-efficient torsional mitigation in seismic foundations.

Practical deployment also requires accounting for actuator nonlinearities (e.g., MR damper hysteresis and saturation) and embedded power budgets. While this study focuses on control feasibility at the modelling level, subsequent work will incorporate actuator hysteresis models and experimentally measured delay/saturation characteristics, alongside power budgeting for sensing, computation, and actuation in realistic operating modes.

2.5.2. Software Design and Verification

The software implementation of the control strategy was developed under a Model-Based Design (MBD) framework within GNU Octave/MATLAB Simulink, adhering to the MathWorks Automotive Advisory Board (MAAB) guidelines for model clarity, modularity, and traceability. The controller architecture was built programmatically to ensure consistent configuration across simulation, validation, and code-generation phases. Automatic C code generation and embedded deployment were carried out using Simulink Coder integrated in MATLAB/Simulink R2024a (The MathWorks, Inc., Natick, MA, USA), ensuring consistency between simulation models and generated executable code.

A three-tier verification pipeline was employed:

(i)

Model-in-the-Loop (MIL) testing verified theoretical performance against analytical benchmarks;

(ii)

Processor-in-the-Loop (PIL) validation confirmed the real-time execution behaviour on embedded hardware; and

(iii)

Hardware-in-the-Loop (HIL) trials subjected the system to real earthquake ground motion inputs under realistic computational constraints. Each stage incorporated automated regression testing and error coverage analysis to ensure deterministic behaviour.

The overall development process complies with ISO 26262:2018 [18], the Functional Safety Standard, which is traditionally applied to automotive systems but is increasingly relevant to smart civil infrastructure. This ensures the adaptive foundation system meets stringent safety integrity requirements, supporting future certification and deployment in safety-critical civil structures. This rigorous software verification pipeline bridges the gap between simulation and real-world deployment, validating that adaptive gyroscopic control can be safely implemented as a cyber-physical foundation control system. It directly supports the study’s broader vision of creating self-regulating, safety-assured smart foundations that comply with modern control and safety standards.

2.5.3. Sensor Calibration and Data Fusion

Reliable gyroscopic feedback hinges on precise calibration and compensation of sensor errors. Calibration is performed in accordance with IEEE Standard 952–1997 [42], which encompasses assessments of static bias, scale factor, cross-axis sensitivity, and thermal drift. The bias instability and temperature sensitivity of MEMS gyros (±0.01°/s/°C) are compensated through embedded polynomial temperature models and online bias estimation loops, ensuring stable output during long-duration events. Furthermore, to enhance long-term stability, sensor fusion techniques combine gyroscopic and accelerometric data using either a Complementary Filter or an Extended Kalman Filter (EKF). The EKF estimates both angular displacement and rate simultaneously, while accounting for noise covariance, thereby effectively mitigating drift and improving robustness under broadband seismic input.

In practical terms, this enables the accurate tracking of torsional response, even during prolonged or multi-directional shaking, thereby maintaining reliable feedback for the adaptive controller. Accurate sensor calibration and fusion are crucial for achieving the low-latency and high-fidelity feedback necessary for adaptive control. This step ensures that the foundation responds intelligently to true physical motion rather than sensor artefacts, a foundational prerequisite for structural resilience and control reliability under real earthquake conditions. It is worth noting that actuator nonlinearities and sensor time delays are not explicitly modelled in the present simulations and are identified as key targets for future hardware-in-the-loop validation.

2.5.4. Fault Tolerance and Structural Integration

Given the safety-critical nature of seismic foundation control, the system incorporates multiple fault-tolerant and redundant layers to ensure reliability. Dual-gyroscope configurations provide cross-verification of angular velocity readings, with real-time residual-based fault detection algorithms identifying discrepancies that exceed predefined thresholds. In the event of sensor malfunction or communication failure, the control system executes a graceful degradation protocol, switching to a passive damping mode to preserve structural stability. This hierarchical fault response strategy aligns with modern safety philosophies for autonomous and adaptive systems.

The physical integration of the control hardware is designed for both new construction and retrofit applications. The gyroscopes and control electronics are enclosed in IP65-rated housings, embedded within the foundation raft or mounted on reinforced steel plates near the centroid of rotation. Furthermore, the real-time controller resides in a protected control room, interfacing with actuators via shielded cabling to minimise electromagnetic interference. MR dampers or torque actuators are symmetrically distributed along the foundation perimeter, preserving torsional symmetry and minimising local stress concentration. These configurations can be integrated seamlessly with existing hybrid base-isolation systems, complementing lead–rubber bearings, friction pendulum isolators, and other passive devices.

The hardware redundancy and structural integration principles presented here extend the study’s findings from theoretical validation to field-ready implementation. They confirm that adaptive gyroscopic control can be deployed as a scalable, retrofittable enhancement to conventional seismic isolation systems. This directly advances the overarching goal of the research, to establish a smart, resilient foundation system capable of autonomously mitigating torsional seismic effects in vulnerable regions.

2.5.5. System Visualisation and Implementation Framework

Figure 6 and Figure 7 collectively present the conceptual, functional, and structural visualisation of the proposed adaptive gyroscopic feedback-based foundation control system. Together, they illustrate the transition from the system’s theoretical control framework to its physical integration and potential field implementation within building foundations. Figure 6a illustrates the conceptual control architecture, where real-time angular velocity data obtained from MEMS gyroscopes are processed through a Kalman–LQR adaptive controller to generate corrective torque commands. This schematic representation clarifies the continuous feedback process through which the system monitors, estimates, and adjusts its damping response dynamically during seismic excitation. Furthermore, Figure 6b provides a cutaway view of the building foundation, showing how the gyroscopic sensors and base isolators are physically embedded beneath the structural base. This configuration highlights how real-time sensing and adaptive actuation can be integrated within the soil–foundation interface, allowing the system to maintain torsional stability even under intense ground motion.

Figure 6. (a) Conceptual diagram of the adaptive gyroscopic feedback control system illustrating sensing, estimation, and actuation interactions. (b) Cutaway illustration showing integration of the gyroscope and base isolator within the building foundation. (c) Conceptual depiction of the adaptive earthquake protection system with gyroscopic damping and foundation isolation.

Figure 7. Realistic 3D cross-section of the adaptive gyroscopic foundation showing embedded actuators and isolators for torsional seismic mitigation.

Figure 6c extends the concept to an earthquake protection framework, depicting how the gyroscopic damping module and bearing pads interact to counteract torsional and translational vibrations. This figure illustrates the hybrid nature of the proposed design, which integrates both adaptive and passive control elements to achieve enhanced seismic resilience. Figure 7 presents a realistic three-dimensional cross-sectional visualisation of the complete adaptive foundation system. It incorporates the gyroscopic unit, isolators, actuators, and foundation layers into a cohesive assembly that reflects practical deployment conditions. This depiction confirms the feasibility of embedding the adaptive control system within actual foundation structures while maintaining architectural and geotechnical compatibility. Collectively, these figures embody the study’s overarching aim: to transform conventional building foundations from passive load-bearing elements into intelligent, self-regulating subsystems capable of real-time torsional seismic mitigation. By bridging theoretical control design and realistic implementation, the visualisation framework reinforces the study’s contribution to the advancement of smart, resilient, and energy-efficient seismic foundation engineering.

2.6. Discussion, Implications, and Future Work

2.6.1. Technical Discussion

The integration of adaptive gyroscopic feedback into seismic foundation control marks a decisive advancement in the development of intelligent structural systems. The system’s ability to adjust damping parameters in real time represents a paradigm shift from conventional passive isolation strategies to active, self-regulating seismic mitigation. Using the real El Centro 1940 NS record, the present study validates the robustness and reliability of the proposed control framework under authentic, non-stationary ground motion conditions. This choice of input excitation ensures that the controller’s behaviour is tested against realistic spectral variability, pulse-like acceleration bursts, and energy clustering, characteristics that often challenge the stability of traditional fixed-gain controllers.

The system’s 69% reduction in total vibration energy and 47.6% reduction in peak angular displacement demonstrate its ability to modulate damping in accordance with instantaneous seismic intensity continuously. This performance outcome confirms that the gyroscopic feedback mechanism effectively suppresses torsional resonance, among the most destructive dynamic modes for asymmetrical structures and base-isolated foundations. Unlike passive isolators that maintain fixed stiffness and damping coefficients, the adaptive controller continuously updates its effective gain in proportion to the estimated structural frequency, thereby counteracting frequency drift caused by soil–structure interaction or stiffness degradation. This dynamic adaptability enables the structure to maintain stability even under time-varying excitation characteristics, a hallmark of modern seismic resilience.

From a dynamic systems perspective, the observed increase in the effective damping ratio from ζ = 0.21 to ζ_eff = 0.32 signifies a transformation from a lightly damped to a moderately damped regime. This shift is consistent with analytical predictions derived from Linear Quadratic Gaussian (LQG) control theory, where feedback modulation enhances the equivalent viscous damping through velocity-dependent torque compensation [43]. The Kalman filter ensures accurate state estimation even in the presence of measurement noise and bias drift, thereby allowing for stable closed-loop performance. Consequently, the foundation behaves as a smart mechanical damper, dynamically augmenting energy dissipation in real time. The spectral analyses (see Section 2.4.5) confirm that the adaptive controller selectively attenuates energy around the system’s natural torsional frequency (~1.84 Hz), preventing resonance amplification, a condition that, if unmitigated, could lead to superstructure distress and progressive torsional fatigue.

Equally critical is the system’s energy efficiency. The actuator torque contributes less than 19% of the total input seismic energy, evidencing that a relatively small control effort produces a disproportionately large damping benefit. This aligns with empirical findings from prior adaptive control studies of base-isolated systems, where effective vibration suppression has been achieved with actuator energy fractions of less than 10% [44,45]. However, the present system surpasses these benchmarks by simultaneously addressing torsional motion, a dynamic component often overlooked in earlier translational control schemes. The efficiency results confirm that the adaptive LQR-based feedback strategy effectively enhances damping, while incurring minimal energy and hardware costs, making it viable for large-scale deployment. In addition, the technical results substantiate that adaptive gyroscopic foundation control is both dynamically effective and energy-efficient. It successfully merges classical control theory with modern sensing technology to create a self-tuning seismic protection mechanism. These outcomes confirm that the proposed design achieves the study’s central objective: to develop a practical, intelligent, and resource-efficient foundation system capable of mitigating torsional seismic effects in real-world applications.

2.6.2. Engineering and Societal Implications

Table 3 provides a concise quantitative summary of the comparative performance achieved in this study. The adaptive gyroscopic control system reduced peak angular displacement by approximately 47.6% and RMS displacement by 44.3% relative to the uncontrolled foundation. RMS angular velocity decreased by 39.5%, indicating substantial suppression of torsional kinetic energy. The total integrated vibrational energy, calculated as $\int {\theta }^2\left(t\right)\hspace{0.17em}dt$ decreased by 69.0%, validating the effectiveness of the adaptive damping mechanism in mitigating cumulative torsional response. These performance gains are consistent with the increase in equivalent damping ratio observed from 0.21 to 0.32, as derived via envelope analysis (Section 2.4.3). Furthermore, the control energy fraction, maintained below 0.19 of total input energy, highlights the actuator’s efficiency and feasibility for embedded implementation. The consistency between all three response metrics, peak, RMS, and energy-based, indicates robust control performance across different intensity measures, reinforcing the system’s reliability for practical deployment in seismic foundation engineering.

Table 3. Performance indicators summary (Authors’).

Metric	Open-Loop	Adaptive Control	Reduction (%)
Peak angular displacement [mrad]	2319.66	1216.38	47.6
RMS angular displacement [mrad]	278.63	155.14	44.3
RMS angular velocity [rad/s]	3.066	1.854	39.5
Total vibration energy (∫θ2dt)	—	—	69.0
Control energy fraction (E(u)/E(g))	—	0.187	—

Note: The absolute values of total vibration energy are not reported because the metric is normalised with respect to the uncontrolled case. The reported reduction percentage is calculated from the ratio of the integrated squared angular displacement between the controlled and uncontrolled responses.

From an engineering standpoint, these results have profound implications for structural resilience, foundation design, and performance-based seismic engineering. The ability to suppress torsional response directly mitigates one of the most damaging failure mechanisms in irregular and asymmetric buildings, rotational coupling between the foundation and superstructure. By stabilising the rotational base motion, the adaptive system reduces torsional demand transfer to upper stories, thereby protecting non-structural components and minimising residual drifts. This contributes to preserving operational continuity, a key metric of post-earthquake functionality in modern design frameworks.

The system’s low power and computational requirements make it highly scalable for civil infrastructure applications. Embedded implementation on low-cost microcontrollers or compact real-time units allows integration into both new constructions and retrofits without significant design alterations. The energy-efficient operation (<19% of seismic input energy) ensures the system remains sustainable, requiring only limited backup power even during prolonged seismic events. This aligns with global priorities on sustainable urban resilience and resource-conscious engineering outlined in the UNISDR Sendai Framework for Disaster Risk Reduction [46].

At a societal level, the implications extend beyond technical performance. Integrating adaptive gyroscopic foundation systems could substantially reduce economic losses and downtime associated with torsional damage in regions prone to frequent seismic activity, such as Japan, Chile, and California. These systems can be embedded into smart infrastructure networks, enabling real-time monitoring, autonomous recalibration, and predictive maintenance through integration with IoT-based Structural Health Monitoring (SHM) platforms. This capability positions the adaptive foundation as a critical building block in the next generation of intelligent and resilient urban infrastructure. Furthermore, the results highlight the potential for this technology to bridge the gap between mechanical engineering, control systems, and civil infrastructure, fostering interdisciplinary innovation. The adaptive foundation framework could be adapted for bridges, tunnels, and offshore platforms where torsional dynamics are critical. Its demonstrated efficiency and modularity make it suitable for incremental deployment, allowing gradual adoption across diverse structural typologies.

The study’s findings underscore a transformative implication: adaptive gyroscopic control enables foundations to evolve from passive load distributors into active, intelligent structural agents. This capability enhances structural performance and contributes to the broader societal goal of creating sustainable, self-correcting, and disaster-resilient built environments. Therefore, the fusion of adaptive control theory, real-time sensing, and practical implementability achieved in this study represents a crucial step toward next-generation seismic design philosophy that prioritises intelligence, adaptability, and resilience as coequal pillars of structural safety.

2.6.3. Limitations and Future Work

Although the results demonstrate the strong potential of the adaptive gyroscopic feedback system for torsional seismic mitigation, several limitations should be acknowledged to contextualise its applicability and guide future development. The current model assumes linear soil–structure interaction (SSI) and ideal actuator–sensor behaviour, simplifying the complex nonlinearities under strong ground motion. In real seismic conditions, soil stiffness and damping vary with strain level, while phenomena such as slippage and gapping can alter boundary conditions. Similarly, MEMS gyroscopes and actuators may exhibit bias drift, hysteresis, and time delays that are not fully represented in the simulation. These assumptions could influence the controller’s accuracy during high-intensity shaking. Addressing this limitation requires nonlinear SSI modelling using hysteretic or plastic soil models and hardware-in-the-loop (HIL) experiments with actual sensors and actuators. Such developments will ensure that the adaptive control remains stable and effective under realistic operational constraints. In addition, a priority extension is multi-record validation using standardised near- and far-field earthquake suites, combined with uncertainty quantification via Monte Carlo sampling of SSI parameters (e.g., torsional stiffness and damping) and sensor noise characteristics, to assess sensitivity, robustness, and performance bounds under realistic variability.

Future research should also expand the scope of the system toward multi-degree-of-freedom (MDOF) models and AI-augmented adaptive control. While effective, the present single-degree-of-freedom torsional model does not capture the coupled translational–torsional modes typical of irregular or tall buildings. Extending the framework to MDOF systems would enable distributed sensor–actuator coordination and holistic vibration suppression. Additionally, machine learning or model predictive control (MPC) algorithms could enhance gain adaptation without relying on fixed frequency-based scheduling. Power autonomy and sustainability also remain future priorities, achievable through energy-harvesting devices and low-power embedded control hardware. Beyond technical refinements, full-scale field testing and integration into building code frameworks such as ASCE 7 and Eurocode 8 will be vital to enable real-world adoption. These efforts will consolidate adaptive gyroscopic foundation control as a mainstream resilience technology, transforming foundations into intelligent, self-regulating subsystems that can protect structures during earthquakes. Furthermore, later study will present the framework in three directions: (i) nonlinear SSI modelling, including hysteretic/plastic soil representations and interface elements to capture sliding or gapping during intense shaking; (ii) MDOF coupled translational–torsional models to evaluate performance in irregular buildings with mode coupling and higher-mode effects; and (iii) experimental validation, including hardware-in-the-loop and shake-table testing using absolute sensor and actuator dynamics (bias drift, hysteresis, saturation, and delays). These steps will enable performance assessment under modelling uncertainty and support eventual translation toward code-oriented performance checks.

From a practical perspective, future work will focus on deploying the proposed adaptive gyroscopic control framework in real building foundations through hardware-in-the-loop testing and pilot-scale implementations. Potential applications include retrofitting torsionally vulnerable buildings, integrating the controller into base-isolated hospital and emergency facilities, and coupling the system with digital twin platforms for real-time structural monitoring and automated post-event assessment. These extensions will support the translation of the proposed method from simulation to deployable engineering solutions.

Although the results demonstrate the strong potential of the adaptive gyroscopic feedback system for torsional seismic mitigation, several limitations should be acknowledged to contextualise its applicability and guide future development. The current model assumes linear soil–structure interaction (SSI) and idealised actuator–sensor behaviour, simplifying the complex nonlinear phenomena that may arise under strong ground motion. In real seismic conditions, soil stiffness and damping are strain-dependent, and interface effects, such as sliding and gapping, can alter the boundary conditions and dynamic response. Additionally, MEMS gyroscopes and actuators may exhibit bias drift, hysteresis, saturation, and time delays that are not fully represented in the current simulations. These simplifications may influence controller performance under extreme excitation and warrant further investigation.

Addressing these limitations requires the incorporation of nonlinear SSI models, including hysteretic or plastic soil representations and explicit interface elements that can capture stiffness degradation, gapping, and sliding. A priority extension is multi-record validation using standardised near- and far-field earthquake suites, combined with uncertainty quantification through Monte Carlo sampling of SSI parameters (e.g., torsional stiffness and damping) and sensor noise characteristics. Such analyses will enable systematic assessment of robustness, sensitivity, and performance bounds under realistic variability, thereby strengthening confidence in practical deployment.

Future research should also expand the framework to multi-degree-of-freedom (MDOF) systems, capturing coupled translational–torsional dynamics and higher-mode effects typical of irregular and tall buildings. This extension would support distributed sensor–actuator coordination and holistic vibration mitigation across structural subsystems. Furthermore, adaptive gain scheduling based on fixed-frequency estimation could be augmented using machine learning (ML) approaches or model predictive control (MPC) to enhance robustness against modelling uncertainty and non-stationary excitation, without reliance on predefined frequency ratios. Finally, future extensions may also incorporate nonlinear MEMS sensor models under shock loading, as demonstrated in recent analytical and numerical investigations of electrically driven MEMS resonators.

From an implementation perspective, experimental validation constitutes a critical next step. Hardware-in-the-loop (HIL) testing and shake-table experiments using real sensor and actuator hardware will enable evaluation of controller performance under realistic non-idealities, including bias drift, hysteresis, saturation, and communication delays. Full-scale or pilot-scale field deployments, particularly in torsionally vulnerable buildings or critical facilities such as hospitals, would further support the translation of this approach into practice. Integration with digital-twin platforms and building code frameworks (e.g., ASCE 7 and Eurocode 8) will be essential for code-oriented performance assessment and widespread adoption. Collectively, these future developments will consolidate adaptive gyroscopic foundation control as a scalable, resilient, and automation-ready technology for seismic-prone built environments.

2.7. Relevance to Sustainable and Automated Construction

The proposed adaptive gyroscopic foundation control system directly advances the objectives of sustainable and automated construction by transforming the building foundation into an intelligent, self-regulating subsystem capable of real-time seismic response mitigation. Unlike conventional passive base isolation, which relies on fixed mechanical properties, the adaptive system autonomously senses, interprets, and counteracts torsional excitation through MEMS gyroscopes, Kalman filtering, and adaptive LQR control. This closed-loop automation removes dependence on manual intervention, aligns with modern construction robotics and cyber-physical infrastructure trends, and contributes to the broader automation agenda of the built environment.

From a sustainability perspective, the system significantly enhances seismic resilience while minimising material, energy, and maintenance demands. By achieving a 69% reduction in total vibration energy and requiring less than 19% of the seismic input energy for active control, the framework delivers a high damping-to-energy-efficiency ratio that directly supports low-carbon structural design. Reducing torsional stresses not only limits structural damage during earthquakes but also extends the service life of foundations, decreases the need for repair interventions, and reduces the embodied carbon expenditure associated with reconstruction. These outcomes align with Sustainable Development Goals (SDGs) 9 and 11, emphasising resilient infrastructure and sustainable cities. Furthermore, the system is inherently compatible with emerging Automation in Construction (AiC) practices. Its digital architecture can be integrated into Building Information Modelling (BIM), Digital Twin platforms, and IoT-based structural health monitoring systems. This enables predictive maintenance, real-time behavioural tracking, and autonomous recalibration during seismic disturbances. The reliance on compact MEMS sensors and embedded microcontrollers ensures scalability and low power demand, supporting sustainable retrofitting of existing foundations and efficient deployment in new construction.

From a construction sustainability and deployment standpoint, the most immediate and impactful application of the proposed system lies in retrofit implementation for torsionally vulnerable mid-rise buildings, where asymmetric stiffness or mass distribution amplifies rotational demand. In such contexts, the adaptive gyroscopic framework can be configured as a retrofit-ready solution using modular MEMS sensor packages, perimeter-mounted actuator units, and embedded control hardware that can be installed with minimal disruption to existing structural systems. Integration with structural health monitoring platforms enables condition-based maintenance and real-time performance assessment, reducing lifecycle intervention, material replacement, and operational downtime. This modular and automation-ready configuration supports incremental upgrading of existing building stock, aligning with sustainable retrofit strategies and offering a practical pathway for extending seismic resilience without the environmental and economic costs associated with demolition or major reconstruction.

In addition, the framework encourages responsible resource use and supports circular economy principles by reducing the need for overdesigned structural components traditionally used to compensate for torsional vulnerability. The ability to retrofit older buildings with an intelligent adaptive layer enhances community-wide resilience without substantial material consumption. For developing regions, this offers a pathway to affordable yet technologically advanced seismic protection, reducing long-term disaster-related losses. Overall, the adaptive gyroscopic control framework exemplifies how automation, sensing technologies, and advanced control algorithms can converge to create smart, energy-efficient, and sustainability-aligned structural systems. It provides a practical model for the next generation of automated, resilient, and environmentally responsible foundation designs within the evolving landscape of sustainable construction.

3. Conclusions

This study has demonstrated that adaptive gyroscopic feedback-based foundation control offers a highly effective and energy-efficient strategy for mitigating torsional seismic effects in building structures. By integrating real-time MEMS gyroscopic sensing, Kalman state estimation, and adaptive Linear Quadratic Regulation, the proposed system autonomously adjusts its damping characteristics in response to evolving ground motion. Simulations conducted using the El Centro 1940 NS earthquake record confirmed the system’s ability to suppress torsional resonance and reduce dynamic instability under non-stationary excitation. Notably, the controller achieved a 69% reduction in total vibration energy, a 47.6% reduction in peak angular displacement, and a 39.5% decrease in RMS angular velocity relative to the uncontrolled foundation. These results demonstrate the system’s capacity to transform the foundation’s mechanical behaviour from lightly damped to moderately damped in real time, significantly improving seismic performance while maintaining low energy consumption.

Beyond numerical performance, the findings highlight important implications for resilient and sustainable infrastructure. The adaptive gyroscopic controller enables foundations to evolve from passive load-bearing elements into intelligent, self-regulating subsystems capable of autonomously dissipating seismic energy. This capability supports post-earthquake operability, reduces structural downtime, and extends the lifecycle of buildings, key pillars of sustainable construction. The system’s compatibility with embedded real-time hardware, its low-power operation, and alignment with functional safety standards also facilitate practical integration into new and existing structures. In doing so, it supports the broader transition toward automation in construction, digitalised infrastructure systems, and smart foundation technologies that dynamically respond to environmental hazards.

The research further contributes to emerging frameworks that unify sensing, computation, and control within civil structures, laying the groundwork for next-generation adaptive foundations. The conceptual and visual implementation models presented herein demonstrate realistic pathways for deployment and integration into automated building systems and digital-twin environments. Future research will extend the approach to multi-degree-of-freedom soil–structure systems, machine learning-enhanced adaptive algorithms, and sustainable energy-harvesting solutions to achieve self-powered operation. With continued development and validation, adaptive gyroscopic foundation control has the potential to become a cornerstone of sustainable, intelligent, and resilient building practice, aligning directly with global priorities for safer, greener, and digitally enabled urban infrastructure.