Damping Oscillations in a Spherical Pendulum Inclinometer Using Vector-Based Control

Fernando Capes; Mikael Andreas Bianchi; Roberto Gardenghi; Manuel Alò

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Damping Oscillations in a Spherical Pendulum Inclinometer Using Vector-Based Control

Fernando Capes, Mikael Andreas Bianchi, Roberto Gardenghi, Manuel Alò

arXiv:2608.31030v1Aug 31, 2026eess.SY

TL;DR

The paper addresses disturbance-induced oscillations that limit the dynamic usability of high-precision pendulum-based inclinometers despite their gravity-referenced stability. It combines electromagnetic actuator design, spherical-pendulum modeling, observer-based velocity feedback, and constrained force allocation, achieving strong experimental damping with a speed–residual-oscillation trade-off.

Problem
Pendulum-based inclinometers have low intrinsic damping, so external disturbances produce persistent oscillations and increased settling time while dynamic behavior remains comparatively understudied.
Method
The paper integrates a six-coil electromagnetic actuator, a control-oriented spherical-pendulum model, observer-based velocity feedback, and constrained force allocation for feasible planar damping.
Results
31.11 dB attenuation of the dominant oscillatory mode and a lowest cumulative oscillation index of Jr = 3.27 mm s were obtained at Imax = 0.6 A.
Takeaways & Limitations
Active electromagnetic damping substantially improves the dynamic usability of pendulum-based inclinometers without contact-based damping methods.

Abstract

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This paper addresses disturbance-induced oscillations in high-precision pendulum-based inclinometers, which reduce measurement availability despite the long-term stability of gravity-referenced sensing. To actively suppress these oscillations, a contactless six-coil electromagnetic actuation system is developed, together with a control framework that generates, in real time, the planar damping force required by the controller despite the nonlinear, unilateral, and bounded nature of magnetic actuation. The proposed approach combines a control-oriented nonlinear model of the spherical pendulum, observer-based feedback control, and a constrained force-allocation scheme that maps the desired vector force into feasible coil currents. Experimental validation on a prototype demonstrates attenuation of the dominant oscillatory mode up to 31.11 dB and a marked reduction in transient duration, with the 50\% decay time decreasing from 11.10 s without control to 1.01 s at the most favorable operating point. The results also highlight a trade-off between transient speed and residual oscillation, demonstrating both the effectiveness of the proposed damping strategy and its value as a practical design framework for high-precision inclinometer systems.

I. Introduction

Pendulum-based inclinometers provide a gravity-referenced alternative for microradian-level sensing, but weak intrinsic damping leaves them vulnerable to persistent disturbance-induced oscillations. The paper therefore targets active damping while preserving equilibrium and measurement integrity.

I. Introduction: Gravity-referenced pendulum inclinometers reduce drift and support microradian-level resolution, unlike MEMS systems limited by temperature bias, stress, and aging.Applications include structural health monitoring of dams.
I. Introduction: Low intrinsic damping causes persistent oscillations after external disturbances, increasing settling time and reducing dynamic measurement availability.Passive alternatives may introduce mechanical coupling, thermal noise, equilibrium perturbations, or limited low-velocity authority.
I. Introduction: The proposed framework generates feasible planar damping forces in real time while respecting the actuator’s physical limits and preserving measurement integrity.The framework addresses nonlinear, unilateral, and bounded magnetic actuation through force allocation.
I. Introduction: The inclinometer uses pendulum-cable triangulation with sequentially activated LEDs and a pixel array to infer pendulum position.The mechanical system is housed in a metallic capsule with radius 20 mm and height 180 mm.
I. Introduction: The actuator is sized to achieve at least 20 dB attenuation of the dominant oscillatory component at fn under the worst-case condition.This requirement guides preliminary actuator sizing.

A. Actuator architecture and geometric constraints

The actuator design balances magnetic force generation with equilibrium preservation and feasibility across the pendulum workspace. Energy-based evaluation of candidate layouts selects a six-coil configuration for the damping objective.

A. Actuator architecture and geometric constraints: The C-shaped electromagnetic actuator balances flux concentration, usable air-gap force, equilibrium preservation, and measurement stability.Permanent magnets and hard magnetic materials are excluded from the pendulum mass, leaving exclusively attractive actuation.
B. Energy-based actuator sizing and layout selection: The sizing analysis targets 20 dB attenuation by dissipating the energy required to reduce the pendulum amplitude from θmax to θtar = θmax/10.The extracted actuator work must satisfy Wext ≥ ΔEreq.
B. Energy-based actuator sizing and layout selection: Candidate layouts use 3, 4, or 6 equally spaced coils at 120°, 90°, or 60° intervals, respectively.Magnetic force contributions are evaluated over pendulum positions and azimuthal swing orientations.
B. Energy-based actuator sizing and layout selection: Only the six-coil layout satisfies the energy-based damping requirement across the full pendulum workspace.The final design uses six evenly spaced C-shaped electromagnets on a circumference of radius 50 mm.

C. Control-oriented force model of the actuator

The actuator is represented by a compact force model whose spatial decay is fitted in Cartesian or polar coordinates and whose peak force scales with excitation current.

1) Force–position relation:: A second-order rational function models the actuator force magnitude as a function of Cartesian position.The model uses peak force amplitude a and directional decay parameters b and c.
1) Force–position relation:: The Cartesian model is transformed into polar coordinates using x = ρ cos φ and y = ρ sin φ.This yields a force expression in terms of radial position and angle.
1) Force–position relation:: The approximation b = c simplifies the model because the simulated force surface is approximately isotropic in the operating region.The simplification imposes equal force decay along the two spatial directions.
1) Force–position relation:: The fitted force model is compared with simulated force data, while Fig. 5 depicts force variation with current as a parameter.Additional simulations identify the current dependence of the peak force.

2) Force–current relation:

The spherical-pendulum formulation expresses motion through generalized angular coordinates and Cartesian force inputs, with kinematics, energy, damping, and virtual-work relations supporting the nonlinear model.

2) Force–current relation:: The Cartesian position derivative defines the Jacobian of the spherical parameterization and determines the pendulum’s squared speed.The Jacobian connects angular-coordinate rates to Cartesian velocity.
2) Force–current relation:: The model uses kinetic and potential energies with the zero-potential reference at the lower equilibrium position.These energies provide the basis for the Lagrangian formulation.
2) Force–current relation:: Dissipation is represented by a Rayleigh function with an equivalent viscous damping coefficient.The generalized forces from an external Cartesian force are obtained using virtual work, and the equations of motion follow from L = T − U.
2) Force–current relation:: The nonlinear spherical-pendulum equations include force components projected through the angular coordinates.The displayed terms combine f_x, f_y, and f_z with θ and ϕ.
2) Force–current relation:: The spherical-pendulum dynamics are derived from a Cartesian force representation that combines gravity and actuator force.This force-driven form treats gravity as an external vector input and supports varying apparent gravity direction.

A. Coordinate regularization through reference-frame rotation

A fixed reference-frame rotation regularizes the spherical-coordinate model by moving the operating region away from coordinate singularities while retaining a compact two-angle representation.

A. Coordinate regularization through reference-frame rotation: Spherical coordinates are singular at θ = 0 and θ = π because ϕ is undefined and the dynamics contain terms proportional to 1/ sin θ.These singularities impair numerical conditioning and complicate linearization and controller design.
A. Coordinate regularization through reference-frame rotation: The Cartesian coordinates and force vector are expressed in a frame F′ obtained by rotating the original frame F about the y-axis by α.The same spherical-pendulum model is then written using the rotated coordinates and force.
A. Coordinate regularization through reference-frame rotation: For α = π/4, the pendulum’s static equilibrium shifts away from the singular region.The rotation keeps the operating region sufficiently far from the poles of the parameterization.
A. Coordinate regularization through reference-frame rotation: The rotated model improves numerical robustness while preserving a compact two-angle representation for linearization, state estimation, and control design.It is adopted for the remainder of the paper.

V. Control design

The control-design section presents the linearized state-space model and observer-based formulation used for active damping over the limited operating range.

V. Control design: The controller uses a linearized state-space model and observer-based formulation for active damping over a limited operating range.The observer-based formulation supports control design within the selected operating region.

A. Linearization

The spherical pendulum model is linearized around the equilibrium configuration at π/4, yielding a form that simplifies small-oscillation analysis and controller design.

The model is linearized about the equilibrium point using a Taylor expansion, with equilibrium identified at π/4.
For small oscillations, the spherical pendulum can be approximated as two independent mass-spring-damper systems.This approximation simplifies controller design.
Near equilibrium, the vertical force does not affect system acceleration under the rigid, fixed-length pendulum-rod assumption.

B. Velocity estimation

Because slow inclination dynamics make direct velocity estimation inaccurate, the controller uses a disturbance observer and velocity feedback to impose damping without changing static equilibrium.

A Disturbance Observer is introduced because the slow measurement dynamics do not permit accurate velocity estimation.A Luenberger observer was considered unsuitable because inclination deviations would produce incorrect velocity estimates.
The observer model incorporates the estimated physical magnitude of the gravitational force through a newly introduced term in the Baug matrix.
Damping is implemented through velocity-based feedback using velocity components estimated by the observer.No position feedback is introduced for quasi-static measurement, avoiding static-equilibrium modification and measurement bias.
The controller imposes a virtual damping acceleration using positive gains kθ and kϕ, then computes the corresponding reference force through the planar input map.The resulting fref is passed to allocation, which accounts for unilateral actuation, current saturation, and position-dependent force capability.
The force-allocation geometry compares coil-generated maximum force vectors with the controller’s reference force and the allocated force.The figure represents the pendulum projection, coil centers, maximum-force arrows, reference force, and allocated force.

D. Constrained force allocation for the six-coil actuator

The six-coil actuator cannot produce arbitrary planar forces, so a bounded allocator converts the desired force into feasible normalized coil inputs while accounting for position dependence, unilateral attraction, and saturation.

Each coil generates an attractive, position-dependent force toward its center, preventing arbitrary planar force generation.The allocator therefore maps fref into six physically admissible coil currents.
The normalized actuation variable satisfies λi = 0 for zero current and λi = 1 for the maximum allowable current.This normalization follows from magnetic force scaling with the square of current.
For a fixed pendulum position, F(˜x) collects the six maximum coil-force vectors and defines the set of achievable planar forces.
Because exact force reproduction is not always possible, allocation is formulated as bounded least squares over unilateral and saturated actuator inputs.A small regularization term penalizes effort, improves numerical conditioning, and makes the problem strictly convex.
For every pendulum position and reference force, the strictly convex problem has a unique solution λ⋆ within [0, 1]6.The corresponding coil currents are recovered from the normalized allocation variables.
A projected-gradient iteration solves the allocation problem online for embedded implementation with limited computational resources.The iteration uses Euclidean projection onto the hypercube [0, 1]6, equivalent to component-wise saturation.

VI. Experimental validation

The experimental campaign combined parameter identification with closed-loop validation, using a faster measurement update rate to enable real-time testing of the damping strategy.

The campaign comprised preliminary parameter identification followed by closed-loop validation of the proposed damping strategy.
By bypassing legacy filtering, the effective measurement update rate increased from about 77 ms to 7.7 ms.The higher update rate made real-time closed-loop validation feasible.

A. Closed-loop system validation

Experimental validation compares controlled and uncontrolled pendulum responses across maximum control currents using prototype measurements and closed-loop performance indices. The results show faster damping with greater actuation, but an intermediate current offers the best balance between attenuation, cumulative oscillation, and residual motion.

Experimental setup: The prototype integrates optical pendulum-position measurement with a magnetic subsystem for active dynamic conditioning.The closed-loop tests record pendulum responses with and without control after a repeatable coil excitation.
Performance indices: 31.11 dB attenuation and Jr = 3.27 mm s occur at Imax = 0.6 A, identifying the strongest combined frequency-domain and cumulative-oscillation performance.These indices jointly evaluate dominant-mode suppression and cumulative oscillation energy.
Transient response: T50 decreases from 6.71 s at 0.1 A to 0.31 s at 1.0 A as the maximum control current increases.The reduction is monotonic across the reported current range.
Steady-state behavior: Residual RMS oscillation increases from about 0.045 mm at 0.2–0.3 A to 0.205 mm at 1.0 A.Higher current limits accelerate the initial transient while raising the steady-state oscillation floor.
Design trade-off: For the prototype, Imax between 0.4 A and 0.6 A provides the most favorable trade-off between transient suppression and steady-state measurement quality.The conclusion also reports T50 reduction from 11.10 s without control to 1.01 s at Imax = 0.6 A.