Robot-Enabled Air-Gap Flux Mapping in Misaligned Electric Machines: Measurement Method and Harmonic Signatures

Abstract

This study presents an experimental framework for mapping the air-gap magnetic flux in electric machines operating under controlled eccentricity and tilt conditions. A six-degree-of-freedom industrial robotic arm positions the rotor, while the stator accommodates a dense single-axis Hall-sensor array. Synchronous data acquisition at 10 kHz captures magnetic-field dynamics during torque-producing excitation. A coordinate-transformation method synthesises virtual rotor poses from a limited set of physical measurements, eliminating the need for exhaustive mechanical scanning. The proposed approach generates pose-resolved RMS and THD maps, together with harmonic amplitude and phase signatures, thereby revealing localised asymmetries and phase-decoherence effects that are not predicted by idealised finite-element models. In a custom PMSM-like prototype, the local RMS value doubles (from 31 mT to 64 mT), while the THD increases by more than 25% across displacement and tilt grids. These findings provide quantitative experimental evidence of misalignment-induced magnetic-field symmetry breaking, supporting model validation and digital-twin calibration for traction, aerospace, and robotic applications.

1. Introduction

Electric machines play a central role in electrified transport, renewable-energy systems, and industrial automation. Their performance is strongly governed by the air-gap magnetic flux density, which directly determines torque production and position-dependent electromagnetic forces [1]. As machines evolve towards higher power density, tighter efficiency margins, and reduced noise, vibration, and harshness, precise knowledge of internal magnetic fields—particularly under non-ideal conditions—has become essential [2,3]. Accurate flux characterisation underpins fault-tolerant operation, high-fidelity digital twins, and real-time diagnostic frameworks [4,5,6].

Finite-element modelling (FEM) remains the primary design and simulation tool for electric machines, enabling estimation of flux linkage, torque production, and back electromotive force [7,8,9]. These quantities form the basis for control-oriented modelling and condition monitoring [10,11,12]. However, FEM relies on idealised assumptions of perfect geometry and symmetry, often with simplified or linearised material properties, and is typically limited to two-dimensional steady-state analyses because of the computational burden of three-dimensional transients [13,14]. Under realistic operating conditions—including rotor eccentricity, tilt, and localised saturation [15]—these simplifications lead to discrepancies in predicted torque ripple and flux distribution, which degrade controller performance and digital-twin accuracy.

Direct experimental measurement of internal magnetic fields, especially within the air gap, is therefore essential to bridge the gap between simulation and reality. Identifying the true magnetic-field distribution is critical for the validation of analytical and numerical models, which remain an active research area with direct relevance to control synthesis [16,17,18,19,20,21,22].

Several experimental methods have been explored for flux characterisation under non-ideal conditions, each with distinct advantages and limitations:

• Dynamometer-based test benches provide system-level measurements of torque, current, and speed, with indirect field estimation achieved through analytical models or observers [23,24,25]. Some configurations integrate embedded Hall sensors [26,27], but spatial coverage remains limited, and rotor misalignments are often introduced manually, reducing repeatability [28,29].
• Active magnetic-bearing (AMB) platforms enable precise rotor positioning via magnetic suspension [30,31], allowing in situ correlation between rotor displacement and magnetic-field behaviour [32,33,34]. More advanced setups permit controlled geometric misalignment during field measurement [35,36], but AMB systems suffer from cross-axis coupling, limited bandwidth, and restricted load capacity [37,38,39].
• Mechanical scanning rigs use precision stages for accurate repositioning [40,41]. They are effective for static field mapping but unsuitable for torque-producing or dynamic studies and typically lack integration with control systems.

Although these methods have advanced experimental flux characterisation, they are constrained by limited programmability, spatial coverage, and dynamic capability. A system capable of spatially resolved flux mapping under realistic, torque-producing conditions therefore remains desirable.

Industrial robotic arms offer a promising solution. With sub-millimetre accuracy, high repeatability, and programmable motion control, they are ideally suited for precise and repeatable rotor manipulation [42,43,44]. This study introduces a robotic measurement platform that integrates multi-axis rotor positioning with real-time Hall-sensor acquisition for magnetic flux characterisation. In contrast to AMB-based systems, the robot’s mechanical manipulation decouples motion in each axis, eliminating cross-axis coupling. Unlike static scanning tables, the robotic platform supports continuous and dynamic pose transitions, including programmable geometric misalignment during operation. This configuration overcomes the limitations of previous approaches in spatial coverage, dynamic control, and mechanical complexity, thereby providing a comprehensive and practical tool for in situ flux mapping.

The proposed system allows both static and dynamic misalignment while maintaining torque production, employing a dense array of Hall sensors and a transformation-based reconstruction framework to extend spatial resolution without excessive mechanical repositioning. Collectively, these features enable robust validation of finite-element models, enhance digital-twin fidelity, and support the development of misalignment-aware control strategies under realistic operating conditions.

This paper contributes to the experimental study of misaligned rotor–stator systems in four principal aspects:

• Introduction of a robot-enabled flux-mapping method that preserves torque-producing operation under programmable misalignment;
• Development of a reconstruction technique that synthesises virtual rotor poses from embedded sensor arrays, thereby densifying spatial sampling;
• Provision of pose-resolved RMS/THD and harmonic maps revealing symmetry breaking and phase decoherence under misalignment;
• Formulation of a rigorous measurement-uncertainty budget accounting for linearity, temperature drift, ADC quantisation, and placement tolerances.

The remainder of this paper is organised as follows. Section 2 presents the robotic measurement platform and data-processing framework, Section 3 reports the experimental findings, and Section 4 concludes with implications and future work.

2. Materials and Methods

2.1. Robotic Magnetic-Field Scan System: Components and Configuration

System Architecture

The experimental platform was developed for spatially resolved mapping of magnetic flux density within the air gap of electric machines. The setup provides full six-degree-of-freedom (DoF) control of rotor position relative to a fixed stator. Building on earlier work on radial-field scanning [45], an industrial robotic manipulator is integrated to achieve precise, repeatable positioning of the rotor (see Figure 1).

Unlike conventional configurations where sensors traverse a stationary magnetic field, this architecture reverses the relationship: the Hall sensors are fixed within the air gap, and the rotor is manipulated by the robot. This inversion allows programmable and repeatable spatial sampling, supporting both static and motion-resolved field measurements. A custom mechanical interface ensures rigid coupling of the rotor to the robot flange, maintaining coaxial alignment with the tool centre point.

Magnetic flux density is measured using single-axis Hall-effect sensors (Honeywell SS49E, $\pm 100$ mT range) embedded on custom printed circuit boards. Sensor placement follows the machine geometry to capture critical regions of the stator field, with rigid mounting for mechanical stability and repeatability. Signal acquisition is executed in real time via two synchronised data-acquisition subsystems: a Humusoft card for parallel high-priority sampling and a multiplexed National Instruments card for extended channel coverage. Both operate at 10 kHz, providing adequate temporal resolution for transient magnetic behaviour.

A MATLAB/Simulink (R2022b) interface orchestrates robot motion, data acquisition, and stator excitation. Motion trajectories are generated offline and transmitted to the robot controller via the Remote Motion Interface (RMI) for synchronised closed-loop execution. Each of the six stator poles is driven by a 10 kHz sinusoidal voltage source through full-bridge LMD18245 drivers operating in fixed off-time chopper mode, with internal current feedback ensuring stable current regulation.

By decoupling the sensor array from rotor motion, the system achieves mechanical isolation and high repeatability. Combined with dense sensor instrumentation and synchronised data acquisition, the platform enables advanced characterisation of electromagnetic machines under both nominal and perturbed conditions.

2.2. System Operation

The complete operational workflow is illustrated in Figure 2. After integration of the rotor with the robot end effector, a two-step calibration procedure is executed:

• TCP and coordinate calibration: Offsets and geometric misalignments are corrected using CAD-derived models of the rotor and stator. A transformation matrix aligns their poses within the robot’s base coordinate frame.
• Rotor alignment: Fine alignment is performed either using metrology instruments (e.g., laser-displacement sensors) or iteratively via Hall-sensor feedback to minimise asymmetry and offset.

Figure 2. Robotic field-scanning workflow showing calibration, trajectory generation, and scan execution.

Figure 2. Robotic field-scanning workflow showing calibration, trajectory generation, and scan execution.

Two rotor-mounting configurations are supported:

• Rigid mount: The rotor is fixed to the robot flange, with all motion commanded by the manipulator. This configuration is ideal for static measurements and continuous-path scanning.
• Free rotation: The rotor is mounted on a low-friction shaft, allowing natural rotation induced by the electromagnetic torque while the robot maintains the spatial pose. This mode is suited to commutation-driven field mapping.

After calibration, a scan grid is defined based on the workspace and machine geometry. All trajectory points are transformed into the robot’s base coordinate frame, verified for reachability and collisions, and executed automatically via the MATLAB–RMI interface.

2.3. Rotor Alignment and Calibration Procedure

Calibration employed robot-based positioning combined with flux density feedback from sensors distributed uniformly around the stator circumference. Three pairs of diametrically opposed sensors were used to evaluate magnetic symmetry and determine the optimal alignment offset.

The Tool Center Point (TCP) of the robot was first referenced to the geometric center of the stator interface. After the rotor was attached to the flange, the same reference was retained to represent nominal coaxial alignment. Calibration proceeded by displacing the TCP along the X and Y axes from $-2$ mm to +2 mm in 0.25 mm increments. At each offset, all six sensors synchronously measured the magnetic flux density while the rotor completed one full revolution. For each offset, the RMS difference between opposed sensor pairs quantified the magnetic asymmetry. The offset yielding the minimum aggregate RMS error defined the optimal alignment point.

Although precision instruments such as laser-triangulation heads could measure offsets directly, the adopted flux-based calibration emulates realistic assembly conditions. Robot-guided iterative alignment with direct magnetic feedback provides a reproducible, straightforward procedure without specialised metrology. Opposed sensor pairs inherently compensate for minor magnetisation irregularities, providing a robust flux-based criterion for coaxial alignment.

2.4. Measurement Uncertainty Analysis

Measurement uncertainty was estimated using the root-sum-of-squares (RSS) method (Appendix A). Major contributing factors include the following:

• Hall-sensor accuracy: $\pm 2$% of reading (Honeywell SS49E);
• ADC quantisation resolution: 14-bit;
• Robot repeatability: $\pm 0.01$ mm (FANUC LR Mate 200iD);
• Sensor-placement tolerance: $\pm 0.1$ mm (3D-printed mounts).

Mechanical uncertainty affects measurement accuracy through local magnetic-field gradients and depends on machine geometry. Total uncertainty ranges from $\pm 1.95$ mT to $\pm 5.26$ mT, with a typical value of $\pm 4.21$ mT under nominal conditions.

2.5. Transformation of Magnetic-Field Components from Virtual Rotor Reference Frame

High-resolution mapping of the magnetic flux density within the air gap requires extensive experimental data, yet measuring all possible rotor configurations is impractical due to the time required for data acquisition. To address this limitation, an embedded array of Hall-effect sensors is integrated into a data-processing pipeline that employs geometric transformation and interpolation techniques to reconstruct virtual rotor poses from a limited set of measurements.

The method uses the known geometry of the sensor array and the initial rotor alignment. By rotating the global coordinate frame so that one axis passes through a chosen sensor, that sensor becomes a local origin. Because the sensors are fixed relative to each other, their data can be expressed in alternative coordinate systems, each representing a virtual rotor orientation, without changing the physical setup. As shown in Figure 3, a single measured configuration can thus be viewed from several spatial perspectives, increasing the effective measurement coverage and allowing analysis of eccentricity and misalignment within a single experiment.

Rotor orientation is described using a ZYX Euler-angle sequence, and eccentricity is treated as a translational offset. These transformations align all measured data in a common reference frame. To reconstruct the magnetic field over the entire air-gap domain, cubic-spline interpolation is applied to the transformed measurements. The interpolation uses the known spatial arrangement of the sensors to estimate flux density at unsampled locations, producing a smooth and physically consistent field distribution. Regularisation reduces noise and ensures numerical stability. Combining geometric transformation with spline interpolation yields a continuous, high-resolution flux map that captures the effects of eccentricity and rotor tilt.

2.5.1. Tilt-Induced Orientation Transformation

This derivation defines a procedure for updating the Euler angles when the reference frame undergoes a known rotation about the Z axis. The rotation matrix corresponding to a ZYX Euler-angle sequence is expressed as

$$R=R_z\left(\gamma \right)R_y\left(\beta \right)R_x\left(\alpha \right),$$

(1)

where the elemental rotation matrices are defined by

$$R_x\left(\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0& sin\alpha & cos\alpha \end{array}\right],$$

(2)

$$R_y\left(\beta \right)=\left[\begin{array}{ccc}cos\beta & 0& sin\beta \\ 0& 1& 0\\ -sin\beta & 0& cos\beta \end{array}\right],$$

(3)

$$R_z\left(\gamma \right)=\left[\begin{array}{ccc}cos\gamma & -sin\gamma & 0\\ sin\gamma & cos\gamma & 0\\ 0& 0& 1\end{array}\right].$$

(4)

If the system undergoes an additional rotation about the Z axis by an angle ${\gamma }^{\prime }$, the updated rotation matrix becomes

$$R^{\prime }=R_z\left({\gamma }^{\prime }\right)R=R_z\left({\gamma }^{\prime }\right)R_z\left(\gamma \right)R_y\left(\beta \right)R_x\left(\alpha \right).$$

(5)

This operation modifies the orientation of the coordinate frame and implicitly alters the Euler angles. From the resulting matrix $R^{\prime }$, the updated angles can be extracted using inverse trigonometric identities:

$$\beta =arcsin(-R_{31}^{\prime }),$$

(6)

$$\alpha =arctan2(R_{32}^{\prime },R_{33}^{\prime }).$$

(7)

A notable limitation of Euler angles is the phenomenon of gimbal lock, which occurs when $\beta =\pm {90}^{\circ }$. In this condition, two of the rotation axes become aligned, resulting in a loss of one degree of rotational freedom. This introduces ambiguity in the determination of $\alpha$ and $\gamma$ independently. For systems requiring continuous or large-angle orientation tracking, alternative representations such as unit quaternions may be employed to avoid singularities and ensure numerical robustness.

2.5.2. Eccentricity-Induced Translation and Rotation

Beyond rotational misalignment, rotor eccentricity is represented by a translational offset of the rotor centre. The coordinates of a point P are updated using a homogeneous transformation combining rotation and translation.

Let P be a point in space expressed in homogeneous coordinates:

$$P=\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right].$$

(8)

Assuming that the rotor is displaced by a vector $T={(t_x,t_y,t_z)}^{\top }$, the associated translation matrix is as follows:

$$T_t=\left[\begin{array}{cccc}1& 0& 0& t_x\\ 0& 1& 0& t_y\\ 0& 0& 1& t_z\\ 0& 0& 0& 1\end{array}\right].$$

(9)

Following translation, a rotation about the Z axis is applied by angle $\gamma$:

$$R_z\left(\gamma \right)=\left[\begin{array}{cccc}cos\gamma & -sin\gamma & 0& 0\\ sin\gamma & cos\gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

(10)

The composite transformation is applied as

$$P^{\prime }=R_z\left(\gamma \right)T_{t}P.$$

(11)

Expanding the result,

$$P^{\prime }=\left[\begin{array}{cccc}cos\gamma & -sin\gamma & 0& 0\\ sin\gamma & cos\gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x+t_x\\ y+t_y\\ z+t_z\\ 1\end{array}\right].$$

(12)

The resulting vector $P^{\prime }$ represents the new position of the rotor centre relative to the transformed reference frame. This method supports accurate simulation of the eccentricity of the rotor by applying known displacement and orientation offsets without modifying the physical test setup.

2.6. Prototype Drive Design

To validate the proposed framework, a dedicated prototype electric-drive system with an external rotor was built. Its compact design allows systematic investigation of unconventional geometries. The stator, 10 mm thick with an outer diameter of 95 mm, comprises six independently actuated coils, each with a resistance of 1.21 $\mathsf{\Omega }$ and an inductance of 592 $\mathsf{\mu }$H, measured at 10 kHz.

Slots at the centres of the stator poles accommodate the sensors, following [46]. Two printed circuit boards (PCBs), each carrying 12 Hall-effect sensors, are mounted on opposite sides of the stator, half aligned with the pole pieces and half positioned in the interpolar regions, providing dense spatial sampling of the magnetic field. Between adjacent poles, sensor pairs are mounted at two radial heights and oriented to minimise exposure to stator-generated flux, isolating the rotor-induced component. The resulting data form the basis for magnetic-field reconstruction and rotor-pose estimation, enabling the drive to act as a spatial encoder.

The rotor comprises eight magnetic poles, each constructed from five N38-grade permanent-magnet bars arranged circumferentially along an outer diameter of 106 mm. A schematic overview of the stator–rotor assembly and the sensor layout is shown in Figure 4 and

Table 1. Arrangement of Hall sensors in cylindrical coordinates.

Sensor Type	Radius r [mm]	Angle $\mathit{\gamma }$ [$°$]	Height z [mm]
ET1…6 (Top)	47.5	${\varphi }_i={60}^{\circ }·(i-1),i=1\dots 6$	+5
EB1…6 (Bottom)	47.5	${\varphi }_i={60}^{\circ }·(i-1),i=1\dots 6$	−5
S1…6 (Side)	47.5	${\varphi }_i=30+{60}^{\circ }·(i-1),i=1\dots 6$	$z_i=\{-5,0,5,-5,0,5\}$

, respectively.

3. Results and Discussion

The experimental results reported here correspond to the rigid-mount configuration actuated by the robot, in which the rotor was positioned and oriented solely by the robotic manipulator. In this mode, the stator coils were unenergised, and no electromagnetic commutation or self-rotation occurred. This configuration was selected to isolate flux-distribution effects from mechanical dynamics and to validate the performance of the robotic actuation and sensing architecture under controlled rotor-orientation excitation. The definitions and calculation methods for all reported metrics are provided in Appendix B.

The PMSM prototype used in this study features a fractional-slot concentrated-winding (FSCW) stator and a rotor with surface-mounted bar-type permanent magnets. This topology inherently produces magnetic-field harmonics distinct from conventional integer-slot distributed-winding motors with arc-shaped magnets. Consequently, the specific harmonic spectra and asymmetry trends observed in the measured flux data are characteristic of this design and may vary for other motor geometries or winding configurations.

3.1. Waveform-Level Analysis

3.1.1. Effects of Rotor Eccentricity

Rotor eccentricity introduces lateral displacement along the X and Y axes while maintaining a fixed angular orientation. Figure 5 shows magnetic-field measurements at three representative sensors—EB6, EB3, and ET6—during independent translations along the X and Y directions. The centred position, $(x,y)=(0,0)$ mm, serves as the baseline reference. Qualitatively, eccentric rotor positions generate clear amplitude asymmetries and increased waveform distortion, particularly at the displacement extremes of $x=\pm 3$ mm and $y=\pm 3$ mm. Quantitatively,

Table 2. Signal metrics per Hall sensor under varying eccentricity conditions.

Sensor	Rotor State	RMS (mT)	THD (%)	Amplitude A (Harmonics, mT)				Phase $\mathit{\varphi }$ (Harmonics, $°$)
				A1	A3	A5	A7	${\mathit{\varphi }}_{\mathbf{1}}$	${\mathit{\varphi }}_{\mathbf{3}}$	${\mathit{\varphi }}_{\mathbf{5}}$	${\mathit{\varphi }}_{\mathbf{7}}$
EB6	x: −3 y: 0	46.04	43.72	63.21	14.40	4.28	1.82	80.67	55.14	30.31	7.68
	x: 3 y: 0	47.19	41.46	64.65	15.27	4.8	2.05	104.51	136.83	172.80	−151.06
	x: 0 y: 0	44.58	39.78	61.58	12.87	3.38	1.13	91.24	92.55	95.09	94.09
	x: 0 y: −3	63.60	47.19	85.23	26.00	9.47	4.01	91.20	92.85	95.83	93.21
	x: 0 y: 3	31.12	34.21	43.59	5.78	0.87	0.23	90.39	89.78	87.64	76.86
EB3	x: −3 y: 0	47.15	41.08	64.66	15.13	4.63	1.86	104.45	135.88	170.12	−155.05
	x: 3 y: 0	45.84	40.33	62.96	14.25	4.16	1.72	80.36	54.66	29.81	7.07
	x: 0 y: 0	44.72	39.81	61.63	13.48	3.68	1.30	92.30	95.26	100.65	103.84
	x: 0 y: −3	31.81	35.85	44.48	6.19	1.10	0.24	89.53	79.87	68.87	57.66
	x: 0 y: 3	65.18	51.41	86.01	28.90	12.63	7.24	90.97	91.09	94.03	96.49

shows that rotor eccentricity substantially affects both the RMS amplitude and total harmonic distortion.

For example, at sensor EB6, the RMS amplitude ranges from 31.12 mT at $(0,3)$ mm to 63.60 mT at $(0,-3)$ mm, demonstrating strong dependence on rotor position. At EB3, the RMS increases from 31.81 mT at $(0,-3)$ mm to 65.18 mT at $(0,3)$ mm, reflecting the sensors’ symmetrical placement relative to the rotor’s central axis.

Total harmonic distortion increases markedly with eccentric displacement, peaking at 51.41% for EB3 at $(0,3)$ mm and 47.19% for EB6 at $(0,-3)$ mm, compared with 39.78% at the centred position. Third- and fifth-harmonic components are particularly amplified; for instance, EB3 at $(0,3)$ mm exhibits $A_3=28.90$ mT and $A_5=12.63$ mT—more than double their nominal amplitudes.

Phase analysis further highlights the influence of eccentricity. In the nominal configuration, the fundamental phase ${\varphi }_1$ is approximately ${91.2}^{\circ }$ for both EB6 and EB3, indicating symmetry. Lateral displacement ($x=\pm 3$ mm) disrupts this symmetry: ${\varphi }_1$ for EB6 rises to ${104.5}^{\circ }$ at $(3,0)$ mm, while EB3 decreases to ${80.4}^{\circ }$, producing a relative phase shift of about ${24}^{\circ }$. Higher-order harmonics also deviate significantly; for EB3 at $(0,3)$ mm, ${\varphi }_5$ and ${\varphi }_7$ reach ${94.03}^{\circ }$ and ${96.49}^{\circ }$, compared with ${100.65}^{\circ }$ and ${103.84}^{\circ }$ at the centred position.

The third sensor, ET6, mounted on a vertically offset plane, exhibits amplitude and phase trends closely resembling EB6, particularly during X-axis displacements. This indicates localised correspondence in magnetic-field distribution rather than a general effect of vertical positioning.

3.1.2. Effects of Rotor Tilt

Angular rotor tilt produces rotations about the $\alpha$ and $\beta$ axes while maintaining a fixed lateral position within the air gap. Figure 6 shows flux measurements at EB6, EB3, and ET6 under $\pm 4^{\circ }$ tilts. The aligned rotor orientation, $(\alpha ,\beta )=(0^{\circ },0^{\circ })$, serves as the reference.

Qualitatively, tilt produces pronounced asymmetries and deviations from ideal sinusoidal waveforms, particularly at $\alpha =\pm 4^{\circ }$. These distortions reflect local variations in the magnetic-field distribution and reveal a stronger contribution from higher-order harmonics, consistent with the disruption of axial symmetry caused by angular misalignment. Quantitatively,

Table 3. Signal metrics per Hall sensor under varying tilt conditions.

Sensor	Rotor State	RMS (mT)	THD (%)	Amplitude A (Harmonics, mT)				Phase $\mathit{\varphi }$ (Harmonics, $°$)
				A1	A3	A5	A7	${\mathit{\varphi }}_{\mathbf{1}}$	${\mathit{\varphi }}_{\mathbf{3}}$	${\mathit{\varphi }}_{\mathbf{5}}$	${\mathit{\varphi }}_{\mathbf{7}}$
EB6	$\alpha$: −4 $\beta$: 0	27.98	35.54	39.16	5.44	0.88	0.21	87.14	76.96	63.32	48.15
	$\alpha$: 4 $\beta$: 0	54.49	45.16	74.14	19.50	6.46	2.80	90.11	87.67	89.23	90.89
	$\alpha$: 0 $\beta$: 0	44.58	39.78	61.58	12.87	3.38	1.13	91.24	92.55	95.09	94.09
	$\alpha$: 0 $\beta$: −4	55.47	46.59	75.25	19.85	6.64	3.00	83.19	62.88	43.82	26.65
	$\alpha$: 0 $\beta$: 4	41.14	39.19	56.93	11.00	2.60	0.73	99.58	121.42	145.01	167.30
ET6	$\alpha$: −4 $\beta$: 0	35.08	35.87	49.01	7.40	1.30	0.34	91.88	90.31	85.04	74.85
	$\alpha$: 4 $\beta$: 0	49.81	44.88	67.72	17.98	5.99	2.58	91.66	92.38	96.40	99.56
	$\alpha$: 0 $\beta$: 0	44.58	39.78	61.58	12.87	3.38	1.13	91.24	92.55	95.09	94.09
	$\alpha$: 0 $\beta$: −4	51.02	45.98	69.21	18.27	6.13	2.78	85.27	70.06	57.43	41.77
	$\alpha$: 0 $\beta$: 4	43.87	39.08	60.69	11.82	2.84	0.82	100.15	122.69	146.24	167.70

shows substantial THD increases and redistribution of harmonic amplitudes across sensors. For instance, EB6’s THD rises from 39.78% in the nominal orientation to 45.16% at $\alpha =+4^{\circ }$, while ET6 reaches 45.98% under $\beta =-4^{\circ }$, highlighting the air-gap field’s sensitivity to angular misalignment.

The distortion is further reflected in the harmonic amplitude spectrum. In sensor EB6, the fundamental component ($A_1$) decreases from 61.58 mT in nominal orientation to 56.93 mT in $\beta =+4^{\circ }$. Higher order harmonics become more pronounced; at $\beta =-4^{\circ }$, $A_3$ reaches 19.85 mT and $A_5$ increases to 6.64 mT. A similar trend is observed at the ET6 sensor, where $A_3=18.27$ mT and $A_5=6.13$ mT under the same tilt conditions, indicating a consistent enrichment of the harmonic content due to angular misalignment.

In addition to amplitude variations, angular tilt introduces notable shifts in harmonic phase content. In the nominal orientation, the fundamental phase ${\varphi }_1$ for the sensors EB6 and ET6 is approximately 91.24^∘, indicating a strong alignment. When tilted to $\beta =-4^{\circ }$, ${\varphi }_1$ shifts to 83.19^∘ for EB6 and 85.27^∘ for ET6, indicating a consistent phase delay associated with tilt. Higher-order harmonics display more irregular and nonlinear phase behaviour. The fifth harmonic phase ${\varphi }_5$ in EB6 deviates significantly, shifting from 95^∘ at the nominal position to 43.82^∘ at $\beta =-4^{\circ }$ and to 145^∘ at $\beta =+4^{\circ }$, highlighting the sensitivity of phase relationships to angular misalignment.

3.1.3. Synthesis of Waveform-Level Insights

The waveform-level observations presented in the preceding sections provide critical insight into the influence of rotor eccentricity and angular tilt on the magnetic-field distribution within the motor air gap. These analyses reveal how such misalignments disrupt the spatial symmetry of the field, altering its waveform shape, amplitude profile, harmonic composition, and phase behaviour.

Rotor eccentricity is chiefly associated with pronounced amplitude asymmetries and enhanced harmonic content. For instance, the EB6 sensor exhibits RMS values ranging from 31.12 mT to 63.60 mT depending on the direction of displacement. In contrast, angular tilt produces more gradual amplitude variations but introduces substantial phase distortions, particularly in higher-order harmonics. As an example, the fundamental phase measured by the ET6 sensor shifts from ${85.3}^{\circ }$ to ${100.1}^{\circ }$ under different tilt configurations.

While these findings are informative, they are derived from a limited set of three Hall-effect sensors. This sparse spatial sampling constrains the ability to fully resolve the complexity of the magnetic field across the entire air gap. To address this limitation, the subsequent chapter extends the analysis to a denser grid of rotor positions.

3.2. Spatial-Level Analysis

3.2.1. Effects of Rotor Eccentricity

The magnetic flux distribution is evaluated at the location of the EB6 sensor over a spatial grid of rotor displacements spanning $x,y\in [-3,3]$ mm, thereby encompassing both lateral and vertical eccentricities. The results, shown in Figure 7, include contour maps of key signal descriptors: RMS amplitude, total harmonic distortion, symmetry index, and the amplitudes and phases of harmonics $n=1,3,5$. These quantities are visualised as surface plots to highlight trends in field intensity, harmonic composition, and waveform distortion across the displacement space.

The RMS field magnitude increases notably for negative y values, peaking near $(x,y)=(0,-3)$ mm, where the flux reaches approximately 65 mT. Vertical displacement, therefore, amplifies the sensed field more strongly than lateral shifts, which produce shallower gradients along the x axis. The fundamental amplitude $A_1$ closely mirrors the RMS distribution, indicating that rotor eccentricity primarily strengthens the fundamental component rather than generating significant higher-order harmonics.

The third- and fifth-harmonic amplitudes, $A_3$ and $A_5$, exhibit localised and nonlinear spatial patterns. Both reach their maxima near $(x,y)=(0,-3)$ mm, mirroring the RMS trend, but decay more rapidly along the x axis. This behaviour suggests that vertical displacement enhances higher-order content, whereas lateral offset tends to attenuate it. The fifth harmonic, $A_5$, displays a more symmetric and steeper decay than $A_3$, indicating greater sensitivity to combined x–y eccentricity.

The total harmonic distortion follows a pattern similar to $A_3$ and $A_5$, peaking in the lower central region. Although the fundamental amplitude $A_1$ is also strong there, elevated higher-order contributions increase the overall THD. In contrast, THD decreases towards the lateral edges—even where $A_1$ remains moderate—highlighting that harmonic balance, rather than absolute amplitude, governs waveform quality.

The symmetry index is lowest in the lower-right quadrant $(x>0,y<0)$, indicating waveform asymmetry under compound misalignment. Conversely, in the upper-left quadrant $(x,y)=(-3,3)$ mm, the field is more balanced, with lower THD and a higher symmetry index. The direction of eccentricity, therefore, plays a decisive role in preserving waveform integrity, even at comparable RMS levels.

Phase maps reveal complex spatial variation. The fundamental phase, ${\varphi }_1$, increases steadily along x (from approximately ${80}^{\circ }$ to ${105}^{\circ }$) with minimal change in y. Higher-order phases exhibit stronger cross-axis dependence: ${\varphi }_3$ forms curved, multidirectional gradients, while ${\varphi }_5$ varies by more than ${120}^{\circ }$, indicating a pronounced loss of phase coherence under compound eccentricity. Collectively, all harmonics exhibit coupled, nonlinear phase behaviour that contributes to the observed waveform asymmetries.

Overall, the analysis confirms that while the fundamental harmonic governs the general field strength, higher-order components—with their irregular phase behaviour—are principally responsible for waveform distortion and asymmetry.

3.2.2. Effects of Rotor Tilt

The magnetic flux distribution is evaluated at the location of the EB6 sensor across a two-dimensional grid of angular misalignments spanning $\alpha ,\beta \in [-4^{\circ },4^{\circ }]$, thereby capturing the influence of tilt-induced variations in rotor alignment (see Figure 8).

The RMS flux magnitude increases with $\alpha$, reaching a peak near $\alpha =-4^{\circ }$. This trend indicates that tilt about the $\alpha$ axis amplifies the magnetic field at the sensor location. In contrast, variation along $\beta$ remains relatively uniform, suggesting that misalignment in the orthogonal plane has a weaker influence on field strength.

The total harmonic distortion reaches its maximum when rotation is concentrated along the $\alpha$ axis, particularly near $\beta =0^{\circ }$, forming a central ridge within the misalignment grid. This pattern indicates increased waveform asymmetry when tilt occurs predominantly in a single direction. Towards the $\beta$ boundaries, the THD decreases, suggesting that distributed tilt across both axes mitigates distortion. The symmetry index follows a similar trend, reaching a minimum near the grid centre—consistent with elevated harmonic content.

Harmonic-amplitude maps corroborate these observations. The fundamental component $A_1$ mirrors the RMS distribution, whereas higher-order harmonics $A_3$ and $A_5$ exhibit pronounced peaks near $(\alpha =-4^{\circ },\beta =0^{\circ })$, closely matching the THD profile. This behaviour indicates that angular misalignment not only alters overall field strength but also amplifies higher-order harmonic content.

Phase maps reveal increasing spatial irregularity with harmonic order. The phase of the fundamental component, ${\varphi }_1$, varies smoothly with $\beta$, whereas ${\varphi }_3$—and particularly ${\varphi }_5$—display complex, multidirectional gradients. Phase shifts in ${\varphi }_5$ exceed ${120}^{\circ }$, signifying pronounced spatial incoherence and reduced waveform stability under angular misalignment.

4. Conclusions

A measurement-centric, robot-enabled pipeline has been developed for mapping air-gap magnetic fields in misaligned electric machines. By combining six-degree-of-freedom rotor manipulation, a stator-mounted Hall-sensor array, and a transformation-based reconstruction process, the system enables spatially resolved analysis of electromagnetic behaviour under controlled eccentricity and tilt. This approach provides direct experimental access to field asymmetries and harmonic phenomena that extend beyond the predictions of idealised models.

Experimental results show that even moderate rotor pose deviations introduce structured distortions in the magnetic field. Local RMS amplitudes can increase by up to a factor of two, accompanied by pronounced phase shifts and higher harmonic content. These effects are systematic electromagnetic responses to misalignment, not random disturbances, and have direct implications for torque ripple, vibration, and the reliability of position-estimation and control algorithms.

The findings highlight that real machines deviate significantly from the ideal sinusoidal field assumptions underlying most dq-frame control and Hall-based position estimation. Higher-order harmonics and phase variations exhibit diagnostic value, suggesting that estimation and control strategies should explicitly incorporate pose-dependent harmonic information. The resulting flux maps also provide spatially resolved calibration data for finite-element models and digital twins, offering a more realistic basis for electromagnetic simulation and validation.

Although the current study is limited to quasi-static operation with single-axis Hall sensing, the methodology is mechanically agnostic and readily extensible. Future implementations should include dynamic excitation, multi-axis field sensing, and automated calibration with synchronised pose–field acquisition. Integrating torque and force measurements would further link local flux variations to electromagnetic loading, while applying the pipeline to different motor topologies would support generalisation across machine classes.

Beyond its immediate experimental utility, the proposed pipeline provides a foundation for hybrid electromagnetic modelling and data-driven diagnostics. The spatially resolved flux maps can serve as reference datasets for training surrogate models or validating machine learning algorithms that predict field distortions in real time. Extending the pipeline towards dynamic operation and embedded sensing will enable closed-loop monitoring and adaptive control of electric drives under realistic operating conditions, bridging the gap between physical measurements, simulation, and digital-twin technologies.