Low-Profile 6-Axis Differential Magnetic Force/Torque Sensing

Force/torque sensing on hand-held tools enables control of applied forces, which is often essential in both tele-robotics and remote guidance of people. However, existing force sensors are either bulky, complex, or have insufficient load rating. This paper presents a novel 6 axis force-torque sensor based on differential magnetic field readings in a collection of low-profile sensor modules placed around a tool or device. The instrumentation is easy to install but nonetheless achieves good performance. A detailed mathematical model and optimization-based design procedure are also introduced. The modeling, simulation, and optimization of the force sensor are described and then used in the electrical and mechanical design and integration of the sensor into an ultrasound probe. Through a neural network-based nonlinear calibration, the sensor achieves average root-mean-square test errors of 0.41 N and 0.027 Nm compared to an off-the-shelf ATI Nano25 sensor, which are 0.80% and 1.16% of the full-scale range respectively. The sensor has an average noise power spectral density of less than 0.0001 N/ $\sqrt {\text {Hz}}$ , and a 95% confidence interval resolution of 0.0086 N and 0.063 Nmm. The practical readout rate is 1.3 kHz over USB serial and it can also operate over Bluetooth or Wi-Fi. This sensor can enable instrumentation of manual tools to improve the performance and transparency of teleoperated or autonomous systems.

and skill assessment [12], [13], the applied forces are valuable.Autonomous robotic US also uses force control [14], [15], and for learning from demonstration-based approaches to AIguided US, data including forces must be collected from US exams [16], [17].
However, force/torque measurement on an US probe is difficult as radiologists resist added bulk, weight, and cable pull.Moreover, the sensor cannot be placed on the face of the probe, in the direct load path, as it would disrupt image formation.The usual approach for instrumenting US probes has been placing off-the-shelf (OTS) force/torque sensors between the probe and an external shell which is held by the user [18], [19].This approach makes the probe bulky, heavy, and difficult to grasp, is expensive, and introduces cable pull.Because the force sensor cannot be placed near the US transducer's face which is in contact with the patient, it requires high torque capability, leading to high cost and limited availability.A different approach by Huang et al. placed small piezoresistive pressure transducers on either side of an US transducer array [20], but this measures only one or two degrees of freedom (DOFs) of force information and may interfere with the imaging.The same factors are true for any other manual tool, where the tip is used to interact with the environment and should not be covered, and the handle should not be made bulky or heavy.While sensor-less approaches are possible in robotics, these are difficult for general manual tasks where no joint torques are available and the environment is unknown [21], [22], [23].
We thus undertook to design a low-cost, low-profile, easyto-fabricate and use sensing solution for manual tools to enable force feedback and control in remote guidance or robotic teleoperation, as well as the many other applications.There are multiple potential modalities for force/torque sensing, reviewed in [24].These include strain gauges [25], fibre Bragg gratings [26], [27], elastomeric transducers [28], piezoresistive pads [29], optical deflection sensing [30], and capacitive sensing [31].Each modality was evaluated and/or tested [32], but each had issues.In particular, optical sensing requires precise fabrication to align the slit with the LED and photodiode, which involves a relatively complex adjustment step, and the components use more space than desired around a manual tool [33].Piezoresistive transducers require precise pre-loading so all axes contact properly at all times (i.e., they cannot measure negative pressures), and effective elastomeric sensors are not yet commercially available [34].We tested several models of piezoelectric transducers which showed 2576-3202 c 2024 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
large hysteresis and inter-axis coupling [32].Conversely, strain gauges require careful surface preparation, material selection, and manufacture of the flexure, leading to higher cost and complexity [25].Additionally, installation and removal would be time consuming.Capacitive sensing again involves more complex electronics and careful fabrication because high resolution sensing is normally only possible with small distances between the capacitor's plates [31].Other force sensing methods exist but have other shortcomings, such as low force rating [35].Magnetic sensors, including Hall effect and anisotropic magneto-resistance (AMR) sensors, have also seen some use for force measurement.Jones et al. and Kyberd et al. positioned a Hall effect sensor near a magnet on a flexible structure to sense when a single-axis force was applied [36], [37].A similar approach using a 3-axis AMR sensor was introduced by Yu et al. [38].AMRs present some advantages over Hall effect sensors, including better sensitivity, but require a Wheatstone bridge configuration which would add significantly to the size and complexity of a sensor.An interesting human finger tip force sensor using the Hall effect is introduced in [39], but it is specifically usable only with a human finger.Conversely, a 3-axis force sensor with Hall effect sensors and magnets was designed in [40].Again, this uses single Hall effect sensors axially offset from magnets.To achieve sufficient signal, the sensor relies on large deflections and cannot be miniaturized.As a result, the presented device is bulky and relatively heavy, with a small full-scale range, and the response is non-linear.
Instead, we previously introduced a multi-axis force sensing approach using Hall effect sensors in a differential configuration [32].In that preliminary work, the sensing modality was described and shown through simulation, and in one-, and three-axis tests, to be effective for force and torque sensing.The best-case deflection resolution of a single transducer was found to be 856 nm, and for a 3-axis jig with a linear calibration, the RMS error was approximately 10.37%.
In this paper, we build on the prior work through the following contributions, each of which are novel: • A physical model for the integration of differential magnetic force sensors into an arbitrary tool for 6-axis force/torque sensing, • A 6-axis sensor optimization-based design method based on the above physical model, • A soft 6-axis force/torque sensor suspension design, using a creep-free elastic material, • An electromechanical design for sensing 6-axis forces and torques exerted on an ultrasound probe.The design is informed by the above optimization, uses the new suspension, as well as miniaturized electronics enabling a thin shell construction around the tool that needs to be instrumented, and • Experimental results with a prototype ultrasound transducer sensor that was designed and built using the above methods.Results comprise linear and neural network-based calibration, measurements of noise levels, resolution and accuracy, and sensing throughput.In the following sections, we first briefly explain the modality and sensing concept before describing the above  x axis, where it is sensitive and very linear in the ±2 mm range (R 2 = 0.999 [32]).For rotations about y, the response is small and the magnet is highly constrained.The magnet is offset 1-1.5 mm along the z (axial) direction.Motion along that axis can be detected by summing the sensor outputs.
contributions, designs, and tests.The final design and integration into an ultrasound probe is shown in Fig. 1.

A. Differential Magnetic Force Sensing
As previously described [32], differential magnetic force sensing (DMFS) uses a collection of n sensor modules placed around a tool or device.Each sensor module consists of two adjacent Hall effect sensors, mounted on a thin shell around the tool, and a small permanent magnet mounted opposite the sensors on the tool itself.The shell is attached to the tool by a compliant suspension.When no load is present, the magnet of each sensor module is centered between its two Hall effect sensors.However, when the user applies a force to the shell, the tool deflects relative to the shell, and thus the magnets move relative to their sensor pairs, which causes a voltage response.An overview is shown in Fig. 2. By calibrating the sensor module deflection outputs with the stiffness of the suspension, forces and torques can be computed.
In particular, taking the difference between the two Hall effect sensor outputs, the response is highly linear for magnet displacements across the sensors (i.e., the x or measurement axis in Fig. 2).For all other axes, the response cancels out or is very small.Taking the sum of the two sensor outputs gives information about the axial (z axis) offset, but also the lateral (y-axis).Hence, each sensor module is considered a 1-DOF transducer, though there is some information about additional DOFs.Thus, approximately n modules are needed for reliable n-axis force/torque measurement.

B. Physical Model
The DMFS concept was previously modeled and verified through simple tests with one Hall effect sensor, followed by single-sensor module (SM) tests and finally a 3-DOF test [32].This modeling is now extended to a general multi-DOF case including the mechanical components to aid in optimal design of the 6-DOF configuration.
To measure the forces and torques applied to an arbitrary manual tool, we place a thin shell around the handle of the tool, where the user grips it.The shell is attached to and separated from the tool using a compliant suspension consisting of m discrete connection points.For example, a connection point could be a silicone rubber pad.Between the tool and the shell are also n SMs.
The suspension members can be placed in any configuration that supports the shell uniformly and symmetrically.However, the SM locations affect the achievable accuracy and numerical stability of the calibration.Thus, we must intelligently select their positions.To do so, we first develop a mathematical model of the forces and displacements in the suspension when the shell is held fixed and a force is applied to the tool.We then develop a simulation in MATLAB that implements this model, taking in a set of sensor positions and applied forces, performing a calibration, and outputting the meansquared error (MSE) of the simulated "measured forces".This simulation is run repeatedly for different sensor positions, and the errors are used to determine the optimal sensor configuration.
1) System Mathematical Model: In the following, we use coordinate-free notation.Practical considerations of coordinate frames are described in Appendix I.Under an applied wrench w w w j = [f f f j τ τ τ j ] (force f f f j and torque τ τ τ j ), the j th suspension member undergoes a 6-axis deformation, δ δ δ j = [δ δ δ tj δ δ δ θj ] , according to Hooke's law (for small deformations): where K K K j is the rank 2 compliance tensor of the sensor element.The matrix value of K K K in a practical coordinate system is described in Appendix I, and we performed tests to find its elements, described in Section III-B1.The rotational part of the deflection, δ δ δ θ , is a vector of small angles since the motion is very constrained.Additionally, we assume the shell and tool are rigid bodies, so we can express the individual deflections in terms of the overall deflection at the tool tip: where x x x j is the position of the j th suspension element, and [v v v] × is the skew symmetric matrix equivalent to the cross product such that [v v v] × u u u = v v v×u u u.Now suppose a wrench, W W W, is applied at the tool tip.From statics: where I k and 0 k are the k × k identity matrix and zero matrix respectively.Substituting in Equations 1 and 2, we find Equation ( 4) can be solved numerically for δ δ δ t and δ δ δ θ , taking the coordinate frames into account (Appendix I).Having solved this, the result can be substituted into Equation (2) to determine the δ δ δ j 's.These give the suspension member wrenches, w w w j by Equation (1).
In the force sensor, the sensor and suspension element locations are not required to coincide.The sensor locations, {s s s i }, can be arbitrary, though constrained by practical mounting considerations, and the displacement at the sensors can be determined by substituting s s s i into Equation (2) instead of x x x j .Furthermore, displacement sensor elements generally measure along just one axis in addition to the normal direction.Thus, one must define the sensitive axes, ŝi , in addition to the sensor positions.The normal directions, ni , are given by the sensor positions and the tool geometry.To simulate a measurement, the calculated δ δ δ i must be projected onto the normal and sensitive axes.To make this more realistic, however, we used data from the single SM test [32] to create a look up table (LUT) of the actual SM output as a function of magnet displacement, δ δ δ j .This is illustrated in Fig. 6E.
2) Force Measurement Simulation: It is now possible to formulate a simulation algorithm with the model.Given a three dimensional mesh of the tool, a tool-dependent constraint set, D, is defined for the possible SM positions on the mesh.In the example of an ultrasound probe as seen in Fig. 6, one could constrain the configuration to have two sensors per wide face and one on either side, and to have a certain minimum spacing between sensors.Once the SM locations are randomly initialized within D, the program solves the equations in Section II-B1 to simulate an applied force and subsequent measurement.This is shown in Algorithm 1.
The calibration in the second last step of Algorithm 1 can be achieved in a myriad of ways.The simplest, which assumes the system is linear, is to determine a calibration matrix, C, as the least squares linear mapping between the set of SM measurements X and the actual measured forces W: Alternatively, a relatively shallow neural network can be trained to learn the nonlinear mapping, leading to far more accurate results, as shown in [41].This is done in the final design of the instrumented tool, but it is impractical to retrain a neural network repeatedly during the sensor position Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Algorithm 1 Calibration Simulation
Generate a random matrix of N wrenches, W = [w w w 1 , . . ., w w w N ] for w w w k ∈ W do Solve Equation ( 4) for δ δ δ t for i ∈ {1, . . ., n} do Substitute δ δ δ t and s s s i into Equation ( 2) to find δ δ δ i Use the LUT to simulate measurement end for Place the measurements in column k of X ∈ R 2n×N end for Determine mapping, C, from X to W (calibration): C = WX (XX ) −1 return RMS error of calibration and cond(C).optimization step.Hence, we use the linear approach during the optimization.
3) Sensor Position Optimization: With the simulation in place, it is possible to optimize the locations of the sensors to obtain the most accurate and numerically stable calibration.The simulation is run repeatedly for different sensor configurations, S, and measurement axes, {ŝ i }.The calibration matrix, C(S, {ŝ i }), is determined for each trial, and the calibration error and condition number, κ(C), are evaluated for each.
Bicchi describes the importance of the condition number [42], showing that a large condition number can lead to poor accuracy even with careful calibration [43].As a rule of thumb, k digits of accuracy are lost when κ(C) is 10 k [44].
The configuration with the lowest mean-squared error (MSE) and condition number is thus chosen according to the cost function in Equation ( 6).Here x x x k is the k th column of X, i.e., the k th measurement, κ(C) is the condition number, and α, β are weights.

w S, {ŝ
A reasonable worst-case calibration MSE is if C = 0; then the MSE becomes the mean squared force magnitude.In Fig. 7, for example, this is 237.The condition number, on the other hand, may vary by several orders of magnitude, so it is reduced to the scale of the MSE using the logarithm.Though the condition number is important, the assumption of linearity is strong and later sections compare the performance to a non-linear, neural network-based calibration.Hence, we also include the calibration error itself in the cost function.
Thus we arrive at the optimization problem in Equation (7).
This is not possible to solve analytically because there is no analytical expression for X(S, {ŝ i }) or C(S, {ŝ i }), which result from repeatedly solving a large set of equations.Thus, we use a search algorithm to explore the solution space.To reduce complexity, we constrained the sensing axes, ŝj , to be horizontal or vertical, tangent to the probe at the given point.The optimization problem was solved using simulated annealing [45].This is described in Appendix II.

A. Electrical Design
Two PCBs were designed for the force sensors: a sensor module board and a master board.A 6-axis force/torque sensor uses 6 SMs and one master.These are described below, and the PCBs are shown in Fig. 3.
Master PCB: The master board takes in all the readings from the SMs, combines and processes them on an ESP32-WROOM-32e microcontroller, and sends the resultant force/torque measurements to the host PC or other device.An inertial measurement unit (IMU) is also included on the board for pose estimation [46], and communicates over I2C with the ESP32.The microcontroller has two cores and a real-time operating system (FreeRTOS), allowing parallel processing, and two SPI drivers.Each SPI driver handles three of the SMs, thus doubling the measurement rate compared to one SPI driver handling 6 SMs in series.The SPI communication includes MISO and MOSI buses (Master In/Out, Slave Out/In), as well as serial clock (SCLK), power (+5V), and ground (GND).These are shared among the SMs in a multidrop configuration while a separate chip select (CS) line goes to each individually.In this way, the wiring can be greatly simplified (See Section V).The layout is shown in Fig. 5.
A universal asynchronous receiver/transmitter (UART) port is used to connect the sensor to a PC using a UART-USB adapter.The configuration of the master PCB and its communications is shown in Fig. 5.
Sensor Module PCBs: Each SM consists of two Hall effect sensors, used for the differential measurement, a lowpass filter for each sensor, a 12 bit, 100ksps analog-to-digital converter (ADC), and a connector for communication with the master PCB.The filtering and the ADC were placed in close proximity to the sensor to reduce noise.
A passive second order low-pass filter with the component values shown in Fig. 3 was chosen after SPICE simulation  with various passive and active configurations due to its simplicity and effectiveness.The -3 dB cut-off frequency was set to 400 Hz since interactions with stiff environments can lead to high force bandwidths.The phase delay is < 1 • until approximately 10 Hz, which is the maximum typical for forces applied by the human hand [47].
The SMs are designed as squares with the Hall effect sensors centrally located so that the SM can be rotated by multiples of 90 • to change the measurement axis.

B. Mechanical Design
Given the sensor electronics and optimal SM locations, we designed the mechanical components.These were subject to several design objectives, listed below.
• Ease of assembly -the sensor should be easy and fast to install and remove from the object.For ultrasound, this allows disinfection between procedures.• Ergonomics -the final device should be as low-profile as possible, adding minimal bulk.• Force range -the probe should displace 1-1.5 mm (the linear range of the sensor) for a 25 N applied force, as is a typical maximum in ultrasound procedures [32].
• Stiffness -the outer shell must deflect much less than the suspension deflections at the SMs.The ultrasound probe force sensor consists of an outer shell containing the SMs and master PCB, and an inner scaffold containing the magnets.The mechanical parts are shown in Fig. 4. The inner scaffold clamps onto the ultrasound probe while the outer shell connects to the scaffold through the suspension.All three were fabricated with a Form 3+ stereolithography 3D printer (FormLabs, Sommerville, MA).
The scaffold, as it is completely supported by the transducer and has no structural role, was printed from 2 mm thick FormLabs standard white resin.The shell was printed from FormLabs Rigid 10K resin for toughness and rigidity.The main wall thickness was 2.5-3 mm.The rigidity of the shell was tested using finite element analysis (FEA) in SolidWorks (SolidWorks Corp., Waltham, MA), by placing elastic supports with the known suspension stiffness values, and applying a 25 N force on one side of the shell.The results are shown in Section IV-F.
Finally, the suspension was both 3D printed from FormLabs Elastic 50A resin, with 50A Shore durometer, and molded from room temperature vulcanization (RTV) silicone with 30A Shore durometer (BBDINO, Shenzhen, China).The design of the suspension is described in Section III-B1.The shell and scaffold were based on a CAD model of the ultrasound probe.Both were fabricated in two halves.The scaffold halves snap together by friction fit tabs and are held by the suspension elements.Small magnets embedded in the edges of the outer shell hold the two halves in place (see Fig. 4).Throughout testing, the design never opened or shifted unintentionally.It was easy and quick to install, robust, and stiff.

1) Suspension Tests:
Printing the suspension enables easy installation by fabricating the suspension element and mounting flanges as one part.A hollow cylinder shape was used with configurable inner diameter (ID) and outer diameter (OD) to tune the stiffness.A flange with two mounting holes was included on either end of the cylinder so it could be screwed into the scaffold and shell.The flanges were offset by 90 • to facilitate installation with a screw driver.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
To determine the stiffness matrix, K, described in Section II-B, a test jig was created for the suspensions.The measured K value could then be input to the simulation to determine if the deflection response was as desired, and thus a suspension element shape could be selected.The test jig consisted of suspension elements mounted between a fixed plate and an ATI Nano25 force sensor (ATI Industrial Automation, Rochester Hills, MI) on a manual 3-axis stage.The stage was moved by incremental, known offsets, and the resultant force was measured.A similar setup with a rotational stage was used to measure the rotational stiffness.The results are shown in Section IV-A.
It was found during testing, however, that the 3D-printed elastomer had relatively large hysteresis and creep, not returning fully to its unloaded position after a load cycle.A different RTV silicone material was thus also tested.The same geometries were used, but without the hollow inner core to account for the softer material.The silicone was poured into polylactic acid (PLA) 3D printed molds and allowed to set for 6 hours.No de-gassing or heating was necessary.

C. Integration
The integration of the electrical and mechanical parts on a 3D-printed dummy ultrasound probe as part of the calibration jig is shown in Fig. 4. As explained in Section III-A, the wiring has been cleaned up significantly since these tests, so separate cables are not needed for every SM (See Fig. 12).
The installation process on the ultrasound probe is simple and fast, taking approximately 1-2 minutes.First, the inner scaffold (with magnets installed and suspension elements screwed to one half) is snapped onto the probe, and the suspension elements are screwed to the other half.This takes the majority of the time.Next, the probe with the scaffold is placed in one half of the outer shell, and the other half is fitted on top, making sure the suspension elements are sitting properly.This takes just a few seconds.
The master PCB can then be plugged in to any USB port to receive power.A reset button on the back of the PCB, accessible through a small hole in the outer shell, is pressed to start the firmware.In case of an issue with the firmware, the reset button is always accessible, and the PCB can be flashed with new firmware via the USB cable.
The firmware, written in C using the Espressif IDF and ESP libraries, runs a real-time operating system (FreeRTOS) on the dual-core CPU.This enables multi-threaded parallel processing of the sensor readings from the two separate SPI buses as well as the IMU.Thread-safe first-in first-out queues with limited capacity (to avoid keeping old samples) are used to synchronize readings between threads and send them to the destination device.

D. Calibration
As shown in the modeling section and in previous work [32], the differential sensor output is linear and the forces can be computed from linear equations.Thus, a linear calibration may, in theory, convert from the raw Hall effect sensor readings to the applied forces and torques.
However, there are many potential non-linearities that are unmodeled.These include flex in the mechanical parts, nonlinear bending and hysteresis of the suspension elements, saturation of the Hall effect sensors, shifting of the suspension or shell, and slight inconsistencies in the magnets or sensors and how they are mounted, among other factors.Thus, it is likely that a non-linear calibration would outperform the linear one, as has been shown in similar situations [41].
Therefore, both linear and non-linear calibrations were carried out and compared, each separately with the 3D printed and molded suspension.The 2n raw Hall effect sensor measurements are input to the calibration rather than the difference and sum since this is more flexible and the calibration can learn to take the difference if needed.For the linear calibration, linear least squares was used to determine a 6×2n calibration matrix from raw voltages to force/torque.For non-linear calibration, neural networks were used.Hyperparameters of hidden layer sizes, number of hidden layers, activation functions, cost function, optimization algorithm, and momentum were varied to determine best architecture.
For the calibration tests, a jig was built to gather data.A 3D-printed dummy ultrasound probe was mounted directly to an ATI Nano25 sensor which was rigidly attached to a table.The sensing hardware was installed on the dummy probe as seen in Fig. 4.Both the master PCB and the ATI sensor were connected to a Windows PC, and samples were recorded with timestamps using a Python script.The data was aligned using the timestamps, and resampled to line up element-wise.For the numerical calibration, the ATI torques in Nm were first multiplied by 10 so their scale was the same as the forces.The input Hall effect sensor readings were scaled down by 2 12 to fall between 0 and 1.
A random series of forces and torques was applied to the ultrasound probe while 20000 samples were recorded at 60 Hz.This was repeated three times, and the data sets combined.The applied forces and torques were approximately normally distributed which is ideal for learning, with nearzero means of -0.016 N and -0.0017 Nm respectively to help produce an unbiased estimator.The 60000 samples were split 75%/25% into a training and a testing set, and 5-fold crossvalidation was used on the training set for hyperparameter tuning.An additional measurement was taken without applying any forces, to study the noise characteristics, and a further set of forces and torques (20000 samples) was recorded after disassembly and reassembly to evaluate the calibration under potentially slightly different conditions.

A. Suspension Tests
A variety of inner and outer diameters were tested for the suspension elements.In each case, the response was linear, with a much higher K value in axial compression/tension than in bending.The K value for each axis was determined by fitting a line and taking the slope.The results are outlined in Table I.Substituting these values into the simulation, the OD-ID combination of 6-4 mm gave the desired response.This was therefore fabricated in larger quantity and installed in the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.device.For the molded silicone, an outer diameter of 6 mm was also sufficient, with a solid core.

B. Simulation
The optimization procedure described in Section II-B was carried out with the mesh of a C3HD 3 ultrasound probe (Clarius, Vancouver, BC), and the chosen suspension elements.The suspension locations shown in Fig. 6 were used, and the SM locations limited to two per wide face and one per narrow face, for a total of six.The resulting optimal configuration is shown by the pink marks in Fig. 6, and the corresponding calibrated forces and torques are also shown.The RMS linear calibration error for force and torque were [0.703, 1.39, 0.521] N and [0.068, 0.083, 0.029] Nm respectively.In the Discussion section, these are compared to the values obtained in the real system, which are shown below.

C. Calibration
With a linear least squares calibration (performed on the training data), the RMS calibration error between our sensor and the ATI was (in N and Nm, on the testing data): e 3dp = 1.04 1.57 2.06 0.0918 0.0753 0.0545 e rtv = 0.463 0.607 0.915 0.0514 0.0484 0.0131 on the 3D printed (e 3dp ) and RTV silicone (e rtv ) respectively.Here x is to the right in Fig. 4, y is into the page, and z is up.
The tracking appears good, but the RMS error is relatively large.Instead, the neural network calibration was tested.In total, 44 different sets of hyperparameters were checked.It was quickly found that tanh activation significantly outperformed ReLU and logistic functions.An ADAM solver with adaptive learning rate starting at 0.0005 with a Nesterov momentum of 0.9 worked best.L2 regularization was also used.With these settings, different numbers of hidden layers and layer sizes were tested.The results are shown in Fig. 8A.Using the elbow method, a 5-layer, 150 neuron-per-layer architecture was chosen.It outperformed the 4-layer 256-neuron model despite having fewer neurons.Models with 1, 2, or 3 hidden layers were clearly underfitting and hardly outperforming the linear calibration.The chosen architecture achieves RMS errors of (in N and Nm) e train = 0.463 0.506 0.532 0.0373 0.0360 0.0253 e test = 0.544 0.593 0.660 0.0432 0.0403 0.0271 for the 3D printed suspension.In comparison, the RTV suspension performs better with the same architecture: e train = 0.300 0.324 0.432 0.0294 0.0284 0.0120 e test = 0.345 0.386 0.513 0.0356 0.0338 0.0127 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The similarity in performance between the training and testing data sets imply that the model is not overfitting much.This is discussed in Section V.The error is well under 1 N and 0.05 Nm in each axis compared to the ATI sensor.As a percent of the force/torque ranges, the errors of the RTV dataset are: %e rng = 0.71 0.91 0.78 1.35 1.36 0.77 As a further test, the calibration network was applied to the additional, separate measurement.This is plotted in Fig. 7.
Likely the primary reason for the superior performance of the RTV silicone is its relative lack of hysteresis and creep compared to the 3D printed photopolymer.An additional test was thus performed to quantify this by repeatedly applying a force and releasing suddenly.While the RTV suspension sprung back to its original position quickly, the 3D-printed one returned relatively slowly and incompletely.This is shown in Fig. 9. Fitting an exponential curve to the responses, the 3D printed suspension has a time constant of 0.32 s compared to the RTV silicone's 0.091 s.Additionally, the former retains a residual offset of approximately 1 N whereas the latter returns to 0 N.

D. Rate and Noise Characteristics
The rate of SPI data transfer for one SM to the master board was determined to be 12 kHz with our firmware.Given that three SMs are read in series, the sampling rate is 4 kHz.Including synchronization, processing, and communication delays, the practical readout rate at which the sensor delivered samples over UART to the PC was reduced to 1276.8±3.3Hz on average.This is fast enough for haptics applications where a control loop of approximately 1 kHz is commonly employed.
From one measurement in which no forces were applied, we can analyze the noise characteristics of the sensor.The 12 raw sensor measurements are each normally distributed with an average standard deviation of 1.92 LSBs, or 0.047% of the full-scale range of 2 12 .When the raw values are substituted into the calibration found in the previous section, the standard deviation becomes 0.0043 N and 0.0314 Nmm.With a 95% confidence interval of 2σ , the force and torque resolution are Hz for force and torque respectively (Fig. 10).

E. Sensor Module Analysis
Not only the locations of the SMs, but also their orientation is relevant to the sensor performance.The results presented thus far were obtained using the optimal orientations found in Section IV-B.Referencing Fig. 4, the measurement axes of SMs 2 and 4 were vertical (axial), SMs 1 and 3 were horizontal (lateral), and SM 5 and 6's measurement axes come out of the page (elevational).The latter two SMs should stay in this configuration because otherwise there are no other sensors sensitive to the elevational component of force.However, it is possible that rotating SMs 1, 2, 3, and/or 4 by 90 • could improve the performance.Therefore, two further tests were carried out with the 3D printed suspension, in which these SMs were alternately oriented laterally or axially.
Performing the same neural network calibration procedure for each trial, the RMS validation errors in Table II were found.The original configuration of trial 0 outperforms the other two configurations, showing that the optimization algorithm was effective in choosing the SM orientations, and that the SM orientation makes a significant difference in the sensor performance.Table III compares the performance to state-ofthe-art sensors.

F. Shell Rigidity
The rigidity of the shell was tested using FEA by placing elastic supports with the known suspension stiffness values in their chosen positions.A 25 N force, the maximum experienced in typical ultrasound procedures, was applied to the right side of the shell.The setup is shown in Fig. 11.Comparing the deflection on the left and right sides of the shell  shows how much the shell itself deformed versus how much the suspension moved.On average, the shell deflection was 0.083 mm while the suspension deflection was 0.95 mm.Thus, shell deformation constitutes 7.8% of the total deflection.Reducing this with a stiffer material may improve accuracy.

V. DISCUSSION AND CONCLUSION
This paper has introduced a sensor modality and optimization-based design procedure which was tested on an ultrasound probe.Integration into other point-of-care ultrasound devices using the same approaches will constitute future work.Other applications can also be explored, including surgical drills [48], catheterization systems [49], collaborative or cooperative robots [50], [51], for example for opthalmic Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
surgery [52], and even tooth brushes [53].Force reflection has been shown to be very effective in surgical robotics [2], [54], [55], [56].The Hall effect sensors and signal processing circuitry could also be packaged into a single small integrated circuit which could enable much lower-profile hardware integration or a standalone force/torque sensor.
Comparing the results from simulation in Section IV-B and the real system in Section IV-C, we see very similar errors.The real force and torque errors from the linear calibration with 3D printed suspension are on average within 0.69 N and 0.019 Nm of the simulated values, which also used a linear calibration.With the silicone suspension, the errors are within 0.21 N and 0.022 Nm.Moreover, the optimal sensor locations are approximately where one would intuitively place them, and the chosen sensing axes corresponded with the ones that experimentally performed best.Thus, the simulation is realistic and the optimization is effective.It is possible that other orientations, not just vertical or horizontal, could improve performance.Investigation of this is future work.
Given the different SM orientations, the question arises which SMs are actually used for forces and torques in which axis.If only forces in one direction were relevant, some sensors could be removed, or they could all be oriented in the same direction.For example, in the current configuration, only SMs 2 and 4 are sensitive to vertical (z) forces.For x and y forces, two SMs are directly sensitive to the force and two SMs have the magnet move axially closer or farther away from the Hall effect sensors.This could be why the error in z is slightly higher than in the other two axes.Conversely, for torques, each axis of load excites responses in four SMs.However, these are all highly coupled between axes, with overlapping responses from the SMs for different torques.Thus, the system is far from being linearly separable, and the more complex architecture of the neural network is warranted.The larger performance difference between the linear and neural network calibrations with the 3D printed suspension versus the silicone one indicates that the hysteresis and creep account for a large proportion of the complexity.
However, there is always a risk of overfitting.Hadi et al. used a single hidden layer with just 5 neurons to good effect [41].However, from Fig. 8, a single hidden layer architecture with 5 or 8 neurons barely outperformed the linear calibration.Conversely, with 45000 samples, 12 inputs, and 6 outputs, less than approximately 500 to 1250 hidden neurons should not lead to overfitting [57].The chosen architecture (5 hidden layers, 150 neurons per layer) uses 750 neurons, which is unlikely to overfit.Having different numbers of neurons in each layer was also tested, for example having [50,150,256,150,50], but the performance was consistently worse.With the chosen architecture, the validation and test errors are similar to the training error, indicating that the network generalizes well.In comparison, architectures with fewer hidden layers were clearly underfitting.Regularization was also used, which limits overfitting.Hence, the chosen architecture is justified and is most likely not overfitting.
An additional point of concern for instrumenting manual tools and devices is the added weight.The inner scaffold with magnets and suspension elements weighs 31.17g while the full shell with all electronics shown in Fig. 4 weighs 202.98 g.In total, this amounts to 234.15 g.New PCBs have been designed but not yet tested which are substantially smaller (45% reduction in surface area).These also use a single flexible PCB with shared data, clock, power, and ground buses instead of the separate wires in Fig. 4.This will decrease the weight significantly.A rendering of the updated circuitry and integration is shown in Fig. 12.Further improvements to the system are also possible.The ESP32 processor could be replaced by a field programmable gate array (FPGA) chip configured as six cores, so each SM could be processed in parallel for minimum latency.Such a system is described by Hadi et al. for optical force sensing [33].
Table III compares our DMFS sensor to state-of-the-art commercially available miniaturized force/torque sensors.The DMFS sensor shows comparable resolution in force and the same order of magnitude in torque as the ATI and Schunk sensors, and better performance than other sensors.Although the DMFS sensor does not currently outperform top-range sensors in resolution and has slightly higher noise, it is lower profile and an order of magnitude less expensive.The load range is much larger in the DMFS sensor because it does not use strain gauges and is thus not dependent on the strength of a mounting flexure.Therefore, the overload torque is much higher, though we did not measure it to failure.The strength and sensitivity can be modified by changing the suspension material.Furthermore, the DMFS sensor is low-profile and distributed around the tool while off-the-shelf sensors must be placed away from the probe's face.On the ultrasound probe in this paper, such a sensor would be at least 10 cm away from the probe face, leading to torques that easily exceed the rated overload torque when a 20N force is applied on the face.Thus, while some existing force sensors perform the same or slightly better than the DMFS sensor in resolution and accuracy, they are not practical for use on manual tools.The benefits of the DMFS sensor are that it is modular, noncontact, and simple to integrate into tools without making them bulky or difficult to assemble.This enables practical use in medical environments, where the tool can be removed and cleaned or replaced.The sensor itself can be disinfected if the PCBs are sealed with a thin layer of casting resin.Furthermore, the mechanical components can all be 3D-printed and/or cast in 3D printed molds.This cost is negligible since integrating off-the-shelf sensors also involves fabricating a mechanical shell.The electrical components are inexpensive and require no special preparation or treatment, so the whole integration of the sensor can be completed at a small fraction of the cost of existing multi-axis sensors.This may be important for remote guidance in low-resource environments.It is also relatively robust to noise from external magnetic fields due to the use of differential Hall effect readings, though this has not yet been tested.Moreover, the PCB design facilitates flexible wired or wireless integration into a variety of systems.
Achieving good performance through a neural networkbased calibration, with average root-mean-square test errors of 0.80% and 1.16% of the full-scale range for force and torque respectively, the sensor has low noise and a 95% confidence interval resolution of 0.0086 N and 0.0628 Nmm.This will hopefully enable more effective tele-guidance and robotics for manual tasks.

Fig. 2 .
Fig.2.Measurement concept of DMFS sensors.Taking the difference of the Hall effect sensor outputs, all displacements cancel out except along the x axis, where it is sensitive and very linear in the ±2 mm range (R 2 = 0.999[32]).For rotations about y, the response is small and the magnet is highly constrained.The magnet is offset 1-1.5 mm along the z (axial) direction.Motion along that axis can be detected by summing the sensor outputs.

Fig. 3 .
Fig. 3. Sensor module (top) and master (bottom) PCBs.The low-pass filter circuit for one Hall effect sensor on the sensor module is also shown.

Fig. 4 .
Fig. 4. Ultrasound force sensor mechanical and electrical integration and mounting on calibration jig.The SMs are labeled numerically for later reference.The master PCB shown here has a separate port for each SM, and each was wired separately.Significant improvements in the wiring and PCBs are discussed in Section V.

Fig. 5 .
Fig. 5. Communication architecture of sensor modules and master PCB.The pins of the 8-pin SPI connectors are shown on the right.

Fig. 6 .
Fig.6.Simulation results from simulated annealing for SM configuration optimization.E is the look-up-table for sensor output as a function of magnet displacement, where each red box is a Hall effect sensor.C shows the probe mesh with suspension locations (black) and explored sensor configurations (color map by cost value).The final, optimal locations are shown in pink.B shows the associated cost and annealing temperature are plotted, showing convergence to a low error value with an exponential cooling schedule.Two simulation outputs from calibration with the optimal sensor configuration are given in A and D. The three axes of force and torque were similar, so only z is shown as an example.All errors are listed in Section IV-B.

Fig. 7 .
Fig. 7. Calibrated output from the DMFS sensor (dashed lines) compared to the ATI sensor (solid lines) using the neural network calibration applied to unseen data from the separate measurement.(Left) Shows all the forces.(Middle) Shows a subset for better visualization, and (Right) shows a similar subset of the torques.The mean error between the ATI and DMFS is 0 in all axes while the standard deviation or RMS error is 0.415 N and 0.0274 Nm respectively.

Fig. 8 .
Fig. 8. (A) Neural network calibration RMS error for the 3D printed suspension, averaged over the five validation folds, versus hidden layer size with different numbers of hidden layers.(B) The learning curve of the chosen architecture, for both training and validation loss from the five folds.

Fig. 9 .
Fig. 9. Hysteresis and creep of the force sensor after a 20 N load is suddenly released.The RTV silicone suspension shows better response speed (time constant, τ ) and returns to 0N whereas the 3D printed suspension has creep.

Fig. 10 .
Fig. 10.Noise power spectral density of the 6-DOF sensor.TABLE II RMS VALIDATION ERROR (AVERAGED OVER THE x, y, AND z AXES) VERSUS SM ORIENTATION.TRIAL 0, THE OUTPUT OF THE OPTIMIZATION PROCEDURE, PERFORMS SIGNIFICANTLY BETTER THAN THE OTHER TWO

Fig. 11 .
Fig. 11.Finite element analysis of outer shell rigidity.(Left) Shell showing elastic supports (blue) and applied force (pink).(Right) Shell colored according to deflection, with several point measurements made symmetrically on both sides.The difference between the two sides shows the shell deflection due to the maximum design force of 25 N.It is on average 0.08 mm.

Fig. 12 .
Fig. 12. Outer shell with updated PCBs.The master board is miniaturized and has a button on the opposite face for resetting.The messy wiring is replaced by a flexible PCB.

TABLE I SUSPENSION
TEST RESULTS IN UNITS OF N/M.THE TORSIONAL STIFFNESS (K θ ) IS IN NMM/RAD

TABLE III COMPARISON
TO STATE OF THE ART MINIATURE 6-AXIS FORCE TORQUE SENSORS (ATI NANO 17, SCHUNK NANO 17, RESENSE HEX 12, MINEBEAMITSUMI MMS101).THE ATI SENSOR HAS THREE CALIBRATIONS WHICH TRADE OFF RESOLUTION AND RANGE.WE USED THE MIDDLE ONE.THE OVERLOAD TORQUE IS THE TORQUE ABOUT THE X AND Y AXES (Z IS THE LONG AXIS OF THE CYLINDRICAL SENSORS) ABOVE WHICH THE SENSOR IS DAMAGED.WHERE DIFFERENT LOAD RATINGS WERE GIVEN FOR XY VERSUS Z, THE AVERAGE IS LISTED.ACCURACY IS LISTED AS PERCENT OF FULL-SCALE RANGE.