Partitioned Reconstruction of Contact Forces in Tactile Sensor

Arrays for Robotic Sensing Systems

María-Luisa Pinto-Salamanca

and Wilson-Javier Pérez-Holguín

Doctoral Program in Engineering with an emphasis in Electronics Engineering, School of Electronic Engineering, GIRA

Research Group, Universidad Pedagógica y Tecnológica de Colombia UPTC, Colombia

Keywords: Tactile Sensing, Contact Forces Reconstruction, Tactile Sensor Array, Force Sensors, Robot Sensing Systems.

Abstract: The reconstruction of contact forces is essential for the performance of robotic manipulation systems from

the information captured by tactile sensors. This work explores the implementation of a model-driven

approach for the triaxial reconstruction of contact forces in tactile sensor arrays using a partition algorithm

that estimates forces in smaller subarrays on a flat and rigid surface. The validation of the presented approach

depends on a prior verification of compliance with the centroids of traction and compression for each analysed

subarray. Considering the force estimation errors, the proposed approach shows a better behaviour than

similar works for single contacts in the force reconstruction for multiple contact events and when using large

size sensors arrays. In addition, the application of the partitioning approach demonstrates a significant

decrease in response time by reducing the number of operations that are needed for the force reconstruction

calculation. Although the relative errors are still significant, the results obtained allow verifying a clear

contribution to the reconstruction of contact events under processing time restrictions for sensor arrays

ranging from small to large scale, that favors the development of electronic skin in robotic applications.

1 INTRODUCTION

The feedback of forces and the perception of contact

events in real-time play a fundamental role in the

planning of the robot's interactions with the

environment (Lambeta et al., 2020) as well as in the

grip or slip control loops (Masoumian et al., 2020).

Likewise, forces estimation is essential for robotic

manipulation and human-robot interaction since the

obtained force components allow a complete

description of a contact phenomenon (Ciotti et al.,

2019).

To replicate the human sense of touch, tactile

sensing systems employ a tactile sensor layer, an

electronic interface layer, and a tactile data decoding

system (Dahiya, et al., 2010), (Ibrahim et al., 2017).

Tactile sensing systems allow performing tasks as

tactile exploration, object identification, and object

grasping and movement. Tactile perception

contributes to expanding the capabilities of robotic

manipulators, humanoid robots, and biomedical

devices, among other applications. An example of

https://orcid.org/0000-0002-2089-0683

https://orcid.org/0000-0001-5238-4470

that is the combination of robotics with tactile sensing

systems, which provides emulation functions of

fingers perception in sophisticated manipulation of

dexterous grippers in hand robots or manipulators (Y.

Li et al., 2019).

Considering the contact medium, tactile sensors

can be continuous or discrete. In particular, discrete

tactile sensors are usually organized as arrays of

individual sensors that can simultaneously be

activated in response to a contact event (Mohammadi

et al., 2019).

The basic unit in tactile sensor arrays is known as

‘taxel’, which is in charge of measuring a contact

event in a single point (Dahiya et al., 2010). Tactile

sensors can also be configured as arrays of taxels to

cover flat areas (Seminara et al., 2015), hard or soft

surfaces (Yuan, et al. 2017), or deformable areas

(Ciotti et al., 2019). Some sensor arrays offer three or

six-axis force estimation with sensing areas up to

4.7mm × 4.7mm with 24 taxels (XELA Robotics Inc).

However, most sensor arrays measure stress or

normal force. In such cases, additional processing

steps are required to decode triaxial forces.

182

Pinto-Salamanca, M. and Pérez-Holguín, W.

Partitioned Reconstruction of Contact Forces in Tactile Sensor Arrays for Robotic Sensing Systems.

DOI: 10.5220/0010716200003061

In Proceedings of the 2nd International Conference on Robotics, Computer Vision and Intelligent Systems (ROBOVIS 2021), pages 182-189

ISBN: 978-989-758-537-1

The contact forces reconstruction process allow

obtaining the force distribution on a surface from

tactile sensor measures (Seminara et al., 2015) using

analytical models based on physical laws (model-

driven), machine learning frameworks (data-driven)

(Wasko et al., 2019), and mixed approaches. Sensors

employed with such approaches comprehend vision-

based, piezoresistive, magnetic, piezoelectric, Hall

Effect, and biomimetic technologies.

Some applications of forces estimation in the

robotics field include shape-recognition in robot-

objects interaction (X. Li et al., 2020), wearable assist

robots (Ito et al., 2019), grasping object in robotic

hands (Mohammadi et al., 2019), human-robot

interaction (Cirillo et al., 2016), robotic skin (Trueeb

et al., 2020), and soft artificial skin (Duong & Ho,

2021), among others.

In tactile sensing systems, there are different

contact sensing areas for electronic skin applications

covering from small to large scales, depending on the

resolution and number of taxels in the sensor array.

Seminara et al. (2015) cover an area close to 36mm ×

36mm using a 3mm thick elastomer layer and a sensor

grid of 10×10 piezoelectric taxels. Duong & Ho

(2021) pose a vision-based model using a FEM

analysis to establish the relationship between nodal

displacements of the markers and external forces

achieving to cover an area of 49763mm

. However,

in the case of works focused on vision, the sensors’

size, and the dependence on complex image

processing algorithms, make it difficult to extend its

use to large artificial skin development in portable

robotics or biomedical applications.

Proper emulation of the human sense of touch

involves meeting a strict time limit to detect contact

events and process them in less than 1 ms (Dahiya, et

al., 2010). A challenge for applications using a

discrete array of sensors is to achieve a reasonable

compromise between the execution speed and the

accuracy of the results, considering the need for

developing calibrating algorithms and parallel

process scenes of complex contact events in real-

time.

There are few works in the literature (Seminara et

al., 2015), (Cimino, 2016) centered on the real-time

implementation of force reconstruction algorithms

employing sensor arrays. These authors propose the

reconstruction of triaxial contact force distributions

on a soft layer surface from the normal stress data

retrieved from a piezoelectric sensor array. Although

this work could be used with other sensor arrays

whose taxels provide discrete stress data, its

application has not been generalized.

This work analyzes, at the simulation level, the

implementation of the model-driven proposed by

Seminara et al. (2015) to reconstruct contact forces in

tactile sensor arrays with a partitioned approach,

considering smaller subarrays. The partitioning

approach was applied to arrays of sensors of different

resolutions ranging from 10×10 to 48×48 taxels to

covering different contact areas. This approach is

aimed to reduce the computational load that allows

speeding up the calculation times required for the

reconstruction of forces. Although the errors obtained

are relatively high, it is expected that combining this

approach with a hardware implementation (FPGA-

like) will achieve compliance with the 1 ms limit in

tactile sensor applications using large arrays of

normal stress sensors.

2 MATERIALS AND METHODS

2.1 Contact Forces Estimation for

Tactile Sensor Arrays

The model-driven approach proposed by Seminara et

al. (2015) allows the estimation of the intensities and

directions of contact forces (three−dimensional force)

starting from normal stress data of a discrete tactile

sensor (single−dimensional data). This is relevant

because this approach would extend the application

of normal stress tactile sensors to the triaxial forces

estimation. The Seminara et al. (2015) model is based

on the solution to the inverse problem of the

Boussinesq equation for an elastic half-space

(Johnson, 1985). This model estimates force vectors

in the same taxels XY-coordinates, on the sensor cover

layer at a distance h on the z-axis (sensor thickness).

This model establishes that the triaxial forces

components (𝑥



force in x-axis, 𝑥



force in y-axis,

and 𝑥



force in z-axis) are defined as:



𝑥



𝑥



𝑥



=𝐶



𝑏+



I−𝐶





𝑤 (1)

where, b is a normal stress vector sensed by the taxels

in the sensor array, C is a matrix defined by the vector

distances given between the taxels coordinates and

the points where the force vectors are reconstructed,

𝐶



is the Moore-Penrose pseudo-inverse matrix of C

(Albert, 1972), and w is a vector that depends both on

two continuous scalar variables µ

and µ

, as well as

on the geometry and the sensor input data.

Partitioned Reconstruction of Contact Forces in Tactile Sensor Arrays for Robotic Sensing Systems

183

The values for µ

and µ

are defined to maximize

an Π objective function that simultaneously fulfils

three physical restrictions: i) compressive normal

forces, ii) tangential forces and normal forces

similarly distributed over the contact area, and iii) no

pinch.

The Seminara et al. (2015) model proposes two

stages to define the objective function (called

preparative phase) and find the optimal solution

(called iterative phase). The preparative phase

includes calculating the matrices, reading the data

from the tactile sensor, and calculating the centroids

required to establish the physical constraints and the

objective function. The iterative phase allows finding

the values of µ1 and µ2 that optimize the Π function.

By a comparison between a FEM simulation and the

analyzed model, the maximum estimated errors were

about 13% in the resultant tangential forces for

Hertzian contacts and 43% of the resultant force in

the x-axis for non-Hertzian contacts (Seminara et al.,

2015).

The mentioned model was evaluated through a

software implementation conducted in Matlab®

R.2020b, by varying the sensor parameters, the taxels

data and the force coordinates. Figure 1 shows a case

of force reconstruction for a tactile sensor array of

10×10 taxels with 4mm×2mm resolution in the XY-

plane and 3mm thickness. For this example, the

normal stress data values read by the sensor are

between -36000 [N/m

] to 20912 [N/m

]. These were

obtained experimentally by applying the function:

𝑆



𝑥,𝑦



=1x10



∗



𝑦 Sin



100𝑥



+𝑥 Cos



100𝑦





(2)

Figure 1: Contact forces reconstruction applying the model-

driven proposed by (Seminara et al., 2015). Scale factors:

Fx=10000[N/mm], Fy=10000 N/mm], Fz=1 [N/mm],

Normal stress surface = 0.003 + S*20000 [N/m2].

2.2 Partitioning Approach to the

Forces Estimation

The partitioning approach proposed involves

sectioning the tactile sensor array into subarrays of

equal size and applying the force reconstruction

algorithm proposed by Seminara et al. (2015) in each

subarray as if they were independent sensors. Then,

these are grouped together to obtain the overall

response of the force estimates.

Although it is clear that the principle of

superposition cannot be applied to a non-linear

model, the sharp decrease in the size of the operations

of the matrix and its consequent decrease in the

system's response time justify the evaluation of the

proposed approach. This approach should be used

considering the accuracy requirements for force

estimation, which may vary in each case.

The algorithm to implement the model-driven of

Seminara et al. (2015) has a computational

complexity of 𝑂𝑑



 order, where d is the size of the

tactile sensor array (Wasko et al., 2019). The model

application implicates matrix operations of



𝑛



𝑛





×3



𝑛



× 𝑛





order, where 𝑛



and 𝑛



are the

number of horizontal and vertical taxels in the array.

Hence, if the size of the array decreases, the

calculation time also decreases. Figure 2 shows four

partition cases to be considered in the proposed

approach that include:

Case 0: the reconstruction of forces in a sensor

array without a subarray (SA0 1×1)

Case 1: mix of the cases above with four subarrays

SA1-SA4 (2×2 subarrays).

Case 2: two vertical subarrays SA1-SA2 (1×2)

Case 3: two horizontal subarrays SA1-SA2 (2×1)

Figure 2: Partition Cases Analyzed in the Contact Force

Reconstruction Approach.

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The performed tests comprise four tactile sensor

arrays whose characteristics are described in Table 1.

For the cases of 10 × 10 taxel matrices, normal stress

data corresponding to Hertzian and non-Hertzian

contact events were the same used by (Seminara et

al., 2015). In the 20×20 taxel array, the sensor input

data were combinations of Hertzian and non-Hertzian

contacts. The input data for the 48×48 array were

obtained with an FSR Matrix Array Sensor for the

plantar pressure measurement systems (PPMs)

described in (Castro et al., 2020).

Table 1: Included Sensors for the implementation of the

partitioned approach.

Tactile

Sensor

Taxels Array



𝑛



× 𝑛





Size

[mm

× mm × mm]

Taxels

Separation

Resolution

[mm × mm]

Sensor 1 10 × 10 20 × 40 × 3 4 × 2

Sensor 2 10 × 10 40 × 40 × 3 4 × 4

Sensor 3 20 × 20 20 × 40 × 3 4 × 2

Sensor 4 48 × 48 384 × 384 × 0.91 8 × 8

The search for the optimal parameters µ1 and µ2

was carried out using the Matlab® function fmincom.

The objective function Π is conditioned for the

centroids of the contact event (Centroids Condition)

such that: the data detected in the matrix must include

positive and negative values to calculate the

compression and tension centroids simultaneously.

Consequently, the partitioning approach initially

checks for this condition on the data in the subarray.

If this condition is met, the approach try to perform as

much partitionings as possible.

In the proposed approach the preparatory phase of

each partitioning case comprises: i) separate stress

data from each partition, ii) redefine the coordinates

of the taxels and the force estimation, iii) calculate the

C matrix and the 𝐶



pseudo-inverse matrix for each

subarray, iv) determine the centroids of tension and

compression for each subarray, and finally v)

evaluate the Π functions.

The iterative phase for each analyzed partition

includes two stages: i) find the optimal values to the

parameters µ1 and µ2 for each analyzed subarray, and

ii) compare the minimum forces obtained with an

established threshold. Finally, the algorithm groups

the reconstructed forces to present a force vector for

each taxel.

3 ANALYSIS OF RESULTS

The simulations carried out applying the proposed

approach generate the estimation errors presented in

Table 2, according to cases 0 to 3 described in the

previous section and the sensor parameters shown in

Table 1. The resulting forces correspond to the sum

of the estimated forces in each axis (𝑋



∑

𝑥



, 𝑋



∑

𝑥



, 𝑋



∑

𝑥



). The error was calculated as:

𝐸𝑟𝑟𝑜𝑟 =

𝑅𝑒𝑓.𝑣𝑎𝑙𝑢𝑒– 𝑅𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 𝐹𝑜𝑟𝑐𝑒

𝑅𝑒𝑓.𝑣𝑎𝑙𝑢𝑒

(3)

For the force reconstruction using sensors 1 and

2, the reference values for error estimation were

obtained by mean a FEM simulation developed in

COMSOL® by Seminara et al. (2015). In the analysis

carried out with sensors 3 and 4, the estimation error

was similar to those obtained for Case 0 (without

partitioning). Due to the application of Equation 3,

Table 2 contains some negative values for the error.

Table 2: Estimation errors obtained during the partitioning

approach validation.

Sensor

Analysis

Case

Estimation errors

0 12.84% 12.93% 7.54%

1 77.52% 65.69% -23.57%

2 46.32% 71.17% -15.15%

3 64.41% 5.60% -0.46%

0 -39% 16% -1%

1 17.52% 51.22% -11.60%

2 -97% 55% -5%

3 42% -1% -7%

0 0% 0% 0%

1 3.91% -12.67% -1.37%

2 0.11% 0.07% -1.15%

3 5.05% -12.05% -0.23%

0 0% 0% 0%

1 51.04% 68.98% -5.44%

2 49.10% -63.09% -5.21%

3 51.61% -67.72% -4.25%

Figure 3 shows a comparison between the optimal

values µ1 and µ2 for each case. During the approach

validation, the µ values for the partition Case 0 is

taken as reference. For the partitions that do not meet

the centroid conditions, the values of µ1 and µ2 are

null. The response of the partitioning approach is

analyzed by classifying the contact events as ‘simple’

for sensors 1 and 2 and ‘multiple’ for sensors 3 and 4.

3.1 Single Contacts

Figure 4 shows the reconstruction cases when using

sensors 1 and 2. In the case of the sensor 1 (Figure

4(a)), the input data corresponds to a single Hertzian-

Partitioned Reconstruction of Contact Forces in Tactile Sensor Arrays for Robotic Sensing Systems

185

type contact event. As shown in Figure 3, for Case 1,

two subarrays (SA2 and SA4) do not meet the

centroid conditions, so their optimal parameters were

null. Figure 3 also shows that Case 2 do not fully

comply with the contact conditions. During the test

with sensor 1 and Case 3 forces are estimated in the

two subarrays taking into account that µ1 and µ2 are

different to zero. However, it would be noted that this

case present a high estimated error (64.41%) in the

Figure 3: Flow diagram for implementation of the

partitioning approach for the contact forces reconstruction.

resulting forces on the x-axis, while the reference

(Case 0) gives a maximum estimation error of about

13% for the tangential forces. The results obtained

allow verifying that the proposed partition strategy

does not work correctly in the case of single contact

and Hertzian events.

For sensor 2, the input data corresponds to a non-

Hertzian contact (Figure 4(b)). For Case 1, it is

observed that the centroid condition is not fulfilled in

subarray SA4. For partitions with two subarrays, the

estimation errors for Case 3 show a better

performance than Case 2. If Case 3 is compared with

the reference (Case 0), the first one present a smaller

error in the resultant force on the y-axis (1%).

However, for these cases the forces estimation error

with respect to the z-axis is better for Case 0 (-1% vs.

-7%). Since the non-Hertzian contact is a simple

contact located in the center of the sensor, the

centroid condition is fulfilled more easily than in the

Hertzian case.

Figure 4: Results of partitioned force reconstruction.

a) Hertzian contact Case 2, b) Non-Hertzian contact Case 3.

3.2 Multiple Contacts

Figure 5 and 6 present the results of applying some

cases of force reconstruction for sensors 3 and 4,

respectively. The input data used with sensor 3 (see

0,18

0,08

0,27

0,11

0,06

0,19

0,32

0,41

0,06

0,35

0,40

0,23

0,04

0,28

0,19

0,47

0,33

0,19

0,46

0,33

0,52

0,39

0,53

0,23

0,64

0,78

0,48

0,84

0,28

0,36

0,40

0,35

0,42

0,45

0,26

0,29

0,12

0,54

0,34

0,36

0,42

0,34

0,42

0,31

0,33

0,47

0,45

0,42

0,58

0,06

0,09

0,46

0,36

0,00 0,20 0,40 0,60 0,80 1,00

SA1

SA2

SA3

SA4

SA1

SA2

SA1

SA2

SA0

SA1

SA2

SA3

SA4

SA1

SA2

SA1

SA2

SA0

SA1

SA2

SA3

SA4

SA1

SA2

SA1

SA2

SA0

SA1

SA2

SA3

SA4

SA1

SA2

SA1

SA2

SA0

Case 1 Case 2 Case 3

Case

0 Case 1 Case 2 Case 3

Case

0 Case 1 Case 2 Case 3

Case

0 Case 1 Case 2 Case 3

Case

Sensor 1 Sensor 2 Sensor 3 Sensor 4

µ1 µ2

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186

Figure 5) are combinations of two Hertian and non-

Hertzian contacts. Based on the estimation errors, the

input data for sensor 3 exhibits the best performance

for the partitioning approach. Although Case 1

fulfilled the centroid conditions for all partitions, the

errors given for Case 2 are lower, so Case 2 is the best

choice to be employed in the proposed approach with

the multiple Hertzian and non-Hertzian contact event.

Regarding the reconstruction of forces with

sensor 4 (see Figure 6), all the partition cases fulfilled

the centroid conditions. However, Case 3 present the

lowest estimation errors for the resulting force in the

z-axis (-4.25%), so this case is the best for the anlyzed

contact event with sensor 4. For this sensor, in each

case of analysis, the response times of the algorithm

were evaluated, obtaining the data presented in Figure

7. Executing the algorithm with one partition (Case 0)

requires 2705.33s, with two subarrays, Cases 2 and 3,

it takes 354.17s and 330.57s, respectively, while with

four subarrays (Case1) it only requires 109.63s.

Figure 7 also shows three tests for the same

partitioning case which generated similar response

times for each test.

Figure 5: Results of partitioned force reconstruction in

multiple contact events with Hertzian and non-Hertzian

events. Reconstruction cases a) Case 0. b) Case 1 (2×2).

Figure 6: Results of partitioned force reconstruction contact

events for a PPMs application. Reconstruction cases a)

Case 0 (1×1). b) Case 1 (2×2). c) Case 2 (1×2). d) Case 3

(2×1).

4 DISCUSSION AND FUTURE

WORKS

The model-driven on which this work is based was

designed to reconstruct single contact events.

Considering this limitation, it is not possible to

guarantee the proper functioning of the proposed

Partitioned Reconstruction of Contact Forces in Tactile Sensor Arrays for Robotic Sensing Systems

187

partition approach for single contact events. This is

because when dividing the sensor, it is not feasible to

ensure that each subarray meets the centroid

conditions since a single contact produces only one

tension centroid and one compression centroid for the

entire sensor.

Figure 7: Response time for partitioning approach with a

48×48 taxels sensor.

Even for single contact events, the non-Hertzian

contact analyzed had better behavior than the

Hertzian one, since the first one allowed the

partitioning of the sensor into two subarrays. This

implies that non-Hertzian contacts have a greater

expectation of being processed adequately with the

proposed approach.

Applying the proposed approach shows a great

decrease in the time required to rebuild the contact

forces when the number of subarrays increases. This

is because the order of operations in the matrix

decreases for smaller subarrays, which improves the

temporal response. For instance, for a 48×48 taxel

matrix, without partitions (Case 0), the algorithm

requires (3 * 48 * 48) × (6912) matrix operations.

For Case 1 with four subarrays, the algorithm requires

(3 * 24 * 24) × (1728) operations, while in Case 2 and

Case 3, the algorithm involves (3 * 24 * 48 = 3456)

operations. This means that, despite the relatively

high estimation errors obtained for this partitioning

approach, its application for large tactile sensor arrays

becomes attractive, since it significantly reduces the

number of operations required.

The latency of the FPGA-based hardware

implementation of the no partitioned model proposed

by Seminara et al. (2015) is 1.6 × 10

-6

s for processing

an 8 × 8 size sensor. For its part, the software

implementation developed herein for the above-

mentioned model has a response time of 16.43 s,

using the normal stress input data generated by

applying Equation 3 and an 8×8 size sensor.

This allows us to infer that given the reduction in the

number of operations required for the proposed

partitioning approach, the response times of the

hardware implementation are expected to be even

lower than those reported by Seminara et al.

In multiple contact events, sensor 3 produces a

good reconstruction of forces because this

configuration allows meeting the centroid conditions

for each analyzed subarray. This allows obtaining the

optimal parameters µ1 and µ2, which enables to use

of this approach to properly model these contacts.

Future works include evaluating the

reconstruction of single contact events for force

feedback using the proposed partitioning approach on

a hardware implementation considering applications

such as electronic skin, manipulation tasks, and

human-robot interaction.

5 CONCLUSIONS

The application of the proposed partitioning approach

for a 48×48 taxel tactile sensor matrix shows a very

significant decrease in the execution time, which goes

from 2705.33 s to 109.63 s only, when performing the

forces estimation using four subarrays of 24×24

taxels. The proposed partitioning approach to the

contact force reconstruction in arrays of tactile

sensors facilitates the decoding of the properties of

the touched objects by considerably reducing the

response time required to process the information

provided by large tactile sensor arrays.

The validation of the partitioning approach

depends on a prior verification of compliance with the

centers of traction and compression for each analyzed

subarray. Therefore, the proposed partition approach

generates high errors for single contacts, while it

presents tolerable errors for multiple contact events

distributed in the subarrays.

Due to its characteristics of low computing power

required and high execution speed, the proposed

approach can be used in applications of human-robot

interaction and force control loops in robotic

manipulation, in which it is tolerable to work with an

approximated knowledge of the properties of the

touched object.

ACKNOWLEDGEMENTS

Authors thank the valuable support and collaboration

of researchers of the COSMIC group at University of

Genoa, Genova-Italy, and the VIE-UPTC for

providing the funds to the international mobility to

the University of Genoa.

0 500 1000 1500 2000 2500 3000

Case 0

Case 2

Case 3

Case 1

Response time [s]

Partitioning cases

TEST 3 TEST 2 TEST 1

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188

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