Augmenting Neural Networks-Based Model Approximators in Robotic

Force-Tracking Tasks

Kevin Saad

1 a

, Vincenzo Petrone

2 b

, Enrico Ferrentino

2 c

, Pasquale Chiacchio

2 d

Francesco Braghin

1 e

and Loris Roveda

1,3 f

Department of Mechanical Engineering, Politecnico di Milano, 20133 Milano, Italy

Department of Information Engineering, Electrical Engineering and Applied Mathematics (DIEM), University of Salerno,

84084 Fisciano, Italy

Istituto Dalle Molle di Studi sull’Intelligenza Artiﬁciale (IDSIA), Scuola Universitaria Professionale della Svizzera

Italiana (SUPSI), Universit

a della Svizzera Italiana (USI), 6962 Lugano, Switzerland

Keywords:

Force Control, Robot-Environment Interaction, Neural Networks.

Abstract:

As robotics gains popularity, interaction control becomes crucial for ensuring force tracking in manipulator-

based tasks. Typically, traditional interaction controllers either require extensive tuning, or demand expert

knowledge of the environment, which is often impractical in real-world applications. This work proposes a

novel control strategy leveraging Neural Networks (NNs) to enhance the force-tracking behavior of a Direct

Force Controller (DFC). Unlike similar previous approaches, it accounts for the manipulator’s tangential ve-

locity, a critical factor in force exertion, especially during fast motions. The method employs an ensemble

of feedforward NNs to predict contact forces, then exploits the prediction to solve an optimization problem

and generate an optimal residual action, which is added to the DFC output and applied to an impedance con-

troller. The proposed Velocity-augmented Artiﬁcial intelligence Interaction Controller for Ambiguous Models

(VAICAM) is validated in the Gazebo simulator on a Franka Emika Panda robot. Against a vast set of trajec-

tories, VAICAM achieves superior performance compared to two baseline controllers.

1 INTRODUCTION

Modeling and controlling accurate force tracking in

robotic manipulators remains a fundamental chal-

lenge for reliable robot-environment interaction.

Achieving high force-tracking performance is critical

in a broad spectrum of tasks, including contact-rich

manipulation, precision assembly, and surface inter-

action (see Fig. 1).

To address this challenge, impedance controllers

achieve force-tracking accuracy through strategies

such as reference generation (Roveda and Piga, 2021;

Huang et al., 2022; Yu et al., 2024), variable stiff-

ness (Shen et al., 2022; Li et al., 2023), and variable

https://orcid.org/0009-0001-2295-0723

https://orcid.org/0000-0003-4777-1761

https://orcid.org/0000-0003-0768-8541

https://orcid.org/0000-0003-3385-8866

https://orcid.org/0000-0002-0476-4118

https://orcid.org/0000-0002-4427-536X

0 2 4 6 8 10

Time [s]

Force [N]

reference

actual

Figure 1: Simulation setup — the Panda robot performs

a force-tracking task sliding on a wooden table with a

spherical-tip end-effector.

394

Saad, K., Petrone, V., Ferrentino, E., Chiacchio, P., Braghin, F. and Roveda, L.

Augmenting Neural Networks-Based Model Approximators in Robotic Force-Tracking Tasks.

DOI: 10.5220/0013830700003982

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 22nd International Conference on Informatics in Control, Automation and Robotics (ICINCO 2025) - Volume 2, pages 394-401

ISBN: 978-989-758-770-2; ISSN: 2184-2809

damping (Jung et al., 2004; Shu et al., 2021; Duan

et al., 2018; Roveda et al., 2020). Reference gen-

eration methods implement an explicit direct force

control (DFC) loop to follow a desired force trajec-

tory, while variable impedance approaches often rely

on simpliﬁed linear-spring environment models (Jung

et al., 2004; Shen et al., 2022; Li et al., 2023; Shu

et al., 2021; Duan et al., 2018; Yu et al., 2024). How-

ever, these simpliﬁed models typically represent only

an approximation of the real environment dynamics

(Roveda and Piga, 2021; Jung et al., 2004; Matschek

et al., 2023).

To overcome modeling inaccuracies, recent liter-

ature has explored Artiﬁcial Intelligence (AI) tech-

niques (Matschek et al., 2023), particularly Neural

Networks (NNs), to learn a model-less, data-driven

mapping between the manipulator’s end-effector state

and the exerted contact force directly at the control

level. In this context, ORACLE (Petrone et al., 2025)

was proposed as a controller that leverages NN-based

models to optimize force tracking. However, OR-

ACLE’s original formulation neglects the effects of

end-effector velocities tangential to the contact plane,

potentially limiting its prediction accuracy and track-

ing performance, especially at high velocities.

This paper presents an extension of the ORA-

CLE strategy by augmenting its model approxima-

tor’s state with tangential velocity components, result-

ing in a reﬁned controller named Velocity-Augmented

Artiﬁcial Intelligence interaction Controller for Am-

biguous Models (VAICAM). The proposed approach

constructs an accurate model of the environment that

relates the end-effector pose, penetration velocity, and

tangential velocity to the resulting contact forces us-

ing feed-forward neural networks (FFNNs) (Naga-

bandi et al., 2018).

To this aim, a dedicated dataset containing dy-

namic trajectories in both force and position space

is generated to train this model effectively. An im-

proved controller is then designed based on this aug-

mented model, enabling optimal selection of con-

trol actions to minimize force-tracking errors. Ex-

tensive validation is carried out in simulation, using

Gazebo (Koenig and Howard, 2004), on a Franka

Emika Panda manipulator (Haddadin et al., 2022).

Furthermore, the paper conducts a comparative analy-

sis between a standard direct force controller (Roveda

and Piga, 2021) and ORACLE (Petrone et al., 2025)

across varying velocity conditions, concretely attest-

ing ORACLE’s performance degradation as the end-

effector velocity increases.

By integrating tangential velocity information into

the ORACLE framework, VAICAM demonstrates

improved force prediction and enhanced tracking

capabilities, extending the applicability of neural-

network-based force controllers to a wider range of

challenging interaction tasks.

2 METHODOLOGY

This paper introduces VAICAM, an AI-driven tool

to enhance the force tracking capabilities of an

impedance controller used in unknown environments.

It augments the ensemble of FFNNs originally pro-

posed in ORACLE (Petrone et al., 2025) by adding to

its input the tangential velocity v of the end-effector

(EE), resulting in a mapping from the robot state and

the DFC control action x

into its next state. This de-

sign choice yields a more accurate prediction of the

next wrench, that will later be used to compute the

optimal residual action added to the low-level control

action of the DFC. VAICAM’s overall control scheme

is summarized in Fig. 2, whose building blocks will

be detailed in the next sections.

2.1 Base Controller

The low-level impedance controller enforces the de-

sired interaction dynamics on the robot, speciﬁcally,

Cartesian space mass-spring-damper dynamics. Con-

sider the equations of motion for a manipulator with

n Degrees of Freedom (DOFs) performing an m-

dimensional task, with m ≤ 6 ≤ n (Featherstone and

Orin, 2016):

B(q

q +C

C(q

q + τ

(

q) + g

g(q

q) = τ

− J

⊤

q)h

(1)

where B

B(q

q) ∈ R

n×n

is the inertia matrix, C

C(q

q) ∈

n×n

is the matrix accounting for the centrifugal and

Coriolis effects, τ

(

q) ∈ R

accounts for viscous and

static friction, g

g(q

q) ∈ R

represents the torque exerted

on the links by gravity, τ

∈ R

indicates the torque

control action, J

J(q

q) ∈ R

m×n

is the geometric Jaco-

bian, and h

∈ R

is the vector of wrenches exerted on

the environment measured by means of a force/torque

sensor mounted on the manipulator’s ﬂange. The vec-

tors q

q ∈ R

represent joint positions, velocities,

and accelerations, respectively.

The expression of the Cartesian impedance con-

trol law with robot dynamics compensation is (Sicil-

iano and Villani, 1999; Formenti et al., 2022)

= J

⊤

q)h

C(q

q + τ

(

q) + g

g(q

q), (2)

where the task space wrench h

realizing the compli-

ant behavior can be chosen as (Caccavale et al., 1999;

Iskandar et al., 2023)

= K

∆

∆x

x + D

x, (3)

Augmenting Neural Networks-Based Model Approximators in Robotic Force-Tracking Tasks

395

DFC (4) VAICAM (10)

Impedance

control (2)

Robot

dynamics (1)

Environment

MA (7)

Dataset

∗

± ρ

s =

F (s

s,x

) − s

On-line control

Physical plant

Dynamical system F

Off-line training

Figure 2: Control architecture. The MA learns the transition function

F of the dynamical system represented by the

impedance-controlled robot interacting with an unknown environment, given data composed of the system states s

s and control

inputs x

, i.e. the DFC action. After training, VAICAM computes the optimal residual action x

∗

, aiming at minimizing the

force tracking error between h

and the predicted wrench

where K

∈ R

m×m

are diagonal matrices of con-

trol parameters, namely stiffness and damping, re-

spectively, and ∆

∆

∆x

x ≜ x

− x

x ∈ R

is the Cartesian

pose error between the setpoint x

∈ R

and the ac-

tual robot pose x

x ∈ R

. Assuming m = 6, x

x is deﬁned

as x

x ≜ (x,y, z, φ, θ, ψ)

⊤

, where (x, y,z)

⊤

and (φ,θ,ψ)

⊤

are translational and rotational components, respec-

tively.

2.2 Direct Force Controller

Given that the impedance controller solely manages

interaction forces passively, lacking the capability to

track a force reference, a DFC loop can be closed

speciﬁcally along the directions in which force track-

ing is necessary (Roveda and Piga, 2021). The

adopted control law is a simple PI controller having

the following model:

= x

+ Γ



∆

∆h

h + K

∆

∆h

hdt



, (4)

where, if m = 6, Γ

Γ = diag(γ

,γ

) is the

task speciﬁcation matrix (Khatib, 1987), with γ

= 1

if the i-th direction is subject to force control, 0 other-

wise. x

∈ R

is the force controller output, while

∈ R

is the reference pose, whose i-th compo-

nent is tracked when γ

= 0. K

∈ R

m×m

are the

proportional and integral gains of the controller, and

∆

∆h

h = h

− h

∈ R

is the error between the reference

wrench to be exerted h

∈ R

and the actual exerted

wrench h

∈ R

2.3 Model Approximator

The Model Approximator (MA) addresses the inher-

ent complications in accurately modeling the robot-

environment interaction with a rather straightforward

method that only requires the user to set up a handful

of experiments that autonomously train the NN-based

model. The MA deals with the current system state

, aiming to predict the state at the next time step

k + 1 following the equation

k+1

F (s

), (5)

where

k+1

is the predicted next state.

F represents

the transition dynamics approximation which outputs

the next predicted state, where F indicates the ac-

tual dynamical system, i.e. the impedance-controlled

robot (see Fig. 2). Speciﬁcally, instead of predicting

k+1

explicitly, the actual FFNN’s output δ

s is chosen

to be the approximate difference between two subse-

quent states, similarly to (Nagabandi et al., 2018), i.e.:

δs

s = s

k+1

− s

. (6)

This allows (5) to be rewritten as

k+1

= s

+ δ

s(s

), (7)

with δ

s(s

) being the actual NN output, as in

Fig. 3.

As regards the state deﬁnition, in general s

s takes

the form

s ≜ (x

x,h

)

⊤

, (8)

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

396

(9)

input

layer

hidden layers

output

layer

k+1

Figure 3: MA architecture. The hidden layers approximate

the dynamics transition function s

k+1

= F (s

) by pre-

dicting the state variation δ

but it might be specialized according to the task setup.

Indeed, assuming tracking forces only along the z

axis, ORACLE (Petrone et al., 2025) only considers

orthogonal components in s

s, i.e. EE penetration ve-

locity and force along the same direction. This work

introduces a new feature in s

s, namely the EE tan-

gential velocity v, which inﬂuences the evolution of

the exerted force, especially at high speeds (Iskandar

et al., 2023). In summary, we propose the following

augmentation in the MA state representation:

s ≜ (z, ˙z, f

)

⊤

→ s

s ≜ (z, ˙z,v, f

)

⊤

, (9)

where v =

˙x

+ ˙y

is the tangential velocity, with

˙x, ˙y being the velocity components on the EE xy plane,

and f

is the normal force along z, i.e., the contact

direction.

From an architectural standpoint, the FFNN is ac-

tually an ensemble of FFNN as in (Chua et al., 2018),

since using an array of N independently trained NNs

minimizes the errors risen by epistemic and stochastic

uncertainties.

2.4 VAICAM Algorithm

Acting as the glue that joins all items together,

the VAICAM algorithm combines the DFC-enhanced

impedance controller and the MA. It utilizes the con-

trol action output by the DFC x

along with the force

prediction of the MA

, and then outputs the opti-

mal residual action x

∗

that will then be added to the

impedance controller action aiming at minimizing the

force tracking error. VAICAM will search for x

∗

the neighborhood of x

in an area centered around it

with predeﬁned radius ρ

ρ > 0

Then, the control input to the impedance con-

troller is updated by solving an optimization problem,

run at each control step k:

∗

(k) = argmin

∈x

±ρ

s,x

), (10)

where

s,x

) = |h

(k) −

)| + Ω

) (11)

Data: x

from (4), s

s = (x

x,h

)

⊤

Result: x

∗

if k = 1 then

Set α

α, β

β and ρ

ρ;

Set weights of the pre-trained MA NN

ensemble;

end

Build the discretized neighborhood of candidate

optimal actions B

) = [x

± ρ

ρ];

for x

∈ B

) do

Infer the predicted force from the model

approximator with (7):

s,x

) = h

+ δ

s,x

);

Compute the regularizer in (12) as

Ω

) = ∥x

∥

+ |x

− x

∗

(k − 1)|

;

Compute the corresponding loss function in

(11) as L

s,x

) = |h

−

| + Ω

);

end

Call VAICAM by solving the optimization

problem in (10) minimizing (11):

∗

(k) = argmin

∈B

)

s,x

);

Algorithm 1: VAICAM algorithm at every control

step k.

is the cost function to minimize, whose ﬁrst term |h

−

| is the expected force tracking error, and

Ω

) = ∥x

∥

+ |x

− x

∗

(k − 1)|

(12)

is a regularizer that contributes to smoothing-out large

jumps of the control term x

. The ﬁrst term in (12) pe-

nalizes heavy actions so as to avoid deep penetrations

of the EE into the environment, whereas the second

prevents fast variations between subsequent actions.

The two terms are deﬁned as:

∥x

∥

∑

c,i

, (13a)

− x

∗

(k − 1)|

∑

c,i

− x

∗

c,i

(k − 1)|, (13b)

where both α

α,β

β ∈ R

are parameters to be chosen by

the user. This method, summarized in Algorithm 1,

along with the base force controller, is activated as

soon as contact is established between the EE and the

environment.

2.5 Training Procedure

The training and validation of the ensemble of FFNN,

which is done in a preliminary ofﬂine stage, uses col-

lected data resulting from thorough exploration of the

state space. In order to collect the data, reference

forces and positions are provided to the base force

controller, and the output data, consisting of the ac-

tual robot states s

s and control actions x

, are recorded.

During the training stage of the FFNN, their weights

Augmenting Neural Networks-Based Model Approximators in Robotic Force-Tracking Tasks

397

are updated using the Stochastic Gradient Descent al-

gorithm in order to minimize the Mean Squared Error

(MSE) between the actual and estimated states.

In our training procedure, we recommend com-

manding sine waves as references in both force and

position spaces, in order to have a thorough explo-

ration and avoid data gaps in the state domain. Com-

manding a sine wave reference for the position addi-

tionally entails that the EE velocity v is sinusoidal,

allowing for the complete exploration of the velocity

space as well. This is a crucial aspect because, com-

pared to (Petrone et al., 2025), the inclusion of the

new feature v requires a dedicated training. Further-

more, we recommend exaggerating the amplitude of

the sine wave force references: even though they can-

not be perfectly tracked by the controller, this ensures

that the force domain is sufﬁciently explored.

After collecting data, they are then processed fol-

lowing methods used in (Nagabandi et al., 2018; Chua

et al., 2018), i.e. they are normalized by subtracting

the mean of each quantity and then dividing by its

standard deviation. A zero-mean Gaussian noise in

the form N (µ = 0,σ) is applied to the measured data

in order to enhance the robustness of the NN.

3 RESULTS

3.1 Task and Materials

The experimental validation is divided into two main

phases, both conducted on the 7-DOF Franka Emika

Panda robot:

• Experiment I: train, validate, and test the Static

Model Approximator (SMA) used by ORACLE,

which is the MA trained and validated using static

position references (Petrone et al., 2025) without

tangential velocity, and the Dynamic Model Ap-

proximator (DMA) used by VAICAM, which is

the MA trained and validated using dynamic posi-

tion references with tangential velocities;

– the goal is to assess the performance of both

MAs on dynamic position reference trajecto-

ries, and validate that the DMA yields higher

accuracy as the tangential velocity increases;

• Experiment II: execute dynamic trajectories us-

ing the base controller (Roveda and Piga, 2021),

ORACLE (Petrone et al., 2025) and VAICAM,

and compare the force tracking results;

– in this case, the objective is to assess the per-

formance of both control strategies, and val-

idate that the controller that uses the DMA

(VAICAM) performs better than both the base

Table 1: Parameters used in the experiments.

Parameter Value

Coulomb friction coefﬁcient µ 0.2

Time step solver ODE

Impedance control translational stiffness K

d,t

1700

Impedance control orientational stiffness K

d,r

300

Damping ratio ξ 1

DFC Proportional gain K

1· 10

−6

DFC Integral gain K

2· 10

−3

Table 2: Conﬁgurations of the FFNN ensemble.

Parameter Value

Number of estimators N 3

Hidden layers 3

Neurons per layer 200

Activation function ReLU

Learning rate 1· 10

−3

Ensemble type Fusion

Training epochs 50

Loss function MSE

Weight optimizer Adam

controller and the controller that uses the SMA

(ORACLE).

The NN’s algorithm is coded in Python using

PyTorch (Paszke et al., 2019), interfacing with the

other modules via ROS communication mechanisms

(Quigley et al., 2009). The controllers implemented

in ROS are coded in C++.

In order to accurately simulate the robot,

Gazebo (Koenig and Howard, 2004) is used as

the simulation software. A spherical tip EE is

mounted at the ﬂange of the robot (see Fig. 1).

The interaction control parameters are chosen as

= diag(K

d,t

r,t

) and D

diag{ξ

d,i

}

i=1

, where K

d,t

and K

d,r

are the transla-

tional and rotational stiffness gains, respectively, and

ξ is the damping ratio. Table 1 provides a summary

of the parameters. In our experiment, a linear force is

exerted on the z axis, thus Γ

Γ = diag(0,0,1,0,0,0).

3.2 Experiment I: Model Approximator

Validation

For a fair comparison with (Petrone et al., 2025),

we base the FFNNs structure on the ﬁndings therein.

Speciﬁcally, all FFNNs used share the same conﬁg-

uration in terms of depth (number of hidden layers),

width (number of neurons per layer) and learning al-

gorithm parameters, as listed in Table 2. Linear type

layers are used, while the activation function is of the

ReLU type (Agarap, 2019). The optimized conﬁg-

uration of the base estimator that ensures enhanced

inference time without compromising prediction ac-

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398

curacy uses N = 3 FFNNs. Adam (Kingma and Ba,

2015) refers to Adaptive Moment Estimation, an op-

timizer used for NN regression tasks, while “Fusion”

indicates how a single output is retrieved from the en-

semble, i.e. the arithmetic average across the indepen-

dent network estimations is computed.

3.2.1 Static Model Approximator

The dataset is composed of 10 trajectories, with 9 of

them being used in the training set, and the remain-

ing 1 in the validation set, used by Adam (Kingma

and Ba, 2015) to adaptively optimize the learning

rate. The position reference is in the form of a static

waypoint, as this MA, unlike the one proposed in

this work, does not take into consideration v (Petrone

et al., 2025). The force reference is in the form of

a sinusoidal wave with randomized frequency, ampli-

tude, and mean. Raw data in both training and valida-

tion sets are pre-processed as indicated in Sect. 2.5.

After training the MA, a test set is developed in

order to assess its performance. Since the aim of

this work is to assess the MA generalization ability

against dynamic position trajectories, this set is com-

posed of horizontal line references that the EE tries to

track with a constant velocity and a sinusoidal force

reference. The MA is tested against a total of 110 tra-

jectories of about 1.2m executed using the base con-

troller, divided into 11 evenly-spaced velocities in the

range [0.01,0.50]m/s. The performance assessment

is based on the comparison of the force predicted by

the MA with the actual force.

3.2.2 Dynamic Model Approximator

Compared to the SMA discussed in Sect. 3.2.1, we

adopt a different training approach, i.e. both the force

and position references in the training set take a si-

nusoidal form. On the basis of the training proce-

dure outlined in Sect. 2.5, sinusoidal position refer-

ences allow covering the desired velocity span bet-

ter than constant velocity proﬁles. The validation set

is composed of 11 trajectories, each randomly given

a velocity from the 11 velocity samples in the range

[0.01,0.50]m/s.

After executing the trajectories, the data stored in

the dataset are pre-processed according to the proce-

dure reported in Sect. 2.5. Using the DMA, the net-

work now takes into consideration v while predicting

the next state, which ameliorates the performance of

the MA at higher velocities. In order to conﬁrm this

thesis, the DMA is evaluated on the same test set as

the SMA’s, i.e. using the same 110 line trajectories.

0.0 0.1 0.2 0.3 0.4 0.5

Velocity [m/s]

0.02

0.04

0.06

0.08

0.10

0.12

RMSE [N]

Dynamic MA

Static MA

Figure 4: Comparison between the static and the dynamic

model approximators, in terms of RMSE, as the EE velocity

increases.

Table 3: Average RMSE of static and dynamic model ap-

proximators across trajectories — η indicates the improve-

ment factor of DMA over SMA.

Velocity RMSE [N]

[m/s] SMA DMA

0.01 0.0154 0.0397 0.3877

0.05 0.0135 0.0203 0.663

0.1 0.0155 0.022 0.7029

0.15 0.0203 0.0228 0.8893

0.2 0.0312 0.0207 1.511

0.25 0.0456 0.0165 2.7628

0.3 0.0557 0.0225 2.4777

0.35 0.0684 0.0216 3.1683

0.4 0.0819 0.0188 4.3528

0.45 0.0936 0.0196 4.7841

0.5 0.104 0.0188 5.5422

3.2.3 Model Approximators Comparison

Fig. 4 reports the Root Mean Square Error (RMSE)

between the predicted and measured force, when ei-

ther the SMA or DMA is employed, for increasing

EE tangential velocity v. As expected, the plot re-

veals a better generalization ability of the DMA over

the SMA to higher EE velocities. While the results

are comparable at low speed, as v increases the SMA

showcases lower prediction accuracy, in terms of both

mean and variance across trajectories.

This qualitative analysis is quantitatively con-

ﬁrmed by Table 3, which also reports the numerical

improvement factor η of DMA over SMA. As v in-

creases, the former outperforms the latter by up to

454%, in terms of average RMSE recorded across tra-

jectories at the same velocity.

Augmenting Neural Networks-Based Model Approximators in Robotic Force-Tracking Tasks

399

0.0 0.1 0.2 0.3 0.4 0.5

Velocity [m/s]

RMSE [N]

VAICAM

ORACLE

DFC

Figure 5: Comparison between the baseline DFC (Roveda

and Piga, 2021), ORACLE (Petrone et al., 2025), and

VAICAM (ours), in terms of RMSE, as the EE velocity in-

creases.

0 1 2 3 4 5 6 7 8

Time [s]

Force [N]

reference

ORACLE

VAICAM

±1 N error

Figure 6: Force tracking comparison between ORACLE

(Petrone et al., 2025) and VAICAM (ours) at v = 0.2m/s.

3.3 Experiment II: Control Algorithm

Validation

Once the SMA and the DMA are trained, the three

control algorithms — DFC (Roveda and Piga, 2021),

ORACLE (Petrone et al., 2025), and VAICAM (ours)

— can be tested against a test set aiming to prove that

VAICAM outperforms its competitors. The optimizer

parameters are tuned as α = 25, β = 200, and ρ =

0.003m for both ORACLE and VAICAM.

As can be seen from Fig. 5, the average force-

tracking RMSE for VAICAM is approximately con-

stant for v ∈ [0, 0.35] m/s, and starts increasing for

v > 0.35m/s. Thanks to the exploitation of the novel

DMA, VAICAM outperforms both ORACLE and the

base DFC. A sample trajectory is considered in Fig. 6,

visually showing that ORACLE’s uncertainty at mod-

erate velocities (due to the limited SMA) translates

into worse force-tracking capabilities. Furthermore,

Fig. 5 shows that VAICAM also enhances the DFC

performance, although with a lower improvement fac-

tor compared to ORACLE. Lastly, it is worth noticing

that VAICAM’s RMSE standard deviation is lower

than DFC’s, thus manifesting a superior robustness

w.r.t. the experimental conditions.

Table 4: Average RMSE of baseline DFC (Roveda and

Piga, 2021), ORACLE (Petrone et al., 2025), and VAICAM

(ours) across trajectories — η indicates the improvement

factor of VAICAM over DFC or ORACLE.

Velocity RMSE [N] η

[m/s] DFC ORACLE VAICAM DFC ORACLE

0.01 2.0579 2.1222 2.0676 0.9953 1.0264

0.05 2.0602 2.1867 1.8917 1.0891 1.1559

0.1 2.1022 2.2118 2.0259 1.0377 1.0918

0.15 2.1563 3.135 2.0528 1.0504 1.5272

0.2 2.199 3.5081 2.0697 1.0624 1.695

0.25 2.3638 3.8424 2.0362 1.1609 1.887

0.3 2.4995 4.3858 2.1098 1.1847 2.0788

0.35 2.6919 4.6601 2.0553 1.3097 2.2673

0.4 2.973 4.883 2.3837 1.2472 2.0485

0.45 3.1873 4.978 2.5117 1.269 1.982

0.5 3.3923 5.4201 2.8123 1.2062 1.9273

To conﬁrm the improvement of VAICAM over

both DFC and ORACLE, Table 4 displays the numer-

ical results by reporting the average RMSE of these 3

strategies for each velocity. As η conﬁrms, VAICAM

surpasses the DFC by up to 31% (at v = 0.35m/s),

while the improvement over ORACLE at maximum

experimental speed (v = 0.5m/s) reaches 127%.

4 CONCLUSIONS

This paper introduced VAICAM, a controller improv-

ing the state-of-the-art of robot-environment interac-

tion tasks that require dynamic trajectories. It makes

use of an FFNN ensemble that acts as a model approx-

imator considering the tangential velocity of the EE.

The controller is based on an impedance controller to

ensure a compliant behavior, and a DFC to guarantee

the desired force tracking characteristics. An ensem-

ble of FFNNs is developed, trained, and tested along

the work. The networks are able to predict the force

that the manipulator will exert on the environment,

allowing the computation of an optimal residual ac-

tion. The latter is then added to the action of a base-

line DFC to improve its force-tracking capabilities, by

correcting the next commanded position. Tested in a

simulated environment, VAICAM outperformed the

baseline DFC (Roveda and Piga, 2021) and a similar

algorithm from recent literature (Petrone et al., 2025).

Possible future improvements of the proposed

strategy are: (i) extending the comparison with other

controllers from the literature, e.g. (Iskandar et al.,

2023); (ii) assessing whether these controllers can

beneﬁt from a MA like the one developed in this

work to improve their performance; (iii) deploying

the algorithm on real hardware, for a complete val-

idation of VAICAM’s performance; (iv) extending

the state space to include tangential position com-

ponents, to tackle variable-stiffness environments;

(v) including the DMA in optimal planning algo-

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

400

rithms foreseeing robot-environment interaction tra-

jectories (Petrone et al., 2022).

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