# A 6 DOF Parallel Mechanism for Initial Echography Training

Mykhailo Riabtsev, Med Amine Laribi, Jean-Michel Guilhem, Victor Petuya, Mónica Urizar, Marc Arsicault, Said Zeghloul

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Mykhailo Riabtsev, Med Amine Laribi, Jean-Michel Guilhem, Victor Petuya, Mónica Urizar, et al. A 6 DOF Parallel Mechanism for Initial Echography Training. the 9th International Workshop on Medical and Service Robots (MESROB), Jul 2025, Poitiers, France. pp.13-22, 10.1007/978-3-031-96081-9_2. hal-05483457

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# A 6 DOF Parallel Mechanism for Initial Echography Training

Mykhailo Riabtsev¹, ² [0000-0002-1003-7080], Med Amine Laribi²[0000-0003-0797-7669], Jean-Michel Guilhem³, Victor Petuya¹[0000-0002-4120-3838], Mónica Urizar¹[0000-0002-8931-6401], Marc Arsicault², Said Zeghloul²[0000-0003-3590-9532]

¹ University of the Basque Country UPV/EHU Plaza Torres Quevedo 1, 48013 Bilbao, Spain

² Department GMSC, Pprime Institute, CNRS - University of Poitiers - ENSMA, UPR 3346, 86073 Poitiers, France

³ Private Practice, Foix, 09000 Toulouse, France
mykhailo.riabtsev@univ-poitiers.fr

**Abstract.** The growing shortage of specialized sonography professionals and the demand for effective training methods have driven the development of various training solutions. This paper presents a cost-effective sonography training system, featuring a 6-degree-of-freedom (6-DOF) parallel Cartesian robot with two identical limbs mounted on a stationary frame. The limbs are connected by a tool rod that utilizes a high-pitched screw mechanism, which facilitates precise vertical actuation of the system. The system incorporates a functional sonography probe for scanning body phantoms simulating pathologies and includes a self-rotation mechanism based on constant velocity joints. The direct and inverse kinematics are analyzed to determine the probe's position and orientation. Offering a realistic scan simulation at a fraction of the cost of traditional solutions, this system addresses the high expense of existing sonography training tools, making it particularly suitable for resource-limited settings and helping to mitigate the shortage of sonography specialists.

**Keywords:** Sonography, Haptic device, Body phantom, Kinematics, Mechanical design.

## 1 Introduction

The issue of an insufficient number of sonography specialists in developed nations has been acknowledged for decades [1], with some regions experiencing this shortage for over 20 years [2]. Moreover, the situation is worsening over time; for instance, in the UK, the proportion of unfilled positions rose from 10.1% in 2009 to 18.1% by 2014 [3]. Similarly, in the US, the vacancy rate increased from approximately 3% in 2013 to nearly 17% in 2023 [4]. This ongoing problem stems from several factors, such as an escalating demand for diagnostic imaging, the retirement of experienced professionals, and an inability to train new specialists in sufficient numbers. Additionally, there is a pressing need for further skill development among practicing sonographers, with many
requiring additional training to meet current demands [5,6]. This creates additional pres-
sure on educational facilities, further reducing their capacity to produce new graduates.

To address these challenges, innovative approaches in training are essential. One
promising solution is the integration of simulation-based systems into sonography edu-
cation. These simulators provide distinct advantages, such as reducing the reliance on
real patients, thereby alleviating the stress of early-stage training and enabling repeated
practice without logistical or ethical limitations.

Several commercially available simulators have been introduced to the market. For
example, Ultrasim, developed by MedSim [7], provides a virtual scanning environment
using a human mannequin, dummy probe, and associated hardware and software to sim-
ulate scans based on the probe's position [8]. However, the mannequin and probe lack
functional capabilities, and the system does not include force feedback to replicate real-
istic tactile sensations. Another system, CAE Vimedix™, features mannequins repre-
senting male and female anatomy, along with supplementary kits for specific scans such
as cardiac or abdominal examinations [9]. While it allows pressure sensing and supports
palpation techniques, the probe does not produce actual scans, relying instead on soft-
ware-generated images.

A distinct approach is taken by ScanTrainer, developed by Intelligent Ultrasound, which replaces physical mannequins with haptic devices and computer simulations [10]. This system incorporates two separate haptic devices for different scanning techniques, providing flexibility in its applications. It offers notable features such as a virtual tutor, haptic feedback that mimics the tactile sensations of real-world scenarios, and a regularly updated library of clinical cases. Studies indicate that students using this system acquire sonography skills more quickly while experiencing fewer psychological barriers often associated with traditional training methods [11]. However, despite its advantages, simulator-based training cannot entirely replace the experience gained from working with real patients.

While these systems have proven effective in improving sonography training, certain
limitations persist. For instance, their cost often places them beyond the reach of many
hospitals and training institutions, particularly in developing regions. Additionally,
some systems fail to provide adequate force feedback, with most being limited to max-
imum forces of 20–30 N. However, real-world scenarios, such as scanning obese pa-
tients, can require forces as high as 50–60 N [12,13], which impacts the realism of the
simulation.

This paper introduces a novel and cost-effective training system designed to address
these limitations, building upon previous studies [14,15]. The proposed system inte-
grates a parallel haptic device, body phantoms designed for pathology simulation, and a
dedicated software package to deliver a more realistic and accessible sonography train-
ing solution. In our earlier work [15], we developed a Cartesian parallel mechanism for
sonography training, offering six degrees of freedom (DOF), with five actively con-
trolled. In that design, the X and Y coordinates of the sonography probe's end, along
with its Euler angles (ψ, φ, θ), could be specified. However, motion along the Z-axis
was left uncontrolled, allowing the operator to manually move the probe vertically. This
decision was made to simplify the design verification process in the prototype phase and
expedite the research. In this paper, we present an enhanced version of the training sys-
tem, which now provides full 6 DOF control.

This paper is structured as follows: Section 2 describes the architecture and kinemat-
ics of the proposed system, Section 3 explains the kinematic model, and Section 4 con-
cludes the study.

## 2 Design and architecture of the mechanism

To achieve full 6 DOF functionality, several modifications were made to the original
mechanism (see Fig. 1). The robot's limbs (1 and 2) were designed to be symmetrical,
both equipped with equal velocity joints (3 and 4) mounted on X-Y carriage assemblies.
These joints handle the self-rotation of the probe (5) as well as its actuation along the Z-
axis. The actuation is facilitated by a high-pitch screw integrated into the probe rod (6).
Due to the high pitch of the screw, it avoids self-locking, enabling forces applied to the
probe to be transmitted back to the corresponding actuators. By regulating the current
in the actuators, the system provides the haptic feedback essential for effective training.
The mechanism incorporates a functional sonography probe. During operation, the
probe is held by a person who performs a specific motion pattern guided by the mecha-
nism while scanning the body phantom (7). Further details about the body phantom can
be found in our previous research [14].

<figure><img src="image_2.png" /><figcaption>Fig. 1. Design of the proposed Cartesian mechanism.</figcaption></figure>

Fig. 2 illustrates the design of a single limb of the proposed device. It consists of a
frame (1), with the Y-axis (2), X-axis (3), and constant velocity joint (4) actuators lo-
cated in its rear section. This configuration minimizes inertia-related forces within the
mechanism. Motion is transmitted via a system of belts and pulleys. The actuation sys-
tem for the Y-axis carriage (5) uses two belts, driven by shafts (6), to pull the carriage
from both sides and prevent jamming. The X-axis carriage (7) and the constant velocity
joint (8) are each controlled by a single belt, which passes through the Y-axis carriage
and a tensioner (9).

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Design of a single limb of the manipulator.</figcaption></figure>

Fig. 3 provides a more detailed depiction of the probe rod. The assembly includes a screw rod (1) connected to a probe holder (2). At the opposite end of the screw rod, a pin (3) transfers torque between the screw rod and a tube (4). This tube features an elongated slot, which allows the pin to move along its longitudinal axis while preventing it from rotating around the tube. A nut (5) is used to control the vertical motion of the probe. Both the tube and the nut are mounted in constant velocity joints (6 and 7, respectively), which enable the transmission of rotational motion from the actuators to the screw rod and nut. If both components rotate by the same angle, the resulting motion of the probe will be self-rotation. However, if only the nut rotates, the probe will move along the longitudinal axis of the probe rod.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Probe rod design.</figcaption></figure>

The kinematic diagram of the manipulator is shown in Fig. 4. In Fig. 4, point C represents the probe's tip, while points A and B are the centers of the limbs' constant velocity joints, defining the probe's position and orientation. The mechanism includes six active joints ($A_1-A_6$) and four passive joints ($P_1-P_4$). Prismatic joints $A_1$ and $A_2$ locate point A along the X and Y axes, respectively, while $A_3$ and $A_4$ perform the same function for point B. Constant velocity joints combine $A_5$ with $P_1$ at point A and $A_6$ with $P_3$ at point B. Finally, $P_2$ enables self-rotation, and $P_3$ (a screw) adjusts vertical positioning by varying the scalar length of vector CA.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Kinematic diagram of the manipulator.</figcaption></figure>

In our previous work [15], Euler angles (ψ, φ, θ) were used to define the probe's ori-
tentation. While intuitive, these angles introduce a mathematical singularity at φ = 0
due to division by zero, though this has no practical impact on the mechanism's opera-
tion. To address this, we adopted Cardan angles (φ₁, φ₂, φ₃) instead. Subsequent anal-
ysis shows that only rotations around the X-axis (φ₁) and Y-axis (φ₂) are required, while
φ₃ = 0, as the probe's longitudinal rotation (ω) is kinematically decoupled from the
Cardan angles.

Fig. 5 provides a vector representation of the mechanism. The vector **CA** defines the axis of the echography probe, aligned with vectors **CB** and **BA**. Point C remains in contact with the X-Y plane, corresponding to the body phantom surface. The orientation of **CA** is determined by the positions of points A and B, while the position of C depends on A and B as well as the scalar length of vector **CA**.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Vectorial diagram of the mechanism: (a) general representation; (b) detailed characterization of points A and B.</figcaption></figure>

The unit vector **n**, shared by vectors **CB**, **BA**, and **CA**, is expressed as:
$$ \mathbf{n} = \frac{\mathbf{CB}}{|\mathbf{CB}|} = \frac{\mathbf{BA}}{|\mathbf{BA}|} = \frac{\mathbf{CA}}{|\mathbf{CA}|} \qquad (1) $$

The precise positions of points A and B depend on their actuation. The mechanism comprises two symmetrical components, each defined by vectors $\mathbf{Z}_A$ and $\mathbf{Z}_B$. Fig. 5b details one component, where the $\mathbf{X}_{(A,B)}$ and $\mathbf{Y}_{(A,B)}$ directions are controlled by actuators $A_X$ and $A_Y$, respectively. These actuators drive a belt transmission system with pulleys of radii $r_X$ and $r_Y$. The pulley rotations, $\alpha_{(A,B)}$ and $\beta_{(A,B)}$, determine the vectors $\mathbf{X}_{(A,B)}$ and $\mathbf{Y}_{(A,B)}$, allowing the positions of points A and B to be calculated as,

$$ \begin{aligned} \mathbf{OA} &= \mathbf{X}_A + \mathbf{Y}_A + \mathbf{Z}_A \\ \mathbf{OB} &= \mathbf{X}_B + \mathbf{Y}_B + \mathbf{Z}_B \end{aligned} \qquad (2) $$

# 3 Kinematic model

## 3.1 Direct kinematic model

Using equations (2), vectors OA and OB, are derived by calculating $\mathbf{X}_A$, $\mathbf{Y}_A$, $\mathbf{X}_B$, and $\mathbf{Y}_B$ as:

$$ \begin{aligned} \mathbf{X}_A &= \left[ \left( \alpha_A + \beta_A \frac{r_y}{r_x} \right) r_x \quad 0 \quad 0 \right]^T \\ \mathbf{Y}_A &= \left[ 0 \quad \beta_A r_y \quad 0 \right]^T \\ \mathbf{X}_B &= \left[ \left( \alpha_B + \beta_B \frac{r_y}{r_x} \right) r_x \quad 0 \quad 0 \right]^T \\ \mathbf{Y}_B &= \left[ 0 \quad \beta_B r_y \quad 0 \right]^T \end{aligned} \qquad (3) $$

Here, $\alpha_A$, $\beta_A$, $\alpha_B$, and $\beta_B$ are actuator angles, and $r_x$ and $r_y$ are pulley radii. Due to the mechanism's design, Y -axis motion is coupled with X -axis actuation, minimizing inertial forces by actuator arrangement.

Probe self-rotation is controlled by actuator angle $\lambda$ and influenced by X and Y motion as:

$$ \omega = \lambda \frac{r_e}{r_s} + \alpha_B \frac{r_x}{r_s} + \beta_B \frac{r_y}{r_x} \qquad (4) $$

Where $r_s$ and $r_e$ are pulley radii of the self-rotation actuator and constant velocity joint, respectively.

With constant Z -coordinates for points A and B, vector BA is:

$$ \mathbf{BA} = [A_x - B_x \quad A_y - B_y \quad A_z - B_z]^T \qquad (5) $$
Knowing that **CA** is collinear with **BA**, the unit vector **n** is:

$$ \|\mathbf{BA}\| = \sqrt{\mathbf{BA}_x^2 + \mathbf{BA}_y^2 + \mathbf{BA}_z^2} \qquad (6) $$
$$ \mathbf{n} = \frac{1}{\|\mathbf{BA}\|} \cdot \mathbf{BA} $$

Scalar **CA** is calculated as:

$$ \|\mathbf{CA}\| = L_i + \frac{(\mu \frac{r_e}{r_s} - \omega)}{2\pi} P \qquad (7) $$

Where $\mu$ is the nut actuator angle, $L_i$ is the initial rod length in the vertical position, and $P$ is the screw pitch. Thus:

$$ \mathbf{CA} = \|\mathbf{CA}\| \cdot \mathbf{n} \qquad (8) $$

Point C coordinates are:

$$ \begin{align} C_x &= A_x - \mathbf{CA}_x \nonumber \\ C_y &= A_y - \mathbf{CA}_y \tag{9} \\ C_z &= A_z - \mathbf{CA}_z \nonumber \end{align} $$

The probe's orientation is derived using the Cardan rotation matrix:

$$ R = X_{\phi_1} \cdot Y_{\phi_2} \cdot Z_{\phi_3} = \begin{bmatrix} \cos \phi_2 \cos \phi_3 & -\cos \phi_2 \sin \phi_3 & \sin \phi_2 \\ \cos \phi_1 \sin \phi_3 + \sin \phi_1 \sin \phi_2 \cos \phi_3 & \cos \phi_1 \cos \phi_3 - \sin \phi_1 \sin \phi_2 \sin \phi_3 & -\sin \phi_1 \cos \phi_2 \\ \sin \phi_1 \sin \phi_3 - \cos \phi_1 \sin \phi_2 \cos \phi_3 & \sin \phi_1 \cos \phi_3 + \cos \phi_1 \sin \phi_2 \sin \phi_3 & \cos \phi_1 \cos \phi_2 \end{bmatrix} \quad (10) $$

The orientation vector **n** is obtained by multiplication of the equation (10) to the vector of the rod in its initial state (vertical):

$$ \mathbf{n} = R \cdot \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \sin \phi_2 \\ -\sin \phi_1 \cos \phi_2 \\ \cos \phi_1 \cos \phi_2 \end{bmatrix} \qquad (11) $$

Equating (6) and (11):

$$ \begin{bmatrix} \sin \phi_2 \\ -\sin \phi_1 \cos \phi_2 \\ \cos \phi_1 \cos \phi_2 \end{bmatrix} = \begin{bmatrix} \frac{A_x - B_x}{\|\mathbf{BA}\|} \\ \frac{A_y - B_y}{\|\mathbf{BA}\|} \\ \frac{A_z - B_z}{\|\mathbf{BA}\|} \end{bmatrix} \qquad (12) $$

Thus:
$$ \phi_2 = asin\left(\frac{A_x - B_x}{\|\mathbf{BA}\|}\right) \qquad (13) $$

Angle $\phi_1$ is derived from the ratio of the X and Y terms in equation (12):

$$ \phi_1 = atan2(B_y - A_y, A_z - B_z) \qquad (14) $$

Angle $\phi_3$ is omitted as it does not affect the probe's kinematics, leaving $\phi_1$ and $\phi_2$ as the primary orientation parameters.

## 3.2 Inverse kinematic model

In the inverse kinematic problem, the location of point C and the angles $\phi_1$, $\phi_2$, and $\omega$ are given.

Using equations (1) and (11), $\|\mathbf{CA}\|$ can be found from their Z -components as,

$$ \|\mathbf{CA}\| = \frac{A_z - C_z}{\cos \phi_1 \cos \phi_2} \qquad (15) $$

The nut actuator angle $\mu$ is then calculated using equation (7):

$$ \mu = \frac{r_s}{r_e} \left( \frac{2\pi(\|\mathbf{CA}\| - L_i)}{P} + \omega \right) \qquad (16) $$

The vector **CA** is obtained as:

$$ \mathbf{CA} = \|\mathbf{CA}\| \cdot \mathbf{n} \qquad (17) $$

With the coordinates of point C and the vector **CA**, the position of point A is determined as:

$$ \begin{aligned} A_x &= C_x + \mathbf{CA}_x \\ A_y &= C_y + \mathbf{CA}_y \end{aligned} \qquad (18) $$

To locate point B, the vector **BA** must first be derived. Its magnitude and direction are determined as:

$$ \begin{aligned} \|\mathbf{BA}\| &= \frac{A_z - B_z}{\mathbf{n}_z} \\ \mathbf{BA} &= \|\mathbf{BA}\| \cdot \mathbf{n} \end{aligned} \qquad (19) $$

The coordinates of point B are then calculated:

$$ \begin{aligned} B_x &= A_x - \mathbf{BA}_x \\ B_y &= A_y - \mathbf{BA}_y \end{aligned} \qquad (20) $$

Once points A and B are located, the vectors **OA** and **OB** are derived. Since the Y - components of vectors **Z**<sub>A</sub>, **Z**<sub>B</sub>, **X**<sub>A</sub>, and **X**<sub>B</sub> in the equations (1) and (2) are zero, the angles $\beta_A$ and $\beta_B$ are determined using equations (4) and (6):
$$ \beta_A = \frac{\mathbf{OA}_y}{r_y} = \frac{A_y}{r_y} \qquad (21) $$

$$ \beta_B = \frac{\mathbf{OB}_y}{r_y} = \frac{B_y}{r_y} $$

The rotation angles of the X-actuators are calculated using the same approach, considering that the X-components of $\mathbf{Z}_A$, $\mathbf{Z}_B$, $\mathbf{Y}_A$, and $\mathbf{Y}_B$ are zero. The angles $\alpha_A$ and $\alpha_B$ are obtained as:

$$ \alpha_A = \frac{\mathbf{OA}_x - \beta_A r_y}{r_x} = \frac{A_x - \beta_A r_y}{r_x} \qquad (22) $$

$$ \alpha_B = \frac{\mathbf{OB}_x - \beta_B r_y}{r_x} = \frac{B_x - \beta_B r_y}{r_x} $$

Finally, the position of the self-rotation actuator is determined using equation (7):

$$ \lambda = \left( \omega - \left( \alpha_B \frac{r_x}{r_s} + \beta_B \frac{r_y}{r_x} \right) \frac{r_s}{r_e} \right) \qquad (23) $$

## 4 Conclusions

In this paper, a cost-effective sonography training system was presented. This innovative system incorporates a 6-degree-of-freedom parallel robot, as a haptic device using body phantoms for simulating pathologies.

The mechanism design, including its components and kinematics, has been described in details. The solutions of the direct and inverse kinematic problems were obtained.

The proposed cost-effective sonography training solution offers a more affordable alternative to existing systems, potentially bridging the gap in specialized sonography expertise. This system can improve training outcomes and provide a realistic scan representation, making it a valuable tool for both medical professionals and students. Future directions include the development of a prototype and comprehensive testing to validate the system's functionality and its potential impact on improving the training and expertise of sonography professionals worldwide.

## Acknowledgements

The authors wish to acknowledge the financial support received from the Programa Posdoctoral de Perfeccionamiento de Personal Investigador Doctor and the Project IT1480-22 provided by the Regional Basque Government. Also, the funding received from the Spanish Government through the Ministerio de Ciencia e Innovación (Project PID2020-116176GB-100) financed by MCIN/AEI/ 10.13039/501100011033.
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