A Cartesian Parallel Mechanism for Initial Sonography Training

Abstract

This paper presents the development and analysis of a novel 6-DOF Cartesian parallel mechanism intended for use as a haptic device for initial sonography training. The system integrates a manipulator designed for delivering force feedback in five degrees of freedom; however, in the current stage, only mechanical architecture and kinematic validation have been conducted. Future enhancements will focus on implementing and evaluating closed-loop force control to enable complete haptic feedback. To assess the kinematic performance of the mechanism, a detailed kinematic model was developed, and both the Kinematic Conditioning Index (KCI) and Global Conditioning Index (GCI) were computed to evaluate the system’s dexterity. A trajectory simulation was conducted to validate the mechanism’s movement, using motion patterns typical in sonography procedures. Quasi-static analysis was performed to study the transmission of force and torque for generating realistic haptic feedback, critical for simulating real-life sonography. The simulation results showed consistent performance, with dexterity and torque distribution confirming the suitability of the mechanism for haptic applications in sonography training. Additionally, structural analysis verified the robustness of key components under expected loads. In order to validate the proposed design, the prototype was constructed using a combination of aluminum components and 3D-printed ABS parts, with Igus^® linear guides for precise motion. The outcomes of this study provide a foundation for the further development of a low-cost, effective sonography training system.

1. Introduction

The current shortage of sonography specialists or the insufficient skills in this area is widely recognized in the medical community [1,2]. One of the proposed solutions is the implementation of tele-robotics, which allows healthcare professionals to provide diagnostic services to patients remotely. In such systems, the sonographer can participate in the examination via an online platform, allowing communication with the patient and remote control of the ultrasound machine. Some tele-robotic systems (see Figure 1), such as MELODY [3], Medirob [4,5], ROSE [6], and MGIUS-R3 [7,8] are already commercially available.

Beyond commercially available systems, significant research has been dedicated to the development and optimization of tele-echography manipulators. For instance, Paper [9] introduces an optimization process for a 4-DOF manipulator, proposing a structure specifically tailored for tele-ultrasound applications. The design aims to minimize singularities near the boundaries of the workspace, thereby enhancing control and stability. Building on this, ref. [10] explores the use of a 3-DOF spherical manipulator, optimized using Genetic Algorithms. This approach improves the manipulator’s performance as a slave unit in a tele-echography system, demonstrating that structural and algorithmic refinements can synergistically improve system responsiveness and precision.

In terms of system-level development, the HaptiScan platform presented in [11] exemplifies an integrated tele-echography solution, incorporating a UR5 robot on the patient side and a Phantom Omni haptic device on the operator side. While promising in concept, this system has yet to undergo experimental validation, leaving its clinical applicability unconfirmed. Complementing these individual system designs, a broader overview is provided in [12], which reviews existing tele-robotic echography solutions based on real-world applications. The review generally reports favorable outcomes and concludes that tele-ultrasound can match the diagnostic quality of conventional ultrasound. However, it also underlines persistent challenges, including high initial deployment costs, the necessity for operator training, and reliance on robust internet connectivity for consistent performance.

Further advancing the field, ref. [13] presents an autonomous ultrasound robotic system capable of integration into tele-robotic configurations. Central to this system is a 7-DOF serial manipulator outfitted with a force/torque sensor embedded in the probe holder. An accompanying image analysis algorithm enables semi-autonomous scanning. Experimental results using a body phantom indicate that the system can produce clinically relevant, high-quality images with a rapid response time, highlighting the feasibility of automation in remote diagnostic workflows.

A different strategy to mitigate the shortage of sonography specialists is to enhance the training process. Several robotic training systems are available on the market. Ultrasim (see Figure 2a), developed by MedSim, is a training simulator for ultrasound procedures that provides a broad range of simulated scans and evaluates the learning process [14]. The system includes a mannequin resembling the human body, a dummy probe, and hardware and software that generate a virtual scan based on the probe’s position [15]. Since neither the mannequin nor the probe can replicate real scanning functions, the entire imaging process is software-based, and there is no force feedback simulating the real tactile experience due to the absence of a haptic interface.

Among mannequin-based ultrasound training simulators, CAE Vimedix™ (see Figure 2b) and Ultrasound Mentor (see Figure 2c) represent comprehensive commercial solutions. Vimedix™ is available in both male and female versions, with modular kits for various scan types such as cardiac, abdominal, and gynecologic exams [16]. Its mannequins offer realistic tactile feedback, including palpable bones in the head and torso, and are equipped with pressure-sensitive probes and motion tracking for spatial positioning. In contrast, the Ultrasound Mentor system developed by Surgical Science follows a similar physical configuration but lacks haptic feedback in the probe [17]. Instead, it emphasizes software-driven learning with a computer-based simulation and a virtual advisor that guides probe positioning. Both systems provide structured training environments, though Vimedix offers a more immersive tactile experience, while Ultrasound Mentor focuses on instructional support through digital feedback.

Shifting away from physical mannequins, a distinct group of simulators emphasizes haptic and virtual simulation technologies. ScanTrainer by Intelligent Ultrasound eliminates the need for full-body phantoms, combining a haptic interface with advanced software that simulates anatomical responses to probe manipulation [18]. Its core features include a virtual tutor, context-aware force feedback aligned with the simulated pathology, and an extensive clinical case library. The system has been shown to accelerate skill acquisition and reduce psychological barriers associated with traditional methods [19]. Similarly, GynoS™ by VirtaMed targets gynecological ultrasound education using a partial pelvic mannequin with virtual imaging and haptic-enabled probes [20]. Its effectiveness has been validated in [21], showing improved accuracy, confidence, and diagnostic capability among students compared to conventional training. Both systems demonstrate how virtual and haptic simulations can enhance realism and learning efficiency without the logistical complexity of full mannequins.

Complementing these high-end simulators are systems designed to support specific procedures or provide accessible alternatives. For example, the Perk Tutor introduced in [22] focuses on needle insertion training using a body phantom, tracked instruments, and a feedback-enabled software interface. It offers the advantage of quantitative data collection during training, enhancing learner evaluation. In contrast, study [23] presents a low-cost solution using a smartphone as a dummy ultrasound probe connected to a PC that emulates the ultrasound console. Although this configuration yielded positive results in experimental trials, the absence of force feedback limits its applicability for tactile skill development. These examples illustrate the spectrum of design priorities in ultrasound simulation—from procedural precision to cost-effective accessibility.

In the realm of remote instruction, tele-training has gained traction as a viable alternative to in-person mentorship. Study [24] describes the use of a standard Lumify Ultrasound system, where a probe connected to a smartphone transmits real-time scan images for remote supervision. The experimental results revealed no significant difference in training outcomes compared to traditional teaching, highlighting tele-sonography’s potential to democratize ultrasound education. Similarly, an innovative, self-directed simulator presented in [25] employs a 3D-printed probe with a stylus pencil, interacting with a touchpad in conjunction with a custom training game called “UnderWater.” Tested with 42 participants, this system demonstrated strong user engagement and skill development, underscoring the value of interactive, low footprint learning platforms.

Beyond simulation and instruction, recent research has explored autonomous ultrasound systems as a long-term solution to workforce shortages. Study [26] proposes an intelligent robotic platform that autonomously identifies and scans target anatomical regions using a UR5e 6-DOF robotic arm, guided by an artificial neural network trained on expert-provided images. The system was validated experimentally and achieved high-quality imaging with precise probe positioning and force control. A similar autonomous framework is introduced in [27], which combines skeletal reference point recognition, real-time force feedback, and reinforcement learning to conduct fully autonomous echography scans. In most cases, diagnostic results closely matched those of experienced clinicians, pointing to the growing viability of autonomous ultrasound systems in clinical practice.

Finally, a broader review of the role of simulation in ultrasound education is provided in [28], which consolidates evidence across multiple platforms and methodologies. The findings consistently affirm that simulation-based training enhances clinical performance, accelerates skill development, and improves learner confidence, while simultaneously reducing the burden on healthcare facilities and instructors. Collectively, the reviewed systems reflect the diversity and evolution of ultrasound education technologies, from tactile mannequins and haptic simulations to low-cost tools, tele-training models, and autonomous platforms.

This paper proposes a concept for a mechanism intended to serve as a haptic device within a sonography training system. For effective training, it is essential to establish correct movement patterns for future specialists. To achieve this, the system incorporates a haptic device designed to accommodate various ultrasound probes, enabling indication of incorrect maneuvers through force feedback. The device provides three rotational and three translational degrees of freedom (3R DOF and 3T DOF), five of which are capable of generating force feedback to correct the operator’s movements (excluding the Z direction). In the future it is planned to implement haptic feedback in all 6 DOF. The proposed device is intended to be used with the body phantoms that are designed to represent vascular structures with realistic wall thicknesses and adjustable depths. Although the Z-axis is not actively actuated in the current prototype, realistic resistance along this axis is achieved through the use of a silicone-based body phantom, which provides force feedback representative of actual tissue contact. This setup allows meaningful simulation of probe–tissue interactions even in the absence of active vertical force control.

To clarify how the proposed system advances beyond existing tele-sonography and training platforms, we offer a comparative analysis focusing on functionality, cost, and design versatility. A comparative overview of the proposed mechanism alongside existing tele-robotic sonography systems is presented in

Table 1. Comparison of the proposed system with existing tele-robotic sonography platforms.

System	Number of DOF		Master Type	Haptic Feedback	Slave Device	Master Architecture	Master Sensors Type	Approximate Cost, Euros
	Master	Slave
Proposed device	5 + 1 *	-	This device	Intended 5 DOF, 60 N	-	Parallel Cartesian	Torque, position	Under 10k
MELODY	6	3 + 3 **	Mock probe	None	Custom mechanism	-	Position, orientation	No data
Medirob	6	6	3D mouse	None	Serial robot	-	Position, orientation	No data
ROSE	6	6	Virtuose Desktop	6 DOF, 10 N (35 N max)	Virtuose 6D	Serial	Force, torque, position	>100k
MGIUS-R3	6	6	Robotic US probe	None	UR 5 serial robot	-	pressure, position, orientation	~150k–200k
HaptiScan	6	6	Phantom Omni	3 DOF, 3.3 N	UR 5 serial robot	Serial	Force, torque, position	50k–100k

* One DOF is passive. ** 3 DOF are provided by human assistance.

, while a corresponding comparison with current training platforms is provided in

Table 2. Comparison of the proposed system with sonography training simulators.

System	Haptic Device	Haptic Feedback	Simulation Technology	Approximate Cost, Euros	Sonography Area
Proposed device	+	Intended 5 DOF *, 60 N	Haptic + Body phantom + software	Under 10k	Abdominal, musculoskeletal, vascular
MedSim	None	-	Mannequin + software	~100k	Abdominal, gynecologic, cardiac, vascular
CAE VimedixTM	None	-	Mannequin + software	~100k	Abdominal, gynecologic, cardiac, vascular
Ultrasound Mentor	None	-	Mannequin + software	50k–100k	Abdominal, gynecologic, cardiac, vascular
ScanTrainer	Phantom	3 DOF, 3.3 N	Software	20k–90k	Abdominal, gynecologic, vascular
GynoSTM	+ (No data)	No data	Software	~100k	Gynecologic
Perk Tutor	None	-	Body phantom + software	Low (No data)	US-guided needle insertion
Persimus	None	-	Smartphone + software	Low (No data)	Abdominal, musculoskeletal, vascular
“UnderWater”	None	-	Stylus pencil + Software	Low (No data)	US-guided needle insertion

* With future extension to 6 DOF.

.

Unlike several established systems such as MELODY, Medirob, and MGIUS-R3, which either lack haptic feedback entirely or incorporate only limited force reflection, the proposed device is intended to provide full 5 DOF haptic feedback—a significant advancement in interactivity and training realism. Moreover, the system is designed with future extensibility to 6 DOF, aligning it with high-end platforms like ROSE and HaptiScan, but at a fraction of the cost.

The maximum force output of 60 N provides a significant advantage over many existing training and tele-sonography systems. For instance, commonly used haptic devices such as the Phantom Omni or ScanTrainer are limited to peak forces below 4 N, while some robotic arms integrated into commercial systems generate maximum forces in the range of 20–30 N. However, clinical sonography procedures—particularly in abdominal examinations—may require forces up to 50–60 N to ensure adequate tissue contact, especially in patients with higher body mass index [29]. Additionally, motion capture studies of clinical sonography have quantified typical probe orientation angles during abdominal and carotid artery scans. For abdominal examinations, average Euler angles of approximately ψ = 43°, θ = 17°, and φ = 98° have been reported, while carotid scans involve slightly higher inclinations (ψ = 67°, θ = 11°, φ = 92°) [30]. Systems that fail to replicate these requirements may struggle to simulate realistic probe–tissue interaction or perform effective scans in such conditions. The higher force capability of the proposed device thus enhances tactile realism and extends its applicability across a wider patient demographic.

Additionally, the proposed system is intended to be multi-functional, supporting both training and potential future tele-sonography applications. This 2-in-1 design not only reduces costs but also enhances adaptability. In contrast, most commercial platforms are strictly partitioned into either clinical or educational use, leading to duplicated hardware investments.

Cost-efficiency in the proposed system is achieved through several deliberate design decisions. The use of a Cartesian parallel kinematic structure reduces mechanical complexity and moving mass, while base-mounted, belt-driven actuators lower both inertia and maintenance demands. Though belt transmission introduces some elasticity, this is negligible for sonographic applications, which do not require sub-millimeter precision. Notably, belt-driven actuation has also proven effective in prior haptic systems—for example, the high-transparency, high-force interface [31] and the compact joystick mechanism [32], both of which demonstrate that belt-based designs can offer a favorable balance between mechanical simplicity and functional haptic performance. In addition, the incorporation of a body phantom—smaller and less expensive than mannequin-based setups or high-end simulation software—further reduces hardware size and cost. The system’s sensing approach, combining motor-side torque estimation with absolute encoders, provides adequate feedback resolution while avoiding the expense of high-end force–torque sensors used in systems like ROSE or MGIUS-R3.

Finally, the proposed platform supports a wide range of sonographic applications, including abdominal, musculoskeletal, and vascular fields—broader than GynoS (gynecologic only) and more cost-effective than systems like CAE Vimedix or MedSim, which exceed €100,000 and still do not offer active haptic feedback.

Although the mechanical structure supports haptic interaction, active force-feedback control is not yet implemented. The system currently permits passive interactions, and no quantitative validation of force-feedback performance has been performed at this stage. Future iterations will incorporate a closed-loop haptic feedback interface and sonographic simulation software to enable a full training experience.

The structure of the paper is as follows: Section 2 provides a detailed description of the system components; Section 3 presents the solution of the direct and inverse kinematic problems; Section 4 outlines the kinematic analysis of the mechanism; Section 5 demonstrates the quasi-static analysis; Section 6 is dedicated to the kinematic verification via trajectory simulation; Section 7 is focused on the structural test of the main elements of the device; in Section 8 the prototype manufacturing process is described; and Section 9 concludes the study.

2. Design and Architecture of the Mechanism

The proposed mechanism is a 6-DOF (Degrees of Freedom) parallel robot (see Figure 3) with 5 active DOF comprising two asymmetrical Cartesian limbs (1 and 2), which are attached to a fixed frame (3). These limbs position an equal velocity joint (4) and a spherical joint (5), which together define the position and orientation of the probe (6). The probe is mounted on a rod (7) that passes through these joints. This setup enables the probe to scan a body phantom (8), which is positioned at the center of the robot’s workspace.

In this haptic device, the orientation angles of the end-effector as well as its displacements along X and Y axes are actively controlled. The Z displacement is passive in the current design iteration but will be added in the future after practical verification of the chosen technical solutions. Currently, the Z displacement is controlled manually by the user, which results in variations in the length of the probe rod. To monitor these changes, an encoder (9) has been integrated into the mechanism. The encoder has a spring-loaded winch with the cable one end of which is attached near the probe. As the probe is able to move along the longitudinal axis of the rod, the length of the cable attached to it will change, and this difference will be registered by the encoder. This measurement is necessary to determine the real position of the probe end.

In Figure 4, the design of a single Cartesian limb is presented. The limbs share a similar structure, differing only in the type of joint in the center of the X carriage (ball joint in the bottom limb and equal velocity joint in the top limb). Each limb comprises a frame (1), which houses the actuators (2, 3, and 4) on its rear side. Actuator (2) controls the displacement of the Y carriage (5), while actuator (3) drives the X carriage (6), both via corresponding belt transmissions. At the center of the X carriage, either a spherical joint (7) or an equal velocity joint is installed. In the current configuration, the equal velocity joint is installed only on the upper limb of the mechanism and is actuated by motor 4 (self-rotation actuation). To enable active control of vertical (Z-axis) motion in future iterations, equal velocity joints will be incorporated into both limbs. However, the corresponding actuators have already been installed in anticipation of this upgrade, facilitating a smoother transition to the advanced version of the mechanism.

As shown in Figure 4, belt drives are used to actuate the X and Y carriages as well as the equal velocity joint. For Y-axis actuation, the belt from motor 2 passes through pulleys on two equalizing shafts (the first marked in red), is fixed to a tensioner on the Y carriage, and then returns to the motor. On the opposite side, the belt loops through the shaft pulleys and is fixed to another carriage-mounted tensioner. This dual-belt configuration prevents jamming caused by the carriage’s length.

The X-axis actuation starts from motor 3, where the belt passes through a tensioner (magenta in Figure 4), runs through pulleys on the Y carriage, passes a fixation point on the X carriage, and returns to the motor. The equal velocity joint is actuated similarly: the belt from motor 4 runs beneath the X-axis belt and drives the joint via a pulley, as shown in Figure 5.

Due to the routing, X actuation depends on Y carriage motion, and equal velocity joint actuation depends on both X and Y positions.

The equal velocity joint is illustrated in Figure 5. It is based on a KGLM-10 spherical joint manufactured by the Igus^® company. For the current application, the joint was modified by cutting two symmetrical slots (1) (only one is visible in Figure 5) to accommodate pins (2) with spherical ends, which transmit torque from the pulley (3) to the sphere (4) of the joint. The modified joint is fixed inside the pulley using epoxy glue, with additional pressure applied by a fixation ring (5). The section of the probe rod that contacts the joint’s sphere is also secured with glue. This connection is sufficient, as high torque transmission is not required for haptic applications.

The kinematic representation of the mechanism is shown in Figure 6.

The mechanism consists of four prismatic active joints ($A_1$ to $A_4$) and a revolute active joint ($A_5$), which, together with a passive joint ($P_1$), forms an equal velocity joint (see Figure 6b). Points A and B represent the centers of the passive joints $P_1$ and $P_3$, respectively. The probe rod extends through points A and B, terminating at point C, which marks the tip of the probe.

The position and orientation of the probe are controlled by the active joints $A_1$–$A_4$, which adjust the positions of points A and B. The revolute joint $A_5$ is responsible for the rotational motion of the probe, which, while designed to operate independently of other movements, is nonetheless influenced by them due to the chosen design solutions. The passive joints $P_2$ and $P_4$ accommodate variations in the length of vector CA during operation and are integrated into the probe rod in the prototype design.

In our previous work [33], we used Euler angles ($\psi , \phi , \theta$) to define the orientation of the probe. Although these parameters offer a more intuitive method of operation, the kinematic analysis revealed that they introduce a mathematical singularity when $\phi =0$. This issue is not a mechanical singularity of the system, but rather a numerical instability inherent to the Euler representation, resulting in ambiguity or discontinuity in derivative computations. To eliminate this issue, we opted to use Cardan angles (${\varphi }_1$, ${\varphi }_2$, ${\varphi }_3$) instead (see Figure 7), which provide a more stable and interpretable orientation framework for the analysis. Indeed, our Kinematic Conditioning Index (KCI) analysis presented in Section 4.2 shows that this configuration corresponds to a position of maximum dexterity, and no practical performance issues were observed during experimental testing. In the subsequent calculations, it will be seen that only the rotations around X axis (angle ${\varphi }_1$) and the rotation around Y axis (angle ${\varphi }_2$), are used (angle ${\varphi }_3=0$), as the rotation around the longitudinal axis of the probe (angle $\omega$) is kinematically decoupled from Cardan angles.

A vector representation of the mechanism is provided in Figure 8.

In Figure 8a, vector CA defines the axis along which an echography probe is mounted, aligning with both vectors CB and BA. Since point C must remain in contact with the $X-Y$ plane—corresponding to the body phantom surface being examined—the orientation of vector CA is determined by the positions of points A and B. If point C loses contact with this plane, the scalar length of vector CA becomes a necessary parameter for fully describing the vector (it is implemented with an encoder in the prototype design). Vector n in the diagram is a unit vector that is the same for the vectors CB, BA, and CA and can be found as the following:

$$\mathbf{n}={\displaystyle \frac{\mathbf{C}\mathbf{B}}{\left|\mathbf{C}\mathbf{B}\right|}}={\displaystyle \frac{\mathbf{B}\mathbf{A}}{\left|\mathbf{B}\mathbf{A}\right|}}={\displaystyle \frac{\mathbf{C}\mathbf{A}}{\left|\mathbf{C}\mathbf{A}\right|}}.$$

(1)

To determine the precise positions of points A and B, it is essential to account for their actuation. The mechanism is composed of two symmetrical components, each characterized by the vectors ${\mathbf{Z}}_{\mathbf{A}}$ and ${\mathbf{Z}}_{\mathbf{B}}$. Figure 8b provides a detailed vector representation of one of these components, where the ${\mathbf{X}}_{(\mathbf{A},\mathbf{B})}$ and ${\mathbf{Y}}_{(\mathbf{A},\mathbf{B})}$ directions are governed 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 rotational positions of the pulleys, denoted as ${\alpha }_{(A,B)}$ and ${\beta }_{(A,B)}$, define the vectors ${\mathbf{X}}_{(\mathbf{A},\mathbf{B})}$ and ${\mathbf{Y}}_{(\mathbf{A},\mathbf{B})}$. Consequently, the locations of points A and B can be derived as

$$\begin{gathered} \mathbf{O}\mathbf{A}={\mathbf{X}}_{\mathbf{A}}+{\mathbf{Y}}_{\mathbf{A}}+{\mathbf{Z}}_{\mathbf{A}}, \\ \mathbf{O}\mathbf{B}={\mathbf{X}}_{\mathbf{B}}+{\mathbf{Y}}_{\mathbf{B}}+{\mathbf{Z}}_{\mathbf{B}}. \end{gathered}$$

(2)

3. Kinematic Model

3.1. Direct Kinematic Model

Based on Equations (2), vectors $\mathbf{O}\mathbf{A}$ and $\mathbf{O}\mathbf{B}$, can be determined by first finding vectors ${\mathbf{X}}_{\mathbf{A}}$, ${\mathbf{Y}}_{\mathbf{A}}$, ${\mathbf{X}}_{\mathbf{B}}$, and ${\mathbf{Y}}_{\mathbf{B}}$ as follows:

$$\begin{gathered} {\mathbf{X}}_{\mathbf{A}}={\left[\begin{array}{ccc}\left({\alpha }_A+{\beta }_A{\displaystyle \frac{r_y}{r_x}}\right)r_x& 0& 0\end{array}\right]}^T, \\ {\mathbf{Y}}_{\mathbf{A}}={\left[\begin{array}{ccc}0& {\beta }_A{r}_y& 0\end{array}\right]}^T, \\ {\mathbf{X}}_{\mathbf{B}}={\left[\begin{array}{ccc}\left({\alpha }_B+{\beta }_B{\displaystyle \frac{r_y}{r_x}}\right)r_x& 0& 0\end{array}\right]}^T, \\ {\mathbf{Y}}_{\mathbf{B}}={\left[\begin{array}{ccc}0& {\beta }_B{r}_y& 0\end{array}\right]}^T. \end{gathered}$$

(3)

Here, ${\alpha }_A$, ${\beta }_A$, ${\alpha }_B$, and ${\beta }_B$ denote the rotation angles of the respective actuators, while $r_x$ and $r_y$ are the radii of the pulleys for the $X$ and $Y$ actuators, respectively. The $Y$-axis actuation is coupled with the $X$-axis due to the mechanism’s design, where all actuators are mounted on the frame to minimize inertial forces.

The probe self-rotation $\nu$ is implemented via an equal velocity joint and is directly controlled by its actuator angle $\lambda$. However, it is influenced by the motions along the $X$ and $Y$ axes, given by:

$$\nu =\lambda {\displaystyle \frac{r_e}{r_s}}+{\alpha }_B{\displaystyle \frac{r_x}{r_s}}+{\beta }_B{\displaystyle \frac{r_y}{r_x}},$$

(4)

where $r_s$ is the radius of the self-rotation actuator pulley and $r_e$ is the radius of equal velocity joint pulley.

Since the Z-coordinates of points $A$ and $B$ are known and remain constant, vector $\mathbf{B}\mathbf{A}$ can be expressed as:

$$\mathbf{B}\mathbf{A}={\left[\begin{array}{ccc}{A}_x{-B}_x& A_y{-B}_y& A_z-B_z\end{array}\right]}^T.$$

(5)

Given that the tool vector $\mathbf{C}\mathbf{A}$ is collinear with vector $\mathbf{B}\mathbf{A}$, the unit vector $\mathbf{n}$ is defined as:

$$\begin{gathered} \left|\mathit{B}\mathit{A}\right|=\sqrt{{BA}_x^2+{BA}_y^2+{BA}_z^2}, \\ \mathbf{n}={\displaystyle \frac{1}{\left|\mathit{B}\mathit{A}\right|}}·\mathbf{B}\mathbf{A}. \end{gathered}$$

(6)

The vector $\mathbf{C}\mathbf{A}$ can be found using the scalar length $\left|\mathit{C}\mathit{A}\right|$ provided by encoder readings:

$$\mathbf{C}\mathbf{A}=\left|\mathit{C}\mathit{A}\right|·\mathbf{n}.$$

(7)

The coordinates of point C are then calculated using vector components:

$$\begin{gathered} C_x=A_x-{CA}_x, \\ {C}_y=A_y-{\mathrm{C}\mathrm{A}}_y, \\ {C}_z=A_z-{\mathrm{C}\mathrm{A}}_{\mathrm{z}}. \end{gathered}$$

(8)

The rotation matrix for the Cardan angles can be written as follows:

$$\begin{gathered} \mathrm{R}={\mathrm{X}}_{{\mathsf{\varphi }}_1}{·\mathrm{Y}}_{{\mathsf{\varphi }}_2}·{\mathrm{Z}}_{{\mathsf{\varphi }}_3}= \\ \left[\begin{array}{ccc}\mathit{cos}{\varphi }_2\mathit{cos}{\varphi }_3& -\mathit{cos}{\varphi }_2\mathit{sin}{\varphi }_3& \mathit{sin}{\varphi }_2\\ \mathit{cos}{\varphi }_1\mathit{sin}{\varphi }_3+\mathit{sin}{\varphi }_1\mathit{sin}{\varphi }_2\mathit{cos}{\varphi }_3& \mathit{cos}{\varphi }_1\mathit{cos}{\varphi }_3-\mathit{sin}{\varphi }_1\mathit{sin}{\varphi }_2\mathit{sin}{\varphi }_3& -\mathit{sin}{\varphi }_1\mathit{cos}{\varphi }_2\\ \mathit{sin}{\varphi }_1\mathit{sin}{\varphi }_3-\mathit{cos}{\varphi }_1\mathit{sin}{\varphi }_2\mathit{cos}{\varphi }_3& \mathit{sin}{\varphi }_1\mathit{cos}{\varphi }_3+\mathit{cos}{\varphi }_1\mathit{sin}{\varphi }_2\mathit{sin}{\varphi }_3& \mathit{cos}{\varphi }_1\mathit{cos}{\varphi }_2\end{array}\right]. \end{gathered}$$

(9)

The orientation of the probe of the haptic device can be obtained with multiplication of the probe initial position vector by the rotation matrix $R$ and can be calculated as

$$\mathbf{n}=\mathrm{R}·\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}\mathit{sin}{\varphi }_2\\ -\mathit{sin}{\varphi }_1\mathit{cos}{\varphi }_2\\ \mathit{cos}{\varphi }_1\mathit{cos}{\varphi }_2\end{array}\right].$$

(10)

Using Equations (6) and (10) we obtain

$$\left[\begin{array}{c}\mathit{sin}{\varphi }_2\\ -\mathit{sin}{\varphi }_1\mathit{cos}{\varphi }_2\\ \mathit{cos}{\varphi }_1\mathit{cos}{\varphi }_2\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{A_x{-B}_x}{\left|\mathit{B}\mathit{A}\right|}}\\ {\displaystyle \frac{A_y{-B}_y}{\left|\mathit{B}\mathit{A}\right|}}\\ {\displaystyle \frac{A_z-B_z}{\left|\mathit{B}\mathit{A}\right|}}\end{array}\right].$$

(11)

From here, it can be seen that

$${\varphi }_2=asin\left({\displaystyle \frac{A_x{-B}_x}{\left|\mathit{B}\mathit{A}\right|}}\right).$$

(12)

Finally, angle ${\mathit{\varphi }}_1$ is determined by the ratio of the $\mathit{X}$ and $\mathit{Y}$ components of Equation (11):

$${\varphi }_1=atan2\left(B_y-A_y,A_z-B_z\right).$$

(13)

Now, the angles ${\varphi }_1$ and ${\varphi }_2$ are determined. As can be seen from Equation (10) after the multiplication only the third column of the rotation matrix, presented in Equation (9) remains. This column does not contain the angle ${\varphi }_3$, thus it is not used in the calculations.

3.2. Inverse Kinematic Model

In the inverse kinematic problem, the coordinates of point C and the angles ${\varphi }_1$, ${\varphi }_2$, and $\omega$ are known. Since the magnitude of $\left|\mathit{C}\mathit{A}\right|$ is provided by encoder readings, the vector $\mathbf{C}\mathbf{A}$ can be determined using the corresponding equation:

$$\mathbf{C}\mathbf{A}=\left|\mathit{C}\mathit{A}\right|·\mathbf{n}.$$

(14)

With the known coordinates of point C and the vector $\mathbf{C}\mathbf{A}$, the location of point A can be obtained as

$$\begin{gathered} A_x={C_x+CA}_x, \\ {A}_y=C_y+{\mathrm{C}\mathrm{A}}_y. \end{gathered}$$

(15)

To find the coordinates of point $B$, the vector $\mathbf{B}\mathbf{A}$ must first be derived, which involves determining its magnitude and direction.

$$\begin{gathered} \left|\mathit{B}\mathit{A}\right|={\displaystyle \frac{A_z-B_z}{{\mathit{n}}_{\mathit{z}}}}, \\ \mathbf{B}\mathbf{A}=\left|\mathit{B}\mathit{A}\right|·\mathbf{n}. \end{gathered}$$

(16)

Now, the location of point B can then be calculated:

$$\begin{gathered} B_x=A_x-{\mathrm{B}\mathrm{A}}_{\mathrm{x}}, \\ {B}_y=A_y-{BA}_y. \end{gathered}$$

(17)

Next, the coordinates of the points determined, vectors $\mathbf{O}\mathbf{A}$ and $\mathbf{O}\mathbf{B}$ can be found. In Equations (1) and (2), the $Y$ components of vectors ${\mathbf{Z}}_{\mathbf{A}}$, ${\mathbf{Z}}_{\mathbf{B}}$, ${\mathbf{X}}_{\mathbf{A}}$, and ${\mathbf{X}}_{\mathbf{B}}$ are zero. Using the relevant Equations (4) and (6), ${\beta }_A$ and ${\beta }_B$ can be found as

$$\begin{gathered} {\beta }_A={\displaystyle \frac{{OA}_y}{r_y}}={\displaystyle \frac{A_y}{r_y}}, \\ {\beta }_B={\displaystyle \frac{{OB}_y}{r_y}}={\displaystyle \frac{B_y}{r_y}}. \end{gathered}$$

(18)

The rotation angles of the $X$ actuators are found similarly, considering that the $X$ components of vectors ${\mathbf{Z}}_{\mathbf{A}}$, ${\mathbf{Z}}_{\mathbf{B}}$, ${\mathbf{Y}}_{\mathbf{A}}$, and ${\mathbf{Y}}_{\mathbf{B}}$ are zero. The angles of rotation of the $X$ actuators at points $A$ and $B$ are then determined as

$$\begin{gathered} {\alpha }_A={\displaystyle \frac{{OA}_x-{\beta }_A{r}_y}{r_x}}={\displaystyle \frac{A_x-{\beta }_A{r}_y}{r_x}}, \\ {\alpha }_B={\displaystyle \frac{{OB}_x-{\beta }_B{r}_y}{r_x}}={\displaystyle \frac{B_x-{\beta }_B{r}_y}{r_x}}. \end{gathered}$$

(19)

Finally, using Equation (7), the position of the self-rotation actuator is calculated:

$$\lambda =\left( \nu -\left({\alpha }_B{\displaystyle \frac{r_x}{r_s}}+{\beta }_B{\displaystyle \frac{r_y}{r_x}}\right)\right){\displaystyle \frac{r_s}{r_e}}.$$

(20)

4. Kinematic Analysis

4.1. Jacobian Characterization

In order to evaluate the kinematic behavior of the mechanism, the corresponding analysis was performed. By performing de partial derivatives from Equations (4), (8), (12) and (13), the Jacobian matrix of Equation (21) is obtained:

$$\mathrm{J}=\left[\begin{array}{cccccc}{J}_{v11}& J_{v12}& J_{v13}& J_{v14}& J_{v15}& 0\\ J_{v21}& J_{v22}& J_{v23}& J_{v24}& J_{v25}& 0\\ J_{v31}& J_{v32}& J_{v33}& J_{v34}& J_{v35}& 0\\ J_{\omega 11}& J_{\omega 12}& J_{\omega 13}& J_{\omega 14}& 0& 0\\ J_{\omega 21}& J_{\omega 22}& J_{\omega 23}& J_{\omega 24}& 0& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right].$$

(21)

To simplify the Jacobian, the input variables were chosen as $A_x$, $A_y$, $B_x$, $B_y$, $\left|\mathit{C}\mathit{A}\right|$, and $\lambda$, while the output variables were $C_x$, $C_y$, $C_z$, ${\varphi }_1$, ${\varphi }_2$, and $\nu$. The Jacobian matrix in Equation (21) is structured accordingly, ensuring a square form with six input and six output variables. The first three rows of the Jacobian matrix represent the partial derivatives of the position equations. Since these equations do not include a self-rotation component, the last column of these rows contains zeros. The fourth and fifth rows correspond to the orientation equations, which are also independent of self-rotation and the length of vector $\left|\mathit{C}\mathit{A}\right|$, leading to zeros in the last two columns.

The self-rotation of the tool is kinematically decoupled from the other motions due to the use of an equal velocity joint. Additionally, the equation that relates $\lambda$ to $\nu$ does not contain the variables $A_x$, $A_y$, $B_x$, $B_y$, or $\left|\mathit{C}\mathit{A}\right|$. Consequently, the sixth row of the Jacobian matrix contains zeros in columns 1–5. The partial derivative with respect to $\lambda$ in the self-rotation equation is equal to 1, as for simplification, we assume that the pulleys of the actuator and the equal velocity joint have a ratio of 1.

It consists of a translation Jacobian matrix ${\mathrm{J}}_{\mathrm{v}}$ and rotation Jacobian matrix ${\mathrm{J}}_{\mathsf{\omega }}$:

$${\mathrm{J}}_{\mathrm{v}}=\left[\begin{array}{ccccc}{J}_{v11}& J_{v12}& J_{v13}& J_{v14}& J_{v15}\\ J_{v21}& J_{v22}& J_{v23}& J_{v24}& J_{v25}\\ J_{v31}& J_{v32}& J_{v33}& J_{v34}& J_{v35}\end{array}\right],$$

(22)

and

$${\mathrm{J}}_{\mathsf{\omega }}=\left[\begin{array}{cccc}{J}_{\omega 11}& J_{\omega 12}& J_{\omega 13}& J_{\omega 14}\\ J_{\omega 21}& J_{\omega 22}& J_{\omega 23}& J_{\omega 24}\end{array}\right].$$

(23)

The equations that compose $\mathrm{J}$ are complex, making analytical singularity analysis inconvenient. It was decided to perform a numerical simulation instead. To identify potential singular positions of the mechanism, it is necessary to examine the determinant values of its Jacobian matrix. A zero determinant at any point within the workspace indicates a singular position of the mechanism at that location. In order to obtain correct simulation results, the components ${\mathrm{J}}_{\mathrm{v}}$ and ${\mathrm{J}}_{\mathsf{\omega }}$ should be studied separately as the Jacobian matrix $\mathrm{J}$ is not homogeneous and contains different units. In the case of the translation determinant values calculation, the probe rod maintained constant orientation in vertical position and point C was translated along the workspace in X and Y direction with a step of 1 mm. The initial conditions for the rotation simulation were the following: the probe was placed in the center of the workspace in a vertical position, then the inclination angle was gradually increased with $1°$ step, after that the probe was rotated around the vertical axis, making conical motion also with the step of $1°$. The inclination angle was increased to the maximum value of $20°$, which is the limit of the joint that is used in the prototype. It is not possible to obtain the determinant of a non-square matrix, so instead, we use the square root of the determinant of a square matrix formed by multiplying the non-square matrix by its transpose (manipulability index). The obtained values for the Jacobian matrix components can be seen in Figure 9.

In Figure 9a, it can be seen that the determinant values remain constant throughout the workspace and not equal zero. In Figure 9b, it can be seen that the values of the determinant are very small, but they are not equal to zero within the workspace of the mechanism. The results of this simulation do not kinematically depend on the translation and can be repeated in any part of the workspace producing the same values. Considering these results, it can be said that the haptic device does not have any singular positions within its workspace.

4.2. Kinematic Conditioning Index

The next phase of the kinematic analysis involved evaluating performance quality of the mechanism by its dexterity [34]. To achieve this, the Kinematic Conditioning Index (KCI) was calculated. The first step in determining the KCI is to compute the conditioning number of the Jacobian matrix. One efficient approach is to use the maximum (${\sigma }_{max}$) and minimum (${\sigma }_{min}$) singular values:

$$\kappa \left(J\right)={\displaystyle \frac{{\sigma }_{max}}{{\sigma }_{min}}}.$$

(24)

As the conditioning number varies between 1 and infinity, it is more practical to use its reciprocal, ${1/\kappa }_H$, which ranges from 0 to 1 [35]. In this scale, 0 represents a singular position, while 1 corresponds to a configuration with maximum dexterity. Therefore, the KCI can be expressed as:

$$KCI={\displaystyle \frac{1}{\kappa \left(J\right)}}.$$

(25)

The dexterity index values were obtained using the same simulation configuration as for the determinant calculation. The results are presented in Figure 10.

As shown in Figure 10, the mechanism does not exhibit singularities within the workspace, consistent with the analysis of the Jacobian determinant. The translational dexterity index remains constant throughout the workspace, with a value of 0.343. This moderate value suggests potential issues with precision. This is expected, as the probe’s positional resolution is directly dependent on the maximum actuator resolution. However, in applications such as sonography, where precise probe positioning is not critical, this value is acceptable.

The rotational KCI values are significantly higher, reaching the maximum possible value of 1 at the center of the workspace, which corresponds to the vertical orientation of the probe rod. This is consistent with the observation that initiating motion in any direction from this position is convenient. Furthermore, the distances CA and CB act as levers, meaning that small movements of the platform result in minor changes in orientation angles, thereby enhancing the rotational resolution. This makes the mechanism particularly well-suited for tasks that demand precision, flexibility, and reliable motion control. However, while this level of rotational precision exceeds the requirements of the current application, it provides a useful margin for potential future use.

4.3. Global Conditioning Index

The Conditioning Index (GCI) is an average estimation of the dexterity through whole workspace of the mechanism [36]. In the case of translational analysis, the obtained KCI value is effectively a GCI since it remains constant throughout the workspace. The GCI can be computed as the average of the KCI values across all computed points, as given by the following equation:

$$GCI={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{KCI}_i,$$

(26)

where $n$ is the number of points within the workspace where the dexterity index was evaluated.

The GCI value computed during the rotational Jacobian matrix analysis is 0.959, reflecting excellent performance and high dexterity.

5. Quasi-Static Analysis

To assess the mechanism’s performance in generating haptic force feedback, a study of the force and torque transmission within the device was conducted. For this analysis, forces were applied at point C in the X and Y directions and torques were applied around the axes corresponding to the angles ${\varphi }_1$ and ${\varphi }_2$. The simulation proceeded as follows: the probe was initially positioned vertically, with point C fixed at the center of the workspace. The probe was then incrementally inclined to a maximum angle of $20°$. After each inclination step, the probe was rotated $360°$ around the vertical axis passing through point C. This procedure simulates the variations in actuator torque for every possible probe orientation. Due to the kinematics of the mechanism, this torque distribution is applicable to any position of point C within the workspace.

The torque and force calculations were performed using the following equation:

$$\left[\begin{array}{c}{A}_{X\tau }\\ A_{Y\tau }\\ B_{X\tau }\\ B_{Y\tau }\\ {\left|\mathit{C}\mathit{A}\right|}_f\\ {\lambda }_{\tau }\end{array}\right]={\mathrm{J}}^{\mathrm{T}}\left[\begin{array}{c}{F}_X\\ F_Y\\ F_Z\\ {\tau }_{\varphi 1}\\ {\tau }_{\varphi 2}\\ {\tau }_{\nu }\end{array}\right],$$

(27)

where $F_X$, $F_Y$ and $F_Z$ are output forces; ${\tau }_{\varphi 1}$, ${\tau }_{\varphi 2}$ and ${\tau }_{\nu }$ are output torques; $A_{X\tau }$, $A_{Y\tau }$, $B_{X\tau }$, $B_{Y\tau }$ and ${\lambda }_{\tau }$ are input torques and ${\left|\mathit{C}\mathit{A}\right|}_f$ corresponds to input force.

It is important to note that in the current configuration, there is no actuator for the ${\left|\mathit{C}\mathit{A}\right|}_f$ input, as motion along the Z-axis is passive. This results in some undesirable forces in the vertical direction that cannot be completely avoided with the present design. However, this issue will be resolved in future iterations of the mechanism, where 6 active degrees of freedom (DOF) will be implemented.

In this analysis, the chosen values for the forces were 1 N, and for the torques, 1 Nm. The $F_Z$ force was set to zero, given that actuation in the Z direction is not currently possible. The resulting torque distribution plots from the simulation are shown in Figure 11, Figure 12 and Figure 13. It can be seen that the values of the torque in the actuators are low, which allows us to install smaller and cheaper actuators. The actuators currently intended to be used in the prototype are capable of creating 4.78 Nm torque that is almost 100 times bigger than the maximum required for this simulation (0.053 Nm), excluding self-rotation.

Since the self-rotation of the probe is kinematically decoupled from the other motions, the torque for this actuator remained constant throughout the simulation and was equal to the specified output torque of 1 Nm.

6. Trajectory Simulation

To verify the kinematic model of the mechanism, a trajectory simulation was conducted using Motion Study in SolidWorks 2022. For the simulation, the CAD model of the proposed mechanism was simplified by reducing it to the probe rod to decrease computational load and improve simulation speed. The end of the probe rod was modified into a sharp cone, with the tip defining point C. Points A and B were also assigned to their appropriate locations on the probe rod.

The trajectories of points A, B, and C were precomputed using MATLAB R2022a Update 5, and a Python 3.9 script was used to process the data, generating a .dxf file for insertion into the CAD model. In this process, the desired motion of the end-effector (point C) was first defined in Cartesian space, and the inverse kinematic model was used to compute the corresponding trajectories of points A and B. A uniform time increment was assigned to each trajectory point. The resulting dataset, including the time-based position profiles for A and B, was then imported into SolidWorks Motion Study as motion inputs. No constraint was applied to point C; its motion was determined solely by the mechanism’s geometry and the driven positions of A and B. To assist in verification, a reference curve representing the intended trajectory of point C was also included in the simulation. Visual verification was performed by observing the alignment between the simulated and reference trajectories of point C to confirm the validity of the kinematic model.

Three scenarios were simulated: orientation change, position change, and a combined change. To describe the simulation results, Euler angles were used as they are more intuitive regarding the described motions. The first two trajectories reflect typical motions used in sonography tests, while the third scenario was selected to fully challenge the mechanism’s performance under complex motion conditions.

Orientation Change: In the first case, the position of the end effector ($C_x$, $C_y$, $C_z$) was kept constant, while the angle $\phi$ changed, causing the probe to trace a conical surface with a $20°$ inclination angle. This inclination corresponds to the maximum allowable tilt angle for the Igus^® joints in the prototype. Although for making the simulations the intuitive Euler angles are used, when performing the kinematic analysis, they are transformed into the Cardan angles so as to maintain the previously established procedure.

Position Change: In the second simulation, the probe’s orientation remained constant while its position varied, with point C moving in a circular trajectory with a radius of 25 mm around the workspace center. This radius was selected to ensure the probe’s movement remained within the dimensions of a typical body phantom (60 × 130 × 50 mm). To describe this simulation, a new angle, R, was introduced, representing the rotational angle of point C about the center of the workspace.

Combined Change: The third simulation involved simultaneous changes in both position and orientation. The trajectory of point C followed a sinusoidal path along the X-axis with an amplitude of 50 mm and a displacement of 120 mm along the Y-axis. Meanwhile, the probe’s inclination angle (Euler angle $\theta$) varied from $-20°$ to $20°$, and the probe was rotated around the vertical axis (Euler angle $\phi$) from 0 to 2π.

Screenshots of these simulations are provided in Figure 14, where 1 denotes the position or trajectory of point C, 2 represents the trajectory of point B and 3 shows the trajectory of point A.

In the trajectory calculation, additional parameters were determined, including the actuator angles ${\alpha }_A$, ${\beta }_A$, ${\alpha }_B$, and ${\beta }_B$, as well as the translational and rotational dexterity indices. The actuator angle plots are illustrated in Figure 15, Figure 16 and Figure 17. The dexterity indices for the first and third simulations are shown in Figure 18 and Figure 19, respectively. In the second simulation, both the translational and rotational Kinematic Condition Indices (KCI) remained constant, with values of 0.343 and 1, respectively.

During the simulations, the end of the rod was precisely following the expected trajectory of point C that confirmed the correctness of the kinematic calculations. At this stage, the prototype lacks integrated motion tracking for quantitative error analysis. However, initial visual validation confirms trajectory-following capability. Future iterations will include encoder-based logging and comparison against simulated paths to quantify tracking accuracy. The simulations can be seen in a video format provided as Supplementary Materials to the current paper.

7. FEM Structural Analysis

To validate the mechanical design, a preliminary FEM structural analysis was performed using SolidWorks software (Student Edition 2024 SP2.0). This analysis focused on the most critical components of the mechanism: the frame and the assembly of the X and Y carriages. Analyzing the frame is essential, as it serves as the foundation for all subsequent modifications and must support the maximum anticipated load. In the carriage assembly, the linear guides of the X carriage have a relatively small cross-section and are fixed only at their ends, making them susceptible to significant stress and deformation. Given that the maximum expected load is relatively low (60 N), structural failure of any component is unlikely; therefore, the primary evaluation criterion should be deformation, which could potentially impact the precision of the setup.

The primary objective of conducting these simulations separately is to facilitate troubleshooting, identify the most deformed components, and enhance efficiency by reducing computational power requirements. Additionally, to expedite this preliminary evaluation, it was assumed that the assembly components were bonded together in both simulations. Future research will involve more detailed calculations. It is also noteworthy that during the prototype construction, several frame connections were reinforced with epoxy resin, thereby aligning the prototype more closely with the simulation conditions.

In the first simulation, the bare frame of the mechanism was analyzed. The frame is constructed from aluminum alloy 6060 T66, a material commonly used for profile manufacturing. The bottom surface of the frame was designated as the support and specified as a fixed geometry feature. A load was applied to the front section of the frame, corresponding to the positions of all Y carriages at their limits, with the X carriage positioned centrally within its guides. Although this configuration is not practically utilized, it is theoretically possible and is expected to induce the maximum deformation due to the extended force lever arm. The applied force simulates haptic feedback in the Z direction, which is currently unachievable in this design due to the lack of actuation in that direction. Nonetheless, this simulation is crucial for future developments, as there are plans to incorporate force feedback in this direction following practical verification of the design features. The mesh was generated using standard adaptive settings suitable for preliminary calculations. The simulation results are illustrated in Figure 20, revealing a maximum deformation of 0.348 mm, a value that is considered small and satisfactory.

In the second simulation, the assembly of the X and Y carriages was utilized. This assembly was simplified by excluding the equal velocity joint. This component comprises a modified Igus^® spherical joint capable of withstanding a maximum short-term axial load of 1400 N [37], along with an aluminum pulley equipped with bearings. Given that these parts are unlikely to experience significant deformation and have complex geometries that would necessitate a finer mesh and longer computation times, it was determined that the equal velocity joint would not be included in the simulation.

The bottom surfaces of the Igus^® TW-14 carriages, utilized for Y displacement, were designated as fixed supports in this calculation. A load of 30 N was applied to the surface of the X carriage, which accommodates the lower bearing of the equal velocity joint. This force represents half of the maximum expected load, as the force feedback in the Z direction, once implemented, will be evenly distributed between the two limbs of the parallel mechanism. The position of the X carriage was centered within the corresponding linear guides, as this position is anticipated to yield the maximum deformation of the guides. The material specified for the linear guides and their carriages was 6060 T66 aluminum alloy. Acrylonitrile butadiene styrene (ABS) was selected for the body of the carriages, as this material was used for 3D printing the prototype components. Additionally, auxiliary parts, such as the bearing shafts, were specified to be made from AISI 1045 steel.

The results of the simulation are illustrated in Figure 21. The maximum deformation of the X linear guide is recorded at 0.2427 mm, which is considered very small and practically insignificant. Notably, one guide exhibits greater deformation than the other, potentially resulting in a slight twisting of the carriage. While this is not expected to pose a significant issue, future improvements could be made by designing the X carriage to be symmetrical and incorporating two Igus^® TW-14 linear carriages on each side.

8. Prototype Manufacturing

The prototype was constructed using various types of aluminum tubes. The limb frames and bottom frames were made from tubes with a cross-section of 35 × 15 mm and a wall thickness of 2 mm. The side support tubes measured 100 × 20 mm, also with a wall thickness of 2 mm, while the back plate was cut from 3 mm aluminum. For motor mounting, an L-shaped aluminum profile measuring 40 × 40 mm with a thickness of 4 mm was used. The bodies of the X and Y carriages, along with the belt tensioners and several connecting components, were 3D printed using ABS material. Igus^® TW-14 pre-loaded miniature guides with corresponding carriages were employed for both X and Y displacement. Actuation is achieved through belt drives using T2.5 belts, with motors employing a 40-teeth pulley and the equal velocity joint utilizing a 60-teeth pulley. The prototype features P80 III KV100 BLDC motors with torque ratings ranging from 0.97 to 4.78 Nm, providing more than sufficient capacity for haptic feedback, while also allowing for a force margin. The equal velocity joint is based on a modified Igus^® KGLM-10 spherical joint, which was adapted by cutting a slot in the central sphere and introducing two pins for torque transmission. The probe rod consists of two aluminum tubes that can slide within each other and are connected by a pin that transmits rotation from the equal velocity joint to the probe. The probe itself is secured to the probe rod with a specially designed clamp that mimics the upper profile of the probe. This clamp includes a rubber belt to ensure adequate fixation of the probe. The prototype is illustrated in Figure 22.

The motors are managed by six Odrive Pro controllers powered by a 24 V supply. These controllers are able to provide torque control and data regarding the current consumed by the motors. This data will be used as feedback to implement the haptic functions of the mechanism. If this solution proves insufficient, force sensors can be installed in the probe clamp in the future.

The Odrive controllers are linked to a Raspberry Pi 4 with 4 GB of RAM via a CAN bus, utilizing an MCP2515 module. This setup enables fast data exchange between the controllers and the single-board computer. The Raspberry Pi operates on ROS2 Humble, which communicates with the controllers through the odrive_ros2 package. A custom package has also been developed to facilitate communication with the mechanism. Currently, the software package responsible for the haptic device’s operation is in the early stages of development and requires further work, testing, and optimization. The prototype hardware serves as a proof of concept; the geometric parameters and design solutions will be thoroughly analyzed and refined in the future to create a viable market-ready product.

Preliminary tests of the prototype were conducted using both conical and square trajectories. Representative videos of these experiments are available in the Supplementary Materials. The parameters for the trajectories were as follows.

The initial position of point C was set at (140, 220, 0). For the conical trajectory, the rod followed a conical path defined by a 20-degree angle between its longitudinal axis and the initial vertical axis. The cone was traced by rotating the rod about the center point of the workspace.

In the case of the square trajectory, the rod also started from a vertical orientation at the center of the workspace. It then moved to the first point of the square path, corresponding to the top-right corner of the square, located at (190, 270, 0). The square had a side length of 100 mm, and its corners were defined by the following coordinates: (190, 270, 0), (190, 170, 0), (90, 170, 0), and (90, 270, 0). Upon completing the square path, the end-effector returned to its initial position.

The experimental results validate the correctness of the implemented kinematic model. However, the observed motion exhibits deviations from the ideal trajectories. These deviations are primarily attributed to non-uniform friction within the Igus^® linear guides and the compliant (spring-like) behavior of the BLDC motors employed in the system. In certain positions, these factors cause the carriages to decelerate, thereby affecting the accuracy of the end-effector’s location.

Future improvements will involve replacing the Igus^® guides with linear bearings, aiming to minimize changes to the existing mechanical design. Additionally, refinements to the motor control algorithms and calibration are planned to further enhance trajectory fidelity in order to facilitate more meaningful performance evaluations.

9. Conclusions

In this paper the design of a 6 DOF Cartesian parallel mechanism with 5 DOF haptic feedback for initial sonography training has been proposed. Insights on its direct and inverse kinematic models have been provided. The kinematic analysis demonstrated that the device operates efficiently within its workspace, with no singularities detected. The calculation of the performance indices showed excellent rotational dexterity and satisfactory translational dexterity. It verifies that the mechanism is capable of performing sonography training operations with the required precision. A trajectory simulation was performed to validate the kinematics, with the device following predefined motion paths typical in sonography. The results of these simulations validated the theoretical kinematic models, demonstrating the reliability of the mechanism’s motion control. The quasi-static analysis examined the transmission of forces and torques during probe motion, highlighting the limitations of the current design, particularly in the Z-axis where motion is passive. While this limitation introduces some unwanted vertical forces, the analysis provides valuable insights for future design iterations, where active control in all 6 degrees of freedom will be implemented. Structural analysis was conducted on critical components of the device, showing minimal deformation under expected loads, confirming that the device’s structure is robust enough to withstand operational stresses while maintaining the necessary precision. Finally, the prototype was manufactured in order to demonstrate the practical feasibility of the proposed design. It serves as a foundation for further optimization and testing, particularly in integrating software controls and improving overall performance.

While the prototype does not yet incorporate a fully implemented closed-loop force-feedback control system, the mechanical design prioritizes force interaction capabilities. The current work focuses on validating the kinematic architecture and establishing the platform’s mechanical feasibility. Future development will extend these results by integrating and experimentally verifying active force control to assess the system’s performance in realistic haptic training scenarios. As such, the references to force-feedback performance in this paper should be interpreted as describing design intent rather than finalized, validated functionality.

Future work will focus on experimental verification of the theoretical results, refining the device’s performance, expanding its force feedback capabilities, and enhancing its usability in practical sonography training scenarios.