Design Optimization of a Parallel Robot for Laparoscopic Pancreatic Surgery Using a Genetic Algorithm

Abstract

Background: Laparoscopic pancreatic surgery demands high precision and minimal invasiveness, yet conventional robotic systems often face challenges due to complex anatomical environments and uncertainties inherent in surgical procedures. Optimizing key design parameters such as the Remote Center of Motion (RCM) and robotic link lengths is critical for enhancing workspace accessibility and instrument maneuverability. Methods: An integrated optimization framework combining genetic algorithms (GA) with fuzzy logic was developed to determine the optimal RCM position and the ideal lengths of crucial robotic links in a 3-DOF parallel robotic system. The GA explored a large design space based on 6951 tracking points recorded during manual instrument manipulation, while the fuzzy logic system refined fitness evaluations by incorporating expert-defined membership functions and heuristic rules to manage uncertainties and ensure robust performance. Results: Simulation studies demonstrated that the optimized RCM position shifted from an initial [100, 0, 300] to [119.003337, −146.610801, 269.07376], yielding improved workspace coverage and enhanced instrument maneuverability. The GA further determined optimal link lengths of approximately 213.5 mm, 248.5 mm, and 48.6 mm for the primary, tertiary, and minimum secondary links, respectively, which were rounded to practical values of 215 mm, 250 mm, and 50 mm. The optimized design exhibited significant improvements in workspace reachability, precision, and operational stability, as validated by detailed 3D workspace plots and time history diagrams of the instrument tip and joint trajectories. Conclusions: The integrated GA–fuzzy optimization approach effectively enhances the design of a 3-DOF parallel robot for laparoscopic pancreatic surgery by achieving superior kinematic performance. The optimized parameters contribute to improved surgical precision and workspace accessibility, indicating strong potential for clinical application and further experimental validation.

1. Introduction

Pancreatic cancer or pancreatic ductal adenocarcinoma is the second cause of cancer-related deaths in the last decade [1]. Risks that can lead to developing pancreatic cancer include family history, obesity, type 2 diabetes, and tobacco use. Pancreatic cancer can be diagnosticated into 4 stages: resectable, borderline resectable, locally advanced, and metastatic, [2]; but due to the lack of relevant symptoms, it is usually diagnosed in advanced stages. Classical treatment methods for pancreatic cancer usually include surgery, chemotherapy, radiotherapy, and palliative care [3]. Radical resection of the pancreas is preferred in most cases to eliminate the tumor in the case of patients with a borderline or locally advanced tumor [4]. Along with traditionally open surgery of the pancreas, several minimally invasive approaches were developed. Minimally invasive pancreatic surgery was considered not feasible until 1996 when Cuschieri et al. performed the first successful laparoscopic distal pancreatectomy [5]. Two minimally invasive techniques distinguished over time: minimally invasive distal pancreatic resection (MIDP) and minimally invasive pancreatoduodenectomy (MIPD). MIPD presents technical challenges, especially during the complex reconstructing phase, limiting its widespread adoption. Although current studies suggest that it is both feasible and safe for carefully selected patients, there is not enough evidence to show its advantages over the open pancreaticoduodenectomy. MIDP can lead to shorter lengths of hospitalization and better quality of life than open pancreaticoduodenectomy. However, its effectiveness for larger tumors that involve blood vessels or other organs remains under-researched, and thus further studies are necessary. Despite these limitations, MIDP is still considered an effective treatment option for smaller tumors that do not involve vascular or organ complications [6]. Robotic distal pancreatectomy offers oncologic outcomes comparable to those of laparoscopic and open approaches. However, it is associated with a shorter hospital stay and a lower rate of postoperative morbidity and complications [7]. The first robotic-assisted pancreatic surgery was documented by Giulianotti et al. [8] in 2003 when 13 robotic minimally invasive pancreas surgeries were performed using the da Vinci (Intuitive Surgical, California, CA, USA) robot, using articulated instruments with up to seven degrees of freedom (DOF), allowing for enhanced dexterity and precision that achieved outcomes comparable to those of conventional open surgery.

While the well-known commercial robotic systems for surgery like da Vinci (Intuitive Surgical, Inc., Sunnyvale, CA, USA) [9], Hugo™ (Medtronic, Minneapolis, MN, USA) [10], Versius (CMR Surgical, Cambridge, UK) [11], Senhance (Asensus Surgical, Inc., Durham, NC, USA) [12], and Revo-i™ (MEEREcompany, Hwaseong, Republic of Korea) [13] dominate the clinical applications, several research prototypes have been developed for pancreatic surgery, particularly for pancreatoduodenectomy or distal pancreatectomy. Hannaford et al. [14] introduced Raven-II, an open-architecture robotic platform characterized by a modular hardware design and an open-source software framework. This configuration enables the extensive customization of the kinematic chain, leading to adjustments in the number and configuration of degrees of freedom to suit specialized surgical tasks. Preliminary evaluations demonstrated robust performance in simulating standard surgical maneuvers, indicating the system’s potential for refined spatial control. Berthet-Rayne et al. [15] presented the Intuitive Imaging Sensing Navigated and Kinematically Enhanced robot (i2Snake), designed to advance endoscopic surgery. The i2Snake features a snake-like robotic endoscope equipped with a camera, light source, and two articulated instruments, supported by a robotic arm for precise global positioning and insertion. Its master–slave teleoperation interface emphasizes ergonomic control and intuitive workflows, with a design that incorporates tailored DOF to navigate complex anatomical pathways effectively. Although initial testing validated its clinical potential, issues such as tendon backlash and gradual elongation over time were identified, highlighting the need for further technical refinements.

Robotic-assisted pancreatic surgery presents significant complexity due to the pancreas position and complex vascular network around it, requiring surgeons to navigate narrow anatomical workspaces with high precision and low marginal error. Optimization of the surgical robots for pancreatic surgery reduces intraoperative complications, enables targeted and minimally invasive techniques, improves patient outcomes and recovery times, enhances surgeon control and procedural accuracy, complies to safety regulations while decreasing error rates, enhances instrument maneuverability in restricted anatomical workspaces, improves system reliability during critical operative steps, and complies the design of the robot with the clinical needs. However, the precision required for robotic-assisted minimally invasive pancreatic procedures necessitates robotic systems capable of exceptional accuracy, extensive maneuverability, and reliable operation within confined anatomical spaces. Therefore, design optimization of robotic-assisted surgical systems is essential to maximize their effectiveness in handling precise and delicate maneuvers required in laparoscopic pancreatic surgery, ultimately aiming to enhance surgical outcomes and patient safety.

Several studies employed different optimization strategies aiming to improve the design and the ergonomics of surgical robots. Wang et al. [15] addressed the structural design optimization problem of continuum manipulators by establishing load-carrying capacity and secondary deformation as the optimization objectives based on the manipulator’s structure. A non-dominated sorting genetic algorithm with an elite strategy (NSGA-II) is employed to handle multi-objective optimization. To validate the effectiveness of the proposed approach, an experimental prototype was developed based on the optimization results. Load-carrying and space-traversal capacity tests confirm the viability of the optimized design. Kumar et al. [16] introduced a compliant mechanism design for a laparoscopic surgical instrument, where an optimal topology (configuration) is developed to achieve a predefined input–output force-displacement relationship. Two distinct optimization approaches are employed. The first approach formulates an objective function that balances compliance—allowing the mechanism to undergo the required deformation (kinematic requirement)—with sufficient stiffness to withstand external loads (structural requirement) once the desired configuration is attained. The second approach focuses on minimizing the volume fraction while adhering to design constraints that restrict node translation. The optimization process is carried out using MATLAB and Altair OptiStruct software. The resulting optimized topology is then transformed into a 3D CAD model, followed by Finite Element Analysis (FEA) to evaluate stress distribution and deformations.

3-DOF parallel robots are being developed for minimally invasive surgery due to their accuracy and compact design. This type of robots usually provides two rotational and one prismatic DOFs, enabling dexterous manipulation within confined surgical spaces. A dexterous endoscopic manipulator with 3-DOF was proposed by [17]. The manipulator is composed of 3 identical kinematic limbs of type PUU. The kinematic architecture allows large bending angles of +/−90 degrees in any direction and provides a workspace nearly free of internal singularities. The complexity of the design of the robot complicates the analysis of the manipulator, especially in singularity analysis requiring special geometrical and analytical approaches. Advancements have also been made in control methodologies; Bach et al. [18] proposed an optimal trajectory tracking control method using Particle Swarm Optimization (PSO) based on inverse kinematics in order to improve the precision of the 3-DOF surgical robot. The proposed controller exhibited faster convergence to the desired trajectory and reduced steady-state errors, indicating its potential efficacy in enhancing the accuracy and reliability of surgical robots

Several challenges of the 3-DOF robots for surgery imply singularity issues (presence of singularities within the robot′s workspace can lead to control difficulties and reduced accuracy during the surgical procedure) and safety concerns (focused on developing secure robotic solutions for surgical procedures, emphasizing the identification and mitigation of safety issues from the early design stage) [19].

Efforts are underway to design parallel robots with large, singularity-free workspaces and the development of accurate dynamic models for effective control and performance [20].

This paper presented the optimization of a 3-DOF parallel robot for laparoscopic pancreatic surgery, integrated with a passive spherical mechanism to ensure a stable Remote Center of Motion (RCM) at the trocar. The novelty of this robotic structure is given by the fact that the robot employs a novel 3-DOF parallel mechanism combined with a passive spherical mechanism that specifically constrains the Remote Center of Motion (RCM). This hybrid approach ensures that instrument motions are precisely pivoted about the RCM—a critical requirement in minimally invasive surgery—while maintaining a compact and efficient structure. The scope of this research is the development and optimization of the parallel robotic system for laparoscopic pancreatic surgery utilizing a sophisticated optimization framework that integrates genetic algorithms and fuzzy logic. The study aimed to determine optimal configurations of critical robotic parameters. Initially, the focus lies on optimizing the Remote Center of Motion (RCM), essential for enhancing workspace accessibility and improving instrument maneuverability. Subsequent optimization steps include identifying optimal lengths for crucial robotic links, significantly impacting the surgical robot′s performance. The study further validates the optimized robotic design through detailed simulation studies and graphical evaluations, demonstrating improvements in workspace and reachability. The scope also extends to practically integrating these optimized parameters into a functional robotic system within an operating room setting, including performing virtual simulations to test and validate robotic performance under realistic surgical conditions.

The paper is structured as follows: Section 1 introduces the research context and objectives. Section 2 outlines the genetic algorithm–fuzzy logic optimization framework and robotic design methods. Section 3 details the results of the optimization and the mechanical design implementation. Section 4 discusses these findings comprehensively, while Section 5 concludes by summarizing the key insights and potential implications of this research.

2. Materials and Methods

2.1. Parallel Robot for Laparoscopic Pancreatic Surgery

The concept of the parallel robot designed for laparoscopic pancreatic surgery is presented in Figure 1 [21]. The kinematic model of the robot has been previously presented in [22]; thus, only the final equations of the inverse geometric model are presented here in Equation (1).

$$\left\{\begin{array}{ll}{q}_1=& Y_p-a_{02}-\sqrt{-{(X_p+a_{01})}^2-{(Z_p+a_{03})}^2+a_1^2-a_4^2+2\sqrt{{(X_p{a}_4-a_{01}{a}_4)}^2+{(Z_p{a}_4-a_{03}{a}_4)}^2}}\\ q_2=& Y_p-a_{02}+\sqrt{-{(X_p+a_{01})}^2-{(Z_p+a_{03})}^2+a_1^2-a_4^2+2\sqrt{{(X_p{a}_4-a_{01}{a}_4)}^2+{(Z_p{a}_4-a_{03}{a}_4)}^2}}\\ q_3=& -a_{2\min }-a_5+1/2(-4a_4\cos (\lambda )\sqrt{4{a_3}^2-{(q_1+q_2)}^2}-8X_p{a}_4\cos (\lambda )+8a_{01}{a}_4\cos (\lambda )\\ & -8\sin (\lambda )Z_p{a}_4+8\sin (\lambda )a_{03}{a}_4+4X_p\sqrt{4{a_3}^2-{\left(q1+q2\right)}^2}-4a_{01}\sqrt{4{a_3}^2-{(q_1+q_2)}^2}\\ & +4{X_p}^2-8X_p{a}_{01}+4{Z_p}^2-8Z_p{a}_{03}+4{a_{01}}^2+4{a_{03}}^2+4{a_3}^2+4{a_4}^2-{(q1+q2)}^2{)}^{1/2}\end{array}\right.$$

(1)

The parallel robot consists of two main modules that work together to support and manipulate the surgical instrument. The first module is a 3-DOF active parallel mechanism, while the second is a passive spherical mechanism that constrains the Remote Center of Motion (RCM) of the instrument. The active module is designed with two passive universal joints (U1P and U2P), three actively controlled prismatic joints (q₁, q₂, and q₃), and a set of nine passive revolute joints (R₁P through R₉P). A global coordinate system is defined at the RCM, which serves as the pivot for the instrument, and an additional coordinate system (OXYZ) is fixed directly to the robot body. The critical geometric parameters of the system include the link lengths (a₁ through a₅), the minimum stroke (a₂min) of the prismatic joint q₃, as well as the instrument’s total length (a) and the inserted portion length (l_ins) relative to the RCM.

Complementing the active module, the spherical mechanism is designed with five passive revolute joints (R₁S through R₅S) and one passive cylindrical joint (C₁S). This arrangement enables not only the rotational movements needed to adjust the instrument but also facilitates its insertion during surgery. The spherical mechanism is connected to the active parallel robot via a linkage that features two additional spherical joints (S₁ and S₂). These joints are crucial for fine-tuning the positioning and orientation of the passive module, ensuring the mechanism can be appropriately aligned with respect to the active robot. Together, the coordination of these modules enables enhanced accuracy and dynamic control in surgical applications. The robot uses the pivot point of the passive mechanism as the RCM (the trocar inserted into the patient); from this point the instrument can only perform three motions: insertion of the instrument along its longitudinal axis, as well as rotations around X and Y axes. The passive spherical mechanism supplements the active system by enforcing a strict RCM constraint, thereby decoupling the translational and rotational motions. This dual-module approach allows for enhanced dexterity and precision.

Equation (1) will be further used in determining the optimal length of several geometric parameters; notations q₁, q₂, and q₃ represent the active joints of the 3-DOF robot; Xp, Yp, and Zp represent the coordinates of the tip of the instrument; length a₀₁, a₀₂, and a₀₃ represents the distance from the frame reference system to the RCM on the X, Y, and Z axis, respectively; a₁ represents the length of the link connecting universal joints U1P and U2P to revolute joint R9P; a₃ represents the length of the link connecting revolute joint R1P to revolute joint R8P; λ = atan (Z_P-a₀₃, X_P-a₀₁).

2.2. Optimization Framework

The first part of the optimization process deals with finding the optimal position of the RCM point by using a genetic algorithm integrated with fuzzy logic for fitness evaluations. Optimization represents a critical step in designing a medical robot, especially when dealing with complex, multidimensional systems that require balancing numerous objectives, constraints, and uncertainties. Traditional optimization methods often have to handle incomplete and not-accurate-enough information that characterize real-world scenarios, a challenge that motivates researchers to integrate evolutionary algorithms with heuristic evaluation techniques. One promising strategy is the optimization framework that combines genetic algorithms with fuzzy logic for fitness evaluation. In this framework, genetic algorithms serve as the primary search mechanism, while fuzzy logic refines the evaluation process by incorporating expert knowledge and heuristic rules to assess candidate solutions in a nuanced manner.

In the case of genetic algorithms, a population of solutions evolved over successive generations. Each member of the population (individual) is represented by a chromosome, a string of values encoding a potential solution to the optimization problem. The algorithm is initiated with a random population and then employs selection, crossover, and mutation operators to generate new solutions. The selection process selects individuals based on their fitness values, and the evolutionary operations encourage diversity while guiding the search toward regions of the solution space that exhibits high performance. This approach is particularly effective when the search space is large and complex. Genetic algorithms do not require gradient information or continuity in the design space, making them suitable for problems with discontinuities, non-linearities, and multiple local optima [23]. By iteratively exploring and exploiting the design space, genetic algorithms can converge on near-optimal solutions that traditional gradient-based methods might miss. However, the performance of a GA heavily depends on its fitness evaluation, which is the process of quantifying how good each candidate solution is.

Fuzzy logic has emerged as a powerful tool for optimizing mechanical systems [24], providing a flexible and intelligent approach to decision-making in uncertain and complex environments. Traditional binary logic operates on crisp values (true/false) while fuzzy logic allows partial thrust and enables more flexibility and adaptive control over the optimization process. This method is useful for mechanism optimization, where multiple variables, constraints, and uncertainties must be analyzed and balanced to achieve optimal performance. Traditional optimization techniques rely on precise mathematical models [25], but real-world scenarios often include imprecise and incomplete information, making deterministic methods less effective. With the help of fuzzy logic, these challenges can be addressed by incorporating expert knowledge and heuristic methods to evaluate and optimize mechanical parameters [26]. For mechanism optimization, fuzzy logic is often used in areas such as kinematic [27] and dynamic analysis [28,29]. A key advantage of fuzzy logic in mechanism optimization is its ability to handle multiple-objective problems often required by engineering applications, especially in the case of engineering solutions for surgery. Fuzzy logic provides a systematic way to assign weight to different performance criteria and derive optimal design based on real-world constraints.

Figure 2 provides a general view on the genetic algorithm using fuzzy logic for fitness evaluation. Problem formulation and variable identification deals with defining the objectives of the optimization (identify the performance goals); determine the physical, geometrical, and operational constraints of the design; identify the decision variables like key mechanical parameters that influence the performance of the design. Encoding of candidate solutions refers to chromosome representation, where each individual of the population is represented as a chromosome by using several binary digits (genes) to numerically represent the individual. Initialization refers to generating the initial population of the design where a diverse set of candidate solutions are randomly distributed across the designed space to ensure adequate workspace. Fuzzy design refers to defining the key performance metrics that serve as inputs for the fuzzy logic system; the design of the membership functions for each input parameter handle uncertainties and formulate fuzzy rules to connect the inputs with the outputs of the fuzzy system. Fitness evaluation implies that each individual is analyzed and evaluated with respect to the current population using the fuzzy logic system to derive the fitness score. Selection process refers to using selection techniques based on fuzzy fitness scores to choose individuals for reproduction (crossover). Crossover implies combining pairs of selected individuals to produce offsprings that inherit from both parents. By mutation, it is implied that random changes are introduced in offsprings to explore new regions of the design space and maintain diversity. Iterative evolution deals with replacing the population based on new offsprings by replacing less fit individuals with newly generated ones. After replacing some individuals, the fitness evaluation is performed on the new population. The iterative process stops when a maximum number of generations is obtained, or improvements of fitness values fall below a set of thresholds over successive generations. After the selected optimal solution is implemented into the design, it is validated by simulations or experimental testing.

For the genetic algorithm of the parallel robot for pancreatic surgery, the optimization problem is formulated as follows: “starting from a set of points recorded during manual operation of the medical instrument inside of a phantom by a medical doctor, determine the optimal position for the RCM point (center for the spherical passive mechanism) to assure that all the recorded points fit inside the workspace of the robot”.

The points were recorded with the help of an optic motion tracking equipment (Optitrack-Natural Point Inc., Corvalis, OR, USA). A total number of 6951 points were recorded and used as initial data for the population.

The first step of the optimization algorithm is to establish chromosome representation. For this, each candidate solution for the optimal RCM is represented as a three-dimensional vector. Let the set of tracking points be:

$$\{P_i\in {ℝ}^3\mid i=1,2,\dots ,N\}$$

(2)

where N = 6951 and the length of the medical instrument L = 450 mm. The chromosome representation for a candidate RCM position yields Equation (3). The initial population is established in 100 randomly generated RCMs.

$$R=(R_x,R_y,R_z)\in {ℝ}^3$$

(3)

with

$$R_z\ge {\max }_i\left\{P_{i,z}\right\}+\delta , i=1\dots N$$

(4)

where δ > 0 ensures that the RCM is placed above all input points. The distance from R to P_i(P_ix, Pi_y, P_iz) is given by Equation (5).

$$d_i=\Vert P_i-R\Vert =\sqrt{{(P_{i,x}-R_x)}^2+{(P_{i,y}-R_y)}^2+{(P_{i,z}-R_z)}^2}, i=1\dots N$$

(5)

The second step of the algorithm implies raw performance evaluation using measured data. For each reachable point P_i, the inverse kinematic model computes the center of the joint R_1P (Figure 1) using Equation (6).

$$T_i=R-{\displaystyle \frac{L}{d_i}}(P_i-R), i=1\dots N$$

(6)

From the computed points, three metrics are derived, which are reachability ratio, convex hull volume, and spread. Equation (7) denotes the set of instrument tip points while Equation (8) represents the convex hull volume if T_t ≥ 4, otherwise, convex hull volume is 0.

$$T_t=\left\{T_i:d_i<L\right\}, i=1\dots N$$

(7)

$$V=Volume(conv(T_t))$$

(8)

The reachability ratio is defined as Equation (9), where #{i:d_i < L} denotes the number of tracking points for which the distance d_i is less than L. The centroid of the tip points is represented in Equation (10) and the spread of the points is defined in Equation (11).

$$r={\displaystyle \frac{\#\left\{i:d_i<L\right\}}{N}}$$

(9)

$$C={\displaystyle \frac{1}{\left|T\right|}}{\displaystyle \sum _{Ti\in T}{T}_i}$$

(10)

$$S={\displaystyle \frac{1}{\left|T\right|}}{\displaystyle \sum _{T_i\in T}‖T_i-C‖}$$

(11)

The third step of the algorithm implies the fuzzification of performance metrics. The Fuzzy Inference System (FIS) maps the three inputs (r, V, and S) to a penalty value P. The objective functions are defined in Equation (12); thus, the multi-objective optimization problem is presented in Equation (13), subject to Equation (14). The goal of multi-objective optimization is to find the optimal RCM that maximizes both the reachable workspace volume and reachability ratio r.

$$\left\{\begin{array}{l}{f}_1(R)=-V+P\\ f_2(R)=-r\end{array}\right.$$

(12)

$$\underset{R\in {ℝ}^3}{\min }f(R)=\left(f_1(R),f_2(R)\right)$$

(13)

$$R\in \{R:R\in [x_{\min },x_{\max }]\times [y_{\min },y_{\max }]\times [z_{\min },z_{\max }],R_z\ge {\max }_i\{P_{i,z}\}+1\}.$$

(14)

The optimization algorithm was implemented in MATLAB using Global Optimization Toolbox and Fuzzy Logic Toolbox. For the genetic algorithm, the multi-objective function gamultiobj was used while for the fuzzy system, the Type II Mamdani FIS [30] was preferred.

The FIS used to assess the fitness of the population during the genetic algorithm I presented in Figure 3. The FIS uses three input functions defined as Ratio, Volume, and Spread, with each function defined by three membership functions, and an output function named Penalty presented in Figure 4. The membership functions were determined based on the input points. Each input/output is a linguistic term for the FIS. A linguistic term in fuzzy logic is characterized by a variable whose values are words or sentences rather than numbers. A common and detailed mathematical formulation for a linguistic term in fuzzy logic is to define it as a five-tuple:

$$X=(X,T(X),U,G,M)$$

(15)

where X is the name of the linguistic variable (“Ratio”, “Volume”, “Spread”, and “Penalty” in our case); T(X) is the term set, i.e., the collection of linguistic values (words) used to describe X (“Low”, “Medium”, or “High” in our case); U is the universe of discourse over which the variable is defined (Ratio [0,1], Volume [0 mm³,1,000,000 mm³], Spread [0 mm, 450 mm], Penalty [0,100], Figure 4); G is the syntactic rule or grammar that specifies how the terms in T(X) can be generated or combined. In our case, these values were determined based on input point set of the genetic algorithm. M is the semantic rule that assigns meaning to the linguistic term by mapping each term to a fuzzy set over U. For each A $\in$ T(X), there is an associated membership function, ${\mu }_A:U\to [0,1]$, which quantifies the degree to which any $x\in A$ is interpreted as the degree ${\mu }_A(x)$. Thus, for a given $x\in U$ and a linguistic term $A\in T(X)$, the semantic evaluation is $“x \mathrm{is} A”\iff {\mu }_A(x)\in [0,1]$.

The fourth step of the algorithm actually contains the use of the FIS and development of the rules. The fuzzy rules define how the input membership functions influence the output membership functions. These rules are based on expert knowledge and ensure optimal robotic performance. The FIS rules are built as a series of IF- AND/OR -THEN expressions based on the expert’s opinion. If the expert’s opinion is not available, then the FIS requires a number of rules equivalent to the cubic number of inputs (no_of_inputs³). But through computing, several rules can be eliminated if they do not influence the output.

The established fuzzy rules for fitness evaluation are the following:

Rule 1—Optimal Performance:

IF r is High AND V is High AND S is Low THEN the system quality is Excellent, and the Penalty is Very Low (a candidate RCM that achieves high reachability and workspace with low dispersion is ideal).

Rule 2—Good Performance with Minor Deviations:

IF r is High AND V is Medium AND S is Low/Medium THEN the system quality is Good, and the Penalty is Low to Moderate (although reachability is excellent, a slightly lower workspace may indicate a small compromise).

Rule 3—Inadequate Reachability:

IF r is Low THEN the system quality is Poor, and the Penalty is High (even if V or S are acceptable, a low reachability ratio severely limits functionality).

Rule 4—Insufficient Workspace Volume:

IF V is Low THEN the system quality is Poor, and the Penalty is High (a low convex hull volume suggests that the candidate RCM does not offer a sufficiently large workspace, regardless of the reachability ratio).

Rule 5—Excessive Dispersion:

IF S is High THEN the system quality is Poor, and the Penalty is High (a high spread indicates that the instrument tip positions are inconsistent, which could compromise surgical precision).

Rule 6—Intermediate Conditions:

IF the performance metrics are in intermediate ranges (r and V are Medium and S is Medium/Low) THEN the system quality is Moderate, and the Penalty is Moderate (This rule covers scenarios where none of the criteria is optimal, but the solution is not entirely poor either)

For each candidate, the activation level for each rule is computed. This process is called aggregation.

$$\left\{\begin{array}{l}{\alpha }_1=min\{{\mu }_r,High(r),{\mu }_V,High(V),{\mu }_S,Low(S)\}\\ {\alpha }_2=min\{{\mu }_r,High(r),{\mu }_V,Medium(V),max({\mu }_S,Low(S),{\mu }_S,Medium(S))\}\\ {\alpha }_3={\mu }_r,Low(r)\\ {\alpha }_4={\mu }_V,Low(V)\\ {\alpha }_5={\mu }_S,High(S)\\ {\alpha }_6=min\{{\mu }_r,Medium(r),{\mu }_V,Medium(V),max({\mu }_S,Medium(S),{\mu }_S,Low(S))\}\end{array}\right.$$

(16)

After the aggregation of all fuzzy rules, the crisp value of the penalty is obtained using Mamdani inference–centroid method. This process is called defuzzification.

$$P(R)={\displaystyle \frac{{\displaystyle \sum _{k=1}^M{\mu }_k}(R)\cdot w_k}{{\displaystyle \sum _{k=1}^M{\mu }_k}(R)}},$$

(17)

The fifth step of the algorithm implies the formulation of the final fitness function. Raw fitness function is defined based on performance metrics to maximize r and V while minimizing S.

$$f(R)=-\alpha r-\beta V+\gamma S$$

(18)

where $\alpha , \beta \mathrm{and} \gamma$ are weighting factors. Then the overall fitness function is obtained.

$$F(R)=f(R)+P(R)$$

(19)

A lower value of F(R) indicates a better candidate RCM.

The last step of the algorithm implies evolution. Based on the fitness evaluation F(R), the best candidate RCM is added to the population while the least fit is removed. New offsprings are obtained by crossover (combination of different candidate RCM) and mutation (random changes in the chromosomes to create new candidates).

The algorithm is repeated for 200 generations but achieved optimal RCM after 115 generations.

After running the genetic algorithm with fuzzy fitness evaluation, an optimal position of the RCM is obtained in such a manner to be able to achieve almost all the input points with minimum motion beyond the RCM point inside the patient. After locking the RCM, the following geometric parameters are optimized: the length of the link a₁, the length of the link a₃, and the length of the link a_2min. The decision vector for the optimization algorithm is defined in Equation (16).

$$x=\left[\begin{array}{l}{a}_1\\ a_3\\ a_{2\min }\end{array}\right]$$

(20)

The bounds for each link are: $l_1\in [30,300],l_3\in [30,300],l_{2min}\in [30,150]$. By imposing the tracking or target point to be $P^i=\left[\begin{array}{l}{P}_x^i\\ P_y^i\\ P_z^i\end{array}\right],i=1\dots N$ and locking the coordinates of the RCM, we compute the Euclidean distance between Pⁱ and RCM using Equation (5) and define the term $t_i={\displaystyle \frac{L}{d_i}}-1,i=1\dots N$ that is used to compute the determine the target coordinates of the instrument holder using Equation (21).

$$\left\{\begin{array}{l}{X}_i=a_{01}-t_i(P_x^i-a_{01})\\ Y_i=a_{02}-t_i(P_y^i-a_{03})\\ Z_i=a_{02}-t_i(P_z^i-a_{03})\end{array}\right.,i=1\dots N$$

(21)

For each target point determined with Equation (21), the inverse kinematic model is computed. To compute the terms q_{1_i} and q_{2_i}, the terms from Equation (22) are derived using Equation (1).

$$\left\{\begin{array}{l}{A}_i=-X_i^2+2X_i{a}_{01}-Z_i^2+2Z_i{a}_{03}-a_{01}^2-a_{03}^2+a_1^2-a_4^2\\ B_i=\sqrt{X_i^2{a}_4^2-2X_i{a}_{01}{a}_4^2+Z_i^2{a}_4^2-2Z_i{a}_{03}{a}_4^2+a_{01}^2{a}_4^2+a_{03}^2{a}_4^2}\end{array}\right.$$

(22)

yielding

$$\left\{\begin{array}{l}{q}_{1\text{\_}i}=Y_i-a_{02}-\sqrt{A_i+2B_i}\\ q_{2\text{\_}i}=Y_i-a_{02}+\sqrt{A_i+2B_i}\end{array}\right.,i=1\dots N$$

(23)

For computing the value of q3, the λ first needs to be defined using Equation (24):

$${\lambda }_i=\arctan \left({\displaystyle \frac{X_i-a_{01}}{Z_i-a_{03}}}\right),i=1\dots N$$

(24)

and the value of q₃ joint is computed using Equation (25):

$$q_{3\text{\_}i}=-a_{2\min }-a_5+{\displaystyle \frac{1}{2}}\sqrt{\begin{array}{l}-4l_4\cos {\lambda }_i{C}_i-8X_i{a}_4\cos {\lambda }_i+8a_{01}{a}_4\cos {\lambda }_i-8a_4\sin {\lambda }_i(Z_i-a_{03})\\ +4X_i{C}_i-4a_{01}{C}_i+4(X_i^2-2X_i{a}_{01})+4(Z_i^2-2Z_i{a}_{03}+a_{01}^2+a_{03}^2)+\\ 4a_3^2+4a_4^2-{(q_{1i})}^2+2q_{1i}{q}_{2i}-{(q_{2i})}^2\end{array}},i=1\dots N$$

(25)

where term C_i is computed using Equation (26).

$$C_i=\sqrt{4l_3^2-{(q_{1\text{\_}i})}^2+2q_{1\text{\_}i}{q}_{2\text{\_}i}-{(q_{2\text{\_}i})}^2},i=1\dots N$$

(26)

For the genetic algorithm, the range of each active joint is given as a constraint q₁ ∈ [100,400], q₂ ∈ [100,400], and q₃ ∈ [100,500]; if a computed value falls outside the constraint interval, the algorithm adds a penalty to the objective function.

The multi-objective optimization problem is formulated as follows:

$$\min f(x)=\left[\begin{array}{l}{f}_1(x)\\ f_2(x)\\ f_3(x)\end{array}\right]=\left[\begin{array}{l}{a}_1+penalty\\ a_3+penalty\\ a_{2\min }+penalty\end{array}\right],\forall i=1\dots N:\left\{\begin{array}{l}{L}_{AB\text{\_}i}=\parallel P_i-RCM\parallel <L\\ q_{1\text{\_}i},q_{2\text{\_}i}\in [100,400]\\ q_{3\text{\_}i}\in [100,500]\\ x\in \left[\begin{array}{l}[30,300]\\ [30,300]\\ [30,150]\end{array}\right]\end{array}\right.$$

(27)

The penalty is a sum of deviation that increases the objective values if the computed q values violate the given bounds.

The genetic algorithm used for both optimization problems are based on gamultiobj function and use a population of 100 individuals, with 200 generations and a mutation rate of 1/1000 individuals.

3. Results

Following the two-step multi-objective optimization algorithm, several geometric parameters of the parallel robot for pancreatic surgery were optimized and further implemented into the constructive design of the robot. The instrument trajectories for the input values of the points, recorded by manipulating manually the surgical instrument with respect to an un-optimized RCM point given as input data for the optimization algorithm, are presented in Figure 5a,b. The figure presents the trajectories of the instrument using the same input points but constrain the instrument trajectory to the optimized RCM position. The points below the RCM position inside the patient are the same, but as can be observed, the RCM position is more central than the initial one. The initial position of the RCM was [100, 0, 300] and the optimized position of the RCM is [119.003337, −146.610801, 269.07376].

The result of the second step of optimization implied the determination of the optimal length for links a₁, a₃, and a_2min. The optimal length for these links given by the genetic algorithm was 213.5455 mm for link a₁, 248.48918 for link a₃, and 48.5886 mm for link a_2min. For constructive reasons, the values of these links have been rounded to 215 mm, 250 mm, and 50 mm, respectively, and a graphical simulation using Siemens NX was performed to test the validity of the design with the rounded length of the links. The result of the workspace simulation using optimized link lengths is presented in Figure 6. The input points from the genetic algorithm were integrated into a cone shape of yellow color in Figure 6 while the workspace of the instrument holder is represented with grey color. As can be observed, the tip of the instrument is not limited only to the input points, as it can go deeper in some cases.

Using the values of the geometrical parameters optimized in the previous section, the constructive design of the parallel robot for laparoscopic pancreatic surgery is presented in Figure 7. The parallel robot is integrated within the operating room, and the patient is placed on the operating table, ready for the medical procedure. The robot has been equipped with a supplementary independent DOF to allow vertical sliding of the entire robot, thus providing the possibility to vertically displace the RCM of the passive mechanism. Each geometric parameter analyzed has been implemented into the constructive design of the robot. The integration method into the mechanisms of the robot is presented in Figure 8, where the lengths of the components resulting from the genetic algorithm optimization were used. To provide better access of the instrument to the operating area, the sliding axis for q₁ and q₂ was placed as front-facing as possible to the passive RCM mechanism.

To further validate the optimization of the design, a virtual simulation using the constructive design of the parallel robot for pancreatic surgery. The graphical simulation was performed using Siemens NX 2406. The starting point of the simulation was P₁ (107, −102, −140) and the target point was P₂ (158, −160, 190). The results of the simulation are presented in Figure 9 for the coordinates at the tip of the instrument and in Figure 10 for the active joints of the parallel robot. In Figure 10, the first graphic represented with cyan color represents the displacement of the tip of the instrument on X, Y, and Z, the second graph represents the velocity (blue color) of the tip of the instrument on X, Y, and Z, and the third graph represents the acceleration (red color) of the tip of the instrument on the X, Y, and Z.

The first graph in Figure 10 represents the displacement (green color) of the active joints corelated with the motion of the tip of the instrument from Figure 10, the second graph represents the velocity (blue colour) of the active joints, and the last graph represents the acceleration (red color) of the active joints.

Figure 9 and Figure 10 show the time history diagrams for both the end-effector and active joints. These diagrams display smooth trajectories, consistent velocities, and controlled accelerations, which indicate that the system can follow the desired trajectory accurately and with minimal jitter. The acceleration profiles and joint speed data suggest that the system responds smoothly to control inputs without excessive overshoot or oscillation. This dynamic consistency is crucial for maintaining operational precision during surgical maneuvers, where any abrupt motion could compromise safety and accuracy. While the simulation studies strongly support the asserted advantages, we acknowledge that further experimental validation using phantom models is necessary to fully verify the operational precision and stability in real surgical environments.

In order to confirm the validity of the optimization process several comparisons regarding the robot before are analyzed.

Table 1. Comparison of optimized and unoptimized parameters.

Optimization Target	Evaluated Parameter	Before Optimization	After Optimization	Result
RCM	Reachability	5241/6951 points	6906/6951 points	+23.96%
	Workspace	9,366,060.48 mm3	12,226,455.35 mm3	+30.54%
Links	Reachability	6142/6951 points	6899/6951 points	+10.89%
	Workspace	8,224,547.69 mm3	9,240,279.33 mm3	+12.35%

presents the results of the comparison of the robot before and after the optimization. Reachability and workspace were analyzed for the 6951 points recorded in the first step of the algorithm. With the initial configuration of the RCM [100, 0, 300], 5241 points out of 6951 yielded real values for the active joint of the robot, while for the optimized RCM [119.003337, −146.610801, 269.07376], the inverse geometric model returned real solution for 6906 points, revealing a 23.96% increase in reachability. Using the same values for the RCM, the workspace of the robot (handle of the instrument) was analyzed, the analysis revealed a 30.54% increase in the workspace of the robot. The second comparison refers to the optimization of the kinematic links. The value before optimization of the links was [380, 295, 80], and after optimization [215, 250, 50], the comparison revealed a 10.89 increase in reachability and 1.35 increase in the workspace of the robot.

The genetic algorithm combined with fuzzy logic successfully determined optimal robotic configurations based on critical performance metrics, such as robot workspace, surgical instrument insertion depth, and the position of the instrument outside the patient by determining the optimal position of the RCM and the optimal length of some structural links. Simulation tests demonstrated robust system performance, confirming that the optimized parameters resulted in superior workspace, better reachability, and thus, better manoeuvrability and reliability. Additionally, virtual simulations validated the practical feasibility of integrating the optimized parameters into a functional parallel robotic system within an operating room setting.

4. Discussion

The optimization of a 3-DOF parallel robotic system for laparoscopic pancreatic surgery, using an integrated genetic algorithm and fuzzy logic approach, has demonstrated significant potential in addressing critical challenges inherent in surgical robotics. This study illustrates the distinct advantages of leveraging fuzzy logic to optimize the design parameters of a robotic system, effectively handling uncertainties and complexities associated with minimally invasive surgical procedures.

Initially, the robotic design optimization started from an unoptimized Remote Center of Motion (RCM) point located at coordinates [100, 0, 300]. Using the genetic algorithm coupled with fuzzy logic evaluation, the system determined an optimized position for the RCM at coordinates [119.003337, −146.610801, 269.07376]. This new location was identified to ensure superior workspace coverage and enhanced manoeuvrability of the surgical instrument, addressing recorded points obtained from manual manipulation of the surgical instrument during preliminary tests.

Fuzzy logic’s incorporation into the optimization framework provided a notable advantage over traditional deterministic methods. By employing fuzzy membership functions, the approach successfully captured the inherent uncertainties of surgical environments, converting these uncertainties into quantifiable measures to guide optimization. This flexibility enabled the optimization framework to incorporate linguistic and heuristic knowledge, significantly enhancing the robot’s performance concerning critical surgical factors such as workspace accessibility, instrument manoeuvrability, and surgeon ergonomics.

The optimized design demonstrated substantial improvements in key performance metrics, particularly in robot workspace accessibility and instrument insertion precision. The genetic algorithm determined optimal lengths for critical robotic links: specifically, 213.5455 mm for link a₁, 248.48918 mm for link a₃, and 48.5886 mm for link a_2min. These values were practically adjusted to 215 mm, 250 mm, and 50 mm, respectively, to facilitate mechanical implementation. By converting geometric parameters into fuzzy membership functions and aggregating them into a coherent suitability score, the robot’s configurations were optimized to maximize operational effectiveness. Simulation results underscored that the optimized robotic system exhibited superior capabilities in navigating narrow anatomical pathways critical for pancreatic surgery. This was evidenced by the improved reachability ratio and increased convex hull volume, both of which directly translate to enhanced manoeuvrability and surgical precision.

While this study uses a genetic algorithm based on fuzzy logic exclusively for offline preoperative optimization of the design parameters of the robot, future clinical implementation requires that the robotic system meet real-time performance standards during the surgery. In order to achieve this requirement, several strategic improvements can be systematically integrated into the robotic control architecture. Development of a fuzzy inference system for the control logic of the robot constitutes an effective approach. By carefully optimizing the fuzzy rules and membership functions in such a manner to limit the number of linguistic variables used by the fuzzy logic controller, it is possible to reduce computational complexity. Implementing high-performance computational modules like Graphics Processing Units (GPU_s) may lead to a considerable increase in the computational speed. Another promising approach involves employing adaptive and incremental fuzzy inference methods, which dynamically adjust their outputs by only recalculating altered inference paths. This incremental updating dramatically decreases computational load and latency without compromising accuracy, making the robotic system highly responsive to dynamic surgical environments.

The simulation results further validated the optimized design. Specifically, simulations performed with Siemens NX 2406 highlighted the practical feasibility and potential reliability of the robotic system in a real clinical context. The virtual test from initial to target positions exhibited accurate and stable operation, emphasizing the optimized robot’s capability to navigate precise paths required during surgical interventions. The time history diagrams for the instrument tip and active joints further confirmed the system’s operational efficiency and stability.

Despite the positive outcomes, some considerations remain for future improvements. Firstly, the current optimization framework should be complemented by empirical validation through phantom trials to confirm practical performance and patient outcomes. Secondly, further refinement of fuzzy rules and membership functions based on extensive experimental feedback could enhance the robot’s adaptability to a broader range of surgical scenarios. Additionally, the potential integration of machine learning techniques could further refine the fuzzy optimization framework, automating parameter adjustments based on large datasets obtained from real-world operations.

The genetic algorithm based on fuzzy logic fitness evaluation has proven itself an effective method for enhancing robotic surgical designs, offering superior flexibility, precision, and operational relevance compared to traditional deterministic methods. This approach presents a promising direction for further advancements in robotic surgical systems, particularly for complex procedures like pancreatic surgery.

Diversity in patient anatomy and tissue characteristics is a major challenge in robotic surgery. The variability in the stiffness of tissues, their shapes, and their relative positions adds uncertainty to the actual surgical environment as well as the workspace and the interaction mechanisms between the surgical robotic instruments and the tissues. This problem is manageable with imaging and modeling specific to the patient prior to the surgery, as this enables optimization processes geared to every unique situation. Furthermore, the incorporation of intraoperative feedback mechanisms such as force feedback allow for modifications to the robot’s configuration in real time, which in turn accommodates the variability in anatomy. Issues such as mechanical tolerances and the deformation of instruments also impact the accuracy of positioning. While performing the surgery, certain components like flexible shafts and joints have the potential to undergo deformation or backlash, which affect precision and repeatability. To overcome these issues, high-fidelity mechanical modeling with detailed tolerance design is essential. Additionally, closed-loop control approaches that make use of feedback from the robots in real time enhance the ability to compensate for mechanical changes that deviate from the ideal design, which guarantees that the instruments stay on the pre-defined path. Further development of the robotic system presented within this paper focuses on the development of real-time control strategies to mitigate environmental disturbances and integrating machine learning techniques to dynamically adapt the robotic system to the operating field.

5. Conclusions

The integration of fuzzy logic and genetic algorithm methodologies has significantly enhanced the design and performance of the robotic system for minimally invasive pancreatic surgery. By optimizing the RCM position and essential robotic link parameters, the approach yielded measurable improvements in surgical precision and workspace accessibility. The optimized position of the RCM and lengths of the robotic links substantially increased the robotic system’s precision and reachability, verified through simulation studies that demonstrated enhanced manoeuvrability, as well as adherence to intended surgical paths. Results from the simulations validated the practicality and operational reliability of the optimized robotic design, indicating strong potential for clinical adoption. Future research incorporating empirical validation, experimental model development, phantom-based experimental studies, and advanced machine learning techniques for dynamically refining fuzzy logic parameters could further enhance the system’s clinical effectiveness, adaptability, and overall performance, significantly contributing to the evolution and broader adoption of robotic-assisted surgical technologies.