Introduction

Amid the continuous growth and advancement of international trade and maritime transportation, ship pilotage has become a cornerstone in ensuring the safe navigation of vessels through complex and challenging waterways. Its impact extends beyond the mere safety of the vessels during navigation, encompassing factors such as the efficiency of port operations, the demand for environmental preservation, and the impetus behind holistic economic advancement. Recent research highlights that the majority of ship-related accidents can be attributed to negligence or improper actions during the ship pilotage phase. As a result, enhancing safety and efficiency in ship pilotage, coupled with the unequivocal determination of its reliability, is important in ensuring secure and sustainable vessel navigation1,2,3.

Ship pilotage plays a vital role in ensuring the safe passage of vessels through narrow, complex, and hazardous waterways. These waters may pose challenges such as navigational obstacles, shallows, and rapid changes in water currents. Ship Pilot guidance by pilots can mitigate the risks of accidents like collisions and groundings, safeguarding both the lives of crew members and vessel properties. Furthermore, Ship Pilots guide vessels entering and leaving ports in waters adjacent to the harbor, ensuring safe berthing and departure. Effective pilotage operations help minimize waiting times, improve operational efficiency, ensure timely cargo loading and unloading, and facilitate the seamless flow of trade. Piloting also plays a significant role in environmental protection. Besides, Ship Pilots can plan navigational routes based on vessel characteristics and water conditions, avoiding sensitive ecological areas and minimizing the impact on marine ecosystems. Competent pilotage services can attract more vessels to ports, fostering trade and economic development. Simultaneously, reducing accidents and navigational delays can mitigate related economic losses4,5,6. Ship pilots form the backbone of the ship pilotage system, playing a central role in the system’s safety dynamics while remaining highly sensitive to varying system conditions and external factors. As vessel reliability continues to improve, pilot reliability takes precedence in ensuring ship pilotage safety. Therefore, research on pilot human reliability and the impact of reliability on ship pilotage conditions is of paramount significance7–8.

Human Reliability Analysis (HRA) is a systematic research methodology designed to analyze, predict, mitigate, and prevent human errors. It involves qualitative and quantitative analysis and assessment of human reliability as a core research objective. Human reliability pertains to maintaining high performance, efficiency, and safety in various work and life situations9–10. The study of human reliability investigates the interaction and influence between humans, machines, and the environment. It primarily focuses on examining human behavior, identifying potential risks and errors in operational and decision-making processes, and implementing preventive and corrective measures to ensure safe and efficient workflows. Human reliability is an interdisciplinary field involving psychology, human factors engineering, management, and more. Its significance is widely recognized in various sectors such as industry, transportation, healthcare, and beyond.

Many scholars have proposed numerous models for HRA. For instance, Zhang Ailin from Jimei University proposed an improved CREAM method11, which integrates fuzzy numbers into the CREAM-based evaluation system, thereby minimizing the influence of expert subjectivity and enabling the quantification of human reliability. Hongjun Fan used the CREAM model as the basic model and modified it to make it suitable for high-energy physics evaluations during the LNG refueling process12. H. Pei proposed a reliability analysis model for marine transportation that combines quantitative and qualitative analysis, which is applicable to both pre-accident and post-accident analyses of marine transportation incidents. Unlike traditional reliability analysis models, the quantitative analysis section employs an optimized CREAM method combined with the GDM classification approach to separately analyze human factor events and non-human factor events13. BAFANDEGAN proposed a human reliability calculation method combining the Bayesian Best-Worst Method and the CREAM method, which reduces the probability of human error to only 0.00017214. Yaju Wu proposed a human reliability analysis model for high-temperature molten metal operations in the metallurgy industry, based on CREAM15, fuzzy logic theory, and Bayesian networks (BN). This model provides comprehensive rules for commonly used performance conditions in conventional CREAM methods to evaluate various conditions in high-temperature molten metal operations in the metallurgical industry. Li Na and colleagues proposed a method that extends the CREAM framework by integrating the Decision-Making Trial and Evaluation Laboratory (DEMATEL) to assign weights to Common Performance Conditions (CPC), thereby refining the calculation formula for the total weighting factor in human error rates16. Ma Yanhui and others proposed a model combining an improved weighted Bayesian network (BN) and the CREAM method to enhance the accuracy of method evaluation and expand its applicability17. This model first uses the grey relational analysis and the Decision-Making Trial and Evaluation Laboratory method to obtain the weights of factors in the scenario, then uses Bayesian network inference to get the adjusted probability distribution of the CPC factor nodes. The weighted factors are then sampled and simulated to obtain the probability distribution of the control mode under the scenario. Yang Zhiyong and others proposed a method to improve the accuracy of CREAM analysis18, reducing the subjectivity of certain system input data. They employed Interval-Valued Intuitionistic Fuzzy Sets (IVIFS) and the Analytic Network Process (ANP) to develop an optimized method for quantitatively calculating CPC weights, accounting for the interdependencies among CPCs and enhancing the alignment of reliability analysis with real-world environments. BAFANDEGAN et al. proposed using the Decision Analysis Network Process (DANP) method to weight CPC factors19, thereby improving the accuracy of human reliability calculations. Xi Yongtao and others analyzed the impact of human factors in piloting20.

However, most human reliability studies focus on computational methods and have not been utilized as influencing factors in ship pilotage. Addressing this gap is critical, as incorporating human reliability into ship pilotage control models bridges a significant void in maritime safety research. This integration recognizes the pivotal role of human performance in navigating complex maritime environments, offering a more comprehensive perspective on factors affecting pilotage safety and efficiency. It offers a more holistic understanding of the variables that affect pilotage safety and efficiency, potentially leading to enhanced training protocols, better risk management strategies, and ultimately, a reduction in maritime accidents attributable to human error. In order to solve the above problems, this paper employs the PID control model to establish the relationship between human reliability and PID parameters, develop a ship pilotage model based on human reliability, and analyze the pilotage control performance under varying levels of human reliability.

The remainder of the paper is structured as follows: “Human reliability model based on improved CREAM” section presents the theoretical foundation of the CREAM method. “Development of a pilotage control model based on human reliability” section details the calculation of human factor reliability and the construction of a pilotage model based on human factor reliability under yaw conditions. “Simulations results and analysis” section examines the control performance of the pilotage model under varying levels of human factor reliability. Finally, the conclusions are drawn and summarized.

Human reliability model based on improved CREAM

This paper incorporates human reliability as a critical influencing factor in ship pilotage and analyzes it using the Cognitive Reliability and Error Analysis Method (CREAM). Human reliability is quantified by an improved CREAM method. The technological framework of this paper is illustrated in Fig. 1.

The basic method of CREAM

The CREAM method categorizes the factors influencing human reliability into nine groups, referred to as CPC factors, as summarized in Table 1. These factors, including organizational completeness, working conditions, and human-machine interface perfection, capture various dimensions of human performance, such as task complexity, environmental conditions, and team collaboration. Each CPC factor is evaluated based on three potential effects—improvement, significance, and reduction—which assess its ability to enhance, significantly impact, or diminish human reliability. Utilizing the evaluation results of these CPC factors, CREAM defines four control modes: strategic, tactical, opportunistic, and chaotic, as illustrated in Fig. 2. These control modes correspond to decreasing levels of human reliability, with ‘Strategic’ representing high reliability and stable performance, and ‘Chaotic’ indicating low reliability and error-prone operations. Table 1; Fig. 2 together provide a structured framework for understanding how CPC factors influence human reliability and control system performance.

Fig. 1
figure 1

Technological framework.

Full size image
Table 1 Parameters of GA method.
Full size table
Fig. 2
figure 2

Schemes follow the same formatting.

Full size image

The basic method of CREAM

Although the CREAM method has been widely applied, it is limited by its strong subjectivity and imprecise evaluation intervals, with overlapping boundaries often reducing its accuracy and reliability. To address these limitations, this paper employs an improved CREAM method for quantifying human reliability.

Calculation of environmental indicators

The impact of CPC factors on the context is divided into three types: improvement, insignificance, and reduction. We quantify these three impact effects, and the definition function is as follows:

$${C_i}=\left\{ {\begin{array}{*{20}{l}} {-1}&{improve} \\ 0&{Not significant} \\ 1&{reduce} \end{array}} \right.$$
(1)

Among them, improve means “if the factor improves human reliability”, not significant means “if the factor has no significant effect”, and reduce means “if the factor reduces human reliability”.

To integrate the combined effects of all CPC factors, the situational environment index \(\beta \) is defined as:

$$\beta =\sum\limits_{{i=1}}^{9} {9{\omega _i}{C_i}} {\text{ }}$$
(2)

where \(\:{\omega\:}_{i}\) represents the weight of the \(\:i-th\) CPC factor affects, indicating its relative significance in influencing human reliability, and \(\:{C}_{i}\) denotes the corresponding effect of the factor (improve, not significant, or reduce). These weights \(\:{\omega\:}_{i}\) can be determined using methods such as DEMATEL or expert evaluation. This index provides a quantitative measure of the situational environment’s overall impact on human reliability.

Calculation of the CPC factor weight

The Decision-Making Trial and Evaluation Laboratory (DEMATEL) method is employed to calculate the weights of CPC factors. This method determines the logical relationships between system elements by integrating expert experience and knowledge. Initially, a direct impact matrix is constructed to represent the influence relationships between system elements. This matrix is then normalized to produce a comprehensive impact matrix, which quantifies the overall interactions among the elements.

From the comprehensive impact matrix, the cause degree \(\:{H}_{i}\) and center degree \(\:{F}_{i}\) of each element are calculated, providing insights into the interrelationships between the system elements and the relative status and weight of each CPC factor. The cause degree reflects the extent to which a CPC factor influences others, while the center degree indicates its overall importance within the system. This method considers both direct and indirect relationships among indicators to comprehensively evaluate their influence.

Using the expert evaluation system, a basic scale is established between factors. By standardizing this scale, the influence degree \(\:{H}_{i}\) and center degree \(\:{F}_{i}\) of each CPC factor are quantified. The weight of each CPC factor \(\:{\omega\:}_{i}\) is then determined using the DEMATEL method as follows:

$${\omega _{\text{i}}}={{\left( {{H_i}+{F_j}} \right)} \mathord{\left/ {\vphantom {{\left( {{H_i}+{F_j}} \right)} {\sum\limits_{{i=1}}^{9} {\left( {{H_i}+{F_j}} \right)} }}} \right. \kern-0pt} {\sum\limits_{{i=1}}^{9} {\left( {{H_i}+{F_j}} \right)} }}$$
(3)

where \(\:{H}_{i}+{F}_{j}\) denotes centrality, which indicates the degree of influence of a certain type of CPC factor and the environmental indicators.

Calculation of the CPC factor weight

After considering the weights of the nine CPC factors, the error probability of ship operators during the pilotage process, referred to as the Collision Avoidance Failure Probability (CFP), is calculated as:

$$CFP=0.0056 \times {10^{0.25\beta }}{\text{ }} $$
(4)

where \(\:\beta\:\) is the situational environment index derived from the weighted sum of CPC factors. The human reliability, defined as the complement of the CFP, is expressed as:

$$R=1 - CFP=1-0.0056 \times {10^{0.25 \cdot \sum\limits_{{I=1}}^{9} {{\omega _i}{C_i}} }}{\text{ }} $$
(5)

This formula quantifies the overall human reliability based on the combined effects of the CPC factors and their weights.

Development of a pilotage control model based on human reliability

From “ Human reliability model based on improved CREAM” section, the human reliability of the pilot is obtained, and the size of the pilot’s reliability under different scene conditions is obtained by calculating human reliability. However, calculating human reliability involves the pilot’s reliability and does not consider the impact on the pilot under different reliability. Therefore, the PID model21,22,23 is used to simulate the pilotage process of the pilot, establish the functional relationship between human reliability and PID parameters, and construct a pilot model based on human reliability. Through simulations, the pilot’s planned path under varying levels of reliability is tracked to analyze its influence on the pilotage process.

Establishment of the pilotage control mode

The pilotage control model is developed through the ship’s kinematic and mechanical analysis, incorporating both position and attitude control. The position error, representing the difference between the planned and actual positions, is calculated as:

$${e_p}={r_p}-{x_p}$$
(6)

This error serves as the input to the PID controller, whose output is given by:

$${u_p}={k_p} \cdot e\_p+{k_i} \cdot \int {e\_p\left( t \right)} dt+{k_d} \cdot \frac{{de\_p\left( t \right)}}{{dt}}$$
(7)

where \(\:{k}_{p}\), \(\:{k}_{i}\) and \(\:{k}_{d}\) are the proportional, integral, and derivative gains, respectively. Similarly, the attitude error is calculated as:

$${e_h}={r_h}-{x_h}$$
(8)

These equations collectively enable the precise control of the ship’s position and attitude during pilotage operations.

Taking the attitude error as the PID input, the attitude controller output is expressed as follows:

$${u_h}={k_p}*e\_h+{k_i}*\int {e\_h(t)} dt+{k_d}*\frac{{de\_h(t)}}{{dt}}$$
(9)

where \(\:e\_h\) represents the error between the planned and actual headings, and \(\:{k}_{p}\), \(\:{k}_{i}\) and \(\:{k}_{d}\) are the proportional, integral, and derivative gains, respectively. This PID equation enables precise adjustments to the ship’s heading by compensating for immediate errors, cumulative deviations, and predicted future discrepancies.

To further analyze the forces acting on the ship, the moment in water is calculated as:

$$M=-D(l)\sin (\varphi )-D(r)\sin (\varphi )+X(l)\cos (\varphi )-X(r)\cos (\varphi )+Y(l)+Y(r)$$
(10)

where \(\varphi \) is the ship’s heading, \(D(l)\) and \(D(r)\) are the ship’s flow resistance on its left and right side, \(X(l)\) and \(X(r)\) represent the hydrodynamic forces on its left and right side, \(Y(l)\) and \(Y(r)\) represent the direction-finding force on its left and right side.

The propulsion control, which ensures the ship maintains its desired speed, is defined as:

$$T={K_T}({r_T}-{x_T}){\text{ }}$$
(11)

where \(\:{r}_{T}\) is the desired velocity, \(\:{x}_{T}\) is the actual velocity, and \(\:{k}_{T}\) is the propulsion coefficient.

Considering that rudder angle control is nonlinear24,25, this paper adopts a linear model to simulate the control of rudder angle, i.e., it constructs the relationship between the rudder angle and roll angle:

$$\theta = - {K_\theta } \cdot \varphi {\text{ }}$$
(12)

where\(\:\:\theta\:\) is the rudder angle, \(\:{K}_{\theta\:}\) is the scale factor of the rudder angle control and \(\varphi \) is the heading angle. This linear approximation simplifies the nonlinear dynamics of rudder control while maintaining practical accuracy.

In summary, the PID model of ship pilotage control based on kinematics and mechanical analysis is obtained.

The rudder angle control of the ship is:

$${u_\varphi }= - {K_\theta } \cdot {e_h}$$
(13)

The propulsion control of the ship is:

$${u_T}={K_T}\left( {{r_T} - {x_T}} \right)$$
(14)

Finally, the position control of the ship is represented as:

$${u_h}={k_p} \cdot {e_h}+{k_i} \cdot \int {{e_h}\left( t \right)} dt+{k_d} \cdot \frac{{d{e_h}\left( t \right)}}{{dt}}$$
(15)

This complete PID model integrates attitude, propulsion, and position control to achieve accurate and stable ship pilotage. The proportional link addresses real-time adjustments, the integral link compensates for accumulated errors, and the derivative link anticipates future deviations to ensure smooth navigation.

Functional relationship between human reliability and PID parameters

The PID control model constructed above does not consider human reliability, which significantly affects system performance. To address this, the paper integrates human reliability into the PID control model by establishing a functional relationship between human reliability and PID parameters. This relationship enables dynamic adjustments of the PID parameters based on varying levels of human reliability.

Functional relationship between K p and human reliability

Through engineering experiments, it has been observed that excessively large \(\:{\varvec{K}}_{\varvec{p}}\) values lead to significant steady-state errors, while overly small \(\:{\varvec{K}}_{\varvec{p}}\) values slow the system’s response speed. Therefore, having too large or too small thresholds \(\:{\varvec{K}}_{\varvec{p}}\) is problematic. Thus, the optimal value is between the two thresholds. The functional relationship between human reliability and \(\:{\varvec{K}}_{\varvec{p}}\) is constructed as follows:

$$ K_{d} = \left\{ {\begin{array}{*{20}l} {K_{d} < K_{{d\min }} ,K_{d} > K_{{d\max }} } \hfill & {R = 0} \hfill \\ { - \frac{{(P - K_{{d\min }} )^{{}} + (P - K_{{d\min }} )}}{{R + 1}} + p + (P - K_{{d\min }} )} \hfill & {P \le K_{d} \le K_{{d\max }} } \hfill \\ {\frac{{(K_{{d\max }} - P)^{{}} + (K_{{d\max }} - P)}}{{R + 1}} + p + (K_{{d\max }} - P)} \hfill & {K_{{d\min }} \le K_{d} \le P} \hfill \\ \end{array} } \right. $$
(16)

Here \(\:{K}_{Pmin}\) and \(\:{K}_{Pmax}\) represent the minimum and maximum values of \(\:{K}_{P}\), \(\:P\) denotes human reliability, and \(\:R\) is an adjustment factor controlling the range of \(\:{K}_{P}\). This relationship ensures that the PID control system adapts dynamically to different human reliability levels.

The functional relationship is illustrated in Fig. 3.

Fig. 3
figure 3

Relationship between \({K_P}\) and human reliability.

Full size image

As shown in Fig. 3, the curve demonstrates that at lower human reliability levels, \(\:{K}_{P}\) is relatively high to compensate for potential delays or errors. As human reliability improves, \(\:{K}_{P}\) decreases and stabilizes at an optimal value, minimizing overshooting and enhancing system stability. This trend highlights the importance of incorporating human reliability into PID parameter adjustments to optimize system performance.

Functional relationship between K i and human reliability

The integral parameter \(\:{K}_{i}\) in a PID controller plays a vital role in correcting steady-state errors by accumulating past deviations. However, its value must be carefully controlled to avoid excessive oscillations or insufficient correction. Through engineering experiments, it has been observed that when \(\:{K}_{i}\) is too small, the steady-state error remains uncorrected, resulting in reduced accuracy, and when \(\:{K}_{i}\) is too large, the system oscillates, negatively impacting stability. To ensure optimal performance, \(\:{K}_{i}\) must be dynamically adjusted within a threshold range. The mathematical relationship between \(\:{K}_{i}\) and human reliability (R) is defined as:

$$ K_{i} = \left\{ {\begin{array}{*{20}l} {K_{i} > K_{{i\max }} } \hfill & {R = 0} \hfill \\ {K_{i} = \frac{{ - (K_{{i\max }} - I)}}{{R + 1}} + K_{{i\max }} } \hfill & {R \in (0,1]} \hfill \\ \end{array} } \right. $$
(17)

where \(\:{K}_{imax}\) is maximum allowable value for \(\:{K}_{i}\), \(\:R\) is human reliability, \(\:I\) is reference value representing initial integral conditions. This function ensures that \(\:{K}_{i}\) decreases with improving human reliability, stabilizing at \(\:{K}_{imax}\) when \(\:R=1\).

The functional relationship is illustrated in Fig. 4.

Fig. 4
figure 4

Relationship between \({K_i}\) and human reliability.

Full size image

As shown in Fig. 4, the mathematical relationship between the I parameter and the human factor reliability is constructed according to the relevant law of PID parameter adjustment, and the function curve shows that the I parameter tends to be optimal with the continuous improvement of the human factor reliability.

Functional relationship between K d and human reliability

Through the engineering experimental method26,27, it is known that \(\:{K}_{d}\) adjusts the system’s dynamic performance. A too-small \(\:{K}_{d}\) overshoots the system and reduces human stability, while a too-large \(\:{K}_{d}\) causes the system to break in advance, leading to an excessive system response time. Therefore, \(\:{K}_{d}\) has a too-large and too-small parameter threshold, and from the two thresholds to the optimal value, the functional relationship between \(\:{K}_{d}\) and human reliability is constructed as follows:

$$ K_{d} = \left\{ {\begin{array}{*{20}l} {K_{d} < K_{{d\min }} ,K_{d} > K_{{d\max }} } \hfill & {R = 0} \hfill \\ { - \frac{{(P - K_{{d\min }} )^{{}} + (P - K_{{d\min }} )}}{{R + 1}} + p + (P - K_{{d\min }} )} \hfill & {P \le K_{d} \le K_{{d\max }} } \hfill \\ {\frac{{(K_{{d\max }} - P)^{{}} + (K_{{d\max }} - P)}}{{R + 1}} + p + (K_{{d\max }} - P)} \hfill & {K_{{d\min }} \le K_{d} \le P} \hfill \\ \end{array} } \right. $$
(18)

where \(\:{K}_{dmin}\) and \(\:{K}_{dmax}\) represent the minimum and maximum allowable values for \(\:{K}_{d}\), R denotes human reliability, and P indicates the current level of system reliability. Figure 5 presents the dynamic adjustment of \(\:{K}_{d}\) based on human reliability:

Fig. 5
figure 5

Relationship between \({K_d}\) and human reliability.

Full size image

As shown in Fig. 5, The red curve highlights the higher \(\:{K}_{d}\) values required at lower reliability levels to enhance error correction, and the blue curve shows the stabilization of \(\:{K}_{d}\) as human reliability improves, preventing excessive adjustments and ensuring smooth system performance.

Through mathematical modeling, human reliability is combined with the P, I, and D parameters representing the PID control system’s rapidity28,29,30, accuracy, and stability. Human reliability is a factor affecting the system to construct a pilot control model based on human reliability31,32. By integrating human reliability into the tuning of \(\:{K}_{d}\), the PID control system achieves optimal dynamic performance across varying conditions. This mathematical relationship ensures that the system remains adaptive and robust, meeting the demands of both low and high human reliability scenarios.

Simulations results and analysis

A pilotage control model based on human reliability is constructed by establishing the functional relationship between human reliability and PID parameters. This model is simulated to evaluate pilotage control performance under varying reliability conditions, and the system’s characteristics are analyzed.

PID parameter tuning

By tuning the PID parameters, the corresponding parameter thresholds are obtained to construct the specific functional relationship between the PID parameters and human reliability. This allows us to obtain the PID parameter values under different human reliability21. In this paper, the method of engineering experiment was used to adjust the PID parameters21. In this study, the engineering experimental method involves evaluating system response to predefined error conditions and adjusting PID parameters accordingly. This approach ensures a balanced performance between system stability, accuracy, and responsiveness under varying reliability scenarios. Tables 2 and 3 present the thresholds and optimal values of PID parameters for linear and complex paths, respectively.

Table 2 Threshold and optimal values of the PID parameters for a linear path.
Full size table

where Table 2 provides the parameter thresholds for a linear path, P (Proportional gain) controls the system’s reaction to current errors, ensuring proportional corrections, I (Integral gain) addresses steady-state errors by accumulating past errors, and D (Derivative gain) predicts future errors based on their rate of change, enhancing dynamic performance.

Table 3 Threshold and optimal values of the PID parameters for a complex path.
Full size table

where Table 3 outlines the parameter thresholds for a complex path, the higher maximum values reflect the need for greater correction in dynamic and unpredictable conditions.

The analysis indicates that the relatively smaller threshold ranges of PID parameters in linear paths reflect the stability and predictability of system dynamics in such scenarios. In contrast, the wider range of parameter values in complex paths highlights the system’s need to address more dynamic and uncertain conditions, where higher P, I and D values are required to ensure robust error correction and control.

Ship linear path simulation

Next, this paper set a straight path and simulate the pilotage effect of the ship under the straight channel through the path tracking situation33. Then, we substitute the PID parameter threshold obtained by the engineering experiment method into the constructed functional model of human factors and PID parameters to obtain the PID parameters34 under different R (Human reliability) under the straight path (Table 4).

Ship pilotage effect under a straight path

The PID parameters under the straight path are substituted under the pilotage control model, and the pilotage control effect is obtained as shown in Fig. 6. The destination path (red line) represents the ideal trajectory, while the actual path (green line) shows the ship’s trajectory under different human reliability conditions.

Table 4 PID parameter values under different human reliability.
Full size table
Fig. 6
figure 6

Straight path following cases.

Full size image

The analysis of Fig. 6 demonstrates that at lower R values (R = 0.5 or 0.6) significant deviations occur between the actual and destination paths, indicating reduced pilotage accuracy and stability caused by lower human reliability. As R increases (R = 0.9 or 1.0), the actual path aligns more closely with the destination path, reflecting improved control accuracy and minimized deviations. These findings underscore the critical role of human reliability in determining the precision of ship pilotage, where higher R values ensure greater trajectory stability and accuracy. By quantifying the relationship between PID parameters and human reliability, the simulation effectively validates the pilotage control model’s performance under straight path conditions.

Analysis of ship pilotage characteristics under a straight path

Under the straight path scenario, the pilotage control system’s performance in terms of accuracy, stability, and response speed is evaluated under varying levels of human reliability R. The accuracy of the pilotage is assessed by calculating the route difference between the actual path and the target path, while stability is measured using the maximum overshoot of the system path. Figure 7 illustrates the straight-line path tracking error for different R values.

As shown in Fig. 7, lower R values (R = 0.5 or 0.6) result in significant deviations between the actual and target paths, indicating reduced accuracy and stability. Conversely, higher R values (R = 0.9 or 1.0) exhibit minimal deviations, reflecting improved pilotage precision.

Figure 8 depicts the maximum overshoot analysis, demonstrating that higher R values significantly reduce the overshoot, thereby enhancing system stability. The results confirm that increasing human reliability not only improves trajectory tracking accuracy but also stabilizes the dynamic response of the pilotage control system.

Figure 9 presents the response time analysis, indicating the time required for the system to stabilize under different levels of human reliability. As R values increase, the response time decreases significantly, with R = 0.5 requiring the longest stabilization time of approximately 0.25 h and R = 1.0 stabilizing in the shortest time. This trend highlights the dynamic improvement in system responsiveness as human reliability improves, ensuring faster adaptation to target paths.

By systematically evaluating accuracy, stability, and response speed, this analysis validates the critical role of human reliability in determining the effectiveness of pilotage control under straight path conditions.

Fig. 7
figure 7

Straight-line path tracking error.

Full size image
Fig. 8
figure 8

Maximum overshoot analysis.

Full size image

As shown in Fig. 9, The rapidity of ship pilotage under different reliability is obtained by calculating the response speed of the path followed by the system.

Fig. 9
figure 9

Response time analysis.

Full size image

Ship complex path simulation analysis

By intercepting a certain section of the path data of the incoming vessel in Qingdao Port, we simulate and analyze the pilotage control of ships under complex paths. For this trial, the PID parameter thresholds are obtained (Table 5) by the engineering experiment method. They are substituted into the proposed functional model of human factors and PID parameters to obtain the PID parameters under different R (human reliability) in complex paths.

Table 5 PID parameter values under different human reliability.
Full size table

Tracking effect under complex paths

The PID parameters under the complex path are substituted under the pilotage control model, and the pilotage control effect is obtained as shown in Fig. 10. As illustrated in Fig. 10, the tracking performance improves significantly with increasing R values. At lower R values (e.g., 0.5 and 0.6), the actual paths exhibit considerable deviations from the target path, indicating reduced pilotage accuracy. However, at higher R values (e.g., 0.9 and 1.0), the actual path closely aligns with the target path, demonstrating superior trajectory tracking precision. The results highlight that higher human reliability ensures more accurate and consistent tracking along complex paths.

Fig. 10
figure 10

Complex path following cases.

Full size image

Analysis of ship pilotage characteristics under complex path

Under a straight path, the pilotage control system’s accuracy, stability, and rapidity under different human reliability are analyzed, i.e., the pilotage effect of ships under different reliability.

The accuracy of the pilotage effect under different reliability is obtained by calculating the route difference between the actual path and the target path, as depicted in Fig. 11. The figure presents the path tracking error analysis. The error curves show that lower R values (e.g., 0.5) correspond to large and unstable deviations, whereas higher R values (e.g., 0.9 and 1.0) result in smoother error curves with reduced magnitude and frequency. This demonstrates that increased human reliability significantly reduces tracking errors, contributing to improved system accuracy and robustness.

Fig. 11
figure 11

Complex path tracking error.

Full size image

The stability of ship pilotage under different reliability is obtained by calculating the maximum overshoot of the system path following. As shown in Fig. 12, the maximum overshoot decreases with increasing R values. For low R values (e.g., 0.5 and 0.6), the system experiences greater overshoot, which compromises stability. In contrast, higher R values (e.g., 0.9 and 1.0) effectively minimize overshoot, enhancing the stability of the pilotage system. This indicates that human reliability plays a crucial role in ensuring system stability under complex path conditions.

The rapidity of ship pilotage under different reliability is obtained by calculating the response speed of the path following the system. Figure 13 illustrates the response time analysis. As R values increase, the response time decreases significantly, highlighting faster system adjustments and improved dynamic response capabilities. At low R values (e.g., 0.5), the system response is slow and less efficient, while higher R values (e.g., 0.9 and 1.0) ensure quicker and smoother responses, essential for effective pilotage in dynamic environments.

The combined analysis of tracking accuracy, stability, and response speed demonstrates that higher human reliability leads to a more effective pilotage control system under complex path scenarios. Specifically, increased R values result in more accurate trajectory tracking, improved stability through reduced overshoot, and faster dynamic response, ensuring smoother and more precise ship control. These findings underscore the critical role of human reliability in optimizing the performance of pilotage control systems, particularly in challenging and dynamic environments.

Fig. 12
figure 12

Maximum overshoot analysis.

Full size image
Fig. 13
figure 13

Response time analysis.

Full size image

Conclusion

This study integrates human reliability analysis into ship pilotage control to enhance maritime safety. First, an improved CREAM method is utilized to quantify human reliability, addressing the subjectivity and variability of traditional approaches. Then, a PID-based pilotage control model is constructed, establishing functional relationships between human reliability and PID parameters. The model is validated through simulations of ship pilotage under straight-line and complex path scenarios, using entry data from Qingdao Port.

The results demonstrate that human reliability is a critical factor influencing the effectiveness of pilotage control. Higher reliability significantly reduces pilotage deviations and response times, thereby improving control accuracy and stability. Specifically, in low-reliability scenarios, the response time for navigation control is four times longer, and the route error is twice as large as that in high-reliability scenarios. The pilotage control model constructed in this study provides a quantitative and visual representation of the impact of human reliability on ship navigation, offering valuable insights for improving port safety and operational efficiency. Key findings from this study include:

  1. 1)

    Faster Response to Emergencies: Pilots with higher reliability demonstrate quicker reactions to potentially hazardous situations, enhancing navigation safety.

  2. 2)

    Reduced Pilotage Deviations: High-reliability pilots exhibit fewer deviations from the intended navigation path, ensuring smoother and more precise ship handling.

  3. 3)

    Improved Stability and Accuracy: Higher pilot reliability results in reduced route errors and overshoot, contributing to the stability and robustness of the pilotage control system.

Future studies will focus on incorporating additional data sources to enhance the applicability and accuracy of the proposed model. Integrating real-time data from advanced sensors, such as environmental conditions (e.g., wind, waves, visibility) and vessel dynamics, could improve adaptability to diverse maritime scenarios. Expanding to multi-source information, including acoustic signals, visual monitoring systems, and pilot physiological indicators, would provide a more comprehensive understanding of pilot behavior and reliability. Validation across diverse ports with varying geographical and operational conditions would help generalize the model, while employing machine learning techniques to dynamically optimize PID parameters based on real-time inputs could significantly enhance the precision and adaptability of pilotage control systems.

In summary, this study highlights the crucial role of human reliability in ship pilotage control and provides a foundation for further research in integrating multi-source data and advanced technologies to improve navigation safety and efficiency. By addressing these future directions, the model can be refined to support intelligent, adaptive, and real-time pilotage control systems.