Extended fault-pair Boolean table based test points selection for robotic systems

1. INTRODUCTION

With the rapid advancements in robotics, internal circuit systems, particularly analog circuits, have become critical components prone to frequent failures due to their nonlinear characteristics and the complexity of operating environments^[1,2]. Research indicates that more than 80% of circuit failures originate from analog circuits, which directly affect the stability and functionality of robotic systems^[3,4]. As a result, efficiently isolating analog circuit faults has emerged as a pivotal challenge in ensuring the reliability of robotic systems. Design for testability (DFT) plays a crucial role in fault diagnosis of analog circuits^[5]. By introducing test points within the circuit to capture critical node signals, DFT significantly enhances the observability of the circuit, thereby improving its fault diagnosability. This approach not only increases test coverage but also effectively reduces fault localization time and lowers maintenance costs. Moreover, the effective implementation of testability design supports real-time condition monitoring, enabling the system to perform self-detection and self-diagnosis during operation, thereby enhancing equipment reliability and maintenance efficiency. However, an excessive number of test points can result in information redundancy, increased system complexity, and elevated testing costs, necessitating optimization of test point selection^[6]. The optimization of test point selection refers to the process of strategically identifying and selecting the most effective set of test points in a system or circuit to achieve accurate fault diagnosis or isolation while minimizing cost, complexity, and redundancy. This process aims to balance the trade-off between diagnostic accuracy and the number of test points^[7].

In recent years, significant research has been conducted on test point selection for analog circuits. Fault dictionary-based approaches to test point selection have emerged as a relatively mature method (with integer coding tables being a specific form of fault dictionary, which will not be differentiated here). A tolerance handling method was proposed in^[8] for diagnosing soft faults in analog circuits, addressing diagnostic inaccuracies caused by component tolerances. This method introduced a slope fault model and combines optimization theory with threshold coefficients, leading to a novel fault dictionary-based test point selection approach. A depth-first graph search-based test point selection method was given in^[9] for analog circuit fault dictionaries, maximizing information gain during the graph node expansion process. This ensures that each additional test point provides the maximum information, facilitating the achievement of a globally minimal test point set. Subsequently, a heuristic graph search approach was proposed by integrating the concept of entropy from information theory and combining fuzzy sets caused by component tolerances with fault voltage distributions in^[10] to evaluate the optimality of test points. A clustering discretization method was applied to overcome the limitations of traditional fault dictionaries in handling component tolerances and continuous-value monitoring variables in^[11]. This method represented fault modes as a set of single or multiple integer codes by introducing an extended fault dictionary. Building on this, an extended integer-coded dictionary method was provided to address fault localization issues in switch-mode power supplies in^[12], utilizing optimal boundary determination techniques to enhance feature separation. To improve the computational efficiency of fault dictionaries, an efficient test point selection method was proposed in^[13] that optimizes the test point set by progressively eliminating isolated faults and test points, thereby resolving the issues of excessive computation time and high dimensionality encountered in traditional approaches.

The above analog circuit test point selection algorithms are all based on fault dictionary technology. However, fault dictionary technology is not a highly precise method. When the number of test points increases, not only does the diagnostic accuracy decrease, but the computational complexity also increases. A more precise fault pair Boolean table technique was proposed in^[14,15], overcoming the limitations of traditional integer-coded table techniques, which cannot isolate all faults. The fault pair Boolean table is constructed based on node voltage values. For two faults, if the voltage difference at the same test point exceeds a given threshold^[16], the two faults are considered isolable, and the corresponding position is marked as 1. Otherwise, if the voltage difference is smaller than the threshold, the faults are deemed non-isolable, and the corresponding position is marked as 0. The given threshold involves a degree of subjectivity and lacks general applicability. Thus, an ambiguity model based on the normal distribution assumption was proposed using the overlap area between probability density curves as a criterion for judgment in^[17]. Considering the diversity of sample distributions, a kernel density estimation (KDE) as a substitute for the aforementioned normal distribution was presented in^[15], offering a more practical approach. However,^[17] and^[18] are still considered as extensions of the fault dictionary technique, which limits the effectiveness of the methods. To improve and overcome the shortcomings of existing approaches,^[19] combined the concept of fault pairs with the ambiguity gap calculation method, defining a new approach for calculating the fault pair isolation capability at test points. Based on this, a new test point selection method was developed using the fault pair isolation table. Subsequently, a new similarity coefficient criterion was put forward to determine fault isolation degree in^[20], taking into account the fact that component tolerance circuit output responses approximately follow a normal distribution.

The methods in^[19] and^[20] can effectively improve isolation accuracy and select more reasonable test points. However, they assume that circuit output responses follow a normal distribution, which may not hold in real-world scenarios. In real-world scenarios, fault data often originate from heterogeneous devices or distributed sensors, resulting in significant distribution shifts due to measurement deviations, environmental noise, and varying operating conditions. To address this issue, federated transfer learning (FTL) has emerged as a promising solution. By enabling cross-domain knowledge transfer through encrypted parameter exchange, FTL eliminates the need for sharing raw data. For example, Yang et al. proposed a targeted transfer learning framework based on distribution barycenter mediation (TTL-DBM)^[21]. This approach employs optimal transport theory to construct the Wasserstein barycenter between the source and target domains as an intermediary, and dynamically aligns marginal and conditional distributions via federated learning. The method achieved high-precision feature adaptation under decentralized data conditions in mechanical fault diagnosis, demonstrating the effectiveness of distribution alignment strategies in complex system diagnostics. This provides important insights for analog circuit fault diagnosis: by adopting non-parametric distribution modeling, it is possible to overcome the limitations of traditional methods that rely on specific distribution assumptions, thereby enhancing diagnostic robustness in complex environments. Therefore, this paper proposes a novel test point selection method that employs the Bhattacharyya coefficient (BC) to quantify distribution overlap and utilizes KDE to model circuit output responses. This approach eliminates the dependence on specific distribution assumptions, making test point selection more general and adaptable to complex environments. The BC measures the similarity between probability distributions by calculating the overlap between two distributions, which enhances fault distinguishability and improves fault isolation accuracy. Meanwhile, KDE, as a non-parametric probability density estimation method, models circuit output distributions without assuming a specific distribution form. This allows the proposed method to accommodate a broader range of data distributions beyond the traditional normal distribution assumption, enhancing its applicability and flexibility in real-world circuit testing scenarios. The main contributions of this paper are as follows.

1. The BC is utilized to calculate overlap areas, replacing the traditional distance computations based on individual samples in fault-pair Boolean tables. This distribution-based approach effectively mitigates inaccuracies arising from reliance on single sample calculations.

2. KDE is employed to model the data distribution in fault-pair Boolean tables, eliminating the traditional normal distribution assumption and providing a more flexible and accurate distribution estimation.

3. The Grey Wolf optimization (GWO) algorithm is utilized to estimate the bandwidth parameter in KDE, achieving a globally optimal solution and ensuring the adaptability and accuracy of KDE across various datasets.

The remaining sections are organized as follows. A BC inspired fault-pair Boolean table is established in Section 2. Then, KDE is stated for KDE approximation and bandwidth is estimated by the GWO algorithm in Section 3. Section 4 gives a whole process of test point selection based on the extended fault-pair Boolean table. Finally, a conclusion is drawn in Section 5.

2. FAULT-PAIR BOOLEAN TABLE ESTABLISHMENT

In this section, the BC is incorporated to improve the fault isolation capability of the fault-pair Boolean table. Firstly, a fault-pair Boolean table is created for fault isolation. Then, the BC is applied to calculate the values within the table.

2.2. Fault-pair Boolean table based on BC

The distance in Equation (2) is initially calculated based on a single sample. This single-sample approach doesn't fully capture the underlying variations in the data, potentially resulting in incorrect fault isolation. To overcome this limitation, the BC is employed to calculate the overlap area between different distributions derived from multiple samples^[22-24]. The BC is expressed as

(3) $$ \begin{equation} BC\left( {{f}_{u}}, {{f}_{v}} \right)=\int{\sqrt{{{p}_{u}}\left( x \right){{p}_{v}}\left( x \right)}dx} \end{equation} $$

where $$ {{p}_{u}}\left( x \right) $$ and $$ {{p}_{v}}\left( x \right) $$ denote the corresponding probability density distributions of fault $$ {{f}_{u}} $$ and $$ {{f}_{v}} $$, respectively. The value of the BC ranges from 0 to 1. When $$ {{p}_{u}}\left( x \right)={{p}_{v}}\left( x \right) $$, BC = 1. Then, Equation (2) can be modified based on the BC as follows.

(4) $$ \begin{equation} d({{f}_{u}}, {{f}_{v}})=1-BC\left( {{f}_{u}}, {{f}_{v}} \right) \end{equation} $$

which is further illustrated in Figure 1. As given in Figure 1A, if $$ d=1 $$, then $$ {{f}_{u}} $$ and $$ {{f}_{v}} $$ are completely separable. In Figure 1B, if $$ d=0 $$, that is the curves of $$ {{p}_{u}}(x) $$ and $$ {{p}_{v}}(x) $$ almost overlap, then $$ {{f}_{u}} $$ and $$ {{f}_{v}} $$ cannot be separated. In Figure 1C, if $$ 0<d<1 $$, then $$ {{f}_{u}} $$ and $$ {{f}_{v}} $$ are partially separable.

Figure 1. The curves of probability density function $$ {{p}_{u}}(x) $$ and $$ {{p}_{v}}(x) $$. (A) The distribution of fault $$ f_u $$ and fault $$ f_v $$ corresponds to no overlap, and fault $$ f_u $$ and fault $$ f_v $$ can be completely separated; (B) The distributions corresponding to fault $$ f_u $$ and fault $$ f_v $$ overlap completely, and fault $$ f_u $$ and fault $$ f_v $$ cannot be separated; (C) The distributions corresponding to fault $$ f_u $$ and fault $$ f_v $$ overlap partially, and fault $$ f_u $$ and fault $$ f_v $$ can be partially separated.

Based on the fault-pair Boolean table derived using BC, if the values calculated by Equation (4) between two faults at a given test point $$ {{t}_{j}} $$ exceed a certain threshold $$ \varepsilon $$ within the range [0, 1], the faults can be isolated at that test point; otherwise, they cannot be isolated.

3. BC APPROXIMATION VIA KDE

The probability density function in Equation (4) is unknown in the circuits of robotic systems. Typically, it is assumed to follow a specific probability distribution for inference, which introduces subjectivity. To address this situation, this paper applies KDE to infer the probability density function from available samples^[25,26]. Let $$ X_{u}^{j}=\left( x_{u, 1}^{j}, x_{u, 2}^{j}, \cdots , x_{u, q}^{j} \right) $$ be $$ q $$ random samples collected from test point $$ {{t}_{j}}(j=1, 2, \cdots n) $$. The probability density function $$ {{p}_{u}}(x)(u=0, 1, 2, \cdots , m) $$ of fault $$ {{f}_{u}} $$ at test point $$ {{t}_{j}} $$ can be derived by the KDE as:

where h is a bandwidth parameter, $$ K $$ is the kernel function and $$ \int{K\left( x \right)dx=1} $$.

As is noted that the bandwidth $$ h $$ plays a key role in KDE for determining the smoothness of the estimate. A bandwidth that is too small results in an estimate that is overly 'rough', whereas a bandwidth that is too large may excessively smooth the data, causing important details to be overlooked^[27]. Therefore, an appropriate bandwidth is crucial for ensuring an accurate approximation of BC. The GWO, a global optimization method simulating the hunting behavior of grey wolves^[28], is adopted to estimate the bandwidth parameter.

The GWO has very few parameters, and the initial search does not require any derivation information. Gray wolves are typically classified into four types: $$ \alpha $$ wolves, $$ \beta $$ wolves, $$ \delta $$ wolves, and $$ \omega $$ wolves. The $$ \alpha $$ wolf is the leader of the pack, responsible for guiding the group in hunting prey. In optimization algorithms, the $$ \alpha $$ wolf represents the optimal solution. $$ \beta $$ wolves assist the $$ \alpha $$ wolf and correspond to the second-best solutions in the algorithm. $$ \delta $$ wolves follow the commands and decisions of the $$ \alpha $$ and $$ \beta $$ wolves, playing the role of scouts and guardians for the pack. Wolves with lower fitness levels may be demoted from $$ \alpha $$ or $$ \beta $$ to the $$ \delta $$ level. $$ \omega $$ wolves are the lowest-ranking wolves, following the actions of the higher-ranking wolves.

Gray wolves gradually approach and encircle their prey. The following update equation is proposed to model this behavior.

where $$ U(t+1) $$ denotes the position of the gray wolf at time $$ t+1 $$; $$ U(t) $$ represents the position of the gray wolf at time $$ t $$; $$ {{U}_{P}} $$ denotes the position of the prey; $$ A $$ and $$ C $$ are two synergy coefficient vectors, which are further expressed by

where $$ a $$ is a vector and its components are linearly decreased from 2 to 0 over iterations; $$ {{r}_{1}} $$ and $$ {{r}_{2}} $$ are random variables in [0, 1].

To further simulate the hunting behavior of grey wolves, it is assumed that $$ \alpha $$, $$ \beta $$, and $$ \delta $$ possess strong capabilities to identify the position of potential prey. During each iteration, the best three wolves ($$ \alpha $$, $$ \beta $$, and $$ \delta $$) in the current population are retained, and the positions of $$ \omega $$ are updated based on the position information of these top three wolves. Therefore, the wolves of $$ \omega $$ should be obliged to update their positions as follows:

where $$ {{U}_{1}}(t+1), {{U}_{2}}(t+1), {{U}_{3}}(t+1) $$ denote the step sizes and directions that the $$ \omega $$ moves towards $$ \alpha $$, $$ \beta $$, and $$ \delta $$ at time $$ t+1 $$, respectively. They are derived by

where $$ {{U}_{\alpha }}\left( t \right) $$, $$ {{U}_{\beta }}\left( t \right) $$, and $$ {{U}_{\delta }}\left( t \right) $$represent the positions of the $$ \alpha $$, $$ \beta $$, and $$ \delta $$ at time t, respectively; the $$ {{A}_{1}} $$, $$ {{A}_{2}} $$, and $$ {{A}_{3}} $$ denote random vectors inferred by Equation (7); the $$ {{C}_{1}} $$, $$ {{C}_{2}} $$, and $$ {{C}_{3}} $$ indicate random vectors given in Equation (8).

It is worth noting that the position of $$ U $$ in Equation (9) corresponds to the bandwidth parameter $$ h $$ that requires optimization. The detailed estimation process is thoroughly defined in Algorithm 1.

4. TEST POINT SELECTION

5. CASE STUDY

In this section, negative feedback circuit (NFC) and active filter circuit (AFC) are used to verify the validity of the proposed method.

5.1. Experiment on NFC

The NFC feeds the output signal back to the input signal in proportion to the controller in a robotic system. This feedback mechanism enables the system to dynamically adjust its behavior by comparing the actual output with the desired input, thereby minimizing errors and enhancing performance. Studying the test point selection for NFC is essential to ensure the robotic system's reliability, efficiency, and fault tolerance^[29]. The schematic diagram of the NFC is given in Figure 2.

Figure 2. NFC schematic diagram. NFC: Negative feedback circuit.

As illustrated in Figure 2, the input signal is a sinusoidal wave with a frequency of 1 kHz and an amplitude of 7 mV. The supply voltage is set to 15 V, and the tolerances of the resistors and capacitors are set at 5%. Data collection is performed using six test points $$ T=\left\{ {{t}_{1}}, {{t}_{2}}, {{t}_{3}}, {{t}_{4}}, {{t}_{5}}, {{t}_{6}} \right\} $$. Additionally, as presented in Table 1, ten fault types are employed for experimental validation.

Table 1

Fault types of the NFC

Fault type	Fault mode
NFC: Negative feedback circuit.
$$ f_1 $$	$$ Q_1 $$ B–C short
$$ f_2 $$	$$ Q_1 $$ C–E short
$$ f_3 $$	$$ Q_2 $$ C–E short
$$ f_4 $$	$$ R_8 = 130 \Omega $$
$$ f_5 $$	$$ R_9 = 39 \mathrm{k}\Omega $$
$$ f_6 $$	$$ Q_1 $$ B–C short & $$ Q_2 $$ C–E short
$$ f_7 $$	$$ Q_1 $$ C–E short & $$ R_8 = 130 \Omega $$
$$ f_8 $$	$$ Q_1 $$ C–E short & $$ R_9 = 39 \mathrm{k}\Omega $$
$$ f_9 $$	$$ Q_1 $$ B–C short & $$ R_8 = 130 \Omega $$ & $$ R_8 = 39 \mathrm{k}\Omega $$
$$ f_{10} $$	$$ Q_1 $$ C–E short & $$ R_8 = 130 \Omega $$ & $$ R_9 = 39 \mathrm{k}\Omega $$

5.1.1. Experimental results of the proposed method

A total of $$ {{10}^{3}} $$ test data samples are collected for each fault type using the Monte Carlo method from six test points. In this study, the threshold is set to $$ \varepsilon =0.95 $$^[30]. By traversing all fault types and test points, the fault-pair Boolean table is constructed using the BC and KDE methods, as given in Table 2.

Table 2

Fault-pair Boolean tables based on BC

No.	Fault pair	$$ t_1 $$	$$ t_2 $$	$$ t_3 $$	$$ t_4 $$	$$ t_5 $$	$$ t_6 $$	$$ NT_i $$
BC: Bhattacharyya coefficient.
1	$$ (f_0, f_1) $$	0	1	0	0	0	0	1
2	$$ (f_0, f_2) $$	0	1	0	0	0	0	1
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
11	$$ (f_1, f_2) $$	1	0	0	0	0	0	1
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
20	$$ (f_2, f_3) $$	0	1	0	0	0	1	2
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
28	$$ (f_3, f_4) $$	0	1	0	0	1	0	2
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
35	$$ (f_4, f_5) $$	0	1	0	0	0	0	1
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
41	$$ (f_5, f_6) $$	0	1	0	1	1	0	3
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
46	$$ (f_6, f_7) $$	1	0	0	0	1	0	2
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
50	$$ (f_7, f_8) $$	0	1	0	0	0	0	1
$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$	$$\vdots$$
55	$$ (f_9, f_{10}) $$	1	0	0	1	1	0	3
	$$ NI(t_i) $$	14	41	0	8	19	8	-

According to Table 2, considering the fault-pair isolation results, test points $$ {{t}_{1}} $$ and $$ {{t}_{2}} $$ are initially added to the optimal test point set $$ {{S}_{opt}} $$ due to their ability to isolate key fault pairs effectively. Subsequently, the fault pairs that can be isolated by $$ {{t}_{1}} $$ and $$ {{t}_{2}} $$ are eliminated from further consideration. After completing this process, it is observed that all fault pairs are successfully isolated. Therefore, $$ {{t}_{1}} $$ and $$ {{t}_{2}} $$ are identified as the optimal test points for the NFC system, ensuring accurate and efficient fault isolation.

5.1.2. Comparison results of the test points selection

Comparisons are made with two types of fault-pair tables presented in^[19] and^[20], both of which assume that the data follows a normal distribution. Additionally,^[19] further requires calculations to be performed within a specified range $$ \left[ {{u}_{i}}-w\cdot {{\sigma }_{i}}, {{u}_{i}}+w\cdot {{\sigma }_{i}} \right] $$, where the value of w influences the precision of the results. In this study, thresholds of $$ \varepsilon =0.95 $$ and $$ \varepsilon =0.98 $$ are selected for comparison. To evaluate the effectiveness of GWO in optimizing the kernel bandwidth, a comparison is also made with the results obtained using particle swarm optimization for kernel bandwidth optimization (PSO-KDE). The detailed results are presented in Table 3.

Table 3

Comparison results of the proposed method with the methods in^[19] and^[20]

Method		Sopt	Fault isolation degree	Number of remainder fault pairs
PSO-KDE: Particle swarm optimization for kernel bandwidth optimization.
$$ \varepsilon=0.95 $$	[19] ($$ w=1.96 $$)	$$ t_3, t_4 $$	4	7
	[19] ($$ w=2.576 $$)	$$ t_3, t_4 $$	4	7
	[19] ($$ w=3 $$)	$$ t_3, t_4 $$	4	8
	[20]	$$ t_2, t_3, t_4 $$	7	2
	PSO-KDE	$$ t_1, t_2 $$	11	-
	The proposed method	$$ t_1, t_2 $$	11	-
$$ \varepsilon=0.98 $$	[19] ($$ w=1.96 $$)	-	-	55
	[19] ($$ w=2.576 $$)	$$ t_3, t_4 $$	4	8
	[19] ($$ w=3 $$)	$$ t_3, t_4 $$	4	8
	[20]	$$ t_2, t_3, t_4 $$	7	2
	PSO-KDE	$$ t_1, t_2, t_5 $$	11	-
	The proposed method	$$ t_1, t_2 $$	11	-

In Table 3, the column labeled $$ {{S}_{opt}} $$ represents the optimal set of test points. The fourth column indicates the fault isolation degree, which refers to the total number of faults that can be fully isolated using the selected optimal test points. The final column shows the number of faults pairs that cannot be isolated. The test point selection performance is compared under two thresholds ($$ \varepsilon $$ = 0.95 and $$ \varepsilon $$ = 0.98). For $$ \varepsilon $$ = 0.95, test points $$ {{t}_{3}} $$ and $$ {{t}_{4}} $$ are selected in^[19] under different $$ w $$ values yield a fault isolation degree of 4, with 7, 7, and 8 remaining fault pairs, respectively, indicating limited isolation performance. In contrast, the test points selected $$ {{t}_{2}}, {{t}_{3}}, {{t}_{4}} $$ in^[20] achieve a fault isolation degree of 7, reducing the number of remaining fault pairs to 2. Both the PSO-KDE algorithm and the proposed method select test points $$ {{t}_{1}} $$ and $$ {{t}_{2}} $$, achieving the highest fault isolation degree of 11 with zero remaining fault pairs, demonstrating superior performance.

For the threshold $$ \varepsilon $$ = 0.98,^[19] selects the same test points as for $$ \varepsilon $$ = 0.95, maintaining a fault isolation degree of 4 with 55, 8, and 8 remaining fault pairs, respectively. Similarly, $$ {{t}_{2}}, {{t}_{3}}, {{t}_{4}} $$ are selected in^[20] achieving a fault isolation degree of 7 with two remaining fault pairs. The proposed method selects the test points $$ {{t}_{1}}, {{t}_{2}} $$ once again, achieving the highest fault isolation degree of 11 with zero remaining fault pairs, demonstrating a best performance. Although the PSO-KDE method can also achieve the highest fault isolation degree of 11 with zero remaining fault pairs, it selects three test points $$ {{t}_{1}}, {{t}_{2}}, {{t}_{5}} $$, which is more than the number of test points required by the proposed method.

In summary, the proposed method consistently achieved complete fault isolation across different thresholds, while selecting the fewest test points. It significantly outperformed the methods in^[19] and^[20], and PSO-KDE, demonstrating its superior fault isolation capability and adaptability.

5.2. Experiment on AFC

The AFC can filter and purify signals in robotic systems, eliminating noise interference and providing high-precision feedback signals for the controller. It can also monitor and control the current in the motor drive circuits in real-time, ensuring the stability and accuracy of the motor current, preventing abnormal motor operation or damage caused by excessive or insufficient current. This improves the control precision and operational efficiency of the motor, enabling precise control of robot movements. The schematic diagram of the AFC is given in Figure 3.

Figure 3. AFC schematic diagram. AFC: Active filter circuit.

In the AFC circuit, the tolerance of resistors and capacitors is set to 5% and 10%, respectively. The second circuit experiment mainly simulates hard faults in the circuit. A total of 19 hard faults were simulated, and ten test points ($$ {{t}_{1}} $$ to $$ {{t}_{10}} $$) were used to collect fault data. The fault modes are shown in Table 4.

Table 4

Fault types of the AFC

Fault type	Fault mode	Fault type	Fault mode
AFC: Active filter circuit.
$$f_0$$	Normal	$$f_{10}$$	$$R_7$$ open
$$f_1$$	$$R_1$$ short	$$f_{11}$$	$$R_8$$ open
$$f_2$$	$$R_1$$ open	$$f_{12}$$	$$R_9$$ open
$$f_3$$	$$R_2$$ short	$$f_{13}$$	$$R_{10}$$ open
$$f_4$$	$$R_2$$ open	$$f_{14}$$	$$R_{11}$$ open
$$f_5$$	$$R_3$$ open	$$f_{15}$$	$$R_{12}$$ open
$$f_6$$	$$R_5$$ short	$$f_{16}$$	$$C_1$$ open
$$f_7$$	$$R_5$$ open	$$f_{17}$$	$$C_2$$ open
$$f_8$$	$$R_6$$ short	$$f_{18}$$	$$C_3$$ open
$$f_9$$	$$R_6$$ open	$$f_{19}$$	$$C_4$$ open

A total of $$ {{10}^{3}} $$ test data samples are collected for each fault type using the Monte Carlo method from ten test points. Based on the collected data, a fault pair Boolean table is constructed and used to perform optimal test point selection. As in Section 5.1, the proposed method is compared with the methods in^[19] and^[20]. In the experiment, thresholds $$ \varepsilon =0.95 $$ and $$ \varepsilon =0.98 $$ are selected to determine whether two faults can be isolated. To evaluate the effectiveness of GWO in optimizing the kernel bandwidth, a comparison is also made with the results obtained using PSO-KDE. The simulation and comparison results are presented in Table 5.

Table 5

Comparison results of the proposed method with the methods in^[19] and^[20]

Method		Sopt	Fault isolation degree	Number of remainder fault pairs
PSO-KDE: Particle swarm optimization for kernel bandwidth optimization.
$$ \varepsilon=0.95 $$	[19] ($$ w=1.96 $$)	$$ t_3, t_4, t_7, t_8, t_9, t_{10} $$	7	74
	[19] ($$ w=2.576 $$)	$$ t_3, t_4, t_7, t_8, t_9, t_{10} $$	7	74
	[19] ($$ w=3 $$)	$$ t_3, t_4, t_7, t_8, t_9, t_{10} $$	9	68
	[20]	$$ t_3, t_4, t_7, t_9, t_{10} $$	13	20
	PSO-KDE	$$ t_1, t_2, t_3, t_4 $$	16	6
	The proposed method	$$ t_1, t_2, t_3, t_4 $$	16	6
$$ \varepsilon=0.98 $$	[19] ($$ w=1.96 $$)	-	-	190
	[19] ($$ w=2.576 $$)	$$ t_3, t_4, t_7, t_8, t_9, t_{10} $$	7	74
	[19] ($$ w=3 $$)	$$ t_3, t_4, t_7, t_8, t_9, t_{10} $$	7	74
	[20]	$$ t_3, t_4, t_7, t_9, t_{10} $$	13	20
	PSO-KDE	$$ t_1, t_2, t_3, t_4 $$	16	6
	The proposed method	$$ t_1, t_2, t_3, t_4 $$	16	6

In Table 5, the test point selection performance is compared under two thresholds ($$ \varepsilon $$ = 0.95 and $$ \varepsilon $$ = 0.98). For $$ \varepsilon $$ = 0.95, test points $$ t_3, t_4, t_7, t_8, t_9, t_{10} $$ are selected in^[19] under different $$ w $$ values yield a fault isolation degree of 7, 7, and 9, with 74, 74, and 68 remaining fault pairs, respectively, indicating limited isolation performance. In contrast, the test points selected $$ t_3, t_4, t_7, t_9, t_{10} $$ in^[20] achieve a fault isolation degree of 13, reducing the number of remaining fault pairs to 20. Both the PSO-KDE algorithm and the proposed method select test points $$ t_1, t_2, t_3, t_4 $$ and $$ t_1, t_2, t_3, t_4 $$, achieving the highest fault isolation degree of 16 with six remaining fault pairs, demonstrating superior performance.

For the threshold $$ \varepsilon $$ = 0.98,^[19] selects the same test points as for $$ \varepsilon $$ = 0.95, maintaining a fault isolation degree of 7 with 190, 74, and 74 remaining fault pairs, respectively. Similarly, $$ t_3, t_4, t_7, t_9, t_{10} $$ are selected in^[20] achieving a fault isolation degree of 13 with 20 remaining fault pairs. The proposed method selects the test points $$ t_1, t_2, t_3, t_4 $$ once again, achieving the highest fault isolation degree of 16 with six remaining fault pairs, demonstrating a best performance. Although the PSO-KDE algorithm and the proposed method select test points $$ t_1, t_2, t_3, t_4 $$ and $$ t_1, t_2, t_3, t_4 $$, both achieve the highest fault isolation degree of 16 with six remaining fault pairs, demonstrating superior performance.

In summary, it can be seen from the above two circuit case studies that the proposed method isolates the maximum number of faults at different thresholds while selecting the minimum number of test points. It significantly outperformed the methods in^[19] and^[20], and PSO-KDE, demonstrating its superior fault isolation capability and adaptability.

6. CONCLUSIONS

This paper has presented a novel test point selection method that extends the fault-pair Boolean table into a distribution-based framework. By integrating BC to quantify distributional overlap and KDE to model circuit response distributions without normal distribution assumptions, the method has effectively improved test point selection. The GWO algorithm has been employed to optimize the bandwidth parameter in KDE, ensuring accurate estimation. Experimental results on a NFC have demonstrated that the proposed method consistently achieves the highest fault isolation degree of 11 with zero remaining fault pairs, even under different thresholds ($$ \varepsilon $$ = 0.95 and $$ \varepsilon $$ = 0.98). Furthermore, the experimental results further show that the method on the AFC consistently achieves the highest fault isolation degree of 16 with six remaining fault pairs under different thresholds ($$ \varepsilon $$ = 0.95 and $$ \varepsilon $$ = 0.98). In contrast, traditional methods in^[19] and^[20] resulted in multiple unresolved fault pairs. These findings confirm the superior accuracy, adaptability, and efficiency of the proposed method, making it highly suitable for fault isolation in analog circuits.

However, there are still several areas that warrant further exploration for fault-pair Boolean tables. As robotic systems are dynamic and subject to frequent changes in operational conditions, it is crucial to develop adaptive fault-pair Boolean tables that can update in real-time as new fault data is collected. Another important direction for future research is the integration of uncertainty modeling into fault-pair Boolean tables, as real-world fault data is often noisy and incomplete.

DECLARATIONS