High-precision navigation system construction method based on five-sensor conical configuration

Abstract

The invention relates to the technical field of navigation systems, and particularly discloses a high-precision navigation system construction method based on five-sensor conical configuration, and the method specifically comprises the following steps: S1, defining a carrier coordinate system and a sensor position matrix; s2, determining the optimal sensor space configuration of the five sensors; and S3, fault detection and isolation of the five sensors. The method has the beneficial effects that by adopting the non-orthogonal conical configuration of the five sensors, the data of the five sensors are mutually fused, so that the error of the whole navigation system is effectively reduced; the sensor fault is detected by using the multi-threshold moving average residual method, so that the misjudgment probability can be reduced while the sensor fault is effectively detected; by constructing the position matrix for the sensor and performing variance analysis, the optimal mounting angle of the sensor is found, and the mounting efficiency and the precision of the navigation system are improved.

Description

Technical Field

[0001] The present invention relates to the field of navigation system technology, and specifically to a method for constructing a high-precision navigation system based on a five-sensor conical configuration, which can be widely used in scenarios such as drones and robots that require high navigation accuracy. Background Art

[0002] Inertial sensors are important sensors used to measure the motion and posture of an object. They primarily include accelerometers and gyroscopes. Accelerometers calculate acceleration by measuring the deformation or displacement of sensitive components based on Newton's second law, while gyroscopes measure angular velocity using the principles of conservation of angular momentum or the Coriolis force. They offer strong autonomy, real-time performance, high accuracy, and robust interference immunity. They are widely used in a wide range of fields, including aerospace, automotive, and robotics.

[0003] A three-sensor orthogonal navigation system arranges multiple sensors perpendicular to each other, forming a structure similar to an orthogonal coordinate system. Leveraging the independent and non-interference characteristics of each sensor axis, it can accurately measure physical quantities in different directions. For example, in three-dimensional space, it can independently obtain information about an object's x, y, and z axes. This system is widely used in fields requiring precise perception of an object's motion. A four-sensor non-orthogonal navigation system incorporates a redundant sensor, with the sensor input axis perpendicular to the four faces of a regular tetrahedron, which can improve navigation accuracy and fault tolerance to a certain extent.

[0004] However, existing navigation systems with three orthogonal sensors lack fault tolerance. When one sensor fails, the remaining two sensors can only measure acceleration and angular velocity in one plane, and cannot measure acceleration and angular velocity in the entire three-dimensional space. While systems with four non-orthogonal sensors have a certain degree of fault tolerance, the system error increases significantly when a sensor fails. Four-sensor non-orthogonal systems only introduce one redundant sensor, which has limited improvement in system navigation accuracy. Furthermore, existing navigation system fault detection generally uses a single-threshold detection mechanism (such as the chi-square test), directly determining whether the system is abnormal based on a preset confidence threshold. Once the chi-square statistic exceeds the threshold, it is determined that there is a system fault. This conventional fault detection method uses a single-level threshold trigger, which cannot distinguish the fault level. It is very sensitive to transient noise, easily causing misjudgment, and has low fault location efficiency. This conventional fault detection method lacks preprocessing of the residual vector and does not eliminate the influence of transient interference. At the same time, the conventional determination of the specific installation angle of the sensor in the navigation system generally relies on personal experience, and a better installation angle is determined through multiple experiments, or the sensor installation angle is determined based on the mechanical structure of the navigation equipment. This method lacks theoretical analysis and cannot accurately find the optimal installation angle. The experimental process requires a lot of time and energy, and is inefficient.

[0005] Therefore, it is necessary to further solve the above problems. The applicant has proposed a method for constructing a high-precision navigation system based on a five-sensor conical configuration. Summary of the Invention

[0006] In view of the shortcomings of the existing technology, the present invention provides a method for constructing a high-precision navigation system based on a five-sensor conical configuration, which solves the problems raised by the above-mentioned background technology.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] A method for constructing a high-precision navigation system based on a five-sensor conical configuration, the method specifically comprising the following steps:

[0009] S1. Define the carrier coordinate system and sensor position matrix;

[0010] S2, determine the optimal sensor space configuration of the five sensors;

[0011] S3, five sensor fault detection and isolation;

[0012] The specific steps of step S1 are as follows:

[0013] S11, take the center of mass of the carrier as the origin of the coordinate system, and establish an orthogonal coordinate system (x b ,y b ,z b ), where x b The axis is along the horizontal forward direction of the carrier, y b The axis is along the horizontal right direction of the carrier, z b The axis is perpendicular to the carrier plane and points vertically upward;

[0014] S12. The direction of each sensor is defined by the relative position of the sensor in the orthogonal coordinate system, using the unit vector v i Indicates the direction of the sensitive axis of the i-th sensor, i = 1, 2, 3, 4, 5, vector v i The definition is as follows:

[0015] v i =cos(yaw i )cos(pitch i )·i+cos(yaw i )sin(pitchi ) ·j+sin(yaw i )·k

[0016] Where i is the direction of the coordinate axis x in the orthogonal coordinate system b The same unit vector, j is the direction of the coordinate axis y in the orthogonal coordinate system bThe same unit vector, k is the direction of the coordinate axis z in the orthogonal coordinate system b Same unit vector, yaw i Indicates that the i-th sensor is around z b Axis azimuth, pitch i Indicates that the i-th sensor is around y b The pitch angle of the axis;

[0017] S13. Construct the sensor position matrix H, project the sensors onto the x, y, and z axes of the orthogonal coordinate system, and use a matrix to represent the projection relationship of the five sensors. This projection relationship is represented by the position matrix H, and the expression is as follows:

[0018]

[0019] As a further solution of the present invention: the specific steps of step S2 include:

[0020] S21, symmetrically distribute the five sensors on the base, with the azimuth angles evenly spaced 72 degrees, and the first sensor and the y of the orthogonal coordinate system b Axis alignment, assuming that the elevation angle of each sensor is α, the sensor position matrix H can be expressed as follows:

[0021]

[0022] As a further solution of the present invention, the specific steps of step S2 also include:

[0023] S22. Determine the sensor elevation angle that minimizes the overall system error, and construct a measurement matrix m of the five sensors:

[0024] m=[m1,m2,m2,m4,m5] T =H·[w x ,w y ,w z ] T +[v1,v2,v3,v4,v5] T

[0025] Among them, w x ,w y ,w z The vector represents the components of the real physical input acceleration and angular rate in the three coordinate axis directions in the orthogonal coordinate system. The multiplication of H and w represents the linear combination of the projection of the physical quantity in the direction of the sensor's sensitive axis, reflecting the theoretical measurement contribution of the physical quantity on each sensor. i It represents the measurement error vector of the i-th sensor, reflecting the inevitable random interference factors in the measurement process, m i represents the measurement value of the i-th sensor.

[0026] As a further solution of the present invention, the specific steps of step S2 also include:

[0027] S23. The least weighted squares method can be used to fuse the measurement data of multiple sensors. The goal of the weighted least squares method is to minimize the weighted residual sum of squares. The objective function is as follows:

[0028] J(w)=(m-Hw) T W(m-Hw)

[0029] Where W is the weight matrix, are the noise variances of the five sensors respectively;

[0030] Derivative the above formula to find the estimated value with the minimum residual The expression is as follows:

[0031]

[0032] Finding the covariance of the estimated values yields:

[0033]

[0034] Assume that the noise of each sensor is independent of each other and the standard deviation of the measurement noise of each sensor is σ v , then the covariance of the estimated value can be simplified as:

[0035]

[0036] When the covariance is minimum, the system noise error is minimum, that is, when the elevation angle α makes det(H T When H) is the largest, the navigation system error is the smallest.

[0037] After calculation, the optimal elevation angle of the sensor is:

[0038]

[0039] det(H T H)=31.25sin 4 αcos 2 α

[0040]

[0041] As a further solution of the present invention: the specific steps of step S3 include:

[0042] S31. Establish the residual history records of the last five moments for each sensor to form a time series buffer. This step disperses the transient interference to multiple time points for analysis. The buffer sequence expression is as follows:

[0043] Q i =[r i (t-5+1),....r i (t-1),r i (t)]

[0044]

[0045] Among them, Q i Represents the residual buffer queue of the i-th sensor, r i Represents the residual of the i-th sensor. Next, the filtered residual vector is constructed by sliding average calculation. The formula is as follows:

[0046]

[0047] in, is the residual vector after processing, Q i (tk) is the k-th historical residual value, and the chi-square statistic is constructed as follows:

[0048]

[0049] As a further solution of the present invention, the specific steps of step S3 also include:

[0050] S32. Select the appropriate threshold value by looking up the table based on the confidence level and the degree of freedom of the statistic. The subscript α is the significance level. When α = 0.05, it corresponds to 95% confidence level, and when α = 0.01, it corresponds to 99% confidence level. n represents the degree of freedom of the statistic. In this system, five sensors are used to estimate three parameters, so the degree of freedom n = 2. When When , the system is judged to be normal and no operation is triggered;

[0051] when When the system enters the warning state, the secondary residual analysis is started, the local residual of each sensor is calculated, and a sliding window is constructed with a window length of N = 50. The sliding window length N = 50 means that the system intercepts the latest 50 consecutive data points of the sensor as the analysis unit each time, and counts the residuals of each sensor in the window that exceed 3σ. i If a sensor has abnormality for ≥10 times in three consecutive windows, it is considered to be faulty.

[0052] when When the system fails, it is immediately judged that the fault occurs. At this time, the faulty sensor is eliminated by traversal, and the i-th sensor is removed from the position matrix H in turn to construct the position matrix H after dimensionality reduction. i (H i ∈R 4×3), construct the chi-square statistic after removing the i-th sensor

[0053]

[0054] Among them, m i , W i are the measurement matrix, estimated value and weight matrix after the i-th sensor respectively. If the new chi-square statistic is satisfied after removing the i-th sensor It is determined that the i-th sensor is faulty.

[0055] Compared with the prior art, the present invention has the following beneficial effects:

[0056] 1. The system constructed by this method introduces two redundant sensors and uses the weighted least squares method to fuse the data of five sensors, thereby reducing the impact of external noise on the entire system, effectively solving the problem of navigation inaccuracy, and reducing the error of the overall navigation system.

[0057] 2. The system constructed by this method introduces redundant sensors and proposes a multi-threshold sliding average residual analysis method to build a fault detection system. When a sensor in the navigation system fails, the faulty sensor can be discovered and eliminated in a timely manner, thereby improving the system's fault tolerance and reducing the probability of misjudgment.

[0058] 3. The system constructed by this method finds the optimal installation angle of the sensor by constructing a position matrix for the sensor and performing variance analysis, thereby improving the installation efficiency and the accuracy of the navigation system. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 A diagram showing the sensor configuration of the navigation system of the present invention;

[0060] Figure 2 This is a flow chart of sensor fault detection according to the present invention;

[0061] Figure 3 A comparison chart of navigation paths generated for different configurations of the system of the present invention;

[0062] Figure 4 This is a navigation speed error diagram for the five-sensor system of the present invention;

[0063] Figure 5 This is a navigation speed error diagram of the three-sensor system of the present invention. DETAILED DESCRIPTION

[0064] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0065] Example

[0066] See also Figure 1-2 The present invention provides a method for constructing a high-precision navigation system based on a five-sensor conical configuration, the method comprising the following specific steps:

[0067] Step 1: Define the carrier coordinate system and sensor position matrix

[0068] 1. Take the center of mass of the carrier as the origin of the coordinate system and establish an orthogonal coordinate system (x b ,y b ,z b ), where x b The axis is along the horizontal forward direction of the carrier, y b The axis is along the horizontal right direction of the carrier, z b The axis is perpendicular to the carrier plane and points vertically upward;

[0069] 2. The orientation of each sensor is defined by its relative position in the orthogonal coordinate system, using the unit vector v i Indicates the direction of the sensitive axis of the i-th sensor, i = 1, 2, 3, 4, 5, vector v i The definition is as follows:

[0070] v i =cos(yaw i )cos(pitch i )·i+cos(yaw i )sin(pitch i )·j+sin(yaw i )·k

[0071] Where i is the direction of the coordinate axis x in the orthogonal coordinate system b The same unit vector, j is the direction of the coordinate axis y in the orthogonal coordinate system b The same unit vector, k is the direction of the coordinate axis z in the orthogonal coordinate system b Same unit vector, yaw i Indicates that the i-th sensor is around z b Axis azimuth, pitch i Indicates that the i-th sensor is around y b The pitch angle of the axis;

[0072] 3. Construct the sensor position matrix H, project the sensors onto the x, y, and z axes of the orthogonal coordinate system, and use a matrix to represent the projection relationship of the five sensors. This projection relationship is expressed by the position matrix H, as follows:

[0073]

[0074] Step 2: Determine the optimal sensor space configuration of the five sensors

[0075] 1. Distribute the five sensors symmetrically on the base, with the azimuth angles evenly spaced 72°. The first sensor is aligned with the y-axis of the orthogonal coordinate system. b Axis alignment, assuming that the elevation angle of each sensor is α, the sensor position matrix H can be expressed as follows:

[0076]

[0077] 2. Determine the sensor elevation angle that minimizes the overall system error and construct the measurement matrix m of the five sensors:

[0078] m=[m1,m2,m2,m4,m5] T =H·[w x ,w y ,w z ] T +[v1,v2,v3,v4,v5] T

[0079] Among them, w x ,w y ,w z The vector represents the components of the real physical input acceleration and angular rate in the three coordinate axis directions in the orthogonal coordinate system. The multiplication of H and w represents the linear combination of the projection of the physical quantity in the direction of the sensor's sensitive axis, reflecting the theoretical measurement contribution of the physical quantity on each sensor. i It represents the measurement error vector of the i-th sensor, reflecting the inevitable random interference factors in the measurement process, m i represents the measurement value of the i-th sensor;

[0080] 3. The least weighted squares method can be used to fuse the measurement data of multiple sensors. The goal of the weighted least squares method is to minimize the weighted residual sum of squares. The objective function is as follows:

[0081] J(w)=(m-Hw) T W(m-Hw)

[0082] Where W is the weight matrix, are the noise variances of the five sensors respectively;

[0083] Derivative the above formula to find the estimated value with the minimum residual The expression is as follows:

[0084]

[0085] Finding the covariance of the estimated values yields:

[0086]

[0087] Assume that the noise of each sensor is independent of each other and the standard deviation of the measurement noise of each sensor is σ v , then the covariance of the estimated value can be simplified as:

[0088]

[0089] When the covariance is minimum, the system noise error is minimum, that is, when the elevation angle α makes det(H T When H) is the largest, the navigation system error is the smallest.

[0090] After calculation, the optimal elevation angle of the sensor is:

[0091]

[0092] det(H T H)=31.25sin 4 αcos 2 α

[0093]

[0094] Step 3: Five sensor fault detection and isolation

[0095] 1. For each sensor, establish the residual history of the last five moments to form a time series buffer. This step disperses the transient interference into multiple time points for analysis. The buffer sequence expression is as follows:

[0096] Q i =[r i (t-5+1),....r i (t-1),r i (t)]

[0097]

[0098] Among them, Q i Represents the residual buffer queue of the i-th sensor, r i Represents the residual of the i-th sensor. Next, the filtered residual vector is constructed by sliding average calculation. The formula is as follows:

[0099]

[0100] in, is the residual vector after processing, Q i (tk) is the k-th historical residual value, and the chi-square statistic is constructed as follows:

[0101]

[0102] 2. According to the confidence level and the degree of freedom of the statistic, select the appropriate threshold by looking up the table The subscript α is the significance level. When α = 0.05, it corresponds to 95% confidence level, and when α = 0.01, it corresponds to 99% confidence level. n represents the degree of freedom of the statistic. In this system, five sensors are used to estimate three parameters, so the degree of freedom n = 2. When When , the system is judged to be normal and no operation is triggered;

[0103] when When the system enters the warning state, the secondary residual analysis is started, the local residual of each sensor is calculated, and a sliding window is constructed with a window length of N = 50. The sliding window length N = 50 means that the system intercepts the latest 50 consecutive data points of the sensor as the analysis unit each time, and counts the residuals of each sensor in the window that exceed 3σ. i If a sensor has abnormality for ≥10 times in three consecutive windows, it is considered to be faulty.

[0104] when When the system fails, it is immediately judged that the fault occurs. At this time, the faulty sensor is eliminated by traversal, and the i-th sensor is removed from the position matrix H in turn to construct the position matrix H after dimensionality reduction. i (H i ∈R 4×3 ), construct the chi-square statistic after removing the i-th sensor

[0105]

[0106] Among them, m i , W i are the measurement matrix, estimated value and weight matrix after the i-th sensor respectively. If the new chi-square statistic is satisfied after removing the i-th sensor It is determined that the i-th sensor is faulty.

[0107] For details, see Figure 3As shown in the figure, a navigation system with a five-sensor configuration and a navigation system with a three-sensor configuration are used to navigate a path with a total length of 150m, and the generated trajectories are compared with the ideal trajectories. The red trajectory in the figure is the ideal trajectory, that is, the accurate trajectory recorded manually, the blue trajectory is the trajectory generated by the navigation system with a five-sensor configuration, and the green trajectory is the trajectory generated by the navigation system with a three-sensor configuration. The final blue trajectory has an eastward position error of 0.8m and a northward error of 0.5m; the green trajectory has an eastward position error of -0.6m and a northward error of -7.2m. It can be seen that the blue trajectory is closer to the ideal trajectory, and the navigation system with a five-sensor configuration has a smaller error.

[0108] For details, see Figure 4-5 As shown, Figure 4 is the navigation speed error diagram of the five-sensor system, Figure 5 This is the navigation speed error diagram of the three-sensor system. From this comparison, we can see that the speed error of the five-sensor system is significantly lower than the speed error of the three-sensor system.

[0109] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A method for constructing a high-precision navigation system based on a five-sensor conical configuration, characterized by: The specific steps of the method include: S1. Define the carrier coordinate system and sensor position matrix; S2, determine the optimal sensor space configuration of the five sensors; S3, five sensor fault detection and isolation; The specific steps of step S1 are as follows: S11, take the center of mass of the carrier as the origin of the coordinate system, and establish an orthogonal coordinate system (x b ,y b ,z b ), where x b The axis is along the horizontal forward direction of the carrier, y b The axis is along the horizontal right direction of the carrier, z b The axis is perpendicular to the carrier plane and points vertically upward; S12. The direction of each sensor is defined by the relative position of the sensor in the orthogonal coordinate system, using the unit vector v i Indicates the direction of the sensitive axis of the i-th sensor, i = 1, 2, 3, 4, 5, vector v i The definition is as follows: v i =cos(yaw i )cos(pitch i )·i+cos(yaw i )sin(pitch i )·j+sin(yaw i )·k Where i is the direction of the coordinate axis x in the orthogonal coordinate system b The same unit vector, j is the direction of the coordinate axis y in the orthogonal coordinate system b The same unit vector, k is the direction of the coordinate axis z in the orthogonal coordinate system b Same unit vector, yaw i Indicates that the i-th sensor is around z b Axis azimuth, pitch i Indicates that the i-th sensor is around y b The pitch angle of the axis; S13. Construct the sensor position matrix H, project the sensors onto the x, y, and z axes of the orthogonal coordinate system, and use a matrix to represent the projection relationship of the five sensors. This projection relationship is represented by the position matrix H, and the expression is as follows:

2. The method for constructing a high-precision navigation system based on a five-sensor conical configuration according to claim 1, characterized in that: The specific steps of step S2 include: S21, symmetrically distribute the five sensors on the base, with the azimuth angles evenly spaced 72 degrees, and the first sensor and the y of the orthogonal coordinate system b Axis alignment, assuming that the elevation angle of each sensor is α, the sensor position matrix H can be expressed as follows:

3. The method for constructing a high-precision navigation system based on a five-sensor conical configuration according to claim 2, characterized in that: The specific steps of step S2 also include: S22. Determine the sensor elevation angle that minimizes the overall system error, and construct a measurement matrix m of the five sensors: <h2 style=";text-align:left;direction:ltr">m = [m1,m2,m2,m4,m5]<h2 style=";text-align:left;direction:ltr"> T <h2 style=";text-align:left;direction:ltr"> =H·[w<h2 style=";text-align:left;direction:ltr"> x <h2 style=";text-align:left;direction:ltr"> ,w<h2 style=";text-align:left;direction:ltr"> y <h2 style=";text-align:left;direction:ltr"> ,w<h2 style=";text-align:left;direction:ltr"> z <h2 style=";text-align:left;direction:ltr"> ]<h2 style=";text-align:left;direction:ltr"> T <h2 style=";text-align:left;direction:ltr"> +[v1,v2,v3,v4,v5]<h2 style=";text-align:left;direction:ltr"> T Among them, w x ,w y ,w z The vector represents the components of the real physical input acceleration and angular rate in the three coordinate axis directions in the orthogonal coordinate system. The multiplication of H and w represents the linear combination of the projection of the physical quantity in the direction of the sensor's sensitive axis, reflecting the theoretical measurement contribution of the physical quantity on each sensor. i It represents the measurement error vector of the i-th sensor, reflecting the inevitable random interference factors in the measurement process, m i represents the measurement value of the i-th sensor.

4. The method for constructing a high-precision navigation system based on a five-sensor conical configuration according to claim 3, characterized in that: The specific steps of step S2 also include: S23. The least weighted squares method can be used to fuse the measurement data of multiple sensors. The goal of the weighted least squares method is to minimize the weighted residual sum of squares. The objective function is as follows: J(w)=(m-Hw) T W(m-Hw) Where W is the weight matrix, are the noise variances of the five sensors respectively; Derivative the above formula to find the estimated value with the minimum residual The expression is as follows: Finding the covariance of the estimated values yields: Assume that the noise of each sensor is independent of each other and the standard deviation of the measurement noise of each sensor is σ v , then the covariance of the estimated value can be simplified as: When the covariance is minimum, the system noise error is minimum, that is, when the elevation angle α makes det(H T When H) is the largest, the navigation system error is the smallest. After calculation, the optimal elevation angle of the sensor is: det(H T H)=31.25sin 4 acos 2 a α=arcsin(√(2 / 3))≈54.73°.

5. The method for constructing a high-precision navigation system based on a five-sensor conical configuration according to claim 4, characterized in that: The specific steps of step S3 include: S31. Establish the residual history records of the last five moments for each sensor to form a time series buffer. This step disperses the transient interference to multiple time points for analysis. The buffer sequence expression is as follows: Q i =[r i (t-5+1),…r i (t-1),r i (t)] Among them, Q i Represents the residual buffer queue of the i-th sensor, r i Represents the residual of the i-th sensor. Next, the filtered residual vector is constructed by sliding average calculation. The formula is as follows: in, is the residual vector after processing, Q i (tk) is the k-th historical residual value, and the chi-square statistic is constructed as follows:

6. The method for constructing a high-precision navigation system based on a five-sensor conical configuration according to claim 5, characterized in that: The specific steps of step S3 also include: S32. Select the appropriate threshold value by looking up the table based on the confidence level and the degree of freedom of the statistic. The subscript α is the significance level. When α = 0.05, it corresponds to 95% confidence level, and when α = 0.01, it corresponds to 99% confidence level. n represents the degree of freedom of the statistic. In this system, five sensors are used to estimate three parameters, so the degree of freedom n = 2. When When , the system is judged to be normal and no operation is triggered; when When the system enters the warning state, the secondary residual analysis is started, the local residual of each sensor is calculated, and a sliding window is constructed with a window length of N = 50. The sliding window length N = 50 means that the system intercepts the latest 50 consecutive data points of the sensor as the analysis unit each time, and counts the residuals of each sensor in the window that exceed 3σ. i If a sensor has abnormality for ≥10 times in three consecutive windows, it is considered to be faulty. when When the system fails, it is immediately judged that the fault occurs. At this time, the faulty sensor is eliminated by traversal, and the i-th sensor is removed from the position matrix H in turn to construct the position matrix H after dimensionality reduction. i (H i ∈R 4×3 ), construct the chi-square statistic after removing the i-th sensor Among them, m i , W i They are the measurement matrix, estimated value and weight matrix after removing the i-th sensor. If the new chi-square statistic is satisfied after removing the i-th sensor It is determined that the i-th sensor is faulty.

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