Underwater transponder calibration method based on stage Aitken accelerated gradient descent method

Abstract

The invention discloses an underwater transponder calibration method based on a stage Aitken accelerated gradient descent method, and belongs to an underwater robot integrated navigation and integrated positioning technology. Firstly, an equivalent sound velocity method is used for establishing a system equation and a target function; a parameter is defined for each measurement value, the probability that the measurement value is a normal value is reflected, the deviation degree of the measurement value is judged by comparing the residual sum of squares corresponding to each measurement value with a set threshold value, and therefore abnormal measurement values are removed; solving an optimal value of a gradient descent method based on stage Aitken acceleration; and finally, adjusting the dynamic step length based on a self-adaptive method. According to the method, the operation efficiency of the gradient descent method is improved by introducing a stage Aitken acceleration method, the problem of local optimum is avoided, the step length of each iteration is dynamically adjusted by adopting a self-adaptive link, and a solution is provided for accurate and efficient calibration of the underwater transponder.

Description

Technical Field

[0001] The present invention belongs to the underwater robot combined navigation and positioning technology, and specifically relates to a calibration method for underwater transponders using the staged Aitken accelerated gradient descent method. Background Art

[0002] Underwater robots can replace humans in performing dangerous underwater operations, and thus have seen great development in recent years. However, when an underwater robot operates in water, it must always know its own position information. Due to the limitations of the underwater environment, navigation methods such as optical signals and radio are difficult to apply in water, and acoustic navigation has become a current research hotspot. The ultra-short baseline (USBL) is widely used due to its small size, high precision, and easy portability. The USBL takes the position of the underwater transponder as a reference and continuously outputs the position information of the carrier. Therefore, the accuracy of the underwater transponder position has a great impact on the navigation accuracy of the USBL.

[0003] The underwater acoustic environment is complex, so the USBL measurement values based on acoustic signals are inevitably contaminated. Abnormal measurement values will greatly affect the calibration accuracy of the transponder. At the same time, as the water depth increases, the sound speed also changes continuously. In traditional methods, the influence of sound speed error is not considered, and the sound speed is regarded as a constant value, which does not conform to the true state of the system. Traditional calibration algorithms include the elimination method and the Gauss-Newton method. The elimination method is less sensitive to error changes, while the Gauss-Newton method linearizes the nonlinear system by using the Taylor expansion and ignoring the high-order terms, so the estimation result is not accurate. Summary of the Invention

[0004] Object of the Invention: The object of the present invention is to provide a calibration method for underwater transponders using the staged Aitken accelerated gradient descent method in view of the above problems. First, model the underwater transponder calibration system, establish the observation equation, and at the same time, use the effective sound speed method to incorporate the sound speed error into the state quantity to construct the objective function. Second, in order to reduce the contamination of the measurement values by noise, an outlier rejection method is proposed to preprocess the USBL measurement values, improving the robustness of the calibration system. Third, use the staged Aitken accelerated gradient descent method for optimization and solution. Compared with the Gauss-Newton method, since it avoids linearization approximation, it has higher accuracy. At the same time, the staged Aitken acceleration link can effectively improve the calculation speed and solve the drawbacks of the traditional gradient descent method, such as slow iteration speed and easy to fall into local optimum. Finally, introduce an adaptive link into the staged Aitken accelerated gradient descent method to dynamically adjust the step size at each iteration, widen the response frequency band, and better conform to the true system state.

[0005] The above object is achieved by the following technical solutions:

[0006] Underwater transponder calibration method based on the staged Aitken accelerated gradient descent method, which is based on the following known quantities:

[0007] Three-dimensional position information in the device coordinate system output by RTK , where respectively represent the coordinates of the three axes in the device coordinate system at time , and the superscript represents the transpose of the matrix;

[0008] Time interval recorded by the USBL transducer for the acoustic signal to propagate from the carrier to the underwater transponder ;

[0009] Angle between the acoustic ray measured by the USBL and the horizontal direction ; The method specifically includes the following steps:

[0010] Step 1: Establish a system equation and an objective function using the equivalent sound speed method;

[0011] Step 2: Outlier rejection: Define a parameter for each measurement value to reflect the probability that the measurement value is a normal value, and then judge the deviation degree of the measurement value by comparing the sum of squared residuals corresponding to each measurement value with a predefined threshold, so as to reject abnormal measurement values;

[0012] Step 3: Solve the optimal value based on the gradient descent method accelerated by the staged Aitken;

[0013] Step 4: Adjust the dynamic step size based on the adaptive method.

[0014] Furthermore, the specific method of Step 1 is as follows:

[0015] The USBL underwater transponder calibration method is: Place the transponder in water, and set its coordinates as , where respectively represent the coordinates to be determined of the three axes of the transponder in the navigation coordinate system , the carrier makes a "field" - shaped movement around the transponder, and use RTK to record the carrier position at each moment , then the distance between the carrier and the transponder is: ,

[0016] At the same time, the distance between the carrier and the transponder is represented by the time interval and the sound speed as:

[0017] Since the speed of sound varies with water depth, the actual propagation path of the acoustic signal is an arc. Therefore, there is a relationship between the straight slant range and the actual acoustic ray: where, is the proportionality coefficient between the actual arc-shaped acoustic ray and the required straight slant range, is the actual arc-shaped acoustic ray;

[0018] Then the effective speed of sound is: where, is the effective speed of sound; is the height difference between the carrier and the transponder; is the vertical height value; is the angle between the acoustic ray varying with height and the horizontal direction; is the speed of sound value varying with depth. Then the observation equation is: , where for convenience of calculation, the reciprocal of the effective speed of sound is denoted as , , is the time residual, and the objective function is constructed using the sum of squared residuals : , where, is the number of measurement values.

[0019] Furthermore, the specific method of step two is as follows:

[0020] First, the observation equation in step one is Taylor-expanded at the initial estimate value , ignoring higher-order small terms: , where, represents the slant range between the transponder and the carrier when the transponder coordinates are at the initial estimate value, ; are respectively the position errors in the direction; is the error of , , , , where, is the observable; is the system matrix; is a state quantity; is an observation noise vector;

[0021] Next, the residual and deviation index are defined: , , , where, is a projection matrix; is the residual; is the identity matrix; is the th deviation index of the measurement value, following a chi-square distribution with 1 degree of freedom; is the th element in; is the th diagonal element of the projection matrix. Next, set the threshold, and detect the abnormal measurement value by comparing with the threshold; Set the confidence level to 85%. By querying the chi-square distribution table, the quantile at this time is 2.072, so the threshold is: , where, is the threshold under the condition of 85% confidence level; represents the mean sum of squared residuals of unit degrees of freedom. If then it is determined that the th measurement value is an outlier and is removed.

[0022] Furthermore, Step 3 specifically includes the following sub-steps:

[0023] First, calculate the gradient for the objective function obtained in Step 1: , , , ,

[0024] Then the gradient vector is: , where, is the same as , is the gradient vector, and the state quantity is updated and iterated along the direction of the gradient vector until the established accuracy is met;

[0025] The parameter vector to be optimized is , then the formula for the update phase from time to time is: , where is the parameter vector to be optimized at time, and is the step size updated at time. Next, perform the Aitken acceleration process for the update phase at , ,

[0026] where represents the parameter vector to be optimized at the th iteration at time; after 3 iterations, is obtained; then the calculation of the optimized value of the final parameter vector at time is as follows: where is the optimized value of the final parameter vector at

[0027] Set the iteration stop precision , the specific value depends on the experimental environment. When the condition is met, the iteration stops. At this time, is the optimal solution; where is the objective function at

[0028] Furthermore, step four specifically includes the following sub-steps:

[0029] Introduce the adaptive method into the gradient descent method to achieve dynamic changes in the step size during the iteration process and widen the system's response frequency band. The process is as follows: , where represents the initial step size value of the iteration at is the step size value finally adopted at time, is the step size value finally adopted at time, To fix the empirical value and meet .

[0030] Beneficial effects:

[0031] In view of the problem that the traditional transponder calibration method does not consider the sound speed error, the present invention proposes an observation model based on the three-axis position and the sound speed error. The effective sound speed method is used to take the dynamically changing sound speed as a state quantity and estimate it in real time through an optimization algorithm, improving the accuracy of transponder calibration.

[0032] In view of the problem that the measured values are prone to anomalies due to the complex underwater acoustic environment, the present invention proposes a method for eliminating abnormal measured values. By solving the corresponding sum of squared residuals for each measured value and comparing it with a predetermined threshold, the reliability of the measured value is judged, and then the abnormal value is eliminated, improving the credibility of the measured values during transponder calibration.

[0033] In view of the problems that the elimination method in the traditional transponder calibration method is insensitive to system error changes and the accuracy decreases due to ignoring high-order terms after Taylor expansion of the Gauss-Newton method, the present invention introduces a gradient descent method based on improved stage Aitken acceleration. The stage Aitken acceleration link is used to overcome the disadvantages of slow calculation and easy to fall into local optimum of the traditional gradient descent method. At the same time, an adaptive method is introduced to dynamically adjust the step size of each iteration, broaden the response frequency band, and improve the accuracy and efficiency of underwater transponder calibration. Brief description of the drawings

[0034] Figure 1 is the flowchart of the method of the present invention;

[0035] Figure 2 is the flowchart of the method for eliminating abnormal values in the present invention;

[0036] Figure 3 is the simulation result of the present invention. Detailed implementation manners

[0037] Refer to Figures 1 - 3 As shown, the underwater transponder calibration method of the stage Aitken acceleration gradient descent method of the present invention is characterized in that:

[0038] The three-dimensional position information in the device system output by RTK , where respectively represent at the moment the coordinates of the three axes in the device system, and the superscript represents the transpose of the matrix;

[0039] The time interval recorded by the USBL transducer for the sound signal to propagate from the carrier to the underwater transponder ;

[0040] The angle between the acoustic ray measured by USBL and the horizontal direction ; The method specifically includes the following steps:

[0041] Step 1: Establish a system equation and an objective function using the equivalent sound speed method:

[0042] The USBL underwater transponder calibration scheme is as follows: Place the transponder in water, and set its coordinates as , where respectively represent the coordinates to be determined of the transponder in the three axes of the navigation coordinate system. The carrier moves in a "field" shape around the transponder, and the RTK is used to record the carrier position at each moment . Then the distance between the carrier and the transponder is : , At the same time, the distance can be represented by the time interval and the sound speed : ,

[0043] Since the sound speed changes with the water depth, the actual sound signal propagation route is an arc. Therefore, there is a relationship between the straight line slope and the actual acoustic ray: , where is the proportionality coefficient between the actual arc-shaped acoustic ray and the required straight line slope, is the actual arc-shaped acoustic ray.

[0044] The effective sound speed is , where is the effective sound speed; is the height difference between the carrier and the transponder; is the vertical height value; is the angle between the acoustic ray changing with height and the horizontal direction; is the sound speed value changing with depth. Then the observation equation is: , For the convenience of calculation, the reciprocal of the effective sound speed is expressed as , , is the time residual. The objective function is constructed using the sum of squared residuals , where is the number of measurement values.

[0045] Step 2. Outlier rejection:

[0046] The underwater acoustic environment is complex, and USBL measurement values are vulnerable to noise pollution. Therefore, it is very necessary to detect and reject abnormal measurement values. The present invention proposes an outlier rejection method that defines a parameter for each measurement value to reflect the probability that the measurement value is a normal value, and then determines the deviation degree of the measurement value by comparing the sum of squared residuals corresponding to each measurement value with a predefined threshold.

[0047] Perform Taylor expansion of the observation equation in Step 1 at the initial estimate , neglecting high-order small terms: , where ; are the position errors in the directions respectively; is the error of . Then the observation equation is simplified to: , , , , where is the observable quantity; is the system matrix; is the state quantity; is the observation noise vector. Next, define the residual and deviation index.

[0048] , , , where is the projection matrix; is the residual; is the identity matrix; is the deviation index of the th measurement value, following a chi-square distribution with 1 degree of freedom; is the th element in ; is the th diagonal element of the projection matrix . Next, set the threshold and detect abnormal measurement values by comparing with the threshold.

[0049] To reduce the mis-removal of normal data, the confidence level is set to 85%. By querying the chi-square distribution table, the quantile at this time is 2.072, and the threshold is: , where is the threshold under the condition that the confidence level is 85%; represents the mean square error of residuals per degree of freedom. If then it is determined that the th measurement value is an outlier and is removed.

[0050] Step 3: Solving the optimal value based on the stage Aitken accelerated gradient descent method:

[0051] As an optimization algorithm, the principle of the gradient descent method is to search for the minimum value of the objective function along the negative direction of the gradient at the current position. It has the advantage of simple structure, and at the same time avoids the error problem caused by linearization approximation, and is more suitable for solving in the transponder calibration system. However, the iterative speed of the traditional gradient descent method is slow, which affects the calibration efficiency. The Aitken method is an acceleration algorithm that can increase the convergence speed from linear convergence to at least quadratic convergence, and can effectively improve the calculation rate.

[0052] First, for the objective function obtained in Step 1 calculate the gradient: , , , ,

[0053] Then the gradient vector is: , where is the same as , is the gradient vector, and the state quantity is updated and iterated along the direction of the gradient vector until the established accuracy is met.

[0054] The parameter vector to be optimized is , then the formula for the update stage from time to time is: , where is the parameter vector to be optimized at time, is the step size updated at time. Next, perform the stage Aitken acceleration process. For In the update phase of the moment, three iterative calculations are performed according to the following formula: , , where, represents the vector of parameters to be optimized in the th iteration at the moment of . After three iterations, is obtained. Then the calculation of the optimized value of the final parameter vector at the moment of is as follows: where, is the optimized value of the final parameter vector at the moment of

[0055] Set the iteration stop precision , and the specific value depends on the experimental environment. When the condition is satisfied, the iteration stops, and at this time is the optimal solution. Among them, is the objective function at the moment of

[0056] Step 4. Dynamic step size adjustment based on the adaptive method:

[0057] As can be seen from Step 3, the process of solving the optimal value by the gradient descent method is an iterative process. In the traditional gradient descent method for optimization, the step size is fixed. However, the most suitable step size is required for each iteration. Therefore, the fixed cannot fully reflect the real changes of the system. The present invention introduces the adaptive method into the gradient descent method to achieve the dynamic change of the step size during the iteration process and broaden the system's response frequency band. The process is as follows: , where, represents the initial value of the step size at the moment of is the value of the step size finally adopted at the moment of is the value of the step size finally adopted at the moment of is the initial value of the step size at the moment of is a fixed empirical value, satisfying .

[0058] Finally, substitute the obtained step size value into Step 3 to solve for the optimal value.

[0059] Verification of the invention:

[0060] In the simulation experiment, first determine the motion trajectory and state of the carrier, and use the trajectory generator to generate the corresponding inertial device output and the position data of the carrier, and set the position of the transponder. Calculate the true slant range and azimuth data based on the positions of the carrier and the transponder. Finally, add Gaussian white noise according to the set sensor parameters to obtain the simulated slant range and azimuth data. To ensure that the system has one and only one solution, set the carrier to move in an ellipse around the transponder. Starting from Figure 3 It can be seen from the simulation results that the underwater transponder calibration method using the improved stage Aitken accelerated gradient descent method proposed in the present invention has a more accurate estimation result in the horizontal direction compared with the traditional gradient descent method. The horizontal position error is 0.177 m, which has a better calibration effect compared with the position error of 0.85 m of the traditional gradient descent method.

Claims

1. A method for underwater transponder calibration using the staged Aitken accelerated gradient descent method, which is based on the following known quantities: RTK outputs the three-dimensional position information of the device system ,in, Respectively Time equipment system Coordinates of the three axes, superscript Represents the transpose of a matrix; The USBL transducer records the time interval between the acoustic signal propagating from the carrier to the underwater transponder. ; USBL measures the angle between the sound line and the horizontal direction ; The method is characterized in that it specifically comprises the following steps: Step 1: Use the equivalent sound velocity method to establish the system equation and objective function; Step 2: Outlier elimination: define a parameter for each measurement value to reflect the probability that the measurement value is a normal value, and then compare the residual sum of squares corresponding to each measurement value with the established threshold to determine the degree of deviation of the measurement value, thereby eliminating abnormal measurement values; Step 3: Calculate the optimal value based on the gradient descent method accelerated by Aitken in the stage; Step 4: Dynamic step size adjustment based on adaptive method.

2. The underwater transponder calibration method of the stage Aitken accelerated gradient descent method according to claim 1 is characterized in that: The specific method of step one is as follows: The calibration method of the USBL underwater transponder is as follows: Place the transponder in water, and set its coordinates as , where respectively represent the coordinates to be determined of the three axes of the transponder in the navigation coordinate system. The carrier moves in a "field" shape around the transponder, and uses RTK to record the position of the carrier at each moment . Then the distance between the carrier and the transponder is: ​ ， At the same time, the distance between the carrier and the transponder By time interval The speed of sound express: ， Since the speed of sound changes with the depth of water, the actual propagation path of the sound signal is an arc. There is a relationship between the straight line slope and the actual sound line: ， in, is the proportionality coefficient between the actual arc sound line and the required straight line slope, It is the actual arc sound line; The effective speed of sound for: ， in, is the effective speed of sound; is the height difference between the carrier and the transponder; is the vertical height value; is the angle between the sound line that changes with height and the horizontal direction; is the sound velocity value that changes with depth; then the observation equation is: ， For the convenience of calculation, the effective sound speed The reciprocal of is expressed as , , is the time residual, and the residual sum of squares is used to construct the objective function : ， in, is the number of measured values.

3. The underwater transponder calibration method of the stage Aitken accelerated gradient descent method according to claim 2 is characterized in that: The specific method of step 2 is as follows: First, the observation equation in step 1 is set at the initial estimate Taylor expansion is performed at , and higher-order small terms are ignored: ， in, It represents the skew moment between the transponder and the carrier when the transponder coordinates are the initial estimated values, ; They are Position error in direction; for The observation equation is simplified to: ， ， ， ， in, is the observed quantity; is the system matrix; is the state quantity; is the observation noise vector; Next, define the residual and deviation indicators: ， ， ， in, is the projection matrix; is the residual; is the identity matrix; For the The deviation index of the measured value follows a chi-square distribution with 1 degree of freedom; for Middle elements; is the projection matrix No. diagonal elements; next, set the threshold by comparing and thresholds to detect abnormal measurements; Set the confidence level to 85% and query the chi-square distribution table to find that the quantile is 2.

072. The threshold is: ， in, is the threshold value when the confidence level is 85%; represents the average residual sum of squares per unit degree of freedom, if Then determine The measured values ​​are outliers and are removed.

4. The underwater transponder calibration method of the stage Aitken accelerated gradient descent method according to claim 3 is characterized in that: Step 3 specifically includes the following sub-steps: First, for the objective function obtained in step 1 Compute the gradient: ， ， ， ， Then the gradient vector is: ， in, and same, is the gradient vector, and the state quantity is updated and iterated along the direction of the gradient vector until the given accuracy is met; The parameter vector to be optimized is , then from Time has come The formula for the update phase at time is: ， in, for The parameter vector to be optimized at time , for The step size updated at all times; the following stage Aitken acceleration processing is performed, for During the update phase of the moment, three iterations are performed according to the following formula: ， ， in, represent The moment The parameter vector to be optimized for the iteration, ; After 3 iterations, we get ;but The final parameter vector optimization value at time is calculated as follows: ， in, for The final parameter vector optimization value at time; Set the iteration stop precision , The specific value depends on the experimental environment. The iteration stops when is the optimal solution; for Time objective function.

5. The underwater transponder calibration method of the stage Aitken accelerated gradient descent method according to claim 4 is characterized in that: Step 4 specifically includes the following sub-steps: The adaptive method is introduced into the gradient descent method to realize the dynamic change of the step size during the iteration process and broaden the system's responsive frequency band. The process is as follows: ， in, express The initial value of the step size for the moment iteration; for The step size used at the last iteration of the moment, for The step size used at the last iteration of the moment, for The initial value of the step size for the moment iteration; is a fixed experience value, satisfying .

Images