Underwater Transponder Calibration Method Based on the Aitken Accelerated Gradient Descent Method

Abstract

The present invention discloses an underwater transponder calibration method using the staged Aitken accelerated gradient descent method, belonging to the underwater robot integrated navigation and positioning technology. First, the system equation and objective function are established using the equivalent sound velocity method; then a parameter is defined for each measurement value to reflect the probability that the measurement value is a normal value, and further, the deviation degree of the measurement value is judged by comparing the sum of squared residuals corresponding to each measurement value with a predetermined threshold, so as to eliminate abnormal measurement values; then the optimal value is solved based on the staged Aitken accelerated gradient descent method; finally, the dynamic step size is adjusted based on the adaptive method. The present invention improves the operation efficiency of the gradient descent method by introducing the staged Aitken acceleration method, avoids the local optimum problem, and dynamically adjusts the step size of each iteration by using an adaptive link, providing a solution for the accurate and efficient calibration of underwater transponders.

Description

Technical Field

[0001] The present invention belongs to the technical field of integrated navigation and positioning of underwater robots, and particularly relates to a calibration method for underwater transponders based on the staged Aitken accelerated gradient descent method. Background Art

[0002] Underwater robots can replace humans to perform dangerous underwater operations, and thus have obtained 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 be used in water, and acoustic navigation has become a current research hotspot. The ultra-short baseline (USBL) is widely used due to its small volume, 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 is also constantly changing. In traditional methods, the influence of the sound speed error is not considered, and the sound speed is regarded as a constant value, which does not conform to the real state of the system. Traditional calibration algorithms include the elimination method and the Gauss-Newton method. The elimination method has a low sensitivity to error changes, while the Gauss-Newton method linearizes and approximates the nonlinear system by using the method of Taylor expansion to ignore high-order terms. Therefore, 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 based on the staged Aitken accelerated gradient descent method for the above problems. First, model the underwater transponder calibration system, establish an observation equation, and at the same time, adopt the effective sound speed method to incorporate the sound speed error into the state quantity to construct an objective function. Secondly, in order to reduce the contamination of the measurement value by noise, an outlier rejection method is proposed to preprocess the USBL measurement value, improving the robustness of the calibration system. Thirdly, use the staged Aitken accelerated gradient descent method for optimization and solution. Compared with the Gauss-Newton method, since linearization approximation is avoided, it has higher accuracy. At the same time, the staged Aitken acceleration link can effectively improve the calculation speed, solving the disadvantages of the traditional gradient descent method with 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 during each iteration, widen the response frequency band, and better conform to the real 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] The time interval recorded by the USBL transducer for the acoustic signal to propagate from the carrier to the underwater transponder ;

[0009] The angle between the acoustic ray measured by the USBL and the horizontal direction ; The specific steps of this method are as follows:

[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 the 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 an 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] ,

[0017] At the same time, the distance between the carrier and the transponder is represented by the time interval and the sound speed as:

[0018]

[0019] 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-line slant range and the actual acoustic ray:

[0020]

[0021] Among them, is the proportionality coefficient between the actual arc-shaped acoustic ray and the required straight-line slant range, is the actual arc-shaped acoustic ray;

[0022] Then the effective speed of sound is:

[0023] Among them, 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:

[0024] ,

[0025] Among them, for the convenience of calculation, the reciprocal of the effective speed of sound is expressed as , , is the time residual, and the objective function is constructed using the sum of squared residuals:

[0026] ,

[0027] Among them, is the number of measurement values.

[0028] Furthermore, the specific method of step two is as follows:

[0029] First, the observation equation in step one is Taylor-expanded at the initial estimate , and the high-order small terms are ignored:

[0030] ,

[0031] Among them, represents the slant range between the transponder and the carrier when the transponder coordinates are at the initial estimate, ; are respectively the position errors in the direction; is the error of

[0032] ,

[0033] ,

[0034] ,

[0035] ,

[0036] Among them, is the observed quantity; is the system matrix; is the state quantity; is the observation noise vector;

[0037] Next, define the residual and deviation index:

[0038] ,

[0039] ,

[0040] ,

[0041] Among them, is the 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 in the th element; is the projection matrix of the th diagonal element. Next, set the threshold, and detect the abnormal measurement value by comparing with the threshold;

[0042] Set the confidence level to 85%. Querying the chi-square distribution table shows that the quantile is 2.072 at this time, so the threshold is:

[0043] ,

[0044] Among them, 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 excluded.

[0045] Furthermore, step three specifically includes the following sub-steps:

[0046] First, calculate the gradient for the objective function obtained in step one:

[0047] ,

[0048] ,

[0049] ,

[0050] ,

[0051] Then the gradient vector is:

[0052] ,

[0053] where, is the same as , and is the gradient vector. The state quantity is updated iteratively along the direction of the gradient vector until the established accuracy is met;

[0054] The parameter vector to be optimized is . Then, from time to time, the formula for the update stage is:

[0055] ,

[0056] 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 stage at time. Perform 3 iterations according to the following formula:

[0057] ,

[0058] ,

[0059] where, represents the parameter vector to be optimized at the time and the th iteration, and . After 3 iterations, is obtained; then the optimized value of the final parameter vector at time is calculated as follows:

[0060] ,

[0061] where, is the optimized value of the final parameter vector at time;

[0062] Set the iteration stop accuracy , The specific value depends on the experimental environment. When the condition is satisfied, the iteration stops. At this time, is the optimal solution; where is the objective function at time

[0063] Furthermore, step four specifically includes the following sub-steps:

[0064] Introduce an adaptive method into the gradient descent method to achieve dynamic changes in the step size during the iteration process, and broaden the system's response frequency band. The process is as follows:

[0065] ,

[0066] where represents the initial value of the step size at time is the step size value finally adopted at time is the step size value finally adopted at time is the initial value of the step size at time is a fixed empirical value that satisfies .

[0067] Beneficial effects:

[0068] 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 dynamically changing sound speed is used as a state quantity by the effective sound speed method and is estimated in real time through an optimization algorithm, improving the accuracy of transponder calibration.

[0069] 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.

[0070] In view of the problems that the traditional transponder calibration method, the elimination method, is insensitive to system error changes, and the Taylor expansion of the Gauss-Newton method ignores high-order terms, resulting in a decrease in accuracy, 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 the traditional gradient descent method, such as slow calculation and easy to fall into local optimum. 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. Description of the Drawings

[0071] Figure 1 It is the flowchart of the method of the present invention;

[0072] Figure 2 It is the flowchart of the outlier rejection method in the present invention;

[0073] Figure 3 It is the simulation result of the present invention. Specific embodiments

[0074] Refer to Figures 1 - 3 As shown, the underwater transponder calibration method of the Aitken accelerated gradient descent method in stage A of the present invention is characterized in that:

[0075] The three-dimensional position information under the device system output by RTK , where respectively represent the coordinates of the three axes under the device system at time , and the superscript represents the transpose of the matrix;

[0076] The time interval recorded by the USBL transducer for the acoustic signal to propagate from the carrier to the underwater transponder ;

[0077] The angle between the acoustic ray measured by the USBL and the horizontal direction ; The method specifically includes the following steps:

[0078] Step 1: Establish a system equation and an objective function by using the equivalent sound speed method:

[0079] 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 three axes of the transponder in the navigation coordinate system , and the carrier makes a "field" - shaped movement around the transponder. Use RTK to record the carrier position at each moment , then the distance between the carrier and the transponder is:

[0080] ,

[0081] At the same time, the distance can be represented by the time interval and the sound speed as:

[0082] ,

[0083] Since the sound speed changes with the water depth, the actual acoustic signal propagation path is an arc, so there is a relationship between the straight - line slope and the actual acoustic ray:

[0084] ,

[0085] Among them, is the proportionality coefficient of the actual arc sound ray to the required straight-line oblique distance, is the actual arc sound ray.

[0086] Effective sound speed is:

[0087] ,

[0088] Among them, 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 sound ray varying with height and the horizontal direction; is the sound speed value varying with depth. Then the observation equation is:

[0089] ,

[0090] Among them, 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:

[0091] ,

[0092] Among them, is the number of measurement values.

[0093] Step 2. Outlier rejection:

[0094] 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, which defines a parameter for each measurement value to reflect the probability that the measurement value is a normal value, and then judges the deviation degree of the measurement value by comparing the sum of squared residuals corresponding to each measurement value with a predetermined threshold.

[0095] Perform Taylor expansion of the observation equation in Step 1 at the initial estimate value , and neglect the high-order small terms:

[0096] ,

[0097] Among them, ; are the position errors in the directions respectively; is The error. Then the observation equation is simplified to:

[0098] ,

[0099] ,

[0100] ,

[0101] ,

[0102] where, is the observed quantity; is the system matrix; is the state quantity; is the observation noise vector. Next, the residual and deviation index are defined.

[0103] ,

[0104] ,

[0105] ,

[0106] 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, the threshold is set, and the abnormal measurement value is detected by comparing with the threshold.

[0107] To reduce the mis-elimination 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. Then the threshold is:

[0108] ,

[0109] where, is the threshold under the condition of 85% confidence level; represents the mean square sum of residuals per unit degree of freedom. If then it is determined that the th measurement value is an outlier and is eliminated.

[0110] Step 3: Optimal value solution based on the Aitken acceleration of the gradient descent method in stage:

[0111] 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.

[0112] First, for the objective function obtained in Step 1 Calculate the gradient:

[0113] ,

[0114] ,

[0115] ,

[0116] ,

[0117] Then the gradient vector is:

[0118] ,

[0119] Among them, 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 satisfied.

[0120] The parameter vector to be optimized is , then from time to time, the formula for the update stage is:

[0121] ,

[0122] Among them, is the parameter vector to be optimized at time, is the step size updated at time. Next, perform the Aitken acceleration process. For the update stage at time, perform 3 iterative calculations according to the following formula:

[0123] ,

[0124] ,

[0125] Among them, represents time's The parameter vector to be optimized in the next iteration . Obtained after 3 iterations . Then The calculation of the optimized value of the final parameter vector at time is as follows:

[0126] ,

[0127] where is the optimized value of the final parameter vector at time.

[0128] Set the iteration stop precision , The specific value depends on the experimental environment. When the condition is satisfied, the iteration stops. At this time is the optimal solution. Where is the objective function at time.

[0129] Step 4. Dynamic step size adjustment based on the adaptive method:

[0130] 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, each iteration requires an optimal step size. 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 dynamic changes in the step size during the iteration process and broaden the system's response frequency band. The process is as follows:

[0131] ,

[0132] where represents the initial value of the step size at time during the iteration; is the step size value finally adopted at time during the iteration, is the step size value finally adopted at time during the iteration, is the initial value of the step size at time during the iteration; is a fixed empirical value, satisfying .

[0133] Finally, substitute the obtained step size value into Step 3 to solve for the optimal value.

[0134] Verification of the invention:

[0135] In the simulation experiment, first determine the motion trajectory and state of the carrier, use the trajectory generator to generate the corresponding inertial device output and the position data of the carrier, set the position of the transponder, calculate the true slant range and azimuth data based on the positions of the carrier and the transponder, and 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 unique 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. An underwater transponder calibration method using the stage Aitken accelerated gradient descent method, which is based on the following known quantities: The three-dimensional position information P under the RTK output device system i = [x i y i z i T , where x i , y i , z i respectively represent the coordinates of the x, y, and z axes of the device system at time i, and the superscript T represents the transpose of the matrix; ​ The USBL transducer records the time interval τ during which the acoustic signal propagates from the carrier to the underwater transponder i ; The angle θ between the acoustic ray and the horizontal direction measured by USBL; It is characterized in that the method specifically includes the following steps: Step 1: Use the equivalent sound speed method to establish the system equation and the objective function; 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 predetermined threshold, so as to reject the abnormal measurement values; Step 3: Solve the optimal value based on the gradient descent method accelerated by stage Aitken; Step 4: Dynamic step size adjustment based on the adaptive method; The specific method of Step 1 is as follows: The calibration method of the USBL underwater transponder is as follows: Place the transponder in water, and set its coordinates as P t =[x t y t z t T , where x t , y t , z t respectively represent the coordinates to be determined of the transponder on the x, y, and z axes in the navigation coordinate system. The carrier moves in a "field" shape around the transponder, and the position P i of the carrier at each moment is recorded using RTK. Then, the distance l i between the carrier and the transponder is as follows:​ At the same time, the distance l between the carrier and the transponder i is represented by the time interval τ i and the speed of sound c: l i = c·τ i Since the sound speed changes with the water depth, the actual acoustic signal propagation path is an arc, so there is a relationship between the straight line slope and the actual acoustic ray: where λ is the proportionality coefficient between the actual arc-shaped sound ray and the required straight-line oblique distance, is the actual arc-shaped sound ray; The effective sound speed is as follows: wherein, is the effective sound speed; R is the height difference between the carrier and the transponder; r is the vertical height value; θ(r) is the angle between the sound ray varying with height and the horizontal direction; c(r) is the sound speed value varying with depth; then the observation equation is: For the convenience of calculation, the reciprocal of the effective sound speed is expressed as δτ i is the time residual, and the sum of squared residuals is used to construct the objective function Where N is the number of measurement values; Step 3 specifically includes the following sub-steps: First, for the objective function obtained in Step 1 calculate the gradient: Then the gradient vector is: Among them, is the same as and is the gradient vector. The state quantity is updated iteratively along the direction of the gradient vector until the established accuracy is met; ​ The parameter vector to be optimized is Then the formula for the update phase from time i to time i + 1 is: where S i is the parameter vector to be optimized at time i, and α(i) is the step size updated at time i; Next, perform the stage Aitken acceleration process. For the update stage at time i, perform 3 iterative calculations according to the following formula: Among them, represents the parameter vector to be optimized for the j-th iteration at time i, where j = 1, 2, 3; after 3 iterations, we obtain Then the optimized value of the final parameter vector at time i is calculated as follows: Among them, is the optimized value of the final parameter vector at time i; Set the iteration stop precision s0. The specific value of s0 depends on the experimental environment. When the condition ‖(J| i+1 )-(J| i )‖ ≤ s0 is satisfied, the iteration stops. At this time, is the optimal solution; where J| i is the objective function at time i.

2. The underwater transponder calibration method of the stage Aitken accelerated gradient descent method according to claim 1, characterized in that, The specific method of Step 2 is as follows: First, perform a Taylor expansion of the observation equation in Step 1 at the initial estimate value and neglect the high-order small terms: where l i0 represents the slant range between the transponder and the vehicle when the transponder coordinates are the initial estimated values, Δx, Δy, and Δz are the position errors in the x, y, and z directions respectively; is the error of, then the observation equation is simplified to: Z = HX + V Where Z is the observable; H is the system matrix; X is the state quantity; V is the observation noise vector; Next, define the residual and the deviation index: W = H(H T H) -1 H T Φ=(I - W)Z where W is the projection matrix; Φ is the residual; I is the identity matrix; ε i is the deviation index of the i-th measurement value and follows a chi-square distribution with 1 degree of freedom; is the k-th element in Φ; w k is the k-th diagonal element of the projection matrix W; Next, set a threshold to detect abnormal measurement values by comparing ε i with the threshold; Set the confidence level to 85%, and query the chi-square distribution table to know that the quantile at this time is 2.072, then the threshold is: Among them, ε0 is the threshold value under the condition of a confidence level of 85%; represents the mean square error of residuals per degree of freedom. If ε i > ε0, then it is determined that the i-th measured value is an outlier and is excluded.

3. The underwater transponder calibration method using the stage Aitken accelerated gradient descent method according to claim 1, characterized in that, Step 4 specifically includes the following sub-steps: Introduce the adaptive method into the gradient descent method to realize the dynamic change of the step size during the iteration process and broaden the system's response frequency band. The process is as follows: α(i + 1)=α(i)0 + γ[α(i)-α(i + 1)0] Where α(i)0 represents the initial step size value of the iteration at time i; α(i) is the final step size value adopted at time i during the iteration, α(i + 1) is the final step size value adopted at time i + 1 during the iteration, α(i + 1)0 is the initial step size value of the iteration at time i + 1; γ is a fixed empirical value, satisfying 0 < γ < 1.

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