Adaptive transition probability matrix parallel interaction multi-model method based on likelihood function

Abstract

The invention relates to an adaptive transition probability matrix parallel interaction multi-model method based on a likelihood function. Firstly, aiming at the problems that the target maneuvering characteristic is stronger and stronger and the model matching degree is reduced due to the fact that a traditional IMM algorithm adopts a prior transition probability matrix, a transition probability correction method based on an exponential function is provided, real-time correction of the transition probability matrix is achieved, and the model matching degree is improved; then, for the problem that the model jump response time is prolonged due to the fact that a large amount of past information is introduced into a transition probability correction function, a transition probability correction method based on a likelihood function is provided, and the transition probability correction function with higher adaptability is constructed by introducing current model information and historical model information; and the problem of model jump lag is solved. According to the method, under the condition that the maneuvering model is not completely matched, the maneuvering target can be effectively tracked, and the tracking precision and the model matching degree are obviously improved.

Description

Technical Field

[0001] The invention belongs to the technical field of target tracking, and in particular is a parallel interactive multi-model method of adaptive transfer probability matrix based on likelihood function. Background Art

[0002] The Interacting Multiple Model (IMM) algorithm is a widely used filtering algorithm in the field of target tracking, especially for maneuvering target tracking tasks. With the increasing complexity and mobility of target motion, a single motion model is difficult to accurately describe the real motion state of the target, and the traditional single model filter is prone to large estimation errors or even target loss. The IMM algorithm describes the motion state of the target as a Markov process by introducing multiple motion models, realizes soft switching between models through the Markov transfer probability matrix (TPM), and dynamically adjusts the model weights according to the likelihood function of each model (i.e., the degree of matching between the model and the actual motion state), performs weighted fusion on the estimation results of each model, and finally provides adaptive tracking of the target state. The IMM algorithm has the advantages of strong adaptability, superior performance, high computational efficiency and flexible expansion. It can effectively cope with the state changes of the target under different motion modes and provide more accurate and robust state estimation. In addition, the IMM algorithm adopts a modular structure of parallel computing, and the filtering process of each model is carried out independently, which is suitable for real-time target tracking tasks and is widely used in the tracking problem of complex maneuvering targets. It is one of the important research directions in the current target tracking field.

[0003] However, the IMM algorithm also has some shortcomings. When the system model remains unchanged, maintaining the dominant position of the matching model can effectively improve the performance of the IMM algorithm. The probability of matching the model is affected by the likelihood function and the prediction probability of the model. Since noise will weaken the likelihood function of the matching model, the probability of matching the model will decrease, resulting in a decrease in the performance of the IMM algorithm. Therefore, it is necessary to modify the transition probability to suppress the impact of noise on model matching. In the case where the model does not jump, the error mainly comes from the process noise of the actual motion model. Reasonable use of the information of the system model that was less affected by noise in the past can increase the transition probability of the matching model and reduce the transition probability of the unmatched model, thereby helping the system improve its anti-noise ability. However, since the jump of the actual system model is instantaneous, and the IMM algorithm is a soft switching algorithm, the IMM algorithm needs a certain amount of time to respond to the model jump. During this period, the matched model is not the true model of the system.

[0004] In view of the above problems, the present invention proposes a method of defining a transfer probability correction function based on an exponential function, using past model information to correct the transfer probability correction function and improve the probability of matching the model; then, on this basis, the transfer probability correction function is optimized using a likelihood function, making full use of past information and current information to achieve effective tracking of maneuvering targets. The present invention can more accurately reflect the degree of adaptation between the system model and the actual motion mode, improve the tracking performance of the target in complex motion scenes, and improve the smoothness and robustness of the tracking system. Summary of the invention

[0005] First, in view of the increasingly strong maneuverability of the target, the traditional IMM algorithm uses a priori transfer probability matrix, which leads to a decrease in model matching. A transfer probability correction method based on an exponential function is proposed, which realizes the real-time correction of the transfer probability matrix and improves the model matching. Then, in view of the problem that the transfer probability correction function introduces a large amount of past information, which leads to the extension of the model jump response time, a transfer probability correction method based on the likelihood function is proposed. This method constructs a more adaptable transfer probability correction function by introducing current model information and historical model information, and solves the problem of model jump lag. The present invention can effectively realize the tracking of maneuvering targets when the maneuver model is not completely matched, and the tracking accuracy and model matching are significantly improved.

[0006] The present invention adopts the following technical solution: a parallel interactive multi-model method of adaptive transition probability matrix based on likelihood function, comprising the following steps:

[0007] Step 1: Input Interaction

[0008] Design two IMM parallel algorithms, one is adaptive IMM and the other is conventional IMM. Assume that the transition probability of model i to model j is π ij , the transition probability matrix is ​​shown in formula (1):

[0009]

[0010] In the formula, π A is the transition probability matrix of adaptive IMM, π C is the transition probability matrix of the conventional IMM, and m is the number of models.

[0011] The output values ​​of each model at the previous moment are interacted based on the Markov state transfer matrix to obtain new filter state values ​​and covariance values. The output after interaction is:

[0012]

[0013] In the formula, is the state estimate of model i at time k, is the state covariance matrix of model i at time k, is the target mixed state at time k, is the mixed state covariance matrix at time k, u k|k (i|j) is the model probability, which is calculated as follows:

[0014]

[0015] In the formula, Normalization constant when inputting interaction probabilities from model i to model j.

[0016] Step 2: Filter estimation

[0017] The state vector obtained by input interaction and its state covariance matrix and the sensor’s measurement Z at time k+1 k+1 Use the filtering algorithm to filter each filter, and get the filtering output of the jth filter at time k+1: and

[0018] Step 3: Model probability update

[0019] The probability of model j can be updated as:

[0020]

[0021] In the formula, is the likelihood function, which is calculated as follows:

[0022]

[0023] In the formula, For the new news, is the innovation covariance matrix.

[0024] Step 4: Update the transition probability matrix

[0025] The present invention proposes a transfer probability correction function defined based on an exponential function, as shown in equations (10) and (11).

[0026]

[0027] In the formula, the gradient of the model probability is Used to indicate the probability change trend of the model. The upper limit of the transition probability correction function is the natural logarithm e. When the model does not jump, the model gradient is close to 0. At this time, the correction function takes a value of approximately 1, which meets the correction requirements. The transition probability correction is as follows:

[0028]

[0029] Normalization yields:

[0030]

[0031] The present invention optimizes the transition probability correction function by using the likelihood function and proposes a transition probability correction function based on the likelihood function. The likelihood model ratio is defined as:

[0032]

[0033] In the formula, Λ r Represents the likelihood function of model r. Normalizing it yields:

[0034]

[0035] Using Lambda r The transition probability correction function is modified again. The modified transition probability correction function based on the likelihood function is as follows:

[0036]

[0037] In the formula, is the probability of ATPM-IMM (LikelihoodATPM-IMM, LATPM-IMM) model j using the transition probability correction function based on the likelihood function, is the probability of the common IMM (CIMM) model j.

[0038] Finally, the corrected transfer probability matrix is ​​calculated and normalized using equations (12) and (13) to complete the update of the transfer probability matrix.

[0039] Step 5: State Estimation Fusion

[0040] Multiply the output of all filters at time k+1 by their corresponding model probabilities to get the interactive output at time k+1:

[0041]

[0042] The beneficial effects of the present invention are as follows:

[0043] 1. Based on the IMM algorithm, the present invention uses an adaptive transfer probability correction matrix to achieve accurate tracking of the target. First, in view of the increasingly strong maneuverability of the target, the traditional IMM algorithm uses a priori transfer probability matrix, which leads to a decrease in model matching. A transfer probability correction method based on an exponential function is proposed, which realizes the real-time correction of the transfer probability matrix and improves the model matching.

[0044] 2. In order to solve the problem that the model jump response time is prolonged due to the introduction of a large amount of past information into the transfer probability correction function, the present invention proposes a transfer probability correction method based on the likelihood function. This method constructs a more adaptable transfer probability correction function by introducing current model information and historical model information, thereby solving the problem of model jump lag.

[0045] 3. The present invention can effectively track maneuvering targets when the maneuvering model is not completely matched, and the tracking accuracy and model matching degree are significantly improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 The figure is a flow chart of the method of the present invention.

[0047] Figure 2 : is a model probability comparison diagram of the three algorithms in the embodiment of the present invention in the case of no measurement noise, where Figure 2 (a) is a probability comparison diagram of uniform motion model. Figure 2 (b) is a comparison chart of the probability of a uniform left turn model. Figure 2 (c) is a comparison chart of the probability of uniform right turn model.

[0048] Figure 3 : is a comparison diagram of the model probability errors of the three algorithms in the embodiment of the present invention without measurement noise, where Figure 3 (a) is a comparison chart of the probability error of the uniform motion model. Figure 3 (b) is a comparison chart of the probability error of the uniform left turn model. Figure 3 (c) is a comparison chart of the probability error of the uniform speed right turn model.

[0049] Figure 4 : is a model probability comparison diagram of the three algorithms in the embodiment of the present invention in the case of measurement noise, where Figure 4 (a) is a probability comparison diagram of uniform motion model. Figure 4 (b) is a comparison chart of the probability of a uniform left turn model. Figure 4 (c) is a comparison chart of the probability of uniform right turn model.

[0050] Figure 5 : is a comparison diagram of the model probability errors of the three algorithms in the embodiment of the present invention under the condition of measurement noise, where Figure 5 (a) is a comparison chart of the probability error of the uniform motion model. Figure 5 (b) is a comparison chart of the probability error of the uniform left turn model. Figure 5 (c) is a comparison chart of the probability error of the uniform speed right turn model.

[0051] Figure 6 : is a comparison diagram of the position root mean square error of the three algorithms in the embodiment of the present invention under the condition of measurement noise, where Figure 6(a) is the root mean square error of the x-axis position, Figure 6 (b) is the root mean square error of the y-axis position.

[0052] Figure 7 : is a speed root mean square error comparison diagram of the three algorithms in the embodiment of the present invention under the condition of measurement noise, where Figure 7 (a) is the velocity root mean square error of the x-axis, Figure 7 (b) is the RMS error of the velocity on the y-axis. DETAILED DESCRIPTION

[0053] The present invention is further described below in conjunction with embodiments and drawings.

[0054] like Figure 1 As shown, a parallel interactive multi-model method of adaptive transition probability matrix based on likelihood function includes the following steps:

[0055] Step 1: Input Interaction

[0056] The IMM algorithm uses two or more models to predict and track the target. The parameters between the models need to interact to obtain a unified estimate and then filter. The present invention designs two IMM parallel algorithms, one of which is an adaptive IMM and the other is a conventional IMM. The constant velocity (CV) model, the constant turn left (CT / L) model and the constant turn right (CT / R) model are used to track the target. The state transfer matrix between the models is:

[0057]

[0058] In the formula, π A is the transition probability matrix of LATPM-IMM, π C is the transition probability matrix of CIMM.

[0059] The output values ​​of each model at the previous moment are interacted based on the Markov state transfer matrix to obtain the new target state value and covariance value. The output after interaction is:

[0060]

[0061] In the formula, u k∣k (i|j) is the model probability, which is calculated as follows:

[0062]

[0063] In the formula, Normalization constant when inputting interaction probabilities from model i to model j.

[0064] The applicable model set of the present invention is not limited to such specific statements and embodiments. In different environments, the model set adopted by the present invention can be appropriately changed.

[0065] Step 2: Filter estimation

[0066] Filtering is a method of estimating the state of a system based on the current state and measurement data. Its core goal is to minimize the error caused by noise or uncertainty and obtain the best estimate of the system state. In multi-sensor fusion or complex dynamic systems, filters can combine historical information and real-time measurements to dynamically correct state estimates, thereby improving prediction accuracy and system reliability.

[0067] The state vector obtained by input interaction and its state covariance matrix and the sensor’s measurement Z at time k+1 k+1 Use the filtering algorithm to filter each filter, and get the filtering output of the jth filter at time k+1: and

[0068] Step 3: Model probability update

[0069] The probability of model j can be updated as:

[0070]

[0071] In the formula, is the likelihood function, which is calculated as follows:

[0072]

[0073] In the formula, For the new news, is the innovation covariance matrix.

[0074] Step 4: Update the transition probability matrix

[0075] The present invention proposes a transfer probability correction function defined based on an exponential function, as shown in equations (28) and (29). When the model does not jump, the error mainly comes from the process noise of the actual motion model. By rationally utilizing the information of the system model that was less affected by noise in the past, the transfer probability of the matching model can be increased and the transfer probability of the unmatched model can be reduced. Therefore, using past model information to define the transfer probability correction function can help the system improve its noise resistance.

[0076]

[0077] In the formula, the gradient of the model probability is Used to indicate the probability change trend of the model. The upper limit of the transition probability correction function is the natural constant e. When the model does not jump, the model gradient is close to 0. At this time, the correction function takes a value of approximately 1, which meets the correction requirements. The transition probability correction is as follows:

[0078]

[0079] Normalization yields:

[0080]

[0081] After normalization, the modified transition probability still meets the requirements of the Markov chain. By incorporating the information of the past model into the transition probability, the TPM is modified using the transition probability correction function to obtain the adaptive TPM-IMM (ATPM-IMM). As shown in formula (28), when hour, At this time, the transition probability of model j increases. After correction, the transition probability At this time, the transition probability from model i to model j increases; when the model gradient When the corrected transition probability At this time, the probability of transition from model i to model j decreases.

[0082] After being corrected by the transfer probability correction function, the probability of matching model j is improved. In addition, the corrected TPM also improves the algorithm's anti-noise ability when the system model does not jump. However, since the actual system model jump is completed instantaneously, and the IMM algorithm is a soft switching algorithm, the IMM algorithm requires a certain amount of time to respond to the model jump, during which the matched model is not the true model of the system. Since the model transfer probability corrected by the transfer probability correction function contains a large amount of past information, even if the maximum value of the likelihood function of each model matches the actual model, it will be treated as noise and the model transfer process will be suppressed, resulting in a prolonged model jump response time. In order to minimize this response lag, it is necessary to optimize the transfer probability correction function.

[0083] Likelihood function Indicates the degree of match between model j and the actual model of the system, the likelihood function The larger the value, the closer the model j is to the actual model. Therefore, the likelihood function Contains the current information of the system and makes full use of the likelihood function The effective information can effectively distinguish whether the system model is in a constant or jumping state, realize the timely adjustment of TPM, and control the response lag phenomenon. The present invention optimizes the transfer probability correction function by using the likelihood function and proposes a transfer probability correction function based on the likelihood function.

[0084] Define the likelihood model ratio as:

[0085]

[0086] In the formula, Λ r Represents the likelihood function of model r.

[0087] Normalizing it gives:

[0088]

[0089] At this time λ r represents the normalized ratio of each likelihood function, λ r The larger the value, the larger the likelihood function of model r, that is, the greater the probability that model r is the true model of the system. By defining the likelihood model ratio and normalizing it, the transition probability correction function can be adjusted in time, so that the model can respond quickly when the target state jumps, reducing system lag. r The transition probability correction function is modified again. The modified transition probability correction function based on the likelihood function is as follows:

[0090]

[0091] In the formula, is the probability of ATPM-IMM model j using the transition probability correction function based on the likelihood function, is the probability of CIMM model j.

[0092] Finally, the corrected transfer probability matrix is ​​calculated and normalized using equations (30) and (31) to complete the update of the transfer probability matrix.

[0093] Step 5: State Estimation Fusion

[0094] Multiply the output of all filters at time k+1 by their corresponding model probabilities to get the interactive output at time k+1:

[0095]

[0096] The effects of the present invention will be further described below in conjunction with the accompanying drawings.

[0097] 1. Experimental conditions

[0098] The experimental environment is Intel(R) Core(TM) i3-8350 CPU@3.4GHz, memory is 16GB, and GPU processor is NVIDA GeForce GTX 1080Ti. The present invention selects CIMM algorithm and ATPM-PIMM algorithm as comparison algorithms to verify the effectiveness and robustness of the method of the present invention.

[0099] 2. Experimental content

[0100] The present invention realizes accurate tracking of the target by using an adaptive transfer probability correction matrix based on the IMM algorithm. First, in view of the increasingly strong maneuvering characteristics of the target, the traditional IMM algorithm uses a priori transfer probability matrix, which leads to a decrease in model matching. A transfer probability correction function based on an exponential function is proposed; then, in view of the problem that the correction function introduces a large amount of past information, which leads to a lag in model transfer, the likelihood function is used and the transfer probability correction function is optimized in combination with current information; finally, the parallel algorithm flow of LATPM-PIMM and CIMM is adopted to complete the tracking of the maneuvering target. Experimental analysis shows that the present invention can effectively realize the tracking of maneuvering targets when the maneuvering model is not completely matched, and the tracking accuracy and model matching are significantly improved.

[0101] Assume that the target moves in a two-dimensional space, its initial position is (2000m, 1000m), its initial speed is (20m / s, 35m / s), and the acceleration interference of the x-axis and y-axis of the target is set to obey the standard deviation of 1m / s 2 The target motion time is 150s and the sampling period is 1s. The target state is [x,v x ,y,v y ], this experiment only considers the impact of IMM on target estimation, so a linear sensor is used for measurement, the measurement state is [x, y], and the Kalman filter (KF) algorithm is used to filter the measurement value. The sensor measurement error has a mean of zero and a variance of (30m) 2 Gaussian distribution. In 0-50s, the target moves in a straight line at a uniform speed. In 51-100s, the target makes a left turn at a uniform speed of 0.05rad / s in the xy plane. In 101-150s, the target makes a right turn at a uniform speed of -0.05rad / s in the xy plane.

[0102] Table 1 Target movement mode

[0103]

[0104] The state transfer matrix is:

[0105]

[0106] The measurement model of the sensor is:

[0107] Z k =HX k +V k (39)

[0108]

[0109] The transition probability matrix between models is:

[0110]

[0111] The initial probability of each model is [1 / 3, 1 / 3, 1 / 3]. The target moves in two-dimensional space, including three modes: linear motion, left turn and right turn. The target state is estimated by three algorithms: CIMM, ATPM-PIMM and LATPM-PIMM. A comparative analysis is conducted to verify the superior performance of LATPM-PIMM algorithm in multi-model target tracking.

[0112] 3. Evaluation indicators

[0113] In order to compare the performance of different filters, the root mean square error (RMSE) and average root mean square error (ARMSE) are selected as performance metrics. For example, the RMSE and ARMSE of the position are defined as follows:

[0114]

[0115]

[0116] In the formula, N represents the sampling time, M represents the number of Monte Carlo times, x and y represent the actual position of the target, Represents the estimated target position.

[0117] 4. Simulation test

[0118] In order to verify the effectiveness of the method of the present invention, a tracking experiment is carried out when the sensor does not have measurement noise. The probability of each model is as follows: Figure 2 As shown in Figure 2. Since the comparison algorithm ATPM-PIMM requires the use of a priori threshold parameters, this experiment uses the threshold 1 with better performance for experimental comparison. Figure 2CT / L (Constant Turn Left) indicates the left turn model, and CT / R (Constant Turn Right) indicates the right turn model. Within 0-50s, the target moves in a straight line at a constant speed, and the true CV probability of the target should be 1; within 51-100s, the target makes a left turn, and the true CT / L probability should be 1; within 101-150s, the target makes a right turn, and the true CT / R probability should be 1. Figure 2 It can be seen that CIMM has the lowest model probability matching, while LATPM-PIMM is closest to the true probability, which means that LATPM-PIMM improves the matching of the model. Figure 3 The real-time probability error of each model is given. It can be seen from the figure that during the model transfer, the error of LATPM-PIMM is greater than that of CIMM and ATPM-PIMM. When the model does not jump, the model probability error of CIMM is the largest, while the model probability error of LATPM-PIMM is the smallest, which is smaller than the model probability error of ATPM-PIMM.

[0119] Depend on Figure 2 , Figure 3 It can be seen that LATPM-PIMM does not extend the lag time of the model. By reasonably introducing the model change gradient information of CIMM, the model jump lag time of LATPM-PIMM is very close to CIMM. Since the sensor error in the actual environment is not zero, the above ideal situation will not occur. In order to further examine the effectiveness of the algorithm, when there is measurement noise, the model probability and model probability error are as follows Figure 4 , Figure 5 shown.

[0120] When there is observation noise, the model probability will not remain stable at a certain value, but will jump within a certain range, but its probability should be much greater than the non-dominant model probability. Figure 4 The real-time probability of each model in the presence of measurement noise is given. From the figure, it can be observed that the model probability matching of LATPM-PIMM is better than that of CIMM and ATPM-PIMM. Figure 6 It also confirms that the overall model probability error of LATPM-PIMM is smaller than that of CIMM and ATPM-PIMM. Therefore, it can be concluded that LATPM-PIMM improves the correct matching rate of the model through the adaptive transition probability matrix based on the likelihood function.

[0121] The position and velocity RMSE of each algorithm are as follows Figure 6 and Figure 7As shown, it can be seen that the errors of LATPM-PIMM and ATPM-PIMM are reduced. This is because the model matching degree of LATPM-PIMM and ATPM-PIMM is higher, which reduces the impact of model error on target state estimation. In addition, both the position RMSE and speed RMSE of LATPM-PIMM are lower than those of ATPM-PIMM, indicating that the filtering accuracy of LATPM-PIMM is higher.

[0122] To illustrate the consistency of the algorithms, after 500 Monte Carlo simulation experiments, the ARMSE of each algorithm is given, as shown in Table 2. From the data in the table, it can be seen that the position and velocity RMSE of LATPM-PIMM are lower than those of LATPM-PIMM and CIMM, indicating that LATPM-PIMM has good consistency.

[0123] Table 2 Unknown measurement noise simulation ARMSE

[0124]

Claims

1. A parallel interactive multi-model method of adaptive transition probability matrix based on likelihood function, characterized in that: The following steps are involved: Step 1: Adopting the interactive multi-model method of adaptive IMM and conventional IMM in parallel to interact the input parameters of each model; Assume that the transition probability from model i to model j is π ij , the transition probability matrix is ​​shown in formula (1): In the formula, π A is the transition probability matrix of adaptive IMM, π C is the transition probability matrix of the conventional IMM, and m is the number of models. The output values ​​of each model in the previous moment are interacted based on the Markov state transfer matrix to obtain new filter state values ​​and covariance values. The output after interaction is: In the formula, is the state estimate of model i at time k, is the state covariance matrix of model i at time k, is the target mixed state at time k, is the mixed state covariance matrix at time k, u k|k (i|j) is the model probability; Step 2: Use a filter to perform filtering estimation on the interaction results of each model in step 1; Step 3: Based on the filtering results obtained in step 2, the model set is updated probabilistically, as shown in formula (4): In the formula, is the likelihood function; Step 4: Update the transition probability matrix; First calculate the likelihood model ratio: In the formula, Λ r Represents the likelihood function of model r, and normalizing it yields: Then calculate the transition probability correction function value based on the likelihood function: In the formula, is the probability of ATPM-IMM (LikelihoodATPM-IMM, LATPM-IMM) model j using the transition probability correction function based on the likelihood function, is the probability of the conventional IMM (CIMM) model j; the final transition probability is modified as follows: Normalization yields: Step 5: Use the filtering result obtained in step 2 and the model probability obtained in step 3 to calculate the final fusion result. Combine the Markov probability transfer matrix to calculate the state estimation output value and covariance estimation output value of each model at the next moment. Multiply the output of the filter at the next moment by its corresponding model probability to obtain the interaction output at the next moment.

2. The method of parallel interactive multi-model based on likelihood function adaptive transition probability matrix according to claim 1, characterized in that: The model probability u of step 1 k|k The calculation method of (i|j) is: In the formula, Normalization constant when inputting interaction probabilities from model i to model j.

3. The method of parallel interactive multi-model based on likelihood function adaptive transition probability matrix according to claim 1, characterized in that: The filter in step 2 can be flexibly selected according to actual needs, and supports the use of most conventional filters, such as Kalman filter (KF), extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), etc.

4. The method of parallel interactive multi-model based on likelihood function adaptive transition probability matrix according to claim 1, characterized in that: The likelihood function of step 3 is The calculation method is as follows: In the formula, For the new news, is the innovation covariance matrix.

5. The method of parallel interactive multi-model based on likelihood function adaptive transition probability matrix according to claim 1, characterized in that: The target state fusion result of step 5 is:

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