Improved IMM algorithm based on support vector regression for UAV tracking

ZENG Yuan ,LU Wenbin ,YU Bo ,TAO Shifei ,ZHOU Haosu ,and CHEN Yu

1.Shanghai Spaceflight Electronic and Communication Equipment Research Institute,Shanghai 201109,China;2.Science and Technology on Near-Surface Detection Laboratory,Wuxi 214035,China;3.School of Electronic and Optical Engineering,Nanjing University of Science and Technology,Nanjing 210094,China

Abstract: With the development of technology,the relevant performance of unmanned aerial vehicles (UAVs) has been greatly improved,and various highly maneuverable UAVs have been developed,which puts forward higher requirements on target tracking technology.Strong maneuvering refers to relatively instantaneous and dramatic changes in target acceleration or movement patterns,as well as continuous changes in speed,angle,and acceleration.However,the traditional UAV tracking algorithm model has poor adaptability and large amount of calculation.This paper applies support vector regression (SVR)to the interacting multiple model (IMM) algorithm.The simulation results show that the improved algorithm has higher tracking accuracy for highly maneuverable targets than the original algorithm,and can adjust parameters adaptively,making it more adaptable.

Keywords: interacting multiple model (IMM) filter,constant acceleration (CA),unmanned aerial vehicle (UAV),support vector regression (SVR).

1.Introduction

Maneuvering targets are the current difficulty in target detection and tracking.Due to their strong maneuverability,it is difficult for the single-model method to track them with high precision and it is easy to lose targets.The interactive multi-model (IMM) method proposed by Blom et al.[1] is one of the methods widely used in maneuvering target tracking algorithms.This method simulates the movement of the targets by establishing a set of models and assumes that the switching of different models follows the Markov process,and the final state of the target is finally weighted and determined by the filter values of different models [2].

The IMM algorithm uses multiple motion models in parallel and uses a Markov process to simulate the transition between models,and finally weights the filtered values of multiple models as an estimator of the target state.For maneuvering targets,the algorithm’s tracking effect is better than that of the single model algorithm.Its adaptability is also better.Compared with the single-model algorithm,the IMM algorithm has many advantages [3]:

(i) The IMM algorithm contains multiple models,and the number and types of models in the model set can be set according to actual needs.

(ii) In the process of target tracking,the accuracy of target tracking can be improved by adjusting the probability of the model.

(iii) Each model has its own filter,and the filters of each model can be adjusted according to actual needs to improve the filtering performance.

However,the IMM algorithm has some shortcomings.First of all,similar to the early multi-model algorithm and some other prior models,the performance of the interactive multi-model algorithm depends to a large extent on the model used.Considering the amount of computation and the competition between models,the size of the model set should not be set too large,and it should be set in advance because different models have different processing methods.Moreover,too many models in the model set will lead to competition between the models,thereby reducing the accuracy of the algorithm [4].Therefore,a model set of proper scale should be established in advance.Once the model set is determined,the model set will not be changed in the tracking process.However,with the development of control technology,the mobility of various targets is getting better and better,and the preset fixed number of model sets is difficult to meet actual needs.

To solve the problems caused by the fixed model set,Li et al.proposed a variable structure multi model algorithm in 1999 [5],and then proposed several practical algorithms based on variable structure multiple models:possible model set algorithm,maximum likelihood function algorithm,and grid algorithm [6] in 2019.Based on the above methods,most of the variable structure multi-model algorithms have been researched [7-9].Many experts and scholars have also proposed improved algorithms to solve the problem of low tracking accuracy of maneuvering targets due to poor adaptive filter.In [10],a fuzzy adaptive controller was designed to join the nonlinear system to adjust the adaptiveness of parameters.In [11],the fuzzy membership function was introduced into the current statistical model to achieve the adaptive adjustment of the target acceleration.In [12],a fuzzy inference system was introduced into the tracking algorithm to reduce the tracking error by adaptively adjusting the maneuvering frequency.

To solve the problem of poor IMM adaptability,this paper applies support vector regression (SVR) to the IMM algorithm.The IMM algorithm using SVR is tested through simulation.The improved IMM algorithm not only has low model complexity,but also improves the tracking accuracy of maneuvering targets.

This paper is organized as follows.Section 2 mainly introduces the detailed principles and steps of the IMM algorithm,the SVR algorithm,and the improved IMM algorithm.Section 3 mainly introduces the IMM algorithm and the improved IMM algorithm simulation results.Section 4 presents conclusions.

2.Methodology

2.1 IMM algorithm

The basic idea of the multi-model (MM) algorithm is to give a model set containing one or more motion models(including non-motion model and motion model) before the state estimation,and then use multiple models to filter and estimate separately in parallel.Finally,the state filter values of each model are weighted and summed according to the probabilities of each model in the current period [13].

The IMM algorithm uses multiple different motion models to match the different motion states of the target.The transition probability between different models is a Markov chain [14],and the target state is estimated using a Kalman filter.It mainly has the following components:

Step 1Input interaction module.

Input the target state filter value and covariance filter value of each model in the previous cycle period into the input interaction module and perform input interaction according to the model probability transition matrix and the model probability.Then put interaction values into each model [15].

Step 2Model filter estimation module.

Each model uses its own filter to filter the input state interaction value and input covariance interaction value to obtain the state estimate and covariance estimate.is the state estimate of theith filter at timek.K(k-1) is the filter gain atk-1.V(k-1) is the innovation sequence at timek-1.

Step 3Update of model probability module.

Step 4Estimation fusion module.

IMM is an algorithm with data fusion as the core.Probability weighted summation of the estimated value of each model is used as the target state estimation for each cycle [16].are the interactive output at timekrespectively.

A filtering cycle of the IMM algorithm mainly includes four steps: input interaction module,model filter estimation module,update of model probability module,estimation fusion module.The algorithm block diagram is shown in Fig.1.

Fig.1 IMM algorithm flow chart

2.2 SVR

Support vector machine (SVM) originally came from classification and then is used to solve regression problems,called ε-SVR of SVR machine model [17].

wheref(x) is the predicted value of the fitting function,yis the predicted value of the fitting function.Disregarding the small errors that fall within some tolerance,say ε,may lead to a better generalization ability achieved by utilizing an ε-insensitive loss function.

The significance of the insensitive loss function reference is that if the difference betweenf(x) andyis within the allowable error range,thenf(x) has no loss.

Introduce slack variable ξ and establish the following constraints:

where 1 denotes them×1 all-one vector,maxthat represent the fitting errors and the positive control parameterChere weights the tradeoff between the fitting errors and the flatness of the linear regression functionf(x).Rewrite the above formula into the following formula:

Step 2Lagrange dual problem.

Introduce Lagrange coefficients μiand transform them into dual form:

The constraints are

In the above process,Karush-Kuhn-Tucker (KKT) conditions must be met,that is,

From (16),we know

(i) When αi＞0,there must be

These points are located at the upper boundary of the pipe,or above the pipe.The predicted value is smaller than the true value.

These points are located at the lower boundary of the pipe,or below the pipe.The predicted value is greater than the true value [18].

At the same time,from (16) we know that for any data point,since ε ＞0,it is impossible for bothaandbto be greater than 0 at the same time,and to get a point inside the pipeline,there must be αi=0,=0.

Step 3Hyperplane computing.

According to the previous calculations,we can get

From the above analysis,the point that affects the hyperplane parameters is located at the pipe boundary or outside the pipe.Regarding the calculation ofb,it can be considered that a point at the upper boundary of the pipeline must have

It can be solved that

Then the prediction function is

wherexis a point on the plane boundary of the hyperplane pipe.

Step 4Proof of convergence.

In our smooth approach,we change the model slightly and solve it as an unconstrained minimization problem directly without adding any new variable and constraint.That is,the squares of 2-norm ε-insensitive loss.In addition,we add the termb2/2 in the objective function to induce strong convexity and to guarantee that the problem has a unique global optimal solution [19].

Therefore,(11) can also be rewritten as

Inspired by smooth support vector machine (SSVM)for classification,the squares of ε-insensitive loss function in the above formulation can be accurately approximated by a smooth function which is infinitely differentiable and defined below:

To prove the convergence,three lemmas and a theorem are introduced as follows:

whereρis a constant.

Bring (39) into (38) to get

It is proved that the support vector regression machine is globally convergent.

2.3 Improved IMM algorithm

Finally,an improved IMM algorithm is introduced,which is improved based on the IMM algorithm [20].As shown in Fig.2,on the basis of the original IMM algorithm,an SVR machine network is added as a feedback network.The model set is still preset,but the feedback network will monitor the matching degree of different models with the actual movement of the target in real time.Then,adjust the process noise covariance matrix coefficients of different model filters according to the degree of matching.Finally,the tracking error can be reduced when the model does not match.

Based on the above discussion,we should first determine the matching degree of the model in real time.The simulation shows that in target tracking,when the target is maneuvering,the filtering residual becomes larger and the greater the maneuvering intensity,the larger the filtering residual [21].Therefore,the filtered residual information is an important parameter for testing the strength of the target’s maneuverability,and it is also an important parameter for testing the degree of model matching.The following formula is now used as the real-time monitoring of the network.

In addition,simulation shows that when the model does not match,reducing the process noise covariance coefficient can reduce the overall tracking error.Therefore,different matching degrees must correspond to a coefficient.Taking innovation as the input of the network,the optimal coefficient as the output value,supplemented by a large amount of training data,the feedback network can be obtained.

The module of coefficient adjustment based on SVR is responsible for receiving the relevant parameters of the IMM model,thus output the coefficient of the error covariance matrix (ECM),and then return this parameter to the IMM model for estimating the current state of the target.The coefficient about ECM is actually used as an intermediate coefficient of the overall tracking model.Therefore,the real state of the target at the current moment can be used as the label,and the loss function for the training of the constraint network is set as the mean square error (MSE) between the estimated state of the target based on the ECM and the real state.The formula for the loss function is as follows:

In the simulation conditions,there are cases of model matching and mismatch,with different adjusting coefficients.Adjusting coefficient and model training are constrained by the above loss function.Thus as long as Monte Carlo simulation is carried out for several times,the corresponding training set and label can be generated.The model training set is the innovation corresponding to different models for several times,and the label is the real state of the target at the current moment.

Since the simulation scenario and model set only involves two simple models,the size of adjustment coefficients is only two.Therefore,the function of adjustment about ECM (SVR module) is not a very complex function.Therefore,a limited training set can fully train a well-performing SVR module.In addition,the training and test scenarios are separate.In the training scenario,the time series involved in motion is 400,which contains scenarios with different model matching degrees.After several Monte Carlo simulations,1 000 sets of data are collected as the training set.

Now suppose that two models are set up in the model set,then the improved IMM algorithm schematic diagram is shown in Fig.2.

Fig.2 Improved IMM algorithm flow chart

It mainly has the following components.

Step 1Input the filtered value of each model in the previous cycle to the input interaction module for input interaction,and then input the interaction value of each model into each model for a new round of filtering.

Step 2Each model performs filtering according to the measurement of the current period,inputs interaction value,and outputs the state estimate and estimate of covariance.

Step 3Each model updates the model probability value according to the relevant result parameters.

Step 4The prediction errord1(k) andd2(k) of two models are used as the input of SVR,and the coefficientsf1(k) andf2(k) of the covariance matrix are output,which can feed back to Step 2 to adjust the estimation results.

Step 5Calculate the overall state output using the current target state estimates of each filter and the probability of each model.

The specific steps of the improved IMM algorithm are basically the same as those of the standard IMM algorithm,which are also divided into: (i) filter initialization and data interaction; (ii) model filter estimation;(iii) model probability update;(iv) data estimation fusion.Among them,Step 1,Step 3,and Step 4 are the same as the original algorithm,and the covariance part prediction of the original algorithm in Step 2 is replaced by

wheref(i)(k-1) represents the output component of the SVR system,and its initial value is 1.

3.Experiments

Assume that the target moves in two-dimensional space,the sampling periodT=1 s,the measurement noise is a Gaussian sequence with a mean value of 0 and a standard deviation of 100 m,and the number of simulations is 200.If the IMM is composed of two single models,one is the constant velocity model,the other is the constant turning model.The initial probabilities of the models are 1/2,and the model transition probability matrix is

A total of 200 Monte-Carlo simulations are used,and the evaluation index is the target root mean square error(RMSE) of position and speed,which can be represented as

whereNis the number of Monte-Carlo simulations;jis thejth simulation;xandyrespectively represent the true value and filtered estimated value of the tracking target state at timek.

In addition,the velocity RMSE is treated by taking relative method,and the formula is as follows:

In (45),the square root values of the two dimensions of velocity are calculated at each moment,and then the value is used to normalize the RMSE of velocity.

Scenario 1Target turning maneuver: The target movement process lasts 400 s,initial state:(1 000,10,0,1 000,10,0).The target starts to move in a straight line at a constant velocity (CV),with an initial speed of 10 m/s.It then executes2a maneuver with constrant acceleration (CA) of 1 m/s from 101 s-191 s.Then it executes a constant turning (CT) motion in 191 s-270 s,and the centripetal acceleration is -π/270 m·s-2.Finally,it goes back to constant velocity motion.The target trajectory is shown in Fig.3.

Fig.3 Track diagram in Scenario 1

It can be seen from Fig.4 and Fig.5 that in the case of turning maneuver,the RMSE of position and speed of the improved IMM algorithm is obviously lower than that of the IMM algorithm.Since the target moves at a constant speed in the initial stage,the model set can match it,the position and velocity RMSEs of the two algorithms are relatively low.In 101 s-190 s,the target moves at a constant acceleration,and the model sets cannot match.Therefore,the RMSE of the position and velocity of the two algorithms increases greatly,and reaches the peak after a period of time,but the improved IMM algorithm is lower than the IMM algorithm.In 191 s-270 s,the target makes a constant turning motion and the model sets match,and the RMSE of the position and velocity errors of the two algorithms begins to decrease.In 271 s-400 s,the target moves at a constant speed and the model sets match,and the RMSE of position and velocity continues to decrease and then floats at a lower level.Under the premise of retaining the low RMSE when the model is matched,it can be seen that the improved IMM algorithm can effectively reduce the RMSE when the model does not match.Finally,the variation trend of adaptive parameters is shown in Fig.6.

Fig.4 RMSE of position in Scenario 1

Fig.5 Root mean square error of speed in Scenario 1

Fig.6 Adaptive parameter changes in Scenario 1

Scenario 2Target continuous maneuver: The target starts to move in a straight line at a constant velocity,with an initial speed of 10 m/s.It then executes a maneuver with an acceleration of 1 m/s2from 51 s-100 s.Then it executes a constant turning motion in 101 s-300 s,and the centripetal acceleration is -π/270 m·s-2.After that,conduct constant acceleration motion in 300 s-350 s,X-axis acceleration is 1 m·s-2andY-axis acceleration is-1 m·s-2.Finally,it goes back to constant velocity motion.The target trajectory is shown in Fig.7.

Fig.7 Track diagram in Scenario 2

It can be seen from Fig.8 and Fig.9 that when the target is continuously manipulated,the improved IMM algorithm can effectively reduce the RMSE when the model does not match.At the same time,the RMSE is kept small when the model is matched.Since the target moves at a constant speed in the initial stage and the model sets match,the position and the velocity RMSE of the two algorithms are relatively low.Within 51 s-100 s,the target moves at a constant acceleration,and the model set cannot be matched.Therefore,the RMSE of the position and the velocity of the two algorithms are greatly increased,and they reach a peak after a period of time,but the RMSE of the improved IMM algorithm is lower than that of the IMM algorithm.In 101 s-300 s,the target makes a constant turning motion,and the model set can be matched.Therefore,the RMSE of the position and the velocity errors of the two algorithms begins to decrease.Within 300 s-350 s,the target moves at a constant acceleration and the model set does not match,so the RMSE of the position and the velocity rises again.Within 350 s-400 s,the target moves at a constant speed and the model set matches,so the RMSE of the position and the speed drops again.Finally,the variation trend of adaptive parameters is shown in Fig.10.

Fig.8 RMSE of position in Scenario 2

Fig.9 RMSE of speed in Scenario 2

Fig.10 Adaptive parameter changes in Scenario 2

The above two examples illustrate that the improved IMM algorithm mainly plays the following roles.

(i) When the model is matched,the tracking accuracy of the traditional IMM algorithm is guaranteed.

(ii) When the model does not match,the feedback network adjusts the degree of trust to the measured value,corrects the covariance prediction value in real time,and eliminates or reduces the filtering tracking accuracy drop and filtering divergence caused by the sudden change of the target state.

4.Conclusions

This paper proposes an improved IMM algorithm based on support vector regression.The traditional IMM algorithm,the SVR method and the improved IMM method based on the above are discussed.

Two types of scenarios are simulated for the traditional IMM algorithm and the improved IMM algorithm.From the experimental results,the following conclusions can be drawn.By using SVR to improve the traditional IMM algorithm,the estimated RMSE of the traditional IMM algorithm in the case of model mismatch is reduced,and the adaptability of the algorithm is improved.The ideas presented in this paper can also optimize and improve single model tracking algorithms and other multi-model tracking algorithms.