A target tracking method based on Gaussian mixture unscented Kalman filter

Abstract

The application provides a target tracking method based on Gaussian mixture unscented Kalman filtering, comprising the following steps: decomposing measurement noise by a Gaussian mixture model to obtain a plurality of Gaussian noise component models; the plurality of Gaussian noise component models are Gaussian distributions with different mean values, variances and mixing ratios; determining state equations and measurement equations of a filtering system under each Gaussian noise component respectively based on the plurality of Gaussian noise component models; fusing a fusion state and a fusion covariance of a target based on a plurality of the state equations and the measurement equations; and suppressing the influence of noise and improving the accuracy of target tracking by decomposing the measurement noise and weightedly fusing the target state.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of target tracking, in particular to a target tracking method based on Gaussian mixture unscented Kalman filtering. BACKGROUND

[0002] Target tracking technology has been widely applied in practice, and sensor data of a target is crucial for target tracking, but there are still some problems to be solved in sensor data of a target in the process, for example, data is usually contaminated by noise or there is a data loss condition. Due to a large amount of dust in a grain warehouse, information measured by a sensor of a flat grain robot is often contaminated by non-Gaussian noise, in order to reduce the influence of non-Gaussian measurement noise on the performance of a target tracking algorithm, a particle filter, Huber's M-estimation method, a maximum correlation entropy criterion, maximum likelihood estimation and a student t-based method can be used to construct a robust target tracking algorithm to improve the performance of the algorithm to a certain extent. However, in practical application, the non-Gaussian measurement noise distribution is usually not only heavy-tailed or peak-biased, but also some more general non-Gaussian distributions such as multi-peak and light-tailed distribution are also common. If the current method is used to model the general non-Gaussian distribution, the modeling accuracy will inevitably be reduced, thereby reducing the estimation accuracy of the algorithm.

[0003] Therefore, the application provides a target tracking method based on Gaussian mixture unscented Kalman filtering, which decomposes measurement noise by a Gaussian mixture model and obtains a plurality of Gaussian noise component models, suppresses the influence of noise and improves the accuracy of target tracking. SUMMARY

[0004] The application aims to provide a target tracking method based on Gaussian mixture unscented Kalman filtering, which comprises: decomposing measurement noise by a Gaussian mixture model to obtain a plurality of Gaussian noise component models; the plurality of Gaussian noise component models are Gaussian distributions with different mean values, variances and mixing proportions; determining state equations and measurement equations of a filtering system under each Gaussian noise component based on the plurality of Gaussian noise component models; and fusing to obtain a fused state and a fused covariance of a target based on a plurality of the state equations and the measurement equations.

[0005] Further, the plurality of Gaussian noise component models obtained by decomposing the measurement noise by the Gaussian mixture model comprise: decomposing the measurement noise by the Gaussian mixture model to obtain an approximate probability density function of the measurement noise; and taking a plurality of Gaussian noises corresponding to a plurality of Gaussian components in the approximate probability density function as the plurality of Gaussian noise component models of the measurement noise.

[0006] Further, the expression of the approximate probability density function of the measurement noise is:

[0007]

[0008] where p(r k ) represents the probability density function of the measurement noise; n represents the Gaussian component variable; N represents the total number of Gaussian components; represents the Gaussian component; β n , and represent the proportion, mean and variance of the Gaussian component ; r k represents the measurement noise with variance R k .

[0009] Further, the expression of the Gaussian component is:

[0010]

[0011] where [·] T represents the transpose operation; q represents the process noise.

[0012] Further, the expressions of the state equation and the measurement equation of the determined filtering system under each Gaussian noise component are:

[0013]

[0014] where x k represents the state of the target at time k; f k (·) represents the process function; x k-1 represents the state of the target at time k-1; q k-1 represents the process noise with variance Q k-1 ; z k represents the measurement state of the target; h k (·) represents the measurement function; represents the measurement noise of the nth Gaussian component model, with covariance

[0015] Further, the fusion state and the fusion covariance of the target obtained by the fusion include: obtaining a mixed state estimate value and a mixed covariance based on the model state and the model probability of the nth Gaussian component model; obtaining an updated state and an updated covariance based on the measurement state of the target; determining a model update probability of the result output by each Gaussian component model changing; obtaining the fusion state and the fusion covariance based on the mixed state estimate value, the mixed covariance, the updated state, the updated covariance and the model update probability.

[0016] Further, the expression of the mixed state estimate value and the mixed covariance is:

[0017]

[0018]

[0019] wherein, denotes the mixed state estimate; n denotes the Gaussian component model variable; N denotes the total number of Gaussian component models; denotes the state of the nth model; denotes the mixing probability from model n to model u; denotes the mixed covariance; (·) T denotes the transpose; denotes the covariance of the nth model.

[0020] Further, the expression for obtaining the updated state and the updated covariance is:

[0021]

[0022]

[0023] wherein, denotes the updated state of the nth Gaussian component model; denotes the predicted state of the nth model; denotes the innovation matrix of the nth Gaussian component model; z k denotes the measured state of the target; denotes the predicted observation; denotes the updated covariance of the nth Gaussian component model, denotes the covariance of the nth Gaussian component model at the previous step; denotes the covariance matrix; (·) T denotes the transpose.

[0024] Further, the expression for determining the model update probability of each Gaussian component model output whose result has changed is:

[0025]

[0026] wherein, denotes the model update probability; denotes the normalization factor; denotes the likelihood function of each Gaussian component model; u denotes the updated model variable; N denotes the total number of Gaussian component models.

[0027] Further, the expression for obtaining the fused state and the fused covariance is:

[0028]

[0029]

[0030] wherein, denotes the fusion state; n denotes a Gaussian component model variable; N denotes a total number of Gaussian component models; denotes the updated model probability; denotes the state of the nth Gaussian component model; P k| denotes the fusion covariance; denotes the covariance of the nth Gaussian component model; (·) T denotes the transpose.

[0031] The technical solution of the embodiments of the present application has at least the following advantages and beneficial effects:

[0032] The present application suppresses the influence of noise and improves the accuracy of target tracking by measuring the decomposition of noise and the weighted fusion of target state. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 An exemplary flowchart of a target tracking method based on Gaussian mixture unscented Kalman filtering is provided for some embodiments of the present application.

[0034] Figure 2 The performance of different algorithms in terms of minimum mean square error in the case of target tracking is shown.

[0035] Figure 3 The real trajectory of the target and the trajectories obtained using different algorithms are shown. DETAILED DESCRIPTION

[0036] To make the objectives, technical solutions, and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described below in detail with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. The components of the embodiments of the present application described and shown in the drawings herein can be arranged and designed in various different configurations.

[0037] Figure 1 An exemplary flowchart of a target tracking method based on Gaussian mixture unscented Kalman filtering is provided for some embodiments of the present application. As shown in FIG. 1, the flowchart 100 includes the following contents: Figure 1

[0038] Step 110, decompose the measurement noise by a Gaussian mixture model to obtain a plurality of Gaussian noise component models; the plurality of Gaussian noise component models are Gaussian distributions with different mean values, variances, and mixing proportions.

[0039] ​The trajectory of the target can be reconstructed using the model information, and a mathematical model describing the relationship between the state of the target, the observation of the sensor, and the time can be established.

[0040] The trajectory reconstruction problem can be modeled as the following state space model:

[0041]

[0042] where, xk represents the state vector of the target at time k; zk represents the measurement vector; f k (·) and h k (·) represent the process function and the measurement function; q k-1 and r k are the process noise and the measurement noise with variances Q k-1 and R k . However, due to the influence of impulse noise, outliers, and other factors, the distribution of q k and r k is usually non-Gaussian. Non-Gaussian distribution will reduce the performance of the unscented Kalman filter algorithm and the URTS is more smooth, because they are originally designed under the Gaussian assumption.

[0043] The present application estimates the state of the target under non-Gaussian noise conditions using observation data By decomposing the noise into several noise components, the combination of these noise components can approximately replace the original noise. The present application uses GMM (Gaussian Mixture Model) to decompose the process noise and the measurement noise to obtain the approximate PDF (Probability Density Function) of the original measurement noise.

[0044] In some embodiments, the decomposition of the measurement noise by the Gaussian Mixture Model to obtain a plurality of Gaussian noise component models comprises:

[0045] The decomposition of the measurement noise by the Gaussian Mixture Model obtains an approximate probability density function of the measurement noise.

[0046] In some embodiments, the expression of the approximate probability density function of the measurement noise is:

[0047]

[0048] where p(r k ) represents the probability density function of the measurement noise; n represents the Gaussian component variable; N represents the total number of Gaussian components; β n , and represent the Gaussian component the proportion, mean and variance of the Gaussian components, and the measurement noise r k with variance R k .

[0049] In some embodiments, the Gaussian component has an expression as follows:

[0050]

[0051] where [·] T denotes a transpose operation; q denotes process noise.

[0052] The Gaussian noise corresponding to each Gaussian component in the approximate probability density function is taken as a Gaussian noise component model of the measurement noise.

[0053] Step 120: determining state equations and measurement equations of the filtering system under each Gaussian noise component based on the plurality of Gaussian noise component models.

[0054] In some embodiments, the expression of the determination of the state equations and the measurement equations of the filtering system under each Gaussian noise component is as follows:

[0055]

[0056] where x k denotes a state of the target at time k; f k (·) denotes a process function; x k-1 denotes a state of the target at time k-1; q k-1 denotes process noise with variance Q k-1 ; z k denotes a measurement state of the target; h k (·) denotes a measurement function. denotes measurement noise of the nth Gaussian component model with covariance R

[0057] Step 130: fusing to obtain a fused state and a fused covariance of the target based on the plurality of state equations and measurement equations.

[0058] In some embodiments, the fusing to obtain the fused state and the fused covariance of the target comprises:

[0059] obtaining a mixed state estimate value and a mixed covariance based on a model state and a model probability of the nth Gaussian component model.

[0060] The mixing probability from model n to model u can be obtained as follows:

[0061]

[0062] wherein, denotes the mixing probability from model n to model u; denotes the nth row and the u-th element of denotes the model probability; denotes the normalization factor; u denotes the Gaussian model variable; and

[0063]

[0064] wherein, the initial value of

[0065] In some embodiments, the expression for obtaining the mixing state estimate and the mixing covariance is:

[0066]

[0067]

[0068] wherein, denotes the mixing state estimate; n denotes the Gaussian component model variable; N denotes the total number of Gaussian component models; denotes the state of the nth model; denotes the mixing probability from model n to model u; denotes the mixing covariance; (·) T denotes the transpose; denotes the covariance of the nth model.

[0069] Based on the measurement state z of the target k , the updated state and the updated covariance

[0070] Sigma points can be generated by

[0071]

[0072] wherein, denotes the 1st sigma point generated at time k-1; denotes the estimated state of the nth model at time k-1; denotes the covariance of the state of the nth model at time k-1; i = 1, 2, …, 2p, is the i-th column vector of , and the parameter λ = α2 (p+o)-p represents the mixing ratio coefficient, where 0 < a < 1 determines the range of sigma points, and o can be set as 3-p; p represents the dimension of the state.

[0073] Then, the predicted state of the previous step and the covariance The formula is as follows:

[0074]

[0075]

[0076]

[0077] wherein, represents the predicted state of the previous step sigma point; represents the i-th sigma point of the n-th model at the k-1 time; represents the predicted state of the n-th model; represents the predicted covariance of the previous step; Q k represents the covariance matrix of the process noise; and respectively represent the weight of the mean value and the covariance of the sigma point; and The specific form is:

[0078]

[0079]

[0080] wherein, λ represents the mixing ratio coefficient; a 2 represents the range parameter; parameter is a non-negative weighting factor, which can integrate the dynamic differences of high-order terms in the equation, so as to include high-order effects. For Gaussian distribution, it is set to 2.

[0081] The formula for calculating is as follows:

[0082]

[0083] wherein, represents the sigma point of the first predicted state of the n-th model; represents the predicted state of the n-th model; represents the predicted covariance of the previous step.

[0084] The formula for calculating is as follows:

[0085]

[0086] wherein, represents the simulated observation value; represents the observation value obtained by the sigma point.

[0087] The specific formula of the covariance matrix and the cross-covariance matrix is as follows:

[0088]

[0089]

[0090] wherein, represents the covariance matrix; R k represents the covariance matrix of the observation noise; represents the cross-covariance matrix; represents the sigma point of the i th predicted state of the n th model.

[0091] The expression of the innovation matrix of the n th model can be as follows:

[0092]

[0093] In some embodiments, the expression of the updated state and the updated covariance is as follows:

[0094]

[0095]

[0096] wherein, represents the updated state of the n th Gaussian component model; represents the predicted state of the previous step of the n th model; represents the innovation matrix of the n th Gaussian component model; z k represents the measurement state of the target; represents the simulated observation value; represents the updated covariance of the n th Gaussian component model, represents the covariance of the previous step of the n th Gaussian component model; represents the covariance matrix; (·) T represents the transpose.

[0097] The model update probability of the result output by each Gaussian component model is determined to change.

[0098] The model probability is updated using the likelihood function ​The likelihood function of each model is:

[0099]

[0100] wherein, represents the likelihood function; represents the covariance of the u-th model; E Z represents the intermediate variable, represents the observation after transmission; represents the observation obtained by simulation.

[0101] In some embodiments, the expression for determining the model update probability of the result of each Gaussian component model output changing is:

[0102]

[0103] wherein, represents the model update probability; represents the normalization factor, represents the likelihood function of each Gaussian component model; u represents the update model variable; N represents the total number of Gaussian component models.

[0104] Based on the mixed state estimation value, the mixed covariance, the update state, the update covariance and the model update probability, the fusion state and the fusion covariance are obtained.

[0105] In some embodiments, the expression for obtaining the fusion state and the fusion covariance is:

[0106]

[0107]

[0108] wherein, represents the fusion state; n represents the Gaussian component model variable; N represents the total number of Gaussian component models; represents the probability after the n-th model update; represents the state of the n-th Gaussian component model; P k| represents the fusion covariance; represents the covariance of the n-th Gaussian component model; (·) T represents the transpose.

[0109] Figure 2 The minimum mean square error for applying multiple algorithms to track the target. Figure 3 The tracking trajectory for applying multiple algorithms to track the target.

[0110] The performance of GMUKF and GMURTS smoother is compared with UKF, MCUF, MEE-UKF, URTSS and MC-RTS. In this study, all simulations are averaged over 200 independent Monte Carlo runs, where 100 samples are used to compute the mean square deviation (MSD) which measures the steady state error of the proposed algorithms. The concept of MSD is as follows:

[0111]

[0112] where x k represents the true state of the system, represents the filtered state.

[0113] For a vehicle tracking model, the process equation can be:

[0114]

[0115] where I2 represents the identity matrix, and ΔT = 0.1s. The state includes the position p 1； and velocity v 1； of the target on the x-axis, and the position p 2； and velocity v 2； on the y-axis. The measurement equation for the range and angle of the target is:

[0116]

[0117] The process noise and measurement noise are set as: and

[0118] The initial values of parameters and P 0|0 are set as:

[0119]

[0120] where x is the true state of the target.

[0121] As can be seen from Figure 2 and Figure 3 , the proposed GMUKF algorithm performs much better than the existing filters under non-Gaussian noise. The UKF and MC-RTS smoother even fail to track the target.

[0122] The above only describes the preferred embodiments of the present application and is not used to limit the present application. For those skilled in the art, the present application can have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A target tracking method based on Gaussian mixture unscented Kalman filter, characterized in that, The method comprises the following steps: decompose the measurement noise by a Gaussian mixture model to obtain a plurality of Gaussian noise component models; the plurality of Gaussian noise component models are Gaussian distributions with different mean values, variances and mixing proportions; the step of decomposing the measurement noise by the Gaussian mixture model to obtain the plurality of Gaussian noise component models comprises the following steps: decompose the measurement noise by the Gaussian mixture model to obtain an approximate probability density function of the measurement noise; take a plurality of Gaussian noises corresponding to a plurality of Gaussian components in the approximate probability density function as the plurality of Gaussian noise component models of the measurement noise; determine state equations and measurement equations of a filtering system under each Gaussian noise component based on the plurality of Gaussian noise component models; fuse the target fusion state and the fusion covariance based on the plurality of state equations and the measurement equations, which comprises the following steps: obtain a mixed state estimation value and a mixed covariance based on a model state and a model probability of the nth Gaussian component model; obtain an update state and an update covariance based on a measurement state of the target; determine a model update probability of a result output by each Gaussian component model; obtain the fusion state and the fusion covariance based on the mixed state estimation value, the mixed covariance, the update state, the update covariance and the model update probability.

2. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 1, characterized in that, The expression of the approximate probability density function of the measurement noise is: wherein denotes the probability density function of the measurement noise; denotes the Gaussian component variable; denotes the total number of Gaussian components; denotes a Gaussian component; , and denotes the proportion, mean and variance of a Gaussian component ; denotes the measurement noise with variance .

3. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 2, characterized in that, The Gaussian component The expression for the Gaussian component is: wherein denotes a transpose operation; denotes process noise.

4. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 1, characterized in that, The expression of the state equations and the measurement equations of the filtering system under each Gaussian noise component is: wherein, represents the state of the target at time k; represents the process function; represents the state of the target at time k-1; represents the process noise with variance . represents the measurement state of the target; represents the measurement function; represents the measurement noise of the nth Gaussian component model with covariance .

5. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 1, characterized in that, The expression of the mixed state estimation value and the mixed covariance is: wherein, represents a mixture state estimate; represents a Gaussian component model variable; represents a total number of Gaussian component models; represents a state of the nth model; represents a mixture probability from model n to model u; represents a mixture covariance; represents a transpose; represents a covariance of the nth model.

6. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 1, characterized in that, The expression of the update state and the update covariance is: wherein, represents the updated state of the n-th Gaussian component model; represents the predicted state of the n-th model; represents the innovation matrix of the n-th Gaussian component model; represents the measured state of the target; represents the predicted observation; represents the updated covariance of the n-th Gaussian component model, represents the covariance of the n-th Gaussian component model of the previous step; represents the covariance matrix; represents the transpose.

7. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 1, characterized in that, The expression of the model update probability of the result output by each Gaussian component model is: wherein, denotes the updated model probability; denotes a normalization factor; denotes the likelihood function of each Gaussian component model; denotes the updated model variable; denotes the total number of Gaussian component models.

8. The target tracking method based on Gaussian mixture unscented Kalman filter according to claim 1, characterized in that, The expression of the fusion state and the fusion covariance is: wherein, denotes a fusion state; denotes a Gaussian component model variable; denotes a total number of Gaussian component models; denotes an updated model probability; denotes a state of the n-th Gaussian component model; denotes a fusion covariance; denotes a covariance of the n-th Gaussian component model; denotes a transpose.

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