Interactive multi-model tracking method based on ellipse constraint and straight line constraint

Abstract

The invention provides an interactive multi-model tracking method based on elliptic constraint and linear constraint, which comprises the following steps: analyzing the constraint of an elliptic track and the constraint of a linear track based on the interactive multi-model tracking of the elliptic constraint and the linear constraint, and respectively modeling the elliptic track constraint and the linear track constraint; constraint information is used as pseudo measurement to be augmented to the filtering process of the interactive multi-model, so that the accuracy of selection of different motion state models and the precision of target tracking are improved.

Description

Technical Field

[0001] This invention belongs to the field of radar technology, specifically relating to an interactive multi-model tracking method based on elliptic constraints and linear constraints. Background Technology

[0002] Interactive multiple model (EMM) is an adaptive algorithm with Markov chain transition probabilities. This algorithm assumes that the target's motion follows a Markov chain with known, finite transition probabilities across different models, and considers the interactions of multiple models to estimate the target's state. In real-world target motion, targets often obey specific constraints. For example, a vehicle's trajectory follows the road shape, a satellite follows its orbital constraints, and a specific jamming aircraft's flight path may be runway-shaped or horseshoe-shaped. Incorporating constraint information into the EMM can improve target tracking accuracy.

[0003] Interactive multiple model algorithms (EMAs) typically employ multiple filters processed in parallel, each corresponding to a different state space, which describes different target motion patterns. Introducing prior constraint information during the filtering process can improve target tracking accuracy. Existing EMA algorithms have undergone extensive research on different types of target maneuvers, but they have failed to incorporate prior constraint information into the EMA algorithms.

[0004] Current constraint information is mainly linear, introducing constraints as pseudo-measures into the tracking system to improve target accuracy. However, linear constraints are too idealistic and rarely used in practical applications. Nonlinear constraints, on the other hand, are difficult to express accurately mathematically and suffer from problems such as filter divergence due to their strong nonlinearity.

[0005] In existing interactive multi-model algorithms, different filters process targets with the same dimension. However, real-world target motion may involve multiple models, each with different dimensions. A more comprehensive and effective solution has yet to be proposed for filtering and weighted fusion of different target state dimensions to obtain the final estimate. Summary of the Invention

[0006] To address the current lack of trajectory constraint-based interactive multi-model (DMM) tracking, this invention proposes an DMM tracking method based on elliptical and linear constraints. This method is designed for scenarios where the target may undergo elliptical and linear motion, incorporating elliptical and linear trajectory constraints as pseudo-measurements into the DMM filtering process. Based on the characteristics of elliptical and linear motion, this invention designs unscented Kalman filter models for elliptical and linear motion, respectively. By introducing constraint information into the DMM algorithm, the specificity of model selection for different motion states and the accuracy of target tracking are improved. The specific steps include:

[0007] Step 1: Analyze the constraints satisfied by the target motion of the elliptical trajectory.

[0008] Assuming the target moves along an elliptical trajectory in a two-dimensional plane, with the major axis of the ellipse being a, the minor axis being b, and the center of the ellipse being (x0, y0), and the angle between the line connecting any point (x, y) on the elliptical trajectory and the center of the ellipse and the x-axis being θ, we can obtain the standard equation and parametric equation of the ellipse:

[0009]

[0010] By differentiating the standard equation and parametric equation of the ellipse respectively, the position and velocity constraints of the elliptical trajectory are obtained as follows:

[0011]

[0012] in and Let ω represent the x-axis velocity and y-axis velocity of any point on the ellipse, and let ω represent the angular velocity of the target moving along the elliptical trajectory.

[0013] The three expressions for the elliptical trajectory constraint are as follows:

[0014] b 2 (x-x0) 2 +a 2 (y-y0) 2 =a 2 b 2

[0015]

[0016] Elliptical trajectory constraints include constraints on position, velocity, angular velocity, ellipse center, and major and minor axes of the ellipse.

[0017] Step 2: Model the motion of the target object with an elliptical trajectory

[0018] This invention employs a constant-speed turning motion (CT) model to describe the target's motion along an elliptical trajectory. The target state vector for a traditional constant-speed turning motion model is as follows:

[0019]

[0020] The target state equation is:

[0021] x k+1 =F(x) k )+Γ(v k (7)

[0022] Where F(x) k ) is the state transition equation, Γ(vk ) is the noise distribution equation.

[0023] Traditional two-dimensional target tracking uses position and velocity as the states, and the target follows an elliptical trajectory, increasing its angular velocity in the state vector. This invention assumes the observation sensor is radar, which can observe the target's range and azimuth. Therefore, target observation is expressed as follows:

[0024]

[0025] Where x k and y k Represents the x-axis and y-axis positions of the target at time k. For example, noise measurement. This represents the noise in azimuth measurement.

[0026] The pseudo-measurement method expands the state dimension, allowing constraints to be introduced as observations into the observation system. Therefore, the target state vector... The dimensions are expanded as follows:

[0027]

[0028] Correspondingly, the target observation equation is also expanded in dimension:

[0029]

[0030] For an objective subject to an elliptical trajectory shape constraint, the constraint relationship always holds, therefore the noise corresponding to the constraint is zero, i.e. i is the pseudo-measurement corresponding to the i-th constraint.

[0031] The zero-mean Gaussian white noise involved in this invention assumes that the target states after dimension expansion are uncorrelated. The corresponding noise covariance matrix is ​​diagonally augmented from the traditional noise covariance matrix, and the augmented noise covariance matrix is ​​as follows:

[0032]

[0033] Where R k It is a diagonal matrix, and the elements on the diagonal are the observation distance noise variance and the observation angle noise variance, respectively.

[0034] Step 3: Analyze the constraints satisfied by the target motion along the straight trajectory.

[0035] A target moving along a straight line in a two-dimensional plane can be described by its slope and intercept. More generally, a straight line perpendicular to the x-axis can be described by the general equation of a straight line:

[0036] ax + by + c = 0 (12)

[0037] Where a, b, and c are constants, and a and b are not both 0. Differentiating the general equation of the straight line yields the velocity constraint at any point:

[0038]

[0039] Therefore, the constraint equations for the straight line trajectory are as follows:

[0040]

[0041] The linear constraint equations constrain the position and velocity relationships at any point on a straight line by introducing three additional parameters.

[0042] Step 4: Model the motion of the target on a straight trajectory

[0043] This invention uses a uniform motion CV model to model the motion of a target on a straight trajectory. The target state vector for a conventional uniform motion model is as follows:

[0044]

[0045] The target vector after dimension expansion is as follows:

[0046]

[0047] The target observation equation, after dimensional expansion, is as follows:

[0048]

[0049] Similarly to the elliptical trajectory constraint, the noise covariance matrix is ​​augmented.

[0050] Step 5: Perform an interactive multi-model algorithm based on elliptic and linear constraints.

[0051] The interactive multi-model algorithm consists of four parts: interaction, filtering, model probability update, and state estimation fusion.

[0052] Interaction refers to the interactions between models, aiming to rationally allocate the state estimates of each model from the previous time step to each model at the current time step. This process involves probability mixing between models, thereby creating a mixture state estimate and covariance for each model. Specifically, based on the model transition probabilities of each model and the model probabilities from the previous step, a mixture probability is calculated. This probability is then used to weighted average the state and error covariance of each model to generate new initial conditions. Let... This is the state estimate of filter j at time k-1.

[0053] Let u be the state covariance transition matrix.k-1 (j) represents the probability of model j at time k-1. The outputs of the N filters after the interaction are as follows:

[0054]

[0055] Each model is filtered independently; this method uses unscented Kalman filtering. Unscented Kalman filtering is developed based on unscented transformation, which is as follows:

[0056] Assume the mean of the target state vector is Covariance is P x First, calculate 2n. x +1 sampling point ξ i and the corresponding weight W i

[0057]

[0058] κ is the scaling parameter, n x For system state x k The vector dimension. By using the state equation to obtain the one-step prediction and weights of the sampling points, the state prediction estimate and the state prediction covariance can be obtained.

[0059]

[0060] Q(k) is the process noise covariance matrix.

[0061] The update process is as follows:

[0062]

[0063] Where x represents the sampling point of the target state.

[0064] The predicted measurement values ​​for the sampling points are as follows:

[0065]

[0066] Here, z represents measurement prediction, and h(x) represents the sampling point x after being processed by the observation function h().

[0067] Measurement prediction and covariance are respectively:

[0068] Weighted summation of measurement predictions:

[0069] Covariance of measurement predictions:

[0070] Cross-covariance of measurement and state vectors:

[0071] The gain is:

[0072]

[0073] State update and state covariance update:

[0074]

[0075] Model probability updates are performed after obtaining the filtering results for each model. This step is based on the likelihood of each model generating the observed data (calculated using the likelihood function) and the previous model probabilities. Using Bayesian rules and considering the current observations' support for each model, the probability of each model is updated.

[0076] If the filtering residual of model j is The corresponding covariance is The model probability of each filter output can be obtained. Updated to:

[0077]

[0078] In the formula:

[0079]

[0080] Where Z(k) is the actual observed value at time k, and H j (k) is the measurement matrix of model j. These are the observed values ​​predicted by model j. R(k) is the observation-prediction covariance of the model, and R(k) is the observation-noise covariance matrix. k (j) represents the weights of model j. The update probability of model j is:

[0081]

[0082] in It represents the sum of all states under the influence of the state transition matrix.

[0083] State estimation fusion involves weighting the state estimates of all models according to their latest model probabilities to form a comprehensive state estimate. This comprehensive state estimate integrates information from all possible models to the greatest extent possible, providing an optimal estimate that comprehensively considers all possible scenarios.

[0084] In this invention, due to the different conditions and the different number of constraint equations between elliptical and linear constraints, the state vectors of the two models exhibit inconsistent dimensionality after dimensional expansion. To address this, the invention employs a method of dimensionality reduction before fusion, that is, retaining only the common state components from each model for fusion, while using the estimation results of the individual models as the final estimates for other components. Before filtering, the initial mixed values ​​are then expanded using the estimation results of the individual models.

[0085] The beneficial effects of this invention are as follows:

[0086] The interactive multi-model tracking method based on elliptic and linear constraints proposed in this invention introduces elliptic and linear constraints into the interactive multi-model algorithm filtering, which can improve the tracking accuracy of targets with linear and elliptical trajectory motion characteristics. Furthermore, this invention establishes a robust framework for incorporating constraints into the interactive multi-model algorithm, allowing for more targeted model selection for different target motion states. Attached Figure Description

[0087] Figure 1 This is a flowchart of the method of the present invention.

[0088] Figure 2 The target's motion trajectory and tracking trajectory diagram.

[0089] Figure 3 This is a schematic diagram of the root mean square error curve for target tracking position.

[0090] Figure 4 This is a schematic diagram of the root mean square error curve for target tracking speed.

[0091] Figure 5 This is a probability graph of an interactive multi-model system with constraints.

[0092] Figure 6 This is a standard interactive multi-model probabilistic plot. Detailed Implementation

[0093] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0094] This invention compares the differences between interactive multi-models with and without elliptic and linear constraints. In the conventional interactive multi-model, no constraints are considered, there is no state augmentation, and both models employ a uniform velocity model and a constant velocity turning model, with unscented Kalman filtering used for filtering. In the interactive multi-model with both elliptic and linear constraints, the elliptic and linear constraints are considered, and corresponding state augmentation is performed. The two constraints are used as pseudo-measures during observation for augmentation. Both models employ a uniform velocity model and a constant velocity turning model, with unscented Kalman filtering used for filtering.

[0095] Assuming the trajectory of a target conforms to both an elliptical trajectory and a straight line trajectory, such as... Figure 1 As shown, the target's trajectory is generated first. The target's position and velocity information are first generated in one dimension, and then mapped onto a two-dimensional trajectory. The variance of the target's one-dimensional motion noise is 0.5 meters. The trajectory begins as an ellipse and ends as a straight line, with the target's trajectory shape as shown. Figure 2 As shown. The major semi-axis of the elliptical trajectory is 1000 meters, and the minor semi-axis is 700 meters. Assume the sensor used is radar, located at the origin, with a range observation noise variance of 5 meters and an azimuth observation noise variance of 0.05 degrees. The radar scan interval is 0.1 seconds, the simulation duration is 300 seconds, and 40 Monte Carlo simulations are performed.

[0096] To evaluate the accuracy of the unconstrained interactive multi-model algorithm, the root mean square error (RMSE) is used to measure the deviation between the true values ​​and the observed values. The formula for calculating the root mean square error (RMSE) is as follows:

[0097]

[0098] Where n is the observation step size, y i For the target truth value, This is the sensor's predicted value.

[0099] like Figure 3 , Figure 4 As shown, the constrained interactive multiple model algorithm has a smaller root mean square error than the conventional interactive multiple model algorithm, proving that the constrained interactive multiple model algorithm has higher accuracy.

[0100] To assess the impact of constraints on model selection in interactive multiple models Figure 5 , Figure 6 The probabilities of the two models during the observation process were plotted. When the target moves along an elliptical trajectory, the model with constant-speed circular motion has a higher probability in the constrained interactive multi-model, while the model with uniform speed has a higher probability in linear motion. There is no significant difference in the probabilities of the two models in the conventional interactive multi-model. This demonstrates that the elliptical and linear constraint-based interactive multi-model tracking method proposed in this invention can effectively improve the tracking accuracy and model selection accuracy of the target under elliptical trajectories and linear motion.

[0101] This invention is not limited to the specific embodiments described above, and various modifications and variations are possible. Any modifications, equivalent substitutions, or improvements made to the above embodiments based on the technical essence of this invention should be included within the scope of protection of this invention.

Claims

1. An interactive multiple model tracking method based on elliptical constraints and linear constraints, characterized in that: Introducing elliptical trajectory constraint and straight line trajectory constraint into the interactive multiple model tracking algorithm, comprising the following steps: Step 1: analyzing the constraint conditions satisfied by the elliptical trajectory target motion; Step 2: modeling the elliptical trajectory target motion; Step 3: analyzing the constraint conditions satisfied by the straight line trajectory target motion; Step 4: modeling the straight line trajectory target motion; Step 5: performing the interactive multiple model algorithm based on the elliptical constraint and the straight line constraint.

2. The interactive multiple model tracking method based on elliptical and linear constraints according to claim 1, characterized in that: The step 1 specifically comprises the following steps: Suppose that the target moves in an elliptical trajectory in a two-dimensional plane, the long semi-axis of the ellipse is a, the short semi-axis is b, the coordinates of the center of the ellipse are (x0, y0), the angle between the line connecting any point (x, y) on the elliptical trajectory and the x-axis and the x-axis is θ, the standard equation and the parametric equation of the ellipse are obtained as follows: x = x0 + a cos(θ) y = y0 + b sin(θ); Derivatives are taken for the standard equation and the parametric equation of the ellipse respectively, and the position and velocity constraints of the elliptical trajectory are obtained as follows: wherein and denote the x-axis and y-axis direction velocities of an arbitrary point on the ellipse, and ω denotes the angular velocity of the target moving along the elliptical trajectory. The three expressions contained in the elliptical trajectory constraint are obtained as follows: b 2 (x-x0) 2 +a 2 (y-y0) 2 = a 2 b 2 The elliptical trajectory constraint contains the constraint relationship of the position, velocity, angular velocity, center of the ellipse, long and short axes of the ellipse.

3. The interactive multiple model tracking method based on elliptical constraints and linear constraints according to claim 2, characterized in that: The step 2 specifically comprises the following steps: The constant speed turning motion model is used to describe the target state vector of the target moving along the elliptical trajectory as follows: The target state equation is: x k+1 = F(x k ) + Γ(v k ) where F(x k ) is the state transition equation, and Γ(v k ) is the noise distribution equation. For the target moving in an elliptical trajectory, the state vector of the target increases the angular velocity, and based on the distance and azimuth angle of the target observed by the sensor, the target observation expression is as follows: where x k and y k represent the x-axis and y-axis positions of the target at time k, represents an example measurement noise, represents an azimuth measurement noise; The pseudo-measurement method introduces the constraint condition as an observation into the observation system by state extension, so that the target state vector is extended as follows: Correspondingly, the target observation equation is also expanded as follows: Suppose that the expanded target states are not related, the noise covariance matrix is diagonally augmented, and the augmented noise covariance matrix is as follows: where R k is a diagonal matrix, and the elements on the diagonal are the observation range noise variance and the observation angle noise variance, respectively.

4. The interactive multiple model tracking method based on elliptical constraints and linear constraints according to claim 3, characterized in that: The step 3 specifically comprises the following steps: The target moving along a straight line in a two-dimensional plane is described by using the equation of the straight line as follows: ax + by + c = 0 Where a, b, and c are constants, and a and b are not zero at the same time; derivatives are taken for the equation of the straight line, and the velocity constraint of any point is obtained as follows: Then the straight line trajectory constraint equation is as follows: ax + by + c = 0 5. The interactive multiple model tracking method based on elliptical constraints and linear constraints according to claim 4, characterized in that: The step 4 specifically comprises the following steps: The constant speed motion model is used to model the target motion along the straight line, and the target state vector is as follows: The expanded target vector is as follows: The expanded target observation equation is as follows:

6. The interactive multiple model tracking method based on elliptical constraints and linear constraints according to claim 5, characterized in that: The step 5 specifically comprises the following steps: The interactive multiple model algorithm comprises four parts: interaction, filtering, model probability updating, and state estimation fusion; Interaction: based on the model transition probability of each model and the model transition probability of the last step, the mixed probability is calculated, and then the mixed probability is used to weight average the state and error covariance of each model; Filtering: the unscented Kalman filter is used to filter each model independently; Model probability updating: according to the probability of each model generating observation data and the previous probability, the support degree of each model is updated through the Bayesian rule combined with the current observation; State estimation fusion: the state estimations of all models are weighted and averaged according to the latest model probabilities to form a comprehensive state estimation, forming the optimal estimation.

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