CN107886058A - Noise related two benches volume Kalman filter method of estimation and system - Google Patents
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Abstract
Description
技术领域technical field
本发明涉及领域，具体地说，特别涉及一种噪声相关的两阶段容积Kalman滤波估计方法及系统。The present invention relates to the field, in particular to a noise-related two-stage volumetric Kalman filter estimation method and system.
背景技术Background technique
目前推导的各类两阶段滤波算法的前提都是假设非线性高斯系统是噪声无关的，即状态方程噪声和量测方程噪声不相关且均为高斯白噪声，这是在理想状态下的噪声情况。但在实际应用中，系统噪声相关的情况普遍存在，例如受到系统内部元器件和外部环境变化的双重影响，会出现噪声相关；量测噪声为有色噪声的系统，进行了噪声扩维，扩充为状态后，也会将原系统转化为噪声相关系统；在机动目标跟踪等需要进行多传感器信息融合的系统中，也大量存在着噪声相关情况。目前对于噪声相关系统，常规的解决方式是忽略相关噪声，使用传统的两阶段容积Kalman滤波进行估计，这样必然会降低估计的精度。本方案通过引入转换系数矩阵，将噪声相关系统转换为不相关系统且得到两者之间的转换关系，再进行估计，充分考虑相关噪声，实现了噪声相关系统的准确跟踪。The premise of various two-stage filtering algorithms currently derived is to assume that the nonlinear Gaussian system is noise-independent, that is, the noise of the state equation and the noise of the measurement equation are uncorrelated and both are Gaussian white noise. This is the noise situation in an ideal state. . However, in practical applications, system noise correlations are ubiquitous. For example, due to the dual influence of system internal components and external environment changes, noise correlations will occur; the measurement noise is colored noise system, and the noise dimension is expanded to After the state, the original system will also be converted into a noise-related system; in systems that require multi-sensor information fusion, such as maneuvering target tracking, there are also a large number of noise-related situations. At present, for noise-related systems, the conventional solution is to ignore the relevant noise and use the traditional two-stage volumetric Kalman filter for estimation, which will inevitably reduce the estimation accuracy. By introducing the conversion coefficient matrix, the scheme converts the noise-correlated system into an uncorrelated system and obtains the conversion relationship between the two, and then estimates it, fully considering the correlated noise, and realizes the accurate tracking of the noise-correlated system.
纯方位跟踪系统是通过两个传感器来跟踪移动目标的状态，得到非线性的测量值，每个传感器只能获得目标状态的角度观察值，两个角度观测值记为αi，k和βi，k，两个角度的观测值形成平面坐标中的交叉点的位置。对于直角坐标系的两个传感器Si1和Si2(i＝1，2，…，N)分别固定在平台P1和P2上，它们之间的距离为d。有很多传感器固定在平台Pj(j＝1，2)上，记为{(S1，j，Pj)，(S2，j，Pj)，…，(SN，j，Pj)}，对应的非线性量测值为{(α1，k，β1，k)，(α2，k，β2，k)，…，(αN，k，βN，k)}。The azimuth-only tracking system uses two sensors to track the state of the moving target to obtain nonlinear measurement values. Each sensor can only obtain the angular observation value of the target state, and the two angular observation values are recorded as α i, k and β i , k , the observations of the two angles form the location of the intersection point in planar coordinates. Two sensors S i1 and S i2 (i=1, 2, . . . , N) of the Cartesian coordinate system are fixed on platforms P 1 and P 2 respectively, and the distance between them is d. There are many sensors fixed on the platform P j (j=1, 2), denoted as {(S 1, j , P j ), (S 2, j , P j ),..., (S N, j , P j )}, the corresponding nonlinear measurement value is {(α 1, k , β 1, k ), (α 2, k , β 2, k ),…, (α N, k , β N, k )} .
其动力学模型是一个四维的非线性系统，xk＝[x1，k x2，k y1，k y2，k]T，其中x1，k和x2，k是东、北方向的位移分量，y1，k和y2，k是和位移分量相对的速度分量，将目标的移动作为CV模型，状态方程和偏差方差如下：Its dynamic model is a four-dimensional nonlinear system, x k = [x 1, k x 2, k y 1, k y 2, k ]T, where x 1, k and x 2, k are east and north directions The displacement component of y 1, k and y 2, k is the velocity component relative to the displacement component, and the movement of the target is used as a CV model. The state equation and deviation variance are as follows:
其中过程噪声方差in process noise variance
跟踪周期T＝1s。 Tracking cycle T = 1s.
根据交叉原则，观测函数According to the intersection principle, the observation function
在多传感器系统中，量测方程为：In a multi-sensor system, the measurement equation is:
其中h1，k(xk)＝h2，k(xk)＝…＝hN，k(xk)＝hk(xk)。where h 1,k (x k )=h 2,k (x k )=...=h N,k (x k )=h k (x k ).
设υi，k＝ciωk，k-1，则有 Suppose υ i, k = c i ω k, k-1 , then we have
初始状态估计值和协方差矩阵为： The initial state estimates and covariance matrix are:
仿真时间为200秒，对两种算法进行了1000次Monte Carlo仿真。算法误差使用均方根误差(RMSE误差)如下进行计算：The simulation time is 200 seconds, and 1000 Monte Carlo simulations are performed on the two algorithms. The algorithmic error is calculated using root mean square error (RMSE error) as follows:
其中M为Monte Carlo次数，和分别表示第n次Monte Carlo仿真下的x*的状态值和估计值。Where M is the Monte Carlo number, and Respectively represent the state value and estimated value of x * under the nth Monte Carlo simulation.
综上所述，现有技术存在的问题为：现有技术没有分析相关噪声，降低了估计的精度；跟踪结果差；在估计过程中没有考虑相关噪声，将相关噪声按照无关噪声的情况考虑，解决的难度在于在估计飞行器的位置的同时，不能将相关的噪声考虑在内；会出现滤波发散现象，根本无法进行进一步跟踪估计。In summary, the problems existing in the existing technology are: the existing technology does not analyze the correlated noise, which reduces the estimation accuracy; the tracking result is poor; the correlated noise is not considered in the estimation process, and the correlated noise is considered as the irrelevant noise. The difficulty in solving this problem lies in that while estimating the position of the aircraft, the relevant noise cannot be taken into account; there will be filtering divergence, and further tracking and estimation cannot be performed at all.
发明内容Contents of the invention
为了解决现有技术的问题，本发明实施例提供了一种。本发明可以用于单个或者多个飞行器的目标跟踪领域，具体以纯方位跟踪系统为例进行说明。In order to solve the problems in the prior art, an embodiment of the present invention provides a method. The present invention can be used in the field of target tracking of single or multiple aircraft, and is specifically described by taking a purely azimuth tracking system as an example.
本发明是这样实现的，一种噪声相关的两阶段容积Kalman滤波估计方法，包括：The present invention is achieved in this way, a noise-related two-stage volumetric Kalman filter estimation method, comprising:
采用恒等变形的方法对噪声相关系统进行变换，建立系统模型；Use the method of constant deformation to transform the noise-related system to establish a system model;
通过加入系数，将所述系统模型由噪声相关系统转化为噪声不相关系统，建立新系统模型；By adding coefficients, the system model is converted from a noise-correlated system to a noise-uncorrelated system, and a new system model is established;
再用所述新系统模型中的噪声参数与两阶段滤波的递推计算，得到噪声相关的两阶段容积Kalman滤波估计器。Using the noise parameters in the new system model and the recursive calculation of the two-stage filtering, a noise-related two-stage volumetric Kalman filter estimator is obtained.
进一步，所述采用恒等变形的方法对噪声相关系统进行变换，建立系统模型的步骤具体如下：Further, the method of adopting the constant deformation to transform the noise-related system, and the steps of establishing the system model are as follows:
所述噪声相关系统为非线性高斯系统：The noise-dependent system is a nonlinear Gaussian system:
xk+1＝fk(xk)+ωk+1，k； (1)x k+1 = f k (x k )+ω k+1, k ; (1)
zk＝hk(xk)+υk； (2)z k =h k (x k )+v k; (2)
其中k是离散时间序列，xk∈Rn×1是系统的状态向量，zk∈Rm×1是量测向量，f(·)和h(·)为已知的非线性状态转移函数和量测函数且在xk处连续可微，过程噪声序列ωk+1，k和量测噪声序列υk均为高斯白噪声序列，其中均值为E(ωk+1，k)＝qk，E(υk)＝rk，方差Qk+1，k和Rk满足如下条件：where k is a discrete time series, x k ∈ R n×1 is the state vector of the system, z k ∈ R m×1 is the measurement vector, f( ) and h( ) are known nonlinear state transition functions and measurement function and are continuously differentiable at x k , the process noise sequence ω k+1, k and the measurement noise sequence υ k are both Gaussian white noise sequences, where the mean value is E(ω k+1, k )=q k , E(υ k )=r k , variance Q k+1, k and R k satisfy the following conditions:
初始状态x0与ωk+1，k、υk无关，且满足：The initial state x 0 has nothing to do with ω k+1, k , υ k and satisfies:
进一步，所述通过加入系数，将所述系统模型由噪声相关系统转化为噪声不相关系统，建立新系统模型的步骤具体为：Further, by adding coefficients, the system model is converted from a noise-correlated system to a noise-uncorrelated system, and the steps of establishing a new system model are specifically:
将噪声相关系统进行变换，通过恒等变形，将噪声相关系统转换为不相关系统，再进行滤波估计；Transform the noise-related system, convert the noise-related system into an uncorrelated system through identity deformation, and then filter and estimate;
由模型公式(2)得：From the model formula (2):
zk-hk(xk)-υk＝0；z k -h k (x k )-υ k = 0;
设Δk为待定系数，则有：Let Δ k be the undetermined coefficient, then:
Δk(zk-hk(xk)-υk)＝0 (3)；Δ k (z k -h k (x k )-υ k )＝0 (3);
带入公式(1)并整理得：Bring into the formula (1) and sort it out:
其中in
Fk(xk)＝fk(xk)+Δk(zk-hk(xk)) (5)；F k (x k ) = f k (x k )+Δ k (z k -h k (x k )) (5);
公式(1)和(2)所示模型转化为：The models shown in formulas (1) and (2) transform into:
zk＝hk(xk)+υk (8)；z k =h k (x k )+υ k (8);
其中 in
将噪声相关系统转换为噪声无关系统，则有：Converting the noise-dependent system into a noise-independent system, we have:
展开得：expands to:
当满足公式(9)时，噪声无关系统过程噪声和量测噪声不相关；When formula (9) is satisfied, the process noise and measurement noise of the noise-independent system are not correlated;
使用转换模型方法得到的噪声相关非线性高斯滤波公式均加角标t表示；The noise-related nonlinear Gaussian filter formulas obtained by using the conversion model method are all indicated by subscript t;
进一步，再用所述新系统模型中的噪声参数与两阶段滤波的递推计算，从而得到噪声相关的两阶段容积Kalman滤波估计器的步骤具体为：Further, using the noise parameters in the new system model and the recursive calculation of the two-stage filtering, the steps of obtaining the noise-related two-stage volume Kalman filter estimator are specifically as follows:
带有随机偏差的非线性高斯系统：Nonlinear Gaussian system with random biases:
其中k是离散时间序列，xk∈Rn×1是系统的状态向量，bk∈Rp×1是系统偏差向量，zk∈Rm×1是量测向量，fk(·)和hk(·)为已知的非线性状态转移函数和量测函数且在xk处连续可微，过程噪声序列偏差噪声序列和量测噪声序列υk均为高斯白噪声序列，偏差噪声与过程噪声、量测噪声互不相关，其中均值为E(υk)＝rk，方差 和Rk满足如下条件：where k is the discrete time series, x k ∈ R n×1 is the state vector of the system, b k ∈ R p×1 is the system bias vector, z k ∈ R m×1 is the measurement vector, f k ( ) and h k (·) is the known nonlinear state transition function and measurement function and is continuously differentiable at x k , the process noise sequence biased noise sequence and the measurement noise sequence υ k are both Gaussian white noise sequences, and the deviation noise is not correlated with the process noise and measurement noise, where the mean is E(υ k )=r k , variance and R k satisfy the following conditions:
初始状态x0、b0与ωk+1，k、υk无关，且满足：The initial state x 0 , b 0 has nothing to do with ω k+1, k , υ k and satisfies:
令make
Hk(Xk)＝hk(xk)+Fkbk；H k (X k )=h k (x k )+F k b k ;
公式(15)给出的系统模型改写为如下形式：The system model given by formula (15) is rewritten as follows:
Xk+1＝Γk(Xk)+ωk X k+1 ＝Γ k (X k )+ω k
Zk＝Hk(Xk)+υkZ k ＝H k (X k )+υk
其中 in
根据所做恒等变换，模型(15)变为如公式(7)和(8)所示的噪声无关系统，如公式(17)所示：According to the identity transformation, the model (15) becomes a noise-independent system as shown in formulas (7) and (8), as shown in formula (17):
其中Fk(Xk)＝Γk(Xk)+Δk(Zk-Hk(Xk))， Where F k (X k )=Γ k (X k )+Δ k (Z k -H k (X k )),
初始化状态条件：Initialize state condition:
for k＝1，2，…，N do；for k=1, 2, ..., N do;
步骤一，时间更新：Step 1, time update:
1)假设已知k-1时刻的后验密度函数对Pk-1|k-1做Cholesky分解，得到 1) Assume that the posterior density function at time k-1 is known Perform Cholesky decomposition on P k-1|k-1 to get
2)计算容积点 和传播容积点其中，i＝1，2，...，m＝2nx；2) Calculate the volume point and propagation volume points Wherein, i=1, 2, . . . , m=2n x ;
3)令mk-1＝qk-1-Δkrk，根据状态向量维数和偏差向量维数将mk进行分块，则有：3) Let m k-1 = q k-1 -Δ k r k , divide m k into blocks according to the dimension of the state vector and the dimension of the deviation vector, then:
同理对rk进行分块有：Similarly, block r k as follows:
借助mk-1估计噪声相关的无偏滤波器状态预测值Estimating noise-dependent unbiased filter state predictors with the aid of m k-1
和有偏滤波器状态预测值and biased filter state predictors
4)令4) order
根据两阶段变换公式中分块矩阵的维度，将进行分块：Depending on the dimensions of the block matrix in the two-stage transformation formula, the To chunk:
同理，令对状态噪声方差矩阵进行分块：In the same way, make Block the state noise variance matrix:
借助耦合关系估计噪声相关的无偏滤波器状态误差协方差和有偏滤波器状态误差协方差 With the help of coupling relationship Estimating noise-dependent unbiased filter state error covariance and the biased filter state error covariance
步骤二，量测更新：Step 2, measurement update:
A)分解得到 A) decomposition get
B)计算容积点和经过量测方程传播的传播容积点其中，i＝1，2，...，m；B) Calculate the volume point and propagation volume points propagated through the measurement equation Wherein, i=1, 2, ..., m;
C)估计噪声相关的量测预测值 C) Estimate noise-dependent measurement predictions
D)估计噪声相关的量测误差协方差和噪声相关的交叉协方差 D) Estimate noise-related measurement error covariance Cross-covariance related to noise
E)将公式按照相应的维度进行分块得到分块增益矩阵： E) Apply the formula Block according to the corresponding dimension to get the block gain matrix:
估计噪声相关的无偏滤波器卡尔曼增益和噪声相关的有偏滤波器卡尔曼增益 Estimate noise-dependent unbiased filter Kalman gain Kalman Gain of Biased Filters Dependent on Noise
F)借助mk-1和rk计算噪声相关的无偏滤波器状态估计值F) Calculating the noise-dependent unbiased filter state estimate with the aid of m k-1 and r k
和噪声相关的有偏滤波器状态估计值Biased Filter State Estimates Dependent on Noise
G)借助计算噪声相关的无偏滤波器估计误差协方差G) by means of Calculate noise-dependent unbiased filter estimation error covariance
噪声相关的有偏滤波器估计误差协方差 Estimation Error Covariance of Noise-Dependent Biased Filters
结束。Finish.
本发明另一目的在于提供一种噪声相关的两阶段容积Kalman滤波估计系统。Another object of the present invention is to provide a noise-related two-stage volumetric Kalman filter estimation system.
本发明实施例提供的技术方案带来的有益效果是：本发明提出的一种噪声相关的两阶段容积Kalman滤波算法(Two-stage Cubature Cubature Filter with correlatednoises，TSCKF-CN)，该算法基于最小方差估计准则提出了变换模型的两阶段容积Kalman滤波算法，通过引入转换系数矩阵，将噪声相关系统转换为不相关系统且得到两者之间的转换关系。The beneficial effect brought by the technical solution provided by the embodiment of the present invention is: a noise-related two-stage volumetric Kalman filter algorithm (Two-stage Cubature Cubature Filter with correlated noises, TSCKF-CN) proposed by the present invention, the algorithm is based on the minimum variance The estimation criterion proposes a two-stage volumetric Kalman filtering algorithm for the transformation model. By introducing the transformation coefficient matrix, the noise-correlated system is transformed into an uncorrelated system and the transformation relationship between them is obtained.
在实际应用中，噪声相关的非线性系统非常普遍，如果不考虑相关噪声，仍旧应用常规的两阶段滤波算法，必然会降低估计的精度。本发明基于最小方差估计准则提出了变换模型的两阶段容积Kalman滤波算法，通过引入转换系数矩阵，将噪声相关系统转换为不相关系统且得到两者之间的转换关系，在实际应用中，将噪声相关作为条件考虑在内，用于单个或者多个飞行器的目标跟踪领域，跟踪精度均优于忽略噪声相关不计的情况，具有较好的跟踪结果。In practical applications, noise-related nonlinear systems are very common. If the correlation noise is not considered, the conventional two-stage filtering algorithm will inevitably reduce the estimation accuracy. The present invention proposes a two-stage volumetric Kalman filtering algorithm for the transformation model based on the minimum variance estimation criterion. By introducing the transformation coefficient matrix, the noise-correlated system is transformed into an uncorrelated system and the transformation relationship between the two is obtained. In practical applications, the Considering the noise correlation as a condition, it is used in the field of single or multiple aircraft target tracking. The tracking accuracy is better than that of ignoring the noise correlation, and it has better tracking results.
本发明的TSCKF-CN算法在噪声相关的非线性系统中，各个时刻位置的估计值均优于不考虑相关噪声的两阶段容积Kalman算法，估计精度优势明显，且在某些时刻，两阶段容积Kalman滤波因为不考虑相关噪声，会出现滤波发散现象，根本无法进行进一步跟踪估计。In a noise-related nonlinear system, the TSCKF-CN algorithm of the present invention can estimate the position at each moment better than the two-stage volumetric Kalman algorithm that does not consider the correlation noise, and has obvious advantages in estimation accuracy, and at some moments, the two-stage volumetric Because the Kalman filter does not consider the relevant noise, there will be filter divergence, and further tracking and estimation cannot be performed at all.
附图说明Description of drawings
图1是本发明实施例提供的噪声相关的两阶段容积Kalman滤波估计方法流程图。FIG. 1 is a flowchart of a noise-related two-stage volumetric Kalman filter estimation method provided by an embodiment of the present invention.
具体实施方式Detailed ways
为使本发明的目的、技术方案和优点更加清楚，下面将结合附图对本发明实施方式作进一步地详细描述。In order to make the object, technical solution and advantages of the present invention clearer, the implementation manner of the present invention will be further described in detail below in conjunction with the accompanying drawings.
现有技术没有分析相关噪声，降低了估计的精度；跟踪结果差。The prior art does not analyze the correlation noise, which reduces the estimation accuracy; the tracking result is poor.
下面结合附图对本发明作详细描述。The present invention will be described in detail below in conjunction with the accompanying drawings.
如图1所示，本发明实施例提供的噪声相关的两阶段容积Kalman滤波估计方法包括：As shown in Figure 1, the noise-related two-stage volumetric Kalman filter estimation method provided by the embodiment of the present invention includes:
S101：采用恒等变形的方法对噪声相关系统进行变换，建立系统模型；S101: transforming the noise-related system by using the method of constant deformation, and establishing a system model;
S102：通过加入系数，将所述系统模型由噪声相关系统转化为噪声不相关系统，建立新系统模型；S102: Convert the system model from a noise-correlated system to a noise-uncorrelated system by adding coefficients to establish a new system model;
S103：再用所述新系统模型中的噪声参数与两阶段滤波的递推计算，从而得到噪声相关的两阶段容积Kalman滤波估计器。S103: Using the noise parameters in the new system model and the recursive calculation of the two-stage filtering to obtain a noise-related two-stage volumetric Kalman filter estimator.
下面结合具体实施例对本发明作进一步描述。The present invention will be further described below in conjunction with specific embodiments.
可选地，所述采用恒等变形的方法对噪声相关系统进行变换，建立系统模型的步骤具体如下：Optionally, the noise-related system is transformed by using the method of constant deformation, and the steps of establishing a system model are as follows:
考虑如下的非线性高斯系统：Consider the following nonlinear Gaussian system:
xk+1＝fk(xk)+ωk+1，k (4-1)x k+1 = f k (x k )+ω k+1, k (4-1)
zk＝hk(xk)+υk (4-2)z k ＝h k (x k )+υ k (4-2)
其中k是离散时间序列，是系统的状态向量，是量测向量，f(·)和h(·)为已知的非线性状态转移函数和量测函数且在xk处连续可微，过程噪声序列ωk+1，k和量测噪声序列υk均为高斯白噪声序列，其中均值为E(ωk+1，k)＝qk，E(υk)＝rk，方差Qk+1，k和Rk满足如下条件：where k is the discrete time series, is the state vector of the system, is the measurement vector, f(·) and h(·) are known nonlinear state transition functions and measurement functions and are continuously differentiable at x k , the process noise sequence ω k+1, k and the measurement noise sequence υ k are all Gaussian white noise sequences, where the mean value is E(ω k+1, k )=q k , E(υ k )=r k , and the variance Q k+1, k and R k satisfy the following conditions:
初始状态x0与ωk+1，k、υk无关，且满足：The initial state x 0 has nothing to do with ω k+1, k , υ k and satisfies:
可选地，所述通过加入系数，将所述系统模型由噪声相关系统转化为噪声不相关系统，建立新系统模型的步骤具体为：Optionally, by adding coefficients, the system model is converted from a noise-correlated system to a noise-uncorrelated system, and the step of establishing a new system model is specifically:
将模型进行变换，通过恒等变形，将噪声相关系统转换为不相关系统，再进行滤波估计。The model is transformed, and the noise-correlated system is converted into an uncorrelated system through the identity deformation, and then the filtering estimation is performed.
由模型公式(4-2)可得：From the model formula (4-2), we can get:
zk-hk(xk)-υk＝0z k -h k (x k )-υ k ＝0
设Δk为待定系数，则有：Let Δ k be the undetermined coefficient, then:
Δk(zk-hk(xk)-υk)＝0 (4-3)Δ k (z k -h k (x k )-υ k )＝0 (4-3)
带入公式(4-1)并整理可得：Bring into the formula (4-1) and sort it out:
其中in
Fk(xk)＝fk(xk)+Δk(zk-hk(xk)) (4-5)；F k (x k )＝f k (x k )+Δ k (z k -h k (x k )) (4-5);
公式(4-1)和(4-2)所示模型转化为：The models shown in formulas (4-1) and (4-2) are transformed into:
zk＝hk(xk)+υk (4-8)；z k =h k (x k )+v k (4-8);
其中 in
将模型从噪声相关系统转换为噪声无关系统，即让系统过程噪声和量测噪声不相关，则有：Converting the model from a noise-dependent system to a noise-independent system, that is, making the system process noise and measurement noise uncorrelated, then:
展开即得：Expand to get:
即当满足公式(4-9)时，系统模型中过程噪声和量测噪声不再相关，可以使用非线性高斯滤波算法进行计算，为了区分非线性高斯滤波公式，使用转换模型方法得到的噪声相关非线性高斯滤波公式均加角标t表示。That is, when the formula (4-9) is satisfied, the process noise and measurement noise in the system model are no longer correlated, and the nonlinear Gaussian filtering algorithm can be used for calculation. In order to distinguish the nonlinear Gaussian filtering formula, the noise correlation obtained by using the conversion model method Non-linear Gaussian filter formulas are all marked with a subscript t.
本发明提出的一种噪声相关的两阶段容积信息滤波估计算法(Two-stageCubature Information Filter with correlated noises，TSCIF-CN)，该算法利用交叉协方差与误差协方差的乘积和Jacobian矩阵的近似关系，将Jacobian矩阵消去，保证了算法在高维非线性系统中的应用。A noise-related two-stage volume information filter estimation algorithm (Two-stageCubature Information Filter with correlated noises, TSCIF-CN) proposed by the present invention, the algorithm utilizes the approximate relationship between the product of the cross covariance and the error covariance and the Jacobian matrix, Eliminating the Jacobian matrix ensures the application of the algorithm in high-dimensional nonlinear systems.
以上所述仅为本发明的较佳实施例而已，并不用以限制本发明，凡在本发明的精神和原则之内所作的任何修改、等同替换和改进等，均应包含在本发明的保护范围之内。The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principles of the present invention should be included in the protection of the present invention. within range.
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