Arma 2 where is lz jersey

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Wings: Over The Reich - Mods. Warbirds Armor Simulations - General. Tactical Simulations - General. ARMA Series. Arma3 - Saturday Sampler. DayZ Mod. EA Battlefield Series. When the stochastic portion of the time series has been identified, it is combined with the deterministic part to form a complete model of the system.

Numerical Example In this section a numerical example is presented which illustrates the major results discussed in previous sections. The steady-state solution to equation is Substituting equations and into equations and yields The normalized eigenvector of 1l associated with e0 is 0.

The coefficients found are 1. Carrying out the operations called for in equation yields 5 8. A complete description of the system is displayed in figure 2. Block diagram of system in numerical example. If we make the same transformation that we did in the ARMA modeling procedure, i.

For this specific example we find A target tracker located on the missile is called a seeker. In many surface-to-air defense systems, the tracker is located on the ground and missile and target inertial position are maintained by tne tracker. In either case, the function of the tracker is to piurov. The control system then commands aerodyna- mic control surface deflections to maneuver the missile onto a collision course with the target.

Many guidance schemes developed using modern optimization techniques rely on estimates of future target states to be effective [37]. This naturally leads to the utilization of a tracking filter. At present, tracking filters use target dynamic models which have been devised assuming a priori statistical knowledge of the target's behavior.

These models work well on the average but show severe deficiencies in certain specific cases. The research presented here involves the use of sta- tistical modeling techniques developed in the preceding chapter to synthesize a target dynamic model.

This idea is novel in that the model 48 49 can be made to adapt to changes in the target's trajectory by utilizing a time history of past measured kinematic variables.

The measurements come, quite naturally, from the tracker. Engagement Geometry The definition of the coordinate frame in which the missile-target engagement occurs is a rather important aspect of the problem formulation. The coordinate frame chosen dictates the cinematic variables which describe the target's motion and also defines the frame of reference for the measurements made by the tracker.

The target trajectory is described by a time-varying range vector in three-space. The range magnitude is defined relative to the origin of an inertial reference frame. Inertial, in this case, means neither translating nor rotating with respect to a point fixed on the earth's surface. Three mutually perpendicular unit vectors, i,j and k, which are colinear with the inertial x-axis, y-axis and z-axis, respectively, are used to establish the coordinate system.

Henceforth, underlined quan- tities are vectors. The x-y plane is tangent to the earth's surface at the origin of the inertial frame. The z-axis is directed away from the earth's surface and completes the right-handed triad.

Orientation of the range vector is determined by two angles: azimuth and elevation. Azimuth is the angle between the inertial x-axis and the projection of the range vector onto the x-y plane. Elevation is the angle between the range vector and the range vector projected onto the x-y plane. These quantities are shown in figure 3. Target trajectory 51 Trajectory Generation To illustrate the use of the modeling technique in the tracking problem, a maneuvering target trajectory was generated using elementary kinematic equations.

For simplicity we assume a point-mass target and therefore model only the motion of the center-of-gravity. Aircraft flight can be categorized as 1 non-accelerating or 2 accelerating. The resulting trajectory in each case is quite different. In non-accelerating flight we obtain a trajectory which is a straight line in inertial space.

If the target is accelerating, as in the case of a target performing evasive maneuvers, two effects are observed. The component of acceleration along the velocity vector increases the target speed magnitude of velocity and the component normal to the velocity vector rotates the velocity vector relative to the inertial frame, thus changing the flight path.

The result is a circular arc in inertial space for constant speed targets. At this point we assume that the target acceleration vector is normal to the velocity vector. This is a good assumption for aircraft targets and also simplifies the kinematics. Furthermore, assume that the non-accelerating portions are at constant altitude z component of posi- tion in the inertial frame.

The accelerating portion is in a maneuver plane defined by the velocity and acceleration vectors. In general the maneuver plane is skewed to the principal inertial planes. The flight segments are defined as non-accelerating 0. Transition from one flight segment to the next occurs instantaneously.

The aircraft completes the turn in This corresponds to an acceleration of The trajectory is shown in figures 4 through 7. Target Dynamic Model Synthesis Off-line A statistical model of the target trajectory presented in the pre- ceding section will be developed. We begin by noting that the measure- ments made by the tracker can be used to create a time history of target 53 inertial position.

The transformation is relatively simple for trackers with fixed inertial position. However, for missile-borne seekers the transformation is more complicated since missile attitude and position relative to the inertial frame must be taken into account.

These quan- tities are measured by an inertial navigation systems INS onboard the missile. For the present example let us assume that we have available a time series of target inertial position. The position vector is three- dimensional with components xt, Yt and zt. Let us further assume that we have the entire time series available at the outset. This will allow us S-- r-t 0 -1 -1 0 1 2 3 5 Inertial x-axis km.

Figure 4. Target trajectory in the inertial x-y plane 54 to develop a dynamic model off-line. Of course, off-line modeling is not practical for real-time tracking but is useful when investigating model characteristics such as model order, error covariances and coefficient values. Later in this chapter an on-line modeling technique will be demonstrated.

Finally, for this example we assume that the measurements are deterministic. This means that the errors in prediction are due to modeling errors and not measurement errors.

The error covariance iden- tified will therefore approximate the process noise covariance. Target position versus time x-axis 55 Since the target motion is planar we can describe the trajectory in a new basis in which one component of the position vector is constant.

That is, one axis of the new coordinate frame will be normal to the maneuver plane. Interpreted physically this means 5. Target position versus time z-axis 57 that the velocity component normal to the maneuver plane is zero. Equation can be found using the ARMA modeling algorithm for dege- nerate time series if we assume we can precisely, or at least with insignificant error, identify the model for a constant.

Let us consider the problem of modeling a constant scalar using the autocorrelation method with biased estimates of the autocorrelation sequence. Note that the expressions for yl and I given by equations and converge to the true values 58 as N oo However, the error associated with using a finite length data sequence is excessive.

Box and Jenkins [ suggest a technique to handle nonstationary situations such as this. The time series now consists of vectors sampled at 4 Hz. The biased sample autocorrelation matrices for the zeroth through nine- teenth lag were computed. We find that CO is singular. This means that a linear combination of the components of A Rt k is zero. Comparing equations and we see that we have found the deter- ministic relationship that we expected to discover earlier.

This normal unit vector uniquely defines the plane of the maneuver in the original coordinate frame. The fact that we can determine the maneuver plane by processing position measure- ments in the fashion described here is significant. Performing the transfor- mation on the autocorrelation sequence -- which in this case amounts to no rxre than picking out the 1,1 element -- we find 1 0.

Therefore, it may be prudent to again difference the data and model the resulting time series. The model found is 1 1. Our objective is to model xt, Yt and zt but the above equations involve Axt, Ayt and Azt. Using equation we can write A A.

Therefore, for this particular example, we can predict At k from measurements of Qt k-l, Rt o with no error.

Several points concerning this model should be mentioned. First, the trajectory modeled is very idealistic. We have assumed that the aircraft can accelerate instantaneously and that constant speed and acceleration are maintained in the turn. These assumptions lead to a perfectly cir- cular trajectory when the aircraft is accelerating. Second, we have assumed that the maneuver occurs in a plane.

This is actually a rather good assumption for piloted aircraft executing a sustained high acce- leration maneuver. Also, deterministic measurements were used to build the model. This, of course, helps significantly in determining the pre- sence of degeneracy in the time series and the correct model order.

Finally, the computations were done on a Cyber a 60 bit wordlength digital computer. Computational precision is extremely high. With these facts clearly in mind, we should not be surprised that the model found is deterministic. We assume that the measurements are corrupted by additive, white, zero-mean Gaussian noise. Our goal here is not to model or analyze a par- ticular radar system but rather to illustrate a tracking concept.

We shall therefore choose noise characteristics which approximately repre- sent state-of-the-art radar performance. The standard deviation of the noise in the range measurement is typically on the order of 10 meters.

Angular measurement noise has a standard deviation on the order of a few tenths of a degree. These statistics are assumed to remain constant during the entire tracking process.

The noisy measurements are presented in figures 8 through In order to make the proposed tracking scheme practical for real- time use, we must first convert the measurements of range, azimuth and elevation to inertial position. To do this let us assume that the tracker is located at the origin of the inertial frame.

This is by no means restrictive since a simple coordinate frame translation can be made to put the origin at the tracker. Now that we have a time series of position vector components we can create a dynamic model of the trajectory in the same manner that we did 7 6 5 d3 2 1 0 0 10 20 30 0O 50 time seconds Figure 8. Range measurement versus time. The difference now is that the output sequence is corrupted with measurement noise -- not deterministic.

Since the measurement noise is relatively small we might expect to observe the same degenerate time series phenomenon that we did in the deterministic case. Before proceeding any further with real-time model identification, let us model the entire noisy output time series to see if we can identify the same deterministic relationship that we found in the deterministic model.

As before, we can check for linear dependence in the difference data by computing the eigenvalues of CO. Using noisy 1. Azimuth angle measurement versus time.

What we must do in this case is compare the relative magnitudes of the normalized eigenvalues of Co. Elevation angle measurement versus time. Displayed in figure 11 are the normalized eigenvalues versus time for the noisy trajectory data.

We see that prior to the initiation of the maneuver t that there are two directions in which the velocity components are zero.

We readily understand this result since the trajectory is rectilinear for t for 10 1. Normalized eigenvalues of CO versus time. This direc- tion is obviously normal to the maneuver plane. The direction cosines of the unit vector normal to the maneuver plane are the components of V0, the eigenvector associated with e0.

Plots of the components of V0 are shown in figures 12 through The true values shown in figures 12 through 14 are the deterministic values given by equation First component of V0 versus time. Second component of Vo versus time.

If the time series is degenerate we transform the autocorrela- tion sequence to the new basis using MI, thus reducing the dimension of the time series to be modeled. We must now transform the AR model to the original basis because the measurements 1. Third component of VO versus time. The transformation matrix relating the two coordinate frames is M. Maneuvering Target Tracking Algorithm Using the equations developed in the preceding sections we can for- mulate the following target tracking algorithm: 1.

Convert measurements to Rt. Update Cj k. Use equation Compute eg and el using equations and and com- pute M. Use equations and Increment k. Measure azimuth, elevation and range at time k.

Numerical Results and Predictor Performance Using the algorithm just presented with the noisy measurements displayed in figures 8 through 10, we can adaptively change the tracker model and thereby more accurately predict target inertial position.

Models computed for a few discrete-time points along the trajectory 74 and the prediction errors resulting from use of the tracker are pre- sented in this section. When the autocorrelation sequence found after each measurement update is used in the Levinson algorithm, we find that an AR 2 model fits the data well. This determination is made by using the ABMA model identification algorithm presented earlier.

Referring again to figure 11, we note that for k e0 the data and the remaining stochastic variable to model is a scalar. Diagonal elements of yl. Diagonal elements of Y2.

We see from figures 15 and 16 and from equations through that the model changes very little in the last 20 to 25 seconds of the trajectory. The transient phase which starts with maneuver initiation is relatively short in duration; lasting approximately 5 seconds. At a 4 Hz sample rate this represents 20 samples. Model parameters are quite dyna- mic during the transient period.

Plotted in figures 17 through 19 are the inertial position predic- tion errors versus time. The first 10 measurements were used only to compute the autocorrelations to get the model identification started.

No tracking was done during this period. We observe from the error plots that the tracker is working well until maneuver onset. At that time the errors begin to increase and there is a noticable bias in the error sequence, par- ticularly in the x-axis and y-axis.

As the tracker adapts the model parameters in response to the maneuver, the errors once again decrease and the mean is once again near zero. Error in predicted position versus time x-axis 79 The error sequence bias which began at maneuver initiation results from the same phenomenon that Bullock and Sangsuk-Iam [71 used to detect the occurrence of maneuvers.

The non-maneuvering target model used to do the tracking prior to maneuver onset is no longer valid after the maneuver starts and we observe a bias in the innovations sequence. The difference is, of course, that here we are not applying any statistical tests to determine if a maneuver has occurred. The important point to note here is that the tracker adjusted the model parameters in response to an unanticipated abrupt maneuver.

Error in predicted position versus time y-axis. However, by using information already needed to do the tracking for example, the noticable increase in el and the accompanying change in VO or by monitoring the mean of the error sequence, we should be able to detect maneuvers with a reasonable degree of confidence. Of course, the objective of this research was not maneuver detection but the close relationship between detecting maneuvers and tracking highly agile eva- sive targets makes the maneuver detection function an attractive poten- tial by-product.

Error in predicted position versus time z-axis. This is because we can reinitialize the tracker and include only the measurements made subsequent to the detected maneuver.

We therefore remove the premaneuver measurements and totally delete the influence of these data from the model. Implementation is rather straightforward.

An auxiliary, constant length data file of posi- tion vectors is maintained to perform the reinitialization. Prior to maneuver detection the oldest position vector in the auxiliary file is deleted each time a new measurement is added.

This is typically referred to as the "sliding window" concept. The length of the window depends on the number of points required to detect the maneuver after the maneuver is actually initiated. Subsequent to maneuver detection the tracking algorithm is reinitialized using only the auxiliary file and then con- tinues to operate as described earlier.

To do so let us assume that 10 postmaneuver measurements are required to detect the maneuver. The auxiliary file is therefore 10 measurements long. The performance of the two methods is compared using the sample mean and sample standard deviation of the position errors in each of the three inertial axes. Tais result is counterintuitive but can be attributed to the small number of samples used to reinitialize the tracker. Ten postmaneuver points are not enough data to obtain a good sample autocorrelation estimate.

All of the dif- ferences in the models and in the error sequences for the two cases were in the first few seconds following maneuver detection. By the end of the trajectory the model parameters AR coefficients and M and prediction errors for both cases were essentially indistinguishable. The effect of discarding the premaneuver points soon became negligible. There is apparently a trade-off which can be made between the length of the sliding window and tracker perfor- mance.

Too many premaneuver points in the window will lead to a sluggish tracker and too few will not be enough to provide a good estimate of the autocorrelation sequence. A major result is the development of an algorithm for finding the coef- ficients and order of the minimum order ARMA model of a multivariable system from the output autocorrelation sequence. The model order is found by examining the ranks of submatrices within a block Toeplitz of output autocorrelation matrices.

Once the order is known, the AR coef- ficients are determined in the usual way by solving the modified, or extended, Yule-Walker equations. The algorithm is a new result.

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