# Line of Sight Tracking Errors

Purpose

This article treats periscope line of sight tracking errors when the LOS is slewed from the zenith to the horizon and points below. Tracking errors are particularly important in periscopes systems for navigation and sighting for obvious reasons and therefore an understanding of their cause  and correction is of importance.

This article also illustrates the vector ray trace method which is used here  to calculate the LOS tracking errors. This is an analytic approach which results in equations that relate tracking errors to LOS viewing angle and head mirror  and  mirror spindle misalignments.  An understanding of the error types and their relative magnitudes and directions allows the setting  of meaningful tolerances on the manufacture and alignment  of the  periscope head mirror system components.

LOS tracking errors can also be determined by modeling of the misaligned mirror using optical design software, This is a purely numerical approach which gives results only for the specific case modeled and therefore makes a generalized study of tracking errors somewhat tedious.

The vector ray trace method determines ray direction after reflection from the mirror for a given incident ray and mirror orientation. The mirror orientation is expressed by the normal vector to the plane of the mirror. Thus, to establish the reflected ray direction one needs only to know the mirror normal vector. So, the basic analysis here centers on expressing the mirror normal as a function of LOS scan angle, mirror and mirror spindle misalignments.

Periscope Line-of-sight (LOS)

The direction in object space defined by the periscope reticle midpoint and periscope objective lens, is referred to here as the periscope line-of-sight (LOS). It is folded 90 degrees from the vertical center line of the periscope by a head mirror or prism. The prism or mirror is rotatable about a horizontal axis to provide slewing of the LOS in the elevation plane. The LOS angle change is twice the rotation angle of the prism.

Periscope Line of Sight  Tracking Errors

When the LOS is slewed in elevation, the actual direction of the LOS can differ from the apparent direction if there is misalignment in the head mirror mount or spindle. The diagram below illustrates tracking error. The graph depicts a typical variation between the actual and apparent lines of sight in the presence of misalignment. The misalignment can be due to non- perpendicularity between the head mirror  spindle axis of rotation and the periscope optical axis, or if the mirror mount is skewed in its nominal  position.

Ideal alignment exists when the mirror spindle  axis is not only  perpendicular to the periscope axis but also when it is parallel to the reflecting plane of the mirror and when the periscope reticle is centered on the  optical axis. The overall periscope is assumed to be plumb.  Deviation from any one of. these condtions results in tracking errors similar to the one illustrated above.

Periscope Tracking Error Analysis

The LOS tracking error characteristic of an operational periscope can be determined by slewing an angle measuring collimator in front of the periscope head window  in a plane containing the plumb line. The recorded data can be used to calibrate the periscope to achieve the actual operational LOS position. If  tracking  errors  exceed the allowed limits, disassembly of the periscope and adjustment of the head mirror mount position, scan spindle  and reticle positions  may be required  to achieve an acceptably low tracking error characteristic

Adjustment can be a formidable challenge due to “cross talk” between the adjustments and difficulty in discerning the impact of each in its makeup of the resultant tracking error characteristic. Therefore, avoiding or minimizing  the adjustment process system is highly desirable. Development of  a meaningful set of tolerances for the head mirror component  fabrication and assembly leads to this goal. The present analysis provides the basis for such.

Vector Ray Trace Method

A ray coaxial with the periscope optical axis is traced from within the periscope out into object space.after reflection from the head mirror. Its direction in the presence of head mirror misalignment is  compared to the direction  when there is no misalignment. The difference  yields the line of sight tracking error.

The tracking error will be seen to vary with periscope elevation angle. The mathematical function. which describes this variation can be conveniently determined by employing the vector ray trace method. (See Ref. 1 for an excellent treatment of this subject.)

The incident and reflected rays are represented by vectors. The miiror plane is represented by a vector normal to the reflecting surface:

The incident ray can be represented by the unit vector

$\fn_phv&space;\mathbf{S}_{0}=S_{0x}\textbf{i}+S_{0y}\textbf{j}+S_{0z}k$

and the  reflected ray by the unit vector

$\fn_phv&space;\mathbf{S}_{1}=S_{1x}\textbf{i}+S_{1y}\textbf{j}+S_{1z}k$

where i, j, and k are unit vectors along the X,Y and Z axes, respectively;  S0x, S0y, and S0z are the corresponding scalar quantities for the incident ray;  and S1x, S1y,and S1z  the scalar quantities for the reflected ray.

The normal to the mirror defines its angular orientation in three-dimensional  space. It is represented, similarly, by the unit vector M.

$\fn_phv&space;\mathbf{M}={}M_{x}\textbf{\textbf{i}}+M_{y}\textbf{\textbf{j}}+M_{z}\textbf{k}$

The mirror normal components are used to  generate the “Reflection Matrix”, R:

$\fn_phv&space;\begin{bmatrix}R&space;\end{bmatrix}=$:$\fn_phv&space;\begin{bmatrix}&space;1-2M_{x}^{2}&-2M_{x}M_{y}&space;&-2M_{x}M_{z}&space;\\&space;-2M_{x}M_{y}&1-2M_{y}^{2}&space;&-2M_{y}M_{z}&space;\\&space;-2M_{x}M_{z}&-2M_{y}M_{z}&space;&1-2M_{z}^{2}&space;\end{bmatrix}$

Multiplication of the input ray vector S0 by the reflection matrix R yields the reflected ray vector S1::

$\fn_phv&space;\begin{bmatrix}\textbf{S}_{1}&space;\end{bmatrix}=\begin{bmatrix}R&space;\end{bmatrix}\begin{bmatrix}\textbf{S}_{0}&space;\end{bmatrix}$

that is,

$\fn_phv&space;\begin{bmatrix}s_{1x}&space;\\s_{1y}&space;\\s_{1z}&space;\end{bmatrix}=\begin{bmatrix}&space;1-2M_{x}^{2}&-2M_{x}M_{y}&space;&-2M_{x}M_{z}&space;\\&space;-2M_{x}M_{y}&1-2M_{y}^{2}&space;&-2M_{y}M_{z}&space;\\&space;-2M_{x}M_{z}&-2M_{y}M_{z}&space;&1-2M_{z}^{2}&space;\end{bmatrix}\begin{bmatrix}s_{0x}_{}&space;\\s_{0y}&space;\\s_{0z}&space;\end{bmatrix}$

Carrying out the matrix multiplication yields equations for each of the reflected ray components::

$\fn_phv&space;s_{1x}=s_{0x}\left&space;(&space;1-2M_{x}^{2}&space;\right&space;)+s_{0y}\left&space;(&space;-2M_{x}M_{y}&space;\right&space;)+s_{0z}\left&space;(&space;-2M_{x}M_{z}&space;\right&space;)$

$\fn_phv&space;s_{1y}=s_{0x}\left&space;(&space;-2M_{x}M_{y}&space;\right&space;)+s_{0y}\left&space;(&space;1-2M_{y}^{2}&space;\right&space;)+s_{0z}\left&space;(&space;-2M_{y}M_{z}&space;\right&space;)$

$\fn_phv&space;s_{1z}=s_{0x}\left&space;(&space;-2M_{x}M_{z}&space;\right&space;)+s_{0y}\left&space;(&space;-2M_{y}M_{z}&space;\right&space;)+s_{0z}\left&space;(&space;1-2M_{z}^{2}&space;\right&space;)$

Thus, for a given input ray aligned to the Y-Axis, the periscope optical axis, in this example, the direction of the reflected ray can be calculated for any mirror orientation. This is a very useful tool for the present analysis wherein the mirror normal orientation can vary greatly depending on the direction and magnitude of the mirror spindle tilt or mirror position skew angle direction and amount..

Consider the following case where the incident ray is aligned with the periscope optical axis, and  the mirror normal lies at 45 degrees to the  Z-Axis in the Y-Z plane, as illustrated above. The incident ray is represented by the unit vector S0 which has only a   Y-Axis component of unit magnitude:

:$\fn_phv&space;\textbf{S}_{0}=\textbf{j}$                 $\fn_phv&space;s_{0x}=0$     $\fn_phv&space;s_{0y}=1$       $\fn_phv&space;s_{0z}=0$

The mirror normal is represented by the vector M:

$\fn_phv&space;M=1/\sqrt{2}\textbf{j}-1/\sqrt{2}\textbf{k}$

$\fn_phv&space;M_{x}=0$        $\fn_phv&space;M_{y}=1/\sqrt{2}$    $\fn_phv&space;M_{z}=-1/\sqrt{2}$

Substitution of the above values for the incident ray and mirror normal vector components into the equations for the reflected ray components gives:

$\fn_phv&space;s_{1x}=0$      $\fn_phv&space;s_{1y}=0$      $\fn_phv&space;s_{1z}=1$

Thus the reflected ray is shown to be a unit vector lying on the + Z-Axis, as expected.

As a second example, consider the case with the same incident ray but with the mirror rotated about the X-axis to achieve elevation of the periscope  line of sight.

The mirror normal after rotation can be determined by application of the equations of coordinate transformation. For a mirror rotation angle α about the X-Axis the mirror normal vector becomes:

$\fn_phv&space;\begin{bmatrix}&space;M_{x}\\M_{y}&space;\\&space;M_{z}&space;\end{bmatrix}=\begin{bmatrix}&space;1&space;&&space;0&space;&0&space;\\&space;0&&space;cos\alpha&space;&sin\alpha&space;\\&space;0&-sin\alpha&space;&&space;cos\alpha&space;\end{bmatrix}\begin{bmatrix}&space;0\\1/\sqrt{2}&space;\\&space;-1/\sqrt{2}&space;\end{bmatrix}$

$\fn_phv&space;M_{x}=0$

$\fn_phv&space;M_{y}=cos\alpha&space;/\sqrt{2}&space;-sin\alpha&space;/\sqrt{2}$

$\fn_phv&space;M_{z}=-sin\alpha&space;/\sqrt{2}&space;-cos\alpha&space;/\sqrt{2}$

The new mirror normal components are used to construct a corresponding reflection matrix.  The reflected ray components given an input ray along the Y-Axis can now be calculated:

$\fn_phv&space;\begin{bmatrix}&space;s_{1x}\\&space;s_{1y}\\&space;s_{1z}&space;\end{bmatrix}=&space;\begin{bmatrix}&space;0&space;&0&space;&0&space;\\&space;0&sin2\alpha&space;&cos2\alpha&space;\\&space;0&cos2\alpha&space;&-sin2\alpha&space;\end{bmatrix}\begin{bmatrix}&space;0\\&space;1\\&space;0&space;\end{bmatrix}$

$\fn_phv&space;s_{1x}=0$

$\fn_phv&space;s_{1y}=sin2\alpha$

$\fn_phv&space;s_{1z}=cos2\alpha$

Thus, the above calculated components of the reflected ray show that the ray lies in the   Y-Z plane,  is inclined at an angle 2α relative to the Z-Axis and points in the direction of the positive Z-Axis, as would be expected.

Tracking Error Nomenclature

The nomenclature for periscope tracking errors is illustrated below. The mirror normal (not shown) and the reflected ray are nominally located in the Y-Z plane. The departure of the reflected ray from the Y-Z plane is defined by its X-direction cosine, S1x. A viewer looking in the direction  of the reflected ray will experience a tracking error, τ, which is directly proportional to the X-direction cosine, S1x. Alternately, the error can be expressed in terms of θ, an azimuth plane quantity.  Error in the elevation direction is the difference between the elevation angle of the reflected ray and the angle  2α, which is twice the rotation angle of the head mirror spindle.

Tracking Error Calculation Method

The reflected ray is determined only by the mirror normal vector for a given incident ray. Therefore,  one has to only calculate the mirror normal vector in order to establish the tracking error characteristic for any mirror /spindle configuration of interest..

.As shown above, the mirror normal components are used to calculate the reflection matrix, which in turn is applied to the incident ray to arrive at the reflected ray components. The mirror normal orientation depends not only on the elevation angle input to the mirror spindle but also on the alignment of the mirror  spindle axis relative to the plane of the mirror,

Tracking Error Example: Mirror Spindle Skew

Consider the case where the mirror spindle is skewed  from its nominal position, as illustrated to the right.

In this example, the spindle is constrained to lie entirely in the    X-Y plane, tilted through angle δ about the Z-Axis.

The components of the reflected  ray which yield the desired tracking error information can be calculated as follows:

The mirror normal coordinates in the perturbed system are first calculated from the  mirror normal coordinates in the fixed reference coordinate system. This is achieved  by the sequential application of the equations of coordinate transformation  in sequence, as follows:

$\fn_phv&space;\small&space;\begin{bmatrix}M_{x}&space;\\M_{y}&space;\\M_{z}&space;\end{bmatrix}=&space;\begin{bmatrix}&space;cos\delta&space;&sin\delta&space;&0&space;\\&space;-sin\delta&space;&&space;cos\delta&space;&&space;0\\&space;0&&space;0&space;&&space;1&space;\end{bmatrix}\cdot&space;\begin{bmatrix}&space;1&0&space;&0&space;\\&space;0&cos\alpha&space;&sin\alpha&space;\\&space;0&-sin\alpha&space;&cos\alpha&space;\end{bmatrix}\cdot&space;\begin{bmatrix}&space;cos\delta&space;&&space;-sin\delta&space;&0&space;\\&space;sin\delta&space;&&space;cos\delta&space;&&space;0\\&space;0&&space;0&space;&1&space;\end{bmatrix}\cdot&space;\begin{bmatrix}&space;0\\&space;1/\sqrt{2}\\&space;-1/\sqrt{2}&space;\end{bmatrix}$

The resultant mirror normal components of the perturbed system  expressed in the reference coordinate system become:

Mx = 1/√2 (-sinδ cosδ + cosα sinδ cosδ – sinδ sinα )

My = 1/√2 (sin2δ + cosα cos2δ – sinα cosδ )

Mz = 1/√2 ( -sinα cosδ – cosα )

The reflected ray components are calculated by multiplying the incident ray vector by the reflection matrix formed from the altered mirror normal components:

S1x = δ ( cosα – sinα – sin 2α – 1 )

S1y = sin 2α

S1z = cos 2α

The above equations are valid for the typically small values of δ encountered in slightly misaligned systems. The math operations to arrive at these equations  are detailed elsewhere.

Since the direction of the reflected ray in elevation is twice the input rotation to the mirror spindle, i.e., 2α, the angle indicated by the S1y and S1z components, there is no line of sight tracking error in the elevation direction. In a system free of misalignment, there would be no component to the reflected ray in the X-direction, i.e., Six would be zero. The deviation in the X-direction indicated by S1x is therefore proportional to the line of sight tracking angle error in the cross scan direction, illustrated above as τ in the traverse plane. The tracking error referenced to the azimuth plane, θ, the arctangent of the ratio of the    X-Axis and Z-Axis components, S1x and S1z and τ are shown in the following graph as a function of the periscope line of sight angle from the horizon, 2α.

The tracking error in radians is plotted for a 0.01 radian tilt of the mirror spindle in the X-Y plane, i.e., a 0.01 radian rotation about the Z-Axis. Near the horizon, the tracking error is about 25% of the input error, varying fairly linearly over a ± 25° region.

The τ and θ graphs differ from one another,  particularly at the extremes of the scan, due  to the differences in definition. Throughout the following discussion attention is focused on the traverse plane error since this correlates with the periscope viewer’s visual experience.

Other Tracking Error Sources: Mirror Skew

Mirror misalignment leads to another tracking error source of a slightly different nature. Consider the case where the mirror is skewed via twist about the Y-Axis through angle δ.   The tracking error function  for this case is determined by  a procedure similar to the one used for the tilted mirror spindle.  The resultant tracking error characteristic for the traverse plane is:

τ = sin-1 ( δ (sinα – cosα))

The graph below shows tracking error for mirror skew with the previous one for tilt of the spindle. Input misalignment is 0.01 radians in each case.

At zero degrees elevation  the tracking error due to mirror twist about the Y-Axis  is exactly equal to the mirror twist angle  while no error is generated by the skewed spindle.

Reticle Offset

The error at zero elevation angle created by mirror twist about the Y-Axis can be offset  by redirecting the periscope line of sight by an equal amount. This would be accomplished in an actual system by adjustment of the position of the reticle. This is adjustment is modeled here by altering the direction of the input ray vector. Nulling of the effect of mirror Y-Axis twist at zero degrees elevation is achieved by rotating the input ray about the horizontal line of sight through an angle equal to the mirror twist angle.

The plot below shows the net LOS tracking error for the combination of mirror spindle tilt in the cross-scan (X-Y) plane,  mirror twist about the Y-Axis and reticle offset.  Equal misalignments of 0.01 radians have been introduced in each case.

The following observations can be made:

The line of sight tracking errors are largely reduced to zero for elevation angles near the horizon, and are maintained to less than 0.001 rad. (10% of any of the  input misalignments) over the elevation range -30 to 30 degrees. Depending on the application this may be sufficiently accurate and would make unnecessary the task of elimination of  the errors over the total 180 degrees elevation range.

This example shows the interplay between three specific sources of  tracking errors each of defined magnitude and direction. In general  mirror skew, and spindle tilt  can result from rotation about any axis. Likewise,” reticle placement” error and the resultant change in the  LOS direction as  simulated by rotation of the nominal input ray can occur about

It is evident that the overall tracking error function can be quite complex,. Modeling and analysis using the vector ray trace method is a very useful  tool in dealing with the problem.. The calculations for the above examples have been performed using Mathcad computer software. A numerical approach versus purely analytical one becomes necessary as the number of misalignments treated  increases beyond two or three..

By studying the effects on tracking errors of periscope head mirror misalignments, a meaningful methodology for tolerancing,  fabricating and aligning the scan mirror system components  can be established..

By setting appropriate tolerances, minimizing the number of hardware adjustments at assembly, and accurately fabricating the hardware components, the problem  of achieving sufficiently small periscope line of sight  tracking errors can be overcome.

References

1. MIL-HDBK-141, 5 October 1962, Chapters 2 and 13