Pointing an Instrument on an Airborne Platform

In general, when a remote-sensing research instrument is mounted on an airborne platform, the direction that it's "beam" is pointing, and therefore the radiation that it senses, will depend on both the attitude of the instument with respect to the platform and the attitude of the platform with respect to a horizon reference system. This paper describes in detail how to measure and apply appropriate pointing corrections, so that the instrument's "beam" will be at a specified elevation angle above the horizon. This will be done by describing how these pointing corrections are measured and applied for the Microwave Temperature Profiler (MTP), an instrument which must scan to fixed elevation angles from near-zenith to near-nadir.

Executive Summary

Because an instrument's attitude is generally non-zero with respect to the platform on which it is mounted and because a platform's attitude is changing in both space and time, the elevation angle Ec that an instument must scan to, in order to obtain a required elevation angle E with respect a horizon reference plane, will also change. The equation for calculating the commanded elevation angle Ec as a function of the required elevation angle E is:

where

and

.  

In these expressions, the s and c operators are abbreviations for the trigometric sine and cosine functions, respectively, and P _a and R_a are the platform's pitch and roll with respect to the horizon reference plane. The instrument attitude matrix, Mi, whose elements appear in the expressions for and , is given by:

where Y_i, P _i, and R _i are the yaw, pitch and roll, respectively, of the intrument with respect to the platform.The elements of Mi are normally constant for any particular platform, so that only the platform pitch ( P_a) and roll (R _a ) with respect to the horizon reference system need to be updated in calculating and and then evaluating the expression for the commanded elevation angle Ec . These equations assume the normal aviation standard that yaw is applied first, followed by pitch and then roll. The roll axis (x-axis) is in the direction of flight, the pitch axis (y-axis) is along the left wing, and the yaw axis (z-axis) is up. Positive roll is right wing down, positive pitch is nose up, and positive yaw is east when heading north.

The conventional coordinate system used for aircraft attitude (yaw, pitch and roll).

Introduction

In general, when a remote-sensing research instrument is mounted on an airborne platform, the direction that it's "beam" is pointing, and therefore the radiation that it senses, will depend on both the attitude of  the instument with respect to the platform, and the attitude of the platform with respect to a horizon reference system. In the figure to the right, the red coordinate axes represent a set of three unit basis vectors for a horizon reference system; that is, the x-y plane is parallel to the horizon, and this is the plane from which we want to measure elevation angles for the instrument beam, which is denoted by "Beam" in the figure. This system is often referred to as the "world" coordinate system, which is why the x, y and z axes are denoted Xw, Yw and Zw, respectively. (Note that the Xw axis is yawed 90 degrees west (left) from the figure above.) The blue or primed coordinate system is the attitude of the MTP on the ER-2 right engine cheek with no aircraft pitch or roll. The instrument attitude is a combination of it's mounting attitude and the aircraft attitude, and the coordinate system attached to the instrument is normally called the body coordinate system, hence the subscript b on the axes. The beam is shown scanned +30 degrees from the Xb axis (in the Xb-Zb plane, since the scan is about the Yb axis). Note that in both the world and body coordinate systems, a positive angle is clockwise about an axis when viewed from the origin. This positive roll is right wing down, positve pitch is nose up, and positive yaw is nose right.

The first order of business in determining how to correctly point the beam of an instrument is to determine what the attitude of the instrument is with respect to the aircraft. This is not always an easy matter! For the purpose of this discussion it is assumed that the instrument has two sides (a plate) or three sides (a box) whose elevation angles with respect to the horizon can be measured (for example, by a digital level). Once this has been done, the measured elevation must be converted to corresponding values of instrument yaw ( Y_i), pitch (P _i) and roll ( R_i). The reason for this is that the attitude corrections will be applied using rotation matrices.

Instrument Attitude Determination

As shown in Appendix A , the attitude of an instrument can be represented by a rotation matrix: , where Rr(r) is a roll of r about the x-axis (direction of flight), Rp(p) is a pitch of p about the y-axis (left wing), and Ry(y) is a yaw of  y about the z-axis (up). Using the notation Y_i, P_i , R_i for the instrument yaw, pitch and roll, respectively, and Mi for the instrument attitude matrix, and applying Equations (A2) , we have:

(1)       

where, to reduce the size of the matrix, we use c(x) = cos(x) and s(x) = sin (x). Since the instrument attitude does not change with respect to the aircraft, the Mi components can be calculated once for any given platform. The next matter of  business is then to express the aircraft attitude in matrix form. Again we use Equations (A2) , but this time for the aircraft yaw (Y_a ), pitch (P_a) and roll (R_a ). For the MTP we do not care about the aircraft yaw, so we will set Y _a = 0 to reduce the size of the resulting rotation matrices. There is however no loss of generality if  Y _a is not zero. The aircraft attitude matrix is then:

(2)       

Armed with these attitude matrices, we are now in a position to determine the instrument yaw, pitch and roll from measurements of the elevation angles of two or more sides of the instrument package. For simplification we will first derive this coordinate transformation by assuming that the aircraft attitude is zero when the measurements are made. (The more general case will be presented later.) If we take the dot product of  each of the rotated instrument axes with respect to the z-axis of the reference coordinate system; that is,

(3)       

(4)        , and

(5)       

we obtain the three direction cosines of the z-axis of the horizon reference system with respect to the rotated instrument axes. If we denote T (roll-like), B (pitch-like) and Z (yaw-like) as the measured elevation angles of the instrument's rotated x, y and z axes, respectively, it is clear that these angles are just the complements of the direction cosines calculated in Equations (3)-(5). That is, for X = T, B and Z, . We can then recaste Equations (3)-(5) as:

(6)       

(7)       

(8)       

These equations represent three equations in three unknowns, so at first glance we should be able to solve for Y_i , P _i and R_i. Unfortunately, this is not the case, since the direction cosines are not independent; they are related by the expression:

(9)       

This should not have been unexpected since it is clear that if the instrument in rotated about the horizon reference system zenith, the measured angles T, B and Z do not change. Note that if T , B, and Z can all be measured, Equation (9) can be used to get a feeling for how well the angles have been measured. (These measurements are generally not easy, so it is good to have this check!) The yaw of the instrument must be measured by some other means, for example, by moving the instrument beam to zero elevation angle in the horizon reference system and then measuring the angle between the longitudinal axis of the aircraft and the instrument beam. Given that the instrument yaw has been measured, any two of equations (6) through (8) can be used to numerically solve for the instrument pitch (P _i ) and roll ( R _i). 

As pointed out earlier, the foregoing assumed that the aircraft attidude was zero during the measurements. If this is not the case, which is generally the case, Equations (3) through (5) must be multiplied by the aircraft attitude matrix Ma given in Equation (2). When this is done, the resulting equations are:

(10) 

(11)  

(12)  


These equations can be equated to the direction cosines of   T , B, and Z just as we did above, and numerically solved for R_i and P_i using any pair of the resulting equations.

Pointing to Specific Elevation Angles

Now that we known the instrument attitude with respect to the aircraft, and assuming that we can obtain the aircraft pitch (P_a ) and roll (R_a) from the avionics system, it is a simple matter to determine what commanded elevation angle ( Ec ) is needed to achieve the required elevation angle (E ). The projection of the basis vector representing the instrument's beam at a commanded position Ec on the horizon reference system are given by:

(13)       

In these equations, A is the azimuth in the horizon reference system of the instrument's x axis. Note that the instruments beam has no y-component since the rotation is assumed to be about the instrument's y-axis. When this expression is expanded, we obtain for the z-component:

(14)       

where
                     
and
              .

Letting   and   , and rearranging, we obtain:

(15)       

which is simply a quadratic equation for , the sine of the commanded elevation angle Ec which is needed to be at elevation angle E in the horizon reference system. The two possible solutions are:

(16)       

(17)       

They correspond to the commanded elevation angles needed to have the backward beam at elevation angle E and forward beam at elevation angle E . Therefore, the second solution (Equation (17)) is the one that should be used.

Finally, it should be noted that because of the non-zero instrument and aircraft attitude, the instrument will not be able to reach very high and very low elevation angles. This will occur whenever the radical in Equation (17) is less than zero. Alternaltively, the maximum achievable elevation angle Emax can easily be determined by taking the first derivative of Equation (14) with respect to the commanded angle Ec and setting the result to zero. This results in:

(18)       

which on substitution back into Equation (14) gives:

(19)        .

This equation can be used to calculate the maximum achievable elevation angle for any instrument and aircraft attitude.

Demonstration

The ER-2 MTP scans to 11 elevation angles, 5 above the horizon, the horizon, and 4 positions below the horizon, and a calibration target which is opposite the horizon view in the instruments coordinate system. MathCad has been used to implement Equation (17) to produce a near realtime AVI file of the ER-2 MTP scan sequence; it begins and end at the calibration target position. It is 184 kB in size. Internet Explore will automatically open Windows Media Play when you click here to see it; in Netscape you will have to right click and select Save Link As to save the file ER2_MTP.AVI so it can be seen in a program that likes AVI files .

Laboratory Checkout

We have built at JPL a double, three-axis, gimbal apparatus (shown to the right) to allow us to verify the MTP pointing algorithms. Historically, the MTP pointing algorithm was derived by Bruce L. Gary using spherical trigonometry, which required several pages of computer code to implement. Since the author was more comfortable with the rotation matrix approach, we wanted to verify that the two approaches were equivalent. This is especially important for the ER-2 research aircraft since the instrument's attitude is extreme in this case.

In order to verify the pointing, we had to convert the angles measured on the airplane (B, T, and Z) to the pointing apparatus. The manner in which this is done depends on how the instrument is mounted to the platform representing the aircraft. Ideally this would be just like the yaw, pitch and roll axes on an aircraft gimbal, involving three axes of rotation in the proper order. Recall that it is absolutely essential that rotations be implemented in the order yaw, pitch, roll -- progressing from the instrument to the aircraft and finally the horizon reference system. This is because rotation matrices do not commute. If for some reason this cannot be done, as was originally the case for our apparatus, because the gimbals representing the instrument attitude were upside down (that is, the order progressing from the instrument was roll, pitch, yaw), it is still possible to get the attitude right, but it takes a bit more work.

To this point all rotations have been about the yaw, pitch and roll axes of the horizon reference system, and never about the rotated instrument axes. If the test apparatus does not have the correct order for yaw, pitch and roll, we can still get it oriented properly by defining a sequence of  rotations about the rotated axes. Take, for example, our original set up where the first rotation is about the roll axis (x-axis); this moves the yaw axis (z-axis) to a new location and the pitch axis (y-axis) to a new location. We now apply a rotation to the rotated y-axis to implement pitch, which now rotates the yaw axis (z axis) as well as the original roll axis (x-axis). Finally, we yaw about the rotated z-axis to implement yaw. Now only the first of these three rotations can be implemented using the rotation matrices given by Equations (A2) , which brings us to why we developed theory for rotation about an arbitrary axis. To simplify matter, we define a new function:

(18)       

which produces applies a rotation  about axis n to vector r. For the case just discussed, we calculate the effect of a rotation about the roll axis as:

 for the new y-axis, and   for the new z axis; the x axis is of course unchanged. Armed with the new y axis we can now perform a rotation about it. This rotates the x axis to: , and the z axis to: .

Finally, we perform a yaw about the new z axis to obtain the final resting place for the x axis:

 

and the y axis:



Note that the final positions for the z axis is just its last position before the rotation about it, namely:



As with the normal rotation matrices, the dot products of the final axes can be performed with the original z axis of the horizon reference system to determine an equivalence with the direction cosines of the complement of the measured instrument attitude.  This leads to:

,    and  

which as before can be solved numerically for and  once , the yaw like angle, has been independently measured.

Appendix A:

The basic tool used to deal with attitude corrections will be rotation matrices. For reasons that will become apparent later on, we start off by specifying a general rotation of a vector a about an arbitray axis specified by the unit vector n to produce a vector b. This is done by the expression:

(A1)   

where the identity matrix I is given by:                                

the symmetric matrix Sym is given by:                     

and the skew symmetric matrix Skew is given by:   

Equation (A1) is known as Rodriques' Equation, and it states that a rotation  about an axis n by an amount decomposes an arbitrary rotation into three contributions due to the identity, symmetric, and skew symmetric matrices generated from the rotation axis n. It is clear that the first term is parallel to the vector being rotated (a) (this is the result of the identity rotation), the second term is parallel to the rotation axis (n) (this is the result of the symmetric rotation), and the third term is normal to both the rotation axis and the vector being rotated (this is the reult of the skew symmetric rotation). In this equation, (n . a) is a dot product and (a x n) is a cross-product.

If we now adopt the normal aviation convention that a rotation about the x-axis is roll, that a rotation about the y-axis is pitch, and that a rotation about the z-axis is yaw, that is, a in Equation (A1) is successively to (1,0,0), (0,1,0) and (0,0,1), then for a rotation of  , it is trivial to show that:

(A2)                       

Note that the convention adopted here is that clockwise rotations looking towards the coordinate system origin along the axis of rotation are positive, so for positive , Rr( ) is left-wing down, Rp( ) is nose up, and Ry( ) is yaw (or heading) to from north to east. Because rotation matrices do not commute, it is very important to note another aviation convention, and that is that the order of  rotation is first yaw, then pitch and finally roll. Using this convention, an arbitrary combination of  yaw (y), pitch (p) and roll (r) can be represented by matrix:

(A3)       

Although Equation (A1) can be used to rotate any vector about an arbitrary axis, Equations (A2) always operate about the original x, y and z axes. These three unit basis vector are what we shall refer to as the horizon reference system; the x-axis is in the direction of flight, the y-axis is along the left wing and the z-axis is up.