Abstract:
An attitude determination system and method employs first and second baseline determinations comparing the relative phase of carrier signals received from GPS satellites. Attitude determinations include determination of roll, pitch and azimuth of ships, aircraft, vehicles and other devices.

Description:
FIELD OF INVENTION 
     This invention relates generally to attitude determination with global positioning system (GPS) satellite signals and more specifically to the resolution of carrier cycle integer ambiguities in the carrier phase difference measurement between separate GPS antennas attached to a rigid structures. 
     BACKGROUND OF THE INVENTION 
     A current attitude determination system is disclosed in U.S. Pat. No. 5,296,861, showing a flat, triangular antenna array mounted horizontally level in the frame of reference of a selected vehicle. U.S. Pat. No. 5,296,861 is expressly referenced and incorporated herein in its entirety. The antenna array of this U.S. patent may be installed on a plate, which in turn may be mounted on a television antenna rotor. The antenna array includes first and second antenna baselines between a reference antenna and selected first and second other antennas. This patent recites a method at column 22, line 3, of: 
     &#34;determin[ing] . . . the range of integers that are possible for the first baseline, given only the known constraints on vehicle roll and pitch . . . &#34; 
     For satellites nearly overhead, only a small range of integers is possible if the vehicle is constrained from significant rolling and pitching. In particular, U.S. Pat. No. 5,296,861 shows step (f) at column 22, line 22, including 
     &#34;calculat[ing] . . . the arc of coordinates that are possible for the second antenna, assuming that the first antenna coordinate is true . . . [and determining] the range of integers that are possible for the second baseline&#34; 
     The flat antenna array method patented, however, cannot operate, if the array is mounted in three dimensions rather than in the plane of the vehicle which carries it. The methodology of the invention requires users of the attitude determination systems to attach antennas at selected points of a vehicle which are not likely to be aligned with the plane of the transporting vehicle. Instead, the points of attachment of the antennas on the vehicle are likely to be out-of-plane. On aircraft, a common configuration is to mount two antennas on the fuselage, and two more out on wings, producing substantial slope between antenna axes and the vehicle-level frame of reference. Accordingly, the vehicle carrying the antennas can have any azimuth between 0 and 360 degrees, and it can tilt in any direction about level up to some maximum angle, such as 15 degrees, but, notably, this range of motion pertains to the vehicle, not to the antenna sites. 
     According to a current approach, the antennas are mounted in flat, planar arrays in the vehicle frame of reference, leading to a particular solution for determining the range of possible carrier cycle integers. However, for antennas that are substantially out-of-plane, no method for determining the range of integers is known to be in use. 
     It is accordingly desirable to develop a system and method for attitude determination which is not dependent upon co-planarity of antenna disposition with the plane of the vehicle on which the antennas are mounted. 
     SUMMARY OF THE INVENTION 
     According to the present invention, the range of carrier cycle integers that are possible for first and second antenna baselines are determinable for a vehicle that has an azimuth between 0 and 360 degrees and can tilt in any direction about level up to some maximum angle &#34;tilt max  &#34; such as 15 degrees according to one embodiment of the present invention. The antennas of the first and second baselines are settable at arbitrary points on the vehicle which carries them, and are not necessarily co-planar with the plane of the vehicle itself. The present invention has the advantages that (1) it will allow the use of antennas substantially out of plane, and (2) that it does not produce a range of carrier cycle integers any greater than necessary. 
     It is very desirable not to produce a range of integers any greater than necessary, because doing so would increase the computation burden when a minimum likelihood estimation (MLE) direct integer search method processes useless cattier cycle integer combinations. A further advantage of the present invention is that it is highly efficient, computationally. The method of the present invention finds sines and cosines of angles without having to explicitly resort to computer execution of sine/cosine evaluation routines. It does require the evaluation of square roots, but since a math co-processor is being used, square root evaluation takes only about 50% more computation time than a simple floating-point multiply. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1a is a diagram of an attitude determination system of the prior art including a flat, triangular antenna array mounted horizontally level in the frame of reference of a selected vehicle; 
     FIG. 1b is a diagram of a first antenna baseline deviating from satellite direction by an angle theta, wherein the dot product of the first antenna baseline with the satellite direction vector is the projection of the first antenna baseline upon the satellite direction vector, consistent with the present invention; 
     FIG. 1c is a block diagram of an attitude determination system according to the present invention. 
     FIG. 2 is a graph of sine theta and cosine theta as a function of theta, consistent with the present invention, where theta is the angle of deviation from the first antenna baseline from satellite direction; 
     FIG. 3 is a diagram of a first antenna baseline deviating from a candidate vector describing a possible first baseline wherein the angle between the first antenna baseline and the candidate vector is gamma in a two dimensional reference frame represented by coordinates x&#39; and z&#39;, consistent with the present invention; 
     FIG. 4 is a diagram showing the method of pointing the first baseline into the candidate vector and then swinging the antenna array around the first baseline by an angle phi, permitting determination of arc subject to tilt limitations and determining minimum and maximum projections of the second antenna baseline onto all satellite directions of incident signal arrival, according to the present invention; 
     FIG. 5 is a diagram laying FIG. 4 on its side, turning it around, and re-labeling the axes, establishing z&#34; by vehicle twisting until the first baseline lies directly under or over the candidate vector, twisting the vehicle by an angle phi and projecting the first baseline onto according to the present invention; 
     FIG. 6 is a diagram of a coordinate system R with axes r1, r2, and r3, in which the first baseline is aligned with r3, according to the present invention; and 
     FIG. 7 is a diagram of a satellite line of sight vector making an angle delta in the r1×r2 plane, adding to delta the vehicle rotation phi about r3, normalizing c to unit length to provide the sines and cosines of delta, permitting determination of the range summing delta and phi to find minimum and maximum values of c2d2 as phi is varied over an allowable range, and providing the range pmin . . . pmax of possible integer values, according to the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     As noted above, FIG. 1a shows a current attitude determination system 1 amply disclosed in U.S. Pat. No. 5,296,861. Attitude determination system 1a includes a flat, triangular antenna array 1a mounted horizontally level in the frame of reference of a selected vehicle 2. U.S. Pat. No. 5,296,861 is expressly referenced and incorporated herein in its entirety. Antenna array 1a of U.S. Pat. No. 5,296,861 is installed on a plate 1b, which in turn may be mounted on a television antenna rotor for example (not shown). FIG. 1a further shows a diagram of first and second antenna baselines 3 and 3&#39; respectively between a reference antenna 4 and a selected first and second other antennas 5 and 8. Reference antenna 4 and first other antenna 5, and reference antenna 4 and second other antenna 8 are respectively connected to first and second pairs of first and second phase coherent GPS correlators which are shown in U.S. Pat. No. 5,296,861. The correlators have respective first, second, third, and fourth correlator outputs derived from said GPS satellite radio carrier signal and said first, second, third, and fourth correlator outputs respectively include first and second phase differences. A single computer or a network of computers is connected to said respective first and second and first and third phase coherent GPS correlators for determining first through fourth pluralities of possible attitudes to first through fourth respective GPS satellites based upon respective first and second phase differences between said first and second and said third and fourth correlator outputs for a first radio carrier signal from said respective first through fourth GPS satellites, wherein said first through fourth pluralities of possible attitudes includes more than one possible attitude due to whole cycle carrier phase ambiguities. 
     FIG. 1b is a diagram of a first antenna baseline 3 between a reference antenna 4 and a selected other antenna 5 and deviating from the direction of a selected GPS satellite 6 indicated by a satellite direction vector 7 by an angle theta, i.e., &#34;θ.&#34; In particular, the dot product of first antenna baseline 3 with satellite direction vector 7 is the projection of first antenna baseline 3 upon satellite direction vector 7, consistent with the present invention. First antenna 6 baseline is described by a vector a=b·a B , where &#34;b&#34; is the length of first antenna baseline 3 in units of GPS carrier wavelengths, (e.g., 19.02 centimeters) and where a B  is a vector of unit length that points along the first baseline in vehicle body coordinates with its origin at reference antenna 4. Further, u N  is a vector of unit length which points from reference antenna 4 into the direction of incident signal arrival from a GPS satellite. u N  points to the satellite, and is expressed in the local navigation coordinates, (e.g., East, North, Up). 
     FIG. 1c is a block diagram of an attitude determination system according to the present invention. Attitude determination system 1 particularly includes reference antenna 4 and first and second antennas 3 and 5. Attitude determination system 1 further includes radio frequency (RF) stages 16, 16&#39;, and 16&#34; which are connected to respective antennas 3, 4, and 5. The attitude determination system further includes correlators 18, 18&#39;, 19, and 19&#39;. Correlator 18 is connected with RF stage 16. Correlators 18&#39; and 19 are connected to RF stage 16&#39;. Correlator 19&#39; is connected with RF stage 16&#34;. Correlators 18, 18&#39;, 19, and 19&#39; receive input signals from the connected one of RF stages 16, 16&#39;, and 16&#34;. Attitude determination system 1 further includes computer 17, computer 17&#39;, and attitude solution computer 17&#34;. Computer 17 is connected to correlators 18 and 18&#39;. Computer 17&#39; is connected to correlators 19 and 19&#39;. Computer 17 compares the relative phases of carrier signals from antennas 4 and 5. Computer 17&#39; compares the relative phases of carrier signals from antennas 3 and 4. Computer 17&#34; determines an attitude solution based upon information received from computers 17 and 17&#39;. 
     As indicated in connection with FIG. 1a, vehicle 2 is defined to have its X axis at the right, for someone in the vehicle facing forward, its Y axis pointing forward, and its Z axis pointing up, relative to the vehicle body. The navigation frame for FIG. 6b as described below, is defined to have its X axis pointing East, its Y axis pointing North, and its Z axis pointing up, relative to the Earth. 
     If the orientation of vehicle 2 were known, it would be possible to use the antenna baseline vector in body frame a B  to express the baseline vector in navigation frame a N . Further, the &#34;whole value&#34; carrier phase detected or sensed by vector phase measurement hardware would be given by the &#34;vector dot product&#34;, φ=b a N  ·u N  +ε, where ε is measurement error due to receiver noise, and due to error that remains after removing the effects of cable electrical path length. 
     Unfortunately, a GPS receiver cannot measure &#34;whole value&#34; or correct carrier phase values. Instead, it can only measure phase with a carrier cycle ambiguity. &#34;Whole value&#34; phase is the phase detected by vector hardware, as a fraction of a carrier wavelengths, plus the carrier cycle integer. For example, the whole value phase of 2.25 corresponds to an ambiguous phase measurement of 90 degrees, (0.25 wavelengths,) plus two whole carrier cycles. The angle theta, i.e., &#34;θ&#34; between the first antenna baseline a N  and satellite line-of-sight u N  is given by the vector dot product of a N  and u N , according to the following relationships: cos(θ)=a N  ·u N  =a x   N  ·u x   N  +a y   N  ·a y   N  u N  +a z   N  ·u z   N , where cos(θ) means the cosine of angle theta, or θ. See, for example: Murray H. Protter, and Charles B. Morrey, College Calculus with Analytic Geometry, Addison-Wesley Publishing Company, Inc., Reading Mass., 1966, page 249. FIG. 1 particularly shows that a N  ·u N  is the &#34;projection&#34; of a N  onto u N . 
     For purposes of analysis, let us assume that vehicle 2 is in an upright position and that it has roll, pitch and azimuth values of zero. In that case, the vehicle body axes, and the local navigation axes coincide and a B  =a N . Now, rotate vehicle 2 in azimuth only, leaving roll and pitch to be zero. The X and Y elements of a N  will clearly vary with the rotation, but the Z component will not be changing. In other words, a z   N  =a z   N  regardless of azimuth, as long as roll and pitch are zero. 
     Further, it is notable that the dot product, 
     
         p=a.sub.x.sup.N ·u.sub.x.sup.N +a.sub.y.sup.N ·a.sub.y.sup.N u.sub.y.sup.N +a.sub.z.sup.N ·u.sub.z.sup.N, 
    
     where 
     
         q1=a.sub.z.sup.N ·u.sub.z.sup.N, 
    
     and 
     
         q2=a.sub.x.sup.N ·u.sub.x.sup.N +a.sub.y.sup.N ·a.sub.y.sup.N u.sub.y.sup.N. 
    
     Further, the dot product &#34;p&#34; has a part q 1  that remains constant, if the vehicle is rotated only in azimuth. Since q 1  is constant, its contribution to the projection can be subtracted, leaving q 2  in two dimensions. This simplifies analysis, because two-dimensional problem is easier to solve that a three-dimensional problem. Additionally, q 2  can have a magnitude up to: 
     
         [((a.sub.x.sup.B).sup.2 +(a.sub.x.sup.B).sup.2)((u.sub.y.sup.N).sup.2 +(u.sub.y.sup.N).sup.2)].sup.1/2, 
    
     because a z   N  =a z   B , with the result that: 
     
         ((a.sub.x.sup.B).sup.2 +(a.sub.y.sup.B).sup.2)=((a.sub.x.sup.N).sup.2 +(a.sub.y.sup.N).sup.2). 
    
     Further, q 2  is at a maximum when vectors a and u align in two dimensions. The vector dot product, p, can lie in the range q 1  -q 2max  when vehicle roll and pitch are zero. 
     For zero values of vehicle roll and pitch, the maximum angle θ max  that can be achieved in FIG. 1a is 
     
         cos(θ.sub.max)=a.sub.z.sup.B ·u.sub.z.sup.N -[((a.sub.x.sup.B).sup.2 +(a.sub.x.sup.B).sup.2)((u.sub.y.sup.N).sup.2 +(u.sub.y.sup.N).sup.2)].sup.1/2. 
    
     and the minimum angle &#34;θ min&#34;   that can be achieved in FIG. 1a is 
     
         cos(θ.sub.min)=a.sub.z.sup.B ·u.sub.z.sup.N +[((a.sub.x.sup.B).sup.2 +(a.sub.x.sup.B).sup.2)((u.sub.y.sup.N).sup.2 +(u.sub.y.sup.N).sup.2)].sup.1/2. 
    
     FIG. 2 is a graph of sine θ and cosine θ as a function of θ, consistent with the present invention, where θ is the angle of deviation from the first antenna baseline from satellite direction. Further, from the relationship, sin 2  ((θ)+cos 2  ((θ)=1, and from FIG. 2, 
     
         sin(θ.sub.max)=[1-(cos(θ.sub.max).sup.2)].sup.1/2, 
    
     and 
     
         sin(θ.sub.min)=[1-(cos(θ.sub.min).sup.2)].sup.1/2. 
    
     Now, imagine turning the vehicle around in azimuth so that θ is at a maximum, θ max , and then additionally tilting the vehicle to maximum tilt, so that the total angle θ=θ max  +tilt max  is as large as possible. If the new angle θ is larger than 180°, then it is possible for the first antenna baseline to lie parallel to u, and pointing away from the GPS satellite. It can accordingly be concluded that the most p can ever be in the negative direction is -b, where b is the first antenna baselength. 
     If θ=θ max  +tilt max  is less than 180°, then the most p can be, in the negative direction, is b·cos(θ max  +tilt max ). Further, the sine and cosine of ((θ max  +tilt max ) follow trigonometric identities, sin(A+B)=sin(A)cos(B)+cos(A)sin(B), and cos(A+B)=cos(A)cos(B)-sin(A)sin(B). This leads to the conclusions, that sin((θ max  +tilt max )=sin((θ max ) cos(tilt max )+cos((θ max ) sin(tilt max ), and that cos((θ max  +tilt max )=cos(θ max ) cos(tilt max )-sin((θ max ) sin(tilt max ). 
     From FIG. 2, the combined angle (θ min  -tilt max  is less than 0° for negative values of sin((θ min  -tilt max ). The foregoing is sufficient to complete an method to determine the range of carrier cycle integers for first baseline. The method, described below, allows for all possible carrier cycle integers, without allowing any extra integers. 
     The method to determine carrier cycle from . . . to range for baseline 1 is expressed in pseudo-code below. This provides min/max limit software for a first baseline, wherein azimuth can be anything about a local vertical position, and roll and pitch are limited, for out-of-plane antennas not in the plane of the vehicle on which they are mounted. ##SPC1## 
     The method above for determining the first baseline can be made more efficient by computing measurement error standard deviation ahead of time. There is little incentive to make the method as fast as possible, however, at the expense of increasing software complexity. The method for first baseline only has to be executed once per integer search, and search time is dominated by the nested search loop. The method for second baseline-may have to execute thousands of times per search, so efficiency is more important for the second baseline. Theoretically, the test for (1-p1*p1)&lt;0 shouldn&#39;t be necessary, but it is included to avoid a possible square root of negative number, in case of a small computer round-off error. Sine and cosine of maximum vehicle tilt are computed ahead of the method, since they will also be needed for the second baseline computations. 
     The method for first baseline was tested under a number of different conditions, according to satellite azimuth and elevation, antenna baseline vector in vehicle body frame, and maximum vehicle tilt. Under each condition tried, the minimum and maximum projection of antenna baseline vector onto the direction of incident signal arrival was obtained using the above method. The projection limits were then tested by randomly generating a sample of 5,000 (five thousand) vehicle orientations satisfying the vehicle tilt limitation, and then computing the actual projections of antenna baseline vector onto the direction of incident signal arrival. It was verified that the maximum projection seen in 5,000 samples was almost as large as the predicted maximum, but never larger. It was also verified that the smallest projection seen in 5,000 samples was almost as small as predicted, but never smaller. 
     FIG. 3 is a diagram of a first antenna baseline deviating from a candidate vector describing a possible first baseline wherein the angle between the first antenna baseline and the candidate vector is gamma in a two dimensional reference frame represented by coordinates x&#39; and z&#39;, consistent with the present invention. FIG. 4 is a diagram showing the method of pointing the first baseline into the candidate vector and then swinging the antenna array around the first baseline by an angle phi, permitting determination of arc subject to tilt limitations and determining minimum and maximum projections of the second antenna baseline onto all satellite directions of incident signal arrival, according to the present invention. The method for second baseline executes at a time, in the overall processing described by U.S. Pat. No. 5,296,861, at which integer search is underway through a range of carrier cycle integers determined for the first baseline, and at which a candidate vector s N  has been generated which describes a possible first baseline. 
     The purpose of the method, at this point, is to determine the range of carrier cycle integers that are possible for the second baseline. Supposing that s N  does indeed describe the first antenna baseline, then the second baseline outer antenna has to lie somewhere along an arc that one would obtain by pointing the first baseline a 1  into s N  and then swinging the antenna array around a 1 . Usually, only a small arc is possible for the second baseline, considering tilt limits of the vehicle. It is necessary to calculate that arc, then find the minimum and maximum projections of second antenna baseline onto all the satellite directions u of incident signal arrival, and then use the minimum and maximum projections to solve for the possible range of carrier cycle integers on second baseline for each satellite. Having the from . . . to limits for carrier cycles on second baseline, the integer search which has begun on first baseline is then continued onto the second baseline. 
     As described in U.S. Pat. No. 5,296,861, integer searches on the first two baselines are first carried out in ordinary Cartesian coordinates, (in the navigation frame; East, North, Up.) Each time a pair of likely antenna coordinates is found, one for each baseline, the pair is used to create a nominal vehicle attitude, and the solution is then linearized about that nominal attitude, meaning that small angular perturbations about a nominal attitude are sought which yield the final attitude solution. 
     The immediate problem for second baseline is to figure out what range of motion is possible for the vehicle, given that first baseline is aligned with s N . Notice that the vehicle is already tilted, somewhat, just by aligning the first baseline with s N . It has to be determined what range of vehicle motion is &#34;left over.&#34; If the vehicle has to be tilted to the maximum just in order to align a 1  with s N , then only one orientation is possible. Usually, there is a range ±φ of angles by which vehicle could be rotated about s N  and still remain within the vehicle maximum tilt. 
     The first step is to begin with the vehicle level and facing North. That aligns the body (B) XYZ coordinate frame axes with the navigation (N) coordinate frame axes (East, North, Up.) Now, twirl the vehicle about vertical, (a change in vehicle azimuth or bearing,) so that antenna baseline 1 is either directly under or over the supposed solution s N  for antenna baseline 1. Create a new coordinate system so that s N , a 1 , and the X&#39; axis of the new coordinate system are all in a vertical plane, as shown in FIG. 3, with γ as the angle between s N  and a 1 . Then, tilt the vehicle up (or down) by γ to bring a 1  into alignment with s N . Finally, rotate the vehicle by φ, pivoting about s N . FIG. 4 shows a Z&#34; axis that has resulted from tilting the Z axis back by γ, to give Z&#39;, and then rotating by an angle φ, pivoting about s N . If φ is rotated through 360 degrees, then the Z&#34; axis sweeps out an inclined circle, as is shown in the figure. Generally, only part of that circle can be reached without exceeding maximum vehicle tilt. To find that part, slice through the figure with a horizontal plane that cuts the Z axis at cos(tilt max ). Vehicle tilt is within its allowable range when Z&#34; is on any portion of the circle that lies above the cos(tilt max ) plane. The limits of φ are given by the two points where Z&#34;=(x&#34;,y&#34;,z&#34;) just touches the plane, or where z&#34;=cos(tilt max ). 
     FIG. 5 is a diagram laying FIG. 4 on its side, turning it around, and re-labeling the axes, establishing z&#34; by vehicle twisting until the first baseline lies directly under or over the candidate vector, twisting the vehicle by an angle φ and projecting the first baseline onto according to the present invention. 
     It simplifies the solution for z&#34; to take the diagram in FIG. 4, lay it on its side, turn it around, and re-label axes as shown in FIG. 5. By similarity of diagrams, z&#34; is given by twisting the vehicle until a 1  lies directly under or over s N , twirling by an additional angle φ, and then projecting a 1  onto s N . The solution for z&#34; can then be written down immediately as ##EQU1## 
     As was done in the previous section, use is made of the fact that a z   N  =a z   B , and ##EQU2## provided that the vehicle is only rotated in azimuth, with zero roll and pitch. 
     The vector dot product can be split up into a vertical part which is independent of azimuth, plus the dot product in two dimensions of the X-Y components of a 1  and s N . 
     The last step is to set z&#34;=cos(tilt max ) and solve for the extremes of φ, subject to the following relationships: ##EQU3## 
     What has been accomplished is to solve for the range ±θ max  of vehicle motion that is possible, or &#34;left over,&#34; given that the first baseline a 1  aligns with a known vector s N . 
     The next step is to solve for the minimum and maximum projection of second baseline a 2   N  onto the incident direction of satellite signal arrival u N , given that first baseline a 1   N  is aligned with s N , and that the allowable vehicle motion is given by a rotation about s N , through any angle up to ±θ max . 
     Rather than solving the dot product a N  ·u N  directly, to obtain the 3-dimensional projection of a 2   N  onto u N , it simplifies the task to take the dot product apart, and then put it back together again. The result will be a two-dimensional problem that is much easier to solve than a corresponding 3-dimensional problem. 
     The key is to recognize that if the vehicle is rotating about a 1   N  =s N , then the dot product a 2   N  ·u N  has a component that is not changing, provided that an appropriate coordinate system is used in which to form the dot product. If a principal axis of the new coordinate system is a 1   N , and if the rotation is only about a 1   N , then the portion of the dot product given by the projection of a 2   N  onto a 1   N , times the projection of u N  onto a 1   N , is constant. That portion can be subtracted, leaving a 2-dimensional problem. 
     FIG. 6 is a diagram of a coordinate system R with axes r1, r2, and r3, in which the first baseline is aligned with r3, according to the present invention. As shown in FIG. 6 according to the present invention, a coordinate system R is constructed with axes, r 1  =a 2   N  ×a 1   N  /|a 2   N  ×a 1   N  |, r 2  =a 1   N  ×r 1  /|a 1   N  ×r 1  |, and r 3  =a 1   N  /|a 1   N  |, where a 1   N  =s N  φ=0. For purposes of constructing R, a 1   N  is available as a given, and a 2   N  is given by twirling and tilting the vehicle to bring a 1   N  into alignment with s N . The vectors r 1 , r 2 , r 3  are all unit length and mutually orthogonal. 
     Satellite line of sight u N  expressed in the R coordinate system is given by c=(c 1 ,c 2 ,c 3 ), and antenna vector 2 in the R system is d=(d 1 ,d 2 ,d 3 ), where c 1  =r 1  ·u N , c 2  =r 2  ·u N , c 3  =r 3  ·u N , d 1  =r 1  ·a 2   N  =0,d 2  =r 2  ·a 2   N , and d 3  =r 3  ·a 2   N , where d 1  is zero because r 1  is orthogonal to a 2   N , having been generated by the cross product of a 2   N  and another vector. The vector dot product in R system is a 2   N  ·u N  =c·d=c 1  d 1  +c 2  d 2  +c 3  d 3 , where c 1  =0 and d2 and c3 are constant. 
     FIG. 7 is a diagram of a satellite line of sight vector making an angle delta in the r1×r2 plane, adding to δ the vehicle rotation φ about r3, normalizing c to unit length to provide the sines and cosines of δ, permitting determination of the range summing δ and φ to find minimum and maximum values of c2d2 as φ is varied over an allowable range, and providing the range pmin . . . pmax of possible integer values, according to the present invention. 
     Looking straight down r 3 , as shown in FIG. 7, the satellite line of sight vector c makes an angle δ in the r 1  ×r 2  plane. To that angle δ will be added the vehicle rotation φ about r 3 . Normalizing c to unit length gives the sines and cosines of δ. The sines and cosines of φ max  are already known from the previous section. From there, it is a straightforward task, using angle addition formulas and tests for φ+δ, to find the sines and cosines of (δ+φ max ), and to test the range of δ+φ to find the minimum and maximum values of c 2  d 2  as φ is varied over the allowable range. Adding the constant c 3  d 3  gives the range (p min  . . . p max ) of possible values for p=c 2   N  ·u N , given that a 1   N  =s N  and that the vehicle can only tilt to a maximum tilt max . 
     The foregoing is sufficient to determine the range of carrier cycle integers for second baseline. The code described below allows for all possible carrier cycle integers, without allowing any extra integers. ##SPC2## 
     The method for baseline 2 can be split into three portions, in order to execute efficiently. One part only has to be executed at program turn-on, since it is only necessary to compute antenna baselengths and some other variables once. The second part only has to be executed once for each first baseline candidate. And the third part has to be executed for each combination of a baseline 1 candidate and a satellite received signal. 
     Most of computer execution time for, the method will be tied up in the third part, and so it is especially important that the third part be computationally efficient. There is an arc-tangent evaluation in the third part, but other than that, it is possible to avoid computer evaluation of transcendental functions. There is a square root evaluation in the third part, but when using a math co-processor, the square root evaluation only takes about 50% longer to evaluate than a simple multiply. 
     It is not necessary to normalize the basis vectors for coordinate system R. In other words, the step of determining r 1  =a 2   N  ×a 1   N  /|a 2   N  ×a 1   N  |, r 2  =a 1   N  ×r 1  /|a 1   N  ×r 1  |, and r 3  =a 1   N  /|a 1   N  |, is modified to avoid the 9 evaluations of a divide. Instead, the un-normalized lengths can be carried along, and then accounted for later. 
     The method to determine carrier cycle from . . . to range for baseline 2 was tested, using the test program shown above. A large number of vehicle random orientations were generated, using a random number generator. Also, 20 random satellite positions were generated, uniformly distributed over the sky. For each satellite and vehicle combination, whole-value carrier phase measurements were simulated, including the simulation of carrier measurement error. It was verified that none of the simulated whole value phase measurements was larger than the maximum determined under test, or smaller than the minimum determined under test.