Method and system for determining a position of a communication satellite utilizing two-way ranging

[0002] This application is related to U.S. patent application Ser. No. 08/803,935, filed Feb. 21, 1997, entitled “Method And System For Determining A Position Of A Transceiver Unit Utilizing Two-Way Ranging in a Polystatic Satellite Configuration”.

[0003] This invention relates to methods and systems for determining a position of a communication satellite utilizing two-way ranging through multiple transceivers.

[0004] Current Automatic Dependent Surveillance (ADS) technology, such as Global Positioning System (GPS), Wide Area Augmentation System (WAAS) or GLONASS, provides positioning information utilizing satellite transmissions. For example, the GPS, developed and deployed by the U.S. Department of Defense, consists of 24 satellites orbiting the earth twice a day at an altitude of 12,000 miles, as well as five ground stations to monitor and manage the satellite constellation. Using atomic clocks and location data, GPS satellites transmit continuous time and position information 24 hours a day to a GPS receiver, which listens to four or more satellites at once to determine a user's position. By measuring the time interval between the transmission and the reception of a satellite signal, the GPS receiver calculates the distance between the user and each satellite, and then uses the distance measurements of at least four satellites to arrive at a position.

[0005] Such systems, however, utilize one-way ranging in which an accurate, synchronized clock is required at each station. Any synchronization error or error regarding the location of one of the satellites results in an error in the determined position of the target vehicle.

[0006] Any type of satellite based navigation system must determine the position and velocity of its ranging satellites to extremely high accuracy. Position and velocity determination are vital parts of the station-keeping function for any satellite, even one which is not part of a navigation system.

[0007] One known system requires the satellite to have simultaneous contact with all of its ground stations in order to obtain position information. This, however, may not always be possible for a satellite in a non-geostationary orbit which moves relative to the Earth and, therefore, periodically loses access to any given point on the Earth's surface.

[0008] It is thus a general object of the present invention to provide a method and system for determining the position of a communication satellite utilizing two-way ranging.

[0009] In carrying out the above object and other objects, features, and advantages of the present invention, a method is provided for determining a position of an orbiting transceiver in a communications network including at least a first and second transceiver at a first and second known location, respectively, on Earth. The first and second transceivers are adapted to transmit and receive communications signals to and from the orbiting transceiver. The method includes determining a first and second range measurement between each of the first and second transceivers and the orbiting transceiver, respectively. The method further includes determining a first and second range rate corresponding to a time rate of change of the first and second range measurements, respectively. The method also includes determining a circle of intersection representative of the set of possible positions for the orbiting transceiver based on the first range and the first range rate wherein the circle of intersection includes a specific orientation in space, a specific radius and a center in a specific, three dimensional position relative to the known position of the first transceiver. Still further, the method includes determining an angular position of the orbiting transceiver along the circle of intersection based on the known position of the second transceiver and the second range measurement. Finally, the method includes determining the position of the orbiting transceiver based on the circle of intersection and the angular position.

[0010] In further carrying out the above object and other objects, features, and advantages of the present invention, a system is also provided for carrying out the steps of the above described method. The system includes at least a first and second transceiver at a first and second known location, respectively, on Earth. The first and second transceivers are adapted to transmit and receive communications signals to and from the orbiting transceiver. A processor coupled to one of the first and second transceivers is operative to determine a first and second range measurement between each of the first and second transceivers and the orbiting transceiver, respectively. The processor is further operative to determine a first and second range rate corresponding to a time rate of change of the first and second range measurements, respectively. Still further, the processor is operative to determine a circle of intersection representative of the set of possible positions for the orbiting transceiver based on the first range measurement and the first range rate wherein the circle of intersection includes a specific orientation in space, a specific radius, and a center in a specific, three-dimensional position relative to the known position of the first transceiver. The processor further determines an angular position of the orbiting transceiver along the circle of intersection based on the known position of the second transceiver and the second range measurement. Finally, the processor determines the position of the orbiting transceiver based on the circle of intersection and the angular position.

[0011] The above object and other objects, features and advantages of the present invention are readily apparent from the following detailed description of the best mode for carrying out the invention when taken in connection with the accompanying drawings.

[0012] FIG. 1 is a diagram illustrating a communication system employing the method and system of the present invention;

[0013] FIG. 2 is a diagram illustrating the geometrical plane containing two ranging nodes of the present invention;

[0014] FIG. 3 is a diagram illustrating an enlarged view of the plane of FIG. 2;

[0015] FIG. 4 is a diagram illustrating a circle of intersection extending above and below the X-Y plane of FIG. 2; and

[0016] FIG. 5 is a diagram similar to FIG. 3, but adding x and y unit vectors, and a point denoted R_qassociated with a later step in the position determination.

[0017] FIG. 1 diagrammatically illustrates a communication system with a typical geometry for practicing the present invention. This system, denoted generally by reference numeral 10, includes a target communication satellite 12 whose position (denoted R₀) is to be determined. The system 10 in FIG. 1 also includes a primary ranging node (RN) 14—such as a satellite ground station—at the position denoted R₁, as well as secondary ranging nodes 11 at positions R₂and R₃. There may be many other secondary RNs at positions R₄, R₅, etc. These additional RNs are not shown in FIG. 1; however, their modes of operation are identical to those described for the RNs at R₂and R₃. A secondary RN may be any device that includes a transceiver such as, for example, automobiles, mobile telephones, aircraft, or the like, as long as they are stationary during operation of the present invention. As shown in FIG. 1, the distances between the satellite 12 at R₀and all the RNs at R₁, R₂, R₃, etc. are denoted a_i=|R_i−R₀|, where i>0.

[0018] For example, a parked commercial airliner can act as a secondary RN. When a commercial airliner is parked at an airport gate, its pilot enters the precise position of the aircraft into the onboard navigation system. This information allows the inertial navigation system to correct for drift and accumulated position error. As long as the aircraft is parked at the gate, its position is known to a high degree of accuracy. If it is equipped with an appropriate transceiver, then it can act as a secondary RN at position R₂or R₃. If a large number of aircraft are thus equipped, and a significant percentage of them are parked at a gate at any given time, then the satellite 12 at R₀can use these parked aircraft at R₂and R₃as RNs.

[0019] The RNs at R₁, R₂, R₃, etc. communicate with each other by relaying messages through the satellite 12 at position R₀. In the course of operation of the present invention, the primary RN 14 receives two distinct kinds of signals: ranging codes broadcast by the satellite 12, and reply codes which also are relayed through the satellite 12, but originate at secondary RNs 11. A processor 16 located at the primary RN 14 at R₁distinguishes the two types of signals, and makes the computation required for the determination of the position of the satellite 12.

[0020] The position of the satellite 12 is determined according to the following sequence of steps. First, at a precisely measured time, and on a precisely measured carrier frequency, the primary RN 14 at R₁transmits a ranging signal to the satellite 12 at R₀, which broadcasts it to all of the RNs, including the primary RN 14 at R₁, and secondary RNs 11 at R₂, R₃, etc. The secondary RNs 11 then transmit their distinctive reply code signals back to the satellite 12 for transmission back to primary RN 14. Immediately upon receiving the ranging and reply code signals, the processor 16 at the primary ranging node 14 (R₁) measures the carrier frequency and arrival time of the ranging signal. It then computes the difference between the ranging code's arrival time and the time of its original transmission from the primary ranging node 14. This time difference is a propagation time, denoted t_p1, and is associated with the distance of the transmission of ranging signals from the primary RN 14 at R₁, to the satellite 12 at R₀, and back to R₁. This distance, denoted D₁, is equal to 2a₁. The processor 16 also measures the difference in carrier frequency between the received reply code signal and that of the originally transmitted ranging code. From that frequency difference the processor computes the Doppler frequency shift, as described below.

[0021] Before continuing with the description of the processor's computations, we describe the measurements made on signals passing through the secondary RNs 11. As mentioned above, immediately upon receiving a ranging code, the secondary RNs 11 at positions R₂, R₃, R₄. etc. each transmit a reply ranging signal to the primary RN 14 at R₁via the satellite 12 at R₀. These reply signals are designed such that they uniquely identify the particular ranging code to which they are replying, and the secondary RN 11 from which they are transmitted. The processor 16 at R₁can then determine the position R_iof the secondary RN 11 which initiated the particular reply code, where R_iis an element of the set of all known secondary RN positions {R₂, R₃, . . . }. The processor 16 also measures the arrival time of each reply signal, and computes the difference between that time and the time of original transmission from the primary ranging node 14 of the ranging code which elicited the reply signal. That difference is denoted as a propagation time, t_pi. Thus, the measured times for round trip communication with secondary RNs at R₂, R₃, etc. are denoted t_p2, t_p3, etc. respectively.

[0022] Referring again to FIG. 1, we see that the paths associated with propagation times t_pifor i>1 are of a different shape than the path associated with t_p1and D₁. For i>1, D_iis the path length from the primary ranging node 14 at R₁, to the satellite 12 at R₀, to the secondary ranging node 11 at R_i, then back to satellite 12 at R₀, and finally, back to primary ranging node 14 at R₁. To a first approximation, D_i=2(a_i+a₁) for i>1. This equality is only approximate because it neglects the fact that the satellite 12 moves during the time of the signal's trip from R₀to R_iand back to R₀. In reality, the satellite 12 will be in a different place when relaying the initial ranging signal from R₁to R_ithan when relaying the reply signal from R_iback to R₁. We can correct for this effect, but in the interest of clarity, we neglect it in the present description of the system, but not in the final implementation.

[0023] We now outline a computational procedure which transforms a set of frequency shifts and propagation time measurements first into a set of communication path lengths (D₁, D₂, D₃, etc.), then into a set of ranges (a₁, a₂, a₃, etc.), and finally, into a three dimensional, vector position of the satellite 12. The measurements of propagation time include time for an electromagnetic wave to travel the prescribed path, and also delay times associated with the electronic detection and retransmission of messages. These intervals are all measured with a single clock 18 at R₁; therefore, secondary RNs at R₂, R₃, etc. do not need clocks, and the system depicted in FIG. 1 does not have any clock bias errors such as those in The Global Positioning System (GPS).

[0024] Signal Delay Times and Distances

[0025] For the path corresponding to each measured propagation time t_p1, t_p2, t_p3, etc., the processor 16 at R₁computes an estimate of the total time delay, t_di, for all transceivers as well as tropospheric and ionospheric propagation along that path. The corresponding delay times t_d1, t_d2, t_d3, etc. are then used in the computation of the corresponding distances between the transceivers.

[0026] The ionosphere and atmosphere are two elements in the signal path that slow the signal to less than the vacuum speed of light, which is defined as c=299,792,458 meters/second. The delay time, t_di, is defined as the propagation time of a signal over the i^thpath minus the time required for an electromagnetic wave to traverse a path of the same length in a perfect vacuum. Given that D_iis the length of the i^thcommunication path, D_i=c(t_pi−t_di). Delay time is the sum of many components, due to the ionosphere, atmosphere, transmitter, receiver, etc., and is never negative.

[0027] The atmospheric and ionospheric signal delays have been thoroughly studied, and mathematical models have been constructed which can predict these components of delay time (for example: Black, H. D., “An Easily Implemented Algorithm for the Tropospheric Range Correction,” JOURNAL OF GEOPHYSICAL RESEARCH, vol. 83, no. B4, p. 1825, (1978); and for the ionospheric effect: Klobuchar, J. A., “Design and Characteristics of the GPS Ionospheric Time Delay Algorithm for Single Frequency Users,” IEEE PLANS '186: Position Location and Navigation Symposium, Las Vegas, Nev., Nov. 4, 1986, p. 280).

[0028] The ionospheric delay is harder to predict with high accuracy than that of the troposphere because the ionosphere undergoes more radical variations over time. If a particular application demands the highest possible accuracy, then the system 10 will rely upon additional measurements rather than a fixed mathematical model in order to determine the ionospheric propagation delay. Because that delay is inversely proportional to the square of the radio carrier frequency, and because it varies with location and time, additional measurements of round-trip signal propagation time through different ground reference points and/or at different carrier frequencies will provide the continuous measurements of the ionosphere which are needed for the highest accuracy range measurements. The availability of a large number of secondary RNs provides extra information which can be used to determine the characteristics of the ionosphere, the troposphere, and other sources of signal delay in the communication path. The present invention can compensate for the time varying tropospheric and ionospheric signal propagation delays more accurately than a system which contains a very small number of RNs or turn-around ranging stations.

[0029] Other sources of delay involving the system's antennas, transmitters, and receivers (including multipath, etc.) are similarly estimated. All of the estimated time delays associated with the i^thcommunication path are estimated, and added together to produce the total estimated time delay, t_di. All such delay times t_d1, t_d2, t_d3, etc. are then used in the computation of the corresponding communication path lengths D₁, D₂, D₃, etc., via the formula

[0030] D_i=c(t_pi−t_di).

[0031] Based on D₁, D₂, D₃, etc., the system 10 computes the ranges a₁, a₂, a₃, etc., via formulas such as: a₁=D₁/2, and a_i=(D_i−D₁)/2. As mentioned previously, the final implementation involves more complicated equations than these, due to the motion of the satellite 12 during the time interval between its broadcast of the ranging signal and its relay of the reply signal. The correction is omitted here to simplify the presentation. The processor 16 then uses those ranges and the known positions of the RNs 14, 11, (a_iand R_ifor i>0) to compute the satellite's position R₀by means of one of the two procedures described below.

[0032] We derive two algorithms for determination of the (three dimensional) vector position of the satellite 12; a “Three Sphere Algorithm” which uses range measurements involving at least three ground transceivers, and a “Range and Range Rate Algorithm” which uses range measurements from only two ground transceivers. “Range Rate” refers to the time rate of change of range (da_i/dt), and is derived from a combination of the time sequence of range measurements and measurements of the Doppler frequency shift of the radio signal, which is described briefly below. The “Range and Range Rate Algorithm” applies only to satellites which are not in geostationary orbit; whereas, the “Three Range Algorithm” applies to geostationary or non-geostationary satellites.

[0033] Doppler Frequency Shift

[0034] If a transmitter at position R₁sends a radio signal at carrier frequency f₁to a receiver at position R₀which is approaching or receding, then the receiver will detect the signal at a different frequency f₀due to the Doppler effect. The size of the Doppler frequency shift depends on the radial velocity, which we denote as v_r, and which equals the time rate of change of the distance between the transmitter and receiver. If the velocity of the receiver relative to the transmitter is much smaller in magnitude than the speed of light, then (approximately),

[0035] f₀/f₁=1−v_r/c.

[0036] If another signal is transmitted at frequency f₀from R₀back to R₁, then the Doppler shift is compounded, so that the frequency detected at R₁is

[0037] f₁₁f₀(1−v_r/c)=f₁(1−v_r/c)²,

[0038] which is approximately

[0039] f₁₁=f₁(1−2v_r/c.

[0040] The Doppler frequency shift (f_d) is the difference between the frequency f₁₁received at point 1 after the round-trip, and that of the signal initially transmitted from point 1, which is f₁; therefore,

[0041] f_d=f₁₁−f₁=−2f₁v_r/c.

[0042] The Doppler shift measurement therefore yields a measurement of the radial velocity,

[0043] v_r=−cf_d/(2f₁),

[0044] and this information contributes to the orbit determination, as described below.

[0045] Geometry

[0046] In this invention, we present two algorithms which may be used to convert a set of scalar measurements into the vector position of the satellite 12. The “Three Sphere” positioning algorithm employs round trip signal propagation between a transceiver at an unknown location (which we denote R₀) and a set of three or more RNs. The “Range and Range Rate” positioning algorithm requires round trip communication between the transceiver at R₀and only two RNs, but also requires that R₀be moving relative to the RNs. In both of these approaches, the round trip pattern of communication implies that all round trip propagation time intervals can be measured from a single clock located at the primary RN. There is no need for highly accurate or synchronized clocks at any other location in the system.

[0047] In contrast, a typical GPS receiver requires measurements of signals from four different transmitters in order to solve for the receiver's position. Due to the one way nature of the communication, the GPS transmitters must have highly accurate and precisely synchronized clocks, and each GPS receiver must solve for the precise local time while determining its three dimensional position.

[0048] An additional advantage of the present invention is that these two algorithms are direct, closed form solutions to the nonlinear system of equations which define the satellite's position; whereas, GPS uses an iterative solution to a linearized approximation of an intrinsically non-linear problem. The direct solution is more computationally efficient, faster, and more accurate.

[0049] “Three Sphere” Algorithm

[0050] The three sphere algorithm transforms the ranges (a₁, a₂, a₃, etc.) computed above and the known positions of three or more RNs (R₁, R₂, R₃, etc.) into the three dimensional, vector position of the satellite 12 relative to the Earth. We use the following notation to aid in the concise expression of the algorithm.

[0051] Vector Notation

[0052] In all of the following equations, upper case letters denote three component vectors, and lower case letters denote scalar (non-vector) quantities.

[0053] A×B=cross product (vector product) of vectors A and B, such that if C=A×B then the three components of C are:

[0054] c_x=a_yb_z−a_zb_y

[0055] c_y=a_zb_x−a_xb_z

[0056] c_z=a_xb_y−a_yb_x

[0057] Note that the resulting C is perpendicular to A and to B.

[0058] A.B=a_xb_x+a_yb_y+a_zb_z=dot product (also known as the scalar product or inner product) of vectors A and B.

[0059] A²=A.A=square of the length of vector A, which equals the sum of the squares of the three components (x, y, z) of the vector (as per the Pythagorean theorem in three dimensions).

[0060] |A|=(A.A)^½=length of vector A.

[0061] R_j=vector which points from the center of the Earth to the position of the j^thRN, where j=[1, 2, 3, . . . ].

[0062] R₀=vector which points from the center of the Earth to the position of the satellite to be located.

[0063] R_jk=R_j−R_k=vector from point k to point j.

[0064] a_j=|R_0j|=distance measurement from the j^thRN to the satellite.

[0065] Intersection of Two Spheres

[0066] The satellite's position, R₀, will be determined relative to the RN positions (R₁, R₂, . . . ). The following geometrical and mathematical derivation of R₀is performed not directly in Earth centered coordinates, but rather, in a coordinate system defined by the positions of the RNs. This choice of coordinate system separates the vector R₀₁(which points from the known R₁to the unknown R₀) into three mutually perpendicular components (denoted R_p1, R_qp, R_0q). In this carefully selected coordinate system, each of these vector components can be computed directly (non-iteratively) from the measurements (a₁, a₂, a₃). The sum of these three vectors and the position vector of the primary RN (R₁) is the desired location of the satellite (R₀) Symbolically:

[0067] R₀=R_0q+R_qp+R_p1+R_1.

[0068] The following discussion describes the geometrical process of locating the intermediate points R_pand R_qin space, and then deriving algebraic formulas for those positions.

[0069] FIG. 2 shows a plane containing the primary RN 14 at R₁and the secondary RN at R₂. It also shows the vector R₂₁=R₂−R₁, which points from R₁to R₂. The measurement al limits the possible positions of R₀to the points on a sphere of radius a₁which is centered at R₁. The intersection of that sphere with the plane of FIG. 2 is a circle of radius a₁. That circle is labeled |R₀₁|=a₁. Although R₀might lie on this circle, it is far more likely to lie above or below the plane depicted in FIG. 2.

[0070] Similarly, the measurement a₂limits the possible positions of R₀to the points on a sphere of radius a₂which is centered at R₂. The intersection of that sphere with the plane of FIG. 2 is the circle labeled |R₀₂|=a₂. Since R₀must lie on both of these spheres, the only possible solution points for R₀in the plane of FIG. 2 are the points labeled R_band R_c, where the two circles intersect; however, there are other possible solution points above and below this plane.

[0071] In order to visualize the location of all possible points which lie on both of the afore-mentioned spheres, one may slowly flip over the picture in FIG. 2, doing so in a way that does not move vector R₂₁. During this slow rotation of the picture, one sees that the circles in the plane trace out spheres in three dimensions, and that R_band R_ctrace out a circle of possible positions for R₀. That circle lies in another plane which is perpendicular to the vector R₂₁. In other words, the circle defined by the intersection of the two spheres sticks out at a right angle from the plane depicted in FIG. 2.

[0072] Having considered the geometrical construction of the circle of intersection, we now develop equations which quantify the position of the structures discussed above. FIG. 3 is an enlargement of the central region of FIG. 2, with additional details added. The vector from R₁to R_bis denoted |R_b1|=a₁, where a₁is its length. Similarly, the vector from R₂to R_bis denoted |R_b2|=a₂, where a₂is its length.

[0073] The vector R_bcwhich points from R_cto R_bis perpendicular to the vector R₂₁. The point of intersection of R_bcwith R₂₁is denoted R_p. This definition of R_pallows us to break the triangle with vertices at R₁, R₂, R_binto one right triangle with vertices at R₁, R_p, R_band another right triangle with vertices at R₂, R_p, R_b. The lengths of the sides of right triangles are related by the Pythagorean theorem: x²+y²=r², where x and y represent the lengths of the two perpendicular sides, and r represents the length of the third side. We now use this theorem to determine the position of R_prelative to R₁and R₂.

[0074] a₁²=R_b1²=R_bp²+R_p1².

[0075] a₂²=R_b2²=R_bp²+R_p2².

[0076] The difference of these two equations is

[0077] a₁²−a₂²=R_p1²−R_p2².

[0078] We note that R_p2=R_p1−R₂₁; therefore,

[0079] R_p2²=R_p1²−2R₂₁.R_p1+R₂₁².

[0080] Combining these two equations gives:

[0081] a₁²−a₂²=2R₂₁.R_p1−R₂₁².

[0082] or equivalently,

[0083] R₂₁.R_p1=(R₂₁²+a₁²−a₂²)/2.

[0084] This means R₂₁.R_p1can be either positive or negative, depending on the sign of R₂₁²+a₁²−a₂². For example, if R₁were inside the sphere denoted |R₀₂|=a₂and a₁were much less than a₂, then R₂₁.R_p1<0. Whether R₂₁.R_p1is positive or negative, R₂₁×R_p1=0, meaning that the two vectors are always either parallel or anti-parallel. In both cases, the vector extending from R₁to the center of the circle of intersection is defined by the following equation: $R_{\mathrm{p1}}=R_{21}\left(R_{21}^2+a_1^2-a_2^2\right)/\left(2R_{21}^2\right).$

[0085] This equation defining R_p1differs from those which preceded it. The earlier equations were general mathematical relationships and partial clues pointing toward a solution, but because this equation contains only known (measured or previously computed) quantities on the right hand side and a single variable on the left, it represents not just a mathematical relationship between a set of known and unknown quantities, but rather, an explicit procedure (neither implicit, recursive, nor iterative) for calculating the left hand unknown in terms of a set of known quantities. Explicit procedural equations such as this will be enclosed in boxes to distinguish them from other equations, which only describe how they are derived. The set of all these explicit, procedural equations evaluated in the specified order, constitute the procedure which determines the satellite position R₀. Now that we know R_p1, we may compute R_bp²from the earlier equation for a₁²:

[0086] R_bp²=a₁²−R_p1².

[0087] As mentioned above, R_bis only one possible solution for R₀, but we have not yet included the a₃data in our calculation. Without that additional information, we could not distinguish between R_band any other point on the circle of intersection lying above or below the plane of FIG. 3. R_bp²is the square of the radius of the circle of intersection, which is also equal to R_0p². $R_{0p}^2=a_1^2-R_{\mathrm{p1}}^2.$

[0088] At this step of the calculation, we know three key things about the circle of intersection: first, the position of its center, R_p=R_p1+R₁; second, its radius, |R_0p|; and third, its orientation in three dimensional space, R₂₁.R_0p=0. We still must determine the angular position of R₀on the circle of intersection in order to obtain the full vector solution: R₀=R_0p+R_p. Determining that angle requires a third measurement.

[0089] Constructing the Coordinate System

[0090] In order to define the angular position of R₀, we define a set of coordinate axes relative to which the angle is measured. We denote the line connecting R₁and R₂as the x-axis, and define the x-directed unit vector as $U_x=R_{21}/\left|R_{21}\right|.$

[0091] (Dividing any vector by its own length produces a unit vector, that is, a vector with length equal to one.) This x unit vector (U_x) is the “unit normal vector” of the circle of intersection, meaning that it is perpendicular to the plane containing that circle.

[0092] The cross product of two parallel (or anti-parallel) vectors is zero. If R₃does not lie on the x-axis, then R₃₁is not parallel (or anti-parallel) to R₂₁and U_x. In that case, R_z1=U_x×R₃₁is non-zero and is normal to the plane containing the RNs. If R₃lies on or very near to the x-axis, then |R_z1|is zero or very close to zero, and the position computation becomes inaccurate. This is true regardless of the computational algorithm used, but this algorithm detects this condition explicitly, so that appropriate action may be taken, and the results of the computation will not be misinterpreted.

[0093] It is sometimes convenient to express R_z1in reference to R_prather than R₁. This can be done because

[0094] R₃₁=R_3p+R_p1

[0095] R_z1=U_x×R_3p+U_x×R_p1

[0096] U_xis parallel to R₂₁and R_p1; therefore, U_x×R_p1=0. We define the remaining two orthonormal vectors (U_yand U_z) of this coordinate system as follows: $\begin{array}{c}{R}_{\mathrm{z1}}=U_x\times R_{3p}\\ U_z=R_{\mathrm{z1}}/\left|R_{\mathrm{z1}}\right|.\\ U_y=U_z\times U_x.\end{array}$

[0097] There is no need to divide U_yby its own length, as we did for U_xand U_z, because U_xand U_zare perpendicular, and both are of length one; therefore, their cross product will be of length one.

[0098] Intersection with Sphere 3

[0099] The measurement a₃defines another sphere upon which R₀must lie. If point 3 is not on the x-axis, then this sphere will cross the previously defined circle of intersection in two points, one above and one below the x-y plane, as shown in FIG. 4. We denote these crossing points R_dand R_e, and note that R_deis parallel to U_z. Denote as R_qthe point of intersection between R_deand the x-y plane (which is the plane containing U_xand U_y, and also the plane depicted in FIGS. 2 and 3).

[0100] We now develop a procedure for locating R_dand R_e. The satellite position R₀is equal to either R_dor R_e. Since one of these will lie below the surface of the Earth, there is no difficulty in choosing the proper one as the last step of the computation.

[0101] As shown in FIG. 4, the vector R_dqis parallel to U_z. FIG. 5, which is similar to FIG. 3, but with point R_qadded, shows that R_p1is parallel to U_x, and R_qpis parallel to U_y. The sum of these three vectors R_d1=R_dq+R_qp+Rp1defines the position of point R_drelative to point R₁, and likewise for R_e.

[0102] The Pythagorean theorem for the right triangle dpq decomposes R_dpinto two components which are aligned so that R_dqis parallel to U_z(see FIG. 4), and R_qpis parallel to U_y(see FIG. 5).

[0103] R_dp²=R_dq²+R_qp².

[0104] The decomposition of R_d3is similar; however, there is an added wrinkle. U_zis perpendicular to R_q3, as shown in FIG. 4; therefore, R_q3has no z component; however, R_q3may contain non-zero components in both the x and y directions. This is in contrast to R_qpwhich has only a y component, as shown in the previous paragraph.

[0105] a₃²=R_d3²=R_dq²+R_q3².

[0106] The difference of these two equations is

[0107] a₃²−R_dp²=R_q3²−R_qp².

[0108] We note that R_q3=R_qp+R_p3; therefore,

[0109] R_q3²=R_qp²+R_p3²+2R_qp.R_p3.

[0110] Combining these last two equations yields:

[0111] a₃²−R_dp²=R_p3²+2R_qp.R_p3.

[0112] Note that R_dis one of the possible solutions for R₀; therefore, R_dp²=R_0p². This is the radius of the circle of intersection, which we calculated previously. Substituting this into the previous equation yields:

[0113] R_qp.R_p3=(a₃²−R_0p²−R_p3²)/2.

[0114] As in a similar equation in the two sphere intersection procedure, this dot product, R_qp.R_p3can be either positive or negative. The same procedure applies to both: $R_{\mathrm{qp}}=U_y\left(a_3^2-R_{0p}^2-R_{\mathrm{p3}}^2\right)/\left(2U_y·R_{\mathrm{p3}}\right).$

[0115] Note that U_y.R_p3=−U_y.R₃₁. This quantity appears in the denominator of the equation above; hence, if it is zero, the algorithm breaks down. This is not an artifact of this particular algebraic derivation, but a fundamental, geometrical constraint. If U_y.R₃₁=0 then point 3 lies on the x-axis. In that case we can determine only that R₀lies on a circle whose unit normal vector is U_x. We can determine the radius of that circle, |R_0p|, and the position of its center, R_p, but not the angular position of R₀around that circle.

[0116] The third RN must lie far enough from the x-axis to break the cylindrical symmetry, and resolve the angular position of R_dand R_e, one of which is equal to R₀. The accuracy of the position computation degrades as the third RN approaches the x-axis. We have applied this analysis to actual orbital configurations of proposed systems in order to assess the accuracy of the algorithm in the face of measurement errors and sub-optimal RN alignments. In almost all cases, U_y.R₃₁is large enough, and the algorithm can continue as follows: The position of point q relative to the center of the Earth is $R_q=R_{\mathrm{qp}}+R_{\mathrm{p1}}+R_1.$

[0117] Knowing R_qp, we can use the Pythagorean theorem for triangle dpq (see FIG. 4) to obtain R_dp²=R_dq²+R_qp², which leads to the two candidate solutions for the satellite position: $\begin{array}{c}\left|R_{\mathrm{dq}}\right|={\left(R_{\mathrm{dp}}^2-R_{\mathrm{qp}}^2\right)}^{1/2};\\ R_d=R_q+U_z\left|R_{\mathrm{dq}}\right|;\\ R_e=R_q-U_z\left|R_{\mathrm{dq}}\right|.\end{array}$

[0118] Notice that the two solutions are located symmetrically above and below the plane containing the RNs. Because the RNs are all on the surface of the Earth, one of these solutions will lie either inside, or on the opposite side of the Earth. In virtually all cases, only one of the solutions, R_dor R_e, will be within the field of view of all of the RNs which supplied the data; therefore, the algorithm can reject the spurious solution, and identify the other as R₀.

[0119] Intersection of Sphere and Concentric Cone

[0120] If only two RNs are in view of a satellite, and provided that the satellite is not stationary with respect to the Earth, then measurements of range and range rate from a single RN can localize the satellite's possible positions to a circle of intersection, just as does the two sphere intersection discussed above. In this case, the circle of intersection corresponds to the intersection of a sphere with a cone whose vertex is located at the center of the sphere. After this circle of intersection is defined by a new procedure, one can apply the same sphere-circle intersection procedure as is used with the Three Sphere positioning algorithm to solve for the satellite's position.

[0121] In the Sphere and Concentric Cone algorithm, the range measurement again constrains possible solution points to the surface of a sphere. The new facet is that the range rate measurement further limits solution points to a cone whose vertex lies at the center of that same sphere. Because the cone is concentric with the sphere, their curve of intersection is a circle. Such a circle is analogous to a line of constant latitude on a globe of the Earth; however, its orientation is defined by the motion of the satellite, and not fixed to the Earth.

[0122] If the satellite were in uniform, linear motion relative to the RN, then the relative velocity vector (V₀₁) divided by its own length would define the axis of the cone (which is U_x), and the ratio of radial to total relative velocity would determine the cone's angle=arccos(v_r/|V₀₁|); however, the actual situation is a bit more complicated. The satellite of interest in the present invention, and also the RNs on the surface of the Earth are in nearly uniform, circular motion relative to inertial space.

[0123] Despite the added complexity in this case, we still obtain a closed form solution. The generalization to elliptical satellite orbits is a trivial extension of this algorithm, and while not discussed in detail, should be considered included in the present invention. Both the satellite and the RNs are moving along trajectories of constant altitude. In the case of the RNs, this is due to the rotation of the surface of the Earth. One can express the velocity in any circular motion as V_j=W_j×R_j; therefore,

[0124] V₀₁=W₀×R₀−W₁×R₁,

[0125] where W_jis the vector angular velocity. For RNs fixed to the surface of the Earth, W₁is identical to the Earth's angular velocity. If the satellite is not geostationary, but is moving in an orbit which is approximately circular, then on the time scale of the calculation process, we consider W₀to be constant.

[0126] As described above, the system measures the distance between the primary RN and the satellite (a₁) from the signal propagation time, and the radial velocity (v_r) from the time history of range measurements and from the Doppler shift. These scalar measurements are related to the satellite's vector position and velocity as follows:

[0127] V₀₁.R₀₁=v_ra₁.

[0128] The range and range rate measurements define V₀₁.R₀₁; however, since R₀is unknown, and is involved in the definitions of both R₀₁and V₀₁, we must do some vector algebra to disentangle R₀from the cross product.

[0129] V₀₁.R₀₁=(W₀×R₀−(W₀−W₀₁)×R₁).R₀₁=(W₀×R₀₁+W₀₁×R₁).R₀₁.

[0130] Note that W₀×R₀₁is perpendicular to R₀₁, while W₀₁×R₁has components both perpendicular and parallel to R₀₁. The dot product (W₀×R₀₁).R₀₁is zero; therefore,

[0131] V₀₁.R₀₁=(W₀₁×R₁).R₀₁.

[0132] We assume that the satellite's angular velocity (W₀) is known to high accuracy, and that what is not known at the time of the measurement is its precise position (R₀). The angular velocity of the Earth (W₁) and the position of the primary RN (R₁) are assumed to be known. This implies that the vector W₀₁×R₁is known, and we use it to define the x unit vector (U_x) in this Range and Range Rate algorithm. It plays the same role here that R₂₁does in the Three Sphere algorithm. $U_x=W_{01}\times R_1/\left|W_{01}\times R_1\right|.$

[0133] The center of the circle of intersection is again denoted R_p, and in this algorithm, $R_{\mathrm{p1}}=U_x\left(R_{01}·U_x\right)=U_x{a}_1{v}_r/\left|W_{01}\times R_1\right|.$

[0134] As in the Three Sphere algorithm, $R_{0p}^2=a_1^2-R_{\mathrm{p1}}^2.$

[0135] Via these equations, i.e., range plus range rate measurements from a single RN, the position, orientation, and radius of the circle of intersection are provided without using a second RN. Once one has the circle of intersection, either from Two Sphere or Sphere and Concentric Cone method, then one can use the same method to intersect that circle with the final sphere to get the satellite's position. In the Range and Range Rate method, all references to the “third RN” in the final sphere intersection method actually refer to the second RN, since no third RN is required. After the assignments R₃=R₂and a₃=a₂, the Range and Range Rate algorithm proceeds in the same way as the Three Sphere algorithm.

[0136] While the best modes for carrying out the invention have been described in detail, those familiar with the art to which this invention relates will recognize various alternative designs and embodiments for practicing the invention as defined by the following claims.