Determination of phase ambiguities in satellite ranges

Claim 9

The method of claim 4, further comprising the step of choosing said integer n equal to four.

Claim 10

The method of claim 4, further comprising the step of (14) solving the n-1 simultaneous equations given by Φ.sub.i,i';1,2 =-ΔR.sub.i,i';1,2 /λ.sub.cp +N.sub.i,i';1,2, (i and i'=1,2, . . . , n; i≠i') to obtain position coordinates (x 2 ,y 2 ,z 2 ) of a roving receiver antenna located at an arbitrary position, when the coordinates (x 1 ,y 1 ,z 1 ) of the reference receiver antenna are known with sufficient accuracy.

Claim 11

The method of claim 4, further comprising the step of choosing said effective frequency, and of said corresponding effective phases β i ,i';j,j';cp, drawn from the group consisting of the L1 carrier frequency f L1 the L2 carrier frequency f L2 the wide lane frequency f w =f L1 -f L2 , and the narrow lane frequency f n =f L1 +f L2 .

Claim 14

The method of claim 8, further comprising the step of choosing said integer n equal to four.

Claim 15

The method of claim 8, further comprising the step of (12) solving the n-1 simultaneous equations given by Φi,i';1,2=-ΔR.sub.i,i';1,2 /λ.sub.cp +N.sub.i,i';1,2, (i and i'=1,2, . . . , n; i≠i') to obtain position coordinates (x 2 ,y 2 ,z 2 ) of a roving receiver antenna located at an arbitrary position, when the coordinates (x 1 ,y 1 ,z 1 ) of the reference receiver antenna are known with sufficient accuracy.

Claim 16

The method of claim 8, further comprising the step of choosing said effective frequency, and of said corresponding effective phases β i ,i';j,j';cp, drawn from the group consisting of the L1 carrier frequency f L1 , the L2 carrier frequency f L2 , the wide lane frequency f w =f L1 -f L2 , and the narrow lane frequency f n =f L1 +f L2 .

Claim 19

The method of claim 12, further comprising the step of choosing said integer n equal to four.

Claim 20

The method of claim 12, further comprising the step of (13) solving the n-1 simultaneous equations given by Φ.sub.i,i';1,2 =-ΔR.sub.i,i';1,2 /λ.sub.cp +N.sub.i,i';1,2, (i and i'=1,2, . . . , n; i≠i') to obtain position coordinates (x 2 ,y 2 ,z 2 ) of a roving receiver antenna located at an arbitrary position, when the coordinates (x 1 ,y 1 ,z 1 ) of the reference receiver antenna are known with sufficient accuracy.

Claim 21

The method of claim 12, further comprising the step of choosing said effective frequency, and of said corresponding phases β i ,i';j,j';cp, drawn from the group consisting of the L1 carrier frequency f L1 , the L2 carrier frequency f L2 , the wide lane frequency f w =f L1 -f L2 , and the narrow lane frequency f n =f L1 +f L2 .

Claim 22

Apparatus for determining and removing the phase integer ambiguities associated with determination of position of a receiver that is part of a Satellite Positioning System (SPS), the apparatus comprising: a reference receiver and associated antenna for receiving SPS signals from a plurality of n satellites (n≧4), numbered i=1, 2, . . . , n, that are part of the SPS, where the position of the reference receiver antenna is known with sufficient accuracy; a roving receiver and associated antenna for receiving SPS signals from the plurality of n satellites that are part of the SPS, where each satellite transmits a predetermined signal at a predetermined sequence of times at each of at least two carrier signal frequencies f L1 =c/λ L1 and f L2 =c/λ L2 , where c is the velocity of light; antenna positioning means for positioning the roving receiver antenna at a first selected position and then at a second selected position on a selected line containing the reference receiver antenna, where the reference receiver antenna lies between and at a distance d from the first and the second selected roving receiver antenna positions, where d need not be determined; signal processing means, connected to the reference receiver antenna and to the roving receiver antenna: (1) for determining an effective frequency f cp =α 1 f L1 +α 2 f L2 =c/α cp , expressed as a selected linear combination of the two carrier frequencies f L1 and f L2 , for the Satellite Positioning System, where λ cp is the corresponding effective wavelength; (2) for receiving the SPS signals received by the reference receiver antenna and by the roving receiver antenna at the first selected position and the second selected position; (3) for determining a plurality of n theoretical distance values R 1 ,1, R 2 ,1, . . . , R n ,1 from each of the n satellites to the roving receiver antenna at the sequence of predetermined times, where R I ,J is the distance between satellite number I, with Cartesian coordinates (x I ,y I ,z I ), and receiver antenna number J, with coordinates (x J ,y J , J ) (I=1, 2, . . . , n; J=1, 2), given by R I ,J =[(x I -x J ) 2 +(y I -y J ) 2 +(z I -z J ) 2 ] 1/2 , where each theoretical distance value is determined from a knowledge of the position of a satellite and of an estimate of a position of the roving receiver antenna, assuming that d≈0 for this determination; (4) for determining a plurality of n theoretical distance values R 1 ,2, R 2 ,2, . . . , R n ,2 from each of the satellites to the reference receiver antenna at the sequence of predetermined times, where each theoretical distance value is determined from a knowledge of the position of a satellite and of the position of the reference receiver antenna, where the position of the reference receiver antenna is known with sufficient accuracy at each of the predetermined sequence of times; (5) for forming theoretical range double difference values ΔR i ,i';j,j' (1) (i,i'=1,2, . . . , n; i≠i'; j,j'=1,2; j≠j') for the reference and roving receiver antenna positions, defined by the relations ΔR.sub.i,i';j,j' (1)=(R.sub.i,j (1)-R.sub.i',j (1))-(R.sub.i,j' (1)-R.sub.i',j' (1)) for the pair of satellites number i and i' and the pair of receiver antennas number j and j', assuming that d≈0 for this determination; (6) for measuring the effective phase β ij;cp (1) from satellite number i of a carrier signal with effective frequency f=f cp received at receiver antenna number j(i=1,2, . . . , n; j=1, 2); (7) for forming double difference phase variables Φ.sub.i,i';j,j';cp (1)=[β.sub.i,j;cp (1)-β.sub.i',j;cp (1)]-[β.sub.i,j';cp (1)-β.sub.i',j';cp (1)] for the satellites number i and i' and the receiver antennas number j and j', using the effective signal frequency f=f cp ; (8) for determining phase integer estimates N e;i ,i';j,j' by the relations N.sub.e;i,i';j,j' (1)=ΔR.sub.i,i';j,j' (1)/λ.sub.cp +Φ.sub.i,i';j,j';cp (1); (9) for performing steps (2)-(8) for the roving receiver antenna located at a second roving receiver antenna position r(rov, -d, L) that also lies on the line L, where the reference receiver antenna position lies between, and equidistant from, the first and second roving receiver antenna positions on the line L, to obtain the double difference range values ΔR i ,i';j,j' (2) defined by ΔR.sub.i,i';j,j' (2)=(R.sub.i,j (2)-R.sub.i',j (2))-(R.sub.i,j' (2)-R.sub.i',j' (2)), measured carrier phase variables β i ,j;cp (2), double difference phase variables Φ i ,i';j,j';cp (2) defined by Φ.sub.i,i';j,j';cp (2)=[β.sub.i,j;cp (2)-β.sub.i',j;cp(2) ]-[β.sub.i,j';cp (2)-β.sub.i',j';cp(2) ], and phase integer estimates N e;i ,i';j,j' (2) given by N.sub.e;i,i';j,j' (2)=ΔR.sub.i,i';j,j';cp (2)/λ.sub.cp +Φ.sub.i,i';j,j';cp (2); and (10) for determining the desired phase integers for the reference and roving receiver antennas by the relations N.sub.i,i';j,j' =floor[(N.sub.e;i,i';j,j' (1)+N.sub.e;i,i';j,j' (2))/2 +0.5].

Claim 23

Apparatus for determining and removing the phase integer ambiguities associated with determination of position of a receiver that is part of a Satellite Positioning System (SPS), the apparatus comprising: a reference receiver and associated antenna for receiving SPS signals from a plurality of n satellites (n≧4), numbered i=1, 2, . . . , n, that are part of the SPS, where the position of the reference receiver antenna is known with sufficient accuracy; a roving receiver and associated antenna for receiving SPS signals from the plurality of n satellites that are part of the SPS, where each satellite transmits a predetermined signal at a predetermined sequence of times at each of at least two carrier signal frequencies f L1 =c/λ L1 and f L2 =c/λ L2 , where c is the velocity of light; antenna positioning means for positioning the roving receiver antenna at a first selected position and then at a second selected position on a selected line containing the reference receiver antenna, where the reference receiver antenna lies between and at a distance d from the first and the second selected roving receiver antenna positions, where d need not be determined; signal processing means, connected to the reference receiver antenna and to the roving receiver antenna: (1) for determining an effective frequency f cp =α 1 f L1 +α 2 f L2 =c/α cp , expressed as a selected linear combination of the two carrier frequencies f L1 and f L2 , for the Satellite Positioning System, where λ cp is the corresponding effective wavelength; (2) for receiving the SPS signals received by the reference receiver antenna and by the roving receiver antenna at the first selected position and the second selected position; (3) for determining a plurality of n theoretical distance values R 1 ,1, R 2 ,1, . . . , R n ,1 from each of the n satellites to the roving receiver antenna at the sequence of predetermined times, where R I ,J is the distance between satellite number I, with Cartesian coordinates (x I ,y I ,z I ), and receiver antenna number J, with coordinates (x J ,y J , J ) (I=1, 2, . . . , n; J=1, 2), given by R I ,J =[(x I -x J ) 2 +(y I -y J ) 2 +(z I -z J ) 2 ] 1/2 , where each theoretical distance value is determined from a knowledge of the position of a satellite and of an estimate of a position of the roving receiver antenna, assuming that d≈0 for this determination; (4) for determining a plurality of n theoretical distance values R 1 ,2, R 2 ,2, . . . , R n ,2 from each of the satellites to the reference receiver antenna at the sequence of predetermined times, where each theoretical distance value is determined from a knowledge of the position of a satellite and of the position of the reference receiver antenna, where the position of the reference receiver antenna is known with sufficient accuracy at each of the predetermined sequence of times; (5) for forming theoretical range double difference values ΔR i ,i';j,j' (1) (i,i'=1,2, . . . , n; i≠i'; j,j'=1,2; j≠j') for the reference and roving receiver antenna positions, defined by the relations ΔR.sub.i,i';j,j' (1)=(R.sub.i,j (1)-R.sub.i',j (1))-(R.sub.i,j' (1)-R.sub.i',j' (1)) for the pair of satellites number i and i' and the pair of receiver antennas number j and j', assuming that d≈0 for this determination; (6) for measuring the effective phase β ij;cp (1) from satellite number i of a carrier signal with effective frequency f=f cp received at receiver antenna number j(i=1,2, . . . , n; j=1, 2); (7) for forming double difference phase variables Φ.sub.i,i';j,j';cp (1)=[β.sub.i,j;cp (1)-β.sub.i',j;cp (1)]-[β.sub.i,j';cp (1)-β.sub.i',j';cp (1)] for the satellites number i and i' and the receiver antennas number j and j', using the effective signal frequency f=f cp ; (8) for determining phase integer estimates N e;i ,i';j,j' by the relations N.sub.e;i,i';j,j' (1)=ΔR.sub.i,i';j,j' (1)/λ.sub.cp +Φ.sub.i,i';j,j';cp (1); (9) for performing steps (2)-(8) for the roving receiver antenna located at a second roving receiver antenna position r(rov, -d, L) that also lies on the line L, where the reference receiver antenna position lies between, and equidistant from, the first and second roving receiver antenna positions on the line L, to obtain the double difference range values ΔR i ,i';j,j' (2) defined by ΔR.sub.i,i';j,j' (2)=(R.sub.i,j (2)-R.sub.i',j (2))-(R.sub.i,j' (2)-R.sub.i',j' (2)), measured carrier phase variables β i ,j;cp (2), double difference phase variables Φ i ,i';j,j';cp (2) defined by Φ.sub.i,i';j,j';cp (2)=[β.sub.i,j;cp (2)-β.sub.i',j;cp(2) ]-[β.sub.i,j';cp (2)-β.sub.i',j';cp(2) ], and phase integer estimates N e;i ,i';j,j' (2) given by N.sub.e;i,i';j,j' (2)=ΔR.sub.i,i';j,j';cp (2)/λ.sub.cp +Φ.sub.i,i';j,j';cp (2); and (10) for determining the desired phase integers for the reference and roving receiver antennas by the relations N.sub.i,i';j,j' =floor[(N.sub.e;i,i';j,j' (1)+N.sub.e;i,i';j,j' (2))/2 +0.5].

Claim 24

The apparatus of claim 16, wherein said integer n is equal to four.

Claim 25

The apparatus of claim 16, wherein said signal processor also solves n-1 simultaneous equations given by Φ.sub.i,i';1,2 =-ΔR.sub.i,i';1,2 /λ.sub.cp +N.sub.i,i';1,2, (i and i'=1,2, . . . , n; i≠i') to obtain position coordinates (x 2 ,y 2 ,z 2 ) of a roving receiver antenna located at an arbitrary position, when the coordinates (x 1 ,y 1 ,z 1 ) of the reference receiver antenna are known with sufficient accuracy.

Claim 26

The apparatus of claim 16, wherein said effective frequency, and said corresponding effective phases β i ,i';j,j';cp, is drawn from the group consisting of the L1 carrier frequency f L1 , the L2 carrier frequency f L2 , the wide lane frequency f w =f L1 f L2 , and the narrow lane frequency f n =f L1 +f L2 .

Claim 27

Apparatus for determining and removing the phase integer ambiguities associated with determination of position of a receiver that is part of a Satellite Positioning System (SPS), the apparatus comprising: a reference receiver and associated antenna for receiving SPS signals from a plurality of n satellites (n≧4), numbered i=1, 2, . . . , n, that are part of the SPS, where the position of the reference receiver antenna is known with sufficient accuracy; a roving receiver and associated antenna for receiving SPS signals from the plurality of n satellites that are part of the SPS, where each satellite transmits a predetermined signal at a predetermined sequence of times at each of at least two carrier signal frequencies f L1 =c/λ L1 and fL2=c/λ L2 , where c is the velocity of light; antenna positioning means for positioning the roving receiver antenna at a selected position spaced apart from the reference receiver antenna position by a distance d that is less than λ cp /2, where d need not be otherwise determined and λ cp is an effective wavelength defined below; signal processing means, connected to the reference receiver antenna and to the roving receiver antenna: (1) for determining a frequency f cp =α 1 f L1 +α 2 f L2 =c/λ cp , expressed as a selected linear combination of the two carrier frequencies f L1 and f L2 , for the Satellite Positioning System, where λ cp is the corresponding effective wavelength; (2) for receiving the SPS signals received by the reference receiver antenna and by the roving receiver antenna at a selected position; (3) for determining a plurality of n theoretical distance values R 1 ,1, R 2 ,1, . . . , R n ,1 from each of the n satellites to the roving receiver antenna at the sequence of predetermined times, where R I ,J is the distance between satellite number I, with Cartesian coordinates (x I ,y I ,z I ), and receiver antenna number J, with coordinates (x J ,y J ,z J ) (I=1, 2, . . . , n; J=1, 2), given by R I ,J =[(x I -x J ) 2 +(y I -y J ) 2 +(z I -z J ) 2 ] 1/2 , where each theoretical distance value is determined from a knowledge of the position of a satellite and of an estimate of a position of the roving receiver antenna, assuming that d≈ 0 for this determination; (4) for determining a plurality of n theoretical distance values R 1 ,2, R 2 ,2, . . . , R n ,2 from each of the satellites to the reference receiver antenna at the sequence of predetermined times, where each theoretical distance value is determined from a knowledge of the position of a satellite and of the position of the reference receiver antenna, where the position of the reference receiver antenna is known with sufficient accuracy at each of the predetermined sequence of times; (5) for forming theoretical range double difference values ΔR i ,i';j,j' (i,i'=1,2, . . . , n; i≠i'; j,j'=1,2; j≠j') for the reference and roving receiver antenna positions, defined by the relations ΔR.sub.i,i';j,j' =(R.sub.i,j -R.sub.i',j)-(R.sub.i,j' -R.sub.i',j') for the pair of satellites number i and i' and the pair of receiver antennas number j and j'; (6) for measuring the effective phase β ij;cp from satellite number i of a carrier signal with effective frequency f=f cp received at receiver antenna number j (i=1,2, . . . , n; j =1, 2); (7) for forming double difference phase variables Φ.sub.i,i';j,j';cp =[β.sub.i,j;cp -β.sub.i',j;cp ]-[β.sub.i,j';cp 1728 β.sub.i',j';cp ] for the satellites number i and i' and the receiver antennas number j and j', using the signal frequency f=f cp ; (8) for determining phase integer estimates N e;i ,i';j,j' by the relations N.sub.e;i,i';j,j' =ΔR.sub.i,i';j,j' /λ.sub.cp +Φ.sub.i,i';j,j';cp ; and (9) for determining the desired phase integers for the reference and roving receiver antennas by the relations N.sub.i,i';j,j' =floor[N.sub.e;i,i';j,j' +0.5].

Claim 28

Apparatus for determining and removing the phase integer ambiguities associated with determination of position of a receiver that is part of a Satellite Positioning System (SPS), the apparatus comprising: a reference receiver and associated antenna for receiving SPS signals from a plurality of n satellites (n≧4), numbered i=1, 2, . . . , n, that are part of the SPS, where the position of the reference receiver antenna is known with sufficient accuracy; a roving receiver and associated antenna for receiving SPS signals from the plurality of n satellites that are part of the SPS, where each satellite transmits a predetermined signal at a predetermined sequence of times at each of at least two carrier signal frequencies f L1 =c/λ L1 and fL2=c/λ L2 , where c is the velocity of light; antenna positioning means for positioning the roving receiver antenna at a selected position spaced apart from the reference receiver antenna position by a distance d that is less than λ cp /2, where d need not be otherwise determined and λ cp is an effective wavelength defined below; signal processing means, connected to the reference receiver antenna and to the roving receiver antenna: (1) for determining a frequency f cp =α 1 f L1 +α 2 f L2 =c/λ cp , expressed as a selected linear combination of the two carrier frequencies f L1 and f L2 , for the Satellite Positioning System, where λ cp is the corresponding effective wavelength; (2) for receiving the SPS signals received by the reference receiver antenna and by the roving receiver antenna at a selected position; (3) for determining a plurality of n theoretical distance values R 1 ,1, R 2 ,1, . . . , R n ,1 from each of the n satellites to the roving receiver antenna at the sequence of predetermined times, where R I ,J is the distance between satellite number I, with Cartesian coordinates (x I ,y I ,z I ), and receiver antenna number J, with coordinates (x J ,y J ,z J ) (I=1, 2, . . . , n; J=1, 2), given by R I ,J =[(x I -x J ) 2 +(y I -y J ) 2 +(z I -z J ) 2 ] 1/2 , where each theoretical distance value is determined from a knowledge of the position of a satellite and of an estimate of a position of the roving receiver antenna, assuming that d≈ 0 for this determination; (4) for determining a plurality of n theoretical distance values R 1 ,2, R 2 ,2, . . . , R n ,2 from each of the satellites to the reference receiver antenna at the sequence of predetermined times, where each theoretical distance value is determined from a knowledge of the position of a satellite and of the position of the reference receiver antenna, where the position of the reference receiver antenna is known with sufficient accuracy at each of the predetermined sequence of times; (5) for forming theoretical range double difference values ΔR i ,i';j,j' (i,i'=1,2, . . . , n; i≠i'; j,j'=1,2; j≠j') for the reference and roving receiver antenna positions, defined by the relations ΔR.sub.i,i';j,j' =(R.sub.i,j -R.sub.i',j)-(R.sub.i,j' -R.sub.i',j') for the pair of satellites number i and i' and the pair of receiver antennas number j and j'; (6) for measuring the effective phase β ij;cp from satellite number i of a carrier signal with effective frequency f=f cp received at receiver antenna number j (i=1,2, . . . , n; j =1, 2); (7) for forming double difference phase variables Φ.sub.i,i';j,j';cp =[β.sub.i,j;cp -β.sub.i',j;cp ]-[β.sub.i,j';cp 1728 β.sub.i',j';cp ] for the satellites number i and i' and the receiver antennas number j and j', using the signal frequency f=f cp ; (8) for determining phase integer estimates N e;i ,i';j,j' by the relations N.sub.e;i,i';j,j' =ΔR.sub.i,i';j,j' /λ.sub.cp +Φ.sub.i,i';j,j';cp ; and (9) for determining the desired phase integers for the reference and roving receiver antennas by the relations N.sub.i,i';j,j' =floor[N.sub.e;i,i';j,j' +0.5].

Claim 29

The apparatus of claim 20, wherein said integer n is equal to four.

Claim 30

The apparatus of claim 20, wherein said signal processor also solves n-1 simultaneous equations given by Φ.sub.i,i'; 1,2=-ΔR.sub.i,i';1,2 /λ.sub.cp +N.sub.i,i';1,2, (i and i'=1,2, . . . , n;i≠i') to obtain position coordinates (x 2 ,y 2 ,z 2 ) of a roving receiver antenna located at an arbitrary position, when the coordinates (x 1 ,y 1 ,z 1 ) of the reference receiver antenna are known with sufficient accuracy.

Claim 31

The apparatus of claim 20, wherein said effective frequency, and said corresponding effective phases β i ,i';j,j';cp, is drawn from the group consisting of the L1 carrier frequency f L1 , the L2 carrier frequency f L2 , the wide lane frequency f w =f L1 -f L2 , and the narrow lane frequency f n =f L1 +f L2 .

Claim 32

Apparatus for determining and removing the phase integer ambiguities associated with determination of position of a receiver that is part of a Satellite Positioning System (SPS), the apparatus comprising: a reference receiver, denoted receiver number 1, and associated antenna for receiving SPS signals from a plurality of n satellites (n≧4), numbered i=1, 2, . . . , n, that are part of the SPS, where the position of the reference receiver antenna is known with sufficient accuracy; a roving receiver, denoted receiver number 2, and associated antenna for receiving SPS signals from the plurality of n satellites that are part of the SPS, where each satellite transmits a predetermined signal at a predetermined sequence of times at each of at least two carrier signal frequencies f L1 =c/λ L1 and fL2=c/λ L2 , where c is the velocity of light, where the roving receiver antenna position r(rov, d, L) is displaced from the reference receiver antenna position by a known distance d in a first direction along a selected line L containing the reference receiver antenna position; antenna positioning means for positioning the roving receiver antenna at a selected position spaced apart from the reference receiver antenna position by a selected distance d along the line L, where the line L makes a known azimuthal angle Ψ with respect to a chosen north direction; signal processing means, connected to the reference receiver antenna and to the roving receiver antenna: (1) for determining, from the known geodetic coordinates (latitude 1 , longitude 1 , altitude 1 ) of the reference receiver antenna position, the geodetic coordinates of the roving receiver antenna (latitude 2 , longitude 2 , altitude 2 ) according to the relations latitude.sub.2 =latitude.sub.1 +(360° d cos Ψ)/(2πR), longitude.sub.2 =longitude.sub.1 +(360° d sin Ψ)/(2πR cos(latitude.sub.1)), altitude.sub.2 =altitude .sub.1 ; (2) for determining an effective frequency f cp =α 1 f L1 +α 2 f L2 =c/λ cp , expressed as a selected linear combination of the two carrier frequencies f L1 =c/λ L1 and f L2 =c/λ L2 , for the Satellite Positioning System, where c is the velocity of light and λ cp is the corresponding effective wavelength; (3) for determining a plurality of n theoretical distance values R 1 ,2, R 2 ,2, . . . , R n ,2 from each of the satellites to the reference receiver antenna at the sequence of predetermined times, where each theoretical distance value is determined from a knowledge of the position of a satellite and of the position of the reference receiver antenna, where the position of the reference receiver antenna is known with sufficient accuracy at each of the predetermined sequence of times; (4) for determining a plurality of n theoretical distance values R 1 ,1, R 2 ,1, . . . , R n ,1 from each of the n satellites to the roving receiver antenna at the sequence of predetermined times, using the geodetic coordinates for this antenna determined in step (1), where R I ,J is the distance between satellite number I, with Cartesian coordinates (x I ,y I ,z I ), and receiver antenna number J, with coordinates (x J ,y J ,z J ) (I=1, 2, . . . , n; J=1, 2), given by R I ,J =[(x I -x J ) 2 +(y I -y J ) 2 +(z I -z J ) 2 ] 1/2 ; (5) for forming theoretical range double difference values ΔR i ,i';j,j' (1) (i,i'=1,2, . . . , n; i≠i'; j,j'=1,2; j≠j') for the reference and roving receiver antenna positions, defined by the relations ΔR.sub.i,i';j,j' =(R.sub.i,j -R.sub.i',j)-(R.sub.ij' -R.sub.i',j') for the pair of satellites number i and i' and the pair of receiver antennas number j and j'; (6) for measuring the effective phase β ij;cp from satellite number i of a signal with effective frequency f=f cp received at receiver antenna number j (i=1,2, . . . , n; j=1, 2); (7) for forming double difference phase variables Φ.sub.i,i';j,j';cp =[β.sub.i,j;cp -β.sub.i',j;cp ]-[β.sub.i,j';cp -β.sub.i',j';cp ] for the satellites number i and i' and the receiver antennas number j and j', using the effective signal frequency f=f cp ; (8) for determining phase integer estimates N e;i ,i';j,j' by the relations N.sub.e;i,i';j,j' =ΔR.sub.i,i';j,j' /λ.sub.cp +Φ.sub.i,i';j,j';cp ; and (9) for determining the desired phase integers for the reference and roving receiver antennas by the relations N.sub.i,i';j,j' =floor[N.sub.e;i,i';j,j' +0.5].

Claim 33

Apparatus for determining and removing the phase integer ambiguities associated with determination of position of a receiver that is part of a Satellite Positioning System (SPS), the apparatus comprising: a reference receiver, denoted receiver number 1, and associated antenna for receiving SPS signals from a plurality of n satellites (n≧4), numbered i=1, 2, . . . , n, that are part of the SPS, where the position of the reference receiver antenna is known with sufficient accuracy; a roving receiver, denoted receiver number 2, and associated antenna for receiving SPS signals from the plurality of n satellites that are part of the SPS, where each satellite transmits a predetermined signal at a predetermined sequence of times at each of at least two carrier signal frequencies f L1 =c/λ L1 and fL2=c/λ L2 , where c is the velocity of light, where the roving receiver antenna position r(rov, d, L) is displaced from the reference receiver antenna position by a known distance d in a first direction along a selected line L containing the reference receiver antenna position; antenna positioning means for positioning the roving receiver antenna at a selected position spaced apart from the reference receiver antenna position by a selected distance d along the line L, where the line L makes a known azimuthal angle Ψ with respect to a chosen north direction; signal processing means, connected to the reference receiver antenna and to the roving receiver antenna: (1) for determining, from the known geodetic coordinates (latitude 1 , longitude 1 , altitude 1 ) of the reference receiver antenna position, the geodetic coordinates of the roving receiver antenna (latitude 2 , longitude 2 , altitude 2 ) according to the relations latitude.sub.2 =latitude.sub.1 +(360° d cos Ψ)/(2πR), longitude.sub.2 =longitude.sub.1 +(360° d sin Ψ)/(2πR cos(latitude.sub.1)), altitude.sub.2 =altitude .sub.1 ; (2) for determining an effective frequency f cp =α 1 f L1 +α 2 f L2 =c/λ cp , expressed as a selected linear combination of the two carrier frequencies f L1 =c/λ L1 and f L2 =c/λ L2 , for the Satellite Positioning System, where c is the velocity of light and λ cp is the corresponding effective wavelength; (3) for determining a plurality of n theoretical distance values R 1 ,2, R 2 ,2, . . . , R n ,2 from each of the satellites to the reference receiver antenna at the sequence of predetermined times, where each theoretical distance value is determined from a knowledge of the position of a satellite and of the position of the reference receiver antenna, where the position of the reference receiver antenna is known with sufficient accuracy at each of the predetermined sequence of times; (4) for determining a plurality of n theoretical distance values R 1 ,1, R 2 ,1, . . . , R n ,1 from each of the n satellites to the roving receiver antenna at the sequence of predetermined times, using the geodetic coordinates for this antenna determined in step (1), where R I ,J is the distance between satellite number I, with Cartesian coordinates (x I ,y I ,z I ), and receiver antenna number J, with coordinates (x J ,y J ,z J ) (I=1, 2, . . . , n; J=1, 2), given by R I ,J =[(x I -x J ) 2 +(y I -y J ) 2 +(z I -z J ) 2 ] 1/2 ; (5) for forming theoretical range double difference values ΔR i ,i';j,j' (1) (i,i'=1,2, . . . , n; i≠i'; j,j'=1,2; j≠j') for the reference and roving receiver antenna positions, defined by the relations ΔR.sub.i,i';j,j' =(R.sub.i,j -R.sub.i',j)-(R.sub.ij' -R.sub.i',j') for the pair of satellites number i and i' and the pair of receiver antennas number j and j'; (6) for measuring the effective phase β ij;cp from satellite number i of a signal with effective frequency f=f cp received at receiver antenna number j (i=1,2, . . . , n; j=1, 2); (7) for forming double difference phase variables Φ.sub.i,i';j,j';cp =[β.sub.i,j;cp -β.sub.i',j;cp ]-[β.sub.i,j';cp -β.sub.i',j';cp ] for the satellites number i and i' and the receiver antennas number j and j', using the effective signal frequency f=f cp ; (8) for determining phase integer estimates N e;i ,i';j,j' by the relations N.sub.e;i,i';j,j' =ΔR.sub.i,i';j,j' /λ.sub.cp +Φ.sub.i,i';j,j';cp ; and (9) for determining the desired phase integers for the reference and roving receiver antennas by the relations N.sub.i,i';j,j' =floor[N.sub.e;i,i';j,j' +0.5].

Claim 34

The apparatus of claim 24, wherein said integer n is equal to four.

Claim 35

The apparatus of claim 24, wherein said signal processor also solves n-1 simultaneous equations given by Φ.sub.i,i';1,2 =-ΔR.sub.i,i';1,2 /λ.sub.cp +N.sub.i,i';1,2, (i and i'=1,2, . . . , n; i≠i') to obtain position coordinates (x 2 ,y 2 ,z 2 ) of a roving receiver antenna located at an arbitrary position, when the coordinates (x 1 ,y 1 ,z 1 ) of the reference receiver antenna are known with sufficient accuracy.

Claim 36

The apparatus of claim 24, wherein said effective frequency, and said corresponding effective phases β i ,i';j,j';cp, is drawn from the group consisting of the L1 carrier frequency f L1 , the L2 carrier frequency f L2 , the wide lane frequency f w =f L1 -f L2 , and the narrow lane frequency f n =f L1 +f L2 .