Inverse differential corrections for SATPS mobile stations

A method for determining two-dimensional spatial location coordinates and clock bias with enhanced accuracy for a mobile user Satellite Positioning System (SATPS) station, the method comprising the steps of:

• providing an SATPS reference station, having location coordinates that are known with high accuracy, the reference station having an SATPS signal antenna and associated SATPS signal receiver/processor for receiving and processing SATPS signals from a plurality of SATPS satellites to determine spatial location and clock bias coordinates for that station;
• providing an SATPS station that is mobile, having an SATPS signal antenna and associated SATPS signal receiver/processor for receiving and processing SATPS signals from a plurality of SATPS satellites to determine spatial location and clock bias coordinates for that station;
• providing the mobile station with a radiowave transmitter and providing the reference station with a radiowave receiver so that the mobile station can transmit signals at a selected frequency that are received by the reference station;
• causing each of the reference station and the mobile station to receive SATPS signals from a plurality of M SATPS satellites (M.gtoreq.3) in common with each other, and to determine the pseudorange values PR(t;i;j) as a function of time t for that station, numbered i (i=reference, i=mobile), from the signals received from the M common SATPS satellites numbered j=j1, j2, . . . , jM;
• causing the reference station to determine pseudorange corrections .DELTA.PR(t;i;j) that, when added to the pseudorange values PR(t;i;j) measured at the reference station, produce spatial location and clock bias coordinates that agree with the known spatial location and clock bias coordinates for the reference station;
• providing the mobile station with the spatial location and clock bias coordinates of a nominal solution of the pseudorange equations
• {(x0(t)-x.sub.j (t)).sup.2 +(y0(t)-y.sub.j (t)).sup.2 }.sup.1/2 +b0(t)=PR(t;i;j),
• where (x0(t),y0(t)) are the spatial location coordinates and b0(t) is the clock bias for this nominal solution, and (xj(t),yj(t)) are the location coordinates of satellite number j;
• determining the spatial location and clock bias coordinates (x',y',b') for the mobile station from the SATPS signals received at that station;
• forming a 3.times.1 matrix .DELTA.W(t;
• mobile)={x'-x0 y'-y0 b'-b0}.sup.tr, having matrix entries that are offset from the mobile station coordinates values (x',y',b') by the nominal solution coordinates values (x0,y0,b0);
• transmitting the entries of the matrix .DELTA.W(t;mobile) to the reference station;
• providing at the reference station an M.times.3 matrix H(t) whose entries are defined by the relations
• H(t)=(h.sub.ji (t)),
• h.sub.j1 (t)=.alpha..sub.j1 (t)=(x0-x.sub.j (t))/r.sub.0j,
• h.sub.j2 (t)=.alpha..sub.j2 (t)=(y0-y.sub.j (t))/r.sub.0j,
• h.sub.j3 (t)=1, (j=1, 2, 3, . . . , M) ,
• r.sub.0j (t)={(x0-x.sub.j (t)).sup.2 +(y0-y.sub.j (t)).sup.2 }.sup.1/2 ;
• augmenting the matrix H(t) by adding M-3 rows of matrix entries to produce an augmented M.times.M matrix H*(t) for the reference station, where three columns of the matrix H*(t) are identical with three columns of the matrix H(t) and each of the columns of H*(t) forms a 1.times.M column matrix that is independent of the other M-1 column matrices of the matrix H*(t);
• forming an inverse matrix H*(t).sup.-1, determining an M.times.1 matrix .DELTA.W* by the matrix equation .DELTA.W*(t)=H*(t).sup.-1 .DELTA.PR(t), where .DELTA.PR(t) is an M.times.1 matrix whose entries are the values .DELTA.PR(t;i;j) (j=1, 2, . . . , M) for the reference station;
• interpreting three entries w.sub.1 *(t), w.sub.2 *(t) and w.sub.3 *(t) of the matrix .DELTA.W*(t) as corrections to the location coordinates and clock bias values (x',y',b') for the mobile station;
• and
• determining the corrected location coordinates (x,y) and corrected clock bias coordinate b for the mobile station using the relations
• x(t)=x'(t)+w.sub.1 *(t),
• y(t)=y'(t)+w.sub.2 *(t),
• b(t)=b'(t)+w.sub.3 *(t).

A method for determining spatial location coordinates with enhanced accuracy for a mobile user Satellite Positioning System (SATPS) station, the method comprising the steps of:

• providing an SATPS reference station, having location coordinates that are known with high accuracy, the reference station having an SATPS signal antenna and associated SATPS signal receiver/processor for receiving and processing SATPS signals from a plurality of SATPS satellites to determine spatial location and clock bias coordinates for that station;
• providing an SATPS station that is mobile, having an SATPS signal antenna and associated SATPS signal receiver/processor for receiving and processing SATPS signals from a plurality of SATPS satellites to determine spatial location and clock bias coordinates for that station;
• providing the mobile station with a radiowave transmitter and providing the reference station with a radiowave receiver so that the mobile station can transmit signals at a selected frequency that are received by the reference station;
• causing each of the reference station and the mobile station to receive SATPS signals from a plurality of M SATPS satellites (M.gtoreq.3) in common with each other, and to determine the pseudorange values PR(t;i;j) as a function of time t for that station, numbered i (i=reference, i=mobile), from the signals received from the M common SATPS satellites numbered j=j1, j2, . . . , jM;
• causing the reference station to determine pseudorange corrections .DELTA.PR(t;i;j) that, when added to the pseudorange values PR(t;i;j) measured at the reference station, produce spatial location and clock bias coordinates that agree with the known spatial location and clock bias coordinates for the reference station;
• providing the mobile station with the spatial location and clock bias coordinates of a nominal solution of the pseudorange equations
• {(x0(t)-x.sub.j (t)).sup.2 +(y0(t)-y.sub.j (t)).sup.2 +(z0(t)-z.sub.j (t)).sup.2 }.sup.1/2 =PR(t;i;j),
• where (x0(t),y0(t),z0(t)) are the spatial location coordinates for this nominal solution, and (x.sub.j (t),y.sub.j (t),z.sub.j (t)) are the location coordinates of satellite number j;
• determining the spatial location coordinates (x',y',z') for the mobile station from the SATPS signals received at that station;
• forming a 3.times.1matrix .DELTA.W(t;
• mobile)={x'-x0 y'-y0 z'-z0 }.sup.tr, having matrix entries that are offset from the mobile station coordinates values (x',y',z') by the nominal solution coordinates values (x0,y0,z0);
• transmitting the entries of the matrix .DELTA.W(t;mobile) to the reference station;
• providing at the reference station an M.times.3 matrix H(t) whose entries are defined by the relations
• H(t)=(h.sub.ji (t)),
• h.sub.j1 (t)=.alpha..sub.j1 (t)=(x0-x.sub.j (t))/r.sub.0j,
• h.sub.j2 (t)=.alpha..sub.j2 (t)=(y0-y.sub.j (t))/r.sub.0j,
• h.sub.j3 (t)=.alpha..sub.j3 (t)=(z0-z.sub.j (t))/r.sub.0j,
• r.sub.0j (t)={(x0-x.sub.j (t)).sup.2 +(y0-y.sub.j (t)).sup.2 +(z0-zj(t)).sup.2 }.sup.1/2 ;
• augmenting the matrix H(t) by adding M-3 columns of matrix entries to produce an augmented M.times.M matrix H*(t) for the reference station, where three columns of the matrix H*(t) are identical with three columns of the matrix H(t) and each of the columns of H*(t) forms an M.times.1 column matrix that is independent of the other M.times.1 column matrices of the matrix H*(t);
• forming an inverse matrix H*(t).sup.-1, determining an M.times.1 matrix .DELTA.W* by the matrix equation .DELTA.W*(t)=H*(t).sup.-1 .DELTA.PR(t), where .DELTA.PR(t) is an M.times.1 matrix whose entries are the values .DELTA.PR(t;i;j) (j=1, 2, . . . , M) for the reference station;
• interpreting three entries w.sub.1 *(t), w.sub.2 *(t) and w.sub.3 *(t) of the matrix .DELTA.W*(t) as corrections to the location coordinates (x',y',z') for the :mobile station;
• and
• determining the corrected location coordinates (x,y,z) for the mobile station using the relations
• x(t)=x'(t)+w.sub.1 *(t),
• y(t)=y'(t)+w.sub.2 *(t),
• z(t)=z'(t)+w.sub.3 *(t).

A method for determining spatial location and clock bias coordinates with enhanced accuracy for a mobile user Satellite Positioning System (SATPS) station, the method comprising the steps of:

• providing an SATPS reference station, having location coordinates that are known with high accuracy, the reference station having an SATPS signal antenna and associated SATPS signal receiver/processor for receiving and processing SATPS signals from a plurality of SATPS satellites to determine spatial location and clock bias coordinates for that station;
• providing an SATPS station that is mobile, having an SATPS signal antenna and associated SATPS signal receiver/processor for receiving and processing SATPS signals from a plurality of SATPS satellites to determine spatial location and clock bias coordinates for that station;
• providing the mobile station with a radiowave transmitter and providing the reference station with a radiowave receiver so that the mobile station can transmit signals at a selected frequency that are received by the reference station;
• causing each of the reference station and the mobile station to receive SATPS signals from a plurality of M SATPS satellites (M.gtoreq.5) in common with each other, and to determine the pseudorange values PR(t;i;j) as a function of time t for that station, numbered i (i=reference, or i=mobile), from the signals received from the M common SATPS satellites, numbered j=j1, j2, . . . , jM, where the satellites j=j1, j2, j3, j4 form a fundamental set of SATPS satellites used for determination of the location of the mobile station;
• determining spatial location and clock bias coordinates (x',y',z',b') for the mobile station from the SATPS signals received at that station;
• causing the reference station to determine pseudorange corrections .DELTA.PR(t;ref;j) that, when added to the pseudorange values PR(t;ref;j) measured at the reference station, produce spatial location and clock bias coordinates that agree with the known spatial location and clock bias coordinates for the reference station;
• providing at the reference and mobile stations an M.times.4 matrix H(t;ref), determined from the reference station location coordinates, whose entries are defined by the relations
• H(t;ref)=(h.sub.ji (t)),
• h.sub.j1 (t)=.alpha..sub.j1 (t)=(x0-x.sub.j (t))/r.sub.0j,
• h.sub.j2 (t)=.alpha..sub.j2 (t)=(y0-y.sub.j (t))/r.sub.0j,
• h.sub.j3 (t)=.alpha..sub.j3 (t)=(z0-z.sub.j (t))/r.sub.0j,
• h.sub.j4 (t)=1, (j=1, 2, 3, . . . , M),
• r.sub.0j (t)={(x0-x.sub.j (t)).sup.2 +(y0-y.sub.j (t)).sup.2 +(z0-z.sub.j (t)).sup.2 }.sup.1/2 ;
• decomposing the matrix H(t;ref) into a 4.times.4 matrix H.sub.f =(h.sub.ji).sub.f (j=j1, j2, j3, j4) and an (M-4).times.4 matrix H.sub.nf =(h.sub.ji).sub.nf (j=j5, . . . , jM);
• forming an M.times.1 matrix v={I-H(H.sup.tr H).sup.-1 H.sup.tr } PR(t;mobile) at the mobile station, where PR(t;mobile) is an M.times.1 vector whose ordered entries are the pseudorange values PR(t;mobile;j) (j=j1, j2, . . . , jM);
• decomposing the matrix v into a 4.times.1 column matrix v.sub.f, consisting of the first four entries of the matrix v, and an (M-4).times.1 column matrix v.sub.nf, consisting of the last M-4 entries of the matrix v;
• transmitting the M-4 entries of the matrix v.sub.nf and the mobile station coordinates (x',y',z',b') to the reference station;
• constructing at the reference station the matrix v.sub.f by the relation v.sub.f =-(H.sub.f.sup.tr).sup.-1 H.sub.nf.sup.tr v.sub.nf ;
• reconstructing the vector v={v.sub.f .vertline.v.sub.nf }.sup.tr at the reference station as a concatenation of the vectors v.sub.f and v.sub.nf ;
• reconstructing the matrix of pseudorange values PR(t;mobile) from the vector v at the reference station;
• adding the pseudorange corrections .DELTA.PR(t;ref;j) to the pseudorange values PR(t;mobile;j) for the fundamental set (j=j1, j2, j3, j4) to form corrected pseudorange values PR(t;mobile;j).sub.cor for the fundamental set;
• and
• determining corrected spatial location and clock bias coordinates for the mobile station, using the corrected pseudorange values PR(t;mobile;j).sub.cor for the fundamental set.