INNER CONSTRAINTS FOR A 3-D SURVEY NETWORK

Transcription

1 eospatial Science INNER CONSRAINS FOR A 3-D SURVEY NEWORK hese notes follow closely the developent of inner constraint equations by Dr Willie an, Departent of Building, School of Design and Environent, National University of Singapore (an 5. Consider a set of n observation equations in u unnowns ( n u v is the x is the B is the f is the n vector of residuals u vector of unnowns (or paraeters n u atrix of coefficients (design atrix n vector of nueric ters (constants n is the nuber of equations (or observations u is the nuber of unnowns > in the atrix for v+ Bx = f ( Enforcing the least squares condition leads to the noral equations where Nx = t ( N = B WB and t = B Wf (3 N is the u u coefficient atrix of the noral equations t is the W is the u vector of nueric ters n n weight atrix If N is non-singular, i.e., atrix inversion N and N exists: there is a solution for the unnowns x given by x = N t (4 If N is singular, i.e., N and N does not exist: there is no solution by conventional eans (i.e., by the usual atrix inverse. 6, R.E. Deain Inner Constraints for 3-D Survey Networs

2 eospatial Science he ai of these notes is to explain how we ight obtain least squares solutions where the noral coefficient atrix is singular. Such solutions, using INNER CONSRAINS are nown as FREE NE ADJUSMENS. In least squares adjustents of survey networs, singular sets of noral equations (singular noral coefficient atrices N are ran deficient sets of equations and arise because design atrices B are ran deficient. his will invariably be due to datu defects, which could be: coordinate origin not defined (no fixed points, the networ orientation not defined (no lines with fixed directions in space, no scale (no easured distances or no height datu defined (no points with fixed heights. he datu defects are directly connected to the ran deficiency of the design atrix B. A way to overcoe this datu deficiency and find a solution is to ipose constraints, and these constraints tae the for of a set of CONSRAIN EQUAIONS C is the c u atrix of coefficients g is the c vector of nueric ters (constants c is the nuber of constraint equations Cx = g (5 Cobining the constraint equations (5 with the observation equations ( and enforcing the least squares condition ( ϕ = vwv Cx g iniu (6 where W is the n n weight atrix and is the c vector of Lagrange ultipliers, gives rise to the set of equations N C x t = C g (7 Equations (7 can be solved provided that C consists of linearly independent rows that overcoe the datu probles. For a particular networ with the axiu nuber of datu defects (e.g., a height networ with no fixed points, a -D networ of distances and directions with no fixed points 6, R.E. Deain Inner Constraints for 3-D Survey Networs

3 eospatial Science or fixed orientation there are any different constraints that can be iposed to enable a solution for x, and aongst this faily of possible constraints there are MINIMAL CONSRAIN sub-sets, i.e., sub-sets containing the iniu nuber of constraints required for a solution; the iniu nuber being equal to the ran deficiency. Within this group of inial constraint sub-sets there will be a single set called INNER CONSRAINS. his set of inner constraints (inner constraint equations has the unique property that Cx = and a solution of a networ using inner constraints yields x such that the nor of x, denoted as x, is a iniu and the trace of Q xx denoted as tr ( Q xx is also iniu. he trace of a (square atrix is the su of the eleents of the leading diagonal, and Qxx is the (square cofactor atrix containing estiates of the variances adjusted quantities (the eleents of x. Hence, a iniu trace solution has a set { s s s } x x x u s x of the whose su of the u eleents is the sallest, i.e., a iniu variance solution. he nor of a colun (or row vector is the square root of the su of squares of the eleents, i.e., the "length" of the vector. A iniu nor solution would have a vector x having the sallest length. Adjustents of survey networs where no datu is defined and INNER CONSRAINS are eployed, are nown as FREE NE ADJUSMENS, and those eploying MINIMAL CONSRAINS are used are nown as MINIMALLY CONSRAINED ADJUSMENS. In free net adjustents (where every point in the networ is free to ove the inner constraints have the unique property that Cx, i.e., certain linear functions of the adjusted quantities x are equal to zero. hese adjusted quantities can be heights in level networs, or coordinates and theodolite orientation constants in conventional -D and 3-D survey networs, or just coordinates in photograetric and PS networs. In conventional adjustents (where there are enough fixed points to properly define the datu and orientation or inially constrained adjustents, or conventional adjustents with additional constraints, the constraint equations are conditions relating to actual points in the networ. In free net adjustents the constraints are applied to, or relate to, a single "fictitious" point; the centroid of the networ. he ey here is to reeber that the ost convenient way to represent position is via height (in level networs or coordinates in - D and 3-D networs, hence ost networ adjustents are actually variation of coordinate adjustents; height being a -D syste. So, coordinates (approxiate or fixed are required for all 6, R.E. Deain 3 Inner Constraints for 3-D Survey Networs

4 eospatial Science points in the networ even if the coordinate datu or height datu is not fixed, or the orientation is not properly nown. Once these coordinates are assigned then the centroid is defined, since it is just the average of the coordinates of all of the points. We can now decide on certain constraints on the centroid and between the centroid and all the points in the networ that will overcoe datu defects in -D, -D and 3-D survey networs. hese three constraints are: the centroid reains unchanged; the average bearing fro the centroid to all points reains unchanged; the average distance fro the centroid to all points reains unchanged. hese constraints give rise to positional constraints, rotational constraints and a scale constraint. In a free networ adjustent of heights (a -D syste there is one datu defect and so one positional constraint is required and this is derived fro the condition that the centroid reains unchanged. In a free net adjustent of a -D networ (easured directions and distances there are three datu defects, so two positional constraints and one rotational constraint are required. hese are derived fro the condition that the centroid reains unchanged and the average bearing fro the centroid to all points reains unchanged. For 3-D networs (easured horizontal directions, zenith angles and distances there are six datu defects, requiring three positional and three rotational constraints. Again, these are derived fro the condition that the centroid reains unchanged and that the average bearing (in space fro the centroid to all points reains unchanged. For -D and 3-D networs without easured distances, there is an additional scale constraint required which can be derived fro the condition that the average distance fro the centroid to all points reains unchanged. he positional, rotational and scale (inner constraints for a 3-D survey networ are derived as follows. 6, R.E. Deain 4 Inner Constraints for 3-D Survey Networs

5 eospatial Science POSIIONAL CONSRAIN Let ( X, Y, Z be the coordinates of an arbitrary point and there are points in the networ. he coordinates of the centroid are P X X Y Z X+ X + X3+ + X = = = = = ; Y = ; Z = (8 If we ipose the condition that does not ove, then there can be no change to the centroid coordinates during an iterative least squares solution, hence δ X = δy = δz. Now since (,,, X = X X X X the otal Increent heore gives X X X δx = δx + δx + + X X X = δx+ δx + + δx = δ X = δx And siilarly, δy = δy and δ Z= δ Z. Reebering that δ X = δ Y = δ Z =, the = positional constraints are = = δx ; δy ; δz = = = (9 6, R.E. Deain 5 Inner Constraints for 3-D Survey Networs

6 eospatial Science ROAIONAL CONSRAINS he three rotational constraints can be derived by iposing the condition that the average bearing fro the centroid to all points reains unchanged. Figure shows a line in 3-D space fro to an arbitrary point (,, P X Y Z. he line P has length s and projections r = P, q = P and p = P on to the X-Y, Y-Z and X-Z planes respectively. Consider the projection r P on the X-Y plane that has bearing θ. Plane trigonoetry gives = θ X X = tan = θ ( X, Y, X, Y Y Y ( Z P''' p ψ φ s q P X, Y, Z P'' X r θ Y Figure. A line in space and its projection on the X-Y, Y-Z and X-Z planes. P' 6, R.E. Deain 6 Inner Constraints for 3-D Survey Networs

7 eospatial Science Noting that X, Y in equation ( are constant (since the centroid cannot ove, the otal Increent heore gives and the partial derivatives are Hence θ X θ Y ( δθ δ + δ X Y θ cosθ Y Y θ sinθ X X = = ; = = X r r Y r r δθ δ δ ( {( Y Y X ( X X Y } r If the average bearing fro to all points in the networ is to reain unchanged then his is a rotational constraint in the X-Y plane. δθ (3 = Let δθ be the vector = [ δθ + δθ + + δθ ] = s and s contains a single scalar quantity. In equation (3 = [ ] = s. Now let s be the product of two vectors s= r d (4 where r ( Y Y δx ( X X δy ( Y Y δx ( X X δy r = r and d = ( Y Y δx ( X X δy r 6, R.E. Deain 7 Inner Constraints for 3-D Survey Networs

8 eospatial Science Pre-ultiplying both sides of equation (4 by r gives rr is a atrix of order rs = rr d ( rr exists. Pre-ultiplying both sides by ( of full ran, since the eleents of r are independent and the inverse rr gives ( ( ( rr rs = rr rr d = Id = d (5 When s on the left-hand-side of equation (5 is zero (enforcing the condition of equation (3 then d L.H.S. and every eleent of on the R.H.S. will be equal to zero, giving {( Y Y δx ( X X δy}= = (6 Expanding (6 gives = = = = YδX Y δx X δy + X δy ( = = = YδX X δy Y δx + X δy But fro equation (9, the positional constraints, constraint in the X-Y plane can be expressed as δx = δy hence the rotational = = ( YδX XδY (7 = [Equations (6 and (7 are identical to an 5 (eq'ns ( and (3, p. 93] Now, referring to Figure, the projection q = P on the Y-Z plane aes an angle φ with the Z- axis. Plane trigonoetry gives φ = Y Y = tan φ,,, Z Z Noting that X, Y the otal Increent heore gives ( Y Z Y Z (8 6, R.E. Deain 8 Inner Constraints for 3-D Survey Networs

9 eospatial Science φ φ δφ δy + δz Y Z and substituting the partial derivatives and siplifying gives δφ δ δ {( Z Z Y ( Y Y Z } q If the average bearing fro (the centroid to all points in the networ is to reain unchanged then the rotational constraint in the Y-Z plane is δφ (9 = In a siilar anner as before, the rotational constraint in the Y-Z plane can be expressed as ( ZδY YδZ ( = Again, referring to Figure, the projection p = P on the X-Z plane aes an angle ψ with the Z-axis. Plane trigonoetry gives ψ X X = tan = ψ ( X, Z, X, Z Z Z ( In an identical anner as before, the rotational constraint in the X-Z plane is which becoes δψ ( = ( ZδX XδZ (3 = Equations (3, (9 and ( are rotational constraints expressed in angular quantities. Equations (7, ( and (3 are the sae rotational constraints expressed in linear quantities (coordinates and coordinate corrections 6, R.E. Deain 9 Inner Constraints for 3-D Survey Networs

10 eospatial Science SCALE CONSRAIN he distance fro the centroid to an arbitrary point (,, P X Y Z is given by = ( + ( + ( = (,,,,, s X X Y Y Z Z s X Y Z X Y Z (4 Noting that X, Y, Z in equation (4 are constant (since the centroid cannot ove, the otal Increent heore gives s s s δ s δx + δy + δ (5 X Y Z Z he partial derivatives are s X X s Y Y s Z Z = ; = ; = X s Y s Z s and substituting into equation (5 and siplifying gives δs X X δx Y Y δy Z Z s {( + ( + ( Z} δ (6 If the average distance fro (the centroid to all points in the networ is to reain unchanged then δ s (7 = In the sae anner as for the developent of rotational constraints, atrix anipulations can be used to turn the su of sall changes in distances to a su of sall changes in coordinates and equations (6 and (7 iply = {( X X δx + ( Y Y δy + ( Z Z δz} (8 Expanding (8 gives X δ X X δ X + Y δy Y δy + Z δz Z δz = = = = = = ( = = = = X δ X + Y δy + Z δz X δ X Y δy Z δz 6, R.E. Deain Inner Constraints for 3-D Survey Networs

11 eospatial Science But fro equation (9, the positional constraints, constraint can be expressed as δ X = δy = δz = = = hence the scale ( Xδ X + YδY + ZδZ (9 = [Equations (8 and (9 are identical to an 5 (eq'ns ( and (, p. 93] SUMMARY OF CONSRAINS Positional constraints Rotational constraints Scale constraint = δx ; δy ; δz = = = = = = = ( Y δ X X δy ( Z δy Y δz ( Z δ X X δz ( X δ X Y δy Z δz + + Equation (7 is the atrix equation for a constrained least squares solution of a survey networ, where C is the coefficient atrix of the constraint equations Cx = g. In the case of INNER CONSRAINS, Cx and equation (7 becoes N C x t = C (3 6, R.E. Deain Inner Constraints for 3-D Survey Networs

12 eospatial Science In the case of a 3-D networ, with no easured distances (no scale and no fixed points there will be seven datu defects requiring seven constraint equations to overcoe the defects in a free net adjustent. he seven constraints cobine (i three positional constraints, (ii three rotational constraints and (iii one scale constraint. If this networ solution has the coordinate corrections in x in the following sequence the (inner constraint equations [ δ X δy δz δ X δy δz δ X δy δz ] x = Cx have the for δ X δy δ Z δ X δy Y X Y X Y X = δ Z Z Y Z Y Z Y Z X Z X Z X δ X X Y Z X Y Z X Y Z δy δ Z (3 In subsequent notes on this topic, several exaples of free net adjustents will be shown. REFERENCES an, W., 5, 'Inner constraints for 3-D survey networs', Spatial Science, Vol. 5, No., June 5, pp , R.E. Deain Inner Constraints for 3-D Survey Networs