e 
4 
A recursive application of Equation (9) to compute the relation between the coordinates 
of vector P in the coordinate spaces Sy and Sy , results in the equation : 
N 
Py = 2 (Rl; N Wii (10) 
i=M 
where, 
[R]; w is defined in Equation 8. 
= 
is the position vector of origin of coordinate space 3i. 4 
relative to the coordinate system S; . (By definition, 
QM-1, M = Py 
i-1,i 
The differentials of vector PN in Equation 11 can be written using the results arrived 
at in Equation 7. 
N 
PN = Di [R]i.N AS ec T 
> SWi-1,N [R]j;N Tiel, ] (11) 
f=} |i T+1 
Notice that in the previous expression : 
A summation of terms is identically zero when the limits of summation are unbalanced; 
i 
S fü) - 0 it re > L 
i= Kk 
[Rly 1, 
H 
Hd 
[= 
[0 
x 
Il 
[" 
= 0," k>L 
6. ASSEMBLY OF AUXILIARY CONSTRAINTS 
  
Figure 2 contains a diagram showing the general steps in the assembly of a typical 
auxiliary constraint. In the first step, the type of the constraint is analyzed in order to identify 
its vector components. The number of these components varies from one in the case of Single 
Point Position Constraint, to three in the case of Line Pair Angular Displacement Constraint. In 
the second step, vector components and their differentials are evaluated in a common reference 
system using Equations 7 or 11 depending whether the vector is free or fixed respectively. The 
presence of a common reference system is guaranteed in an observation adjustment problem. 
The third step is concerned with producing assembly matrices and constants based on the vector 
transforms computed in Step Two. Finally, the condition equation assembly is performed 
according to Formulas 3 or 5. 
The modulation of the process of constraint assembly described above has the distinct 
advantage of requiring the support of a fairly uncomplicated data management system. This 
management system is only required to supply and store information concerning a single vector 
at any time during assembly process no matter what type of constraint is being evaluated. 
References 
[1] Elassal, A.A. : "A Unified Approach to the Problem of Observation Adjustment", Paper 
presented to the ASP Semi-Annual Convention, San Francisco, September, 1971.