Ihe Handbook of 
y. 1977, 
Vision for Robotic 
Society of Japan, 
e + Prolog", Logic 
/, 1986 
, Iskarous, M. and 
ems in Service of 
on Rehabilitation 
mas, E. D.(eds.), 
pp on Biorobotics: 
pan, 1995. 
T. and Ueno, H., 
bot Arm: HARIS, 
on Mechatronics, 
T. 0:Saito, :51Y. 
: Multiple Tactile 
t of 11th Annual 
r Biomechanism, 
; and K. 
Intelligent Vision 
Int. Conf. on 
, 1996. 
, T., Maeda, A. 
)pment of Torque 
ystem, Proc. 20th 
989. 
in Robot Control 
of Robotics and 
87, 1994. 
, Journal of the 
.6, pp. 593-598, 
Deep Knowledge 
te in Economics 
19, 1986. 
mous Robot Arm 
Mechatronics '94, 
[-"Ookuma, Y. 
edge Engineering 
Note on Software 
)5. 
jased Vision and 
nomous Human- 
n Robotics and 
Elsevier Science 
otion Scheduling 
ice Robot Arm, 
Ogy, Systems, 
, Frontieers in 
tions, Vol. 35, 
k Planning and 
-Type Intelligent 
sis, Tokyo Denki 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 5. Hakodate 1998 
SEPARATE ADJUSTMENT OF CLOSE RANGE PHOTOGRAMMETRIC MEASUREMENTS 
X. Wang & T. A. Clarke 
Optical Metrology Centre 
City University 
London EC1V O0HB 
UK 
Commission V, Working Group WG V/1 
KEY WORDS: Separate Adjustment, Bundle Adjustment, Real-Time, Least Squares 
ABSTRACT 
Photogrammetric methods will increasingly be used for real-time applications. For example, in a manufacturing environment the 
position of components must be located quickly and accurately for many assembly tasks. The computational effort should be 
minimal and if possible completely predictable. The conventional bundle adjustment method is unlikely to be used in this context 
because of the speed requirement while the direct linear transform method has modelling deficiencies for high precision 
measurement, furthermore, direct intersection with prior camera calibration does not allow for the dynamic situation found in 
industrial environments. The paper will describe a methodology for solving collinearity equations. Unlike the traditional bundle 
adjustment which solves for the unknown spatial co-ordinates of targets and camera parameters simultaneously, a separate solution 
for least squares estimation is developed which divides the parameters into two different groups, one for camera parameters, and the 
other for the co-ordinates of object points. With camera parameters fixed the 3-D co-ordinates of any spatial target can be located by 
spatial intersection of lines, and with spatial targets fixed the camera parameters can be determined by spatial resection. This process 
is repeated for both sets of parameters which are gradually refined. The final result can be proven to be statistically the same as 
would be achieved using the bundle adjustment but with a considerable time and memory saving. This separate adjustment method 
is found very successful to deal with close range photogrammetric measurements (CRPM), especial for a multistation convergent 
network. 
1. SIMULTANEOUS LEAST SQUARES ADJUSTMENT The cofactor matrix of the estimated parameters is given by 
In surveying and close range photogrammetry, redundant = (A zi 
measurements are always necessary for high precision, g. run (9 
reliability and statistics (Mikhail & Gracie 1981, Cooper 1987). 
This means that the number of observation is more than the 
minimum for a unique solution of the unknown parameters. 
This section will briefly discuss the simultaneous least squares 
estimation for redundant measurements. Let the functional 
model be expressed as 
2. BUNDLE ADJUSTMENT OF CRPM 
In close range photogrammetry, when m cameras are used to 
measure n object points, x will be a vector with (3n+6m) 
unknown parameters and / will be a vector with 2mn image 
observations. The unknown parameters x can be divided into 
two groups, x, for 3D coordinates of the object points and x; 
for the camera parameters. 
fo» 1 (1.1) 
Where x is a vector of the unknown parameters and / is a vector 
of the observations. The linearized observation equations may Therefore in equation (1.2).4 and Ax become 
be expressed as 
Ax 
AAx=b+v :W (1.2) A- [A A;] and x-| | 
Ax, 
where A = a is a Jacobi matrix, v is a vector of residuals for E TF. 
x É ; € where A = — and A, = — 
the observations and W is the weight matrix of the observations. ax, ox, 
There are obviously many possible values for v, to fit the 
functional model. Several methods exist to give a minimum The unknown parameters x, and x; may be solved 
value for the combinations of residuals (Kuang 1996). The least simultaneously as follows 
squares criterion is the most popular which minimises the sum 
of the weighted squares of the residuals. When all the unknown AX A. TA, TT ut 
parameters are considered simultaneously the least squares Ax -| Hl + q "i 
estimation gives the following solution Ax, A An AW 
AW 
ny 
Ax - (A'WA) ! A'Wb (1.3) EN bi: (2.1) 
2 
177