A new approach for the determination of singularities in motion of spatial linkages, displacement functions and the number of linkage configurations is proposed. Singularities in motion occur at positions where an overconstrained group of links become movable at least locally while the driving link is fixed. The determination of displacement functions is based on modeling of the linkage by two open kinematic chains formed: (i) by links of an overconstrained group of links and (ii) the driving link and the frame. Conditions of “assembly” of the open chains provide invariants-scalar products-that yield the appropriate equations that relate the parameters in motion of the driving link and the overconstrained group of links. The number of linkage configurations is proposed to determine through the number of singularity positions that are simulated with a certain combination of the linkage design parameters. Applications of the proposed approach to the RCRCR, RCRRC and RSRC linkages are represented in the following parts 2, 3 and 4 of the paper.