  In this paper we develop theory of sequential parametrized motion planning
which generalises the approach of parametrized motion planning, which was
introduced recently in [3]. A sequential parametrized motion planning algorithm
produced a motion of the system which is required to visit a prescribed
sequence of states, in certain order, at specified moments of time. The
sequential parametrized algorithms are universal as the external conditions are
not fixed in advance but rather constitute part of the input of the algorithm.
The second part of this article consists of a detailed analysis of the
sequential parametrized topological complexity of the Fadell - Neuwirth
fibration. In the language of robotics, sections of the Fadell - Neuwitrh
fibration are algorithms for moving multiple robots avoiding collisions with
other robots and with obstacles in Euclidean space. In the last section of the
paper we introduce the new notion of TC-generating function of a fibration,
examine examples and raise some general questions about its analytic
properties.
