  Autonomous motion of a system (robot) is controlled by a motion planning
algorithm. A sequential parametrized motion planning algorithm \cite{FP22}
works under variable external conditions and generates continuous motions of
the system to attain the prescribed sequence of states at prescribed moments of
time. Topological complexity of such algorithms characterises their structure
and discontinuities. Information about states of the system consistent with
states of the external conditions is described by a fibration $p: E\to B$ where
the base $B$ parametrises the external conditions and each fibre $p^{-1}(b)$ is
the configuration space of the system constrained by external conditions $b\in
B$; more detail on this approach is given below. Our main goal in this paper is
to study the sequential topological complexity of sphere bundles $\dot \xi:
\dot E\to B$; in other words we study {\it \lq\lq parametrized families of
spheres\rq\rq} and sequential parametrized motion planning algorithms for such
bundles. We use the Euler and Stiefel - Whitney characteristic classes to
obtain lower bounds on the topological complexity. We illustrate our results by
many explicit examples. Some related results for the special case $r=2$ were
described earlier in \cite{FW23}.
