  Let $M$ be a subset of vector space or projective space. The authors define
the \emph{generalized configuration space} of $M$ which is formed by $n$-tuples
of elements of $M$ where any $k$ elements of each $n$-tuple are linearly
independent. The \emph{generalized configuration space} gives a generalization
of the classical configuration space defined by E.Fadell. Denote the
\emph{generalized configuration space} of $M$ by $W_{k,n}(M)$. The authors are
mainly interested in the calculation about the homotopy groups of generalized
configuration space. This article gives the fundamental groups of generalized
configuration spaces of $\mathbb{R}P^m$ for some special cases, and the
connections between the homotopy groups of generalized configuration spaces of
$S^m$ and the homotopy groups of Stiefel manifolds. It is also proved that the
higher homotopy groups of generalized configuration spaces $W_{k,n}(S^m)$ and
$W_{k,n}(\mathbb{R}P^m)$ are isomorphic.
