  This paper deals with a new class of random flights $\underline{\bf
X}_d(t),t>0,$ defined in the real space $\mathbb{R}^d, d\geq 2,$ characterized
by non-uniform probability distributions on the multidimensional sphere. These
random motions differ from similar models appeared in literature which take
directions according to the uniform law. The family of angular probability
distributions introduced in this paper depends on a parameter $\nu\geq 0$ which
gives the level of drift of the motion. Furthermore, we assume that the number
of changes of direction performed by the random flight is fixed. The time
lengths between two consecutive changes of orientation have joint probability
distribution given by a Dirichlet density function.
  The analysis of $\underline{\bf X}_d(t),t>0,$ is not an easy task, because it
involves the calculation of integrals which are not always solvable. Therefore,
we analyze the random flight $\underline{\bf X}_m^d(t),t>0,$ obtained as
projection onto the lower spaces $\mathbb{R}^m,m<d,$ of the original random
motion in $\mathbb{R}^d$. Then we get the probability distribution of
$\underline{\bf X}_m^d(t),t>0.$
  Although, in its general framework, the analysis of $\underline{\bf
X}_d(t),t>0,$ is very complicated, for some values of $\nu$, we can provide
some results on the process. Indeed, for $\nu=1$, we obtain the characteristic
function of the random flight moving in $\mathbb{R}^d$. Furthermore, by
inverting the characteristic function, we are able to give the analytic form
(up to some constants) of the probability distribution of $\underline{\bf
X}_d(t),t>0.$
