  We consider the problem of characterising the generic rigidity of bar-joint
frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a
given affine subspace. The special case when $d=2$ was previously solved by I.
Streinu and L. Theran in 2010. We will extend their characterisation to the
case when $d\geq 3$ and each vertex is constrained to lie in an affine subspace
of dimension $t$, when $t=1,2$ and also when $t\geq 3$ and $d\geq t(t-1)$. We
then point out that results on body-bar frameworks obtained by N. Katoh and S.
Tanigawa in 2013 can be used to characterise when a graph has a rigid
realisation as a $d$-dimensional body-bar framework with a given set of linear
constraints.
