Connection on spinor bundle induced by Levi-Civita Connection

Question

Given a Levi-Civita connection $\nabla$ on the tangent bundle with bundle metric $g$, a spinor representation $(\rho, S)$ and a spin structure $P$, what is the induced connection on a spinor bundle $P_{Spin} \times_\rho S$? I am looking for a coordinate-free definition. I understand that $\nabla$ canonically induces a covariant derivative on the tensor algebra, which descends to the Clifford algebra since $\nabla g = 0$. But I do not know how to go on. I would appreciate both book recommendations and direct explanations. Thanks in advance!