Motion of Reference Frames wrt each other

I am currently working on a Python script to simulate motion of reference frames wrt each other, and I need some help with the math.

Here is the problem statement-

Suppose I have a frame A
A frame B
frame B's origin has pos-vector P_B,A (function of time) wrt A in frame A
frame B is oriented at angle τ (function of time) wrt Some axis of A (Call it ψ )-I have the dcm for this transformation

Now, I have these two things-
1) Basis vectors of B
2) a pos vector (x, y, z) in B

How do I map the above 2 to frame A? As in, the math of the complete transformation