differential equations - Finding a 3d curve from torsion and curvature with NDSolve

I'm trying to use the Frenet–Serret formulas to find the curve that matches the torsion and curvature I specify numerically with an InterpolatingFunction.

The problem is that the system is overdetermined, and have been getting all kinds of errors from Mathematica. If I don't tell NDSolve about the definition of N as the normal vector and B as the cross product of T and N, then I have a 9x9 system, which will solve on occasion, but with results don't seem right.

The curve should be a Mobius strip with length $2\pi$, so I also have the boundary conditions of $T(0)=T(2\pi), N(0)=N(2\pi), B(0)=B(2\pi)$, but when I include those conditions, I get

"There are significant errors {0., 0., 0., 0., 0., 0., 2.49764*10^-6, 0., 0., 
-2.06667*10^-6, 0., 0.} in the boundary value residuals. Returning the best solution 
found."

Some errors are to be expected, and $10^{-6}$ is not significant to me, but the resulting curve is a straight line from $(0,0,0)$ to $(\pi/2,0,0)$.

When I try specifying $N(s)$ as $\frac{T'(s)}{\|T'(s)\|}$ and $B(s)$ as $T\times N$, and only using the first 2 equations on the Wikipedia page, I get:

NDSolve::ntdvdae: Cannot solve to find an explicit formula for the derivatives. 

NDSolve will try solving the system as differential-algebraic equations.`

NDSolve::ndcf: Repeated convergence test failure at s == 0.; unable to continue.

As far as I can tell, NDSolve doesn't support vector equations, so I have been breaking everything down into the component equations.

I'd be happy to share the points that form the interpolating functions for torsion and curvature, but I don't know where to post them: it's almost 1500 points.

Answer

First@NDSolve[{
x''[t] == k1[t] nx[t], nx'[t] == -k1[t] x'[t] - k2[t] bx[t], bx'[t] == k2[t] nx[t],
y''[t] == k1[t] ny[t], ny'[t] == -k1[t] y'[t] - k2[t] by[t], by'[t] == k2[t] ny[t],

z''[t] == k1[t] nz[t], nz'[t] == -k1[t] z'[t] - k2[t] bz[t], bz'[t] == k2[t] nz[t],
x[ti] == iniPos[[1]], y[ti] == iniPos[[2]], z[ti] == iniPos[[3]],
x'[ti] == iniDir[[1]], y'[ti] == iniDir[[2]], z'[ti] == iniDir[[3]],
nx[ti] == iniNor[[1]], ny[ti] == iniNor[[2]], nz[ti] == iniNor[[3]],
bx[ti] == iniBin[[1]], by[ti] == iniBin[[2]], bz[ti] == iniBin[[3]]},
{x, y, z, nx, ny, nz, bx, by, bz}, {t, ti, te}]

k1[t] is curvature, k2[t] is torsion, {x[t],y[t],z[t]} is path, {nx[t],ny[t],nz[t]} is normal, {bx[t],by[t],bz[t]} is binormal, arc length is (te-ti)*norm[iniDir].

This is signed curvature...