I am trying to understand the derivation of some formulas in the paper Motion Control of Wheeled Mobile Robots by C. Samson in Springer Handbook of Robotics (2008).
Currently I am stuck at the following formula:
$$ \frac{\partial \mathbf{OP}}{\partial t} = \frac{\partial \mathbf{OP}_s}{\partial t} + \dot d \mathbf{j}_s - dc(s)\dot s \mathbf{i}_s $$
where $c(s)$ is the curvature of curve $C$ parametrized by the curvilinear abscissa $s$.
In the formula above I do not understant how the last term $dc(s)\dot s \mathbf{i}_s$ is obtained.
I think all begins by defining the vector $\mathbf{OP}$ and deriving it using the product rule:
$$ \mathbf{OP} = \mathbf{OP}_s + d \mathbf{j}_s $$
$$ \frac{\partial \mathbf{OP}}{\partial t} = \frac{\partial \mathbf{OP}_s}{\partial t} + \dot d \mathbf{j}_s + d \frac{\partial \mathbf{j}_s}{\partial t} $$
Could you please tell me if my derivation of the formula is correct so far?
My next problem is that I don't know what's the result of deriving the normal unit vector $\mathbf{j}_s$ with respect to time $t$. How does it relate to the curvature $c(s)$ and the velocity in $s$ direction?
The solution is to use the frenet formulas and assuming a torsion $\tau$ of zero because we are only moving in the plane. According to the formulas, the time derivative of the normal vector $\mathbf{j}_s$ is
$$ \frac{\partial \mathbf{j}_s}{\partial t} = (\tau \mathbf{k_s} - c \mathbf{i}_s)\left| \frac{\partial \mathbf{OP}_s}{\partial t} \right| = -c \dot s \mathbf{i}_s $$
where $\mathbf{k_s}$ is the binormal unit vector of the torsion $\tau$, which is zero in the plane.
However, I am not really sure where these formulas come from but this is another question. It is similar to deriving a tangent vector which yields the vector perpendicular to it.