How to calculate the time an object needs to travel along a parameterized curve?

Question

Imagine you have an object that travels along a trajectory parameterized by time in the form:

$$\vec{p} = \begin{pmatrix} x_0 \\ y_0 \end{pmatrix} + \begin{pmatrix} v_x \\ v_y \end{pmatrix} t + \frac{1}{2} \begin{pmatrix} a_x \\ a_y \end{pmatrix} t^2 + \frac{1}{6} \begin{pmatrix} j_x \\ j_y \end{pmatrix} t^3,$$

where $\vec{v}$, $\vec{a}$, and $\vec{j}$ are the velocity, acceleration and jerk, respectively.

Now given a start time $t_0$, I would like to know how long it takes ($\Delta t$) the object to travel a certain distance $d$ along the trajectory. I assume I would need to solve the following integral (or at least something along those lines, pun intended) for $\Delta t$, but I'm not really sure how to do that:

$$d = \int_{t_0}^{\Delta t} |\vec{p}|\text{d}t$$

I would like to do this in Python and I would think that this is a very basic problem, however I haven't found the right tools to do this yet. Numpy for example offers the roots function, and I guess this might help me here, but I haven't figured out how yet.

Also, except for this similar-ish post, I didn't manage to find much on this, probably because I haven't figured out the correct key terms yet.

Answer 1

The correct formula for the distance is $$ d=\int_{t_0}^{\Delta t}\Big|\frac{d}{dt}\vec p\Big|\,dt $$ (add up infinitesimal distances to see this). To invert this in python I would calculate a grid of $d$-values given a grid of $\Delta t$-values and then interpolate reversely: stick the time grid into the slot for the $y$ and the $d$- grid into the slot for $x$ and interpolate to get $\Delta t$ given an arbitrary $d$.

from numpy import linspace, sqrt
from scipy import interpolate
from scipy.integrate import quad

def dp( t, v, a, j ):
    return( v + a*t + j*t**2/2 )

def dp_length( t, v, a, j ):
    return( sqrt( (dp( t, v, a, j )[0])**2 + (dp( t, v, a, j )[1])**2 ) )

def distance( t, v, a, j ):
    return( quad( dp_length, 0, t, args = ( v, a, j ) )[0] )

v = array([[1],[1]])
a = array([[1],[1]])
j = array([[1],[1]])

T = linspace( 0, 10, 10 )
D = linspace( 0, 0, 10 )

for i in range(len(T)):
    t = T[i]
    D[i] = distance( t, v, a, j )

d = 185
time = interpolate.interp1d( D, T )
print( time(d) )