Hello Mathematics Community,
I am looking for some advice/references on the following problem regarding animating the trajectory of rigid body motion. I believe this problem will help give better intuition for studying the motion of rigid bodies.
Suppose that we are given the position and orientation of a rigid object in Euclidean space $(x,y,z)$ which are represented by the coordinates $(\boldsymbol{r},\Lambda)$ respectively. Further, let us assume that the evolution of the position and orientation is governed by an ordinary differential equation of the form $$(\dot{\boldsymbol{r}},\dot{\Lambda}) = f(\boldsymbol{r},\Lambda) $$ where dot's refer to time derivatives $\frac{d}{dt}$.
We then can solve this ODE numerically using, say, the Runge-Kutta algorithm. This then gives us a time indexed sequence of vectors and matrices, namely,
$$\boldsymbol{r}_i = \begin{bmatrix}x_i\\y_i \\ z_i \end{bmatrix} $$ for the object's position and
$$\Lambda_i = \begin{bmatrix}\Lambda_{11}^i& \Lambda_{12}^i & \Lambda_{13}^i\\\Lambda_{21}^i& \Lambda_{22}^i & \Lambda_{23}^i\\ \Lambda_{31}^i& \Lambda_{32}^i & \Lambda_{33}^i \end{bmatrix} $$ for it's orientation. If one were to take enough points $i$, then this sequence approximates the trajectory of the object (its position and orientation in space) by applying these sets of transformations to a chosen $(x,y,z)$ coordinate system.
I would like to animate the trajectory but I need some help/examples as to how to implement. I think a possible solution is as follows:
Step 1: Either solve the ODE in Python and import an array of solution points as a csv file OR set up the problem using C# (I think this is what Blender uses though I am at loss as to how to interface with it), implement Runga-Kutta and get an time indexed array of state-points.
Step 2: Interpolate the missing points to obtain a smooth trajectory.
Step 3: Fix a coordinate system to the animated object
Step 4: Somehow apply coordinate transformations $(\boldsymbol{r},\Lambda)$ to the object
Step 5: run the simulation from which we obtain an animation the trajectory of a rigid body in space.
I know how to solve ODE's in Python or MATLAB (Step 1) but I am at a loss as to how to complete steps 2-5. If anyone has any expertise with such a problem or can suggest any reading material/resources that would be greatly appreciated. I am currently attempting to implement this using the Blender software, but if someone knows of more appropriate software for such a problem please feel free to forward your recommendations.Thank you.