Triple pendulum solver

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the problem of the triple pendulum. See this article for the derivation of the equations of motion.

Figure 1: Diagram of the triple pendulum.

The equations being solved in this simulation are

\begin{aligned} \begin{bmatrix} M_1 l_1^2 & \mu_2 l_1 l_2 \cos{\Delta_{21}} & \mu_3 l_1 l_3\cos{\Delta_{31}} \\ \mu_2 l_1 l_2 \cos{\Delta_{21}} &M_2 l_2^2 & \mu_3 l_2 l_3\cos{\Delta_{32}} \\ \mu_3 l_1 l_3 \cos{\Delta_{31}} & \mu_3 l_2 l_3 \cos{\Delta_{32}} & M_3 l_3^2 \end{bmatrix} \begin{bmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \\ \ddot{\theta}_3 \end{bmatrix} &= \begin{bmatrix} Q_{\theta_1} - \mu_1 g l_1 \cos{\theta_1} + \mu_2 l_1 l_2 \dot{\theta}_2^2 \sin{\Delta_{21}} + \mu_3 l_1 l_3 \dot{\theta}_3^2 \sin{\Delta_{31}} \\ Q_{\theta_2} - \mu_2 l_2(l_1\dot{\theta}_1^2\sin{\Delta_{21}}+g\cos{\theta_2}) + \mu_3 l_2 l_3 \dot{\theta}_3^2 \sin{\Delta_{32}}\\ Q_{\theta_3} - \mu_3 l_3 (l_1\dot{\theta}_1^2 \sin{\Delta_{31}} + l_2\dot{\theta}_2^2 \sin{\Delta_{32}} + g\cos{\theta_3}) \end{bmatrix} \end{aligned}

Where $M_1 = m_{1b} + \dfrac{m_{1r}}{3} + m_{2b} + m_{2r} + m_{3b} + m_{3r}$ , $M_2 = m_{2b} + \dfrac{m_{2r}}{3} + m_{3b} + m_{3r}$ , $M_3 = m_{3b} + \dfrac{m_{3r}}{3}$ , $\mu_1 = m_{1b} + \dfrac{m_{1r}}{2} + m_{2b} + m_{2r} + m_{3b} + m_{3r}$ , $\mu_2 = m_{2b} + \dfrac{m_{2r}}{2} + m_{3b} + m_{3r}$ , $\mu_3 = m_{3b} + \dfrac{m_{3r}}{2}$ and $\Delta_{ij} = \theta_i - \theta_j$ .

Simulation parameter form.
Parameter	Value	Explanation
$g$ :		Gravitational acceleration (in $\mathrm{m}\cdot \mathrm{s}^{-2}$ ).
$l_1$ :		Length of pendulum 1 in metres.
$l_2$ :		Length of pendulum 2 in metres.
$l_3$ :		Length of pendulum 3 in metres.
$m_{1b}$ :		Mass of pendulum bob 1 in kilograms.
$m_{1r}$ :		Mass of pendulum rod 1 in kilograms.
$m_{2b}$ :		Mass of pendulum bob 2 in kilograms.
$m_{2r}$ :		Mass of pendulum rod 2 in kilograms.
$m_{3b}$ :		Mass of pendulum bob 3 in kilograms.
$m_{3r}$ :		Mass of pendulum rod 3 in kilograms.
$b_{1b}$ :		Linear dissipation coefficient of pendulum bob 1.
$b_{1r}$ :		Linear dissipation coefficient of pendulum rod 1.
$c_{1b}$ :		Quadratic dissipation coefficient of pendulum bob 1.
$c_{1r}$ :		Quadratic dissipation coefficient of pendulum rod 1.
$b_{2b}$ :		Linear dissipation coefficient for pendulum bob 2.
$b_{2r}$ :		Linear dissipation coefficient for pendulum rod 2.
$c_{2b}$ :		Quadratic dissipation coefficient for pendulum bob 2.
$c_{2r}$ :		Quadratic dissipation coefficient for pendulum rod 2.
$b_{3b}$ :		Linear dissipation coefficient for pendulum bob 3.
$b_{3r}$ :		Linear dissipation coefficient for pendulum rod 3.
$c_{3b}$ :		Quadratic dissipation coefficient for pendulum bob 3.
$c_{3r}$ :		Quadratic dissipation coefficient for pendulum rod 3.
$t_f$ :		End time for the simulation in seconds.
$\theta_{1,0}$ :		Initial value of $\theta_1$ in radians.
$\dot{\theta}_{1,0}$ :		Initial value of $\dot{\theta}_1$ in radians per second.
$\theta_{2,0}$ :		Initial value of $\theta_2$ in radians.
$\dot{\theta}_{2,0}$ :		Initial value of $\dot{\theta}_2$ in radians per second.
$\theta_{3,0}$ :		Initial value of $\theta_3$ in radians.
$\dot{\theta}_{3,0}$ :		Initial value of $\dot{\theta}_3$ in radians per second.
$\epsilon$ :		Error tolerance.
$\mathrm{Tol}_{\mathrm{type}}$ :		Tolerance type, can be either absolute (0) or relative (1).
$h_{\mathrm{Initial}}$ :		Initial step size.
$h_{\mathrm{Min}}$ :		Minimum allowed step size.
$\Delta t$ :		Time increment for skipping ahead in animation.
$t_1$ :		Time you want to skip ahead to in animation when you press the skip button.
$\mathrm{Width}$ :		Width (in px) of Plotly windows used for plotting and animation below.
$\mathrm{Height}$ :		Height (in px) of Plotly windows used for plotting and animation below.
$t_\mathrm{Scale}$ :		Proportion of animation time passed per real time. $t_{\mathrm{Scale}}=1.0$ means animation and real time match. $t_{\mathrm{Scale}}<1.0$ means the animation is going more slowly than real time. $t_{\mathrm{Scale}}>1.0$ means it is going more rapidly.
$\mathrm{Opacity}$ :		Opacity of the lines in the 3D phase space animation. Customizable in case you need to tweak it in order to see the red dot marker.

© Brenton Horne. I do not claim to be infallible, so feel free to open issues or pull requests on GitHub if you notice flaws or things that can be improved.
I favour Oxford spelling, so if you notice I generally use different spelling to Australian spelling, that is probably why.
Homepage. Source code: https://github.com/fusion809/fusion809.github.io.
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Triple pendulum solver

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