A New Twisting Somersault: 513XD

Abstract

We present the mathematical framework of an athlete modelled as a system of coupled rigid bodies to simulate platform and springboard diving. Euler’s equations of motion are generalised to non-rigid bodies and are then used to innovate a new dive sequence that in principle can be performed by real-world athletes. We begin by assuming that shape changes are instantaneous so that the equations of motion simplify enough to be solved analytically, and then use this insight to present a new dive (513XD) consisting of 1.5 somersaults and five twists using realistic shape changes. Finally, we demonstrate the phenomenon of converting pure somersaulting motion into pure twisting motion by using a sequence of impulsive shape changes, which may have applications in other fields such as space aeronautics.

Appendices

Appendix A: Model Parameters

We combine multiple body parts of Table 5 to produce the three segments denoted by $B_i$ for $i\in \{b,l,r\}$, which represent the body, left arm, and right arm, respectively. The numerical values of the mass and tensor of inertia of segments in our model are

$$\begin{aligned} m_b&= 66.319&\tilde{I}_b&= {{\mathrm{diag}}}{(14.204, 13.867, 0.612)}\\ m_l&=m_r=4.660&\tilde{I}_l&= \tilde{I}_r = {{\mathrm{diag}}}{(0.176, 0.176, 0.005)}. \end{aligned}$$

The collection of $\{\varvec{\tilde{J}}_i^j\}$ that specify the geometry of our model is

$$\begin{aligned} \varvec{\tilde{J}}_b^l&= (0, 0.2, 0.5196)^t&\varvec{\tilde{J}}_b^r&= (0, -0.2, 0.5196)^t&\varvec{\tilde{J}}_l^b&=\varvec{\tilde{J}}_r^b=(0, 0, 0.3647)^t. \end{aligned}$$

In the layout position, the athlete has shape $(\alpha _l,\alpha _r)=(\pi ,\pi )$ and the tensor of inertia is

$$\begin{aligned} I_s={{\mathrm{diag}}}(21.3188, 20.6091, 0.9956). \end{aligned}$$

In the twist position, the shape is either $(\alpha _l,\alpha _r)=(0,\pi )$ or $(\alpha _l,\alpha _r)=(\pi ,0)$, which produces the same diagonalised tensor of inertia

$$\begin{aligned} J_t={{\mathrm{diag}}}(18.3745, 17.6925, 0.9679). \end{aligned}$$

Table 5 Frohlich’s twelve-segment model of a male athlete that is 1.82 m in height and weighs 75.639 kg

Appendix B: Components of I and $\varvec{A}$

Evaluating (4) with the abduction–adduction plane of motion restriction simplifies the tensor of inertia I to the form $I = \displaystyle \left( \begin{array}{ccc} I_{xx} &{} 0 &{} 0\\ 0 &{} I_{yy} &{} I_{yz}\\ 0 &{} I_{yz} &{} I_{zz} \end{array}\right)$. Explicitly, the components are

$$\begin{aligned} I_{xx}&= a_0 - 2a_1\cos _{{}_+}{(\alpha _l,\alpha _r)} + 2a_2\sin _{{}_+}{(\alpha _l,\alpha _r)}- 2a_3\cos {(\alpha _l + \alpha _r)} \\ I_{yy}&= a_5 - 2a_1\cos _{{}_+}{(\alpha _l,\alpha _r)} + a_4\cos _{{}_+}{(2\alpha _l,2\alpha _r)}- 2a_3\cos {\alpha _l}\cos {\alpha _r} \\ I_{zz}&= a_6 + 2a_2\sin _{{}_+}{(\alpha _l,\alpha _r)} - a_4\cos _{{}_+}{(2 \alpha _l,2\alpha _r)}+ 2a_3\sin {\alpha _l} \sin {\alpha _r} \\ I_{yz}&= a_2\cos _{{}_-}{(\alpha _l,\alpha _r)} - a_1\sin _{{}_-}{(\alpha _l,\alpha _r)} + a_4\sin _{{}_-}{(2 \alpha _l,2\alpha _r)} - a_3\sin {(\alpha _l-\alpha _r)}, \end{aligned}$$

where

$$\begin{aligned} \cos _{{}_+}{(\alpha _l,\alpha _r)}= & {} \cos {\alpha _l}+\cos {\alpha _r} \quad \sin _{{}_+}{(\alpha _l,\alpha _r)} = \sin {\alpha _l}+\sin {\alpha _r}\\ \cos _{{}_-}{(\alpha _l,\alpha _r)}= & {} \cos {\alpha _l}-\cos {\alpha _r}\quad \sin _{{}_-}{(\alpha _l,\alpha _r)} = \sin {\alpha _l}-\sin {\alpha _r}. \end{aligned}$$

Similarly, (5) simplifies to $\varvec{A}=\big (A_l \dot{\alpha }_l+A_r \dot{\alpha }_r,0,0\big )^t$ where

$$\begin{aligned} A_l&= a_7 - a_1 \cos {\alpha _l}+ a_2 \sin {\alpha _l} - a_3 \cos {(\alpha _l+\alpha _r)}\\ A_r&= -a_7 + a_1 \cos {\alpha _r}- a_2 \sin {\alpha _r} + a_3 \cos {(\alpha _l+\alpha _r)}. \end{aligned}$$

The constants $a_0, a_1,\dots ,a_7$ are determined by the collection of $\left\{ m_i, \tilde{I}_i,\varvec{\tilde{J}}_i^j\right\}$ and are

$$\begin{aligned} a_0&= 18.298&a_1&= 0.774&a_2&= 0.340&a_3&= 0.038 \\ a_4&= 0.376&a_5&= 16.836&a_6&= 1.748&a_7&= 0.758. \end{aligned}$$

Appendix C: Equations of Orientation

The orientation can be tracked from the solution of the equations of motion (11). We will represent the orientation with unit quaternions as they provide an elegant form of encoding the angle vector information. Consider a clockwise rotation of $\theta$ about the unit vector $\varvec{u}$, which can be presented with the unit quaternion $q=q_0+\varvec{q}=\cos (\theta /2)+\varvec{u}\sin {\theta /2}$, where the vector $\varvec{q}=q_1\varvec{i}+q_2\varvec{j}+q_3\varvec{k}$ specifies the imaginary parts. To rotate an arbitrary vector $\varvec{v}$ by the quaternion q, we first treat the vector as a pure quaternion expressed as $v=0+\varvec{v}$ and then apply the transformation $p=qv\bar{q}$, where $\bar{q}=q_0-\varvec{q}$ is the quaternion conjugate. The result is a pure quaternion $p=0+2(\varvec{v}\cdot \varvec{q})\varvec{q}+(q_0^2-\varvec{q}\cdot \varvec{q})\varvec{v}-2q_0\varvec{v}\times \varvec{q}$, which is linear in $\varvec{v}$ and can therefore be rearranged to obtain the vector

$$\begin{aligned} \varvec{p}=\left[ 2(\varvec{q}\varvec{q}^t+q_0 \varvec{q})+(q_0^2-\varvec{q}\cdot \varvec{q})\mathbbm {1}\right] \varvec{v}. \end{aligned}$$

(35)

Now the coefficient of $\varvec{v}$ is precisely the rotation matrix R, so substituting it in $\hat{\varvec{\Omega }}=R^t\dot{R}$ and removing the hat operator gives

$$\begin{aligned} \varvec{\Omega }=2\left( \begin{array}{cccc} -q_1 &{} q_0 &{} q_3 &{} -q_2\\ -q_2 &{} -q_3 &{} q_0 &{} q_1\\ -q_3 &{} q_2 &{} -q_1 &{} q_0 \end{array}\right) \left( \begin{array}{c} \dot{q}_0\\ \dot{q}_1\\ \dot{q}_2\\ \dot{q}_3\end{array}\right) , \end{aligned}$$

(36)

where $\varvec{\Omega }$ is a known vector obtained from solving the equations of motion (11). As q is a unit quaternion, we can incorporate the constraint $q_0\dot{q}_0+q_1\dot{q}_1+q_2\dot{q}_2+q_3\dot{q}_3=0$ with (36) to derive the equations of orientation

$$\begin{aligned} \dot{q} = \frac{1}{2}\left( \begin{array}{cc} 0 &{} -\varvec{\Omega }^t\\ \varvec{\Omega } &{} -\hat{\varvec{\Omega }}\end{array}\right) q. \end{aligned}$$

(37)

Together with (11) and (37), a complete description of the dynamics for a system of coupled rigid bodies can be given.