Formatted abstract

Although the physical laws governing motion have been known for centuries, there has been very little research aimed at making mathematical predictions of a twisting dive from takeoff. An eleven segment mathematical model of a diver is presented where the human body is described as a series of nested levels. These segments represent the upper torso, middle torso, lower torso, upper arms, lower arms, upper legs and lower legs of the diver.
Traditionally, Euler angles have been used to describe the orientation of a rigid body. However, if orientation is described in terms of Euler angles, the resultant equations of motion contain singularities. For this model, Euler parameters are used to describe orientation so that the resultant equations of motion are universally nonsingular.
Instead of using only the centre of mass of the diver to describe flight, the displacement and orientation of the lower torso were modelled, throughout the dive. Equations of motion were formulated using a Hamiltonian approach, leading to four sets of nonlinear differential equations to describe the flight of the diver. These equations are associated with the linear momentum, displacement and orientation of the lower torso.
The equations of motion were coded into Fortran and integrated numerically using the Merson form of RungeKutta. To evaluate the predictions of rigid body motion, the simulation model was compared with Euler's equations of motion for a single rigid body. Symmetric and asymmetric models were evaluated.
The symmetric case was analysed as it depicts a simple somersaulting dive and the asymmetric case was chosen as it is a good indicator of how well the simulation model handles somersaults and twists at the same time. For each case the rigid diver was given initial angular velocity about its transverse axis and the model closely approximated Euler's equations of motion.
To determine the accuracy of the simulation model for aerial movements, its predictions were evaluated against film data of divers performing various twisting somersaults. Five sets of experimental data were supplied by Yeadon. Two data sets were of a diver performing a forward somersault with two twists, two data sets were of a diver performing a forward somersault with three twists and one set of data was of a diver performing a forward oneandahalf somersault with two twists.
Twelve joint centres were digitised from the film data and threedimensional coordinates of each joint centre were calculated using a subroutine supplied by Yeadon. The experimental values for the translational and rotational motion of the diver were compared with the simulated estimates.
Digitising error played a large role in determining the accuracy of the simulation. The effect of digitising error was twofold: it distorted the input data during a dive and it distorted the initial conditions. To obtain more accurate estimates of the initial conditions, the initial linear momentum and angular velocity estimates were optimised using simulated annealing.
Although the maximum deviations in the somersault angle were comparable to the values reported by Yeadon [71], both the tilt and twist deviations were significantly larger. To date, no estimates for the translational motion of a physical point have been reported in the literature; however it can be assumed that these estimates can also be improved.
Digitising errors are magnified when derivatives are calculated [70]. One reason that this simulation model shows greater deviations than those reported by Yeadon is that our model prescribes both linear and angular velocities for each of several nested level segments, while Yeadon only prescribed angular velocities. Distorted input data also means that error was added to the simulation at each time step. As our equations of motion compute translational and rotational motion, they require more input than those of Yeadon. Thus, our simulation model will be more influenced by experimental error.
Although digitisation error is inevitable, a more accurate calibration method would certainly increase the accuracy of the simulation. Digitising error could also be reduced by increasing the number of cameras that film the dives so that each of the twelve joint centres can be seen through at least two cameras at any instant.
To date, simulation models have only been concerned with the rotational motion of the diver. Formulating equations of motion for the orientation and translation of a physical point makes video analysis and physical interpretation easier while ensuring that applications are relevant for coaches. The advantage of a nested level design is that the complexity of the model can easily be manipulated so that the diver may be described with as much detail as desired, simply by changing the value of an index.
By being able to learn from simulated dives, coaches will be better equipped to help their athletes. This model can be used to examine the effect of different twisting techniques on overall performances. It may also be used to create new dives, without incurring injury to divers. Although this simulation model has been applied to the analysis of twisting dives, it might equally be applied to other types of aerial movement.
