# Dynamics and Simulation of General Human and Humanoid Motion in Sports

Veljko Potkonjak, Miomir Vukobratovic, Kalman Babkovic, Branislav Borovac
DOI: 10.4018/978-1-60566-406-4.ch003
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## Abstract

This chapter relates biomechanics to robotics. The mathematical models are derived to cover the kinematics and dynamics of virtually any motion of a human or a humanoid robot. Benefits for humanoid robots are seen in fully dynamic control and a general simulator for the purpose of system designing and motion planning. Biomechanics in sports and medicine can use these as a tool for mathematical analysis of motion and disorders. Better results in sports and improved diagnostics are foreseen. This work is a step towards the biologically-inspired robot control needed for a diversity of tasks expected in humanoids, and robotic assistive devices helping people to overcome disabilities or augment their physical potentials. This text deals mainly with examples coming from sports in order to justify this aspect of research.
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## Mathematics

### Free-flier motion

We consider a flier as an articulated system consisting of the basic body (the torso) and several branches (head, arms and legs), as shown in Figure 1. Let there be independent joint motions described by joint-angles vector (the terms joint coordinates or internal coordinates are often used). The basic body needs six coordinates to describe its spatial position: , where defines the position of the mass center and are orientation angles (roll, pitch, and yaw). Now, the overall number of degrees of freedom (DOF) for the system is , and the system position is defined by

. (1)
Figure 1.

Unconstrained (free flier) and constrained system

We now consider the drives. It is assumed that each joint motion has its own drive – the torque . Note that in this analysis there is no drive associated to the basic-body coordinates X (this is a real situation with humans and humanoids in “normal” activities, however, in space activities – actions like repairing a space station, etc. – reactive drives are added, attached to the torso; the proposed method for simulation can easily handle this situation). The vector of the joint drives is , and the extended drive vector (N-dimensional) is ; zeros stand for missing basic-body drives.

The dynamic model of the flier has the general form:

Or

(2)

Dimensions of the inertial matrix and its submatrices are: , , , , and . Dimensions of the column vectors containing centrifugal, Coriolis’ and gravity effects are: , , and .

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