Recursive Dynamics - Human Motion Simulation: Predictive Dynamics - page 69

CHAPTER
4

Recursive Dynamics

Never mistake motion for action.
Ernest Hemingway (18991961)

4.1 Introduction
The aim of this chapter is to introduce and develop a rigorous computational
platform for describing the dynamics of a system of segmented links of a
human body. Mass, moments of inertia, velocities, and accelerations will form
a coupled set of nonlinear differential equations called the equations of
motion.
General dynamics equations have been extensively studied in recent decades
in terms of computational efficiency as the advent of computational platforms has
readily enabled the calculation of physics for objects in motion. A convenient
method for formulating the equations of motion, building upon the
Denavit Hartneberg (DH) method and using the Lagrangian equations for a serial
kinematic chain, will be presented. This methodology lends itself well for kine-
matic skeletons of the human body and yields a convenient and compact descrip-
tion of the equations of motion. However, computation of the torques from
equations of motion is of the order Oðn 4

Þ , where n is the number of degrees of

freedom (DOF) for the system.
The human motion prediction optimization problem is usually a large-scale
sparse nonlinear programming (NLP) problem. Accurate sensitivity (gradient
calculation) is a key factor to efficiently achieve an optimal solution.
Although the finite difference approach can be used to approximate gradients,
the computational expense becomes substantial as the number of variables
increases (i.e., the number of DOF). In addition, accuracy of the derivatives
can affect convergence of the optimization process,

thus leading to further

computational expense.
Although different algorithms for sensitivity of dynamic equations have been
studied, limited work is found for inverse recursive Lagrangian formulation with
sensitivity for general motion planning problems. By using 4 4 transformation

matrices (DH method), recursive Lagrangian formulation is more efficient and
convenient to implement compared with recursive Newton Euler formulation
( Hollerbach, 1980 ). In addition, the Lagrangian formulation is the energy con-
cept for the equations of motion typically defined in the joint space; this results

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