SYNTHESIS OF HUMAN MOVEMENT—FORWARD SOLUTIONS - Biomechanics and Motor Control of Human Movement - page 204

computer programs using symbolic computer languages such as LISP, PRO-
LOG, and MAPLE and to generate the equations of motion. It is also possible
to use computer languages such as C or BASIC in writing self-formulating
computer programs general enough to accept model description as an input
and provide model response as an output. In fact, Mechanical Dynamics,
Inc., of Ann Arbor, Mich., has developed a computer program that automates
the dynamic simulation process. Their product is marketed under the name
ADAMS (Automatic dynamic analysis of mechanical systems; Chace, 1984).
Formulating the equations of motion may be done in several ways. The
first and most direct, but possibly the least efficient way, is to apply New-
ton's laws of dynamics to each segment in the model. Although the reaction
forces and torques are obtained as by-products of the solution, the method is
cumbersome and does not lend itself easily to a general dynamic simulation
program. However, if some concepts of graph theory are incorporated into
Newton's laws of motion, the result is a methodical procedure that can be
used for writing self-formulating dynamic simulation programs. Three com-
puter programs that utilize these ideas have been developed at the University
of Waterloo: VECENT (vector network), a three-dimensional package for
particles (Andrews and Kesavan, 1975); PLANET (plane network) for planar
mechanisms; and ADVNET (advanced network) for 3D systems (Andrews,
1977; Singhal and Kesavan, 1983).
The second method to formulate the equations of motion is to utilize
Lagrangian dynamics (Wells, 1967). Lagrange's equations require the con-
cepts of virtual displacement and employ system energy and work as functions
of the generalized coordinates to obtain a set of second-order differential
equations of motion. To a large extent, the method reduces the entire field of
dynamics to a single procedure involving the same basic steps, regardless of
the number of segments considered, the type of coordinates employed, the
number of constraints on the model, and whether or not the constraints are
in motion. Alternative methods have also been used in this research, such as
methods of virtual work extended using D'Alembert principles as in DYMAC
(Paul, 1978).
Which method is more suitable? Each method has its advantages and dis-
advantages. However, the Lagrangian method is characterized by simplicity
and is applicable in any suitable coordinates. The task of this chapter is to
encode a procedure based on Lagrange's dynamics and suitable for computer
implementation using symbolic manipulation language. In doing so, system
elements are described by lists that are linked together. Each list has its name
and index number and a stack of parameters associated with it. The first ele-
ment in a given list is usually an integer that establishes a link between the
list and other relevant lists in the system. The equations of motion are then
obtained by systematic manipulation of these lists. Before we go any further,
a review of Lagrange's method is necessary.