Posture Prediction and Optimization - Human Motion Simulation: Predictive Dynamics - page 47

initial guess is used that satisfies the joint limits but does not necessarily satisfy
the distance constraint, which is discussed below.

3.5.2 Constraints
The first constraint in Equation (3.2) is the distance constraint and requires that
the end-effector contact a predetermined target point (user-specified) in Cartesian
space, where

is a small positive number that approximates zero. This constraint
represents distance squared. The DH method is used to determine the position of
the end-effector after a series of given displacements. In addition, each general-
ized coordinate is constrained to lie between lower and upper limits, represented
by q i and q i , respectively. These limits ensure that the digital human does not
assume an unrealistic position in an effort to contact the target point.

ε

3.5.3 Cost function
For simplicity, and to demonstrate the concept, consider the first objective func-
tion, which represents a simple form of discomfort and is proportional to the devi-
ation from the neutral position. The neutral position is selected as a relatively
comfortable posture, typically a standing position with arms at one's sides, where
q i

N represents the overall posture.
Technically, the displacement from the neutral position is given by

is the neutral position of a joint, and q

j q i 2 q i j ;

2 is used. Because motion in some
joints contributes more significantly to discomfort, a weighting parameter w i is
introduced to stress the importance of a particular joint. Currently, these weights
are determined by trial and error. Then, the cumulative discomfort (of all joints)
is characterized by the following objective function:

however, for computational simplicity ð q i 2 q i Þ

X

n

q i

2

f Discomfort ð q Þ 5 w i ð q i 2 Þ (3.3)

i 5 1

Note that the intent in developing this performance measure is not necessarily
to quantify discomfort. Rather, the function in Equation (3.3) is designed simply
to yield a realistic posture and is effective in that capacity.

3.6 A 3-DOF arm example
Consider, for example, the simplified planar 3-DOF arm shown in Figure 3.3 .
This is an illustrative example to demonstrate our formulation where we have
restricted the arm to planar motion.
The objective of this exercise is twofold:
1. Given the point p 5 [5.67 3.59] and the following joint ranges of motion

A 52 3 , q , 3 , it is required to determine whether this point is reachable.