Biomedical Engineering Reference
In-Depth Information
N
N
where q
;
q
;
q
A
R
are the state variables, and
τA
R
are the generalized forces.
This dynamics problem is defined over
the time domain
Ω
5
ðT
0
;
TÞ
with
U
indicating
derivative with respect to
t
. The superscript
N
represents the number of DOF
of model M.
Forward dynamics calculates the motion (q,
boundary
Γ
5
fT
0
;
Tg;
t
A
Ω
,
t
being the time and the symbol
ð
U
Þ
by inte-
grating
Equation (5.1)
with the specified initial conditions. In contrast, inverse
dynamics computes the associated force
q and
q) from the force
τ
that leads to a prescribed motion for
the system. The two procedures are depicted in
Figure 5.2
. For simplicity, we use
only q to represent kinematics of the system.
In practice, it is difficult to measure complete displacement (q) and force (
τ
)
histories accurately for a biosystem with many DOFs, especially involving a com-
plex motion. This is because the experimental measurement is either not accurate
enough or too expensive to achieve the required accuracy. However, the boundary
conditions and some state response of the system might be available. In this case,
neither forward dynamics nor inverse dynamics can be applied to the biosystem S
directly. As a consequence, the predictive dynamics procedure is proposed to
solve these types of problems. The basic idea is to formulate a nonlinear optimi-
zation problem based on the physics of motion (the dynamics of the motion). An
appropriate performance measure (objective function) for the biosystem is defined
and minimized subject to the available information about the system that imposes
various constraints.
The ultimate intent of predictive dynamics is to enable the study of cause and
effect on a human simulator, whereby a user defines the human body, body type,
strength, and fatigue limits and is able to simulate an entire motion of the human.
We illustrate the predictive dynamics problem using
Figure 5.3
where the joint
variables in this optimization problem are both displacement histories and force
histories, and are both unknown; g are the constraints defined based on the avail-
able information
τ
ϒ
about the system S, such as physical constraints, collision, and
known
unknown
(A)
known
unknown
(B)
FIGURE 5.2
Flowcharts of (A) forward dynamics and (B) inverse dynamics.
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