Alternative formulations for
optimization of dynamic systems and task-based simulation of digital human models
are presented. The basic idea of the formulations is to treat the state
variables also as independent variables in the optimization process; i.e.,
generalized displacements, velocities, accelerations and forces as variables in
addition to the real design variables. Simultaneous formulations based on
different forms for the differential equations (DEs) are developed, namely, the
first and second order forms of the DEs. Central difference, Newmark’s method
and some methods based on collocation are presented. Similar to the
simultaneous analysis and design (SAND) approach used for design and simulation
of systems subjected to static loads, and direct collocation for optimal
control, the governing equations are treated as equality constraints. There are
three main advantages of the new formulations for dynamic systems: (i) the
equations of motion for the system need not be integrated explicitly, (ii)
design sensitivity analysis of the systems is not needed since all the problem
functions are explicit in terms of the variables, and (iii) the available
simulation software can be used quite easily without any modifications to
optimize dynamic systems since design sensitivity analysis of the system is not
needed (which is quite time consuming and difficult to implement). With the
alternative formulations, the optimization problem becomes large; i.e., the
numbers of variables and constraints are large. However, the problem functions
are quite sparse; i.e., each function depends on only a few variables. These
sparse properties of the functions are exploited in the optimization process.
To evaluate the formulations, a five DOF model of a mechanical system is
optimized. The proposed formulations work very well for the sample problem;
their advantages and disadvantages are discussed. Evaluation of the
formulations using two digital human models is in progress and the results of
that investigation will be reported later.
1. Introduction
1.1 Introduction and
Motivation
Simulating human postures and
motion are traditionally complex and difficult problems due to redundancy of
the human musculoskeletal system and other biomechanical factors. The concept
for task-based posture prediction using nearly real-time multi-objective
optimization techniques provides a viable approach for predicting natural
postures and intermediate motions of digital humans represented by a relatively
large number of degrees of freedom (DOF). Various approaches and numerical
algorithms for implementing task based posture prediction using multi-objective
optimization algorithms have been studied by Mi (2004). Although success has
been achieved, the application is limited to a 15-DOF upper body model that is
based purely on kinematics.
Realistic virtual human
modelling, simulation and natural-looking motion require several
considerations, such as external forces, muscle forces, muscle stress and
fatigue. The current model cannot predict human motions and fatigue, when
external loads need to be carried. Therefore, a transient dynamic formulation
for the motion prediction of realistic digital human models is needed.
Basically, in a specified time range, equations of motion (dynamic equilibrium
equations) and several constraints on the state variables need to be satisfied
for the system.
The most common approach for
optimization of systems has been the one where only the design variables are
treated as optimization variables. All other response quantities, such as
generalized displacements, velocities, and accelerations are treated as
implicit functions of the design variables. Therefore, in the optimization
process, a system of differential equations (DEs) is solved to obtain these
response variables and to calculate values of various functions of the
optimization problem. Then the optimization algorithm is used to update the
design. This nested process of solution of DE and design update, also called
the conventional approach, is repeated until a stopping criterion is
satisfied. However, this optimization process is difficult to use in practice
(Arora and Wang 2003). The main drawback with these formulations and the
solution process is that the response quantities, called the state variables,
are treated as implicit functions of the design variables. On the one hand, the
optimization process consists of repeated integration of linear or nonlinear
differential-algebraic equations (DAEs) or just differential equations (DEs).
On the other hand, derivatives of state variables with respect to the design
variables are needed in the optimization process. This is known as design
sensitivity analysis whose implementation into any simulation software is
quite difficult, especially for multidisciplinary problems requiring use of
different discipline-specific analysis software.
Another interesting approach for transient
dynamic optimization is the so-called equivalent static load method (Kang et
al. 2001; Choi and Park 2002), where the problem is transferred to an
equivalent quasi-static problem. The idea is to find a static load set that can
generate the same displacement field as that with the dynamic load at certain
times. Therefore, multiple equivalent static load sets obtained at different
time points can represent various states of the system under the dynamic load.
However, with this approach, DEs must still be integrated a number of times and
design sensitivity analysis must also be performed for the resulting static
problems.
To alleviate the difficulties
mentioned above, a fundamental shift in the direction of research on
computational design optimization formulations and algorithms is needed. It is
useful to develop alternate formulations of the transient dynamic optimization
problems that do not require explicit solution of DEs at each iteration.
Optimization methods need to be developed such that there is no need for design
sensitivity analysis to optimize systems. By formulating the optimization
problem in a mixed space of design variables and state variables, the DEs are
imbedded as equality constraints in one single optimization problem, therefore
no explicit solution of DEs is needed. Design sensitivity analysis is also not
needed, making it easier to implement the optimization process with the
existing simulation software as well. The foregoing ideas have been applied
successfully to optimize structural and mechanical systems subjected to dynamic
loads (Wang and Arora, 2004a,b).
In this research, the foregoing
idea of simultaneous analysis and design (SAND) is utilized and alternative
formulations are developed for the optimization of transient dynamic systems.
Various state variables, such as the displacements, velocities and
accelerations are treated as independent variables in the optimization process.
The equations of motion are treated as equality constraints in the optimization
formulation. Therefore the constraints on displacements and accelerations are
expressed explicitly in terms of the optimization variables. The resulting
optimization problem is sparse and is solved using sparse nonlinear programming
(NLP) algorithms (Gill et al. 2003). An available example of a mechanical
system is optimized and the solutions are compared to those in the literature.
The advantages and disadvantages of different optimization formulations are
also discussed.
The present work presents and
evaluates some simultaneous formulations for transient dynamic systems. The
major differences of the present work from the one in the literature are: (1)
Some simultaneous formulations based on the SAND and direct collocation
concepts are introduced, evaluated and compared for transient dynamic systems,
(2) Some of the formulations directly discretize the second order form of the
equations of motion, which will facilitate use of the existing simulation
software in the optimization process, and (3) a modern and powerful
optimization algorithm and associated software are used that takes full
advantage of the sparsity structure of the proposed formulations.
The
following abbreviations are used throughout the report:
DAE differential-algebraic
equations
DE differential
equations
DOF degrees
of freedom
MPEC mathematical programming with equilibrium
constraints
NLP nonlinear
programming
SAND simultaneous
analysis and design
SQP sequential
quadratic programming
1.2 Objectives of
Research
The main objective of this
research is to propose and evaluate alternative formulations and solution
methods for optimization of transient dynamic systems and task-based motion
prediction of digital humans. The computational effort with these formulations
may be reduced due to the fact that the “solution of equations of motion” - is
avoided. The optimization problem itself becomes quite large although it is
sparse. This sparsity must be exploited in the optimization process. Based on
this discussion, objectives of this research are set-up as follows:
1.
To propose
and develop different alternative formulations.
2.
To develop a
computer-based optimization process. Existing large-scale NLP programs will be
used and their role in the alternative formulations will be evaluated.
3.
To evaluate
the proposed alternate formulations by solving various example problems, and
apply the methods for digital human simulation.
The challenges and difficulties
include treatment of large numbers of variables and constraints in the
optimization process. However, this research will shed light on potential of
alternative formulations for application to digital human models.
1.3 Overview of the
Material
In Section 2, the general
framework for dynamic response optimization formulations is described,
including the dynamic simulation model, objective function and various
constraints. Section 3 contains a review of the relevant literature. Some
topics with respect to MPEC, and optimal control of trajectory design are
discussed. The conventional formulation, where only the design variables are
treated as optimization variables is presented in Section 4. In Section 5, some
simultaneous formulations based on the discretization of the first order DEs
are presented. These formulations are well developed in the optimal control
field. Alternative formulations based on the direct discretization of the
second order DEs are developed in Section 6. Advantages and disadvantages of
these formulations are discussed in Section 7. A numerical example of
optimization of a mechanical system from the literature is solved and the solutions
are compared in Section 8. The sparsity features of different formulations are
discussed and a sparse SQP solver is used as an optimizer. Two digital human
simulation problems are proposed in Section 9. Discussion and conclusions are
presented in Section 10.
2. Dynamic
Response Optimization Problem
The basic dynamic response optimization
problem is to determine design parameters of the dynamic system, to achieve
certain goals (e.g., minimization of a cost function, such as the maximum
acceleration or torques) while satisfying all the performance requirements (Afimiwala and Mayne 1974; Lim and Arora 1987; Kim and
Choi 1998; Arora 1999). In this section, a general problem for
optimization of dynamic systems is presented. In the remaining sections, the
conventional and alternative formulations are presented and implemented for
dynamic problems and numerical results for example problems are discussed.
2.1 Simulation Model
Let 
be an m
dimensional vector to represent the design variables for the problem. They
might include the mass, stiffness, and damping parameters of the dynamic
systems. 
is a d
dimensional vector that represents the state variables, or generalized
displacements for the problem. For a mechanical robotic system or a digital
human model, this represents joint profile, such as joint angles. The equations
of motion for a dynamic system can be derived from the energy principle. A
general form of the equations of motion is:
(2.1.1)
with the initial conditions 
, and 
. 
is the
generalized force vector of dimension 
. Note that for a
robotic or a digital human model, Eq. (1) can in general be expressed as
(2.1.2)
where 
represents the
mass-inertia matrix 
; 
is a 
Coriolis and
centrifugal force vector; 
is a 
friction and damping
vector; 
is the gravity
vector; 
is a 
driving force or
torque vector. For general dynamic systems, the equations of motion can be
solved directly in the second order form in Eq. (2.1.2) (Bathe 1982), or by
converting them to a first order form, the so-called, state space representation of Eq. (2.1.2), which is
given as
(2.1.3)
where 
. Although
considerable study of dynamics and control is based on Eq. (2.1.3), considerable
simulation software for structural and mechanical systems is based on the
second order form of the equations of motion. Therefore, it is worthwhile to
examine both forms of the equations for simulation. Note that the forces are
not taken as optimization variables in the formulations and derivations also in
this report in this report. However, in general problems of dynamic human
motion prediction, the forces such as joint torques will also need to be
discretized and treated as optimization variables in the formulations. It is
noted that this has been done for static structural and mechanical systems
(Wang and Arora 2004a).
2.2 Cost Function
and Constraints
In general, a cost functional
includes the state and design variables, as
(2.2.1)
where T is
the total time interval considered. The objective functional J may be
the cost of the system, performance measures, or any other function of the
state variables. A time-dependent functional,
such as maximum acceleration or maximum displacement, can also be treated as
will be seen later in the example problem.
Design requirements are imposed mostly as
inequality constraints (although equality constraints can also be treated). One
type of constraints that does not depend on time t is
(2.2.2)
The
other type of constraints is the so-called point-wise constraint, which needs
to be satisfied at each point of the entire time interval 
(2.2.3)
or a simplified form as
(2.2.4)
In dynamic systems, an
important problem is to determine the response of the system, such as the
generalized displacements, velocities and accelerations, to given inputs.
However, these values need to satisfy certain requirements, such as in Eq.
(2.2.4), to ensure that the mechanical system performs optimally in some sense.
The time-dependent inequality constraints in Eq. (2.2.4) may include the
following displacement, velocity and acceleration constraints; and the time-independent
constraints on the design variables:
(2.2.5)
(2.2.6)
(2.2.7)
(2.2.8)
where 
and 
are the lower and
upper bounds for the generalized displacements, velocities, accelerations and
design variables, respectively. Other design requirements may also be included.
Five treatments of the point-wise constraints have been discussed in the
literature (Tseng
and Arora
1989; Arora 1995). In this study, the conventional
treatment is used where the time interval is divided into subintervals and the
constraints are imposed at each time grid point.
3. Review
of Literature
Engineering optimization problems are in general ODE- or
PDE-constrained mathematical programming problems (Biegler et al. 2003). For these problems, one
fundamental and promising solution technique is the so-called simultaneous
approach, where the PDE
governing equations (equilibrium equations) are treated as equality constrains
in the formulation, and solved as a large nonlinear programming problem (NLP)
(Biegler et al. 2003; Schulz 2004).
&