ABSTRACT This paper presents a recursive direct differentiation metho

Kishor D Bhalerao

Kurt S Anderson

Mohammad Poursina

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Proceedings of the ASME 2009 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2009
August 30-September 2, 2009, San Diego, USA
DETC2009-87585
DRAFT: LOGARITHMIC COMPLEXITY SENSITIVITY ANALYSIS OF FLEXIBLE
MULTIBODY SYSTEMS
Kishor D. Bhalerao∗
Mohammad Poursina
Kurt S. Anderson
Department of Mechanical, Nuclear and Aerospace Engineering
Rensselaer Polytechnic Institute
Troy, NY 12180
Email: bhalek@rpi.edu
ABSTRACT
This paper presents a recursive direct differentiation method
for sensitivity analysis of flexible multibody systems. Large rota-
tions and translations in the system are modeled as rigid body
degrees of freedom while the deformation field within each body
is approximated by superposition of modal shape functions.The
equations of motion for the flexible members are differentiated
at body level and the sensitivity information is generated via a
recursive divide and conquer scheme. The number of differen-
tiations required in this method is minimal. The method works
concurrently with the forward dynamics simulation of the sys-
tem and requires minimum data storage. The use of divide and
conquer framework makes the method linear and logarithmic in
complexity for serial and parallel implementation, respectively,
and ideally suited for general topologies. The method is applied
to a flexible two arm robotic manipulator to calculate sensitivity
information and the results are compared with the finite differ-
ence approach.
INTRODUCTION
Development of efficient, computationally low cost and
highly parallelizable methods in the multibody systems has
greatly expanded the realms into which these tools may and have
∗Address all correspondence to this author.
been effectively applied. As evidence is the degree to whichar-
ticulated multibody methods now enjoy use in the areas ranging
from the dynamics and control of large, complex, highly flexible
spacecraft, to the modeling, simulation and analysis of molecu-
lar systems at the nano-scale. In the area of modern multibody
systems dynamics, design of highly complex systems, which is
iterative and computationally taxing in nature, can still be chal-
lenging. Sensitivity analysis, can play an important role asso-
ciated with multibody computational problems such as implicit
integration schemes, linearized dynamics, optimal control, and
design optimization.
Although easy implementation and simplicity makes finite
difference approximation perhaps the most broadly adoptedap-
proach to generate sensitivity information, it suffers from critical
shortcomings. This procedure is time-consuming due to the fact
that it requires one additional simulation for each of the perturbed
design parameters. Furthermore, selecting the optimal perturba-
tion size of a set of design variables [1] and sensitivity of the nu-
merical solution to the perturbation size [2–4] are important ( t
times critical) issues, and may significantly influence the success
of this technique. Analytical sensitivity analysis methods such as
adjoint variable, direct differentiation and automatic differentia-
tion are capable of overcoming most of the problems which exist
in finite difference techniques.
In the adjoint variable methods, based on variation prin-
1 Copyright c© 2009 by ASME
Proceedings of the ASME 2009 International Design Engineering Technical Conferences &  
Computers and Information in Engineering Conference 
IDETC/CIE 2009 
August 30 - September 2, 2009, San Diego, California, USA 
DETC2009-87585
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
ciples, explicit calculation of the state sensitivities isavoided
through the introduction of a set of adjoint variables [2,5–9]. Ma-
nipulating these adjoint variables, variations of the system equa-
tions, and the variations of design criteria, produces the required
adjoint relationships. Solving sequence of adjoint relationships
provides the design sensitivity vector. This vector directly or-
responds to the variation (sensitivity) of design criteriain terms
of design variable variations. Implementation of these methods
can be complex, particularly when dealing with variable step ize
or multirate integrations methods, and a large amount of data
(the complete state of the system at each function valuation, for
the duration of the simulation over which the sensitivity isbeing
evaluated) has to be stored for the forward problem. The need
to access this complete set of state data as the adjoint equations
are integrated backwards in time can require a large number of
I/O operations which greatly slows the rate at which the sen-
sitivities may be determined [10, 11]. Another source of error
is the backward temporal integration necessary for the calcula-
tion of adjoint variables. Additionally, numerical stability for the
adjoint variable methods remains an open question as indicate
in [2] and [12].
Automatic differentiation [13, 14] is a computer science
based approach in which the variables are identified within ex-
isting code, and derivative expressions are determined by direct
application of the chain rule of differentiation. This method can
provide results which are numerically unstable and a blind appli-
cation of the chain rule of differentiation can lead to errone us
results, particularly with regard toDAE systems.
The other popular methods for sensitivity analysis are those
based on direct differentiation [10, 15–21]. In these techniques,
direct application of the chain rule of differentiation is ued to ex-
plicitly form the states sensitivities. Mathematically easy to un-
derstand, high numerical stability and relative insensitivity of the
solution accuracy to parameter perturbations have distinguished
direct differentiation algorithms among many competitiveap-
proaches. Besides that, if the number of design variables issmall
and the number of design constraints is large, the direct differ-
entiation method becomes more attractive than adjoint variable
techniques. Direct differentiation however, can be computation-
ally expensive if not performed intelligently when dealingwith
large systems as is clear from the following example.
There are several ways to describe the dynamics of a flexi-
ble body and here we limit our discussion to modal superposition
method (FDCA) as described in [22]. In sensitivity analysis, it
is desired to determine the sensitivity of a specific cost functio
to the variation of the particular design or control variables. In
FDCA, large rotations or translations are modeled asn rigid body
(relative and/or absolute) degrees of freedom associated wi h the
interconnecting kinematic joint free modes of motion. These are
fully described by introducing sets of generalized coordinates
{qi}ni=1 and generalized speeds{ui}
n
i=1. The elastic deformation
of the bodyk in the system may be represented by sets of modal
coordinates{qki }
νk
i=1 and their time derivatives{u
k
i }
νk
i=1. The ob-
jective functionJ is often an explicit function of the design and
state variables, while states of the system, themselves, are implic-
itly dependent on the values of the design parameters. Therefor ,
the sensitivity equation of the objective functionJ with respect
to design variablep can be written as
∇J =
∂J
∂p
+
n
∑
r=1
(
∂J
∂qr
dqr
dpj
+
∂J
∂ur
dur
dpj
+
∂J
∂u̇r
du̇r
dpj
)
+
nb
∑
k=1
νk
∑
r=1
(
∂J
∂qkr
dqkr
dpj
+
∂J
∂ukr
dukr
dpj
+
∂J
∂u̇kr
du̇kr
dpj
). (1)
Herenb represents the number of bodies. In multi-flexible-
body systems, generating the dependencies of highly coupled
states and states derivatives on design parameters is computation-
ally expensive. Fortunately, the state variable sensitivities need
not be solved for directly, but can be determined from temporal
integration of the system state time derivatives as
dqr
dpj
∣
∣
∣
∣
t=τ+dt
=
Z t =τ+dt
t =τ
dq̇r
dpj
∣
∣
∣
∣
t=τ
dt+
dqr
dpj
∣
∣
∣
∣
t =τ
, (2a)
dur
dpj
∣
∣
∣
∣
t=τ+dt
=
Z t =τ+dt
t =τ
du̇r
dpj
∣
∣
∣
∣
t=τ
dt+
dur
dpj
∣
∣
∣
∣
t =τ
. (2b)
In the above equations,qr contains the generalized coordi-
nates for all the kinematical joints and the modal coordinates
in the system. Similarly, all the generalized and modal speeds
are buried in the vectorur , while the vector ˙ur involves the time
derivatives of the generalized speeds. Based on equation (2), the
main task in sensitivity analysis reduces to that of finding effi-
ciently the sensitivity of ˙ur with respect to the design variable(s).
Integrating du̇rdpj over the time domain of interest and substitut-
ing back the results in equation (1) will provide the sensitivity of
the objective function with respect to the desired design parame-
ter(s).
The governing equations of motion of a general multi-
flexible-body system, in the state space form, is represented as,
M m×mu̇m×1 = K m×1. (3)
In this coupled set of equations,m defines the total number of
rigid and flexible degrees of freedom of the system.M is the
known mass matrix andK is a known vector of the applied and
state-dependent inertia forces on the system. Direct differentia-
tion of equation (3) with respect to the desired parameter gives
an expression fordu̇dpj as
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⇒ [M m×m]
du̇
dpj m×1
=
∂K
∂p j
+
∂K
∂q
dq
dpj
+
∂K
∂u
du
dpj
−[
∂M
∂p j
+
∂M
∂q
dq
dpj
+
∂M
∂u
du
dpj
]u̇. (4)
Application of the direct method in this manner incurs large
computational expenses in generating the differentiations which
ranges fromO(m2) to O(m3). Also, solution for the state deriva-
tive sensitivities by direct methods is ofO(m3) expense. These
costs can quickly become prohibitive for larger values ofm.
Therefore, in generating sensitivity information for suchhighly
complex systems, it is necessary to introduce quick and efficient
algorithms.
In this paper, an efficient logarithmic complexity (for par-
allel implementation) direct differentiation method is presented
for the determination of first order design sensitivities offlexible
multibody systems. The governing equations which define the
states and state derivatives of the system are derived basedon
Flexible Divide and Conquer Algorithm (FDCA) as described
in [22]. The use of divide and conquer framework makes the
method suitable for systems with tree or ladder like topologies.
In such complex topologies FDCA-based sensitivity analysis is
expected to be more efficient than the other methods [17, 23] in
which the constraint equations and their sensitivities to the design
variables in forward dynamics and sensitivity equations should
be solved for. Since this methodology works concurrently with
the forward dynamics problem as opposed to the adjoint variable
methods, there is no need for the backward temporal integration
of the equations and consequently requires minimal data stor-
age. The sensitivity equations for the entire system are genrated
via a hierarchicassemblyprocess and as a result the actual dif-
ferentiation required in this method is minimum as comparedto
traditional methods.
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DETC2009-87585 DRAFT: LOGARITHMIC COMPLEXITY SENSITIVITY ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS

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DETC2009-87585 DRAFT: LOGARITHMIC COMPLEXITY SENSITIVITY ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS

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