Dynamic Performance of a SCARA Robot Manipulator With ...

206 IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009
Dynamic Performance of a SCARA Robot Manipulator
With Uncertainty Using Polynomial Chaos Theory
Philip Voglewede, Anton H. C. Smith, and Antonello Monti
Abstract—This short paper outlines how polynomial chaos theory (PCT)
can be utilized for manipulator dynamic analysis and controller design in
a 4-DOF selective compliance assembly robot-arm-type manipulator with
variation in both the link masses and payload. It includes a simple linear
control algorithm into the formulation to show the capability of the PCT
framework.
Index Terms—Monte Carlo methods, polynomials, robot dynamics,
uncertainty.
I. INTRODUCTION
The design of any mass-produced system involves understanding
how manufacturing variation will affect its performance. For robotic
manipulators, this problem is exacerbated by changes in the payload.
While these affects are minimal for highly geared manipulators, the
affects are large for direct driven manipulators. Robotic designers
need to better understand how variation in both product manufacture
and payload affects the dynamic performance of these types of
manipulators.
Unfortunately, there is a lack of appropriate tools to generally assess
the dynamic performance of mechanisms with uncertainty. There
has been significant work on the study of how clearances statically
affect the output of a mechanism [1]–[4]. There are also numerous
computer-aided tolerancing tools that can perform static stack-ups via
linearization or a statistical Monte Carlo (MC) analysis [5]–[8]. However,
there is a lack of appropriate tools to simulate and predict the
window of possible dynamic configurations resulting from the variation.
As such, variational dynamic simulations are typically performed
as an MC of the governing equations that, in turn, results in large run
times.
Polynomial chaos theory (PCT) can be used for uncertain dynamic
analysis, including the controller design. PCT allows one to solve
stochastic differential equations by using orthogonal polynomials in
conjunction with a Galerkin projection. This allows the transformation
of stochastic differential equations into an expanded set of standard
ordinary differential equations [9]–[12]. PCT has been successfully applied
to fluid mechanic [13], general oscillatory [14], heat transfer [15],
and electrical [16] systems. More recently, it has been applied specifically
to multibody dynamic systems [17]–[19], but specific examples
outlining uses in this realm are lacking.
This paper applies PCT to a selective compliance assembly robot
arm (SCARA)-type manipulator to show how the methodology can
Manuscript received May 25, 2007; revised September 2, 2008. First published
January 13, 2009; current version published February 4, 2009. This paper
was recommended for publication by Associate Editor I. Chen and Editor F.
Park upon evaluation of the reviewers’ comments. This work was supported by
the U.S. Office of Naval Research under Grant N00014-02-1-0623.
P. Voglewede is with the Mechanical Engineering Department, Marquette
University, Milwaukee, WI 53132 USA (e-mail: philip.voglewede@marquette.
edu).
A. H. C. Smith is with the Electrical Engineering Department, University of
South Carolina, Columbia, SC 29208 USA (e-mail: smith35@cec.sc.edu).
A. Monti is with the Institute for Automation of Complex Power Systems at
the EON Energy Research Center, Rhenish-Westphalian Technical University
(RWTH) Aachen, Aachen, Germany (e-mail: amonti@eonerc.rwth-aachen.de).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TRO.2008.2006871
be utilized to better understand and predict dynamic performance of
manipulators under feedback control with uncertainty.
II. PCT BASICS
The basis of PCT is the expansion all the generic variables in the
governing equations in terms of a orthogonal polynomial basis. Following
a notation similar to [10], a generic variable u can be expressed
as a infinite sum of orthogonal polynomials as
u(β) =u 0 I 0 +
∞∑
u i1 I 1 (ξ i1 (β))
i 1 =1
+
∞∑
i 1 =1
i 1 ∑
i 2
u i1 i 2
I 2 (ξ i1 (β),ξ i2 (β)) + ··· (1)
where β is the random event, ξ = {ξ 1 ,ξ 2 ,...,ξ n } is the random vector,
I i is the orthogonal polynomial in the Askey scheme, and u i , u ij ,
etc., are denoted as the PC coefficients. Since this notation is rather
cumbersome to deal with, it is usually abbreviated as
u(β) =
∞∑
u j Φ j (ξ(β)) (2)
j =0
where u j are the PC coefficients and Φ j (ξ) are the multidimensional
orthogonal polynomials. For practical purposes the polynomial expansion
is truncated to a finite number of polynomial terms and the
dependence on the uncertain event β is assumed. How many terms are
kept is dependent upon the number of uncertainties in the system n v
and the order of the adopted polynomial n p . 1 The total number of terms
of the expansion P is given by
P =
( (np + n v )!
n p !n v !
)
− 1. (3)
The expansion of the variables leaves the unknown vector ξ that
must be dealt with. However, due to the special nature of the orthogonal
polynomials, the Galerkin projection of the expanded equations will
eliminate the dependence upon the unknown stochastic variable ξ.This
is done by defining the inner product as
∫
〈f(ξ)g(ξ)〉 = f(ξ)g(ξ)W (ξ) dξ (4)
where W (ξ) is the weighting function associated with the chosen basis
in the Askey scheme (see [10] for detailed descriptions). Due to the
special nature of the inner product, many of the inner products will be
zero as
〈Φ i Φ j 〉 = 〈Φ 2 i 〉δ ij (5)
where δ ij is the Kronecker delta. The results are an expanded set
of equations that can be solved using traditional techniques for the
unknown PC coefficients. The PC coefficients can then be backsubstituted
into (2) to yield the distribution of the original variables
in time.
A. PCT in Manipulator Dynamic Analysis
Unfortunately, some nonlinearities cannot be solved using PCT, as
the Galerkin projection does not eliminate the dependence upon the
unknown variables ξ i . For multidimensional serial manipulators, the
general form of the dynamic equations of motion follow [20]
M ¨θ + C(θ, θ) ˙ ˙θ + G(θ) =τ (6)
1 See [10] for a discussion on the convergence of this series.
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IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009 207
TABLE I
SCARA PARAMETERS FOR NUMERICAL STUDY
Fig. 1. Schematic of the SCARA robot showing the geometric parameters and
coordinate frames.
where M is the mass matrix, C is a vector of Coriolis and centrifugal
terms, G is the gravitational terms, and τ is a vector of generalized
forces. These terms, in general, contain polynomial and trigonometric
nonlinearities. Polynomial nonlinearities cause no problems to the formulation.
Trigonometric functions can be solved easily using a Taylor
series expansion with a small-angle approximation, as will be shown
later in this paper.
B. PCT in Control Analysis
The addition of control laws to a serial manipulator simply changes
the right-hand side of (6). For example, using independent potential
difference (PD) control on the joints causes no issues when applying
PCT. Nonlinear control laws can be more vexing if the nonlinearity
is not polynomial or trigonometric based. Care must be exercised to
ensure that additional nonlinearities are not introduced by division as
well.
III. PCT APPLIED TO SCARA ROBOT
In this paper, we will assume a 4-DOF SCARA-type manipulator
having variation in the mass (and subsequently, the inertia) of both the
first two links as well as payload, as shown in Fig. 1 and Table I. Variation
in the link lengths and centers of mass could also be incorporated
into the model but are left out as their effects are assumed to be small
compared to the mass variation. The other mechanical parameters are
set to reasonable values, as given in Table I. The variation is assumed
to be uniformly distributed across the range. 2
A. Dynamics of the SCARA Robot
An open kinematic chain’s dynamics can be derived utilizing either
a Newton–Euler or Lagrangian formulation. Utilizing a Lagrangian
formulation, the equations of motion for the SCARA robot as shown
in Fig. 1 are found to be [21], [22]
⎡
⎤ ⎡ ⎤
p 1 + p 2 c 2 p 3 +0.5p 2 c 2 0 −p 5
¨θ 1
p 3 +0.5p 2 c 2 p 3 0 −p 5
¨θ 2
⎢
⎣ 0 0 p 4 0
⎥ ⎢
⎦ ⎣
¨θ
⎥
3 ⎦
−p 5 −p 5 0 p 5
¨θ 4
} {{ }
M
⎡ ⎤ ⎡
⎤ ⎡ ⎤
τ 1 −p 2 s 2 ˙θ2 −0.5p 2 s 2 ˙θ2 0 0 ˙θ
⎡ ⎤
1 0
τ
=
2
⎢
⎣ τ
⎥
3 ⎦ − 0.5 ∗ p 2 s 2 ˙θ1 0 0 0
˙θ 2
⎢
⎣ 0 0 0 0
⎥ ⎢
⎦ ⎣
˙θ
⎥
3 ⎦ + 0
⎢ ⎥
⎣ p 4 g ⎦
τ 4 0 0 0 0 ˙θ 4
0
} {{ }
F
(7)
where c 2 and s 2 are cos(θ 2 ) and sin(θ 2 ), respectively, and
p 1 =
4∑
I i + m 1 x 2 1 + m 2 (x 2 2 + a 2 1 )+(m 3 + m 4 )(a 2 1 + a 2 2 )
i=1
p 2 =2[a 1 x 2 m 2 + a 1 a 2 (m 3 + m 4 )]
4∑
p 3 = I i + m 2 x 2 2 + a 2 2 (m 3 + m 4 )
i=2
p 4 = m 3 + m 4
p 5 = I 4 (8)
where τ i is the input torque (or force), I i is the moments of inertia
around the centroid, m i is the mass, x i is the mass center, and a i is the
length for link i. In order to include the affect of mass on the inertia,
the inertia terms are further defined as
I i = m i κ 2 i , i =1,...,4 (9)
where κ i is the radius of gyration of the links. Assuming a PD control
law on each joint, the torques are
τ i = K pi (θ ides − θ i )+K di ( ˙θ ides − ˙ θ i ), i =1,...,4 (10)
2 It should be noted that virtually any type of distribution and even mixed
distributions (one variable being normally distributed and one being uniform)
can be incorporated into the framework.
where des denotes the desired values.
In order to use a general nonlinear ordinary differential equation
solver, the equations are expressed in state space form [23]. For this
particular problem, the governing equations can be put into eight
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208 IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009
first-order equations with
⎡ ⎤ ⎡ ⎤
u 1 θ 1
u 2
θ 2
u 3
θ 3
u 4
θ 4
u ≡
=
u 5
˙θ 1
u 6
˙θ 2
⎢ ⎥ ⎢ ⎥
⎣ u 7 ⎦ ⎣ ˙θ 3 ⎦
u 8
˙θ 4
where
⎡
˙u =
⎢
⎣
and M and F are defined in (7).
u 5
u 6
u 7
u 8
M −1 F
⎤
⎥
⎦
(11)
(12)
B. SCARA Dynamic Equations Incorporating Uncertainty
The governing equations can then be expanded using the PCT framework.
The benefit of using PCT is that it allows one to solve the stochastic
differential equations with an expanded differential equation set. In
other words, the random distributions need to be calculated only once
and the ODE solver needs to run only once. This dramatically improves
the efficiency of the solver. The tradeoff is that the number of differential
equations as well as the calculation of numerous inner products
are much larger. Thus, as the number of unknown variables becomes
large, the method becomes cumbersome, and a traditional MC would
be better [24]. However, once the basis is chosen, these inner products
need to be calculated only once.
The first step is to express all our variables in the appropriate basis.
For this particular example, a Legendre polynomial basis is chosen due
to the uniform distribution on the inputs. 3 Each state space variable
(u i ) is expanded in terms of the corresponding PC coefficients (u ij )
u i = u i0 (t)+u i1 (t)ξ 1 + u i2 (t)ξ 2 + u i3 (t)ξ 3
i =1,...,8. (13)
To keep the notation simpler, the use of the hat is omitted and the
dependence of the polynomial coefficients on time will be dropped.
For simplicity, the polynomial basis is truncated at first order. As will
be shown in this example, the results will concur with this assumption.
We can now proceed to substitute the approximations into (12). The
difficulty is that the equations contain trigonometric functions. These
trigonometric functions will cause problems with the methodology and
must be expressed as a linear combination of the stochastic variables
ξ 1 , ξ 2 ,andξ 3 . Using a Taylor’s series approximation method as in [11],
one can approximate the trigonometric functions as
cos u 2 ≈ cos u 20 − sin u 20 (u 21 ξ 1 + u 22 ξ 2 + u 23 ξ 3 )
sin u 2 ≈ sin u 20 +cosu 20 (u 21 ξ 1 + u 22 ξ 2 + u 23 ξ 3 ) (14)
where a first-order polynomial approximation is utilized in this
case. Higher order approximations will not cause problems with the
formulation.
3 It should be noted that the use of the Legendre may not be the most appropriate
as the output variations may be better represented by a Gaussian distribution,
and thus, a Hermite basis may be better [10].
TABLE II
PD CONTROLLER GAINS FOR NUMERICAL STUDY
Now that all the nonlinearities have been eliminated, the equation
can be projected onto the basis. These inner products are cumbersome
to compute as they contain a large quantity of terms. In order to avoid
mistakes, these were computed using an automated script in MAPLE
[25]. Due to the projection, four equations are created for every row in
(12) for a total of 32 equations. These 32 equations are now integrated
using a standard ODE solver (in this case, MATLAB’s ode45 function
was utilized) to solve for the 32 unknown PC coefficients (i.e., u ij )as
functions of time. The time history of these can then be substituted back
into (13) to determine the mean and variation. Due to the uncertain
variables ξ i in (13), a “mini-MC” is performed on the output. An
analytical solution to the problem has also been proposed in [26].
IV. RESULTS
In order to show the power of the PCT formulation, the SCARA
robot was put under closed-loop control with the parameters shown in
Table II. The first two joints were commanded to go to 30 ◦ and 20 ◦ (i.e.,
u 1des =0.5236 rad and u 2des =0.3491 rad). Controller gains were
chosen to show differing types of output (e.g., overdamped versus
underdamped, large variation versus small variation).
The results of this analysis are shown in Fig. 2. An MC analysis
was also performed to compare with the solution. Both the mean and
plus/minus three standard deviations (±3σ), were plotted to show the
resulting variation. A normal plot (not shown) was created at several
time intervals to observe the normality of the output distribution. The
normal plot showed that the distributions are normal with shortened
tails. Thus, the ±3σ curves represent more of the distribution and
slightly overestimate the variation that does not cause a problem for
this study.
Analysis of the plots show that the PCT analysis allows for the
dynamic analysis of the stochastic differential equations of motion,
and in this case, yields nearly identical results to the MC simulation.
For this particular problem, the difference is shown in Table III. The
most drastic difference is shown in joint 3 where some “noise” is seen
in the PCT formulation. More accurate results can be garnered by
adding more terms on the PCT expansion [27]. Also of interest is that
the transition between underdamped and overdamped oscillations in
θ 2 are handled easily. Also, the amount of variation if large or small is
easily accommodated by the framework as shown in the graph for θ 3 .
PCT has traded the number of ODE solver calculations by expanding
the number of differential equations from 8 to 32 and requiring the
calculation of 32 inner products, 16 of which were trivial. Once the
inner products are determined, the analysis takes significantly less time
as compared to the MC analysis. For a sample size of 1000, the MC
simulation took approximately 159.70 s to compute while the PCT
method took a mere 1.27 s. Thus, scenario analyses can be performed
easily in real time without having to run numerous datasets.
It is interesting to note the existence of times where the total variation
decreases significantly. Further numerical analysis shows that at these
points, the percentage error in the standard deviation estimate is the
greatest. These situations (here called variation nodes) could possibly
be useful for system testing and evaluation.
The use of PCT in the controller design allows control analysis
quicker than a traditional MC. Different controller gains can easily be
tried and the resulting dynamics can be simulated. The PCT procedure
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IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009 209
Fig. 2.
Simulation results for all four joints of a SCARA robot with variation in the first two link masses and the payload.
TABLE III
MAXIMUM ABSOLUTE DIFFERENCE BETWEEN PCT
AND MC FOR SCARA ROBOT
6) PCT has no problem with both underdamped and overdamped
type responses.
Additionally, this example also corroborates several of the known
advantages regarding the use of PCT. Specifically, PCT is more efficient
from a simulation time standpoint, allows dynamic changing of
parameters during the simulation, and yields a richer result that can be
utilized in other applications like controller design.
also opens the door for easy optimization of the controller gains for
the small amount of variation. Sensitivity studies can quickly and
effectively be performed for different controller gains.
V. CONCLUSION
This paper has shown in detail how to use PCT to analyze the
dynamic response of an open-loop mechanism by applying it to a
SCARA robot manipulator. Through this particular example, several
items were found.
1) PCT on the SCARA robot is feasible as long as the number of
unknowns are small. Automating the process further aiding in
speeding up the process. However, this even has its limits as
even the automated process is still cumbersome.
2) Using PCT on robotic applications requires a judicious choice of
states and formulation of the problem to make sure that nonlinearities
are not introduced into the equations.
3) PCT on a SCARA robot gives consistent results to a large MC
for a simple one-term expansion. This was true even using a
standard approximation of the trigonometric identities.
4) PCT exhibits “antinodes” (positions in the output where the
variation decreases significantly), even in this robotics example.
However, this was not noted on all joints.
5) PCT can be utilized with feedback control. However, integral
and nonlinear control poses some interesting problems that have
yet to be solved.
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