IEEE
TRANSACTIONS
ON
INDUSTRIAL
ELECTRONICS,
VOL.
40,
NO.
1,
FEBRUARY
1993
105
Modeling and Control Strategies for
a
Variable Reluctance Direct-Drive Motor
Fabio Filicori, Corrado Guarino
Lo
Bianco, Albert0 Tonielli
Abstract-In industrial automation and robotic applications,
direct-drive motors represent a suitable solution to friction and
backlash problems typical of mechanical reduction gears. Vari-
able reluctance
(VR)
motors are well suited for direct-drive
implementation but, because of the strongly nonlinear elec-
tromechanical characteristics, these motors are traditionally
designed as stepper motors.
The main aim
of
the work described in the paper is the design
of
a high-performance ripple-free dynamic torque controller for
a
VR
motor, idtended for trajectory tracking in robotic applica-
tions.
An
original modeling approach is investigated in order to
simplify the design of the high-performance torque controller.
Model structure and parameter estimation techniques are pre-
sented. Different approaches to the overall torque controller
design problein are also discussed and the solution adopted is
illustrated.
A
cascade controller structure is considered. It con-
sists of a feedforward nonlinear torque compensator, cascaded
to a nonlinear flux
or
current closed-loop controller. The feed-
forward compensator is carefully considered and optimization
techniques are used for its design.
Two
optimization criteria are
proposed: the first minimizes copper losses, whereas the second
minimizes the maximum value of the motor-feeding voltage.
Although developed for a specific commercial motor, the pro-
posed modeling and optimization strategies can be used for
other
VR
motors with magnetically decoupled phases, both
ro-
tating and lidear. Laboratory experiments for model validation
and preliminary simulation results of the overall torque control
system are presented.
I. INTRODUCTION
LECTRICAL motors generally produce their maxi-
E
mum power at very high velocity. When the number
of poles is increased, it is possible to obtain higher torque
and lower velocity at constant power. Since increasing the
number
of
poles has practical size and cost limitations, a
mechanical reduction-gear is typically used. In robotic
applications, very low velocity, very high torque, and good
accuracy in an angular position represent typical operat-
ing specifications. Friction and backlash deriving from
mechanical reduction gear impose severe limitations on
Manuscript received August
1,
1991. This work was partially supported
by A.X.I.S. s.p.a.-Tavarnelle
Val
di Pesa (Firenze), M.U.R.S.T.-Pro-
getto Speciale Nazionale “Controllo dei Processi:” CIOC-CNR.
F. Filicori is with the Faculty
of
Engineering, University
of
Ferrara,
44100 Ferrara, Italy.
C.
G.
Lo
Bianco and A. Tonielli are with the Department
of
Electron-
ics, Computer and System Science (DEIS), University
of
Bologna, 40136
Bologria, Italy. A. Tonielli is the author to whom correspondence should
be addressed.
IEEE Log Number 9204611.
the performance
of
positioning servos [ll,
[2].
Direct-drive
connection is the best solution in this application environ-
ment.
Direct-drive motors can be obtained by optimizing the
design and control of standard dc and ac machines [3] or
specifically designing a variable reluctance (VR) [4]-[6] or
switched reluctance (SR)
[7]
motor.
A
VR motor with a
very high number of poles can be designed. Hence, by
means of a proper electromechanical configuration it is
possible to select the desired torque/velocity ratio.
The ideal robotic actuator, besides direct-drive Lonnec-
tion, should reach position with high accuracy, provide
constant torque output for every angular position/veloc-
ity, and keep constant velocity even in the presence of
variable loads. With VR motors, stepping drives
[81
are
not able to satisfy any of these specifications, mainly
because they do
not
consider motor nonlinearities. Dy-
namical performance can be improved by feedback con-
trol, as shown in
[9],
but neglected nonlinearities cause
torque ripple, thus preventing the use of this approach
where accurate trajectory tracking
is
needed. Torque non-
linearities are considered in [lo], where an interesting
controller structure is suggested. The limitations of this
approach are related to the constant velocity and mag-
netic linearity assumptions. To meet the specifications
required for a robotic actuator, it is necessary to carefully
take into account all the motor nonlinearities through a
suitable motor model and then design an advanced con-
trol system that linearizes motor and load characteristics
Our work aims at the design
of
a high-performance
ripple-free dynamic torque controller for a VR motor,
intended for trajectory tracking in robotic applications.
To
this aim, an accurate non-linear dynamical model of the
motor is the basis for any further activity.
An
original
modeling approach is proposed, in order to simplify the
design of the high-performance torque controller. The use
of flux as the selected state variable plays a fundamental
role in the simplification of modeling and control prob-
lems.
A
simple model, which is linear with respect to the
control variables, results from this choice; furthermore,
unlike other proposed models
[5],
[ll], [12], only nonlinear
functions of one independent variable are involved, even
when magnetic saturation is considered.
A
cascade controller structure similar to the one pro-
posed in
[lo]
was selected. It consists of a static feedfor-
ward nonlinear compensator, cascaded to a nonlinear flux
[ill,
[121.
0278-0046/93$03.00
0
1993 IEEE
~
106
IEEE
TRANSACTIONS
ON
INDUSTRIAL ELECTRONICS, VOL.
40,
NO.
1,
FEBRUARY
1993
or current closed-loop controller. The feedforward nonlin-
ear compensator transforms the torque set point into a
corresponding flux or current one, whereas the internal
(flux or current) closed-loop nonlinear controller is based
on estimated or directly measured feedback.
The selection of flux or current as the directly con-
trolled variable in the internal closed-loop controller, de-
pends on different factors [17]. Flux can be the best
choice in a microprocessor-based realization, allowing for
implementation of a flux observer. Current may be pre-
ferred in a sliding mode design, implemented with analog
technologies, because it is directly measurable. According
to the selection of flux as the state variable in the model,
the design of a feedforward compensator
is
performed
with respect to the flux controller. Transformation of a
flux set-point into a current one can be easily effected
using the proposed model.
A
current controller is consid-
ered in this paper.
In the design of the feedforward compensator, the
problem of transforming a scalar torque into an equiva-
lent three-phase flux vector is considered. The proposed
model, with its simple structure and nonlinear functions
of one variable, greatly simplifies the task.
Unlike [lo], optimization techniques are used for the
design of the feedforward precompensator, and the non-
linear closed-loop controller is designed in a stator refer-
ence frame, thus avoiding the use of coordinate trans-
formations. Two different performance indexes are
considered in the optimization procedure; the first, opti-
mal at low velocity, minimizes copper losses, whereas the
other, optimal at high velocity, minimizes the inverter
voltage required to impress the flux.
Even if it is specifically designed for a commercial VR
motor called
NSK
Motornetics RS-1410
[6],
the proposed
solution can be used for other VR motors with magneti-
cally decoupled phases, both rotating [4] and linear
[5].
In the paper, an original procedure for the determina-
tion of a VR motor dynamical model is proposed and
compared with other solutions. Model structure and pa-
rameter estimation techniques are presented. Different
approaches to the overall torque controller design prob-
lem are then discussed and the adopted solutions illus-
trated. Details on the optimal design of the feedforward
torque compensator are presented. Two different perfor-
mance indexes are considered and optimal solutions are
compared. Experimental results are then presented to
validate the model, whereas the optimized flux profile and
the validity of the overall approach are verified by prelimi-
nary simulations of the torque control system.
NOMENCLATURE
Voltage, current, flux linkage
Winding resistance
Torque;
T
denotes
a
particular torque value
Phase index
Rotor angular position and velocity
Rotor dumping factor and inertia
Energy stored in the magnetic field
M
N
Number of teeth in the rotor
Number of turns of a stator winding
11.
DYNAMICAL MODEL
OF
A
VARIABLE RELUCTANCE
MOTOR
The actuator considered in this work is the
NSK-
Motornetics RS-1410 three-phase rotating motor, whose
structure is shown in Fig. 1. The most interesting charac-
teristics are the structure of the teeth, the phase position-
ing, and the double stator. Selection of a structure with a
very
high number
of
teeth
(150)
corresponds to the choice
of a direct-drive realization with very high torque and very
low velocity, as required in robotic applications. Phase
positioning and double stator guarantee optimized mag-
netic paths to the flux. For the motor considered, the
following properties have been experimentally verified:
0
Magnetic hysteresis and Foucault current are negligi-
0
The three phases are almost completely magnetically
Equations describing the stator magnetic circuit are,
ble
decoupled.
independently for the three phases,
In the following, whenever this does not generate mis-
understandings, references to phase and time will be
omitted.
In
order to put (1) in a state space form,
two
different alternatives exist, depending on the selected
state variable.
The current, being directly measurable, is the most
common and natural choice in a model built for control
purposes. This solution, adopted in
[5]
and ill], leads to
the following nonlinear model:
di
dt
with nonlinearities that are functions of
two
variables.
variable, leading to the following nonlinear model:
Alternatively, the flux can be selected as the state
d@
dt
__-
-
-$(e,@)
+
U.
(3)
This simpler formulation, when saturation of magnetic
circuits is being considered, still requires the definition of
a nonlinear function of
two
variables and is based on a
state variable that is not directly measurable. In our
research activity model
(3)
is considered. This paper shows
how the specific nonlinear function in
two
variables
f(0,
@)
contained in
(3)
can be split into the sum of
two
nonlinear functions in one variable and how this model
can simplify the design of a ripple-free dynamic torque
controller.
FILICORI
et
al.:
MODELING
AND
CONTROL STRATEGIES FOR
A
VARIABLE
RELUCTANCE
107
Stators
\Rotor
Fig.
1.
Cross-section
of
the motor and magnetic
flux
path
of
a
phase.
In order to obtain the complete state-space dynamic
model for the motor, the nonlinear function
f(0,
@)
in
(3)
must be specified, as well as torque and mechanical equa-
tions.
As
far as the
f(O,@,>
function is concerned, a
nonlinear periodic dependence on the angular position
must be considered for a
VR
motor.
On
the other hand,
the flux can enter linearly or nonlinearly depending on
motor operating conditions.
A
linear magnetic circuit can
be considered, greatly simplifying the model [lo],
[
141.
However, in several application fields-as, for example,
robotics-high accelerations are required and the motor
must be operated under magnetic saturation conditions to
maximize the torque/mass ratio. This imposes consider-
ing also a nonlinear dependence of function
f(0,
@)
on
the flux.
The procedure for the determination of
f(0,
@)
starts
by considering the magnetomotive force (m.m.f.1 devel-
oped by a single phase:
Ni
=
H(O,@).
(4)
On analyzing the magnetic circuit shown in Fig. 1, it can
be assumed that the m.m.f. is decomposable in
two
com-
ponents: the first, which refers to the iron portion of the
magnetic flux path, nonlinearly depends on flux and is
assumed to be almost independent on the angular posi-
tion; the second, taking into account the air-gap portion,
is assumed to be a linear function of flux and periodically
depends on the angular position. Including the scaling
factor
N
in the nonlinear functions, an expression for the
current is, therefore,
i
=
F(@)
+
R(8)*@
(5)
where
F(@)
describes nonlinear effects in the iron part of
the flux path, whereas
R(8)
represents a position-depen-
dent term associated to the air-gap reluctance. In particu-
lar,
R(0)
can be interpreted as a normalized air-gap
reluctance.
The torque equation can be derived in several different
ways [151, [16]. For example, by means of the D’Alembert
principle, assuming a virtual displacement at constant flux,
the motor torque generated by a single phase is
From
(51,
it follows that
(7)
Equation
(7)
quantifies
two
well-known properties
of
VR
motors. The first property is that torque is
a
quadratic
function of flux. This means that the sign of the torque
developed by a single phase does not depend on the sign
of the flux (and hence of the current) but on the motor
angular position only. The second property is that at
constant flux, the torque
is
proportional to the derivative
of the air-gap reluctance with respect to position. This
explains why a motor with a larger number of teeth must
be designed in order to generate a larger torque. Besides,
(7)
shows that to generate a constant torque indepen-
dently of the angular position, a variable flux must be
generated since
R
is
a periodic function of angular posi-
tion.
Considering also mechanical equations, the complete
dynamical model for the motor is:
A.
State Equations
-
de
dt
--w
-
B. Output Equations
i,=f,(e,@,)
=F(@,)
+R,(8)@,;
j=O,1,2
(lla)
111.
MODEL IDENTIFICATION
AND
PARAMETER
ESTIMATION
In
the previous paragraph, the structure of a model for
VR
motors was introduced. Two different parameter esti-
mation techniques are proposed. The first requires cur-
rent and flux measurements only, whereas the second also
needs the torque.
In
the final paragraph reporting experi-
mental results, it is shown that a direct flux measurement
is not strictly required, since flux can be obtained indi-
rectly by current and feeding voltage measurements.
The only terms to be identified in (8)-(11), because
they are not directly available from motor data sheets, are
F(@)
and
R(8).
Before we enter into estimation details, it
is worth remembering that, due to the complete symmetry
of the three phases of the motor, the unknown functions
must be defined for a single phase only. Hence, to simplify
notations, reference to the phase will be omitted.
I08
IEEE
TRANSACTIONS ON INDUSTRIAL
ELECTRONICS,
VOL.
40,
NO.
1,
FEBRUARY
1993
A.
Estimation Method Based
on
Current and
Flux
Measurement
Remembering that
F(@)
represents nonlinearities re-
lated to magnetic saturation, a monotonic function can be
assumed to be of the kind
H
F(@)
=
FrQr
=
Fl@
+
F*(@)
(
12)
r=
1
where
F,
represents the coefficients
of
the expansion and
F*
is a strictly nonlinear function since it includes all the
polynomial terms of an order greater than one. The term
F,,
has been neglected because magnetic hysteresis
is
negligible. Consequently,
(5)
can be rewritten as:
i
=
F*(@)
+
(F,
+
R(O))@
=
F*(@)
+
R"(0)@.
(13)
Determination
of
R"f0):
Since
F*(@)
is a strictly nonlin-
ear function, it follows from
(13)
that
i
=
R"(
0)@
if
@
-
0. (14)
Term
R"(0)
represents the slope of the curve
i(0,@)
around the operating point
@
+
0
and, hence,
Samples of
R"(0)
at a different angular position
0,
can
be experimentally derived by estimating the slope of the
corresponding flux curve, as a function of the current,
near the origin. To completely define function
R"(0),
a
suitable interpolating function and an approximation cri-
terion must be defined. By defining
0
=
0
as the position
of full stator and rotor teeth alignment for the phase
considered, observing that
R"(
0
is a periodic function
symmetrical with respect to
0
=
0,
the following represen-
tation can be selected:
L
R0(0)=R,+R(0)=R,+ cRrcos(MrO).
(16)
r=
1
Order
L
and coefficients
R,,
defining the trigonometric
series
(161,
can be computed by means of discrete Fourier
transform.
Determination
of
F*f@):
Equation
(13)
gives
F*(@)
=
i
-
R"(
e)@.
(17)
From any
0,
(e.g.,
0
=
01,
samples of
F*(@)
can be
obtained from
(17)
(after the computation of
R"(0))
by
means of experiments at different current and flux.
According to
(121,
order
H
and coefficients
F,,
com-
pletely defining the polynomial function
F*(@),
can be
obtained by means of the least-squares method.
B. Estimation Method Based
on
Current,
Flux,
and Torque
Measurements
ten as
Bearing in mind
(16),
torque equation
(7)
can be rewrit-
dR(0) dR"(0) 2T
(
18)
~____-
-
--
-
d0
d0
oz.
Values of
dR(
0
)/d
0
(or
dR"(
0
)/d
0
),
sampled at dif-
ferent rotor positions
On,
can be obtained by means
of
flux
and torque measurements. Using
(16)
for
R"(O),
it is
possible to obtain
dR"(
0)
L
=
-
R,Mr
sin(
Mr0).
(19)
r=
1
d0
Again, order
L
and coefficients
R,
of
(19)
can be
calculated by means of the discrete Fourier transform.
Coefficients
R,
also parametrize the periodical function
R"(0)
in
(16)
up to constant
R,.
If
R,
is considered equal
to zero, an estimate of function
R(0)
is obtained. This is
not a limitation because the constant term
R,
can be
considered to be the
F,
term of
F(@)
function, (see
(12)
and
(13)).
The procedure proposed in Section
111-A
for the esti-
mation of
F*(@)
is not limited by the hypothesis that the
function is strictly nonlinear. This observation enable the
same procedure to be used for the estimation of function
F(@)
in the most general form
(12).
CONTROLLER
In motor equations
(8)-(11),
strong non-linearities are
present. The design of a closed-loop controller for a
VR
motor is hence quite difficult and requires a precise
definition of the control targets as well as advanced con-
trol algorithms.
In
this paper interest is concentrated
on
robotic appli-
cations. The multiple-axis robot controller is usually de-
signed under the assumption that torque actuators are
available. Electric motors are not directly torque actua-
tors; a suitable drive configuration is required. Standard
dc or ac drives are usually designed as velocity or position
actuators even
if
they internally contain current loops.
In
robotic applications a torque drive could be preferable,
whereas velocity and position control
of
coordinate axes
are dealt with in the main robot controller. These consid-
erations have suggested the design of a torque-controlled
drive for the VR motor. For this particular type of motor,
besides imposing a desired dynamic behavior on the
torque, the controller should also guarantee ripple-free
torque operation independently of rotor position and ve-
locity.
Direct implementation of a closed-loop torque con-
troller would require torque measurement
or
estimation.
This solution is, in general, very expensive and not conve-
nient in a direct-drive application, because a torque sen-
sor between the motor and the load would introduce
undesired elasticities in the link. On the other hand,
strong nonlinearities and very low motor velocity prevent
simple implementation of a motor torque observer.
Iv. GENERAL STRUCTURE
OF A
NONLINEAR
TORQUE
FILICORI
et
al.:
MODELING AND CONTROL STKATEGIES
FOR
A VARIABLE RELUCTANCE
IO0
Fig.
2.
Block
diagram
of
the
controller.
In motor drives based on standard dc and ac motors
(both synchronous and asynchronous), the torque control
problem can be solved by transforming it into an equiva-
lent current control one. This simple solution is possible
because the torque is proportional to the current, or to a
specific component of the current vector in a proper
reference system. Besides, owing to the wide availability
of high-quality and low-cost current transducers, this
solu-
tion is also convenient from an economical point of view.
For
VR
motors, the torque versus current function is
nonlinear, thus preventing the simple solution adopted in
drives for standard motors.
To
overcome this problem
a
cascade controller structure, similar to the one proposed
in
[lo]
and illustrated in Fig.
2,
has been selected. It
consists of an external static feedforward nonlinear com-
pensator, followed by
a
nonlinear flux or current (depend-
ing on design choices) closed-loop controller. The feedfor-
ward compensator transforms the torque set point into a
corresponding flux or current one, whereas the internal
closed-loop controller is based
on
estimated or directly
measured feedback, depending on the controlled variable
selected. Unlike [lo], optimization techniques are used for
the design of a feedforward precompensator, and the
closed-loop controller operates in
a
stator reference frame,
thus avoiding the use of coordinate transformations.
This paper presents motor modeling and control opti-
mization activities. Emphasis is placed on the optimiza-
tion techniques used in the design of the feedforward
compensator. Work related to the design of the closed-
loop flux or current controller is currently in progress
[
171
and is only briefly reported here in order to validate the
design of the feedforward compensator. Before entering
into details about the feedforward compensator design,
some general considerations are worth making.
Direct calculation of a current set point is not conve-
nient because torque dependence on current must also
consider magnetic nonlinearities. Recalling
(6),
a
simpler
relationship exists between torque and flux. Therefore,
the feedforward compensator is designed under the
as-
sumption of an internal flux closed-loop controller. If
current is selected, the flux set point can be directly
converted into
a
current one by means of the model
output equation (lla). It must be pointed out how the
proposed model structure greatly simplifies the design of
the torque controller.
A
critical point
is
the transformation of the scalar
torque request into
a
corresponding three-phase flux vec-
tor. Recalling motor equation
(11,
it can be noted that
fluxes relative to different phases can be impressed inde-
pendently by means of the associated control inputs
U,;
the control problem thus has as many degrees of freedom
as the number
of
phases. These degrees of freedom can
be used for different purposes. In [lo], for
a
four-phase
motor, two adjacent phases, selected according to the
actual rotor position and torque sign, are used to impose
torque dynamics and ripple-free operation. The remaining
two phases are controlled in order to keep their current at
zero. In [ll], for an m-phases motor, the desired dynamics
is imposed
on
motor acceleration by controlling a single
phase, selected as a function of position and torque sign.
The remaining controls must keep the remaining phase
currents at zero or rake them to zero
as
fast as possible.
Both approaches have problems, mainly related to the
need for
a
fast switch-on and switch-off of phase currents
that impose
a
voltage waveform that is strongly impulsive.
Since the voltage is limited in
a
real power inverter,
an
increase in the residual torque ripple occurs. Further-
more, the solution proposed in [ll], while allowing good
dynamic specification of the error between the actual and
the desired acceleration, does not control the torque
ripple explicitly.
The approach considered in this paper attempts to use
as
many degrees
of
freedom as possible in order to get the
best performance from the motor.
As
already mentioned,
the selected controller structure is shown in Fig.
2.
Inde-
terminations in transforming the scalar torque
TG
into
the flux vector
4*,
are used inside the feedforward com-
pensator to obtain optimal ripple-free “torque-sharing”
flux functions. Two optimization criteria are proposed: the
first, optimal
at
low velocity, minimizes copper losses; the
second, optimal at high velocity, minimizes the maximum
value of the motor feeding voltage at maximum velocity
and torque.
Independently of the selected optimization criterion,
due to motor symmetry, “torque-sharing” flux functions
are periodical and the same function is optimal for all
three phases. For every rotor position
8,
the same func-
tion can be used for the three phases with an argument
0,
=
6’
+
j0,
(j
=
0,
1,2)
shifted by one-third or two-thirds
of a motor step
(0,
=
;*(2.rr/M)).
Moreover, optimal
“torque-sharing” flux functions can be computed off-line,
in the worst operating condition,-for a single torque value
(typically,
a
large torque value
T
deliverable without rip-
ple).
In fact, from the optim5l flux function
&(HI
computed
at the maximum torque
T
deliverable without ripple un-
der the specific operating conditions, the -flux function
corresponding to a lower torque value
T
=
TK
(0
5
K
5
1)
can be obtained by
By means of this simple scaling operation, suboptimal