Note: Descriptions are shown in the official language in which they were submitted.
CA 02489181 2004-12-O1
FIELD OF THE INVENTION
The present invention relates generally to control systems and their deign.
More
particularly, the present invention relates to a tongue control system for a
hydraulic
actuator and the design of such system.
BACKGROUND OF THE INY~ENTI41~1
Hydraulic actuators are widely used to drive robotic manipulators in indusuy
for
tads such as earth moving, material handling, construction and manufacturing
sutomapon duo to their large power-to-mass ratio. However, precise control of
hydraulic
actuators is more difficult than control of conventional electric motors due
to the
presence of non-linear flow-pressure characteristics, such as: variations in
the trapped
fluid volume in each actuator chamber; fluid compressibility; friction between
rtloving
parts; variations of hydraulic parameters; presetue of leakage; and
transmission ziott-
linearities.
Much of the work on hydraulic control relies on linear eonttal design
methodology
that is based on local liberalization of the actuator dynaxmcs about a nominal
pperating
paint. However, these methods suffer from two major drawbacks: First, since
the actuator
dynamics are highly non-linear, a single linear time invariant controher can
be only tuned
far a particular operating point and the performance degrades as the system
state moves
away tom the operating point. Second, since the dynamics of actuator and load
are coupled
together, the dynanucs of the load (which can be very complex) are implicitly
embedded irl
the linearized model of the actuator, which complicates the treatment of the
situation; it is
then difficult to achieve precise control_ As a result, these methods rely
highly on The
knowledge of the load chasacteristics and vaTiadon in those characteristics.
Mechatronics systems, such as electro-hydraulic mbot manipulators, are
essentially
multi-dimensional non-linear systems comppsed of mechanical and actuator
subsystems
accounting for load dynamics and actuator dynamics, respdctively. Tine control
problem ran
be greatly simplified in many applications, if actuators behave as do ideal
source of
force/torgue with low impedance, I.e. similar to electric motots_ However,
rite force/totduc
CA 02489181 2004-12-O1
generated by a hydraulic actuaior is a$eeted by its own motion resulting ip a
coup)od
dynamics of the actusttor and load.
Ta account for parametric uncartaanty of estimated and/or somewhat
itia,ccusate
parameters, non-linear adaptive control methods have previously been employed.
An
adaptive robust method can also be used, which takes the nonparametric
unmodeled
dynamics into account by assuming a known bound on the no~parame4ic
uncertainty.
Dynamic feedback lineariaation has been used to attempt jo cancel out the
actuator non-linear dynamics. The advantage of this method is chat for farce
control
purpose, no knowledge of load dynamics is required because iI cancels out the
ef feet of
velocity pextusbatioti. l-Iowever, in practice, exact cancellaTion of tha
actuator dynamics is
not possible due is paratnetric and nonparatnetric modeling uncertainties.
This problem
has been addressed by transforming the linearized system info sta~ard linear
fractional
uncertain structures; however, no method has been presented for cottiputarion
of
uncertainty bounds.
The control of a torque/foree output is very different in nature from known
anea~pts to control motion or position. When a controller is for controlling a
system in
which there is no morion, or negligible motipn, much of the hydraulic
behaviour is
masked, and friction is the main observable factor affecting the system. This
is one
reason why many known systems seek to compensate for friciion~ camponeats. 1t
is
also necessary to consider the dynamics of the whole system together, namely
the
combination of the actuator and the load. In torque/force control situations
such as ihost
discussed according to embodiments of the present invention, ii is necessary
to consider
the eflfect of veiacity on the system; as such, the iotque/force cAntrol
problem is quite
different in nature from the motion and displacement control problems.
It is, therefore, desirable to provide a procedure far idenufic$tion of
actuator non-
linear dynamics arid quantification of modelling error. Mast existing adaptive
methods
deal only with parametric uncesiainties and some robust adaptive schemes
assume a
krsown bound for non-parametrlc uncertainties in actuator non-lincdr dynamics.
No
method is known to estimate ~s bound, and attempts to accost fvr non-linear
dynamics
have drawbacks.
2
CA 02489181 2004-12-O1
5UMMwl~Y OF THE 11IW>ENTION
It is an object of the present invention to obviate or tuitigatc at last one
disadvantage of previous torque/force controllers and methods of their design.
The robot control problem is simplified by minimizing the coupling between the
two sub-systems that can be achieved by minimizing the effect of velocity
disturbance on
actuator torque. Then the robot control probleta is effectively red4ced to the
torque
control of the hydraulic actuator and the control of the mufti-body dynamics
of a
manipulator that traditionally relies on torque control inputs.
In a first aspect, the present invention provides a controller #~or a
hydraulic
actuator, the hydraulic actuator being for generating a maunpulating influence
to be
applied to a load. The controller includes a linearizing controlldr for
storing a linear
model representing non-linear dynamic behaviour of an unloaded hydraulic
actuator. The
linearizing controller is shown as part of an inner loop. A robust linear
controher is also
provided for compensating for non-]inearitirs in the linear model using an
uncertainty
model having a non-linear component. and an estimated bound foT the
uncertainty model.
The robust linear controller is shown as part of an outer loop. preferably
cascaded wish
the inner loop.
In the case where the hydraulic aottiator includes a joint, the non-linear
dynamic
behaviour of the unloaded hydraulic actuator can be obtained by substantially
minimizing
effects of the load on the manipulating influence. This is achieved by
perturbing the
linear model in response to a velocity of the joint. The linear model can be
based on
measured linear parameters of the hydraulic aciuatos. The linearizlng
controller can
include means for obtaining the Imear model based on a lineaiizing control law
for the
hydraulic actuator, or means fpr determining the littearizing control law for
the hydraulic
actuator. The controller can further include means for calculating the
estimated bound.
The manipulating influence can include a torque or a force, and the hydraulic
actuator
can be a rotary hydraulic actuator or a linear hydraulic actuator.
in another aspect, the present invention provides a control architecture for
torque
control of a hydraulic actuator. The control architecture includes a dynamic
ferdback
linearizing Inner loop, and a robust linear feedback outer loop. Tha outer
loop is
CA 02489181 2004-12-O1
cascaded with the inner loop to permit the actuator to generate a desit~ed
torque
irrespective of motion of the hydraulic actuator.
In a farther aspect, the present invention provides a method of designing a
hydraulic actuator conIroller. The method includes the following sups:
determining an
uncertainly model for a linearized model of the hydraulic actuator, the
uncertainty model
including a non-linear component; estimating an uncertainty bound, for the
uncertainty
model, based on identified parameters of non-linear behaviour of the actuator;
and
designing a robust linear controller based on the determined unceiiainty model
and the
estiutated uncertainty bound.
The step of estimating the uncertainty bound can include the following steps:
applying a lineaiizitig control law using control input as an excitation
signal; identify;ng
a linear discrete time model as the uncertainty model, based on measured
values of torque
and control input; and computing a minimum value of the uncertainty bound such
that the
uncertainty model is not invalidated by the tneasttred ~ralues of tpi'tlue and
control input.
The non-linear component can be based on unmodeled actuator dynamics. The
linearizing control law can be a dynamic feedback linearizing control law. The
step of
designing the robust linear controller can include imposing robust stability
and.
performance constraints based on characteristics of the uncertainty model. The
method
can further include the step of calculating a linearizing control law based on
the identified
parameters of non-linear behaviour of tha actuator_ The method cats further
include the
step of extracting the identified parameters based on measured signals.
According to another aspect, the present invention provides a computer-
readable
storage medium, comprising statements and instructions which, when executed,
cause a
computer to perform a method according to embodiments of tie present
invention.
ether aspects and features of the present invention will become apparent to
those
ordinarily skilled in the art upon review of the following description of
specific
ctnbodiments of tha invention in conjunction with the accompanying tlgures.
$>6tIEF DESCRIPTION OF THE DRAW11VGS
F~nbodirneiics of the present lnveritinn will now be described, by way of
example
only, wish reference to the attached Figures, wherein:
4
CA 02489181 2004-12-O1
Fig. lA is a block diagram of a force controller according to an embodiment of
the present invention;
Fig. 1B is a block diagram of a torque controller according to an embodiment
of
the present invention;
Fig. 2 is a flowchart illustrating a method of desiring a hydraulic $ctuator
controller according to an embodiment of the present invention;
Fig. 3 is a black diagram of an uncertainty model structure according to
embodiment of the present invention;
Fig. 4 is a block diagcatn of as uncertainty model structure according to
another
embodiment of the present invention;
Fig. 5 graphically illustrates a validation of a non-linear model according to
an
embodiment of the present izlvention;
Fig_ 6 graphically illustrates a $-equcncy response of a linee~rized tnodcl
represented by an uncertainty model stntcture according to an embodiment of
the present
invention;
Fig. 7 graphically illustrates a limit cycle in actuator dynarrlics in the
presence of
a high gain external conuoller;
Fig. 8 graphically illusuates frequency responses of resulting input and
output
sensitivity functions together with weighing functions;
Flg. 9 graphically illustrates an effect of external torque disturbance ort
generated
torque;
Fig_ 10 graphic$lly illustrates step tracking of actuator torqtyo in the
presence of
velocity;
Fig. 11 graphically illustrates 0.5 Hi sine tracking of actuator torque in
presersce
of velocity;
Fig. 12 graphic$lly illustrates 2 He sine tracking of actuator torque in
presence of
velocity;
Fig. 13 gaphically illustrates 5 H~ sine stacking of actuator torque in
presrnce of
velocity;
S
CA 02489181 2004-12-O1
F~~. la graphically illustrates step response of actuator torque in absence of
velocity;
Fig. 15 graphically illustrates 0.5 Hz sine tracking Qf actu$lor torque in
absence of
velocity;
Fig. 16 ~-aphically illusTrates 3 Hz sine tracking of actuator jorque in
absence of
velocity; and
Fig. 17 graphically illustrates 5 Hz sine tracking of aci~tator torque in
absence of
velocity.
DETwiLED DJESCIg.IPTION
Generally, the present invention provides $ method and system for
identification
and torque~foree control for hydraulic actuators, such as rotary hydraulic
actuators. The
merhodology can be readily applied to linear hydraulic actuators. The
composite
controher consists of a dynamic feedback linearizing inner loop cascaded with
a robust
linear feedback outer loop. The proposed coniroher allows the aciitaior to
generate
desired torque irrespective of the actuator motion. In fact, the controller
reduces
sigtuficantly the impedance of the actuator as seen by its external load,
making the
system an ideal, or substantially ideal, source of torque suitable far rrlaay
robotics and
automation applications. A.ti identification method to extract the parameters
of non-linear
model of actuator dynamics and to estimate a bound far modeling uncertainty,
used for
synthesis ofihe outer optimal controller, is also presented. Results are
illustrated
experimentally on a pitch actuator of a Schilling industrial robot.
In the realm of hydraulic actuators, rotary hydraulic actuators produce a
torque
output, whereas linear hydraulic actuators produce a force output. The term
"manipulating influence" is used herein to represent either a torque or a
force, and is used
as a generic term co cover both possibilities, depending on whether a
hydraulic actuator is
linear or rotary. The terms "torque/force control" and "hydraulic actuator
output control"
are used herein to refer to the control of the output of the hydraulic
actuator, whether ii is
a linear or rotary actuator. The terms "torque/force controller" and
"hydraulic actuator
output controller" as used herein represent a controher for cotitTolling force
(in the case
of a rotary actuator) and/or for controlling torque (in the case of a linear
actuator). Of
course, the term "controller" is also used herein to refer to a torquelforce
controller- The
6
CA 02489181 2004-12-O1
term 'velocity" is used herein to represent a speed and trajectory (either
linear or
angular). Though reference is made to actuator velpcity, joint velocity, and
load velocity,
it is to be understood that each of these velocities is describing the sa=ne
velocity in the
case where an actuator is applying a torquelforee to a load. As Such, a
calculation of load
velocity can be used to determine actuator velocity.
The control of a torque/force output is very different in nature from lmown
attempts to control motion or position. When a controller is for Furetrolling
a system in
which there is no motion, or negligible motion, much of the hydraulic
behaviour is
masked, and friction is the main observable factor affecting the systettt.
This ix one
reason why many known systems seek to compensate for frictional components. Ii
is
also necessary to consider the dynamics of the whole system together, namely
the
combination of the actuator and the load. In torque/force control sltuatioas
such as those
discussed according to embodiments of the present invention, it is necessary
to consider
the effect pf velocity on the system; as such, the torque/force control
problem is quite
different is nature from the motion and displacement control problems. 1n
torqucJforce
control according to embodiments of the present invention, the two subsystems
are
decoupled by way of velocity feedback. This results 1n systettx modularity. It
is no
longer required to model the environment, since the actuator can be controlled
uidependently of the effects of the load.
The robot control problem is greatly simplified by minitttizitig the coupling
between the two sub-systems that can be achieved by minimizing or eliminating
the
effect of velocity disturbance on actuator torque. Then the robot control
problem is
reduced to the torque control of bydraulic actuator and the control of the
multi body
dynamics of manipulator th$t ireditionally relies on torque control inputs.
A combined scheme of identification and torque control is described for
hydraulic
actuators, such as rotary hydraulic actuators. The methodology is readily
applicable fee
linear hydraulic actuators. The composite cantmller includes a dynamic
feedback
linearizing inner loop cascaded with an optimal robust linear feedback outer
loop. The
proposed controller allows the actuator to generate desired torque
irrespective of the
actuator motion. In fact, the controller reduces significantly the itupedanee
of the actuator
as seen by its external load, making the system an ideal source of torque
suitable for
7
CA 02489181 2004-12-O1
many robotics and automatioTl applications. Discussion of and "ideal" source
herein is to
be understood as referring to a source having behaviour that is subs~nually
Ideal.
The controller allows a hydraulic actuator to generate desired torque or force
regardless of actuator motion. It can reduce significantly the impedance of
the actuator as
seen by its external load, making the System an ideal, or neat ideal, source
of torque or
force suitable for many robotics and auioautiion applications. ~'he dynamic
feedback
linearizing controller can be constructed based an an identification procedure
which
identifies paraiueters of actuator non-linear model. Since the feedback-
linearizcd model
is not perfectly "Linear", a novel identification procedure is developed to
fit an
"4ncerrain" model structure to the almost lineari2ed system. The robust outez-
loop
controller permits to consider different types Qf performance objective,
either in time
domain or in frequency domain. The proposed linear robust outer-loop coatmller
presents a very efficient means for attenuating limit-cycle oscillailons in
servo-valve
dynamics, and is discrete and easily implementable. A11 desigp procedure can
he realized
by means of powerful convex optimization algorithms.
The identification and control scheme according to embodiments of The present
invention relies neither on a priori knowledge of load dynamics nor external
forgoes, and
ma1ces the eonuolled actuator as a source of torque with low impedatlee, i.e.
acting virtually
as an electro-motor_ Moreover, no a priori assumption is made on a bound for
actuator
utunodded dynamics. The bound is estimated via an identification procedure.
The coptroller
synthesis procedure allows imposing several performance objectives either in
time-domain
or frequency domain and it comprises numerically tractable convex
optimizations. An
identif canon method to extract the parameters of non-linear model of actuator
dynamics
and to estimate a bound for modelling uncertainty, used for synthesis of the
outer optimal
controller, is also presented.
Fig. IA is a block diagram of a force controller accoto an embodiment of
the present invention. This figure is similar in nature to Fig. '~$, except
that it illusit"ates
the system more gener$lly, and indicates forces rather than forgoes. As
mentioned
earlier, it is the type of actuator (lrnear ac rotary) th;~t determines
whether embodiments
of the present invention will act to control force or to control tprque; the
system itself is
the same in either case. The cositrollet is for a hydraulic actuator, the
hydraulic actuator
8
CA 02489181 2004-12-O1
being for generating a manipulating influence to be applied to a load. The
controller
includes a linearizing controller for storing a linear mcadel representing non-
linear
dynamic behaviour of an unloaded hydraulic actuator. The linearizing
controller is
shown as part of an inner loop. A robust linear controller is also prpvided
for
compens$ting for non-lineanties in the linear model using an uncertainty model
having a
non-linear component, and as estimated bound for the tmGertainty ruodel. The
robust
linear controller is shown as pelt of an outer loop, preferably cascaded with
the inner
loop.
In the case where the hydraulic actuator includes a joirii, the non-linear
dynamic
behaviour of the unloaded hydraulic actuator can be obtained by substantially
minimizing
effects of the load on the manipulating influence. This is achieved by
pemubing the
linear model in response to a velocity of the joint. The linear irtodel can be
based on
measured linear parameters of the hydraulic actuator. The linearizing
controller can
include means for obtaining the linear model based on a linearizi~g control
law for the
hydraulic actuator, or means for determining the lineatizing control law for
the hydraulic
actuator. The conuolltt can further include means for calculating ihc
estimated bound.
The manipulating influence can include a Torque or a force, and the hydraulic
actuavar
can be a rotary hydraulic actu.stor or a linear hydraulic actuator.
Described in a different manner, an cmbodirpent of the present invention
provides
$ control architecture for torque control of a hydraulic actuator. The control
architecture
includes a dynamic feedback linearizing iriner loop, and 3 robust linear
feedback outer
loop. The outer loop is cascaded with the inner loop to permit the actuator to
generate a
desired torque irrespective of motion o f the hydraulic actuator.
.As shown in Fig. 18, the actuator velocity and positiotr ars produced as a
result of
actuator torque z superimposed with external torquz z," . On the other hand,
actuator
velocity and motion, in rum, affect actuator torque through nori-litlear
actuator dynamics.
in this paper the effect of velocity on generated torque is considered as
disturbance. As
such, the control objective is to solve two tasks: c4mpensate for non-linear
dynamics for
achieving accurate reference tracking; and minimize sensitivity tA velocity
disturbance,
that is equiv$lent to increase the back-drivability of the actuator or to
decrease its
impedance.
9
CA 02489181 2004-12-O1
According to an embodiment of the present invention, the proposed conlToher
scheme includes three main stagzs, with an optional initial stage, which can
be
implemented first. In the in1t1a1 stage, parameters of the actuator non-linear
model are
identified. This identification can be achieved using the measured signals
(actuator
position, velocity, and chamber pressures) and by a standard leasT squares
algorithm.
In the first main stage, using the identified parameters of the actuator non-
linear
model, a dynamic feedback lineatizing control law u; is calculated (See Fig-
18)- The
term "control law" as used in this specificatipn is a common term in the art
used to
designate a control relationship that has been determined to be true for a
particular
arrangement. Neglecting the servo-valve dynamics, this command ideally
Transforms the
non-linear system into a simple mtegr$tor if v is considered as a never input
and T as
output. However, in the course of identification experiments it becomes
obvious that
there are non-negligible discrepancies between the non-linear model of
actuator whose
parameters are optimally ldentiFed and the tr4e system_ This implies that the
non-linear
model cannot perfectly capture the dynamical behavior of the actuator arid
that the
dynamic mapping from v to r differs frAm an integrator_ Therefore, having
implemented the feedback linearization, we perform the second round of
identification to
fit an uncertainty model stricture to the dynamic mapping from v to r _ This
uncertainty model structure includes a linear Time-invariant (LTI) irlodel G
and a non-
linear operator o chat represents both the parametric and non-pararrtetric
uncertdipties of
the actuator model (F(~ o) in Fig. 1R). Herein no assumption is made on the
value of
the uncertainty bound; instead, this bound is estimated from the
idantification procedure.
It is worthwhile to notice that the model uncertainty represented by the
operator o 1s non-
l:n~u~ and the standard methods for uncertainly bounding in system
idezttification theory
are not applicable in this case. The proposzd identification method is based
on the recent
resuhs in model vul:durior~.
The third stage of the proposed method is to design the external linear
controller
C satisfying several performance and robust stability requirements.
Specifically, we
translate these requirements imo 1, or H,~ control design speci~tcations that
in turn, are
formulated by some mixed Linear Programming and Linear IVJ:atnx. Inequality
constraints.
CA 02489181 2004-12-O1
»ased on the identified non-linear model, a dynamic feedback lincarizing
conuol
law can be calculated. This command should ideally uansfotm the non-linear
system into
a simple Integrator. However, due to imperfect parameter identification and
the presence
of unmodeled dynamics, the feedback-linearized system may dc~iate from a
simple
integrator model.
To describe the foregoing in a different manner, reference is made to Fig. 2,
which is a flowchart illustrating a method of designing a hydraulic actuator
conuoller
according to an embodiment of the present invention. The method of designing a
hydraulic actuator controller according to an embodiment of the present
invention
includes the fohowing steps: determining an uncertainty model for a linearized
model of
the hydraulic actuator, the uncertainty model including a non-linear
component;
estimating an uncertainty bound, for the uncertainty model, based on
identified
parameters of non-linear behaviour of the actuator; and designing a robust
linear
controller based on the determined uncertainty model and the estimated
uncertainty
bound.
Fig. 2 also illustrates some optional steps, which can be included in the step
of
estimating the uncertainty bound- The optional steps are as follows:
applying a linearizing control law using control input as ran excitation
signal; identif~rin.g
a linear discrete time model as the uncertainty model, based on measured
values of torque
and control input, and computing a minimum value of the uncertainty bound such
Thai the
uncertainty model is not invalidated by the measured values a f torque anal
control input.
The non-linear component can be based an unmodeled actuator dynamics. T$e
lineatzzing control law can be a dynamic feedback linearizing cozitrol law.
The step of
designing the robust linear controller can include imposing robust stability
aad
performance consuaints based on characteristics of the uncertainty model (such
constraints will be described later). The method can further include the step
of calculating
a linearizing control law based on the identified parameters of non-linear
behaviour of the
actuator. The method can fut~ther include the step of extracting the
identtfted parameters
based on measured signals.
Expressed in a slightly different manner, according to an embodiment of the
invention, there is provided a method of designing a hydraulic actuator
controller,
11
CA 02489181 2004-12-O1
cotriprising- deternnining an uncertainty model to compensate for differences
between a
linearizcd model of the hydraulic actuator and actual behavioyr of the
hydraulic actuator,
the unce~rtainry model including a ~a-linear component; estimating an
uncertainty
bound, for the uncertainty model, based on identified parameters of non-linear
behaviour
of the actuator; and designing a robust linear controller based on the
determined
uncertainty model and the estimated uneertaitzty bound. The uucertainry model
can
include a linear time invariant model component.
A method according to an embodiment of the present ~~v~pon can alternatively
be described as an identification method for designing a hydraulic actuator
controller.
The method includes the following steps: detcrznining the paratnetcrs of the
non-linear
model of the actuator dynamics; designing a linearizing cpntrol law based on
the
determined non-lineal model; fliting an "uncertain" model structure to the
almost
lineariied system, and estimating an uncertainty bound for the modellin&
uncertainty; and
designing a robust linear controller based on the determined uncertainty model
and the
estimated uncertainty bouttd-
The model structure can include a linear time-invariant (LTI) model and an
uncertainty block representing all factors that affect linearizaxion quality.
The LTI model
together with an upper-bound for the uncertainty block can be estimated by an
identification procedure. The identification method can handle non--linear
uncertainty
blocks, as opposed to existing mtihods that handle mainly linear
uncertainties. The
identification scheme requires no "a priori" information on the sysTem
dynamics.
The step of estimating the uncertainty bound can include the following steps:
using a linearizing conuol input as an excitation signal and torque as the
output signal;
identii~ring a linear discrete time model as the tmcertainiy model, based on
measured
values of input/oyiput signals; and computing a mdttimutrl value of the
uncertainty bound
such That the uncertainty model is not invalidated by the measured values of
torque and
conuol input signals.
Actuator dynamics
The dynatztics of hydraulic pressure of the chambers assuming co~ipressible
fluid
are described by
12
CA 02489181 2004-12-O1
- C, - fir, Ct cer - D
_1 ~1 _ Y, x) V, (x) P~ t V~ ~x~ t v< <x~ w + c Y' (x~ a (1)
!r p~ Ct - ~, - C~, p: c~.r 'D z
p
Yz (x~ Yz (x~ Vz (x) yz (x)
where Q is the effective bulk modulus, p~, p2 ate pressures inside the two
chambers of
actuator, x is the positipn angle, Y, (z) = V~ t Yx(Yz (x) ~ Vp - Dx) is the
trapped fluid
volume in the first (second) chamber, respectively. l~ is the volume
displacement of
actuator and z ~ (-D-' Va , D-1Vo ) .
The coefficients of the internal and external leakages are denoted by c1 and
c~h
respectively. a is the spool-valve displacement and
K, ~ 0.5(p: - p~ ) - ("1)t 6$n(u)~Q.S(P~ 1' Pa ) ' p, ~. _ =1~?~ (2)
where p, is the supply pressure, p~ is the external pressure and c~, is the
discharge
coefficient of the valve. 1n this embodiment, we assume an identical discharge
coe~cient
cp, for both inlet and outlet po=ts of the valve, although some servovalves
have larger cp,
for the outlet pore than the inlet ports, Generally, cP, depends on liquid
density, however
in this embodiment cP is considered constant. Obviously, it is possible to
change the
equations if different assumptions are made.
In this embodiment we neglect the servo-valve dynamics arid hence the servo-
valve displacement a is treated directly a$ contml input si~ai- The torque
generated by a
mtary hydraulic actuator r is proportional to the pressure difference between
the two
chambers, i.e_
t = D(P~ 'T Pz ) (3)
~t'fective bulk modules D Volume displacement
cP Discharge coefficient of c, Internal leakage
valve
External leakage Vi,2 Trapped fluid volume
in
chambers
Vo Initial fluid volume x #zosition ~gie
13
CA 02489181 2004-12-O1
w Angular velocity g,_., Supplied ~]ows
p,,a a Servo-v~ve displacement
laressure
inside
the
two
chambers
PQ Irxiernal pressure (control input)
z Hydraulic trarque Pp Supply pressure
v Control input for linearized t"f Torque reFerence
system signal
z",External torque disturbance y Upperbourtd for uncertainty
C lrxternal comrcaller B Vector of parameters
luPut-to-torque exact linrarizatiop
hifferentiating the actuator torque in (3) with respect to time and replacing
p1 and
p2 lzom the actuator dynamics equations yields (4) as follows:
'~ ~ -~(y + c~,)P(x)(P, -P.) t L~carp(x)Po -R~'~'(=')fv t ~iDcPQ(P,~Px~x~u)u
where w = x is the angular velocity of the actuator, and P(x), P~(x) and Q(p~,
pz, x, u)
are defined by
P(x) = Y, (x)-' t Vz (s)-~ (5)
P(x) = V' (x) ' - Vi (x) ' (6)
Q(Pi Pa ~ x, u) = Yv (x) ~ ~ t Vz (x)-' ~z (7)
Equation (1) describes the second order dynamics ofthe actuator. The fact That
the command signal a appears in the first derivative of the generated torque
shows that
the relative degree of the system 1s one. It is evident from Equation (4) that
the actuator
torque depends on two inputs: morion variables i.e. position and velocity (x,
~v ] and
spool-valve displacement u. Herein, the former is treated as known
disturbance, while the
latter is considered as control input. The goal of an ideal torque c4r~troller
design is hence
to perform precise torque tracking regardless of actuator motion.
14
CA 02489181 2004-12-O1
From Equation (4), the lineari2ing command can be computed by
x* _ ~(y t cr~ )(P~ - PZ )p(x) - RDc- P~ ~ (x) t Da~(x)~ t v (8)
Racp~(P~ ~ P. ~ x~ ssn(u*))
where v 1s the new command signal. Obviously, this cozitrol law transforms v-T
map into
an integr$tor, i.e. z = v. In order to itnplement the linearizing comtrtand
law in (8) we
need to express the conuol signal u~ explicitly tn terms of the measured
signals p1, pz, x>
w and the nzw input signal v. However, Equation (8) does not express ufi in an
explicit
form because u'~ appears in the right-hand side ()tHS) of (8). This problem
can be easily
solved by observing the definitions of Q and K;. In fact, one can il~fer from
(2) anal (7)
that Q(.) depends only on the sign of u'~
Therefore, by virtue of (8) and noting that scalars Q(.), cp, and (3 arr all
positive
valued, we can say
s$n(u ~ ) = si~(~(r', + c~r )(P. - Pz )p(x) - ~c~r~., p (x) t D3 ~(x)~ + v)
which shows that u* depends only on p~, pz, x, w and y. Note that the
linearizing
command (8) is applicable when Q ~ 0. For (p, , P2 ) _ (P~ ~ Pa ) or (P1 , p~
) _ ( pa , pa ) ~ Q
1s zero and the actuator dynamics becomes uncontrollable from the input. The
vanapons
ofpt and p, in this case, depend only on velocity_
ldeutitlcation of actuator dyn$mics and uncertainty bpunding
In this section we describe a two-stage procedure for pacatnetric
identification of
actuator non-linear dynamics and for quantizaiion of modeling uncertait~iy in
11 Topology.
At the fuse stage of this procedure, the patatneiers c~f the nolt-linear model
of the actuator
ace identified. It is assumed that the measurements of the pressure signals
p~, p2, the
velocity u> > the input signal a and the position x ate available; anc~ the
derivative of the
torque signal z is computed by numerical differentiation. Define
Y(P~ Pa ~ x~ ~~'~ ) _ ~-(PW P= )p(-r)~ - DPa pax), - D' p(x)~, ~Q(-)~~ ~d
CA 02489181 2004-12-O1
B = ~~(c1 t c,, ), ~tc~, , ~, ~co ] ~ , then equation (4) can be expressed in
standard linear
regression form
~' = I'(P> > Pz -x'~ ~~ u)B
The estimated parameter vector B is the solution to the following convex
optimization problem
minllr ~ Y(D~ ~ Pz ~'~. w~ u)BII P (10)
where IL~Ipdenotes signalp-norm.
However, in practice the noise caused by numerical differentiation of s is not
negllgible_ In order to analyze the effect of noise on the ideniific3tion
problem (10), let
denote cr as the noise introduced by numerical differentiation of z and define
p~,p, .
~l:
Also, define y - ~Y(ro )T , Ylr, )'',..., Y(rN ) ']~ . Let K 1 be the
condition nN.rnber of Y _
Obviously, a large K indicates that the regression matrix Y is close to
singularity. if ( 10)
is considered as a least squares problem (p = 2), then it can be shown that
for a ~ k-1,
we have
II YB Ilz ' 2(1 t K)Il er 1l= 1" IITIIz 0(~2 )
where B = B - B.
It is evident from this inequality Thai the propagation of not&e to the
identification
problem can be minimized when x is close to ona_ Since Y is a fWnction of
input signal
u, then x obviously depends on the choice of u_ This fact suggests that the
input signal
has an itnponant role in achieving the minimum possible parametric error B.
From
experiuTental point of view, if a set of feasible input signals is available
for identification
purpose, then one can choose the 'best input signal that minimizes x
Unmodeled dynamics: On the other hand, there is always a part of the actuator
dynamics that are not captuYed by the actuator torque dynamics equation (4);
this part is
referred to as u,nmodeled dynamics. According to an embaditnent of the present
16
CA 02489181 2004-12-O1
invention, we represent the unmodeled dynamics by a perturbation sigzial
d (p, , p z , x, w, ~, r). The unt~aodeled dynamics can be due to the servo-
valve dynamics,
hysteresis in the electromagnetic circuit that derives the valve operation,
deadband in
control valve, delay in the servo-valve, err. Moreover, The actuator can be
a$ected by any
perturbation that is not a function 4f actuator states. As shown in Fjg. l,
the external
torque z ~"~ is such perturbation that affects actuat4r dynamics throNsh the
velocity.
Therefore, the actuator torque dynamics is indeed in the form of
T=Y(p,,p"r,~,u)Btd(p,,pz,x,~,u,~)
which implies that if the identified parameters are used in the linearimng
command law
(8), the resulting dynamics will take the form
r-v =Y(p~,pZ,x~w,u)B td(P>>Pa.x,w,~,r) (12)
The first Term in the RHS of (12) refers to the pasametric uncenainty_ Note
Thai
the perturbation d (p, , p=, x, ~, u, r) is only a function of the estimated
parameters
(through its dependence on u) and not a function of the parametrie error B .
As a result,
we consider d(.) as the non-parametric uncertainty. fn the seduel we present
two methods
for representation of uncertainty. Clearly, there are many other
representations that cart
deal with specific cases of parametric or nnnparametric uncertainties. The
choice of each
representation depends on the n$ture of unce~ttainty as well as the available
foals for
solving the resulting identification problem. In general, uncertainty can be
represented by
linear fractional forms. However, The solution of the resulting model
validation problem
usually leads to non-convex optimizations that are not numerically tractable
(e.g. when
o E ~3tHm one should solve a so-called p problem)
Case A
Fig. 3 is a block dia~atn of an uncertainty model structure according to an
embodiment of the present invention, described as Case A. In pat'ticular, Fig.
3 illustrates
an additive model structure, where G is a model for an integrator.
Assume that both the parametric and non-parametric unce~tainry terms are
bounded and that they can be represented by
17
CA 02489181 2004-12-O1
x(PmP2~x~~~u)B td(PmP=,x. w, u, t)=~t(AY+e(t))
where o is a bounded non-linear operator with II°ly <_ y where the ~,
noun of an operator
like o mapping the sigpal x to the signal y, Is defined by I~oll1 -
sup~,~° ~~xli~ with
(IxIIo = sup, ~x(i)~ . Moreover, a (t) represents any perturbation that is not
a function of
system states. Consequently, equation (12) becomes
r = (D-1 t ~)v t e(r) (13)
where D"'' is the integration operator.
It is observed that the operator o enters as an additive uncertainty to the
integrator system. Now, the main problem is to compute an upperbound far this
opcratar.
The proposed method consists of a new Identification proced4re as follows:
Based on the
estimated parameters B , the linearizlng control law (8) is applied while
input L is
considered as the excitation signal. Then,
by usinb the measured values o f z and v , we identi fy a linear discrete time
model G
and compute the minimum value of y where II4Ij~ 5 y, such fat the following
uncertainty model structure
z = (G~ + WA)v t a (14)
is not iwaJidated by the experimental data z and v.
Here a represents sensor noise or any other external distwbancz to the
actuator
dynamics that is independent of system slates. This signal is assmxted to be
bounded by
I'eIIW 5 0- (15)
In literature, this type of identification scheme has been recognized as model
validation-based identifiedtion and It is based on the model validation
concept. In the
uncertainty model structure (14), W is a laiown weighting transfer function
aad the model
G represents the effect of integrator term in (13)_
18
CA 02489181 2004-12-O1
The main reason for using The l1 norm (or the Induced lm to l~, norm) for
characteri2ation of the uncertainty, is due to the fact that the uncertainty
in the actuator
dynamics has non-linear characteristic. Therefore, unlike many existing
methods for
bounding LTI utlcercainti~s in FI2 or Hm topologies, herein we need to use an
induced
operator norm for characterizing the non-linear model uncertainty. Moreover,
the
advantage of using h norm over the other Induced nouns is that the resulting
identification problem can be solved by a linear pr4gramrning-
For model G in form of a rational transfer function G(g-' ) = B(q-' ) l A(q-'
) ,
where q'' is a unit delay operator, the formulated identificatiotl problem is
tantamount to
solving a non-convex optimization problem. Art iterative algorithm known in
the urt can
then be used to solve the problem. Here it is assumed that the model C is
expressed in
terms of the orthonormal bests functions as
n
G(9-')=~<<~'r(~-')
where Fk (y-' ~ is the k-th known orthonormal basis and l, s are the
parameters to be
identified.
With this model description, the parameters appear linearly and the resulting
optimization will be convex. The ongoing analysis shows that the stated
idetitificatlon
problem can be solved via a linear programming. Let l ~ [l o ,...,1" ~ be the
vector of
parameters and Tr represent the first ~ columns of a lower tri~Lgular Toeplitz
matrix
constructed from lr(0~,...,T(N~T ~_ Moreover, F is a (N+ 1) x n matrix whose k-
th
colutttn is the first N + 1 samples of the Impulse response o f itte basis
function F~ .
Sitnllarly, Tw is a (N -1)x (N -1) lower triangular Toeplitc matrix That is
consTructed
from the f rst samples of the impulse response of the k=IOw~n weighting
fisnction W.
Propo$itiop 1 Suppose that Nt 1 samples of the experimEntal data v and t are
available (vN and ~rN ) and a bound on thz noise signal a as (15) is
available. Then thz
follow~,g linear programming problem identifies the parameters of the model G
and
19
CA 02489181 2004-12-O1
computes the value of the smallest y with IIQIh 5 y such that the model
structure (1.4) is
not invalidated by the given experimental data:
mtn y
subject to : -~ ~' eN > y
T (rN -eN -T ~) ~ yE (17)
w
IeN ~ < Ql (18)
In the above optimization problem, the parameter vector 1 , The noise vector
eH
and the scalar y are the optimization variables and ~ is a vector o f
dimension N + 1 with
unit elements. Moreover, the k th eletneni of the vector function
E~ _ ~E~ (0),..., E, (N)~r is defined by
Ev(k) ~ inax Iv(i)~ (19)
OS.Sk.N
It is worthwhile to note that in the above proposition, the value of y is an
upperbound for the additive model uncertainXy with respect to the given set of
experlinental data. Fitldlng an upperbound for the additive mpdeling error for
all possible
experimental data, known in the literature as the so-called worsr-case
uncertainty
bounding, has been the subject of many researches in the past several years,
particularly
far case of LTl uncertainties. However, the problzm of computing d worst-case
upperbound for non-linear uncertainties sell remains an open issue.
Case B1: when only ~ is uQkn~pwp and load dyn8mics is affable
Fig. 4 is a block diagram of an uncertainty model strucrNre according to
another
embodiment of the present invention, referred to as Case $l. In particular,
Fig. 4
illustrates a model structure for representation of uncertainly in bulk
modulus coeff'acieni
when toad dynamics are stable. G -~ is a model for an integrator.
As mentioned previously, the model structure (14) is a special case for
representing the uncertainty in actuator dynamics. To demansirate another
example, let
us consider the case when the only source of parametric uncertainty 1s the
error in bulk
w
CA 02489181 2004-12-O1
taodulus eoef~icient ~f _ This error indicates that the effect of velocity has
not been
perfectly eliminated by the linearizing controller_ Applying iha lineari2ing
command
signal t.= with nominal ~ implies that
z _ a - ~SDzP(x)rv
Now let the mapping from t t z4, to r~ to w be expressed by
w = L (T + r~, ) where z~,~ represents any external torque and L is an
operator
representing load dynamics. peflne the uncertain block as o = D 2 P(z)~3 ,
then
t=vtoLzttlLz~r (2U)
Note chat A is still a non-linear operator due to the presence of P(x). ,As ii
can be
seen from Fig. 4, thz uncertainty in this case appears in a feedback
connection with an
integrator. if load dynamics is unknown, L should be contained in v . .But, in
the sequel
we assume L is lmown. In this case the transfer function associated with L
(i.e. G(s))
plays The role of a weighting function as W does in (14). Given N+ 1 samples
of v and z,
the following model structure is Then used for identi$cation of the model G ,
and an
upperbaund, y for the uncertainty v , where holy s y
Gz =vtnWr-re (21)
where z a = al, z~, represents the effect of external disturbance to actuator
dynamics
r2sultiag from external torque zu~. Obviously,
~lell.v ~~l4ll~ll~l~~lh~IIW YIILII~NT..,Ip
Note that here G stands for the derivative operator (rather than integrator
operator in model structure ( 14)). Moreover, we assume Ihat a bound on both
IILII~ and
Ilz,~, li~ ara kaown. With the same argument as in proof of Proposition 1, one
can show
that the following linear progamming problem solves the identification problem
rr;in y
sub~eru to : TzFI - vN ~ <_ YIILIy (Fr t IIT,~~ l1~ 1~
21
CA 02489181 2004-12-O1
Case B2: when paly ~ la unlwowu aad load dynamicx sre uastabte
Equation (2U) can be written as
L -'T=L-lvtOTtG~r~, (22)
The following model structure is then proposcd
G,z=G2vtot+e (23)
where G, represents a model for sL(s) and C2 is a modcl for L(s)-' .
Obviously,
G, and GZ include load dynamics and hence, load dynamics is considered
unlaiown_
Similarly, a represents the term ~TQ, and it is bounded by IIe~I~ ~ II~III
~Ira~ Ilp - yIIT,~, ~I~
The models G, and Ciz are parameterized as in (16) with parameter vectors
1, and I~ , respectively_ Therefore, given N+ 1 samples of v, z together with
knowledge
of I~r,~ ~lo , the identification problem is to find l~ and 1Z and minimum
value of y s4ch
that The model structure (23) is not invalidated by data. The following linear
pmgramming can be similarly shown to solve the identification problem
min y
subjecr to : l,.lz,~
TTFW T~Fzl2l s r(E~ + ~~T,~y~~ ~)
Remark I It is possible that the paratnctric and noa-parametric uncertainties
that
are represented by D are time-varying. In other words, if B varies with time
they B will
be also time- dependent_ All previous identification results are still valid
in this case.
However, it should be noted that when parametcr variation is very significant
(and
assuming that the overall identification procedure is long enough to capture
the variation
of B ), the estimated bound for o can be large_ This means that 'the robust
external
controller designed based on this large upperbound will be conservative. Aparc
from an
adaptive approach, one way to resolve this problem is to repeat the
identification-
controller desigi irt some time intervals. Clcarly, these time intervals
should be long
enough to let the identification and controller design procedure be completed
while, on
22
CA 02489181 2004-12-O1
the other hand, short enough to be capable of following parameter variations.
Such a
repetitive identi$cation-robust cozitrol design has been used in literature
for slow-varying
systems- In our particular case the preferable minianum time iritetval turns
to be about 2
minutes. However, our implicit assumption is that the actuator operates in
steady state
and parameters do not vary significantly.
External optimal l, -Hm controller design
The nominal model G together with the uncertainty uppereound y can be used in
a robust control strategy for desi~g the external linear discrete-time
controller C that
maps the torque error signal z to the new input signal v. In the sequel, we
speci#~r
different robust stability and gerfotrnance conditions for the entire closed-
loop system
according to three model structures (14), (21) and (23).
Cure A
The nominal output sensitivity function can be defined as S = (1 t GC)-' and
the
nominal input sensitivity function can be defined as S" = CS . The additive
uncertain
stntcture (14) induces a robust stability condition on the nominal input
setisitzvity
function
(2d)
Moreover, in order to attenuate the effect of high frequency sensor noise on
the
input signal v and to limit the amplitude o f the input signal v, an H.~
constraint should
be imposed on thr input sensitivity function
For the feedback loop shown in Fig. l, the torque trae]cing error i= = z"~ - z
0 and
the reference signal T,~ are related to the output sensitivity function by t =
ST,~ . A
presently desirable performance objective is to minimize the maximum (over all
pos$lble
reference signals) peak-to-peak torque tracking error, This is equivalent to
i2t: The
I_~ norm or the induced Im norm of the output sensitivity function-
23
CA 02489181 2004-12-O1
Since H~ and H~ norm of any LTI system are bounded by its h norm, this
performance objective obviously minimizes an upperbouad for the HZ and ft o
norm of
the output sensitivity function. This is a propert y that no optimal H 2 or
H~, controller
possesses. The minimization of II,S~y also minimizes the effect of external
disturbance a
in z , however one should nose that due to the presence of non-linear operator
o in
model structure (14), a bounded disturbance catz destabilize the system
depending on
initial conditions and nature of non-linearity. Therefore, one should keep in
mind that the
effect of external disturbance is minimized as long as initial cpzidiuons are
sufficiently
close to system equilibrium point.
Csse $1
Given the model structure (21 ), the robust stability copdition becomes
G~ Y~ ( )
W 1 t G-1C < 27
Recalling that W = L it becomes clew that for the loads with high flexibility,
the
frequency response of W ( jt~r) is large in some resonant frequencies.
Therefore, constraint
(27) requires that the closed loop sensitivity function G-' be small in load
resonance
t t G'' C
frequencies- This implies that when the effect of velocity is r;ot perfectly
eliminated by
the linearizing controller, a limitation is imposed on the achievable
perfprmance through
a robust stability constraint. This result is in accordance with known
analysis describing
the limitation effects of lightly damped modes of load on the achievable
performance of
force controllers. It is worthwhile to notice that these lightly damped modes
affect the
performance of our proposed conimller only when the effect of velocity is not
perfectly
compensated. l~.owever, these modes limit the performance of typical P1D force
controllers even in absence of uncertainty in actuator dynarnics, due to
particular
structure of these controllers-
In order to minimize the amplitude of tracking error, the same performance
objective as in (26) can be considered herein.. Note Thai here in definition
of all
sensitivity functions S, S" , one should replace C by GT ~
Case $2
24
CA 02489181 2004-12-O1
Similar to case A and case B1, model structure (23), induces a robust
stability
constraint such as
y-~
c, t c2C ,~
on the external controller C. Note that since G, represents sG(s) and Z(s~ is
unstable, the
transfer function ~ ' is non-minimum phase and this fact eau impose some
llmitariotis
ttc-~c
on the achievable performance of torque coniroller_
Limit-cycle
When the gain of external controller is high, a self exciting oscillation
(liruit-
cycle) is observed in the generated torque. Fig. 7 graphicahy illustrates a
limit cycle in
actuator dynamics in the presence of a high gain external controller.
According to
experimental results, the frequency of the main harmonic of the limit-cycle
can range
from 200 to 400 rad/sec depending on controller gain- Moreover, the frequency
is almost
independent of load inertia_ As seen in Fig. 7, the oscillations sXatt when
velQCity
approaches zero, suggesting that the oscillations may be caused by static
friction. The
valve dead-zone can be also a cause because its effect is dominant near zero
velocity that
eorreaponds to low flow acrd small valve opening. Moreover, the occurrence of
litnit-
cycle depends also on reference torque_ For example, for high frequency
sinusoidal
reference signals (higher than 1Hz), the oscillations do not nortn.ahy occur.
The existence
of limit-cycle phenomenon has been reported and it has been attributed to the
existence of
electromagnetic hysieresis itl the valve dynamics_ Ii has been shr~wn chat the
oscillations
can be caused by lightly damped modes of load dynamics.
If limit-cycle is considered to be caused only by load dyna;nics, its effect
is
injected into torque dynamics (4) through the velociXy signal w . The effect
of velocity
can then be perfectly compensated by the linearizing controller in absence of
uncertainty
and especially in absence of parametric error iti bulk modules ~ . However,
when ~3 is
not zero, a consir$int like (27) should be imposed to ensure ~hust stability
with an
acceptable level of oscillation attenuation. On the other hand, if limit-cycle
onginates
from actuator dynamics, its effect can be represented by an ~eetlainTy term as
in model
structure ( 14). A constraint like (24) can then ensure the stability
rAbustness together with
?5
CA 02489181 2004-12-O1
a level of performance- Another efficient way to attenuate the oscillations is
to impose
point-wise constraints on the input sensitivity function S" (as defined in
case A) in the
frequency raz;ges where the oscillations occur. These ccanstraints aim to
cancel out the
e#fect of the main harmonics of the limit-cycle by preventing there to be
injected into the
system thmugh v. We describe these constraints by
IS"(j~~~~~~., for x=1,...m (28)
In any case, the existence of limit-cycle obviously limits the achievable
performance of the torque controller.
Synthesis
From a synthesis point of view, there are morn cor~str~irlis that should de
imposed
on the output sensitivity function S. For example, if the notttirial model G
has any
unstable pole-zeros or pure delays, the complementary sensitivity function 1-S
should
also contain exactly the same dynamics in order to avoid any unstable pole-
zero
simplification between the controller and the model. These constraints are
usually
referred to as Zero interpolation conditions, and they are transformed into LP
constraints.
A synthesis procedure according to an embodiment of the present invention for
des~giing the external controller C is based on the foxtnulatiQrt of convex
Linear Matrix
lnequahty (LMI) or Linear Programming (LP) constraints for each of the eonlTol
design
specifications (24)-(26). In general, These constraints are
lnf7niteTdimensional but in rr~any
cases they can be reduced to finite-dimensional optimization- For example, it
is laiown
that in SISO case, a pure 1 minimization in (26) has an Finite lmpulse
Response solution
for S. Moreover, the minimization problem (26) together with (2$) has
typically an FJR
solution for S . However, by imposing all the control specifications (24)-
(2S), the
optimal solution for S can be no longer FiR- One way to checb: this property
is to
approximate all infinite-dimensional constraints by finite-dimensional ones
via finitely
many variables and f:nicely Enany equations methods. Here in our problem, we
considered
an FIR structure for the output sensitivity function S. The interpolation
conditions as well
as the control specifications (24) and (26) are consequently transformed into
LP
constraints. Furthermore, by the application of Bounded Real Lemma, the H,~
constraint
26
CA 02489181 2004-12-O1
(2S) is transformed into an LM1 constraint. Also, the point- wise coustrainis
in (28) are
transformed info an LMI by using methods known in the art.
Experimental results
The experimental tests have been conducted on the pitch actuator of the Titan
11
Schi111ng industrial robot which is located at the robotics laboratory of the
Canadian
Space Agency. In a particular experimental result, the joint is drivetz by a
vane type
rotary hydraulic actuator that getzerares a nominal torque of Sp0 IVrR at
nominal supply
pressure of 3000 Psi. The position angle of the actuator can vary between -90~
tp t 900
and it is measured by a 16-bit encoder. The maximum velocity of the actuator
is
192°~sec. The chamber pressure pi and px are measured by two pressure
transducers. All
analog signals are sampled at lkHz.
identification and uncertainty bounding
The parameters of the actuator non-linear model are estimated via a typical
least-
squares optimization (10) using 1000 time-domain data samples. The identified
parameters c~ , f~, t c~, and ,a of the non-linear model are shawu i~ Table 1.
Vo D ~a ~~ 1' ~~~
1.67 x 10~ 2.66 x 10-5 1.49 x 10 1.11 x 1 1.44 x 104
4 Q~~
Table 1
The identification procedure typically needs cotnputauon of z through
numerical
differentiation. in order to decrease noise amplification durin8
differentiation, we
decimated r with a factor of S before differentiation. As discussed earlier,
the choice of
input signal a has a strong impact on the condition number of the regression
m2~trix y
which in its turn can affect the parametxie error. It is Mown that for
persistently exciting
input sigwals with wide frequency bandwidth, this condition number is close to
one and
consequently the parametric error is reduced.1-iowever, when. such persistent
input
signals are applied in a hydraulic actuator they can cause sharp variation of
torquz signal
z , which can complicate the numerical differentiation of z needed for
identification
27
CA 02489181 2004-12-O1
purpose. So it seems that there is a compromise between the degree of
persistency of
input signal and the degree of difficulty in numerical differentiatiots of z .
After idenpfication of actuator parameters and in order to validate the rion-
linear
model, we computed the estimated input signal a from equation (4) using the
identified
parameters and the measured signals T, p, . pa , u' and x . The estimated
input a was
corupared with the measured input signal u_ Fig_ 5 graphically illustrates a
vat;dation of a
non-linear model according to an embodiment of the present invention, which
shows a
satisfactory match between a and a in low frequencies. Taking o as input arid
z as
output to actuator dynamics, this comparison is in fact a meaSUre of matching
between
the ~e and the identified invesse nop-linear dynamics of the actuator.
ldeslly, the identified garameters, which ate used to compute the feedback
linearication control law (8), results 1n an integrator system mapping the new
input v to
actuator torque z . In practice, however, the mapping deviates from an
ittiegrator due to
unmodeled dynamics. Using model structure (14) we identif ed the nominal model
G ,
assuming w= l,
G(9_, ) - 9_z 0.1313 - O.U603q-'
1- 0.994y
via the identification procedure described earlier. The frequency response of
~r is shown
in Fig. b, graphically illustrates a frequency response of a lipearjzed model
represented
by an uncertainty model structure according to an embodiment of the present
invention, It
is evident from the figure that G behaves as an integrator within frequ~tcy
range of 0.17
and 170 Hz. For the given data, the upperbound of the additive uncertainty is
calculated
to be I~4'I~ 50.16.
(Jpdmsl l, -X,~ robust control design
The synthesis of C is based on models structure (14) nerd constraints
presented in
section 5.1. Since the l, norm of the additive tton-linear uncertainty is
bounded by 0.16,
we must have ~~5, (~r ~ 0.16-' =6.25 to maintain robust stability. Moreover,
in order to
attenuate the effect of the noise ozl the new input signal v, we specify high
pass filter W"
as wci,ghting function in constraint (25)
28
CA 02489181 2004-12-O1
1- 0.99q''
~'" (q ) ~ _,
5.22 - 4.23q
Note that in S1S0 case the constraint (25) is equivalent to bound
~Sx~w~bYl~~ ~~1~~
As discussed previously, the system may exhibit limit-cycle if high gain
linear
controller is used. Fig. 7 shows this phenomena when the linear controller is
simply a
proportional gain (C = K = 2). The trrqaency range of the principal harmonic
of the
limit-cycle for different values of K is between 200 and 400 rad/sec. ltr
order Io attenuate
the limit-cycle oscillation, we imposed two point-wise constraints on the
input sensitivity
fttnetion in 200 and 400 rad/sec. with S, _ ~_ ' b dB , as stated in (28).
The controller synthesized based on all these design ~peci~cations is a 2lth
order
discs~etc-time transfer function_ The optimal controller gives an output
sensitivity fv~retion
with IISIy = 2-04 which implies that the amplitude of tracking error for any
reference
signal with I~t,~ ~I~ 51 does not exceed 2.04.
Note th at the complexity of external controller is a n~tufal consequence of
imposing several robust stability and performance constraints. For example, in
case of
pointwise constraints (28), the controller needs to include narrow-band
behavior in two
different frequencies and consequently its order becomes high- Moreover,
unlike ~I,~
control, in 1' case, no direct relationship exists between the order of
optimal controller
and chat of model. This means that the order of optimal 1, coiitrohers can be
arbitrarily
high regardless of model order. However, 1n Hm or HZ case, the contx'oller
order is
bounded. by order o f model and weigk~ting functions.
From implementation point of view, since the resulting controller is designed
1n
discrete-time domain, its implementation does not need any continuous-to-
discrete
transfonnation_ Although in none of our experiments we had an innplemetztation
problem
due to complexity of C, standard model-redtlctlon techniques can be applied
provided
that the reduced order controller does not violate key stability arid
performance criteria.
Performa~ace evaluation
29
CA 02489181 2004-12-O1
In order to demonstrate back-dtivabiliiy (equivalently low sensitivity of the
controlled actuator to velocity and external torque perturbations), we
conducted an
experiurent in presence of two types of controllers. Fig. 8 graphically
illustrates
frequency responses of resulting input and output sensitivity ~unttions
together with
weighing functions. Fig. 9 graphically illustrates as effect of external
torque distwbancr
on generated torque, specifically the response of the hydraulic torque to
external torque
disturbance with and without having inner feedback linearization loop,
respectively.
During the experiment, the end-effector of the robot was moved by hand while
controllers were regulating the torque of pitch actuator to zero. It is
evident from the
figures that the sensitivity ofthe control system to external, torque
disturbance is
substantially reduced when the feedback linearizarion is used. Let us deFne
backdrivability index of an actuator as the ratio of torque amplitude to
velocity amplitude
when z,,~ --- 0 and when actuator is subject to extetzlal torques
li~li~ ~,f.o.~,~.~
Obviously, for an ideal source of torque r~ = D. This index is also a measure
of
impedance of actuator. Without feedback linearizatioa miler loop (a), the
backdrivability
index is 27 but for the proposed cascade controller (b), this quantity
decreases to 7.9.
Low sensitivity to external torque disturbance in case (b) ituplies chat the
hydraulic
system is backdtivdble gad performance of torque controller is not much
affected by load
variations or external torques.
Fig. lp shows the step response of the proposed controller, also described as
step
tracking of actuator tprque in the presence of velocity. The rise-pines and
the settling
times of the coatrol system as well as sensitivity to velocity disturbance are
reported in
Table 2:
- T
r,. r, r, r; r~
2? ms 21 ms 200 ins 100 ms 7.H
Table 2
CA 02489181 2004-12-O1
Figs.11, 12 and 13 show the sinusoidal reference tracking for dif~ere~lt
frequencies and in presence of velocity disturbance. Specifically, the figures
illusuate 0.5
Hz, 2 H2, and S 1-12 sine tracking, respectively, of actuator torque in
presence of velocity.
Note That because of the sxnoothncss of sinusoids, the limit-cycle phenomenon
might not
be observed for high-gain controllers.
In many robotics applications such as contact force control, the hydraulic
actuators motion is negligible. Therefore, the distortion caused by velocity
in the
generating torque is reduced. In order to check the performance of the
designed controller
in absence of velocity, we conducted the previous experiments whey robot end-
effector
was locked.
Cotpp~risou of utep response of system
Ln presence and in absence of velocity in $ig- 10 an$ Fig. ~4, shows a slight
improvement in perfo~manee of controller in absence of velocity. Flg.14
graphically
illustrates step response of actuator torque in absence Qf velocity, while
Figs. 15-17
graphically illustrate 0.5 Iiz, 2 Hz, and 5 Hz sine tracking, respectively, of
actuator
torque in absence of velocity. Obviously, if the effect of velocity is well
compensated by
the linearizing controller one should not expect a signiFicant difference
between the
petfottnance of compasita controller in static and non-static case- The
tracking
performance of controller with respect to sinusoidal reference inputs with and
without
presence of velocity disturbance ate illustrated in Figs. 11, 1~ 13 and Figs.
15, 16, 17,
respectively.
Conclusion
A novel combined scheme for identification and t'obust torque control of
rotary
hydraulic actuators has been presented. The feedback linearization loop has
been used to
linearize the actuator dynamics and to compensate for non-linear effect of
velocity
disturbance, while the outer l, -Ho optincrally loop has beep itnpletnented to
ensure best
degree of achievable robustness and performance for the system in the face of
possible
uncompensated non-linearities. The stability analysis of the internal dynamics
provides
some necessary results that could be considered in developing new methods for
design of
the torque controllers achieving global stability. The experiittental results
described
herein illustrate the implementabiliry and efficiency of the proposed combined
scheme.
31
CA 02489181 2004-12-O1
As will be understood by those of skill in the art, the meibods of design,
methods
and systems relating to torqueJforce controller embodiments of the present
invention Gan
be generally embodied as software tesiding on a general pulposc, 4r other
suitable,
computer having a modem or Internet connection to a desired optical network.
The
application software embodying methods of design, methods and/or systems
relating to
torque/force controller erribodiments of the present invention can be provided
on any
suitable computer-useable medium for execution by the computer, such as CD-
RAM,
hard disk, read-only memory, or random access meutory, or as pad of any
carrier signal
or carrier wave. In a presently preferred embodiment, the application software
is written
in a suitable programming language, such as C-t--r or Matlab/Situtdi:Ak, and
is organized,
as described above, into a p)urality of modules or elements that perform the
method steps
described. The methods could be irapletnentod in a digital sill processor
(DSP) or
other similar hardware-related implemezitation_ When reference j$ made to a
means for
pcrfotrniag steps of methods according to au embodimept of the present
invention, such
means are to be understood to include computer-readable means as described
above.
Therefore, according to an aspect as described above, the present invertuon
provides a computer-readable storage medium, comprising computer instructions
for:
calculating a linearizing control law based on identified parameters of an
actua~toT non-
linear model; determining an uncertainty model for the actuator, the
uncertautry model
including a non-linear component; estimating an uncertainty bound based on the
identified parameters; and designing an external linear controller based on
the cglculated
linearizing control law, the determined uncertainty model, and the estimated
uncertainty
bound_
Although errrbodiments of the present invention have primarily been described
iti
conjunction with a torque/f4rCe contTol)er, it is to be understood that such a
controller cats
be provided separately. or can be provided integrally with op aetl~ator. For
instance, a
manufacturer or rese)ler of hydraulic actuators could include a tnrque/force
controller
according to an embodiment of the present invention integral with the
hydraulic actuator.
As such, in an aspect, there is provided a hydraulic actuator, comprising a
controller
according to embodiments o f the present invention. In another aspect, there
is provided a
hydraulic actuator comprising a computer-readable storage tnedit~m, comprising
staterrrents and insm~ctions which, when exocuted, cause a computer (or a
processor in
32
CA 02489181 2004-12-O1
data communication with the actuator) to perform a method according to
embodiments of
the present invention.
Embodiments of the present invention can find appucaaon in association with
hydraulic actuators used in many applications. For instance, they are widely
used to
drive robotic manipulators in industry for earth moving, mate~tial handlinS,
atui in the
areas of construction, forestry and manufacturing automation. Ocher
appLicaiions include
high power industrial machinery such as machine tools, aircraft, material
handling,
construction, mining, and agricultural equipment. Robotie uses includz
assembly tasks,
mobile robots, and robotic applications in space, for example with Special
Pwtpose
Dexterous Manipulators (SPDMs) such as the CanadarmT~ _ General engiaeering
applications include vibration isolation and autpmobile active suspension.
Military
applications also exist in aerpspace, aviation, submarines and maritime
applications.
The above-described embodiments o f the present invetLtion are intended to be
exaurples only. .Alterations, modifications and variations may be effected to
the
particular embodiments by those of skill in the art without departing from the
scope of
the invention, which is defined solely by the claims appended hereto.
33
