Sensorless Torque Estimation using Adaptive Kalman Filter and ...

Sensorless Torque Estimation using Adaptive Kalman Filter and
Disturbance Estimator
Sang-Chul Lee, Student Member, IEEE, Hyo-Sung Ahn, Member, IEEE
Abstract— This paper presents a stochastic estimation
method and a signal processing based method for estimating
disturbance torques without using any force sensors. The first
method will address a robustness against measurement noises
by estimating noise covariance. The second method will show
several practical merits. By containing system models inside
of the estimator, the total disturbance torque injected into the
plant is estimated. The experimental results conducted using
a master-slave manipulator show the validity of two proposed
methods.
I. INTRODUCTION
NOWADAYS as science advances, the works that are
hard to be executed by human hands and need a
minute control are increasing. Teleoperation and telesurgery
are representative examples of such works. In the field of
telesurgery, improving transparency and haptic feedback is
a very important issue. To achieve them and to be able to
improve in actual situations, proper force control and force
coordination between the human operator and the reaction
force from the environment are necessary. To realize a precise
position control and force feedback, Kodak Tadano et al
suggested a master-slave system that uses force sensor and
pneumatic cylinders [1], and A. Wróblewska summarized
demands of force feedback systems and illustrated surgical
tools [2]. In these ways, reflected force can be easily measured,
because they are based on force sensors. However,
there are several drawbacks related to space, cost, frequency
bandwidth and infection. To overcome disadvantages of
methods using force sensors, several methods have been
proposed for force estimation. One of the popular solutions
is the state feedback observer [3]. By including disturbance
as a state variable, the state feedback observer is designed
based on error dynamics. Another solution is the disturbance
observer [4]–[8]. By using a part of inverse transfer function,
the disturbance torque is estimated. Another famous solution
is the Kalman filter. We can find many reports on states or
parameters estimation using the Kalman filter [9]–[13].
The estimation methods mentioned above can be used
to estimate internal system parameters and exogenous disturbance.
However, there exist some remarkable following
issues. (i) many estimation researches assumed that measurement
data are accurate at every processing time. i.e.,
there is no measurement noise. In practice, disturbance
compensation under the noisy measurement is necessary for
Authors are with Distributed Control and Autonomous Systems Laboratory,
Department of mechatronics, Gwangju Institute of Science and
Technology (GIST), Korea, E-mail: hyosung@gist.ac.kr
more accurate control. (ii) The state feedback observer needs
analytic feedback gain calculation. (iii) Disturbance observer
is designed based on the assumption that the disturbance is
injected into the rotor as a torque. Thus if there exist other
types of disturbances, the estimated torque may be different
from the actual torque that we want to estimate. (iv) The
estimation based on Kalman filter takes a long operation
time; so reducing operation time is desirable.
In order to overcome the problems mentioned above , this
paper proposes two solutions. The first estimation method
is a stochastic approach which uses adaptive Kalman Filter
(AKF). The discrete Kalman filter (DKF) needs a covariance
matrix of the measurement vector properly selected in advance.
However, the AKF based estimation method updates
the measurement covariance matrix at each processing time.
The second estimation method is the signal processing based
disturbance estimator. The purposes of the disturbance estimator
are to estimate the torques asymptotically stably and
to ensure a fast response. This paper is organized as follows.
In section II, basic estimation approaches are described.
In section III the adaptive Kalman filter based estimation
method is proposed. The disturbance estimator is proposed
In section IV. Experimental results are shown in section V,
and finally, in section VI of this paper the conclusion is made.
II. BASIC CONCEPT OF DISTURBANCE ESTIMATION
In the bilateral teleoperation, the transparency is one of
the desired goals. When the transparency has achieved, an
operator feels the massless and infinitely stiff operation, and
experiences the immediate sensation of manipulating the
remote environment. To achieve the transparency, many researchers
proposed varied types of teleoperation architectures
[15]. Fig.1 depicts the structure of the experimental system
which used in experimental test, and it is one of the bilateral
teleoperation systems. The structure has bilateral symmetry
with an opposite device. In order to achieve the transparency
and feel the sensation from the remote site, teleoperation
architectures need properly measured force information. Up
to date, however, many teleoperation systems have been
using force sensors for haptic(force) feedback and control. To
overcome the drawbacks of the force sensors, two methods
proposed in this paper estimate the disturbance torque by
using an input and an output signal of the motor installed
on the each joint of the manipulator. As a result of the
estimation, the estimated disturbance torque information, the
angular position, and(or) the angular velocity of each joint
will be transmitted to the opposite side without any force

Fig. 1.
Structure of the teleoperation system
sensors (see Fig.1). Thus, control designers will be able to
apply many teleoperation architectures and their own control
strategies with the haptic(force) feedback.
A. System model
III. ADAPTIVE KALMAN FILTER BASED
DISTURBANCE ESTIMATION
In this section, we treat the disturbance estimation using
noisy measurement. A linear stochastic model of a DC motor
used for the stochastic approach is represented as follows:
ẋ(t) = F x(t) + Bu(t) (1)
y(t) = Hx(t) + v(t) (2)
where the matrix F ∈ R n×n represents the system matrix
and the matrix B ∈ R p×p represents the control input matrix.
The matrix H ∈ R m×n is the output matrix, and v ∈ R m is
the measurement noise vector. By using an equivalent block
diagram of the DC motor as depicted in Fig.2, the linear
stochastic model is obtained as follows:
ẋ 1 = x 2
ẋ 2 = − B J x 2 + K J I − 1 J D in (3)
y = x 1 + v
where B, K, J, I, and D in denote viscous coefficient, torque
constant, moment of inertia, control signal(armature current),
and the disturbance torque, respectively. Noting that x =
[ θ ω
] T
, we obtain the vector matrix form of the linear
stochastic model.
[ ˙θ
˙ω
] [ 0 1
=
0 − B J
] [ θ
ω
]
+
[ 0
K
J
] [ 0
I +
− 1 J
]
D in
(4)
I
K
Fig. 2.
D in
1
J
motor
B
J
1
1
s s
Block diagram of a DC motor
where θ, and ω represent angular position(x 1 ), and angular
velocity(x 2 ), respectively. Equation (4) shows the disturbance
torque(D in ) is acting as a second input. We assume the
disturbance torque(D in ) is independent to the state variables
and unbounded. Based on the additional assumptions that
the characteristic of the disturbance is nearly constant and
sampling speed is fast enough, the disturbance is included
in the system model as a state variable. By integrating the
disturbance into the system model(4), and discretization, the
extended model obtained as follows :
⎡
⎤ ⎛ ⎡
⎤⎞
⎡
θ(k + 1)
0 1 0
⎣ ω(k + 1) ⎦ = ⎝I + T s
⎣ 0 − B J
− 1 ⎦⎠
J
D in (k + 1)
0 0 0
⎡ ⎤
0
+ T S
⎣ K ⎦ I(k)
J
0
⎡
y(k) = [ 1 0 0 ] ⎣
θ(k)
ω(k)
D in (k)
⎤
⎣ θ(k)
ω(k)
D in (k)
⎤
⎦
(5)
⎦ + v(k) (6)
where T S represent the sampling time. We assume
the measurement noise(v(k)) is the zero mean gaussian
noise(v(k)˜N(0, R k )). The following AKF based disturbance
observer estimates D in by the extended model (5-6).

B. AKF based disturbance observer
In the extended system model (5-6), as the pair of the
system matrix and the output matrix is observable, full
order estimation is guaranteed by the DKF. Even the DKF
guarantees the full order estimation, in order to yield the best
result, the DKF needs an accurately selected measurement
covariance matrix in advance. However, obtaining the
accurate covariance matrix is not easy. Moreover, the
measurement covariance matrix can be changed with time.
To overcome the problems mentioned above, this paper
suggests the AKF based disturbance observer. For an LTI
system, the recursive algorithm of the AKF is described as
follows [14], [16]:
Time update
1 ) Project the state ahead
ˆx k|k+1 = F k−1ˆx k−1|k−1 + B k u k (7)
2 ) Project the error covariance ahead
Measurement update
1) Kalman gain
P k|k+1 = K k P k−1|k−1 F k T + Q ∗ k (8)
K k = P k|k−1 H T k (H k P k|k−1 H T k + ˆR ∗ k) −1 (9)
2) Update estimate with measurement Z k
ˆx k|k = ˆx k|k−1 + K k (z k − H k ˆx k|k−1 ) (10)
3) Update the error covariance
P k|k = (I − K k H k )P k|k−1 (11)
where F ∈ R n×n , B ∈ R n×n , K ∈ R n×m , and P ∈ R n×n
represent the system matrix, the input matrix,the kalman
gain, and the covariance state matrix, respectively. H ∈
R m×n , Q ∈ R n , and z k ∈ R m denote the output vector,
the covariance vector of system noise, and measurement,
respectively. The AKF based disturbance observer is different
to the DKF in terms of an adaptive covariance matrix
ˆR
k ∗ ∈ Rm×m in (9). The covariance matrix is updated by
a covariance uncertainties online estimator represented as
follows [16]:
ˆR ∗ (k i ) = 1 i∑
{[z j − H(k j )ˆx(k
N
j )][z j − H(k j )ˆx(k j )] T
j=i−N+1
+ H(k j )P (k j )H T (k j )}
(12)
The first term in the bracket represents a squared error
matrix between the measurements and the estimated states,
and diagonal elements of the second term are the covariances
of state variables. In the first term, the noisy(clear) measurement
makes the big(small) squared error term. Thus, ˆR∗ k
is
increased(decreased), and it decreases(increases) the Kalman
gain K k . Consequently, the dependence of the estimation
result between previously estimated states and current measurement
will be affected by the quality of measurement. As
we can see in (12), the covariance uncertainties estimator
needs N number of recent measurements. By an average of
the recent N number of terms in the bracket, the AKF based
disturbance observer updates the measurement covariance
matrix ˆR k ∗ at each processing time. In addition, large number
of the measurement will offer a more accurate measurement
covariance matrix( ˆR k ∗) estimation.
A. System model
IV. DISTURBANCE ESTIMATOR
This section proposes a signal processing based estimation
method. For the estimation, the DC motor model is used (see
Fig.2). In Fig.3, the upper part describes the system model,
and the middle and lower part represent the disturbance
estimator. In the system model, an input disturbance D input
and an output disturbance D output are described. The input
disturbance D input can be any kind of disturbance, e.g., load
force(torque), gravitational torque, parametric fluctuation or
any combinations of them. In order to validate the total
disturbance estimation, D output is injected. We assume the
disturbances are independent to the system and unbounded.
Subscript n denotes the nominal value of parameters.
B. Disturbance estimator
The disturbance estimator consists of an extraction part
and an estimation part. The two parts are cascade-connected.
As a first step, the extraction part calculates the system
output generated by the input and output disturbances. After
that, the estimation part generates the final estimation result.
Each process is described as follows:
1) Extraction part:
The composition of the extraction part is same to the
system model, but it does not have any disturbances. The
computation of the extraction part is described as follows:
(
K
θ D = −
s(Js + B) I + 1
s(Js + B) D input + 1 )
s D output
(
)
K n
+
s(J n s + B n ) I
(13)
The term in the upper bracket represents the actual
position result(θ I,D ) generated by the control input(I)
and both the input disturbance(D input ) and the output
disturbance(D output ). The term in the lower bracket
shows the position result((θ I ) generated by the control
input(I) only. Due to the subtraction, where the parameters
K, J, andB are same with K n , J n , andB n , the output of
the extraction part(θ D ) is the angular position generated
by the disturbances only. Otherwise, the output contains
contain the angular position generated by the parametric
fluctuation(uncertainty) also. It implies all the dynamics
not considered in extraction part are considered as the

I
motor
K
T
D input
1
J
B
J
D output
1 1
s I , D s
law is applied:
u = K P e + K I
∫
edτ + K D
de
dt
(16)
The PID control law(15) together with ė 0 = e 1 and error
equations (14), error dynamics described as follows:
D
disturbance.
K n
Fig. 3.
PID
1
J n
u D
u
ˆ
total
B
J
n
n
1
1
s I s
1
J n
B
J
n
n
Extraction
1
1
s *
s
I
I , D
Estimation
*
D
Disturbance Estimator
Block diagram of motor with disturbance
2) Estimation part:
The estimation part calculates the final estimation result
by using a control loop. The reference signal of the control
loop is output of the extraction part(θ D ). Except the torque
coefficient(K n ) of the extraction part and a PID controller
of the estimation part, the open-loop transfer functions are
exactly same. If the estimation part is stable, the feedback
signal(θ ∗ D ) follows the reference(θ D), and the output of
the PID controller approaches the total disturbance. Consequently,
designing the disturbance estimator is concluded by
making the estimation part asymptotically stable. In order to
obtain a stable region of the PID gains, state space equations
is used. Except the PID controller, the open-loop dynamics
of the estimation part is described as follows:
ẋ 1 = x 2
ẋ 2 = − B n
x 2 + 1 u (14)
J n J n
y = x 1
where x = [ θ ∗ D ω∗ ] T
. In order to determine error
equations, let e = θ D − θ ∗ D = θ D − x 1 = e 1 . Then error
equations obtained as follows:
ė 1 = −ẋ 1 = e 2
ė 2 = B n
x 2 − 1 u (15)
J n J n
To determine the control signal(u) in (14), the PID control
ė 0 = e 1
ė 1 = e 2 (17)
( ) ( ) ( )
KI KP Bn
ė 2 = − e 0 − e 1 − + K D e 2
J n J n J n
If the characteristic matrix of the error dynamics is Hurwitz,
error approaches zero asymptotically. It indicates θD ∗ and u
also asymptotically approach θ D and the total disturbance
D total , respectively. In order to calculate stable region of
the PID gains, Routh−Hurwitz theorem is used. Calculated
region of the PID gains are written as follows :
K I
J n
> 0 ,
K P
J n
> 0 ,
B n
J n
+ K D > 0 (18)
As we mentioned above, the estimated total disturbance
contains not only exogenous disturbances(D input and
D output ) but also all the discordance between the real
system model and the model of the extraction part. In
addition, simply removing all the viscous friction loops in
Fig.3, the viscous friction also estimated as the disturbance.
It shows the flexibility of the disturbance estimator.
V. EXPERIMENTAL RESULTS
To validate proposed two estimation methods, experimental
tests were conducted using a master-slave system. The
overall setup is shown in Fig.4, and the specifications of
motor drivers and motors are shown in Tables I and II. For
the tests, the slave manipulator was fixed, and the operator
manipulated the master to give an order to the slave device.
A. AKF based disturbance observer
The experimental test of the AKF based estimation
method is conducted by two ways. At the first, we confirmed
the improvement of the AKF based method in comparison
with the DKF. In order to obtain noisy measurement, same
zero mean gaussian noise having 0.5 of the standard deviation
is added to the measurements of both the AFK and DKF.
In addition, wrong measurement covariance (R k = 0.2 2 )
is given to the DKF. Due to the manipulator is fixed, the
mean of the measurement is ideally zero. Fig.5 shows the
measurements, i.e., control signal(I) of the slave motor,
and the position discordance between the master and slave
manipulators. The AKF based observer also tested under the
same condition. We set the wrong measurement covariance
as (0.2 2 ), and we set the accumulation number(N) in (12)
as 5. The estimation results of the DKF and AKF based
observer are shown in Fig.6. Due to the DKF dose not
have the measurement covariance adaptation algorithm, the

Fig. 5.
Measured armature current and Position difference(DKF)
Fig. 4.
Experimental setup : master-slave manipulator
TABLE I
SPECIFICATION OF MOTOR DRIVERS
Output voltage
Output current
Maxon - ADS 50/5
VCC min. 12 VDC; max. 50VDC
Depending on load, continuous 5A
Fig. 6. Disturbance estimation. (a) DKF, (b) AKF (N=5).
TABLE II
SPECIFICATION OF MOTORS
Maxon - RE30
Assigned power rating(W) 60
Max. continuous current(A) 4
Torque constant(mNm / A) 25.9
Max. continuous torque(mNm) 86.2
estimation is based on the given wrong measurement covariance
matrix. As the Fig.6(a) demonstrates, the estimation
result of the DKF containing a lot of noise. If it used for
the haptic feedback and the position control, it will make
huge vibration on the manipulator. On the other hand, even
the given initial measurement covariance matrix of the AKF
was mismatched to the actual noise, the AKF calculated
relatively clear estimation result(see Fig.6(b)) based on the
covariance uncertainties online estimator(12). Such result
demonstrates the AKF based disturbance observer corrects
the measurement covariance matrix at each processing time.
Moreover, such result indicates possibility of the reliable
estimation under the noisy circumstance. As the second test,
we applied huge number(R k = 1 2 ) of the standard deviation
to the measurement noise. Fig.7(a) shows the estimation
result when the number of accumulation(N) is 5. On the
other hand, when we used 30 number of measurement data,
as shown in Fig.7(b) significantly improved estimation result
is obtained. As depicted above, the disturbance estimation
using noisy measurement is achieved.
B. Disturbance Estimator
This subsection validate the disturbance estimator. For
the test, same bilateral teleoperation system is used. Two
Fig. 7. Disturbance estimation using different number of measurement
accumulation (AKF), standard deviation = 1
measurements of the disturbance estimator, the control
signal and the angular position of the slave manipulator
are depicted in Fig.8. From the two measurements,
extraction part calculates the position(θ D ) by using (13),
and then estimation part calculates the total disturbance. In
the experimental test, PID gains K P = 0.135, K I = 0.01,
K D = 0.008 are used to make estimation part asymptotically
stable. We can find that a set of PID gains satisfies condition
(17) in the continuous time domain. Of course PID gains
can be selected by trial and error also. In Fig.9(a), the
output of the extraction part(θ D : reference of the estimation
part) is denoted as reference, and the feedback signal of
the estimation part(θD ∗ ) is represented as tracking. Tracking
performance can be used as an estimation performance index
in real time. The total estimation result is shown in Fig.9(b).
In the test, the angular position is used for the measurement
of the disturbance estimator, however, even the designer
uses the angular velocity as the measurement but not
position, disturbance estimator will precisely estimate the
total disturbance just by eliminating the last one integrator at
each extraction and estimation part. It shows the flexibility

plant model as its sub-model. In addition, we used a PID
controller for the disturbance estimation. However, any kind
of controller and control technique can be used which can
make the estimator stable.
Fig. 8.
Fig. 9.
Measurement (Disturbance Estimator)
Estimation result of the disturbance estimator
of the disturbance estimator again.
VI. CONCLUSIONS
This paper proposed a stochastic estimation method and
a signal processing based method for the purpose of disturbance
torque estimation without force sensors. The AKF
based method presented robustness against the measurement
noise. When the measurements have a lot of noise, and its
characteristic is unknown, this method can be one of the
reliable methods.
Through the section IV and V, we proposed several merits
of the disturbance estimator, and they can be summarized
as follows: (i) Wherever the disturbance injected between
two measurements(I and θ I,D ), the disturbance estimator
estimates the total disturbance as asymptotically stable. (ii)
The disturbance estimator inherits the structure of the system
model. Thus it provide the instinctive and direct application.
(iii) Unrestricted selection of the measurement type is possible.
(iv) We can determine the types of disturbance included
in the total disturbance by considering sub-model of the
disturbance estimator. In addition to the merits mentioned
above, when the disturbance estimator used for feedback
compensation, the disturbance estimator will try to make
the system act as same with the sub-model. Because all
the effects not considered in the sub-model is estimated
as the disturbance. In consequence, direct and fast loop
shaping could be achieved. Feedback control, loop shaping
and extending to the nonlinear disturbance estimator are our
future works. In this paper, a DC motor model is used
for total disturbance estimation. However, the disturbance
estimator can be used in countless system simply having the
ACKNOWLEDGEMENT
This research was supported by the institute of Medical
System Engineering(iMSE) in the GIST, and by the
MKE(The Ministry of Knowledge Economy), Korea, under
the ITRC(Information Technology Research Center) support
program supervised by the NIPA(National IT Industry Promotion
Agency) (NIPA-2010-C1090-1031-0006)
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