Decentralized robust control of modular robot manipulators with
harmonic drive transmission using RARKF-based joint torque
estimation
2022 34th Chinese Control and Decision Conference (CCDC) | 978-1-6654-7896-0/22/$31.00 ©2022 IEEE | DOI: 10.1109/CCDC55256.2022.10034169
Tianhe Wang1 , Tianjiao An1 , Chongyang Wei1 , Yuanchun Li1 , Bo Dong*1
1. Department of Control Science and Engineering, Changchun University of Technology, Changchun, 130012
E-mail: dongbo@ccut.edu.cn
Abstract: This paper presents a decentralized robust control of modular robot manipulators (MRMs) with harmonic
drive transmission using redundant adaptive robust extended Kalman filter (RARKF)-based joint torque estimation. The
dynamic model of MRMs is formulated via RARKF-based joint torque estimation method. A decentralized robust
control is developed to guarantee the trajectory tracking error is uniform ultimate bounded (UUB). The closed-loop
robotic system is asymptotic stability on the basis of Lyapunov verification. The experiments are performed to clarify
the effectiveness of the proposed method.
Key Words: Modular robot manipulator, Decentralized control, Harmonic drive, Redundant adaptive robust Kalman
filter
1
INTRODUCTION
A modular robot manipulator (MRM) is a robot composed
of many modules with the same structure and function.
Each module has a completely independent structure and
function. The MRM can freely add and remove its own
modules according to different task requirements and specific working conditions which perform different configurations of the structure. MRM is widely used in many
fields, especially in the field of medical and industrial
robots.
In the middle of the 20th time, the harmonic drive was
used, which has the advantages of high gear ratio and light
weight. In the past few years, many researchers have studied the modeling of harmonic transmission devices[1-3].
After that, use these findings and solve the trouble encountered in torque estimation[4,5], motion control of robot system based on harmonic drive[6-9].
Installing torque sensors at the robot joints will complicate
the structure of the joints, which will directly affect the
rigid connection between the joint module and the robot
connecting rod. The strain gauge in the joint torque sensor
will change with the change of the external environment
temperature, which will also directly affect the accuracy of
the joint torque sensor. Therefore, a joint torque estimation
scheme of the robot system is proposed, which uses the
harmonic transmission device to estimate the joint torque
value by using the joint position information, which is of
great significance for improving the accuracy of MRM.
This work is supported by the National Natural Science Foundation
of China (Grant nos. 61773075, 62173047 and 61703055), the Scientific Technological Development Plan Project in Jilin Province of China
(Grant no. 20200801056GH) and the Science and Technology project
of Jilin Provincial Education Department of China during the 13th FiveYear Plan Period (Grant nos. JJKH20200672KJ, JJKH20200673KJ and
JJKH20200674KJ).
c
978-1-6654-7896-0/22/$31.00 2022
IEEE
In order to eliminate the torque measurement fluctuation
caused by gear tooth meshing, a standard Kalman filter is
used for estimation. However, the standard Kalman filter can only guarantee the optimality of the torque estimation, but the robustness is also very important in the
torque estimation.Because robustness and optimality are
contradictory, as robustness increases, optimality will decreasey. However, robustness is not always required, and
stronger robustness will result in loss of optimality. Therefore, in order to balance the robustness and optimality, an
adaptive robust extended Kalman filter [10] algorithm is
designed, which can make the filter perform between the
robust mode and the optimal mode, the filter gain can be
adjusted adaptively. However, the non-linear error in the
measurement model will make the switching function in
AREKF invalid. In response to this problem, we have introduced a redundancy factor to form a redundant adaptive
robust extended Kalman filter (RAREKF). This allows the
switching function to work stably and provides tolerance
for modeling errors in torque estimation. In this paper,
a redundant adaptive Kalman filter algorithm is used for
robot joints with harmonic transmission. It can reduce the
influence of nonlinear modeling errors and noise. The algorithm has adaptive robustness, it can self-adjust the filter
gain and self-switching filter mode according to the load
change, and realize the dynamic balance of robustness and
optimality. The decentralized robust control method is used
to ensure that the trajectory tracking error in the case of
redundant adaptive Kalman filtering is consistent and ultimately bounded, and experiments are used to verify the
effectiveness of the proposed method.
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2
DYNAMIC MODEL FORMULATION
2.1
Harmonic Drive Model
changes. The local elasticity coefficient kf i of the flexspline can be expressed as
Considering an MRM is made up of N modules, each
module is made up of a rotating joint based on harmonic
drive. According to the ideal input/output kinematics relationship, the position of the harmonic drive angle can be
obtained.
θwi = γi θf i
(1)
where i is the ith module, γi stands for deceleration ratio,
θwi is the rotation angle of the wave generator , θf i is the
rotation angle of the flexible wheel, the torque transmission
relationship of the harmonic device is expressed as
τwi =
1
τf i
γi
(2)
where τwi is the input torque of the wave generator. τf i
is the output torque of the flexspline, because the position
of the circular spline in the harmonic drive is fixed and the
circular spline itself is not flexible. So that, θci = 0. The
output and input are not linear due to torsional compliance,
motion error and nonlinear friction in harmonic drive. Describe the kinematic relationship between the motion and
the force constraint in the harmonic drive, the remaining
effects can be combined by introducing friction, flexibility,
and motion error characteristics. θwOi is the external position of the wave generator, θwIi the input position of the
wave generator, θf Oi is the angle position of the flexspline
on the load side, θf Ii is the angle position of the flexspline
measured by the gear. τmri stands for motor friction. τf ri
represents the total friction torque of the harmonic drive.
From the output side of the transmission. The elastic coefficient of the wave generator and the elastic coefficient of
the flexspline are denoted by kwi and kf i respectively. We
consider the friction loss in harmonic drive.
1
τωi = τi − τmri = (τf i − τf ri )
γi
(3)
where τi is the motor torque. The wave generator twist and
the flexspline twist can be expressed as
Δθωi = θωOi − θωIi
(4)
Δθf i = θf Oi − θf Ii
(5)
In the above two formulas, the input position θwIi of the
waveform generator can be measured by the motor side
encoder. The output position of the flexible wheel can be
measured θf Oi by the connecting rod side encoder. Based
on the measured position information, the following relationship can be used to calculate the total torsion angle of
the harmonic device of the joint.
Δθi = θf Oi −
θwIi
γi
(6)
According to the given harmonic drive stiffness and hysteresis performance, the results show that when the torque
of the flexspline changes, the local elastic coefficient also
kf i =
dτf i
dΔθf i
(7)
Considering the symmetry of the stiffness of the harmonic
drive, kf i can be expressed as
2
kf i = kf i0 (1 + (cf i τf i ) )
(8)
where kf i0 and cf i are undetermined constant values. If
kf i0 = 0, the torsion angle of the flexspline can be calculated as follows
τf i
dτf i
arctan(cf i τf i )
=
(9)
Δθf i =
k
cf i kf i0
fi
0
In addition, at rated torque, the harmonic drive deformation
range drops rapidly to zero, which means that the wave
generator stiffness increases rapidly. The elastic coefficient of the wave generator in the harmonic drive can be
expressed as
kwi = kwi0 ecwi |τwi |
(10)
where kwi and cwi are undetermined values. If kwi0 = 0,
the torsion angle of the wave generator can be expressed as
τwi
dτwi
sgn(τwi )
=
(1 − e−cwi |τwi | ) (11)
Δθwi =
k
cwi kwi0
wi
0
According to Eqs. (9), and (11), the total torsion angle of
harmonic drive calculated as
Δθi =
arctan(cf i τf i ) sgn(τwi )
+
(1 − e−cwi |τwi | ) (12)
cf i kf i0
γi cwi kwi0
The total torsion angle Δθi of harmonic drive can be
brought into Eq. (6) by measuring joint position information with dual encoders. The joint torque can be calculated
by the above formula, Therefore, τf i can be expressed as:
τf i =
2.2
1
sgn(τwi )
tan(cf i kf i0 (Δθi −
(1 − e−cwi |τwi | )))
cf
γi cwi kwi0
(13)
RARKF-based joint torque estimation
Consider the filtering model
Xk = F (k, Xk−1 ) + wk
(14)
Yk = Xk + v k
(15)
where k ∈ N , Xk is the system state at time k and
Yk is the measured value at time k.The dynamics of the
system in state Xk are represented by system function
F (k, Xk−1 ), w represents process noise, v means measurement noise. Covariance matrices W and V are Gaussian white. When estimating the torque of the joint,
F (k, Xk−1 ) = Xk−1 ,and the filtered value of τf i is X,
and the measured value of τf i is Y . In the torque model,
the unknown change of the joint torque from k to k − 1
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is represented by w, and the random noise in the torque
calculation result is represented by v. On the time step k
Y k = τf i k
P̄Y k =
(16)
The structure of the RARKF algorithm is as follows:
(1) Predict the state
X̂k|k−1 = f (k, X̂k−1|k−1 )
(17)
Pk|k−1 = Pk−1|k−1 + Wk−1
(18)
(2) Adjust the gain
P̄Y (k) is defined as
k|k−1 = C(Pk|k−1 )
(20)
(21)
−1
k|k−1 P Y k
(22)
X̂k|k = X̂k|k−1 + Kk Ŷk
−1
(23)
−1
k|k−1 + V
−1
k)
(24)
In order to ensure the stability of the filter when there are
modeling errors, the H∞ filtering algorithm is designed. If
the following equation holds
max
||X̂k|k ||22
≤ δ2
2
||wk ||2 + ||vk ||22 + ||E||22
,k > 0
(28)
1/2
(29)
-1
(k|k − 1)
(30)
ARKF filter can adaptively switch the filter mode to
achieve a balance of robustness and optimality. When there
is a modeling error that makes the condition PY k > βPY k
hold, Eq.(27) is the same as Eq.(26), let β > 1. Therefore, the redundancy factor and compensation function are
introduced into ARKF to form a redundant adaptive robust
Kalman filter algorithm. When the modeling error follows
the joint load changes, the redundancy factor β can be introduced to make the result more accurate.
Because the size of the motor current can reflect the size of
the load, the redundancy factor is defined as
σIM (k) 2 , σIM (k) 2 > 1
(31)
β=
1,
otherwise
The compensation function is defined as
(25)
Define δ as the attenuation factor, the gain scheduling operator can be determined as
−1
−1
−2 T
L k Lk )
(26)
k|k−1 = (P
k|k−1 − δ
In order to change the matrix k|k−1 , we need to define a
matrix Lk .The attenuation factor δ is related to estimation
error, measurement noise and modeling error. When the
attenuation factor δ is relatively small, the system needs
robustness. We can change the attenuation factorto change
the robustness and optimality of the system. When the attenuation factor δ becomes larger and larger, the robustness
of the system getting smaller and smaller, when the attenuation factor δ = ∞, the filter at this time is Kalman filter.
When the attenuation factor δ is given, the algorithm may
not reach the optimality of filtering, and it will work in robust filtering mode. Therefore, in order to better switch
the filtering mode, we use the adaptive robust Kalman filter
(ARKF) algorithm.
In the ARKF, the gain scheduling operator C in Eq. (19) is
determined by Pk|k−1 and
−1
(P −1 k|k−1 − δ −2 LT k Lk ) , P̄Y k > βPY k
k|k−1 =
Pk|k−1 , otherwise
(27)
5498
k=0
The compensation factor ε̄−2
max I in L(k) satisfies the following conditions
ε̄−2
max I ≤
(4) Update estimates
Pk|k = (
ρPY k−1 +Ỹk ỸkT
ρ+1
Lk = δ(P −1 k|k−1 − ε̄−2
max I)
(19)
Ŷk = Yk − X̂k|k−1
PY k =
k|k−1 + Vk
Ỹk ỸkT ,
where ρ is a forgetting factor. When the value of P̄Y k is
greater than the measurement error variance PY k , the filter
will be in robust filtering mode at this time. In order to
make the filtering mode stable, let
where C is a gain scheduling operator.
(3) Update measurement value
Kk =
−1
−1
ε̄-2
k|k−1
max I = f β k P
where the compensation factor is expressed as
P̄Y k
fβ k = β −1 diag
PY k
(32)
(33)
This can make the filtering stable when β = 1 .According to Eqs.(28)-(33) and Eqs.(19)-(23), the redundant adaptive robust Kalman filter algorithm can be expressed by
Eqs.(17)–(24).
2.3
Dynamic Model Formulation
Considering RARKF-based MRM dynamic model is
Imi γi q̈i + fi (qi , q̇i ) + Imi
i−1
T
zmi
zj q̈j
j=1
+ Imi
j−1
i−1 j=2 k=1
(34)
τf i
zmi (zk × zj )q̇k q̇j +
= τi
γi
T
where i is the ith subsystem, Imi is the moment of inertia
of the motor, γi represents the reduction ratio of the gear, qi
is the position information of the ith joint, q̇i is the speed
information of the ith joint, and q̈i is the acceleration information of the ith joint. fi (qi , q̇i ) represents the concentrated friction of the harmonic device and the friction of the
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motor. The friction term fi (qi , q̇i ) is a function related to
qi and q̇i .
3
DECENTRALIZED
TROLLER DESIGN
ROBUST
CON-
The overall control of each joint is defined as
No filter
RARKF filter
4
Joint torque estimation(Nm)
fi (qi , q̇i ) = (fci +fsi exp(−fτ i q̇i2 ))sgn(q̇i )+fqi (qi , q̇i )+bi q̇i
(35)
where fci , fτ i , fsi and bi are uncertain parameter values
in the friction model.
5
3
2
1
0
-1
-2
-3
τf i
+ ui
τi =
γi
(36)
i−1
j−1
i−1 T
zmi
zi q̈j
30
2
e = q − qd , r = ė + λe, a = q̈d − 2λė − λ e
(38)
where λ is any positive constant.
Assume that the inertia of the motor is known, but some
system parameters are unknown, and its reference trajectory and the first and second derivatives are bounded.
Use the decomposition-based friction compensation
method to compensate the friction of ith joint, where
Y (q̇i ) and F̃ i are denoted as
Y (q̇i ) = [q̇i , sgn(q̇i ), exp(−fˆτ i q̇i )sgn(q̇i ),
− fˆsi q̇ 2 exp(−fˆτ i q̇i )sgn(q̇i )]
60
70
80
fˆτ i − fτ i ]
τf i
+ b̂i q̇i + (fˆci + fˆsi exp(fˆτ i q̇i2 ))sgn(q̇i )
γi
i−1
Ij2 (uijc + uijv )
+ uiu + Y (q̇i )(uipc + uipv ) +
τi =
j=1
+
j−1
i−1 (41)
i
i
i
Jkj
(Vkjc
+ Vkjv
) + Imi γi ai
j=2 k=1
where Imi is expressed as the inertia of the motor, τf i is the
estimated joint torque, b̂i , fˆci , fˆsi , fˆτ i is the nominal friction parameter, uiu is used to compensate for nonparametric
uncertainty fqi (q, q̇). For parameter uncertainties F̃ci and
F̃vi ,uipc and uipv compensation can be used. For each joint,
friction is compensated in the same way. Compensators
uipc , uipv and uiu are defined as
uiu =
−ρf i |rrii | |ri | > εi
−ρf i |εrii | |ri | ≤ εi
(42)
t
uipc = −k(
T
However, the friction parameter models bi , fci , fsi and
fτ i are not accurate, where b̂i , fˆci , fˆsi , fˆτ i is the estimated
value of bi , fci , fsi , fτ i .
Due to the change of temperature, the uncertainty of the
parameter model is not constant.
(40)
where F̃ci is an uncertain constant vector, F̃vi is a variable
i
| < ρin . An adaptive comand has defined meanings as |F̃vn
pensator is designed to compensate the constant parameter
uncertainty by using the control design method based on
decomposition, robust compensator to compensate for F̃vi .
The control torque τi of the ith joint is
(39)
i
fˆsi − fsi
50
F̃ i = F̃ci + F̃vi
T
zmi
(zk × zj )q̇k q̇j = ui
fˆci − fci
40
F̃ i can be decomposed into
According to the above formula, the control input ui can
be designed for a single joint. Because the motion of the
lower joint causes the uncertainty of the model, a robust
controller can be used to compensate for the uncertainty, in
order to achieve the desired performance with a small feedback gain. So use the decomposition-based control design
method
Using the basic strategy of decomposition-based system
modeling and control methods, a separate compensator is
designed to compensate for each different type of uncertain parameters and variables. For some uncertainties that
cannot be measured in real time, robust compensators can
compensate, and these compensators work together to form
a controller.
Define the error as
F̃ = [b̂i − bi
20
(37)
j=2 k=1
i
10
Figure 1: The estimated torque of joint 1 is unfiltered and
RARKF filtered
j=1
+ Imi
0
Time(s)
where ui is the control input for joint i.
For the ith joint, according to (34) and (36)
Imi γi q̈i + fi (qi , q̇i ) + Imi
-4
Y T (q̇i )ri dτ + ri )
(43)
0
uipvn =
⎧
T
⎨−ρi Y (q̇i )T ri , |Y (q̇i )T ri | > εi
n |Y (q̇i ) ri |
T
pn
⎩ −ρin Y (q̇ii) ri , |Y (q̇i )T ri | ≤ εipn
n = 1, 2, 3, 4
εpn
(44)
where εi and εipn are positive control parameters.
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1.5
0.4
Joint torque estimation error(Nm)
Joint torque estimation(Nm)
No filter
RARKF filter
1
0.5
0
-0.5
-1
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
Time(s)
Time(s)
Figure 2: The estimated torque of joint 2 is unfiltered and
RARKF filtered
Figure 4: Joint torque estimation error of joint 1 after
second-order low-pass filtering
0.3
Joint torque estimation error(Nm)
Joint torque estimation error(Nm)
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0.2
0.1
0
-0.1
-0.2
-0.3
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
Time(s)
Time(s)
Figure 3: Joint torque estimation error of joint 1 after
RARKF filtering
Figure 5: Joint torque estimation error of joint 2 after
RARKF filtering
Decompose Imi
i−1
j=1
T
zmi
zi q̈j into Uzic and Uziv , where Uzic
represents constant uncertainty, Uziv stands for variable parameter uncertainty, using a decomposition-based control
design method, design an adaptive compensator uijc for the
constant uncertainty, and robust compensator uijv for variable part.
t
i
(45)
ujc = -k2 k2 ri dτ + ri
0
uijvn =
−ρDj |kk22 rrii | |k2 ri | > εiDn
,
−ρDj kεi2 ri |k2 ri | ≤ εiDn
n = 1, 2
(46)
Dn
εiDn is the positive control parameter.
i−1
j−1
T
Similarly, for the ith joint,Imi
zmi (zk × zj )q̇k q̇j
j=2 k=1
can be decomposed into Vzic and Vziv , where Vzic stands
for constant uncertainty, Vziv stands for variable parameter
uncertainty. Using the decomposition-based control design
i
is designed for conmethod, an adaptive compensator Vkjc
i
stant uncertainty term Vzic . and a robust compensator Vkjv
5500
0.3
is designed for the variable part Vziv .
t
i
Vkjc
= −k3 (
k3 ri dτ + ri )
(47)
0
i
Vkjvn
=
−ρV k ρV j |kk33 rrii | , |k3 r| > εiV n
,
−ρV k ρV j kεi3 ri , |k3 r| ≤ εiV n
n = 1, 2
Vn
(48)
Theorem: Consider a modular robot manipulator with the
subsystem dynamic model formulated in (34), the model
uncertainties existed in (39). The error of closed-loop
MRM system is UUB under the decentralized robust control law proposed by (41).
Proof: The candidate Lyapunov functions are selected as
follows:
1 2
r
(49)
Vir =
2Bi i
The derivative of (49) can be obtained as:
V̇ir = ri ((−fpi −Y (q̇)(F̃ic +F̃iv )−(Uzic +Uziv )−(Vzic +Vziv )
(50)
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0.6
0.2
0.4
Position tracking(Rad)
Joint torque estimation error(Nm)
0.3
0.1
0
-0.1
Trajectory error
Actual trajectory
Desired trajectory
0.2
0
10-3
-1
-0.2
-2
-0.2
-0.4
-0.3
-0.6
-3
11
0
10
20
30
40
50
60
70
80
0
11.1
11.2
10
11.3
20
11.4
30
11.5
40
50
60
70
Time(s)
Time(s)
Figure 6: Joint torque estimation error of joint 2 after
second-order low-pass filtering
Figure 8: Joint 2 trajectory tracking
5
0.6
10-3
Position tracking(Rad)
1
0
0.2
11.4
11.6
11.8
12
0
-0.2
-0.4
REFERENCES
Trajectory error
Actual trajectory
Desired trajectory
-0.6
0
10
20
30
40
50
60
70
80
Time(s)
Figure 7: Joint 1 trajectory tracking
If the Lyapunov function satisfies the following conditions:
ri | ≥
4
i i
n=1 (ρn εpn )
4
(51)
The position tracking error ei and velocity tracking error ėi
can be UUB.
4
CONCLUSION
A decentralized robust control of MRMs with harmonic
drive transmission using RARKF-based joint torque estimation is proposed in this paper. The dynamic model of
MRMs is formulated via RARKF-based joint torque estimation method. A decentralized robust control is developed to guarantee the trajectory tracking error is UUB and
the experiments are performed to clarify the effectiveness
of the proposed method.
2
0.4
80
EXPERIMENT
In order to verify the effectiveness of the proposed control scheme, a 2-DOF MRM is used to conduct experiments to verify the effectiveness of the algorithm. In the
experiment, redundant adaptive robust Kalman filter and
second-order low-pass filter are used to filter the estimated
joint torque. Experiments show that after the estimated
joint torque passes through the redundant adaptive robust
Kalman filter, the estimated torque of the harmonic drive
device is smoother and more in line with the requirements.
By using decentralized robust control, it can be ensured that
the trajectory tracking error under RARKF is consistent
and ultimately bounded, the effectiveness of the development is proposed through experiments.
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2022 34th Chinese Control and Decision Conference (CCDC)
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