Gain Scheduling based Backstepping Control for Motion Balance Adjusting of a Power-line Inspection Robot DIAN Songyi1, HOANG Son1, 2, PU Ming1, LIU Junyong1, CHEN Lin1 1.. Vietnam Nati
Trang 1Gain Scheduling based Backstepping Control for Motion Balance
Adjusting of a Power-line Inspection Robot
DIAN Songyi1, HOANG Son1, 2, PU Ming1, LIU Junyong1, CHEN Lin1
1 School of Electrical Engineering and Information, Sichuan University, Chengdu 610065, China
E-mail: scudiansy@scu.edu.cn
2 Vietnam National University of Forestry, Hanoi, Vietnam
E-mail: hoangsonbk83@yahoo.com.vn
Abstract: This paper presents a gain scheduling backstepping control (GSBC) for motion balance adjusting of a power-line
inspection (PLI) robot, which is an underactuted mechanical system of two degrees of freedom with one control input First, a dynamic model of the motion balance adjusting of the PLI robot is constructed Second, this model is linearized at a nominal operating point to overcome the computation infeasibility of the conventional backstepping technique Finally, to extend the operation area of the closed-loop system further from the nominal operating point, an equilibrium manifold linearization model (EML model) is developed using a scheduling variable, and then the GSBC scheme is designed based on the EML model The robust stability of the closed-loop system is ensured by the Lyapunov theorem Simulation results show that the GSBC gives much better performance than that of the backstepping control, these results illustrate the feasibility of the proposed control scheme
Key Words: Gain scheduling backstepping, balance adjusting, power line inspection robot
1 Introduction
In recent years, the study on underactuated nonlinear
systems has caught extensive attention in theory and
practical applications [1-5] The underactuated nonlinear
system is a system having fewer actuators than its degrees of
freedom [6] Some well-known underactuated systems with
two degrees of freedom and one actuation have been
considered including the inverted pendulum system [7, 8],
the ball and beam system [9-11] The underactuated systems
have some advantages such as light weight and low energy
consumption, whereas the control design of these systems is
more complex than that addressed in fully actuated systems
To control underactuated systems, there are several main
methods such as the sliding mode control (SMC) [12, 13],
the energy method [8, 14], and the backstepping control (BC)
[15-17] Among these control methods, the advantage of the
BC is the high robustness of a closed-loop system, which is a
popular strategy to design the controller for underactuated
nonlinear systems However, in some cases, the dynamic
model may consist of cross terms or high nonlinearity
functions; therefore, it is necessary to consider the
computational feasibility with backstepping procedures To
resolve this problem, the linearization technique is applied
for the sake of having an approximate linear model of the
nonlinear plant, which guarantees the simplicity needed to
perform backstepping procedures
However, the linearization technique is the conversion of
a linear approximation into a nonlinear function at a given
point, thus a controller design based on linearization model
only achieves efficiency when a system operates around a
nominal operating point So as to resolve this problem, the
researchers focused on expanding the linearization model to
an equilibrium manifold model (EML model) using a
scheduling variable
In this paper, a gain scheduling backstepping control
(GSBC) is proposed for the motion balance adjusting of the
PLI robot [18] The motion balance adjusting process of the
PLI robot in the second step of the Line-leading mode is an
underactuated nonlinear system In this step, the PLI robot moves to the power cable and simultaneously keeps balance
by adjusting the self-balance mechanism, as illustrated in
Fig 1 The proposed GSBC is advantaged in the following aspects The nonlinear dynamic model of the PLI robot is
first converted into a linear model at a nominal operating point The BC which is designed based on this linear model
can overcome computation infeasibility of the conventional backstepping method Next, in order to expand the operation area of the closed-loop system, the linear model of the plant
is transformed to an EML model using a scheduling variable Consequently, the proposed GSBC scheme based on this EML model both extends the operational area of the closed-loop system and overcomes the computational infeasibility of the initial nonlinear model The closed-loop
stability of the controlled system is guaranteed by the
Lyapunov theorem The simulation results demonstrate the efficiency and feasibility of the proposed GSBC scheme
Two bundled cables
Counter-weight box Self-balance
adjusting mechanism
Insulated access cable
Motion arms Pulley
Fig.1: Line-loading mode of the PLI robot
This paper is divided into the following sections The preliminary knowledge of the motion balance adjusting of
July 27-29, 2016, Chengdu, China
Trang 2the PLI robot is discussed in Section 2 In Section 3, the
GSBC for the motion balance adjusting of the PLI robot is
designed The simulation results are presented in Section 4
Finally, Section 5 provides concluding remarks
2 Preliminary knowledge
2.1 Dynamic model for the motion balance adjusting of
the PLI robot
active joint
1
X
2
X
2
Z
2
Y
1
Z
1
Y
1
O
2
O
2 T
l
2 ( )
m kg
1
m
2
m
1 ( )
m kg
(1)
2
Y
1
X
1
Z
1
h h20
1
O
2
O
(1) The center of mass (COM) of the robot body
(2) The COM of the counter-weight box
(a) 3D model (b) model in the X1O1Z1 plane
1
T
1
2
sin
Contact between pulley and cable
d
1
X
1
Y
2
Z
2
Z
2
Y
2
Y
(1)
(2)
2
O
2
O
1
O
(c) model in the X1O1Y1 plane
Fig.2: Balance adjustment parameters of the PLI robot
Table 1: The parameters of the PLI robot*
*m 1 is the mass of the robot body, m 2 is the mass of the
counter-weight box, l is the length of actuator bar (Fig.2(b)), d is
the height of the T-shaped base (Fig.2(a, c))
Consider the motion balance adjusting of the PLI robot in
Fig.2 Let T1( )t be the angle between the robot and the
cable, let T2( )t be the angle of the active joint Let h be 1
the distance between the cable and the center of mass (COM)
of the body, let h2 h20lsinT2 be the distance from the
cable to the COM of the counter-weight box Let u t be 2( )
the torque acting on the active joint T2( )t
We use the Euler-Lagrange equation to obtain the motion
equations of the motion balance adjusting of the PLI robot
The Lagrangian equation is given as follows [19]
1, ,
i
i i
T
»
where U is the external torque acting on the ith i
generalized coordinate L K P , with K and P are the
kinetic and potential energy of the motion balance adjusting
of the PLI robot, respectively:
2 2
d2
,(2)
sin
T
where g is the gravitational acceleration In the Table 1, we
have
1 1 2 20
m h m h , (4)
so the P can be re-written as
2 cos 1 sin 2sin 1
Substituting Eq (2) and Eq (5) into Eq (1), it yields
T T T T
1
T
2
T T11 (6)
2
cos sin
m gd
2 2 2sinsin cosT
2 2 2 22
2cos2
2 2 2 2
cos
22 2
(
2 2
(
2
Remark 1: The Eq (6) and Eq (7) are dynamic equations
of the motion balance adjusting of the PLI robot
2.2 Analysis of the motion balance adjusting of the PLI robot
To investigate the motion balance adjusting of the PLI robot, we define the state variable vector as follows
x x x x x ¬ªT T T T1 T T TT T T1 22 2º¼ºººT (8) The state-space equations are
2
2
cos sin ]
x
x
m l
x1 x
2 2
m g2 x
x3 x
2 4
m g2 x
(9)
Remark 2: When the steady-state input equals zero
equilibrium point at the origin as
Consider the equation system as follows
2
4
2
2 2
0
0
0
cos sin ] 0
x
x
m l
(11)
Trang 3If 1 dtanx1 1
l
this equation, it yields S/ 4dx1dS/ 4), then the roots of
the Eq (11) are
4
2
0
0
t n
d
l x
d
l
x
(12)
satisfying S/ 4dx1dS/ 4, the Eq (11) has respective
yields x and 3 u2(0) Therefore, the state equation (9) has
equilibrium manifold
3 Controller design
3.1 Problem formulation
In this paper, the backstepping technique is utilized to
design the stabilizing controller for the balance of the PLI
robot
Step1.1: We consider a control problem with output x 1
Let
e1 x1 x1d (13)
Where x is a desired goal of the balance angle 1d T1
Taking the derivative of (13) as
e e111 x x x1111 x x x1d1d1dddd x x x2222x x1d1d1 (14)
A Lyapunov function candidate is chosen as
2
1 2
V e ,
V e e e x x
V1 e e11 1e111 e11((( 22 1d1 ) (15)
It can be seen that (15) is negative infinity if we choose
x k e x x1d1d1d k k11111!0 (16)
Step 1.2: We choose virtual control as
x2d k e1 1x x1d1d1 , (17) x x x2d2d2d2d k e k e k e1 11 11 11 1x x1d1d1 (18)
To realize x2 ox2d, we get a new error
e2 x2x2d, (19)
e e2222 x xx2222x x2d2d2 (20)
The now
1 1( 2 1d) 1( 2 1 1)
V1 e e (1( 2 1d1d1ddd)))) 11111(((( 22 (21)
To realize e2o and 0 e1o , a Lyapunov function 0
candidate is chosen as
2
1 2
V V e , (22) then
2
V k e2e e e x x
2 k e2
1 1
1 2222 2d2 ) (23) Substituting x x22 in state-space equation (9) into Eq (23),
it yields
2
2
1 1 1
¸
x
¸
2d
x ¸¸
To realize V2 o The problem of concern here is to find 0 the value of x , making 3 V V222 satisfying
V k e2
1 1
From equation (24), we can see that finding x can lead 3
to computation infeasibility In order to resolve this problem, state-space equation (9) is linearized to make use of the backstepping technique
Next, the linearization model of the PLI robot in Eq (9) is
0
2 21 1 23 3
4 41 1 4 2
x x
x a x a x
x a x b u
x1 x
x2 a x21 12121 121
x3 x
x4 a x a xx41 14141 141
(26)
m gd a
g a l
2
1
b
and
2 2 2
2 20 2
4
m l h
m gl a
Remark 4: From Eq (26), we can see that BC is designed
based on the linearization model of the PLI robot, which can easily overcome the computation infeasibility
3.2 Controller design of GSBC
The linearization model in Eq (26) will not be accurate when the closed-loop system operates outside the nominal
0
x 0 0 0 0 Therefore, to expand the operating area of the closed-loop system, this linearization model will be expanded about an EML model using a scheduling variable
A Equilibrium manifold linearization model
To begin with, the state-space equation of the plant is described in Eq (9) The linearization of Eq (9) about its equilibrium manifold yields the parameterized linearization family [20, 21]
d
Where
x( ), ( )
( )
U
f A
x D D
D ¨ ¸§©ww ·¹ and
x( ), ( )
( )
U
f B
( )
AD and B( )D are the parameterized plant linearization family matrices; x( )D and U( )D are the parameterized steady-state variable and the parameterized input; and D is
a scheduling variable
We choose the x of the PLI robot as a scheduling 1d
variable, then D x1d,
x( )D x 0 xD 0 , (28)
U( )D u2( )D m gl2 sinx1d (cosx3D), (29)
Trang 421 23
41
43
( )
A
D
4
Where
3 arcsin dtan 1d
l
D
21
*
2
tan
a
23
1d
*
2 2
1d
tan
l
a
l
a
l
and 41
1d
*
(tan ) (cos ) 1 d x
l a
l
Now, the EML model Eq (27) can be re-written as
x x
x x
D
x1 x
*
x2 a2121(( 11
x3 x
*
(31)
B Controller design
Theorem: When the EML model of the PLI robot is
defined by Eq (31), the GSBC scheme based on this EML
model is designed The robust stability of the closed-loop
system is ensured, the operational area of the closed-loop
control system is extended, and the computational
infeasibility of initial nonlinear model is overcome as well
Proof:
Step 2.1: Starting with the first equation of Eq (31), we
define an error
e1 x1 x1d, (32)
e e11111 x x x11111 x x x1d1d1d1d1dd x x x2222x x1d1d1 (33)
To realize e1o , a Lyapunov function candidate is 0
chosen as
2
1 2
V e , (34)
V e e e x x
V1 e e11 1e111 e11((( 22 1d1 ) (35)
If we choose x2x x1d1d1d1d1d1d k e k k k e1 11 11 111 11 ˈ1!0, then
2
V k e2
V1 k e2
1
k e1 11 (36)
Step 2.2: To realize x2x x x1d1d1d1dd k e k e1 11 11 11 , that is
x k e x x1d1 , we choose virtual control as
x2d k e1 1x x1d1d1 (37)
To realize x2ox2d, we get a new error
e2 x2x2d, (38)
2d
e2 x x22 x x2d a x a x21( 11 x x1d))) a x a x23((( 3 3 ) (39)
To realize e2o and 0 e1o , we design Lyapunov 0 function as
2
1 2
V V e (40) From Eq (35), Eq (37), and Eq (39), the derivative of
2
V can be obtained as
V k e2e e a xª¬ x a x xD x º¼
2 k e2
1 1
If we choose
23
1
then
V k e2k e d
2 k e2
1 1
1 (43)
Step 2.3: We choose the virtual control as
23
1
To realize x3ox3d, we get a new error
e x x , (45)
e333 x x3333x x3d3d3d x x444x3d3 (46)
To realize e3o , 0 e2o and 0 e1o , we design 0 Lyapunov function as
V V e e e e , (47) then
V V e e
V3 V V22 e e33e3333 (48) From Eq (41), Eq (44), and Eq (46), the derivative of V3
can be obtained as
3( )i 1 1 2 2 3( 23 2 4 3d)
Choosing
*
4 3 3 23 2 3d, 3 0
x k e a e x x3d3d3d k k33333!0 (50) Then we would have
V V3 k e k1 122 2k e2 22k e3 32d0
3 k e2
1 1
1 (51)
Step 2.4: We choose the virtual control as
4d 3 3 23 2 3d, 3 0
x k e a e x x3d3d3d k k33333!0 (52)
To realize x4ox4d, we get a new error
e x x , (53)
4 41( 1 1d) 43( 3 3 ) 4 2 2( ) 4d
e4 4 4 a41 41 41( 1 1 1 1 1 1d 1d 1d 1d 1d 1d) 43 43 43 43 43( 3 3 3 3 3 3 3 )) 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2( )( )) x4d 4 (54)
To realize e4o , 0 e3o , 0 e2o and 0 e1o , we 0 design Lyapunov function as
V V e e e e e , (55) then
V 4 V V V333 e e e e4 444e4444 (56) From Eq (49), Eq (52) and Eq (54), the derivative of V V444
is shown as follows
*
i
V4( ))
]
4d 4
(57)
To realize V V444 dd 00 , we choose
Trang 5* *
4
b
D
D
º¼
4d
x
(58)
Substituting Eq 58 into Eq 57, we would have
V k e2k e k e k e d
4 k e2
1 1
1 (59) From Eq (59), we can see that GSBC law u t in Eq 2( )
(58) is a stabilizing function to be determined for the
closed-loop system The theorem is proved
4 Simulation Results
The physical parameters of the PLI robot are listed in
Table I The parameters for BC and GSBC are initially
chosen as k1 k2 k3 k4 5
The control goal is to stabilize the PLI robot at the desired
balance position Two cases of different conditions are
addressed as follow:
Case 1: The initial condition is
is x1d 0 rad
-0.2
0
0.2
0.4
-1
0
1
Time (s)
T1
GSBC BC Desired angle
GSBC BC
Fig.3: Simulation results of case 1: T1 (top) and TT11 (bottom)
-0.5
0
0.5
-2
0
2
4
GSBCBC
GSBC BC
Time (s)
(2
Fig.4: Simulation results of case 1: T2 (top) and TT22 (bottom)
-200 -100 0 100 200 300
400
GSBC BC
Time (s)
u 2
Fig.5: Simulation results of case 1: control signals
desired balance position is x1d 0.2 rad
-0.2 0 0.2
-0.5 0 0.5 1
T1
Time (s)
GSBC BC Desired angle
GSBCBC
Fig.6: Simulation results of case 2: T1 (top) and TT11 (bottom)
-0.5 0 0.5
-2 0 2 4
GSBC BC
GSBC BC
Time (s)
T2
Fig.7: Simulation results of case 2: T2 (top) and TT22 (bottom)
-200 -100 0 100 200 300
400
GSBC BC
Time (s)
u 2
Fig.8: Simulation results of case 2: control signals
Trang 6In case 1, the control objective is to keep the robot
balanced at the origin x1d 0 rad The simulation results
indicate that, all the two schemes managed to asymptotically
balancing the PLI robot at this point (Fig 3) In this case, the
results corresponding to the BC and GSBC controllers
similarly because the linear model and EML model are same
21 21, , 23 23 41 41
a ) Fig 4 and Fig 5 denote the angles, the angular velocities and the
control signals of the active joint, respectively
Simulation results in case 2 indicate that due to the high
nonlinear dynamic model, the BC does not meet the control
requirement, and there is a steady-state error while GSBC
can asymptotically balance the robot at the desired position
(Fig 6) Fig 7 and Fig 8 denote the angles, the angular
velocities and the control signals of the active joint,
respectively
5 Conclusion
In this paper, the control problem of the motion balance
adjusting of the PLI robot is addressed by a proposed GSBC
Based on the knowledge about dynamic equations of the
motion balance adjusting of the PLI robot, an EML model is
formulated The GSBC is designed based on the equilibrium
manifold linearization model both extending the operation
area of the closed-loop system and overcoming the
computation infeasibility of the initial nonlinear dynamic
model The closed-loop stability of the controlled system is
guaranteed by the Lyapunov theorem Simulation results
illustrate the superiority of the proposed GSBC scheme
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