US 2022/0088786 A1 · Fractional Order Sliding Mode Synchronous Control Method For Teleoperation System Based On Event Trigger Mechanism

Abstract

The present invention provides a fractional order sliding mode synchronous control method for a teleoperation system based on an event trigger mechanism. The method comprises: establishing a dynamics model for the teleoperation system by considering external disturbance and parameter uncertainty, selecting a master robot and a slave robot, interactively establishing the teleoperation system through a communication network, determining system parameters of the dynamics model, designing a fractional order nonsingular rapid terminal sliding mode surface equation by utilizing a position tracking error and a fractional order calculus, setting a trigger event condition of information interaction between the master robot and the slave robot, designing a self-adaptive fractional order nonsingular rapid terminal sliding mode controller based on the sliding mode, designing a Lyapunov function to carry out stability analysis, proving the boundedness of a closed-loop state signal of the system.

A fractional order sliding mode synchronous control method for a teleoperation system based on an event trigger mechanism, characterized in that, the method comprises steps of:

S1, establishing a dynamic model for the teleoperation system by considering external disturbance and parameter uncertainty, wherein the dynamic model is:
Mm(qm){umlaut over (q)}m+Cm(qm,{dot over (q)}m){dot over (q)}m+Gm(qm)+Bm({dot over (q)}m)=τm+Fh
Ms(qs){umlaut over (q)}s+Cs(qs,{dot over (q)}s){dot over (q)}s+Gs(qs)+Bs({dot over (q)}s)=τs+Fe (1)
in which, qi, {dot over (q)}i, {umlaut over (q)}1∈Rnrespectively represent a generalized displacement, a speed and an acceleration of n joints of the robot;
Mi(qi) represents a positive definite inertial matrix of the system;
Ci(qi, {dot over (q)}i) represents a matrix of Coriolis forces and centrifugal forces of the system;
Gi(qi) represents a gravity moment of the system;
Fh, Fe∈Rn respectively represent an external force applied by an operator and an external force applied by the environment;
τi∈Rn represents a generalized input moment;
Bi({dot over (q)}i)∈Rn represents a unknown external interference;
i=m, s, m represents a master robot, s represents a slave robot;
Mi(qi), Ci(qi,{dot over (q)}i) and Gi(qi) have a relationship of:
{
M
i
⁡
(
q
i
)
=
M
oi
⁡
(
q
i
)
+
Δ
⁢
M
i
⁡
(
q
i
)
∈
R
n
×
n
C
i
⁡
(
q
i
,
q
.
i
)
=
C
oi
⁡
(
q
i
,
q
.
i
)
+
Δ
⁢
C
i
⁡
(
q
i
,
q
.
i
)
∈
R
n
×
n
⁢
G
i
⁡
(
q
i
)
=
G
oi
⁡
(
q
i
)
+
Δ
⁢
G
i
⁡
(
q
i
)
∈
R
n
(
2
)
in which, Moi(qi) represents a nominal value of the positive definite inertia matrix of the system;
Coi(qi,{dot over (q)}i) represents a nominal value of the matrix of Coriolis forces and centrifugal forces of the system;
Goi(qi) represents a nominal value of the gravity moment of the system;
ΔMi(qi) represents parameter change of the positive definite inertia matrix of the system;
ΔCi(qi,{dot over (q)}i) represents parameter change of the matrix of Coriolis forces and centrifugal forces of the system;
ΔGi(qi) represents parameter change of the gravity moment of the system;
at the same time, Bi({dot over (q)}i) satisfies:
∥Bt({dot over (q)}i)∥≤Bi (3)
in which, Bi represents an unknown positive number;
then,
{
P
m
⁡
(
q
m
,
q
.
m
,
q
¨
m
)
=
-
Δ
⁢
M
m
⁡
(
q
m
)
⁢
q
¨
m
-
Δ
⁢
C
m
⁡
(
q
m
,
q
.
m
)
⁢
q
.
m
-
Δ
⁢
G
m
⁡
(
q
m
)
-
B
m
⁡
(
q
.
m
)
∈
R
n
P
s
⁡
(
q
s
,
q
.
s
,
q
¨
s
)
=
-
Δ
⁢
M
s
⁡
(
q
s
)
⁢
q
¨
s
-
Δ
⁢
C
s
⁡
(
q
s
,
q
.
s
)
⁢
q
.
s
-
Δ
⁢
G
s
⁡
(
q
s
)
-
B
s
⁡
(
q
.
s
)
∈
R
n
(
4
)
in which, Pm(qm,{dot over (q)}m,{umlaut over (q)}m) represents a parameter matrix of the master robot;
Ps(qs,{dot over (q)}s,{umlaut over (q)}s) represents a parameter matrix of the slave robot;
Pm(qm,{dot over (q)}m,{umlaut over (q)}m) and Ps(qs,{dot over (q)}s,{umlaut over (q)}s) are bounded and satisfies:
∥Pm(qm,{dot over (q)}m,{umlaut over (q)}m)∥≤κm
∥Pm(qm,{dot over (q)}m,{umlaut over (q)}m)∥≤κs (5)
in which, κm represents an upper limit of ∥Pm(qm,{dot over (q)}m,{umlaut over (q)}m)∥, κm=b0m+b1m∥qm∥+b2m∥{dot over (q)}m∥2;
κs represents an upper limit of Ps(qs,{dot over (q)}s,{umlaut over (q)}s), κs=b0s+b1s∥qs∥+b2s∥{dot over (q)}s∥2;
b0i, b1i, b2i(i=m, s) is a positive constant;
accordingly, equation (1) for the system is re-represented as:
{
M
o
⁢
m
⁡
(
q
m
)
⁢
q
¨
m
+
C
o
⁢
m
⁡
(
q
m
,
q
.
m
)
⁢
q
.
m
+
G
∘
m
⁡
(
q
m
)
=
τ
m
+
P
m
⁡
(
q
m
,
q
.
m
,
q
¨
m
)
+
F
h
M
∘
s
⁡
(
q
s
)
⁢
q
¨
s
+
C
o
⁢
s
⁡
(
q
s
,
q
.
s
)
⁢
q
.
s
+
G
o
⁢
s
⁡
(
q
s
)
=
τ
s
+
P
s
⁡
(
q
s
,
q
.
s
,
q
¨
s
)
-
F
e
(
6
)
S2, selecting the master robot and the slave robot, interactively constructing the teleoperation system through a communication network, measuring respectively mass and length information of a master robot connecting rod and a slave robot connecting rod and real-time robot position information of the master robot connecting rod and the slave robot connecting rod, and determining system parameters of the dynamic model;
S3, designing a sliding mode surface equation of a fractional order nonsingular rapid terminal by using position tracking errors of the master robot and the slave robot and Riemann-Liouville fractional order calculus;
S4, setting a trigger event condition of information interaction between the master robot and the slave robot, designing a self-adaptive fractional order nonsingular rapid terminal sliding mode controller which can compensate measurement errors and eliminate the influence of system uncertainty based on the sliding mode, enabling the controller to avoid the Zeno phenomenon, performing stability analysis, proving the boundedness of a closed-loop state signal of the system, enabling the slave robot to track the motion of the master robot within a limited time, and realizing synchronous control of the teleoperation system;
in step S4, the trigger event condition is:
{
τ
i
⁡
(
t
)
=
u
i
⁡
(
t
k
i
)
⁢
,
⁢
∀
t
∈
[
t
k
i
,
t
k
+
1
i
)
t
k
+
1
j
=
i
⁢
n
⁢
f
⁢
{
t
∈
R
❘

z
i
⁡
(
t
)

⁢
z
i
⁡
(
t
)
≥
b
i
}
,
t
1
i
=
0
,
j
=
1
,
2
,
…
⁢
,
n
(
10
)
in which, zi(t) represents a measurement error, zi(t)=[zi1, zi2, . . . , zin]T and zi(t)=ui(t)−τi(t);
τi(t) represents a generalized input moment of the teleoperation system, τi(t)=[τi1, τi2, . . . , τin]T;
bi represents a positive definite matrix, bi(t)=[bi1, bi2, . . . , bin]T ∈Rn;
ui(t) represents actual controller under event trigger mechanism, ui(t)=[ui1, ui2, . . . , uin]T;
tki, k∈z+ represents controller update time;
in a period of time ti∈[tki,tk+1t), the control signal remains constant until ti=tk+1i, minimum internal time of the mentioned event trigger mechanism is strictly greater than 0, which eliminates occurrence of the Zeno behavior;
based on the designed sliding mode, the actual controller is chosen as:
τi(t)=ui(t)−λi(t)bi (11)
in which, λi(t) represents a continuous time-varying parameter satisfying conditions λi(tki)=0,λi(tk+1i)=±1 and |λi(t)|≤1, ∀t∈[tki,tk+1i;
the sliding mode controller of the self-adaptive fractional order nonsingular rapid terminal is designed as:
{
u
m
⁡
(
t
)
=
u
_
m
-
b
_
m
⁢
tanh
⁡
(
s
m
T
⁢
M
om
-
1
⁢
b
_
m
⁢
/
⁢
ɛ
m
)
u
_
m
=
M
om
⁡
(
q
m
)
⁢
q
¨
s
⁡
(
t
-
d
s
)
+
C
om
⁡
(
q
m
,
q
.
m
)
⁢
q
.
m
+
G
om
⁡
(
q
m
)
+
u
_
m
⁢
1
+
u
_
m
⁢
2
u
_
m
⁢
1
=
-
M
om
⁡
(
Θ
m
⁢
s
⁢
i
⁢
g
⁡
(
s
m
)
μ
+
α
m
⁢
1
⁢
D
α1
⁡
[
si
⁢
g
⁡
(
e
m
)
γ
m
⁢
⁢
1
]
+
α
m
⁢
2
⁢
D
α
2
+
1
⁡
[
β
m
⁡
(
e
m
)
]
)
u
_
m
⁢
2
=
{
-
s
m
T
⁢
M
o
⁢
m
-
1

s
m
T
⁢
M
o
⁢
m
-
1

⁢
κ
^
m
,
if
⁢
⁢
s
m
T
⁢
M
o
⁢
m
-
1
≠
0
⁢
0
,
if
⁢
⁢
s
m
T
⁢
M
o
⁢
m
-
1
=
0
(
12
)
{
u
s
⁡
(
t
)
=
u
_
s
-
b
_
s
⁢
tanh
⁡
(
s
s
T
⁢
M
os
-
1
⁢
b
_
s
⁢
/
⁢
ɛ
s
)
u
_
s
=
M
os
⁡
(
q
s
)
⁢
q
¨
m
⁡
(
t
-
d
m
)
+
C
os
⁡
(
q
s
,
q
.
s
)
⁢
q
.
s
+
G
os
⁡
(
q
s
)
+
u
_
s
⁢
⁢
1
+
u
_
s
⁢
⁢
2
u
_
s
⁢
⁢
1
=
-
M
os
⁡
(
Θ
s
⁢
s
⁢
i
⁢
g
⁡
(
s
s
)
μ
+
α
s
⁢
⁢
1
⁢
D
α1
⁡
[
si
⁢
g
⁡
(
e
s
)
γ
s
⁢
⁢
1
]
+
α
s
⁢
⁢
2
⁢
D
α
2
+
1
⁡
[
β
s
⁡
(
e
s
)
]
)
u
_
s
⁢
⁢
2
=
{
-
s
s
T
⁢
M
os
-
1

s
s
T
⁢
M
os
-
1

⁢
κ
^
s
,
if
⁢
⁢
s
s
T
⁢
M
os
-
1
≠
0
⁢
0
,
if
⁢
⁢
s
s
T
⁢
M
os
-
1
=
0
(
13
)
in which, bm represents a positive definite matrix, bm=[bm1, bm2, . . . , bmn]T and satisfies the constant bmj>bmj(j=1, 2, . . . , n);
bs represents a positive definite matrix, bs=[bs1, bs2, . . . , bsn]T and satisfies the constant bsj>bsj(j=1, 2, . . . , n);
εm, εs represents a positive constant to be designed;
Θi represents a positive definite diagonal matrix, Θi=diag[θi1, θi2, . . . , θin] and satisfies the constant θij>0(j=1, 2, . . . , n);
μ represents a positive constant to be designed and satisfies μ∈(0,1);
{circumflex over (κ)} represents an estimated value of an upper bound of ∥Pm(qm, {dot over (q)}n, {umlaut over (q)}m)∥, {circumflex over (κ)}m={circumflex over (b)}0m+{circumflex over (b)}1m∥+{circumflex over (b)}2m∥{dot over (q)}m∥2;
{circumflex over (κ)}s represents an estimated value of an upper bound of ∥Ps(qs, {dot over (q)}s, {umlaut over (q)}s)∥, {circumflex over (κ)}s={circumflex over (b)}0s+{circumflex over (b)}1s∥+{circumflex over (b)}2s∥{dot over (q)}s∥2;
variables {circumflex over (b)}0i, {circumflex over (b)}1i, {circumflex over (b)}2i representing estimated values of parameters b0i, b1i, b2i, respectively;
the self-adaptation law is designed as:
{circumflex over ({dot over (b)})}0i=Λ0i∥siTMoi−1∥
{circumflex over ({dot over (b)})}1i=Λ1i∥siTMoi−1∥∥qi∥ (14)
{circumflex over ({dot over (b)})}2i=Λ2i∥siTMoi−1∥∥qi∥2
in which, Λ0i, Λ1i, Λ2i represents positive constants;
a Lyapunov function is selected as:
V
=
V
1
+
V
2
⁢
⁢
V
1
=
∑
i
=
m
,
s
⁢
1
2
⁢
s
i
T
⁢
s
i
⁢
⁢
V
2
=
∑
i
=
m
,
s
⁢
1
2
⁢
∑
j
=
0
2
⁢
Λ
j
⁢
⁢
i
-
1
⁢
b
~
j
⁢
⁢
i
2
(
15
)
in which, V represents a Lyapunov function;
V1 represents a first Lyapunov function, V2 represents a second Lyapunov function, {tilde over (b)}ji=bji−{circumflex over (b)}ji represents a self-adaptive estimation error, j=1, 2, . . . , n;
limited time stable operation of the nonlinear uncertain teleoperation system is ensured through stability analysis.

1,064 more elements

Fractional Order Sliding Mode Synchronous Control Method For Teleoperation System Based On Event Trigger Mechanism

Life of this patent

Fractional Order Sliding Mode Synchronous Control Method For Teleoperation System Based On Event Trigger Mechanism

Life of this patent

Abstract

Claims3