Therefore, this chapter develops a practical and detailed two-link planar robotic systems modeling and a robust control design for this kind of nonlinear robotic systems with uncertainti
Trang 2above, there is a need of a detailed and practical two-link planar robotic system modeling
with the practically distributed robotic arm mass for control
Therefore, this chapter develops a practical and detailed two-link planar robotic systems
modeling and a robust control design for this kind of nonlinear robotic systems with
uncertainties via the authors’ developing robust control approach with both H∞ disturbance
rejection and robust pole clustering in a vertical strip The design approach is based on the
new developing two-link planar robotic system models, nonlinear control compensation, a
linear quadratic regulator theory and Lyapunov stability theory
2 Modeling of Two-Link Robotic Systems
The dynamics of a rigid revolute robot manipulator can be described as the following
nonlinear differential equation [1, 2, 6, 10]:
),(),()(q q V q q q N q q M
F c (1.a)
)()
(),(q q G q F q F q
N d s (1.b) where M (q) is an n n inertial matrix, V ( q q, an ) n n matrix containing centrifugal and
coriolis terms, G(q) an n1 vector containing gravity terms, q(t) an n1 joint variable
vector, F c an n1 vector of control input functions (torques, generalized forces), F d an
n
n diagonal matrix of dynamic friction coefficients, and F s (q an ) n1 Nixon static
friction vector
However, the dynamics of the robotic system (1) in detail is needed for designing the
control force, i.e., especially, what matrices M (q), V ( q q, and ) G (q) are
Consider a general two-link planar robotic system in Fig 1, where the system has its joint
mass m1 and m2 of joints 1 and 2, respectively, robot arms mass m1r and m2r distributed
along arms 1 and 2 with their lengths l1 and l2, generalized coordinates q1 and q2, i.e.,
their rotation angles, q[q1q2], control torques (generalized forces) f1 and f2,
Fig 1 A two-link manipulator
Theorem 1 A general two-link planar robotic system has its dynamic model as in (1) with
)(
M M M M q
M (2)
2 2 1 2 2 2 2 2 2 2
2 2 2 1 1 1 11
cos)(
2)(
)(
q l m m l m m
l m m m m M
r r
r r
2 2 2 2 21 12
cos)(
)(
q l m m
l m m M M
),(q q m2 2m2 l2q2 q2
) (
) cos(
) (
cos ) (
) (
2 1 2 2 2 2
2 1 2 2 2 2 1 1 2 2 1 1 1
q q l m m
q q l m m q l m m m m g q G
r
r r
r
where g is the gravity acceleration, m1 and m2 are joints 1 and 2 mass, respectively, m r1
and m r2 are total mass of arms 1 and 2, which are distributed along their arm lengths of l1
and l2, the scaling coefficients 1, 2, 1 and 2 are defined as follows:
ri l S l dl
m 0 () () , i1,2 (6) where 1(l) and 2(l) are the arm mass density functions along their length l, S1(l) and )
(
2 l
S are the arm cross-sectional area functions along the length l Proof: The proof is via Lagrange method and dynamic motion equations The mass distribution can be various by introducing the above new scaling coefficients Due to the page limit, detail of the proof is omitted
Remark 1 From (2)–(4) in Theorem 1, M(q)V(q,q) Theorem 1 is also different from the result in [3-6] Especially, there are no corresponding items of i in [3-6]
Corollary 1 A two-link planar robotic system with consideration of only joint mass has its
dynamic model as in (1) and Theorem 1, but with
11 (m m )l m l 2m l cosq
)cos(22 12 2
2 21
M , M 22 m2l22 (8)
Trang 3ROBUST CONTROL DESIGN FOR TWO-LINK NONLINEAR ROBOTIC SYSTEM 553
above, there is a need of a detailed and practical two-link planar robotic system modeling
with the practically distributed robotic arm mass for control
Therefore, this chapter develops a practical and detailed two-link planar robotic systems
modeling and a robust control design for this kind of nonlinear robotic systems with
uncertainties via the authors’ developing robust control approach with both H∞ disturbance
rejection and robust pole clustering in a vertical strip The design approach is based on the
new developing two-link planar robotic system models, nonlinear control compensation, a
linear quadratic regulator theory and Lyapunov stability theory
2 Modeling of Two-Link Robotic Systems
The dynamics of a rigid revolute robot manipulator can be described as the following
nonlinear differential equation [1, 2, 6, 10]:
),
()
,(
)(q q V q q q N q q
M
F c (1.a)
)(
)(
),
(q q G q F q F q
N d s (1.b) where M (q) is an n n inertial matrix, V ( q q, an ) n n matrix containing centrifugal and
coriolis terms, G(q) an n1 vector containing gravity terms, q(t) an n1 joint variable
vector, F c an n1 vector of control input functions (torques, generalized forces), F d an
n
n diagonal matrix of dynamic friction coefficients, and F s (q an ) n1 Nixon static
friction vector
However, the dynamics of the robotic system (1) in detail is needed for designing the
control force, i.e., especially, what matrices M (q), V ( q q, and ) G (q) are
Consider a general two-link planar robotic system in Fig 1, where the system has its joint
mass m1 and m2 of joints 1 and 2, respectively, robot arms mass m1r and m2r distributed
along arms 1 and 2 with their lengths l1 and l2, generalized coordinates q1 and q2, i.e.,
their rotation angles, q[q1 q2], control torques (generalized forces) f1 and f2,
Fig 1 A two-link manipulator
Theorem 1 A general two-link planar robotic system has its dynamic model as in (1) with
)(
M M M M q
M (2)
2 2 1 2 2 2 2 2 2 2
2 2 2 1 1 1 11
cos)(
2)(
)(
q l m m l m m
l m m m m M
r r
r r
2 2 2 2 21 12
cos)(
)(
q l m m
l m m M M
),(q q m2 2m2 l2q2 q2
) (
) cos(
) (
cos ) (
) (
2 1 2 2 2 2
2 1 2 2 2 2 1 1 2 2 1 1 1
q q l m m
q q l m m q l m m m m g q G
r
r r
r
where g is the gravity acceleration, m1 and m2 are joints 1 and 2 mass, respectively, m r1
and m r2 are total mass of arms 1 and 2, which are distributed along their arm lengths of l1
and l2, the scaling coefficients 1, 2, 1 and 2 are defined as follows:
ri l S l dl
m 0 () ( ) , i1,2 (6) where 1(l) and 2(l) are the arm mass density functions along their length l, S1(l) and )
(
2 l
S are the arm cross-sectional area functions along the length l Proof: The proof is via Lagrange method and dynamic motion equations The mass distribution can be various by introducing the above new scaling coefficients Due to the page limit, detail of the proof is omitted
Remark 1 From (2)–(4) in Theorem 1, M(q)V(q,q) Theorem 1 is also different from the result in [3-6] Especially, there are no corresponding items of i in [3-6]
Corollary 1 A two-link planar robotic system with consideration of only joint mass has its
dynamic model as in (1) and Theorem 1, but with
11 (m m )l m l 2m l cosq
)cos(22 12 2
2 21
M , M 22 m2l22 (8)
Trang 4,(q q m21l2q2 q2
)cos(
cos)(
)(
2122
21221121
q q l m
q q l m q l m m g q
),(
2
212221
q q q q l m q q
V (11)
in (1) Note that the Coriolis matrix is different from some earlier literatures in [3, 4]
Theorem 2 Consider a two-link planar robotic system having its robot arms with uniform
mass distribution along the arm length Thus, its dynamic model is as (1) – (6) of Theorem 1
with its scaling factors as follows:
3/12
1
, and 12 1/2 (12) Proof: It can be proved by Theorem 1 and the uniform mass distribution in (6)
Theorem 3 Consider a two-link planar robotic system having its robot arms with linear
tapered-shapes respectively along the arm lengths as:
l k r l
r i() 0i i , 0l l i, S i(l) r i2(l), i1,2 (13) where r i (l) in length is a general measure of the arm cross-section at the arm length l, e.g.,
as a radius for a disk, a side length for a square, abfor a rectangular with sides a and b,
etc., S i (l) is the cross-sectional area of arm i at its length position l, i is a constant, e.g.,
as for a circle and 1 for a square Assume that arm 1 and arm 2 respectively have their two
end cross-sectional areas as:
)0(1
S , S t1S1(l1),S 02 S2(0),S t2S2(l2) (14) where S 0i S ti, i1,2 Their density functions are constants as i(l)i, i1,2 Then,
its dynamic model is as in (1) – (6) of Theorem 1 with its scaling factors:
)(
10
36
0 0
0 0
ti i ti i
ti i ti i i
S S S S
S S S S
4
23
0 0
0 0
ti i ti i
ti i ti i i
S S S S
S S S S
,1
i , and its arm mass:
3/)( 0i ti 0i ti
M 0 , NN0N, VV0V (17) where M0, N0, V0 are known parts, M, N, V are
unknown parts Then, the models in Section 2 can be used not only for the total uncertain robotic systems with uncertain parameters, but also for a known part with their nominal parameters of the systems
Following our [6], we develop the torque control law as two parts as follows:
u q M q q N q q q V q q M
F c 0( )d 0( ,) 0( ,) 0( ) (18) where the first part consists of the first three terms in the right side of (18), the second part is
the term of u that is to be designed for the desired disturbance rejection and pole clustering, d
q is the desired trajectory of q, however, the coefficient matrices are as (2) – (6) in Theorem
1 with all nominal parameters of the system Define an error between the desired q d and
the actual q as:
q q
e d (19)From (1) and (17)–(19), it yields:
])(),(),()
()[
1 q M q q V q q q N q q M q u M
e d
u Fu e E
(20)
),()(
1 q V q q M
E , F M1(q)M(q)
N M q E q F
w d d 1 (21) From [6], we can have the fact that their norms are bounded:
w
w , E e, F f (22) Then, it leads to the state space equation as:
Trang 5sin)
,(q q m21l2q2 q2
)cos(
cos)
()
(
21
22
21
22
11
21
q q
l m
q q
l m
q l
m m
g q
),
(
2
21
22
21
q q
q q
l m
q q
V (11)
in (1) Note that the Coriolis matrix is different from some earlier literatures in [3, 4]
Theorem 2 Consider a two-link planar robotic system having its robot arms with uniform
mass distribution along the arm length Thus, its dynamic model is as (1) – (6) of Theorem 1
with its scaling factors as follows:
3/
12
1
, and 121/2 (12) Proof: It can be proved by Theorem 1 and the uniform mass distribution in (6)
Theorem 3 Consider a two-link planar robotic system having its robot arms with linear
tapered-shapes respectively along the arm lengths as:
l k
r l
r i() 0i i , 0l l i, S i(l)r i2(l), i1,2 (13) where r i (l) in length is a general measure of the arm cross-section at the arm length l, e.g.,
as a radius for a disk, a side length for a square, abfor a rectangular with sides a and b,
etc., S i (l) is the cross-sectional area of arm i at its length position l, i is a constant, e.g.,
as for a circle and 1 for a square Assume that arm 1 and arm 2 respectively have their two
end cross-sectional areas as:
)0
(1
S , S t1S1(l1),S 02 S2(0),S t2S2(l2) (14) where S 0i S ti, i1,2 Their density functions are constants as i(l)i, i1,2 Then,
its dynamic model is as in (1) – (6) of Theorem 1 with its scaling factors:
)(
10
36
0 0
0 0
ti i
ti i
ti i
ti i
i
S S
S S
S S
S S
4
23
0 0
0 0
ti i
ti i
ti i
ti i
i
S S
S S
S S
S S
,1
i , and its arm mass:
3/
)( 0i ti 0i ti
M 0 , N N0N, VV0V (17) where M0, N0, V0 are known parts, M, N, V are
unknown parts Then, the models in Section 2 can be used not only for the total uncertain robotic systems with uncertain parameters, but also for a known part with their nominal parameters of the systems
Following our [6], we develop the torque control law as two parts as follows:
u q M q q N q q q V q q M
F c 0( )d 0( ,) 0( ,) 0( ) (18) where the first part consists of the first three terms in the right side of (18), the second part is
the term of u that is to be designed for the desired disturbance rejection and pole clustering, d
q is the desired trajectory of q, however, the coefficient matrices are as (2) – (6) in Theorem
1 with all nominal parameters of the system Define an error between the desired q d and
the actual q as:
q q
e d (19)From (1) and (17)–(19), it yields:
])(),(),()()[
1 q M q q V q q q N q q M q u M
e d
u Fu e E
(20)
),()(
1 q V q q M
E , F M1(q)M(q)
N M q E q F
w d d 1 (21) From [6], we can have the fact that their norms are bounded:
w
w , E e, F f (22) Then, it leads to the state space equation as:
Trang 6Bw BFu x E B Bu Ax
x [0 ] (23)
][ 1 2 1 2
for manipulators to follow is to be bounded functions of time Its corresponding velocity q d
and acceleration q d, as well as itself q d, are assumed to be within the physical and
kinematic limits of manipulators They may be conveniently generated by a model of the
type:
)()()
()
(t K q t K q t r t
qd vd p d (25)
where r(t) is a 2-dimensional driving signal and the matrices K v and K p are stable
The design objective is to develop a state feedback control law for control u in (18) as
)()(t Kx t
u (26) such that the closed-loop system:
B BK A
x( 0 ) (27)has its poles robustly lie within a vertical strip :
}{
)( 2 1
A c 0 (30) From [6], we derive the following robust control law to achieve this objective
Theorem 4 Consider a given robotic manipulator uncertain system (27) with (1)–(6),
(17)-(22), (24), where the unstructured perturbations in (21) with the norm bounds in (17)-(22), the
disturbance rejection index 0 in (29), the vertical strip in (28) and a matrix Q>0
With the selection of the adjustable scalars 1 and 2, i.e.,
0/
)1( f e 1 , (1f 1e) 20 (31) there always exists a matrix P0 satisfying the following Riccati equation:
0)
/1()/(
)/1
(
2 1
2 1 1
P B PB PA
P A e
e f
I I
(33)
Then, a robust pole-clustering and disturbance rejection control law in (18) and (26) to
satisfy (29) and (30) for all admissible perturbations E and F in (22) is as:
Px B r Kx
u (34)
if the gain parameter r satisfies the following two conditions:
(i) r0.5 and (35) (ii) 2[2 (1 ) (] / ) 0
1 1 2
I PA
P A P
e f
I (37)
It is evident that condition (i) is for the 1-degree stability and -degree disturbance rejection, and condition (ii) is for the 2-degree decay, i.e., the left vertical bound of the robust pole-clustering
Remark 4 There is always a solution for relative stability and disturbance rejection in the
sense of above discussion It is because the Riccati equation (32) guarantees a positive
definite solution matrix P, and thus there exists a Lyapunov function to guarantee the robust
stability of the closed loop uncertain robotic systems The nonlinear compensation part in (18) has a similar function to a feedback linearization The feature differences of the proposed method from other methods are the new nominal model, and the robust pole-clustering and disturbance attenuation for the whole uncertain system family It is further noticed that the robustly controlled system may have a good Bode plot for the whole frequency range in view of Theorem 4, inequality (29) and its H-infinity norm upper bound
Remark 5 The tighter robust pole-clustering vertical strip 1Re(A c)2 has
}]))1(2(
)/([
{5.0
2 / 1 1
1 1
2 / 1 1
I PA
P A P e f
Trang 7ROBUST CONTROL DESIGN FOR TWO-LINK NONLINEAR ROBOTIC SYSTEM 557
Bw BFu
x E
B Bu
Ax
x [0 ] (23)
][ 1 2 1 2
for manipulators to follow is to be bounded functions of time Its corresponding velocity q d
and acceleration q d, as well as itself q d, are assumed to be within the physical and
kinematic limits of manipulators They may be conveniently generated by a model of the
type:
)(
)(
)(
)(t K q t K q t r t
qd vd p d (25)
where r(t) is a 2-dimensional driving signal and the matrices K v and K p are stable
The design objective is to develop a state feedback control law for control u in (18) as
)(
)(t Kx t
u (26) such that the closed-loop system:
B BK
A
x( 0 ) (27)has its poles robustly lie within a vertical strip :
}{
)( 2 1
A
A c 0 (30) From [6], we derive the following robust control law to achieve this objective
Theorem 4 Consider a given robotic manipulator uncertain system (27) with (1)–(6),
(17)-(22), (24), where the unstructured perturbations in (21) with the norm bounds in (17)-(22), the
disturbance rejection index 0 in (29), the vertical strip in (28) and a matrix Q>0
With the selection of the adjustable scalars 1 and 2, i.e.,
0/
)1
( f e 1 , (1f 1e) 20 (31) there always exists a matrix P0 satisfying the following Riccati equation:
0)
/1()/(
)/1
(
2 1
2 1 1
P B PB PA
P A e
e f
I I
(33)
Then, a robust pole-clustering and disturbance rejection control law in (18) and (26) to
satisfy (29) and (30) for all admissible perturbations E and F in (22) is as:
Px B r Kx
u (34)
if the gain parameter r satisfies the following two conditions:
(i) r0.5 and (35) (ii) 2[2 (1 ) (] / ) 0
1 1 2
I PA
P A P
e f
I (37)
It is evident that condition (i) is for the 1-degree stability and -degree disturbance rejection, and condition (ii) is for the 2-degree decay, i.e., the left vertical bound of the robust pole-clustering
Remark 4 There is always a solution for relative stability and disturbance rejection in the
sense of above discussion It is because the Riccati equation (32) guarantees a positive
definite solution matrix P, and thus there exists a Lyapunov function to guarantee the robust
stability of the closed loop uncertain robotic systems The nonlinear compensation part in (18) has a similar function to a feedback linearization The feature differences of the proposed method from other methods are the new nominal model, and the robust pole-clustering and disturbance attenuation for the whole uncertain system family It is further noticed that the robustly controlled system may have a good Bode plot for the whole frequency range in view of Theorem 4, inequality (29) and its H-infinity norm upper bound
Remark 5 The tighter robust pole-clustering vertical strip 1Re(A c)2 has
}]))1(2(
)/([
{5.0
2 / 1 1
1 1
2 / 1 1
I PA
P A P e f
Trang 84 Examples
Example 1 Consider a two-link planar manipulator example (Fig 1) First, only joint link
masses are considered for simplicity, as the one in [3, 6] However, we take the correct
model in Corollary 1 and Remark 2 into account The system parameters are: link mass
kg
m12 , m210kg, lengths l11m, l21m , angular positions q1, q2 (rad), applied
torques f1, f 2 (Nm) Thus, the nominal values of coefficient matrices for the dynamic
equation (1) in Corollary 1 are:
)(
)1(1020222
2 2
C
C C
1210),
g q q
C C C
(39) where C i cosq i, i1,2, C12cos(q1q2), S 2 sin q2, and g is the gravity accelera-
1 t t
q d , and q d2(t)cost1 (40) The initial states are set as q1(0)q2(0)2, and q1(0)q2(0)0, i.e., e1(0)4,
e , e1(0)0, and e2(0)0 The state variable is x[e e] where eq dq
The parametric uncertainties are assumed to satisfy (22) with f 0.5, e40 , N 10
Select the adjustable parameters 10.012,20.0015 from (31), disturbance rejection
index 0.1, the relative stability index 10.1, and the left bound of vertical strip
2000
2
since we want a fast response By Theorem 4, we solve the Riccati equation (32)
to get the solution matrix P and the gain matrix as:
2 2
16431584
158412693
I I
I I
P ,K r BP[950.1823I2985.7863I2]
with r0.6 The eigenvalues of the closed-loop main system matrix A BK are
{-0.9648, -0.9648, -984.8215, -984.8215} Remark 5 gives the result 21873 The
uncertain closed-loop system has its 2Re[(A c)]1 robustly
The total control input (law) is
u M N q q q V q M F
1 2
2 2
2 2
10
10120
1
1210
coscos5.110)1(10
)1(102022
C C C g q
q q
S
t
t C
C C
C
C C
2 2
2
2
2 950.1823 985.786310
)1(10
)1(102022
(41)
A simulation for this example is taken with M(q)0.4M0(q), i.e., 40% disturbance,
5.02857
.0)()(
f q
M q
M , V m ( q q,) 0.2V m0(q,q) with 20% disturbance, and
)(2.0)(q N0 q
Trang 9ROBUST CONTROL DESIGN FOR TWO-LINK NONLINEAR ROBOTIC SYSTEM 559
4 Examples
Example 1 Consider a two-link planar manipulator example (Fig 1) First, only joint link
masses are considered for simplicity, as the one in [3, 6] However, we take the correct
model in Corollary 1 and Remark 2 into account The system parameters are: link mass
kg
m12 , m210kg, lengths l11m, l21m , angular positions q1, q2 (rad), applied
torques f1, f 2 (Nm) Thus, the nominal values of coefficient matrices for the dynamic
equation (1) in Corollary 1 are:
)
1(
10
)1
(10
2022
2
2 2
C
C C
12
10)
110
1012
C C
C
(39) where C i cosq i, i1,2, C12 cos(q1q2), S 2 sin q2, and g is the gravity accelera-
0cos
5
1)
(
1 t t
q d , and q d2(t)cost1 (40) The initial states are set as q1(0)q2(0)2, and q1(0)q2(0)0, i.e., e1(0)4,
e , e1(0)0, and e2(0)0 The state variable is x[e e] where eq d q
The parametric uncertainties are assumed to satisfy (22) with f 0.5, e40 , N 10
Select the adjustable parameters 10.012,20.0015 from (31), disturbance rejection
index 0.1, the relative stability index 10.1, and the left bound of vertical strip
2000
2
since we want a fast response By Theorem 4, we solve the Riccati equation (32)
to get the solution matrix P and the gain matrix as:
2 2
16431584
158412693
I I
I I
P ,K r BP[950.1823I2985.7863I2]
with r0.6 The eigenvalues of the closed-loop main system matrix A BK are
{-0.9648, -0.9648, -984.8215, -984.8215} Remark 5 gives the result 21873 The
uncertain closed-loop system has its 2Re[(A c)]1 robustly
The total control input (law) is
u M
N q
q q
V q
M F
1 2
2 2
2 2
10
10120
1
1210
coscos5.110
)1(10
)1(102022
C C C g q
q q
S
t
t C
C C
C
C C
2 2
2
2
2 950.1823 985.786310
)1(10
)1(102022
(41)
A simulation for this example is taken with M(q)0.4M0(q), i.e., 40% disturbance,
5.02857
.0)()(
f q
M q
M , V m ( q q,) 0.2V m0(q,q) with 20% disturbance, and
)(2.0)(q N0 q
Trang 10
Fig 3 Error signals: (a) e 1 (t), (b) e 2 (t) in Example 1
Example 2 Consider a two-link planar robotic system example with the joint mass and the
arm mass along the arm length The mass of joint 1 is m11kg, and the mass of joint 2 is
5
0
2
m kg The dimensions of two robot arms are linearly reduced round rods The two
terminal radii of the arm rod 1 are r013cm and r t12cm The two terminal radii of the
arm rod 2 are r022cm and r t11cm Their end cross-sectional areas are S019cm2,
.19343.0
2464.19343.04929.27301.5
2
2 2
C
C
),(
1 S 2q2
),(
2464.12464.16579.5
C
C C
g (43)
The desired trajectory q d (t) is the same as in Example 1 The initial states are set as
2)0(
1
q , q2(0)0, q1(0) q2(0)0, i.e.,e1(0)4, e2(0)2, e1(0)0,e2(0)0 The parametric uncertainties in practice are assumed to satisfy (22) with f 0.25, 10
e
, N 10 Select the adjustable parameters 10.0375 and 20.0188 from (31), the disturbance rejection index 0.1, the relative stability index 10.1, and the left bound of vertical strip 2100 By Theorem 4, the solution matrix P to (32) and the gain matrix K are
2 2
47.1255.13
55.1387.898
I I
I I
P ,Kr BP[65.0589I259.8659I2]with r4.8 The eigenvalues of the closed-loop main system matrix A BK are {1.1072,58.7587, 1.1072,58.7587} The uncertain system has
.19343.0
2464.19343.04929.27301.5
2
2 2
C
C C
2464.1
2464.16579.5
C C C
.19343.0
2464.19343.04929.27301.5
2
2 2
C
C C
65.0589I2 59.8659I2 e e (44) The simulation is taken with M(q)0.25M0(q), V m(q,q ) 0.1V m0(q,q), and
)(1.0)(q N0q
The results are shown in Figs 4-5 It is noticed that the error may be
reduced when the gain parameter r is set large
Trang 11ROBUST CONTROL DESIGN FOR TWO-LINK NONLINEAR ROBOTIC SYSTEM 561
Fig 3 Error signals: (a) e 1 (t), (b) e 2 (t) in Example 1
Example 2 Consider a two-link planar robotic system example with the joint mass and the
arm mass along the arm length The mass of joint 1 is m11kg, and the mass of joint 2 is
5
0
2
m kg The dimensions of two robot arms are linearly reduced round rods The two
terminal radii of the arm rod 1 are r013cm and r t12cm The two terminal radii of the
arm rod 2 are r022cm and r t11cm Their end cross-sectional areas are S019cm2,
02464
.1
9343
0
2464
19343
.0
4929
27301
.5
2
2 2
C
C
),
12
2464
1 S 2q2
)
,(
1
2464
12464
.1
6579
5
C
C C
g (43)
The desired trajectory q d (t) is the same as in Example 1 The initial states are set as
2)0(
1
q , q2(0)0, q1(0) q2(0)0, i.e.,e1(0)4, e2(0)2, e1(0)0,e2(0)0 The parametric uncertainties in practice are assumed to satisfy (22) with f 0.25, 10
e
, N 10 Select the adjustable parameters 10.0375 and 20.0188 from (31), the disturbance rejection index 0.1, the relative stability index 10.1, and the left bound of vertical strip 2100 By Theorem 4, the solution matrix P to (32) and the gain matrix K are
2 2
47.1255.13
55.1387.898
I I
I I
P ,Kr BP[65.0589I259.8659I2]with r4.8 The eigenvalues of the closed-loop main system matrix A BK are {1.1072,58.7587, 1.1072,58.7587} The uncertain system has
.19343.0
2464.19343.04929.27301.5
2
2 2
C
C C
2464.1
2464.16579.5
C C C
.19343.0
2464.19343.04929.27301.5
2
2 2
C
C C
65.0589I2 59.8659I2 e e (44) The simulation is taken with M(q)0.25M0(q), V m(q,q ) 0.1V m0(q,q), and
)(1.0)(q N0q
The results are shown in Figs 4-5 It is noticed that the error may be
reduced when the gain parameter r is set large
Trang 12Fig 4 States and their desired states: (a) q1(t) & q1d(t), (b) q2(t) & q2d(t)
Fig 5 Error signals: (a) e 1 (t), & (b) e 2 (t)
5 Conclusion
The chapter develops the practical models of two-link planar nonlinear robotic systems with their arm distributed mass in addition to the joint-end mass The new scaling coefficients are introduced for solving this problem with the distributed mass along the arms In addition, Theorems 2 and 3 respectively present two special cases: a uniform arm shape (i.e., uniform distributed mass) and a linear reduction of arm shape along the arm length Based on the presented new models, an approach to design a continuous nonlinear control law with a linear state-feedback control for the two-link planar robotic uncertain nonlinear systems is presented in Theorem 4 The designed closed-loop systems possess the properties of robust pole-clustering within a vertical strip on the left half s-plane and disturbance rejection with an H-norm constraint The suggested robust control for the uncertain nonlinear robotic systems can guarantee the required robust stability and performance in face of parameter errors, state-dependent perturbations, unknown parameters, frictions, load variation and disturbances for all allowed uncertainties in (22) The presented robust control does always exist as pointed out in Remark 4 The adjustable scalars i, i=1, 2, provide some flexibility in finding a solution of the algebraic Riccati equation The designed uncertain system has 1-degree robust stabilization and -degree
disturbance rejection The controller gain parameter r is selected such that the designed
uncertain linear system achieves robust pole-clustering within a vertical strip The examples illustrate excellent results This design procedure may be used for designing other control systems, modeling, and simulation
6 References
[1]J.J.Craig, Adaptive control of mechanical manipulators, Addison-Wesley (Publishing
Company, Inc., New York, 1988)
[2]J.H Kaloust, & Z Qu, Robust guaranteed cost control of uncertain nonlinear robotic
sys-tem using mixed minimum time and quadratic performance index, Proc 32nd IEEE Conf on Decision and Control, 1993, 1634-1635
[3]J Kaneko, A robust motion control of manipulators with parametric uncertainties and
random disturbances, Proc 34rd IEEE Conf on Decision and Control, 1995, 1609-1610 [4]R.L Tummala, Dynamics and Control – Robotics, in The Electrical Engineering Handbook,
Ed by R.C Dorf, (2nd ed., CRC Press with IEEE Press, Boca Raton, FL, 1997, 2347) [5]M Garcia-Sanz, L Egana, & J Villanueva, “Interval Modelling of a SCARA Robot for
Robust Control”, Proc 10 th Mediterranean Conf on Control and Automation, 2002
[6]S Lin, & S.-G Wang, Robust Control with Pole Clustering for Uncertain Robotic Systems,
International Journal of Control and Intelligent Systems, 28(2), 2000, 72-79
[7]S.B Lin, and O Masory, Gains selection of a variable gain adaptive control system for
turning, ASME Journal of Engineering for Industry, 109, 1987, 399-403
[8]S.-G Wang, L.S Shieh, & J.W Sunkel, Robust optimal pole-clustering in a vertical strip
and disturbance rejection for Lagrange’s systems, Int J Dynamics and Control, 5(3),
1995, 295-312
Trang 13ROBUST CONTROL DESIGN FOR TWO-LINK NONLINEAR ROBOTIC SYSTEM 563
Fig 4 States and their desired states: (a) q1(t) & q1d t), (b) q2(t) & q2d(t)
Fig 5 Error signals: (a) e 1 (t), & (b) e 2 (t)
5 Conclusion
The chapter develops the practical models of two-link planar nonlinear robotic systems with their arm distributed mass in addition to the joint-end mass The new scaling coefficients are introduced for solving this problem with the distributed mass along the arms In addition, Theorems 2 and 3 respectively present two special cases: a uniform arm shape (i.e., uniform distributed mass) and a linear reduction of arm shape along the arm length Based on the presented new models, an approach to design a continuous nonlinear control law with a linear state-feedback control for the two-link planar robotic uncertain nonlinear systems is presented in Theorem 4 The designed closed-loop systems possess the properties of robust pole-clustering within a vertical strip on the left half s-plane and disturbance rejection with an H-norm constraint The suggested robust control for the uncertain nonlinear robotic systems can guarantee the required robust stability and performance in face of parameter errors, state-dependent perturbations, unknown parameters, frictions, load variation and disturbances for all allowed uncertainties in (22) The presented robust control does always exist as pointed out in Remark 4 The adjustable scalars i, i=1, 2, provide some flexibility in finding a solution of the algebraic Riccati equation The designed uncertain system has 1-degree robust stabilization and -degree
disturbance rejection The controller gain parameter r is selected such that the designed
uncertain linear system achieves robust pole-clustering within a vertical strip The examples illustrate excellent results This design procedure may be used for designing other control systems, modeling, and simulation
6 References
[1]J.J.Craig, Adaptive control of mechanical manipulators, Addison-Wesley (Publishing
Company, Inc., New York, 1988)
[2]J.H Kaloust, & Z Qu, Robust guaranteed cost control of uncertain nonlinear robotic
sys-tem using mixed minimum time and quadratic performance index, Proc 32nd IEEE Conf on Decision and Control, 1993, 1634-1635
[3]J Kaneko, A robust motion control of manipulators with parametric uncertainties and
random disturbances, Proc 34rd IEEE Conf on Decision and Control, 1995, 1609-1610 [4]R.L Tummala, Dynamics and Control – Robotics, in The Electrical Engineering Handbook,
Ed by R.C Dorf, (2nd ed., CRC Press with IEEE Press, Boca Raton, FL, 1997, 2347) [5]M Garcia-Sanz, L Egana, & J Villanueva, “Interval Modelling of a SCARA Robot for
Robust Control”, Proc 10 th Mediterranean Conf on Control and Automation, 2002
[6]S Lin, & S.-G Wang, Robust Control with Pole Clustering for Uncertain Robotic Systems,
International Journal of Control and Intelligent Systems, 28(2), 2000, 72-79
[7]S.B Lin, and O Masory, Gains selection of a variable gain adaptive control system for
turning, ASME Journal of Engineering for Industry, 109, 1987, 399-403
[8]S.-G Wang, L.S Shieh, & J.W Sunkel, Robust optimal pole-clustering in a vertical strip
and disturbance rejection for Lagrange’s systems, Int J Dynamics and Control, 5(3),
1995, 295-312
Trang 14[9]S.-G Wang, L.S Shieh, & J.W Sunkel, Robust optimal pole-placement in a vertical strip
and disturbance rejection, Proc 32nd IEEE Conf on Decision and Control, 1993,
1134-1139 Int J Systems Science, 26(10), 1995, 1839-1853
[10]S.-G Wang, S.B Lin, L.S Shieh, & J.W Sunkel, Observer-based controller for robust pole
clustering in a vertical strip and disturbance rejection in structured uncertain
systems, Int J Robust & Nonlinear Control, 8(3), 1998, 1073-1084
[11]J.J Craig, Introduction to Robotics: Mechanics and Control (2nd ed., Addison-Weeley
Publishing Company, Inc., New York, 1988)
Trang 15Role of Finite Element Analysis in Designing Multi-axes Positioning for Robotic Manipulators 565
Role of Finite Element Analysis in Designing Multi-axes Positioning for Robotic Manipulators
T.T Mon, F.R Mohd Romlay and M.N Tamin
x
Role of Finite Element Analysis in Designing Multi-axes Positioning
for Robotic Manipulators
T.T Mon, F.R Mohd Romlay and M.N Tamin
Universiti Malaysia Pahang, Universiti Teknology Malaysia
Malaysia
1 Introduction
Simulation of robot manipulator in Matlab/Simulink or any other mechanism simulator is
very common for robot design However, all these approaches are mainly concerned with
design configuration having little analysis meaning that the robot model is formed by
linking the kinematics and solid description, and simulated for alternative configuration of
movements (Cleery & Mathur, 2008) Indeed comprehensive design should have analysis at
different computational levels Finite element method (FEM) has been a major tool to
develop a computational model in various fields of studies because of its modelling and
simulation capability close to reality Subsequently, modelling and analysis with FEM has
become the most convenient way to economically design and analyze real world problems,
either in static or dynamic As a result, huge amount of reports on this topic can be found in
the literature (Mackerle, 1999) Unfortunately however, this technique has not
comprehensively applied in designing in designing a robot while choosing the best
components for the design is as important as having good performance and no
environmental impact of the machines over its lifetime Building block of a robot
manipulator is electromechanical system in which mechanical systems are controlled by
sophisticated electric motor drives Since energy saving everywhere is a major challenge
now and in future, getting electromechanical design right will significantly contribute to
energy saving
This chapter is dedicated to the application of Finite Element Method (FEM) in designing
multi-axes positioning for robot manipulators Computational model that can predict
physical behaviour of dynamic robot manipulators constructed using FE codes is presented,
and this is major contribution of the chapter FEM tools necessary for modelling and
analysis of multi-axes positioning are presented in large part Rather than a FEM discourse,
FEM is presented by highlighting mathematics behind and an application example as they
relates to practical robotic manipulation It is, however, assumed that the reader has
acquired some basic knowledge of FEM consistent with the expected level of mathematics
Hence, the chapter is organized as follows
In the early part of the chapter, the important terminologies used in robotics are defined in
the background The material is presented using a number of examples as evidenced in the
28
Trang 16published reports Then the chapter will go to mathematics behind finite element modelling
and analysis of kinematics of a structure in 3D space as robotics involves tracking moving
objects in 3D space This will also include mathematical tools essential for the study of
robotics, particularly matrix transforms, mathematical models of robot manipulators, direct
kinematic equations, inverse kinematic technique and Jacobian matrix needed to control
position and motion of a robot manipulator More emphasis will be on how these
mathematical tools can be linked to and incorporated into FEM to carry out design analysis
of robot structure
The rest of the chapter will present application of FEM in practical robot design, detailed
development of FE model computable for multi-axes positioning using a particular FE code
(ALGOR, 2008), and useful results predicted by the computational model The chapter will
be closed by concluding remarks to choice of FE codes and its impact on the computational
model and finally the usefulness of computational model
2 Background in Robotics
Multi-axis positioning meant here is different movements of a point, or a structure in
different directions This term is drawn from the term usually come with computer
numerical controlled machines just as 3-axis, 5-axis and so on, where the 3-axis machine, for
instance, implies that it can make a maximum of three different positioning of the controlled
elements Each axis is alternatively referred to as degree of freedom (DOF) that is something
to do with motion in a system or a structure Since the term ‘axis’ is adopted to represent an
element that creates motion, 3-axis positioning means three DOF’s, for example (Rahman,
2004) In relation to these definitions, one manipulator of a robot can represent one axis or
DOF as the manipulator is the robot’s arm, a movable mechanical unit comprising of
segments or links jointed together with axes capable of motion in various directions
allowing the robot to perform tasks Typically, the body, arm and wrist are components of
manipulators Movements between the various components of the manipulator are
provided by series of joints
The points that a manipulator bends, slides or rotates are called joints or position axes
Position axes are also called the world coordinates The world coordinate system is usually
identified as being a fixed location within the manipulator that serves as an absolute frame
of reference
In general, the manipulator’s motion can be divided into two categories: translation and
rotation Although one can further categorize it in specific term such as a pitch
(up-and-down motion); a yaw (side-to-side motion); and a roll (rotating motion), any of these is fall
into either translation or rotation The individual joint motion associated with either of these
two categories is referred to degree of freedom Subsequently, one degree of freedom is
equal to one axis The industrial robots are typically equipped with 4-6 axes
The power supply provides the energy required for a robot to be operated Electricity is the
most common source of power and is used extensively with industrial robots Payload is the
weight that the robot is designed to lift, hold, and position repeatedly with the same
accuracy Hence, the power supply has direct relation to the payload rating of a robot
Among the important dynamic properties of a robot that properly regulates its motion are:
stability, control resolution, spatial resolution, accuracy, repeatability and compliance To
take these factors into account in the design of a robot is a complex issue Lack of stability
occurs very often due to wear of manipulator components, movement longer than the intended, longer time to reach and overshooting of position
Control resolution is all about position control It is a function of the design of robot control system and specifies the smallest increment of motion by which the system can divide its working space It is the smallest incremental change in position that its control system can measure In other words, it is the controller’s ability to divide the total range of movements for the particular joint into individual increments that can be addressed in the controller This depends on the bit storage capacity in the control memory For example, a robot with 8 bits of storage can divide the range into 256 discrete positions The control resolution would
be also defined as the total motion range divided by the number of increments For example,
a robot has one sliding joint with a full range of 1.0 m The robot control memory has 12-bit storage capacity The control resolution for this axis of motion is 0.244mm The spatial resolution of a robot is the smallest increment of movement into which the robot can divide its work volume
Mechanical inaccuracies in the robot’s links and joint components and its feedback measurements system (if it is a servo-controlled robot) constitute the other factor that contributes to spatial resolution Mechanical inaccuracies come from elastic deflection in the structural members, gear backlash, stretching of pulley cords and other imperfections in the mechanical system These inaccuracies tend to be worse for large robot simply because the errors are magnified by the large components The spatial resolution is degraded by these mechanical inaccuracies
Rigidity of the structure also affects the repeatability of the robot Compliance is a quality that gives a manipulator of a robot the ability to tolerate misalignment of mating parts It is essential for assembly of close-fitting parts In an electric manipulator, the motors generally connect to mechanical coupling The sticking and sliding friction in such a coupling can cause a strange effect on the compliance, in particular, being back-drivable
An Off-line programming system includes a spatial representation of solids and their graphical interpretation, automatic collision detection, incorporation of kinematic, path planning and dynamic simulation and concurrent programming The off-line programming will grow more in the future because of graphical computer simulation used to validate program development It is important both as aids in programming industrial automation and as platforms for robotic research (Billingsley, 1985; Keramas, 1999; Angeles, 2003)
3 Mathematical Foundation
Forward and reverse kinematics methods are the principal mathematics behind typical modeling, computation and analysis of robot manipulators Since the latter is deduced from the former, broader review will focus on some related mathematics of the former Forward kinematic equation relates a pose element to the joint variables The pose matrix is computed from the joint variables The position and orientation of end-manipulator (also-called the end-effector, the last joint that directly touches and handles the object) is computed from all joint variables The position and orientation of the end manipulator are computed from a set of joint variable values which are already known or specified The computation follows the arrow directions starting from joint 1 as illustrated in Fig 1 It should be noted that in kinematics analysis, the manipulators are assumed to be rigid
Trang 17Role of Finite Element Analysis in Designing Multi-axes Positioning for Robotic Manipulators 567
published reports Then the chapter will go to mathematics behind finite element modelling
and analysis of kinematics of a structure in 3D space as robotics involves tracking moving
objects in 3D space This will also include mathematical tools essential for the study of
robotics, particularly matrix transforms, mathematical models of robot manipulators, direct
kinematic equations, inverse kinematic technique and Jacobian matrix needed to control
position and motion of a robot manipulator More emphasis will be on how these
mathematical tools can be linked to and incorporated into FEM to carry out design analysis
of robot structure
The rest of the chapter will present application of FEM in practical robot design, detailed
development of FE model computable for multi-axes positioning using a particular FE code
(ALGOR, 2008), and useful results predicted by the computational model The chapter will
be closed by concluding remarks to choice of FE codes and its impact on the computational
model and finally the usefulness of computational model
2 Background in Robotics
Multi-axis positioning meant here is different movements of a point, or a structure in
different directions This term is drawn from the term usually come with computer
numerical controlled machines just as 3-axis, 5-axis and so on, where the 3-axis machine, for
instance, implies that it can make a maximum of three different positioning of the controlled
elements Each axis is alternatively referred to as degree of freedom (DOF) that is something
to do with motion in a system or a structure Since the term ‘axis’ is adopted to represent an
element that creates motion, 3-axis positioning means three DOF’s, for example (Rahman,
2004) In relation to these definitions, one manipulator of a robot can represent one axis or
DOF as the manipulator is the robot’s arm, a movable mechanical unit comprising of
segments or links jointed together with axes capable of motion in various directions
allowing the robot to perform tasks Typically, the body, arm and wrist are components of
manipulators Movements between the various components of the manipulator are
provided by series of joints
The points that a manipulator bends, slides or rotates are called joints or position axes
Position axes are also called the world coordinates The world coordinate system is usually
identified as being a fixed location within the manipulator that serves as an absolute frame
of reference
In general, the manipulator’s motion can be divided into two categories: translation and
rotation Although one can further categorize it in specific term such as a pitch
(up-and-down motion); a yaw (side-to-side motion); and a roll (rotating motion), any of these is fall
into either translation or rotation The individual joint motion associated with either of these
two categories is referred to degree of freedom Subsequently, one degree of freedom is
equal to one axis The industrial robots are typically equipped with 4-6 axes
The power supply provides the energy required for a robot to be operated Electricity is the
most common source of power and is used extensively with industrial robots Payload is the
weight that the robot is designed to lift, hold, and position repeatedly with the same
accuracy Hence, the power supply has direct relation to the payload rating of a robot
Among the important dynamic properties of a robot that properly regulates its motion are:
stability, control resolution, spatial resolution, accuracy, repeatability and compliance To
take these factors into account in the design of a robot is a complex issue Lack of stability
be also defined as the total motion range divided by the number of increments For example,
a robot has one sliding joint with a full range of 1.0 m The robot control memory has 12-bit storage capacity The control resolution for this axis of motion is 0.244mm The spatial resolution of a robot is the smallest increment of movement into which the robot can divide its work volume
Mechanical inaccuracies in the robot’s links and joint components and its feedback measurements system (if it is a servo-controlled robot) constitute the other factor that contributes to spatial resolution Mechanical inaccuracies come from elastic deflection in the structural members, gear backlash, stretching of pulley cords and other imperfections in the mechanical system These inaccuracies tend to be worse for large robot simply because the errors are magnified by the large components The spatial resolution is degraded by these mechanical inaccuracies
Rigidity of the structure also affects the repeatability of the robot Compliance is a quality that gives a manipulator of a robot the ability to tolerate misalignment of mating parts It is essential for assembly of close-fitting parts In an electric manipulator, the motors generally connect to mechanical coupling The sticking and sliding friction in such a coupling can cause a strange effect on the compliance, in particular, being back-drivable
An Off-line programming system includes a spatial representation of solids and their graphical interpretation, automatic collision detection, incorporation of kinematic, path planning and dynamic simulation and concurrent programming The off-line programming will grow more in the future because of graphical computer simulation used to validate program development It is important both as aids in programming industrial automation and as platforms for robotic research (Billingsley, 1985; Keramas, 1999; Angeles, 2003)
3 Mathematical Foundation
Forward and reverse kinematics methods are the principal mathematics behind typical modeling, computation and analysis of robot manipulators Since the latter is deduced from the former, broader review will focus on some related mathematics of the former Forward kinematic equation relates a pose element to the joint variables The pose matrix is computed from the joint variables The position and orientation of end-manipulator (also-called the end-effector, the last joint that directly touches and handles the object) is computed from all joint variables The position and orientation of the end manipulator are computed from a set of joint variable values which are already known or specified The computation follows the arrow directions starting from joint 1 as illustrated in Fig 1 It should be noted that in kinematics analysis, the manipulators are assumed to be rigid
Trang 183.1 Defining the location of an object in space
As a general case, the object location in 3D space is considered A matrix representation is
widely used to represent the object location as it is convenient and easy to handle especially
when the location is changed Two parameters are needed to define the object location:
position and orientation Basically, a homogenous vector ‘v’ is represented as
(1)
if v is a free vector or
(2)
if v represents the position of a particular point in the usual coordinates system (x, y, and z)
This coordinates system is referred to a frame These frames are used to track an object
location in space As shown in Fig 2, the frame F o is attached to a fixed point while another
F A to an object The object position is described by the vector pA of the origin A of the frame
F A The orientation of the object is given by the homogeneous vectors of each unit vectors
xA, yA and zA of F A with respect to F o
Then the object location is mathematically represented by a post matrix ‘P’ as:
(3) where the matrix
(4) and
(5) formed by the coordinates of the vectors xA, yA and zA is a rotation matrix that holds the
orientation of the object while the p A holds the position of the object In a compact form, the
pose matrix can be written as:
(6) where O refers to a 1x3 vector of zeros
Fig 1 Forward kinematics
Fig 2 The object with respect to frames
When a point, say Q given by its coordinate vector Aq = with respect to frame F A, is
transformed to the frame F O, the transformed vector say oq can be expressed as:
Trang 19Role of Finite Element Analysis in Designing Multi-axes Positioning for Robotic Manipulators 569
3.1 Defining the location of an object in space
As a general case, the object location in 3D space is considered A matrix representation is
widely used to represent the object location as it is convenient and easy to handle especially
when the location is changed Two parameters are needed to define the object location:
position and orientation Basically, a homogenous vector ‘v’ is represented as
(1)
if v is a free vector or
(2)
if v represents the position of a particular point in the usual coordinates system (x, y, and z)
This coordinates system is referred to a frame These frames are used to track an object
location in space As shown in Fig 2, the frame F o is attached to a fixed point while another
F A to an object The object position is described by the vector pA of the origin A of the frame
F A The orientation of the object is given by the homogeneous vectors of each unit vectors
xA, yA and zA of F A with respect to F o
Then the object location is mathematically represented by a post matrix ‘P’ as:
(3) where the matrix
(4) and
(5) formed by the coordinates of the vectors xA, yA and zA is a rotation matrix that holds the
orientation of the object while the p A holds the position of the object In a compact form, the
pose matrix can be written as:
(6) where O refers to a 1x3 vector of zeros
Fig 1 Forward kinematics
Fig 2 The object with respect to frames
When a point, say Q given by its coordinate vector Aq = with respect to frame F A, is
transformed to the frame F O, the transformed vector say oq can be expressed as:
Trang 20(7)
In compact form
where oTA = P
Similarly in alternative representation of frame transforms, simple space translation and
rotation can be conveniently represented as a compact-form matrix as:
T q = Tu q
Rq = Rv,θ q
(9) (10)
where q = is coordinates vector of a point Q, Tu is a translation vector, Rv,θ is a rotation
vector, T q is coordinates vector of a point Q' where Q is translated by Tu and Rq coordinates
vector of a point Q' where Q is rotated around v by an angle θ
Furthermore, the translations and rotations along the reference axes, called canonical
translations/rotations, have the homogeneous matrix of the following form respectively as:
3.2 DH parameters
In the conventional analysis of motion of robot manipulators, the Denavit-Hartenberg (DH) modelling technique is commonly used as a standard technique Reference frames are assigned to each link based on DH parameters, starting from the fixed link all the way to the last link The DH model is obtained by describing each link frame with respect to the preceding link frame The original representation of one frame with respect to another using pose matrix requires a minimum of six parameters The DH modelling technique reduces these parameters to four, routinely noted as:
di, the link offset,
ai, the link length,
θi, the link angle and
αi, the link twist as illustrated in Fig 3
di
Zi-1
Xi-1Fi-1
FiXiZi
αi θi
ai
Fig 3 Illustration of DH parameters