机械臂的自动控制系统
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The output of the system equations:
2.The mechanical arm odel linearization
1) The value of the parameter
• Assignment for(4) (5) (6):
a1=3.77 a2= 2.12 a3= 0.70 a4=80.32 a5=24.07
single‐link flexible robot arm
wangyuan
Arrangement：
M1: Build this system in Simulinkwith D/A and A/D converte rs M2: Design a state‐feedback controller (any form are OK) M3: Design a observer (full‐order or reduced‐order) M4: Design output‐feedback controller directly (e.g. PID)
2.State feedback control
The constitution of the state feedback
state feedback u=- Kx+ v
..
xx Ax BU y Cx
State description of close‐loopsystem
.
x =(A-BK)x+Bv
. 2.5 1 1
1 0
x 2.2 3.1 X 0U Y 0 1 X
solution：
RankQ B 0
AB
2.5 2.2
1 3.1
2.5 2.2
2
2.5
RankQ C
A CA
2.2 2.5
2.2
1
3.1 2 1 3.1
So the system is controlled
Assuming F F1 F 2
Jacobian Matrix：
2) The dynamics equation of mechanical arm:
M(θ) G(θ) M(θ)
:
The total driving moment vector
:
The angle of rotation vector
:
Coriolis Centrifugal torqueInertia
2) The relative order of computing system(计算系统相对阶)
After the two column is 0,
The first two lines of all 0,
And:
• On the output for
, Non singular, The system in the neighborhood of a in x, ，The relative order of the
To the automatic control system accurately, it is necessary to get the accurate values of all status system. However, with the increase of system complexity, rely on the state of sensors to measure information will increase the complexity of hardware, may even lead to system instability. And some state information be restricted conditions, through physical measurement. At this point, the state observer provides a new train of thought: if the analytical model of controlled object is known, can through the design of state observer, then based on the system of external variables (input and output variables) of the actual measured values, the estimate of state variables.
The calculation of observer:
A
2.5 2.2
1 3.1
,B
1 0
,C
1 0
0 1
Two poles : 1 5i
SOLUTION:
.
.ቤተ መጻሕፍቲ ባይዱ
~
~
x (A LC )x Bu Ly
.
.
~
x
5.1 21.81
1 3.1
~
x
1 0
u
7.6 24.01
y
4.The output feedback controller
system is 2.
3) Coordinate transformation
4) System is converted into standard type: The output
5) State feedback matrix,
,The system
become a completely controllable linear system
Group 4: single‐link flexible robot arm
1.The dynamics equation of mechanical arm:
（m2 l2）
（m1 l1）
1) Position equation of manipulator operation end：
State transformation:
The observer design of flexible manipulator, the first to design an auxiliary system, for the flexible manipulator system status, refactoring :
m1,m2,n1,n2,n3,n4 is for the design of real Numbers 。 For type, based on the idea of sliding mode observer and Luenberger observer, construct the following observer :
2 2s 3
k k 1 7.6, 2 9.55
k 7.6 9.55
3.Design a observer (full‐orde
r or reduced‐order)
The flexible mechanical arm vibration force analysis
The observer design and analysis
Simultaneous
F 7.6 10
The output feedback controller
The output feedback controller
The optimal control of a robot arm
机械臂的跟踪误差
跟踪误差曲线
. 2.5 1 1 x 2.2 3.1 X 0U
Y
1 0
0 1
X
Solution:
2.5
RankQ C
A CA
2.2 2.5
2.2
1
3.1 2 1 3.1
RankQ B 0
AB
2.5 2.2
1 3.1
2.5 2.2 2
k k sI A BK
s 2.5
1
21
2.2 s 3.1
SI
A BFC
s
2.5 2.2
F
1
1 S
3F.12
S 2 S F 1 5.6 5.55 3.1F 1 2.2F 2
The output feedback controller
Assuming
1 3I
1,2
SO
S 1 3I S 1 3I
S 2 S F 1 5.6 5.55 3.1F 1 2.2F 2
(7)
The form of state transformation matrix: ]^T
Hypothesis and the assumption that the variables:
4) Mechanical arm model equation of state:
State equation:
:
The gravity vector
:
Inertia matrix
The mathematical expression: Simultaneous : (1) (2) (3)
The mathematical expression and The quantitative value:
3) Defined state variables:
Integrated design of the u Kxv control action u usually depends on the actual response of the system, and that is to say, u can be expressed as the output of a linear function
u Kx v
If the control u is a linear function of the state y, so called linear system output feedback accordingly。
The output feedback controller
The output feedback controller
s k k k 2 s( 5.6) 5.55 3.1 2.2
1
1
2
Because the system can be control completely, so all its poles can be arbitrary configuration.
s s
(1
2i)
*
s
(1
2i)
2.The mechanical arm odel linearization
1) The value of the parameter
• Assignment for(4) (5) (6):
a1=3.77 a2= 2.12 a3= 0.70 a4=80.32 a5=24.07
single‐link flexible robot arm
wangyuan
Arrangement：
M1: Build this system in Simulinkwith D/A and A/D converte rs M2: Design a state‐feedback controller (any form are OK) M3: Design a observer (full‐order or reduced‐order) M4: Design output‐feedback controller directly (e.g. PID)
2.State feedback control
The constitution of the state feedback
state feedback u=- Kx+ v
..
xx Ax BU y Cx
State description of close‐loopsystem
.
x =(A-BK)x+Bv
. 2.5 1 1
1 0
x 2.2 3.1 X 0U Y 0 1 X
solution：
RankQ B 0
AB
2.5 2.2
1 3.1
2.5 2.2
2
2.5
RankQ C
A CA
2.2 2.5
2.2
1
3.1 2 1 3.1
So the system is controlled
Assuming F F1 F 2
Jacobian Matrix：
2) The dynamics equation of mechanical arm:
M(θ) G(θ) M(θ)
:
The total driving moment vector
:
The angle of rotation vector
:
Coriolis Centrifugal torqueInertia
2) The relative order of computing system(计算系统相对阶)
After the two column is 0,
The first two lines of all 0,
And:
• On the output for
, Non singular, The system in the neighborhood of a in x, ，The relative order of the
To the automatic control system accurately, it is necessary to get the accurate values of all status system. However, with the increase of system complexity, rely on the state of sensors to measure information will increase the complexity of hardware, may even lead to system instability. And some state information be restricted conditions, through physical measurement. At this point, the state observer provides a new train of thought: if the analytical model of controlled object is known, can through the design of state observer, then based on the system of external variables (input and output variables) of the actual measured values, the estimate of state variables.
The calculation of observer:
A
2.5 2.2
1 3.1
,B
1 0
,C
1 0
0 1
Two poles : 1 5i
SOLUTION:
.
.ቤተ መጻሕፍቲ ባይዱ
~
~
x (A LC )x Bu Ly
.
.
~
x
5.1 21.81
1 3.1
~
x
1 0
u
7.6 24.01
y
4.The output feedback controller
system is 2.
3) Coordinate transformation
4) System is converted into standard type: The output
5) State feedback matrix,
,The system
become a completely controllable linear system
Group 4: single‐link flexible robot arm
1.The dynamics equation of mechanical arm:
（m2 l2）
（m1 l1）
1) Position equation of manipulator operation end：
State transformation:
The observer design of flexible manipulator, the first to design an auxiliary system, for the flexible manipulator system status, refactoring :
m1,m2,n1,n2,n3,n4 is for the design of real Numbers 。 For type, based on the idea of sliding mode observer and Luenberger observer, construct the following observer :
2 2s 3
k k 1 7.6, 2 9.55
k 7.6 9.55
3.Design a observer (full‐orde
r or reduced‐order)
The flexible mechanical arm vibration force analysis
The observer design and analysis
Simultaneous
F 7.6 10
The output feedback controller
The output feedback controller
The optimal control of a robot arm
机械臂的跟踪误差
跟踪误差曲线
. 2.5 1 1 x 2.2 3.1 X 0U
Y
1 0
0 1
X
Solution:
2.5
RankQ C
A CA
2.2 2.5
2.2
1
3.1 2 1 3.1
RankQ B 0
AB
2.5 2.2
1 3.1
2.5 2.2 2
k k sI A BK
s 2.5
1
21
2.2 s 3.1
SI
A BFC
s
2.5 2.2
F
1
1 S
3F.12
S 2 S F 1 5.6 5.55 3.1F 1 2.2F 2
The output feedback controller
Assuming
1 3I
1,2
SO
S 1 3I S 1 3I
S 2 S F 1 5.6 5.55 3.1F 1 2.2F 2
(7)
The form of state transformation matrix: ]^T
Hypothesis and the assumption that the variables:
4) Mechanical arm model equation of state:
State equation:
:
The gravity vector
:
Inertia matrix
The mathematical expression: Simultaneous : (1) (2) (3)
The mathematical expression and The quantitative value:
3) Defined state variables:
Integrated design of the u Kxv control action u usually depends on the actual response of the system, and that is to say, u can be expressed as the output of a linear function
u Kx v
If the control u is a linear function of the state y, so called linear system output feedback accordingly。
The output feedback controller
The output feedback controller
s k k k 2 s( 5.6) 5.55 3.1 2.2
1
1
2
Because the system can be control completely, so all its poles can be arbitrary configuration.
s s
(1
2i)
*
s
(1
2i)