# Force/Position Hybrid Control of 6PUS-UPU Redundant Driven Parallel Manipulator Based on 2-DOF Internal Model Control

Authors: Kuijing Zheng; Chao Wang
Journal: Journal of Control Science and Engineering
 (2014)

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## Abstract

To improve control performance of parallel manipulator, servo control is optimized according to 2-DOF control with Internal Model Control. Position loop of the controller is redesigned based on the original current loop and speed loop. The hybrid force/position control strategy on the basis of cross-coupling is proposed. With mechatronic cosimulation system, it is proved that the force/position hybrid control on the basis of 2-DOF Internal Model Control has better stability and position precision compared with the traditional PID control. The control precision and stability of the parallel manipulator are improved effectively in actual experiment. Two types of compliance operation including peg in hole and surface tracking are realized in 6PUS-UPU parallel manipulator and both have good applicable effect.


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## Body

## 1. Introduction

Parallel manipulator is becoming increasingly popular in a large number of applications. It is a complex system with characteristics of multidegree of freedom, multi-input multioutput, high nonlinearity and strong coupling, and so forth. Control of parallel manipulator is very complex and researches mostly focus on kinematic and dynamic control. Dynamic control can meet the demands of high performance more than kinematic control. Dynamic feed forward control and computed torque control are common dynamic control methods [1–3]. Müller and Hufnagel [4] proposed the adaptive and singularity-free inverse dynamics models for control of parallel manipulators, Achim and Matthias [5] used the dynamics model as a feed forward on a handheld parallel mechanism to reduce the disturbance, and Farhadmanesh and Rastin [6] designed a new fuzzy model-based controller for complex dynamical systems with application to a 3-RRR spherical parallel manipulator.Classic PID controller is widely used in various kinds of industrial fields for its simple structure and strong robustness. However, classic PID controller cannot meet the different performance requirements simultaneously. 2-DOF control and Internal Model Control are introduced to improve the performance of parallel manipulator [7–10].First, the servo control strategy of 6PUS-UPU parallel manipulator is analyzed. The PID control is combined with 2-DOF and Internal Model Control to redesign servo control loops of the parallel manipulator. Next the hybrid force/position control of parallel manipulator is proposed. Then the entire model of the 6PUS-UPU parallel manipulator is established, and the mechatronic cosimulation is carried out. Finally the control method is verified by actual prototype experiment.


## 2. Control Strategy of 6PUS-UPU Parallel Manipulator

On the basis of the original current loop and velocity loop, the position loop is redesigned to improve control precision and stability of the parallel manipulator.


### 2.1. Overview of 6PUS-UPU Parallel Manipulator

A novel 6-PUS/UPU parallel manipulator is shown in Figure1, including the fixed platform, the moving platform, and connected branches. The fixed platform and the moving platform are connected with six PUS (prismatic pair-universal pair-sphere pair) actuated branches and a passive UPU (universal joint-prismatic pair-universal joint) middle branch. Because one degree of freedom is constrained by the UPU branch, the moving platform only can achieve five degrees of freedom. The parallel manipulator is driven redundantly for its six actuated branches.Figure 1
6PUS-UPU parallel manipulator.


### 2.2. 2-DOF PID Control Based on IMC

For modern control systems, it is hoped to have good position tracking characteristics and good antidisturbance capacity simultaneously. Traditional control methods such as PID control belong to one degree of freedom control and only have one adjustable parameter. The control systems with one adjustable parameter are difficult to meet the requirement of good position tracking characteristics and good antidisturbance capacity. 2-DOF control is designed to solve the problem and two separate parameters are adjusted to have the two above-mentioned performances.The servo control system of 6PUS-UPU parallel manipulator is composed of position loop, velocity loop, and current loop.(
1
) Current loop is the innermost loop in the servo system. Current loop consists of the current controller, SPWM, filters, and other components. Proportional and integral controls are used in current controller to have the fastest response among the three loops. The transfer function of current controller is expressed as
(1)
G
i
s
=
K
i
T
i
s
+
1
T
i
s
,
where K
i is the proportional gain factor and T
i
=
L
/
R is the integral gain factor.The control objects of current loop are inverter and armature circuit. The inverter is a conversion circuit from DC to AC. In PMSM control system, SPWM voltage inverter is used and considered as a first-order damp element. The transfer function of inverter is expressed as(2)
G
ivt
s
=
K
n
T
n
s
+
1
,
where K
n is control gain factor and T
n is time constant.The armature circuit is considered asR-L model and its transfer function is expressed as
(3)
G
r
l
s
=
1
R
+
L
s
,
where R is armature resistance and L is armature inductance.In order to remove the noisy signal, a feedback filter is designed as a first-order damp element. The transfer function of the feedback filter is expressed as(4)
G
c
f
s
=
K
c
f
T
c
f
s
+
1
,
where K
c
f is gain factor and T
c
f is time constant.The signal filter is designed to offset the delay caused by the feedback signal. Its transfer function is a first-order damp element and the transfer function is expressed as(5)
G
x
i
s
=
1
T
c
f
s
+
1
.In current loop,I
r is input current and I
m is output current.(
2
) Velocity loop is used to improve the antidisturbance capacity by reducing error between output velocity and given velocity. PI control is used and its transfer function is expressed as
(6)
G
v
s
=
K
v
T
v
s
+
1
T
v
s
,
where K
v is proportional gain factor and T
v is integral gain factor.Feedback filter is also needed in velocity loop and its transfer function is expressed as(7)
G
w
s
=
K
w
T
w
s
+
1
,
where K
w is gain factor and T
w is time constant.The signal filter is also added to offset the delay and its transfer function is expressed as(8)
G
x
v
s
=
1
T
w
s
+
1
.In velocity loop,K
t is torque constant of the motor, J is inertia of the motor, T
e is torque of the motor, T
L is torque of the load, ω
r is input angular velocity, and ω
m is output angular velocity.(
3
) Position loop is the outermost loop and is redesigned based on current loop and velocity loop. The block diagram of three servo loops is shown in Figure 2. In position loop, θ
′ is input angle and θ is output angle; C
1
(
s
) and C
2
(
s
) are 2-DOF IMC controllers, respectively.Figure 2
Block diagram of three control loops.The mathematical models ofC
1
(
s
) and C
2
(
s
) are expressed as
(9)
C
1
s
=
3
λ
1
s
+
1
3
λ
2
s
+
1
,
C
2
s
=
3
λ
2
s
+
1
T
0
s
+
1
3
λ
2
2
s
(
λ
2
/
3
)
s
+
1
,
where λ
1, λ
2 are control parameters and T
0 is time constant.


### 2.3. Force/Position Hybrid Control Strategy

The force/position hybrid control of 6PUS-UPU parallel manipulator is proposed as shown in Figure3. For five nonredundant branches, position control modes are used, and for the redundant branch force control mode is used.Figure 3
Force/position hybrid control.The cross-coupling relationship between driving axes should be taken into consideration. The coupling position errors of five nonredundant branches from the 1st to the 5th branch in joint space are fed back to the 6th redundant driven branch. The coupling position errors are converted to the force signal to make compensation in synchronous controller. So the position precision and torque balance of the parallel manipulator will be improved simultaneously.It is assumed that the theoretical position of branchi is s
i
(
t
) and the actual position is s
i
′
(
t
) at the time t. Then, the position error of the branch at the time t is as follows:
(10)
e
i
(
t
)
=
s
i
(
t
)
-
s
i
′
(
t
)
i
=
1
~
5
.Only when the error of each branch is close to zero would the synchronous control be realized. Its variablesξ
i
(
t
) (i
=
1
~
10) are defined as
(11)
ξ
i
(
t
)
=
e
n
(
t
)
-
e
m
(
t
)
n
=
1
~
4
,
m
=
n
+
1
~
5
.According to the minimum norm theory, the error of synchronous control is defined as(12)
ρ
t
=
∫
0
t
1
10
∑
i
=
1
10
ξ
i
(
w
)
d
w
.The driving forceF
6 can be divided into two parts for the redundant driven branch. One is the driving force from the optimized inverse redundant torque F
q and the other is the driving force from the compensation of synchronous control error F
q. The total driving force is defined as
(13)
F
6
=
F
q
+
F
ρ
.By means of computing torque method, the relationship between the redundant driving force and the coupling errors can be expressed as(14)
F
ρ
=
m
6
ρ
′′
t
+
c
6
ρ
′
t
+
g
ρ
t
,
where m
6, c
6, and g are mass, coefficient of Coriolis force, and gravitational acceleration of the 6th branch; ρ
′
(
t
) and ρ
′
′
(
t
) are velocity error and acceleration error of synchronous control.Eventually, the hybrid control with position and force is realized effectively.


## 2.1. Overview of 6PUS-UPU Parallel Manipulator

A novel 6-PUS/UPU parallel manipulator is shown in Figure1, including the fixed platform, the moving platform, and connected branches. The fixed platform and the moving platform are connected with six PUS (prismatic pair-universal pair-sphere pair) actuated branches and a passive UPU (universal joint-prismatic pair-universal joint) middle branch. Because one degree of freedom is constrained by the UPU branch, the moving platform only can achieve five degrees of freedom. The parallel manipulator is driven redundantly for its six actuated branches.Figure 1
6PUS-UPU parallel manipulator.


## 2.2. 2-DOF PID Control Based on IMC

For modern control systems, it is hoped to have good position tracking characteristics and good antidisturbance capacity simultaneously. Traditional control methods such as PID control belong to one degree of freedom control and only have one adjustable parameter. The control systems with one adjustable parameter are difficult to meet the requirement of good position tracking characteristics and good antidisturbance capacity. 2-DOF control is designed to solve the problem and two separate parameters are adjusted to have the two above-mentioned performances.The servo control system of 6PUS-UPU parallel manipulator is composed of position loop, velocity loop, and current loop.(
1
) Current loop is the innermost loop in the servo system. Current loop consists of the current controller, SPWM, filters, and other components. Proportional and integral controls are used in current controller to have the fastest response among the three loops. The transfer function of current controller is expressed as
(1)
G
i
s
=
K
i
T
i
s
+
1
T
i
s
,
where K
i is the proportional gain factor and T
i
=
L
/
R is the integral gain factor.The control objects of current loop are inverter and armature circuit. The inverter is a conversion circuit from DC to AC. In PMSM control system, SPWM voltage inverter is used and considered as a first-order damp element. The transfer function of inverter is expressed as(2)
G
ivt
s
=
K
n
T
n
s
+
1
,
where K
n is control gain factor and T
n is time constant.The armature circuit is considered asR-L model and its transfer function is expressed as
(3)
G
r
l
s
=
1
R
+
L
s
,
where R is armature resistance and L is armature inductance.In order to remove the noisy signal, a feedback filter is designed as a first-order damp element. The transfer function of the feedback filter is expressed as(4)
G
c
f
s
=
K
c
f
T
c
f
s
+
1
,
where K
c
f is gain factor and T
c
f is time constant.The signal filter is designed to offset the delay caused by the feedback signal. Its transfer function is a first-order damp element and the transfer function is expressed as(5)
G
x
i
s
=
1
T
c
f
s
+
1
.In current loop,I
r is input current and I
m is output current.(
2
) Velocity loop is used to improve the antidisturbance capacity by reducing error between output velocity and given velocity. PI control is used and its transfer function is expressed as
(6)
G
v
s
=
K
v
T
v
s
+
1
T
v
s
,
where K
v is proportional gain factor and T
v is integral gain factor.Feedback filter is also needed in velocity loop and its transfer function is expressed as(7)
G
w
s
=
K
w
T
w
s
+
1
,
where K
w is gain factor and T
w is time constant.The signal filter is also added to offset the delay and its transfer function is expressed as(8)
G
x
v
s
=
1
T
w
s
+
1
.In velocity loop,K
t is torque constant of the motor, J is inertia of the motor, T
e is torque of the motor, T
L is torque of the load, ω
r is input angular velocity, and ω
m is output angular velocity.(
3
) Position loop is the outermost loop and is redesigned based on current loop and velocity loop. The block diagram of three servo loops is shown in Figure 2. In position loop, θ
′ is input angle and θ is output angle; C
1
(
s
) and C
2
(
s
) are 2-DOF IMC controllers, respectively.Figure 2
Block diagram of three control loops.The mathematical models ofC
1
(
s
) and C
2
(
s
) are expressed as
(9)
C
1
s
=
3
λ
1
s
+
1
3
λ
2
s
+
1
,
C
2
s
=
3
λ
2
s
+
1
T
0
s
+
1
3
λ
2
2
s
(
λ
2
/
3
)
s
+
1
,
where λ
1, λ
2 are control parameters and T
0 is time constant.


## 2.3. Force/Position Hybrid Control Strategy

The force/position hybrid control of 6PUS-UPU parallel manipulator is proposed as shown in Figure3. For five nonredundant branches, position control modes are used, and for the redundant branch force control mode is used.Figure 3
Force/position hybrid control.The cross-coupling relationship between driving axes should be taken into consideration. The coupling position errors of five nonredundant branches from the 1st to the 5th branch in joint space are fed back to the 6th redundant driven branch. The coupling position errors are converted to the force signal to make compensation in synchronous controller. So the position precision and torque balance of the parallel manipulator will be improved simultaneously.It is assumed that the theoretical position of branchi is s
i
(
t
) and the actual position is s
i
′
(
t
) at the time t. Then, the position error of the branch at the time t is as follows:
(10)
e
i
(
t
)
=
s
i
(
t
)
-
s
i
′
(
t
)
i
=
1
~
5
.Only when the error of each branch is close to zero would the synchronous control be realized. Its variablesξ
i
(
t
) (i
=
1
~
10) are defined as
(11)
ξ
i
(
t
)
=
e
n
(
t
)
-
e
m
(
t
)
n
=
1
~
4
,
m
=
n
+
1
~
5
.According to the minimum norm theory, the error of synchronous control is defined as(12)
ρ
t
=
∫
0
t
1
10
∑
i
=
1
10
ξ
i
(
w
)
d
w
.The driving forceF
6 can be divided into two parts for the redundant driven branch. One is the driving force from the optimized inverse redundant torque F
q and the other is the driving force from the compensation of synchronous control error F
q. The total driving force is defined as
(13)
F
6
=
F
q
+
F
ρ
.By means of computing torque method, the relationship between the redundant driving force and the coupling errors can be expressed as(14)
F
ρ
=
m
6
ρ
′′
t
+
c
6
ρ
′
t
+
g
ρ
t
,
where m
6, c
6, and g are mass, coefficient of Coriolis force, and gravitational acceleration of the 6th branch; ρ
′
(
t
) and ρ
′
′
(
t
) are velocity error and acceleration error of synchronous control.Eventually, the hybrid control with position and force is realized effectively.


## 3. Mechatronic Cosimulation

### 3.1. Cosimulation Principle

The 3D model of the parallel manipulator is established in Adams and the control model of the parallel manipulator is established in Simulink module of MATLAB. The mechanical model is combined with the control system of the parallel manipulator to realize mechatronic cosimulation.The cosimulation diagram between Adams and MATLAB is shown in Figure4.Figure 4
Structure of Adams and MATLAB cosimulation.The major step is shown in Figure5.Figure 5
Flowchart of cosimulation.


### 3.2. Examples of Cosimulation

As shown in Figure6, 3D model of the parallel manipulator is established in Adams. MOTION_1 to MOTION_5 are added as position driver of five nonredundant branches and FORCE is added as force driver of the redundant branch.Figure 6
3D model in Adams.The control models of nonredundant branches and the redundant branch are established in Simulink module of MATLAB. Position control is adopted in five nonredundant branches and force control is used in the redundant branch. The parameters in relevant transfer function mentioned above are listed in Table1.Table 1
Parameters list.
Parameter
Data
λ
1
0.018
λ
2
0.15
T
0
0.01475 s
J
0.000188 kg·m2
K
v
0.015
T
v
5 s
K
w
1
T
w
0.01 s
K
i
1
T
i
0.00208 s
K
cf
1
T
cf
0.00016 s
K
n
4.43
T
n
0.0001 s
R
1.3Ω
L
0.0027 HThe given trajectory is a major arc of 200 mm radius in cosimulation. The output response of the moving platform is shown in Figure7.Figure 7
3D comparison chart of output response.In Figure7, the output of the moving platform with 2-DOF IMC is closer to the theoretical value than the traditional control and the accuracy of tracking position is improved effectively.


## 3.1. Cosimulation Principle

The 3D model of the parallel manipulator is established in Adams and the control model of the parallel manipulator is established in Simulink module of MATLAB. The mechanical model is combined with the control system of the parallel manipulator to realize mechatronic cosimulation.The cosimulation diagram between Adams and MATLAB is shown in Figure4.Figure 4
Structure of Adams and MATLAB cosimulation.The major step is shown in Figure5.Figure 5
Flowchart of cosimulation.


## 3.2. Examples of Cosimulation

As shown in Figure6, 3D model of the parallel manipulator is established in Adams. MOTION_1 to MOTION_5 are added as position driver of five nonredundant branches and FORCE is added as force driver of the redundant branch.Figure 6
3D model in Adams.The control models of nonredundant branches and the redundant branch are established in Simulink module of MATLAB. Position control is adopted in five nonredundant branches and force control is used in the redundant branch. The parameters in relevant transfer function mentioned above are listed in Table1.Table 1
Parameters list.
Parameter
Data
λ
1
0.018
λ
2
0.15
T
0
0.01475 s
J
0.000188 kg·m2
K
v
0.015
T
v
5 s
K
w
1
T
w
0.01 s
K
i
1
T
i
0.00208 s
K
cf
1
T
cf
0.00016 s
K
n
4.43
T
n
0.0001 s
R
1.3Ω
L
0.0027 HThe given trajectory is a major arc of 200 mm radius in cosimulation. The output response of the moving platform is shown in Figure7.Figure 7
3D comparison chart of output response.In Figure7, the output of the moving platform with 2-DOF IMC is closer to the theoretical value than the traditional control and the accuracy of tracking position is improved effectively.


## 4. Experiment and Application

The self-developed prototype of 6PUS-UPU parallel manipulator is shown in Figure8.Figure 8
The prototype of 6PUS-UPU parallel manipulator.In the experiment, the trajectory is from the starting point(
0,0
,
0
), through the point (
-
100
,
-
100,0
), to the final point (
100,100,0
).The two experiments were done in PID control mode and hybrid synchronous control mode, respectively. The actual trajectories of the moving platform are measured by the laser interferometer. By comparing the data, the errors of the hybrid control can be reduced by about 15% compared to PID control mode.As shown in Table2, the output torque peak in two control modes of five nonredundent branches is recorded by oscilloscope. The torque peak of the hybrid synchronous control can be reduced by about 10% compared with PID control mode and the output torques are banlanced.Table 2
Output torque peak table (rounded, unit: Nm).
Control mode
Branch one
Branch two
Branch three
Branch four
Branch five
PID control
485
2830
3891
3590
1381
Synchronous control
440
2516
3463
3265
1207Based on this, two types of actual compliance operation are realized in 6PUS-UPU parallel manipulator. As shown in Figure9, peg in hole (a) and surface tracking (b) both have good effects.Application of 6PUS-UPU parallel manipulator.
(a)
Peg in hole
(b)
Surface tracking


## 5. Conclusion

The control method combining 2-DOF control and Internal Model Control was optimized. On the basis of current loop and velocity loop, position loop was redesigned to improve stability and position precision simultaneously. Based on cross-coupling, the force/position hybrid control of parallel manipulator is realized. Mechatronic cosimulation and the experiment of 6PUS-UPU parallel manipulator were carried out and both high stability and good position precision are verified. It is effective for peg in hole and surface tracking by actual operation and it should be also applicable to glass cleaning, egg grasping, component assembly, space docking, and so forth.



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*Source: 857947-2014-12-08.xml*