- Email: [email protected]
Development of Three Dimensional Parallel Mechanism Robot with Pneumatic System
5th IFAC Symposium on Mechatronic Systems Marriott Boston Cambridge Cambridge, MA, USA, Sept 13-15, 2010
Development of Three Dimensional Parallel Mechanism Robot with Pneumatic System * Mao-Hsiung Chiang ** Hao-Ting Lin * Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, 10617 Taiwan (Tel: 8862-33663730; e-mail: [email protected]). ** Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, 10617 Taiwan (Tel: 8862-33663252; e-mail: [email protected]). Abstract: With the development of this technology, more and more robots have been used in the industries. This paper presents a three dimensional parallel mechanism robot with pneumatic systems. This article focuses on designing and implementing the mechanical parts, the pneumatic systems, and the control parts. Concerning the mechanical parts, the system is a three dimensional parallel mechanism robot which includes serial parts and parallel parts, and the end-effector of this system moves in a three dimensional motion in an X-Y-Z coordinate system. Regarding the pneumatic systems, the pneumatic actuators, the pneumatic servo valves and other pneumatic elements are used in the experiments. Concerning the control parts, a Fourier series-based adaptive sliding mode controller with H∞ tracking performance (FSB-ASMC+ H∞) is proposed to control the systems. The experimental results show the end-effector performs properly. In conclusion, the three dimensional parallel mechanism robot with pneumatic systems is implemented and the controller is the most efficient way to implement the systems. Keywords: parallel mechanism robot, pneumatic system, adaptive sliding mode control, tracking control
compressibility, low stiffness and leakage, such that it is a highly nonlinear system and hard to acquire accurate mathematic models [8-9]. Because of the highly nonlinear system and inaccurate mathematic models, the pneumatic servo control becomes much more complicated.
1. INTRODUCTION As labor wages increase, more and more countries have developed various kinds of robots to make our lives easier. A robot is a kind of mechanism which can take motion automatically with its components and programs. Robots are widely used in automobile, mechanical, semiconductor, electronic and food and beverage industries and have gradually replaced the labor force [1]. In these industries the predominant robots used are called “industrial robots,” which can be divided into serial and parallel robots. A serial robot which has an open-link structure is able to operate within a much larger scope, but is not able to handle heavier loads. On the contrary, a parallel robot which has a closed-link structure is only able to operate within a smaller scope because of its limited, more complicated structural design, but is able to handle heavier loads [2-4]. In addition, a parallel robot has more stiffness and reliability, so it is widely used in the industrial areas.
Research in the field of pneumatic servo control has been developed in 1960s. Some control algorithms like PID, state-space and adaptive control in pneumatic servo systems were developed via higher speed microcomputers in the 1980s. In recent years due to the development of modern control theories, the problems of the pneumatic servo control have been gradually solved [10-14]. The aim of the present paper is to develop and implement a three dimensional parallel mechanism robot with pneumatic systems. This paper describes a complex hybrid 3D robot with three vertical rod-less pneumatic cylinders combined with parallel links, such that the endeffector moved in a three dimensional motion in an X-Y-Z coordinate system along the tracking paths. In addition, as to the robot structure, it was designed by parallel structures, so that it was hard to calculate the kinematics of the robot. Thus, a kind of robotic coordinate system- D-H notation coordinate system was proposed for solving the kinematics of the robot which included forward kinematics and inverse kinematics. Besides, because the rod-less pneumatic cylinder has high nonlinearities, the accurate mathematic models of the servo pneumatic system are hard to calculate, so that it is quite difficult for tracking control. Thus, the intelligent controller based on a Fourier Series-based adaptive sliding mode
In the industrial areas, robots are frequently used and the related techniques are important to be considered. One of the related techniques is a drive source and a pneumatic system is one of them. Pneumatic systems which are primarily composed of pneumatic components like an air source, a pneumatic servo valve and a rod-less cylinder have advantages of cleanliness, high response, reliability, easy maintenance and safer operating conditions, it is quite suitable to be applied in robotic fields [5-7]. Nevertheless, pneumatic systems have disadvantages like air 978-3-902661-76-0/10/$20.00 © 2010 IFAC
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The parallel links controlled by pneumatic actuators were used, and thus made the end point move in the three dimensional motion. In addition, linear encoders with resolution of 0.1um were necessary for measuring the loading mass’s position that was the sum displacement of the pneumatic cylinder. The measuring signals of the linear encoders were fed back to pc-based control unit part via a counter card. The control signals of the pneumatic servo valves were from the pc-based control unit part with the sampling time of 1ms via an AD/DA interface card.
controller with H∞ tracking performance was proposed. Our results show that the proposed three dimensional parallel mechanism robot with pneumatic systems was implemented and verified experimentally. 2. TEST RIG LAYOUT The three dimensional parallel mechanism robot with pneumatic systems is schematically shown in Fig. 1. Fig. 2 photographically illustrates the developed three dimensional parallel mechanism robot with pneumatic systems. The robot primarily contained two parts, incorporating both a mechanical part and a PC-based control unit part. In a mechanical part, a three-axes parallel mechanism robot, a pneumatic source, three rod-less pneumatic cylinders, three proportional servo valves, and three optical encoders were included. In addition, the robot basically included two base plates parallel to each other, three parallel links connected with vertical rod-less pneumatic cylinders and an end point. For the design that, the end point basically moved within two base plates and was parallel to them. In a pc-based control unit part, an experimental software, an AD/DA interface card and a counter card were included. More over, in the system generation, the air pressure was set up as five bars and the pcbased control unit part via AD/DA interface card was used to control pneumatic servo valves to command the robot.
Fig. 2. The photo of the three dimensional parallel mechanism robot with a pneumatic system 3. SYSTEM ANALYSIS
1.air source 4.PC-based controller 7.optical encoder
2.valve
3.air preparation unit
5.interface card
6.servo valve
8.rodless pneumatic actuator
9.servo pneumatic threeaxial parallel mechanism robot
Because of the development of high technology, the fast motion and precisely accurate plate is demanded. In this paper, the three dimensional parallel mechanism robot with pneumatic systems was implemented. The robot using rodless pneumatic cylinders made the end-effector move in three dimension motion that showed high precision, stiffness and response. Furthermore, the robot consisted of a moving endeffector connected to three vertical pneumatic cylinders through three parallel links. Each chain contained a parallel link and joints activated by a rod-less pneumatic cylinder. The motion of the end-effector was transmitted through links, joints and actuators. In this section, kinematics of the three dimensional parallel mechanism robot with pneumatic systems and dynamic models of the pneumatic system are presented.
Fig. 1. The layout of the three dimensional parallel mechanism robot with a pneumatic system
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3.1 Kinematics Analysis In order to analyze the kinematic models of the robot, D-H notation was used to resolve the geometric relation and acquired the inverse and forward kinematics. Fig. 3 showed the coordinate frames of the system. P0 to P7 are denoted by different coordinate systems. The inverse kinematics can be stated as that given the end-effector pose and the parameters of the robot, find the actuators moving. The mathematic models of this system in the inverse kinematics as follows (1)
hB b Pz 7 L sin( B 3 ) cos( B 4 )
(2)
hC b Pz 7 L sin(C 3 ) cos(C 4 )
(3)
B3
3 1 Px7 Py 7 Rra 2 2 cos 1 L cos( A4 )
(4)
3 1 Px7 Py 7 Rra 2 2 cos 1 L cos( B 4 )
(5)
R r a Py 7 L cos(C 4 )
(10)
Py 7 R r a L cos(C 4 ) cos(C 3 )
(11)
Pz 7 hC b L sin(C 3 ) cos(C 4 )
(12)
3.2
hA b Pz 7 L sin( A3 ) cos( A4 )
A3
Px7 L sin(C 4 )
Dynamic model of pneumatic servo system
The dynamic models of the pneumatic servo system mainly comprise four parts ： the pneumatic servo valve dynamics, the mass flow rate of the servo valve, the continuity equation and the load dynamics. The state equations of the pneumatic servo system is achieved as follows x1 (t ) x2 (t ) x2 (t )
M K ( x (t )) S ( x2 (t ), x3 (t ), x4 (t )) Mg sgn( x1 (t )) S c 1 M kx2 (t ) x3 (t ) x3 (t ) x1 (t ) kRTs Cd C0 wu (t ) fˆ ( x3 (t ), Ps (t ), Pe (t )) A( x1 (t ) )
x4 (t )
C 3 cos 1
A4
(6)
Px7 3 Py7 sin 2 L
(7)
Px7 3 Py 7 2 L
(8)
Px7 L
(9)
( Ax3 (t ) Ax4 (t )) sgn( x1 (t )) K f x2 (t )
kx2 (t ) x4 (t ) l x1 (t ) kRTs Cd C0 wu (t ) fˆ ( x4 (t ), Ps (t ), Pe (t )) A(l x1 (t ) )
1
B 4 sin 1
C 4 sin 1
where hA , hB , hC are the position of the joint along A, B, C rod-less pneumatic cylinder, b is the distance between the end-effector and the centroid of the load, ( Px7 , Py7 , Pz 7 ) is the pose of the end-effector, L is the length of the parallel link, A3 , B 3 and C 3 are the joint angle of the P3 on each chain, A 4 , B 4 and C 4 are the joint angle of the P4 on each chain, R is the distance between the centroid of the bottom plate and each rod-less pneumatic cylinder, r is the distance between the centroid of the load and the P5 joint, and a is the width of the slide on the rod-less pneumatic cylinder. Fig. 3. D-H notation coordinate of the three dimensional parallel mechanism robot
The forward kinematics can be stated as that given the actuators moving and the parameters of the robot, find the end-effector pose. The mathematic models of the forward kinematics as follows 341
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trajectory by FSB-ASMC+ H∞. Fig. 5(a) presents the targets with a ball trajectory and the system responses of the endeffector. Fig. 5(b) shows the tracking error of the end-effector. Fig. 5(c), Fig. 5(f) and Fig. 5(i) shows the system responses of each cylinder. Fig. 5(d), Fig. 5(g), Fig. 5(j) shows the tracking errors of each cylinder. Fig. 5(e), Fig. 5(h), Fig. 5(k) shows the control signals of each cylinder. As shown, the maximum of the tracking errors of an end-effector is 1.4mm.
4. CONTROLLER DESIGN A FSB-ASMC+H∞ controller was proposed in this paper. This control strategy can solve the time-varying uncertainties, the un-modeled dynamics and the disturbances for realizing the servo pneumatic system. A general nonlinear system is shown as follows y ( n ) (t ) F (t ) g (t )u ( t )
(13)
(a) Real Desired
via the Fourier series-based approximation technique, Eq. (14) can be rewritten as
200 Z axis(mm)
y ( n ) (t ) WFT q F (t ) WgT q g (t )u (t ) w(t )
250
(14)
where w(t) denotes the lumped uncertainty. Let’s define a switch surface as s a1e(t ) a2 e(t ) ... an e
( n 1)
(t ), an 1
150 100 50 0 100 100
50
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0
0
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And the H∞ tracking performance was incorporated with the Fourier series-based adaptive sliding mode controller. Suppose that the control input is chosen as
-50 -100
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(b) 4
3.5
n 1 n 1 s (n) ˆ T q (t ) a e (t ) p W F F i i 1 ( n 1) i ei (t ) ym (t ) 2 2 i 1 i 1 u (t ) T ˆ q (t ) W g g
3
error(mm)
2.5
2
1.5
(16)
1
where ρ>0 is the design constant serving as an attenuation level, s is the sliding surface.
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To ensure the feasibility of the design of the mechanism and the controller for the three dimensional parallel mechanism robot with pneumatic systems, the experiments for the end-effector in different tracking trajectories were implemented. In this section, two path tracking control for the end-effector are presented inclusive of a circle trajectory and a ball trajectory. In a circle trajectory, the end-effector first moved from (X, Y, Z)=(0,0,0) to (X, Y, Z)=(0,0,200) in 2 seconds, and then moved from (X, Y, Z)=(0,0,200) to (X, Y, Z)=(100,0,200) in 1 second. Finally the end-effector moved along a circle trajectory with a diameter of 200mm in 17 seconds. Fig. 4 shows the experimental results of the path tracking control for the end-effector in a circle trajectory by FSB-ASMC+ H∞. Fig. 4(a) presents the targets with a circle trajectory and the system responses of the end-effector. Fig. 4(b) shows the tracking error of the end-effector. Fig. 4(c), Fig. 4(f) and Fig. 4(i) shows the system responses of each cylinder. Fig. 4(d), Fig. 4(g), Fig. 4(j) shows the tracking errors of each cylinder. Fig. 4(e), Fig. 4(h), Fig. 4(k) show the control signals of each cylinder. The maximum of the tracking errors of an end-effector is 3.5mm. In a ball trajectory, the end-effector first moved from (X, Y, Z)=(0,0,0) to (X, Y, Z)=(0, 0, 300) in 3 seconds, and then the endeffector moved along a ball trajectory with a diameter of 100mm in 12 seconds. Fig. 5 shows the experimental results of the path tracking control for the end-effector in a ball
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Fig. 1 Experimental results of the tracking control with a circle trajectory (a) system responses of end point (b) tracking errors of end point (c) system responses of Aaxis (d) tracking errors of A-axis (e) control signals of A-axis (f) system responses of B-axis (g) tracking errors of B-axis (h) control signals of B-axis (i) system responses of C-axis (j) tracking errors of C-axis (k) control signals of C-axis
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[4] M.A. Laribi, L. Romdhane, S. Zeghloul, Analysis and dimensional synthesis of the DELTA robot for a prescribed workspace, Mechanism and Machine Theory 42 (2007) 859870.
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[5] S. Konishi, M. Nokata, O.C. Jeong, S. Kusuda, T. Sakakibara, M. Kuwayama, H. Tsutsumi, Pneumatic Micro Hand and Miniaturized Parallel Link Robot for Micro Manipulation Robot System, in: Proceedings of the 2006 IEEE International Conference in Robotics and Automation, Orlando, Florida, 2006, pp. 1036-1041.
error(mm)
4 2 0 -2 -4 0
5
Time (sec)
[6] H.-S. Choi, C.-S. Han, K.-y. Lee, S.-h. Lee, Development of hybrid for construction works with pneumatic actuator, Automation in Construction 14 (2005) 452-459.
(k)
Voltage(V)
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[7] M.-H. Chiang, Development of X-Y Servo PneumaticPiezoelectric Hybrid Actuators for Position Control with High Response, Large Stroke and Nanometer Accuracy, Sensors 10 (2010) 2675-2693.
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[8] M.-H. Chiang, L.-W. Lee, C.-F. J. Kuo, Y.-N Chen, A H∞ Tracking-Based Adaptive Sliding-Mode Controller for Nonlinear Pneumatic Path Tracking Systems via a Functional Approximation Approach, Journal of the Chinese Society of Mechanical Engineers 24 (5) (2009) 507-515.
Fig. 2 Experimental results of the tracking control with a ball trajectory (a) system responses of end point (b) tracking errors of end point (c) system responses of A-axis (d) tracking errors of A-axis (e) control signals of A-axis (f) system responses of B-axis (g) tracking errors of B-axis (h) control signals of B-axis (i) system responses of C-axis (j) tracking errors of C-axis (k) control signals of C-axis
[9] S. Ning, G. M. Bone, Development of a Nonlinear Dynamic Model for a Servo Pneumatic Positioning System, in: Proceedings of the IEEE International Conference on Mechatronics＆Automation, Niagara Falls, Canada, 2005, pp. 43-48.
6. CONCLUSIONS This paper proposes a three dimensional parallel mechanism robot with pneumatic systems in which the endeffector moves in proper path tracking trajectories via parallel mechanism with three vertical pneumatic rod-less cylinders and parallel links. In order to develop and implement this robot, the kinematics of this robot were proposed to analyze and the pneumatic systems were used in this experiments. In addition, a FSB-ASMC+ H∞ controller was developed to overcome the systematic problems.
[10] Z. Rao, G. M. Bone, Nonlinear modeling and control of servo pneumatic actuators, IEEE Transactions on Control System Technology 16 (2008) 562-569. [11] L. Cai, W. Huang, Fourier series based learning control and application to positioning table, Robotics and Autonomous Systems 32 (2000) 89-100. [12] N. Shu, G. M. Bone, High steady-state accuracy pneumatic servo positioning system with PVA/PV control and friction compensation, in: Proceedings of the 2002 IEEE International Conference on Robotics ＆ Automation, DC, Washington, 2002, pp. 2824-2829.
The experimental results clarify that three dimensional parallel mechanism robot with pneumatic systems can achieve fine path tracking control of the end-effector with the maximum errors of 3.5mm in a circle trajectory and with maximum errors of 1.4mm in a ball trajectory. If the joints could be improved, the tracking accuracy could be improved further.
[13] Y.-C. Tsai, A.-C. Huang, FAT-based adaptive control for pneumatic servo systems with mismatched uncertainties, Mechanical Systems and Signal Processing 22 (2008) 12631273.
REFERENCES
[14] Y.-C. Tsai, A.-C. Huang, Multiple-surface sliding controller design for pneumatic servo systems, Mechatronics 18 (2008) 506-512.
[1] T. Brogardh, Present and future robot control development- An industrial perspective, Annual Reviews in Control 31 (2007) 69-79. [2] J. Wang, X.J. Liu, Analysis of a novel cylindrical 3-DoF parallel robot, Robotics and Autonomous Systems 42 (2003) 31-46. [3] Q. Li, F.X. Wu, Control performance improvement of a parallel robot via the design for control approach, Mechatronics 14 (2004) 947-964. 344
Development of Three Dimensional Parallel Mechanism Robot with Pneumatic System * Mao-Hsiung Chiang ** Hao-Ting Lin * Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, 10617 Taiwan (Tel: 8862-33663730; e-mail: [email protected]). ** Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, 10617 Taiwan (Tel: 8862-33663252; e-mail: [email protected]). Abstract: With the development of this technology, more and more robots have been used in the industries. This paper presents a three dimensional parallel mechanism robot with pneumatic systems. This article focuses on designing and implementing the mechanical parts, the pneumatic systems, and the control parts. Concerning the mechanical parts, the system is a three dimensional parallel mechanism robot which includes serial parts and parallel parts, and the end-effector of this system moves in a three dimensional motion in an X-Y-Z coordinate system. Regarding the pneumatic systems, the pneumatic actuators, the pneumatic servo valves and other pneumatic elements are used in the experiments. Concerning the control parts, a Fourier series-based adaptive sliding mode controller with H∞ tracking performance (FSB-ASMC+ H∞) is proposed to control the systems. The experimental results show the end-effector performs properly. In conclusion, the three dimensional parallel mechanism robot with pneumatic systems is implemented and the controller is the most efficient way to implement the systems. Keywords: parallel mechanism robot, pneumatic system, adaptive sliding mode control, tracking control
compressibility, low stiffness and leakage, such that it is a highly nonlinear system and hard to acquire accurate mathematic models [8-9]. Because of the highly nonlinear system and inaccurate mathematic models, the pneumatic servo control becomes much more complicated.
1. INTRODUCTION As labor wages increase, more and more countries have developed various kinds of robots to make our lives easier. A robot is a kind of mechanism which can take motion automatically with its components and programs. Robots are widely used in automobile, mechanical, semiconductor, electronic and food and beverage industries and have gradually replaced the labor force [1]. In these industries the predominant robots used are called “industrial robots,” which can be divided into serial and parallel robots. A serial robot which has an open-link structure is able to operate within a much larger scope, but is not able to handle heavier loads. On the contrary, a parallel robot which has a closed-link structure is only able to operate within a smaller scope because of its limited, more complicated structural design, but is able to handle heavier loads [2-4]. In addition, a parallel robot has more stiffness and reliability, so it is widely used in the industrial areas.
Research in the field of pneumatic servo control has been developed in 1960s. Some control algorithms like PID, state-space and adaptive control in pneumatic servo systems were developed via higher speed microcomputers in the 1980s. In recent years due to the development of modern control theories, the problems of the pneumatic servo control have been gradually solved [10-14]. The aim of the present paper is to develop and implement a three dimensional parallel mechanism robot with pneumatic systems. This paper describes a complex hybrid 3D robot with three vertical rod-less pneumatic cylinders combined with parallel links, such that the endeffector moved in a three dimensional motion in an X-Y-Z coordinate system along the tracking paths. In addition, as to the robot structure, it was designed by parallel structures, so that it was hard to calculate the kinematics of the robot. Thus, a kind of robotic coordinate system- D-H notation coordinate system was proposed for solving the kinematics of the robot which included forward kinematics and inverse kinematics. Besides, because the rod-less pneumatic cylinder has high nonlinearities, the accurate mathematic models of the servo pneumatic system are hard to calculate, so that it is quite difficult for tracking control. Thus, the intelligent controller based on a Fourier Series-based adaptive sliding mode
In the industrial areas, robots are frequently used and the related techniques are important to be considered. One of the related techniques is a drive source and a pneumatic system is one of them. Pneumatic systems which are primarily composed of pneumatic components like an air source, a pneumatic servo valve and a rod-less cylinder have advantages of cleanliness, high response, reliability, easy maintenance and safer operating conditions, it is quite suitable to be applied in robotic fields [5-7]. Nevertheless, pneumatic systems have disadvantages like air 978-3-902661-76-0/10/$20.00 © 2010 IFAC
339
10.3182/20100913-3-US-2015.00125
Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
The parallel links controlled by pneumatic actuators were used, and thus made the end point move in the three dimensional motion. In addition, linear encoders with resolution of 0.1um were necessary for measuring the loading mass’s position that was the sum displacement of the pneumatic cylinder. The measuring signals of the linear encoders were fed back to pc-based control unit part via a counter card. The control signals of the pneumatic servo valves were from the pc-based control unit part with the sampling time of 1ms via an AD/DA interface card.
controller with H∞ tracking performance was proposed. Our results show that the proposed three dimensional parallel mechanism robot with pneumatic systems was implemented and verified experimentally. 2. TEST RIG LAYOUT The three dimensional parallel mechanism robot with pneumatic systems is schematically shown in Fig. 1. Fig. 2 photographically illustrates the developed three dimensional parallel mechanism robot with pneumatic systems. The robot primarily contained two parts, incorporating both a mechanical part and a PC-based control unit part. In a mechanical part, a three-axes parallel mechanism robot, a pneumatic source, three rod-less pneumatic cylinders, three proportional servo valves, and three optical encoders were included. In addition, the robot basically included two base plates parallel to each other, three parallel links connected with vertical rod-less pneumatic cylinders and an end point. For the design that, the end point basically moved within two base plates and was parallel to them. In a pc-based control unit part, an experimental software, an AD/DA interface card and a counter card were included. More over, in the system generation, the air pressure was set up as five bars and the pcbased control unit part via AD/DA interface card was used to control pneumatic servo valves to command the robot.
Fig. 2. The photo of the three dimensional parallel mechanism robot with a pneumatic system 3. SYSTEM ANALYSIS
1.air source 4.PC-based controller 7.optical encoder
2.valve
3.air preparation unit
5.interface card
6.servo valve
8.rodless pneumatic actuator
9.servo pneumatic threeaxial parallel mechanism robot
Because of the development of high technology, the fast motion and precisely accurate plate is demanded. In this paper, the three dimensional parallel mechanism robot with pneumatic systems was implemented. The robot using rodless pneumatic cylinders made the end-effector move in three dimension motion that showed high precision, stiffness and response. Furthermore, the robot consisted of a moving endeffector connected to three vertical pneumatic cylinders through three parallel links. Each chain contained a parallel link and joints activated by a rod-less pneumatic cylinder. The motion of the end-effector was transmitted through links, joints and actuators. In this section, kinematics of the three dimensional parallel mechanism robot with pneumatic systems and dynamic models of the pneumatic system are presented.
Fig. 1. The layout of the three dimensional parallel mechanism robot with a pneumatic system
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Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
3.1 Kinematics Analysis In order to analyze the kinematic models of the robot, D-H notation was used to resolve the geometric relation and acquired the inverse and forward kinematics. Fig. 3 showed the coordinate frames of the system. P0 to P7 are denoted by different coordinate systems. The inverse kinematics can be stated as that given the end-effector pose and the parameters of the robot, find the actuators moving. The mathematic models of this system in the inverse kinematics as follows (1)
hB b Pz 7 L sin( B 3 ) cos( B 4 )
(2)
hC b Pz 7 L sin(C 3 ) cos(C 4 )
(3)
B3
3 1 Px7 Py 7 Rra 2 2 cos 1 L cos( A4 )
(4)
3 1 Px7 Py 7 Rra 2 2 cos 1 L cos( B 4 )
(5)
R r a Py 7 L cos(C 4 )
(10)
Py 7 R r a L cos(C 4 ) cos(C 3 )
(11)
Pz 7 hC b L sin(C 3 ) cos(C 4 )
(12)
3.2
hA b Pz 7 L sin( A3 ) cos( A4 )
A3
Px7 L sin(C 4 )
Dynamic model of pneumatic servo system
The dynamic models of the pneumatic servo system mainly comprise four parts ： the pneumatic servo valve dynamics, the mass flow rate of the servo valve, the continuity equation and the load dynamics. The state equations of the pneumatic servo system is achieved as follows x1 (t ) x2 (t ) x2 (t )
M K ( x (t )) S ( x2 (t ), x3 (t ), x4 (t )) Mg sgn( x1 (t )) S c 1 M kx2 (t ) x3 (t ) x3 (t ) x1 (t ) kRTs Cd C0 wu (t ) fˆ ( x3 (t ), Ps (t ), Pe (t )) A( x1 (t ) )
x4 (t )
C 3 cos 1
A4
(6)
Px7 3 Py7 sin 2 L
(7)
Px7 3 Py 7 2 L
(8)
Px7 L
(9)
( Ax3 (t ) Ax4 (t )) sgn( x1 (t )) K f x2 (t )
kx2 (t ) x4 (t ) l x1 (t ) kRTs Cd C0 wu (t ) fˆ ( x4 (t ), Ps (t ), Pe (t )) A(l x1 (t ) )
1
B 4 sin 1
C 4 sin 1
where hA , hB , hC are the position of the joint along A, B, C rod-less pneumatic cylinder, b is the distance between the end-effector and the centroid of the load, ( Px7 , Py7 , Pz 7 ) is the pose of the end-effector, L is the length of the parallel link, A3 , B 3 and C 3 are the joint angle of the P3 on each chain, A 4 , B 4 and C 4 are the joint angle of the P4 on each chain, R is the distance between the centroid of the bottom plate and each rod-less pneumatic cylinder, r is the distance between the centroid of the load and the P5 joint, and a is the width of the slide on the rod-less pneumatic cylinder. Fig. 3. D-H notation coordinate of the three dimensional parallel mechanism robot
The forward kinematics can be stated as that given the actuators moving and the parameters of the robot, find the end-effector pose. The mathematic models of the forward kinematics as follows 341
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Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
trajectory by FSB-ASMC+ H∞. Fig. 5(a) presents the targets with a ball trajectory and the system responses of the endeffector. Fig. 5(b) shows the tracking error of the end-effector. Fig. 5(c), Fig. 5(f) and Fig. 5(i) shows the system responses of each cylinder. Fig. 5(d), Fig. 5(g), Fig. 5(j) shows the tracking errors of each cylinder. Fig. 5(e), Fig. 5(h), Fig. 5(k) shows the control signals of each cylinder. As shown, the maximum of the tracking errors of an end-effector is 1.4mm.
4. CONTROLLER DESIGN A FSB-ASMC+H∞ controller was proposed in this paper. This control strategy can solve the time-varying uncertainties, the un-modeled dynamics and the disturbances for realizing the servo pneumatic system. A general nonlinear system is shown as follows y ( n ) (t ) F (t ) g (t )u ( t )
(13)
(a) Real Desired
via the Fourier series-based approximation technique, Eq. (14) can be rewritten as
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y ( n ) (t ) WFT q F (t ) WgT q g (t )u (t ) w(t )
250
(14)
where w(t) denotes the lumped uncertainty. Let’s define a switch surface as s a1e(t ) a2 e(t ) ... an e
( n 1)
(t ), an 1
150 100 50 0 100 100
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And the H∞ tracking performance was incorporated with the Fourier series-based adaptive sliding mode controller. Suppose that the control input is chosen as
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3.5
n 1 n 1 s (n) ˆ T q (t ) a e (t ) p W F F i i 1 ( n 1) i ei (t ) ym (t ) 2 2 i 1 i 1 u (t ) T ˆ q (t ) W g g
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error(mm)
2.5
2
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(16)
1
where ρ>0 is the design constant serving as an attenuation level, s is the sliding surface.
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5. EXPERIMENTS
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To ensure the feasibility of the design of the mechanism and the controller for the three dimensional parallel mechanism robot with pneumatic systems, the experiments for the end-effector in different tracking trajectories were implemented. In this section, two path tracking control for the end-effector are presented inclusive of a circle trajectory and a ball trajectory. In a circle trajectory, the end-effector first moved from (X, Y, Z)=(0,0,0) to (X, Y, Z)=(0,0,200) in 2 seconds, and then moved from (X, Y, Z)=(0,0,200) to (X, Y, Z)=(100,0,200) in 1 second. Finally the end-effector moved along a circle trajectory with a diameter of 200mm in 17 seconds. Fig. 4 shows the experimental results of the path tracking control for the end-effector in a circle trajectory by FSB-ASMC+ H∞. Fig. 4(a) presents the targets with a circle trajectory and the system responses of the end-effector. Fig. 4(b) shows the tracking error of the end-effector. Fig. 4(c), Fig. 4(f) and Fig. 4(i) shows the system responses of each cylinder. Fig. 4(d), Fig. 4(g), Fig. 4(j) shows the tracking errors of each cylinder. Fig. 4(e), Fig. 4(h), Fig. 4(k) show the control signals of each cylinder. The maximum of the tracking errors of an end-effector is 3.5mm. In a ball trajectory, the end-effector first moved from (X, Y, Z)=(0,0,0) to (X, Y, Z)=(0, 0, 300) in 3 seconds, and then the endeffector moved along a ball trajectory with a diameter of 100mm in 12 seconds. Fig. 5 shows the experimental results of the path tracking control for the end-effector in a ball
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Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
(b)
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position (mm)
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Fig. 1 Experimental results of the tracking control with a circle trajectory (a) system responses of end point (b) tracking errors of end point (c) system responses of Aaxis (d) tracking errors of A-axis (e) control signals of A-axis (f) system responses of B-axis (g) tracking errors of B-axis (h) control signals of B-axis (i) system responses of C-axis (j) tracking errors of C-axis (k) control signals of C-axis
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Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010
[4] M.A. Laribi, L. Romdhane, S. Zeghloul, Analysis and dimensional synthesis of the DELTA robot for a prescribed workspace, Mechanism and Machine Theory 42 (2007) 859870.
(i) position (mm)
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[5] S. Konishi, M. Nokata, O.C. Jeong, S. Kusuda, T. Sakakibara, M. Kuwayama, H. Tsutsumi, Pneumatic Micro Hand and Miniaturized Parallel Link Robot for Micro Manipulation Robot System, in: Proceedings of the 2006 IEEE International Conference in Robotics and Automation, Orlando, Florida, 2006, pp. 1036-1041.
error(mm)
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5
Time (sec)
[6] H.-S. Choi, C.-S. Han, K.-y. Lee, S.-h. Lee, Development of hybrid for construction works with pneumatic actuator, Automation in Construction 14 (2005) 452-459.
(k)
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[7] M.-H. Chiang, Development of X-Y Servo PneumaticPiezoelectric Hybrid Actuators for Position Control with High Response, Large Stroke and Nanometer Accuracy, Sensors 10 (2010) 2675-2693.
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[8] M.-H. Chiang, L.-W. Lee, C.-F. J. Kuo, Y.-N Chen, A H∞ Tracking-Based Adaptive Sliding-Mode Controller for Nonlinear Pneumatic Path Tracking Systems via a Functional Approximation Approach, Journal of the Chinese Society of Mechanical Engineers 24 (5) (2009) 507-515.
Fig. 2 Experimental results of the tracking control with a ball trajectory (a) system responses of end point (b) tracking errors of end point (c) system responses of A-axis (d) tracking errors of A-axis (e) control signals of A-axis (f) system responses of B-axis (g) tracking errors of B-axis (h) control signals of B-axis (i) system responses of C-axis (j) tracking errors of C-axis (k) control signals of C-axis
[9] S. Ning, G. M. Bone, Development of a Nonlinear Dynamic Model for a Servo Pneumatic Positioning System, in: Proceedings of the IEEE International Conference on Mechatronics＆Automation, Niagara Falls, Canada, 2005, pp. 43-48.
6. CONCLUSIONS This paper proposes a three dimensional parallel mechanism robot with pneumatic systems in which the endeffector moves in proper path tracking trajectories via parallel mechanism with three vertical pneumatic rod-less cylinders and parallel links. In order to develop and implement this robot, the kinematics of this robot were proposed to analyze and the pneumatic systems were used in this experiments. In addition, a FSB-ASMC+ H∞ controller was developed to overcome the systematic problems.
[10] Z. Rao, G. M. Bone, Nonlinear modeling and control of servo pneumatic actuators, IEEE Transactions on Control System Technology 16 (2008) 562-569. [11] L. Cai, W. Huang, Fourier series based learning control and application to positioning table, Robotics and Autonomous Systems 32 (2000) 89-100. [12] N. Shu, G. M. Bone, High steady-state accuracy pneumatic servo positioning system with PVA/PV control and friction compensation, in: Proceedings of the 2002 IEEE International Conference on Robotics ＆ Automation, DC, Washington, 2002, pp. 2824-2829.
The experimental results clarify that three dimensional parallel mechanism robot with pneumatic systems can achieve fine path tracking control of the end-effector with the maximum errors of 3.5mm in a circle trajectory and with maximum errors of 1.4mm in a ball trajectory. If the joints could be improved, the tracking accuracy could be improved further.
[13] Y.-C. Tsai, A.-C. Huang, FAT-based adaptive control for pneumatic servo systems with mismatched uncertainties, Mechanical Systems and Signal Processing 22 (2008) 12631273.
REFERENCES
[14] Y.-C. Tsai, A.-C. Huang, Multiple-surface sliding controller design for pneumatic servo systems, Mechatronics 18 (2008) 506-512.
[1] T. Brogardh, Present and future robot control development- An industrial perspective, Annual Reviews in Control 31 (2007) 69-79. [2] J. Wang, X.J. Liu, Analysis of a novel cylindrical 3-DoF parallel robot, Robotics and Autonomous Systems 42 (2003) 31-46. [3] Q. Li, F.X. Wu, Control performance improvement of a parallel robot via the design for control approach, Mechatronics 14 (2004) 947-964. 344