论文2G精华版本 ICRA 2018 files 0852

A Synchronization Scheme for Position Control of Multiple Rope Climbing Robots Guangli Sun Xiang Li Peng Li Yu Meng Yang Zhou Enzhi Xu Yun Hui Liu Abstract The ability of rope climbing robots in aloft oper ation is limited by its self supporting and locomotion ability In many applications a given task is also too complex to be achieved by a single rope climbing robot acting alone The solution of multiple rope climbing robots can overcome the limitations However existing control methods for rope climbing robots are limited to single robot and the open issue of coordination between multiple rope climbing robots has not been systematically addressed This paper presents a new synchronization scheme for position control of multiple rope climbing robots such that each robot moves to the correspond ing desired position while synchronizing the heights between each other Maintaining the same height is very important to guarantee the stability of the task oriented manipulator installed among multiple robots when it is performing the manipulation task The development of the proposed controller is based on the singular perturbation approach by treating the fast actuator dynamics as a perturbation of the slow robot dynamics such that the lowest control complexity is achieved The exponential stability of the overall system that consists of the fast and slow subsystems is proved by using Tikhonov s theorem Experimental results are presented to illustrate the performance of the proposed controller Index Terms multiple Rope Climbing Robots climbing robots motion control I INTRODUCTION From the database of skyscraperpage 1 a total of 15 000 highrises have been built up all over the world The demand of cleaning aloft inspection and maintenance works at the highrises is also increasing signifi cantly However these periodical operations are dangerous and complicated only skilled workers are able to carry out the operations To replace the labor intensive and dangerous manual works several climbing robots have been designed and developed including rope climbing robots 2 pillar climbing robots 3 5 wall climbing robots 6 8 Neither the pillar climbing robot nor the wall climbing robot is suitable for the operations on highrises since both have to work in a specifi c environment i e along the pillars or on the walls with desirable surface conditions while the rope climbing robot can provide more fl exibility in the sense it can be deployed everywhere just by setting up some ropes or wires A rope climbing robot is mainly made up of three mod ules i e a self supporting module a locomotion module and G Sun X Li Y Zhou and Y H Liu are with the Department of Mechanical and Automation Engineering The Chinese University of Hong Kong Hong Kong S A R P Li is with the School of Mechanical Engineering and Automation Harbin Institute of Technology Shenzhen Graduate School Y Meng is with the School of Information and Control Engineering Chinese University of Mining and Technology Xuzhou China This work was supported by the HK RGC under Grant no 415011 and CUHK6 CRF 13G a task oriented manipulator The self supporting part keeps the robot stable in the air The locomotion part makes the robot moving in the space The task oriented manipulator can be customized according to the specifi c manipulation task The performance of both the self supporting module and the locomotion module are heavily affected by gravitational forces which also requires that the weight of the task oriented manipulator is light Using multiple rope climbing robots can overcome the limitations caused by gravitational forces because the task oriented manipulator is now carried by multiple robots In addition even in applications where a single rope climbing robot can accomplish the task multiple robots may still be preferred which can provide a certain level of redundancy and hence make the overall system more robust to failure To stabilize the task oriented manipulator installed among multiple rope climbing robots the heights of multiple robots should be synchronized While several controllers have been reported for the position control of rope climbing robots in the literature existing results are limited to a single robot 9 10 and the problem of coordination of multiple robots particularly the problem of synchronization control has not been addressed In parallel various synchronization methods have been proposed for different system In 11 a cross coupling controller was developed by using a composite error rather than an individual error for synchronization In 12 the incorporation of adaptation control and the cross coupling technique was proposed to guarantee the synchro nization of two motion axes In 13 a cross coupling control technique was introduced for mobile robots to maintain the time varying formations In 14 distributed controllers based on the Lyapunov and backstepping methods were de signed for the synchronization motion tracking and collisions avoid among multiple underactuated ships and validated in simulations This paper considers the problem of synchronization con trol and position control of multiple rope climbing robots Note that the overall rope climbing robotic system consisting both the robot dynamics and the actuator dynamics is a high order system with couplings between each other and con trollers for rope climbing robots should be able to stabilize both subsystems 15 In this paper a new synchronization scheme is proposed for position control of multiple rope climbing robots By using singular perturbation method 16 17 the overall dynamics of rope climbing robotic system is decomposed into a slow robot dynamics and a fast actuator dynamics A slow time scale controller is introduced to enable each robot to move to the desired position while 2018 IEEE International Conference on Robotics and Automation ICRA May 21 25 2018 Brisbane Australia 978 1 5386 3080 8 18 31 00 2018 IEEE3736 Suspension Mechanism Compression Mechanism Sproket Link Chain Rubber Teeth a Spur Gear and Sprocket Motor b Fig 1 3D model of the rope climbing car a front view b back view The movement of the robot is realized by using the mechanism of the movement of dual symmetric chain and sprocket and the combination of a compression and a suspension mechanism The compression mechanism compresses the link chain and the rubber teeth against the rope while the suspension mechanism adjusts the movement of the chain and the sprocket and hence makes tight contact with the rope synchronizing the height between each other while a fast time scale controller is developed to stabilize the actuator dy namics Then the stability of the overall closed loop system is analyzed by using Tikhonov s theorem While the overall dynamics is a fourth order system the proposed controller does not require any high order state variables or observers Therefore the main advantages of the proposed controller are simplicity and ease of implementation Experimental results are presented to exhibit the effectiveness of the proposed control method II BACKGROUND A Mechanical Structure A rope climbing robot has been designed and developed in our previous works 23 and the climbing mechanism is based on the structure of caterpillar 19 and illustrated in Fig 1 a The movement of the robot is realized by using the mechanism of the movement of dual symmetric chain and sprocket and the combination of a compression and a suspension mechanism The compression mechanism compresses the link chain and the rubber teeth against the rope while the suspension mechanism adjusts the movement of the chain and the sprocket and hence makes tight contact with the rope A powerful brush DC motor and a lithium ion battery are installed to drive the robot The robot is controlled to move along the full constrained steel rope where the head and the tail of the rope are well fi xed B Dynamic Model The overall dynamic model of a rope climbing robot consists of two subsystems i e the robot and the actua tor In the following analysis the rotary elements of the rope climbing robot are converted to equivalent translational elements without loss of generality Considering n rope climbing robots the overall dynamic model can be described as 15 24 M q D q G K q 1 B K q u 2 where q q1 qn T nis the vector of robot dis placement qi i 1 n is the displacement of the ith robot nis the vector of motor displacement iis the displacement of the ithmotor M diag M1 Mn n nis the diagonal mass matrix of robot where Miare positive constants D diag D1 Dn n ndenotes the diagonal friction matrix where Diare also positive constants and G nis the gravity matrix K diag K1 Kn n ndenotes the stiffness matrix where Kiis the stiffness of the ithrope B diag B1 Bn n nis the diagonal mass matrix of actuator and u ndenotes the input forces exerted on actuators Equation 1 describes the well known rigid body dynam ics and equation 2 describes the actuator dynamics The two subsystems are linked to each other with the stretching force of the rope i e K q In this paper it is assumed the dynamic models for the robot and actuator are well defi ned that is M D G B and K are exactly known III SYNCHRONIZATIONSCHEMEFORMULTIPLE ROPE CLIMBINGROBOTS The mechanical dynamics of robot is slower 1 while the dynamics of robot is faster 2 and hence the overall rope climbing robotic system exhibits the features of two time scales 25 In this paper the singular perturbation theory 26 is introduced to design controllers for two subsystems separately by treating the faster dynamics as perturbation of the slower dynamics A Controller for Actuator Subsystem The overall controller for multiple rope climbing robots can be proposed as u us uf 3 where us uf nrepresent the slow controller for the robot subsystem and the fast controller for the actuator subsystem respectively First the fast controller is specifi ed as uf Kv q 4 3737 where Kv n n is a diagonal and positive defi nite matrix Substituting 3 into 2 yields B Kv q K q us 5 which leads to B q Kv q K q us B q 6 Next a variable y is introduced as the stretching force of the rope as y K q such that equations 1 and 6 can be written as M q D q G y 7 B y Kv y Ky K us B q 8 Expressing K and Kvin terms of as 25 K K1 2 9 Kv K2 10 where is a small parameter Then equation 8 becomes 2B y K2 y K1y K1 us B q 11 The system described by 7 and 11 is singularly perturbed The variables y and y can be viewed as fast time scale variables while the state variables q and q denote slow time scale variables With 0 equation 11 becomes y us B q 12 in which the overbar variable denotes the value of variable at 0 Then substituting equation 12 into equation 7 at 0 yields the quasi steady state model M B q D q G us 13 By assuming that y is constant on a fast time scale t a new variable is introduced as y y Substituting the fast time scale and y y into 11 with d y d d2 y d 2 0 and 0 we have B d 2 d 2 K2 d d K1 0 14 which is referred to as boundary layer system According to Tikhonov s theorem 26 the overall system is stable if both the boundary layer system 14 and the quasi steady state system 13 are exponentially stable The exponential stability of the boundary layer system can be guaranteed by specifying K1and K2from 14 Then the control objective is to design a slow controller uswhich guarantees the exponential stability of the quasi steady state system 13 B Controller for Robot Subsystem In this section a slow controller us is specifi ed such that the quasi steady state system is exponentially stable The slow controller also guarantees the convergence of the position of each robot to the desired position and achieves the synchronization between each other First a sliding vector is introduced as s q qr q qd q q s 15 where q r qd q q s 16 is a reference vector qand sare positive constants q q qdwhere qd qd1 qdn ndenotes the vector of desired positions qdiis the desired position for the ith robot and denotes the synchronization vector which is specifi ed as 1 n 2 1 n n 1 T where 1 q1 q2 n qn q1 The defi nition of synchronization errors was introduced in 27 We can now propose the slow controller usas us Kss kq q M B qr D qr G 17 where Ks n n is positive defi nite and kqis a positive constant In 17 the fi rst two terms include the position control the synchronization control and the velocity control and the last three terms represent the dynamic compensation The key novelty of the proposed control method is to integrate both the position error q and the synchronization error into a single controller such that multiple robots can move to the desired positions while always maintaining the same height when they are climbing along the ropes Both the position control task and the synchronization control task are important for the coordination of multiple rope climbing robots Note that the overall dynamic model described by 1 and 2 is a high order system while the complexity of the proposed controller in 3 is the same as that of the dynamics of the robot subsystem That is the use of singular perturbation approach helps to decrease the complexity and also the computational const of the controller By using the sliding vector 15 the dynamics of the quasi steady state system can be rewritten as M B s Ds M B qr D qr G us 18 Substituting 17 into 18 yields the following dynamic equation M B s D Ks s kq q 0 19 Now we are in the position to state the following theorem Theorem The proposed controller described by 3 4 and 17 for multiple rope climbing robots guarantees the stability of the closed loop system and the convergence of tracking errors and synchronization errors to zero if the control parameters are chosen such that min Ks max 1 2 M B D 20 q 1 2 21 where min and max represent the minimum and the maximum eigenvalues respectively Proof First a Lyapunov like candidate is proposed as V 1 2s T M B s kq 2 qT q 22 3738 Differentiating 22 with respect to time yields V sT M B s kq qT q 23 Substituting 15 and 19 into 23 we have V sT D Ks s sTKq q kq qT q sT D Ks s qkq qT q skq T q 24 Note that T q n P i 1 q i i i 1 q1 1 q1 n q2 2 q2 1 qn n qn n 1 q1 q2 1 q2 q3 2 qn q1 n 2 1 22 2n T 25 where 1 n T n Substituting 25 into 24 yields V sT D Ks s qkq qT q skq T 26 If the control parameters Ksand qare chosen such that conditions 20 and 21 are satisfi ed it is obtained that V sT D Ks s qkq qT q V 0 27 where is a positive constant The inequality 27 indicates that V V 0 e twhere V 0 denotes the initial value of the Lyapunov like candidate Therefore the exponential stability of the quasi steady state system is guaranteed Since the fast controller 4 ensures the exponential stability of the quasi steady state system 13 from the analysis in the previous section we can also conclude that the overall controller 3 guarantees the stability of the rope climbing robotic system described by 1 and 2 In addition the inequality 27 implies that s q 0 as t Then from 15 it is obtained that q s 0 28 In the system of multiple rope climbing robots it is usually specifi ed that qd1 qd2 qdn Hence the conver gence of q 0 leads to the convergence of 0 Then from 28 it is concluded that q 0 That is q qd q qd and qi qjas t Therefore the convergence of tracking errors and synchronization errors to zero is guaranteed IV EXPERIMENT Anexperimentalsetupofsynchronizationofrope climbing robots has been established in The Chinese Uni versity of Hong Kong shown in Fig 2 is made up with two rope climbing robots as the realization of Fig 1 a and Fig 1 b The main devices in each rope climbing robots are a brush DC motor with 40 1 reduction gears which is with 250 walt rating power and 120 round per minute rating speed an embedded system which here is STM32F103C8T6 a brush motor driver which can use PWM to change the motor speed it varies from 5000 to 5000 and an rotary incremental optical encoder with 1200 segments which is installed at the shaft of the DC motor a Lithium ion Battery with 24Voltage rating voltage and 15Ampere Hour capacity Two robots are connected by CAN bus wire and numbered as robot1 and robot2 Robot1 gets the two robots sensors datas from the CAN bus wire and sends out the data to the ground control station which is a personal computer PC of intel i5 windows 7 system through tansparent wireless module which you can think the robot1 is communicated with PC through serial ports To verify the stability of the control scheme setpoint control and tracking control experiments are implemented as follows Rope climbing Robot RopeRope fixed fixed fixed fixed Fig 2 whole system of the rope climbing robots synchronization experi mental setup A setpoint control experiment The target of setpoint control experiment is to observe the setpoint precision of two climbing robots when synchronized The target set here is 0 5m The control frequency of the embedded system is 10ms while the PC gets the wireless data every 200ms which means T 200ms The control gains are Kv 1000 Ks 1 Kq 10000 q 2000 s 60000 Fig 3 a illustrates the trajectory of two rope climbing robots during the setpoint control Fig 3 b illustrates the trajectory errors of two rope climbing robots during the setpoint control which is the q in equation 15 Fig 3 c illustrates the synchronization error of two rope climbing robots during the setpoint control which is the in equation 15 From Fig 3 a and Fig 3 c two robots are closely moved to the target The stable error is 1mm The maxmim division happens at the 25T time and the robot2 left behind robot1 1 4cm Compared to the size of the robot it can be ignored Snapshots of t 0T t 50T t 150T can been seen in Fig 4 B tracking control experiment The target of tracking control experiment is to observe the synchronization effects of two climbing robots The input sine curve is 0 5 sin 210 T t in which the T is the wireless sending period 200ms The initial position of robot1 is 0 22m and robot2 is 0 So two robots can not only 3739 a b c Fig 3 Setpoint control results a trajectory of two robots b trajectory error of t