A Stable Docking Operation by a Group of Space Robots

Transcription

1 A Stable Docking Operation by a Group of Space Robots Vijay Kumar 1, Pushpraj M Pathak 2 Mechanical and Industrial Engineering Department Indian Institute of Technology, Roorkee Roorkee , India 1 vijaydalla@gmailcom, 2 pushpfme@iitracin Abstract This paper presents an attitude control strategy in the three planar space robots having two DOF for docking operation of a free flying hexagonal object using the bond graph technique for modeling and simulation As the highly non linearity, strongly coupled and non-holonomic characteristics of a free flying space robot, the position and posture of the base body changes when the free floating space robot is moving its manipulator to the position to capture a target As the base of space robot is not fixed, it gets disturbed while docking operation in the absence of attitude controller Therefore, it is a great challenge to control the base disturbance To minimize this problem, dynamic model of the three space robots is proposed The simulation results show that the attitude disturbance is minimized by using three space robots in place of the two space robots keeping all parameters same Keywords free flying object; space robot; attitude control I INTRODUCTION To manipulate a tumbling object by space robot is a specially attractive but equally difficult to control its dynamics A space manipulator behaves differently than a ground manipulator mainly due to the absence of gravity and the absence of fixed base in inertial space The motion of manipulator causes dynamic reaction forces and moments which disturb the attitude of space robot base especially in the case of free floating space robot, and it becomes responsible for impulse generation The resulting impulse gets transferred along with the mechanical arm down to the servicing satellite making commotion to the attitude of the satellite and the servicing satellite gets disturbed while docking operation of the captured free floating body, especially in the case of improper control between the of the manipulator and the object The attitude disturbance of the base of robot gradually enhances with the passage of time Therefore, it becomes crucial to sort out the problem of attitude control of a space robot during docking operation By considering the significant role of a group of space robots in future space mission, it becomes necessary to study the different aspects of space robot such as base disturbance, trajectory control, force control and impedance control The main important one is to control base disturbance of a group of space robots successfully, which is comparatively studied in three space robot systems and two space robot systems When two space robots handle a floating object, the control strategy can be designed such that the attitude disturbance is shared between the two robots Hu and Vukovich et al [1], presented an idea to control the position and force of free-flying co-ordinated manipulators using closed kinematic constraints Chang an et al, [2] reported the coordinated control method for a group of free floating space robots and tumbling object in the space micro-gravity environment Shinichi and Taira [] developed a tracking control method using the transpose of the Generalized Jacobian Matrix (GJM to handle a free-flying rigid body cooperatively by several space manipulators Flores-Abad and Ma [4] presented an optimal control strategy for a space manipulator by predicting an optimal future time and motion state for capturing and then control the manipulator to reach the determined motion state having minimal impact to the base satellite during a capturing operation Zhao et al [5], focused on hybrid control: tracking control and active damping control in space robot capturing a floating object and found that it was better than the uniform grasping as a result that the impact force reduced to 0% approximately Huang et al [6] investigated the problem of the dynamic balance control of multi-arm free-floating space robot during capturing an active object in close proximity Yoshida and Sashida [7] paid attention in investigating impact dynamics among space free-floating multi-body systems, by developing an extended inertial tensor concept and to establish the theoretical basis for floating chain collisions It is in order to comprehend the way to preestimate the magnitude of the impulse and the motion after collision from the information of a relative velocity between the hand and target just before contact This provides a useful knowledge in developing a control method to relieve the tumbling motion after contact Papadopoulos [8] developed a kinematic model for mulle manipulator space robot systems as functions of the body-field barycentric vectors Pathak et al [9] discussed a scheme for robust trajectory control of freefloating space robots Patolia et al [10], studied force control in dual arm planar cooperative space robot for cooperative manipulation by two arms In this paper the attitude disturbance in the bases of the three space robot systems and two space robot systems is comparatively studied using the Generalized Jacobian Method It has been assumed that the space robots handle a hexagonal shape free floating object which is lying in the workspace of space robots A group of space robots and floating object are modeled and the study is carried out using simulation and animation 12

2 II MODEL FOR DOCKING BY THREE SPACE ROBOTS USING BOND GRAPH The modeling of the space manipulator can be executed just similar to a ground robot with a distinction that, in case of the space robot, the base is free floating It involves the modeling of translational and rotational dynamics of the links, the bases of the three planar space robots and free floating body To model a group of space robots the following are assumed: first, during docking operation of space robot in free flying mode, the attitude controller of the spacecraft is switched off and hence the spacecraft can translate and rotate in response to the manipulator movement or interaction, if any with the environment Second, it is also assumed that the system has single arm manipulator with revolute joint arm is in the open kinematic chain The general relations used for translational and angular velocity propagation from one link to next be given as follows: ( V ( V R[ ( ( P ] (1 A A A A A i A i i i1 i i i i1 ( R ( ( (2 i 1 A i 1 i A i 1 i i1 i i i1 A Here, the velocities are calculated for the (i+1 th link A ( V is the linear velocity of link i+1 as observed from i 1 the absolute frame {A} and expressed in frame {A} A i R is the rotation matrix describing frame {i} with respect to i i frame {A} ( Pi 1 is the position vector of the frame {i+1} as observed from frame {i} and expressed in frame i1 A {i} coordinates Similarly, ( i1 is the angular velocity of link i+1 as observed from the absolute frame {A} and expressed in frame {i+1} For three planar space robots, the displacement relations can be derived geometrically, and from differentiation of these relations, the velocity relation can be found Figure 1 shows the schematic sketch of a 2 DOF three planar space robots In Fig 1, {A} represents the absolute frame, {V} represents the vehicle frame, {0} frame is located at the base of the robot The frame {0} and frame {1}, the frame {4} and frame {5}, and the frame {8} and frame {9} coincide, respectively Frames {1} and {2} are located at joint 1 and joint 2 of the robot system 1, respectively Frames {5} and {6} are located at joint 1 and joint 2 of the robot system 2, respectively Similarly, Frames {9} and {10} are located at joint 1 and joint 2 of the robot system, respectively The frames {}, {7} and {11} locate the of the robot 1, robot 2 and robot, respectively Let (l 1, l 2, (l, l 4 and (l 5, l 6 be the lengths of the first and second link of robot 1, robot 2 and robot, respectively Let r be the distance between the robot base and center of mass (CM of the vehicle Let 1, 2 and represent the rotation of the base of the robot 1, robot 2 and robot, respectively 1, 2,, 4, 5, and 6, be the joint angles, as shown in Figure 1 Let (X CM1, Y CM1, (X CM2, Y CM2 and (X CM, Y CM be the coordinates of the CM of the base of robot 1, robot 2 and robot, respectively, with respect to the absolute frame {A} Let, (X 2, Y 2 and (X, Y be the distance measured from the absolute frame {A} to the base of the robot system 2 and robot system, respectively The kinematic relations for the displacements: (X 1, Y 1, (X 2, Y 2 and (X, Y of three space robots in X and Y directions measured from the absolute frame can be easily found A robotic manipulator is geometrically modeled by the followings: Fig1 Schematic diagram of 2 DOF three space robots for docking operation 1

3 X X r c( l c( l c( 1 CM ( Y Y r s( l s( l s( 1 CM (4 X 2 X 2 X CM 2 r c( 2 l c( 2 c( Y2 Y2 YCM 2 r s( 2 l s( 2 s( X X X CM r c( l5 c( 5 c( Y Y YCM r s( l5 s( 5 (8 6 s( 5 6 Here, subscript 1, 2 and stand for the robot 1, robot 2 and robot, respectively and s( and c( are used to denote sin( and cos(, respectively The angular displacement with respect to X-axis is given as, (5 (6 ( ( ( (11 From the above set of s displacement equation, the velocities of the s of the robots 1, 2 and, can be established as, X X r s( l ( s( 1 CM ( s( Y Y r c( l ( c( 1 CM ( c( X X r s( l ( s( 2 CM ( s( Y Y r c( l ( c( 2 CM ( c( X X r s( l ( s( CM ( s( Y Y r c( l ( c( CM (12 (1 (14 (15 (16 (17 6 ( 5 6 c( 5 6 The angular velocities of the s of the robot arms are expressed as, ( ( (20 The transformer modulli can be found from the kinematic relations (12 to (17 to draw the bond graph model The transformer modulli used to draw the part of bond graph model for space robot 1 are shown in Table I and similarly it can be presented in the tabular form for the space robot 2 and TABLE I TRANSFORMER MODULLI USED IN BOND GRAPH MODEL OF ROBOT 1 OF THE THREE SPACE ROBOT SYSTEMS Modulus : Expression Space robot 1 Modulus : Expression 1: r cos 1 6: -l 1 sin( : -r sin 1 7: 05 l 2 cos( : 05 l 1 cos( : -05 l 2 sin( : -05 l 1 sin( : l 2 cos( : l 1 cos( : -l 2 sin( A Model of Floating Body The schematic representation of the floating hexagonal object is shown in Figure 2 During the manipulation task, the floating body may have an angular rotation about Z- axis It is presumed that the floating body with side a offers a rotation about Z-axis During docking operation, the points of the gripping on a free flying object will have displacements along X -axis as well as Y- axis Let, (O x, O y be the coordinates of origin O of local frame {L} (attached on the floating body, measured from the absolute frame {A} Let, P 1, P 2 and P are the position of gripping points before rotation and P' 1, P' 2 and P' are the position of gripping points after rotation of the floating body The displacements of the gripping points on the hexagonal floating body of robot 1, robot 2 and robot can be given, respectively, as: x1 OP1 OA Ox acos Ox acos cos x1 acos a cos cos a( / 2(cos 1 (21 ' y1 AP 1 ( Oy acossin ( Oy 0 y1 acos sin a( / 2sin (22 Now the coordinates x 2 and y 2 can be found from P 2 P' 2 C in Figure 2 (b, x2 2acos sin( / 2cos( / 2 x2 a sin( / 2cos( / 2 (2 y2 2acos sin( / 2sin( / 2 y2 a sin( / 2sin( / 2 (24 x BP ( Ox 0 ( Ox acossin ( Ox 0 (a (b Fig 2 Floating object (a before rotation and (b after rotation 14

4 x acos sin a( / 2sin (25 ' y OP OB ( Oy acos ( Oy acos cos y acos a cos cos a( / 2(1 cos (26 From the above Equations, the velocities of the gripping points can be written as, x1 a( / 2 sin (27 y1 a( / 2 cos (28 sin( / 2sin( / 2 x2 a( / 2 cos( / 2cos( / 2 sin( / 2cos( / 2 x2 a( / 2 cos( / 2sin( / 2 (29 (0 x a( / 2(cos (1 y a( / 2(sin (2 The tranformer modulli used for drawing bond graph model of a free floating hexagonal object can be found from (27 to (2 and it is shown in Table II The part of bond graph model are shown in Figure TABLE II TRANSFORMER MODULLI USED IN BOND GRAPH MODEL OF A FREE FLOATING OBJECT Modulus 11 a( / 2 sin Expression 12 a( / 2 cos 1 14 sin( / 2sin( / 2 a( / 2 cos( / 2cos( / 2 sin( / 2cos( / 2 a( / 2 cos( / 2sin( / 2 15 a( / 2(cos 16 a( / 2(sin B Evaluation of Jacobian The Jacobian is evaluated for each space robot system as shown in Figure in the form of signal block diagram The objective of using Jacobian is to convert the effort signals in the workspace into the joint torques The various transformer modulli required to model Jacobian in the bond graph modeling, can be deduced from the forward kinematics relations given as: X 1 X CM 1 r1 s 1 Y Y r c 1 CM s( l s( s( l1c( 1 1 l2c( l2c( s( 1 1 l2s( l1c( 1 1 l2c( ( X 2 X CM 2 r2 s 2 Y2 YCM 2 r2 c2 s( l s( s( lc( 2 l4c( 2 4 l4c( s( 2 l4s( lc( 2 l4c( 2 4 X X CM rs Y YCM r c s( l s( s( l5c( 5 l6c( 5 6 l6c( (4 5s( 5 l6s( 5 6 l5c( 5 l6c( 5 6 (5 In figure, Jacobian is represented as the signal block diagram with gains K 1, K 2, K, K 4, K 5 and K 6 for space robot 1 as shown in Table III can be also extended for robot 2 and in the same way TABLE III GAINS USED IN SIGNAL BLOCK DIAGRAM OF JACOBIAN Modulus Expression K 1 -l 2 sin( K 2 l 2 cos( K -l 1 sin( l 2 sin( K 4 l 1 cos( l 2 cos( K 5 -r sin 1 K 6 r cos 1 A proportional derivative integral (PID controller is used to generate the correct effort for giving the error signal found by subtracting the reference signal and actual velocity This controller feeds a torque to the joint minimizing the error in trajectory The complete bond graph of space robot, Jacobian and PID controller for robot 1 is shown in Figure One can similarly draw the bond graph of space robot, Jacobian and PID controller for robot 2 and shown as blocks in Figure Figure also shows the bond graph of floating body handles by space robot 1, 2 and The bond graph shown in Figure essentially represents three space robot systems interacting with a floating body Figure can also be reduced into a two space robot system by removing bond graph model of space robot The bond graph modeling of 2 DOF three planar space robot systems is drawn using flow map method The transformer modulii employed for the bond graph modeling of the tumbling hexagonal object are deduced using equation (27 through Equation (2 The mass and inertia of the links are modeled using I element The flow activated C elements are used as sensors to assess the rotation of the base and to measure the joint rate of the joint, and the motion of each arm The R element represents the damping present The both R g and K g elements represent damping and stiffness, respectively, to the gripperthe M v is the mass of space robot base and is the angular velocity of the space robot base about Z axis 15

5 Fig Bond graph model with signal block diagram of controller of two DOF three planar cooperative space robots Here, junction-1 and junction-0 represent effort sum junctions which shares same velocity and effort pass junction, respectively The inertial element in the controller is differentially causalled which is removed by adding a Pad The bond graph model has been developed in Bond pad module of SYMBOL SHAKTI software [12] The bond graph is compiled using Borland compiler to create a simulation file III SIMULATION AND RESULTS During the docking operation, the circular reference is used and it can be defined as, X Acos t X (6 0 0 Y Asin t Y (7 where A is the amplitude, is the frequency, X 0 andy 0 are the center of the circle The input reference flow information to the bond graph model, from Equations (6 and (7 can be given in terms of velocity as, X Asin t (8 Y Acos t (9 The simulation is carried out to study the attitude change in the base of both the space robot systems when handling (docking the floating object This simulation has two parts, (i three space robot systems used for docking purposes and (ii two space robot system used for docking purposes for the same input parameters Initially, the of individual arm is in contact with tumbling object, which is to be manipulated along the desired reference circular trajectory The docking function is accomplished by a cooperative action among all the arms The initial configuration of the three planar space robots and the two planar space robots for cooperative operation are shown in the Fig 4 (a and Fig 4 (b, respectively The initial values for the joint angles and the other input parameters are shown in Table IV and Table V, respectively Fig 5 shows the plot of base disturbance ( of three space robots and two space robots for docking operation 16

6 (a (b Fig 4 Initial configuration of (a three planar cooperative space robot systems, and (b two planar cooperative space robot systems TABLE IV INITIAL PARAMETERS Parameters Robot base angle ( 0 Floating body angle ( 0 Joint angle ( 1 20 Values Joint angle ( 2 50 Joint angle ( 11 Joint angle ( 4 74 Joint angle ( Joint angle ( 6 6 In case of three space robot system, the base disturbance values of 1 and 2 change from 0 to radian and 0 to radian with respect to time in the base of robot 1 and 2, respectively, but for the two space manipulators, the same ie 1 and 2 gets disturbed from 0 to radian (max and 0 to radian (max with respect to time, respectively Comparing the values of base disturbance, it is seen that the three space robots get comparatively less disturbed than that of two robots This may be due to the establishment of more stability due to more contact points on the hexagonal free flying object in the case of three space robots The rotation of the floating body with respect to time is shown in Fig 6 for the three space robot systems and the two space robot systems It is seen that rotation of floating bodies in two space robot systems is comparatively greater than three space robot This may be responsible for more attitude disturbance in two space robot systems The trajectory traced out by the of sets of space robots are depicted from Fig 7 (a to Fig 7 (e It is also shown in Fig 8 (a and Fig 8 (b by animation of selected frames It is observed that initially, actual displacement of s track the given input circular path but afterwards it deviates from the given route This may be an account of non holonomic attributes of free floating body Also, it is seen that the s of two space robot systems deviate more than that of three space robot systems which may cause more disturbance of the space robot base TABLE V PARAMETER VALUES USED FOR SIMULATION Arm parameters Space Robot 1 Space Robot 2 Space Robot Length of link 1 (m Length of link 2 (m Location of base of arm from vehicle CM (r (m Mass of link with actuator (kg Mass of link with actuator and gripper (kg Rotary intertia of the link 1 (kg m Rotary intertia of the link 2 (kg m Gripper stiffness (K g (N/m Gripper damping (R g (Ns/m Distance of origin from X 1 =0 X 2 =1208 absolute frame (m Y 2 =0 Y 2 =0112 PID Controller Gain Parameters Proportional gain (K p Integrative gain (K i Derivative gain (K d Common Parameters Length of side of hexagonal object (a 015m Mass of floating body (m b 01 kg Inertia of floating body (I b 1125 kg Space robot base mass (M v 400 kg Rotary inertia of base (I v 6667 kg m 2 Reference Trajectory Parameters (A 005 m Reference Trajectory Parameters ( 1 rad/s Joint resistance (R j 0010 Nm/rad/s Stiffness of spring (K s, K h Damping resistance (R d Pad Parameters Nm/rad 1000 Nms/rad X = 0667 Y =

7 Fig 5 Plot of versus time for three space robot systems and two space robot systems Fig 6 Plot of floating body rotation versus time for three space robot systems and two space robot systems ( a ( b ( c ( d ( e Fig7 Plots of the reference and actual displacement, (a, (b, and (c for robot 1, 2 and, respectively, of three space robot systems and (d and (e for robot 1 and 2, respectively, of two space robot systems 18

8 ( a ( b Fig 8 Animation of frames for (a three space robot systems and (b space robot systems IV CONCLUSION In this paper, the comparative study of the three space robot systems and the two space robot systems is exercised The main aim behind using three robot systems in lieu of two robot systems is to give more stable cooperative manipulating operation As a more stable system bears less attitude disturbance, an optimal control strategy of space robot is offered to mitigate attitude disturbance The proposed control strategy is successfully evidenced by modeling and simulation of both the systems ie two space robot systems and three space robot systems employing the bond graph technique with the PID controller The controller minimized the error between the desired trajectory and real trajectory by adjusting the process control inputs The results show successful control of base attitude disturbance in the three planar space robots which may cause for fuel preserving used by the attitude control device As the space robot has nonholonomic attributes, the nonholonomic system does not inevitably correspond to the given expected path The error in trajectory may be due to nonholonomicity to the system In future work, an appropriate controller design may be suggested to control trajectory properly REFERENCES [1] Yan-Ru Hu, and G Vukovich, Dynamic control of free floating coordinated space robots, Journal of Robotic Systems, Vol05, no 4, pp , 1998 [2] Liu Chang an, Wu Kehe and Xu Yan, Coordinated control of mulle free-floating space robotic system, Proceedings of the IEEE International Conference on Mechatronics and Automation Niagara Falls, Canada, July, 2005, pp [] Shinichi Sagara and Yuichiro Taira, cooperative manipulation of a floating object by some space robots: application of a tracking control method using the transpose of the generalized jacobian matrix, Artif Life Robotics, 2118, pp , 2008 [4] Angel Flores-Abad and Qu Ma, Control of a space robot for minimal attitude disturbance to the base satellite for capturing a tumbling satellite, SPIE Proceedings, Vol 885, 7 May, 2012 [5] Yang Zhao, Cheng Wei and Hongliu Wang, Tracking control in space robot grasping a floating object, Proceeding ROBIO'09 of the International Conference on Robotics and Biomimetics, 2009, pp [6] Panfeng Huang, Yangsheng Xu and Bin Liang, Dynamic balance control of multi-arm free-floating space robots, International Journal of Advanced Robotic Systems, Vol 2, No2, ISSN , pp , 2005 [7] Kazuya Yoshida and Naoki Sashida, Modeling of impact dynamics and impulse minimization for space robots, Proceedings of the 199 IEEE/RSJ International Conference on Intelligent Robots and Systems Yokohama, Japan, July 26-0,199, pp [8] Evangelos G Papadopoulos, Large payload manipulation by space robots, Proceedings of the 099 IEEURSJ International Conference on Intelligent Robots and Systems Yokohama Japan July, 199, pp [9] P M Pathak, R Prasanth Kumar, A Mukherjee, Anirvan Dasgupta, A scheme for robust trajectory control of space robots, Simulation Modelling Practice and Theory pp , 16 June, 2008 [10] Haresh Patolia, P M Pathak, S C Jain, Docking operation by two dof dual arm planar cooperative space robot, 04th National Conference on Machines and Mechanisms (NaCoMM09, NIT Durgapur, India, December 17-18, 2009, pp [11] A Mukherjee, R Karmarkar and A K Samantray, Bond Graph in Modeling Simulation and Fault Identification, I K International Publishing House Pvt Ltd, 2006 [12] Users Manual of SYMBOLS Shakti, High-Tec Consultants, STEP, Indian Institute of Technology, Kharagpur,