Reduced Model Based Control of Two Link Flexible Space Robot

Transcription

1 Intelligent Control and Automation, 0,, 0 doi:0.436/ica.0.03 Published Online May 0 ( Reduced Model Based Control o Two Link Flexible Space Robot Abstract Amit Kumar, Pushparaj Mani Pathak, Nagarajan Sukavanam Department o Mathematics, Indian Institute o Technology Roorkee, Roorkee, India Department o Mechanical and Industrial Engineering, Indian Institute o Technology Roorkee, Roorkee, India tomardma@gmail.com, pushpme@iitr.ernet.in, nsukvma@iitr.ernet.in Received January 6, 0; revised April, 0; accepted April 5, 0 Model based control schemes use the inverse dynamics o the robot arm to produce the main torque component necessary or trajectory tracking. For modelbased controller one is required to know the model parameters accurately. This is a very diicult task especially i the manipulator is lexible. So a reduced model based controller has been developed, which requires only the inormation o space robot base velocity and link parameters. The lexible link is modeled as Euler Bernoulli beam. To simpliy the analysis we have considered Jacobian o rigid manipulator. Bond graph modeling is used to model the dynamics o the system and to devise the control strategy. The scheme has been veriied using simulation or two links lexible space manipulator. Keywords: Flexible Space Robots, Bond Graph Modeling, Reduced Model Based Controller, EulerBernoulli Beam. Introduction Space robots are a blend o a vehicle and a robot. It is mechanically more complex than a satellite. The lexible space robot will be useul or space application due to their light weight, less power requirement, ease o maneuverability and ease o transportability. Because o the light weight, they can be operated at high speed. For lexible manipulators lexibility o manipulator have considerable inluence on its dynamic behaviors. It is observed by many researchers that ixedgain linear controllers alone do not provide adequate dynamic perormance at high speeds or multidegrees oreedom robot manipulators. Out o various schemes studied so ar, those involving the calculation o the actuator torque (orce) using an inverse dynamics model (the computed torque methods), and those applying adaptive control techniques have been extensively studied, and show the greatest promise. The traditional control strategy o robot manipulators is completely error driven and shows poor perormance at highspeeds, when the high dynamic orces act as disturbances. The current trend is towards modelbased control, where the dynamic orces are incorporated in the control strategy as eed orward gains and eedback compensations along with the servocontroller which is required only to take care o external noise and other actors not included in the dynamic model o the robot. As is expected, the modelbased control scheme exhibits better perormance, but demands higher computational load in real time. A particular area o modelbased control is adaptive control, which is useul when the dynamic parameters o the robot are not wellknown a priori. The controller adapts itsel during execution o tasks and improves the values o the dynamic parameters. Murotsu et al. [,] proposed control schemes or lexible space manipulator using a virtual rigid manipulator concept. Samanta and Devasia [3] have discussed modeling and control o lexible manipulates using distributed actuator. A technique is presented to analyze the dynamics o lexible manipulators using bond graphs. The nonlinear coupling o large rigid body motion and small elastic vibration o the lexible arms has been taken into consideration in the model. The concept o using distributed piezoelectric transducers or controlling elastic vibration o arms has been incorporated in the analysis. Fardanesh and Rastegar [4] developed an inverse dynamics model based on trajectory pattern method.

2 A. KUMAR ET AL. 3 Wang and Gao [5] have reported inverse dynamics modelbased control or lexiblelink robots, based on modal analysis, i.e., on the assumption that the deormation o the lexiblelink can be written as a inite series expansion containing the elementary vibration modes. Leahy et al. [6] worked on robust model based control. They also discussed an experimental case. Fawaz et al. [7] developed a model based realtime virtual simulator o industrial robot in order to detect eventual external collision. The method concerns a model based ault detection and isolation used to determine any lock o motion rom an actuated robot joint ater contact with static obstacles. Tso et al. [8] worked on a modelbased control scheme or robot manipulators employing a variable structure control law in which the actuator dynamics is taken into consideration. Zhu et al. [9] attached an additional modelbased parallelcompensator to the conventional modelbased computed torque controller which is in the orm o a serial compensator to enhance the robustness o robot manipulator control. Qu et al. [0] have discussed robust control o robots by the computed torque law with respect to unknown dynamics by judiciously choosing the eedback gains and the estimates o the nonlinear dynamics. The choices or the constant gains depend only on the coeicients o a polynomial bound o the unknown dynamics. Khosla and Kanade [] presented the experimental results o the realtime perormance o modelbased control algorithms. The computedtorque scheme which utilizes the complete dynamics model o the manipulator is compared with the independent joint control scheme which assumes a decoupled and linear model o the manipulator dynamics. Pathak et al. [] worked on a scheme or robust trajectory control o space robot and this work presents a reduced modal based controller or trajectory control o lexible space robot in work space. Berger et al. [3] carried out the application o bond graph modeling to robots. Special emphasis is placed on adjusting the exact bond graph to allow or valid numerical solutions. This paper presents a model based controller which requires the inormation o the base o space robot, link length, joint angle and evaluation o Jacobian. The controller is based on robust overwhelming control o the lexible space manipulator. To illustrate the methodology, an example o a two DOF lexible space robot is considered. The controller is provided with velocity inormation o manipulator. Bond graphs are used to represent both rigid body and lexible dynamics o the link in a uniied manner. The advantage o the bond graph is that it is used to model system, in multi energy domain and various control strategies can be devised using this modeling method. SYMBOLS Shakti [4] sotware is used or bond graph modeling and simulation.. Modeling o Two Arm Flexible Space Robot Let us assume that the space robot has single manipulator with revolute joints and is in open kinematic chain coniguration. Figure shows the schematic sketch o two arm lexible space robot. In this igure {A} represents the absolute rame, {V} represents the vehicle rame, {0} rame is located in space vehicle at the base o the robot, {} rame is located in irst arm at the base o irst arm, {} rames is located second joint. The rame {t} locates the o the robot. Let L be the length o the irst link, L be the length o second link and r is the distance between the robot base and center o mass (CM) o the space vehicle. Let the irst motor is mounted on base and it applies a torque on irst link. Similarly let the second motor is mounted on irst link and it applies a torque on second link. The cross section area o links are assumed to be A. The densities o irst and second links are and respectively. The lexible links are uniorm in cross section with lexural rigidities EI and EI. The lexible link o space robot has been modeled as EulerBernoulli beam. 3. Bond Graph Modeling The kinematic analysis o lexible links is perormed in order to draw the bond graph as shown in Figure. Figure also shows the pad sub model. From the kinematic Figure. Schematic representation o two arm lexible space robot.

3 4 A. KUMAR ET AL. Figure. Complete bond graph model o two arm lexible space robot with reduced model based controller. relations the dierent transormer moduli used in bond graph model are derived. Two type o motion o the links are considered. First motion is the motion perpendicular to the link and second motion is the rotational motion o the links. The velocities in perpendicular to link is resolved in X and Y direction. In Figure, M V and I V represents the mass o space vehicle and inertia o space vehicle. Let X cm and Y cm be the coordinate o the CM o the space vehicle with respect to absolute rame. The irst and second link is divided into our segments o equal length. The segment angles o irst and second link are ψ, ψ, ψ 3, ψ 4 and ω, ω, ω 3, ω 4 respectively. The segment inertias are attached to junction structure corresponding to these velocities. Pads are used or computational simplicity i.e., to avoid the dierential causality. The velocity o the irst link at junction in X and Y direction with respect to absolute rame can be evaluated

4 A. KUMAR ET AL. 5 as X X rsin Y cosπ x cm () Y Y rcos Y sin π () y cm where, and Y is the velocity o the link in direction perpendicular to link at junction. The velocity o the irst link in X and Y direction with respect to absolute rame can be evaluated as X X L 8sin (3) t_ 5x 4 4 8cos Y t_ Y 5y L 4 (4) 4 where X 5x and Y 5 y are the velocity o the irst link at junction 5 in X and Y direction. The velocity o the second link at junction 6 in X and Y direction with respect to absolute rame can be evaluated as X cos 6 _ 6 π x X t Y (5) Y 6 y Y t_y 6sin π (6) where Y 6 is the velocity o second link at junction 6 in the direction perpendicular to link. The velocity o the robot at in X and Y direction with respect to absolute rame can be evaluated as 8sin 8cos X X L (7) t_ 9x 4 4 Y t_ Y 9y L 4 (8) 4 where X 9x and Y 9 y are velocity o the link in X and Y direction at junction 9. From the above kinematic analysis dierent transormer modulli used in drawing bond graph can be evaluated and are shown in Table. 4. Reduced ModelBased Controller Design In work space control the robot is required to track a given endeector trajectory, this is achieved by using a reduced model based controller. The bond graph model o a two DOF lexible planar space robot controller is shown in Figure. The model is represented in the inertial rame. Controller consists o three parts. ) Space robot virtual base model ) Overwhelming controller [] 3) Jacobian. 4.. Space Robot Virtual Base Model A virtual base o the space vehicle is considered. To design the model based controller, drawing analogy rom the space vehicle, the velocity o manipulator in X and Y direction is considered in controller design. It is Table. Transormer moduli or inding velocities at various points in bond graph model. Description X Direction Y Direction Robot Base TF = r sin() TF = r cos() First Segment o First Link Second Segment o First Link Third Segment o First Link Fourth Segment o First Link TF = cos(/ + ψ ) TF = sin(/ + ψ ) TF = cos(/ + ψ ) TF = sin(/ + ψ ) TF = cos(/ + ψ 3 ) TF = sin(/ + ψ 3 ) TF = cos(/ + ψ 4 ) TF = sin(/ + ψ 4 ) First Link Tip TF = (L /8) sin(ψ 4 ) TF = (L /8) cos(ψ 4 ) First Segment o Second Link Second Segment o Second Link Third Segment o Second Link Fourth Segment o Second Link TF = cos(/ + ω ) TF = sin(/ + ω ) TF = cos(/ + ω ) TF = sin(/ + ω ) TF = cos(/ + ω 3 ) TF = sin(/ + ω 3 ) TF = cos(/ + ω 4 ) TF = sin(/ + ω 4 ) Second Link Tip TF = (L /8) sin(ω 4 ) TF = (L /8) cos(ω 4 ) determined by translational inertia M o space vehicle. The displacement relation o the robot are given as X X rcos Lcos L cos Y Y rsin Lsin L sin (9) (0) Here is the rotation o the space vehicle. The velocity o the robot X, and Y due to motion o the space vehicle can be ound as, X X rsin Lsin Lsin () Lsin Lsin Lsin Y Y rcos Lcos Lcos () Lcos Lcos Lcos I it is assumed that the controller overwhelms the ro

5 6 A. KUMAR ET AL. bot dynamics, then neglecting the coeicients o and in Equations () and () we get, X X rsin Lsin Lsin Y Y rcos Lcos Lcos modulli involve only link lengths and joint angles o the manipulator. (3) 4.. Overwhelming Controller (4) Using Equation (3) and (4), the transormer modulli µ 5, and µ 6 shown in bond graph o Figure, can be written as, 5 rsin Lsin Lsin 6 rcos Lcos Lcos One can observe rom these expressions that these The linear overwhelming controller [] is applied to the o the space robot model. The overwhelming controller is provided with reerence velocity and velocity inormation. The eort signal produced by the controller is magniied by high gain and then the joint torques are evaluated with the help o Jacobian. This joint torque inormation is ed to dierent joints Evaluation o Jacobian I we assume that the arms o manipulator are rigid then, kinematic relations or the displacement X, Y in X and Y directions can be written as, X XCM rcos Lcos Lcos Y Y rsinl sin L sin CM (5) The angular displacement with respect to absolute rame X axis is given as, (6) The Jacobian o the orward kinematics can be calculated in bond graph. For the planar case discussed here this relation can be worked out directly rom Equation (5) as ollows, X r sin L sin L sin X CM Y Y r cosl cos L cos CM X X sin sin sin sin CM r L L L Y Y r cos L cos L cos L cos CM L sin L cos L sin L cos (7) (8) For the evaluation o joint torque i we assume that space vehicle rotational velocity, and CM velocity X CM, Y CM are small, then neglecting the irst and last term o Equation (8) we get X L sin L sin L sin Y L cos L cos L cos (9) X 3 Lsin, J Y (0) 4 Lcos. With the help o Equation (0) various transormer modulli shown in bond graph can be evaluated as 5. Simulation and Results Lsin Lsin It is assumed that initially base rotation; irst and second Lcos Lcos joint angles are 0 radian, 0.5 radian and 0 radian respec

6 A. KUMAR ET AL. 7 tively. Figure 3 shows the initial coniguration o the space robot. At the beginning o the simulation overwhelmer initial position is initialized to robot position. Let us assume that the reerence displacement command or the robot is a circle o radius R. Then the coordinates will be given as, re cos 0 sin 0 X t R t X () Y t R t Y re () Here ( X 0, Y 0) is the center o the circular reerence circle. The corresponding reerence velocity command is gi ven by, R t (3) rex sin R cost (4) rey At the start o simulation the reerence trajectory is initialized to trajectory. The parameters used or simulation are shown in Table. The simulation is carried out or.56 seconds. Table. Parameters and values used or simulation. Parameter Value 9 Modulus o E lasticity E = 70 0 N/m Link Length Link Inertia Location o Robot Base rom Vehicle CM Mass o the First Link L = 0.5 m; L = 0.6 m m =.0 kg Cross Section Area o First A = m, and Second Link A = m Joint Resistances Mass o Base Moment o Inertia o Space Vehicle R = 0.00 Nm/(rad/s), R = Nm/(rad/s) I V =.0 kgm I m = 0.00 kg m, I m = 0.00 kg m Controller M C = 0. 00, K C = 0.4, R C = 0.3 Internal Damping Coei cient or Material Input Reerence Angular Velocity 3 4 I = m, I = m 4 r = 0. m M = 5.0 kg Gain Value µ H = 8.0 Motor Inertias Reerence Circle Radius R int = 0.0 N/(m/s) R = m =.0 rad/s Density o Aluminum ρ = 700 kg/m 3 Figure 3. Initial coniguration o two arm lexible space robot. Figure 4(a) shows the plot or reerence trajectory in x direction, Figure 4(b) shows the plot reerence trajectory in y direction, Figure 4(c) shows the plot or trajecry in x direction, and Figure 4(d) shows the plot or to trajectory in y direction. From Figures 4(a) and (c) it is seen that ollows the reerence trajectory in x direction and rom Figures 4(b) and (d) it is also seen that ollows trajectory in y direction eectively. Due to the nature o the lexible link vibration are observed. Figure 5(a) shows the plot o base rotation with respect to time. The base o the space vehicle rotates between the points 0.0 to 0.03 as the space robot is ree loating. Figure 5(b) shows the plot o irst joint angle with respect to time. Figure 5(c) shows the plot o second joint angle with respect to time Figure 5(d) shows the plot o CM o space vehicle in x direction with respect to time. Figure 5(e) shows the plot CM o space vehicle in y direction with respect to time. Figure 5() shows the plot o CM trajectory o space vehicle. Figures 5(d), (e) and () it is seen that there is continues movement in base o space robot as it is ree loating. Figure 6 shows the trajectory error plot. The error is taken as the dierence between reerence and actual position. Figure 6(a) shows the error plot or X position with respect to time. Figure 6(b) shows the error plot or Y position with respect to time. From the simulation we see that initially error increases with time but ater some time it start reducing. 6. Conclusions In this paper we presented a novel reduced model based controller or trajectory control o lexible space robot in work space. Euler Bernoulli beam model is used to model lexible link. Reduced model based controller consists o three parts ) the model o the virtual base o space vehicle ) an overwhelming controller 3) a Jacobian evaluation to evaluate joint torques. The advantage

7 8 A. KUMAR ET AL. Xre (m) Yre (m) (a) (b) X (m) Y (m) (c) Figure 4. (a) Reerence trajectory in x direction; (b) Reerence t rajectory in y direction; (c) Tip trajectory in x direction. ( d) Tip trajectory in y direction. (d) (a) (b)

8 A. KUMAR ET AL. 9 Xcm (m) (c) ( d) Ycm (m) Ycm (m) X cm (m) (e) () Figure 5: (a) Plot o base rotation versus time; (b) Plot o irst joint angle versus time; (c) Plot o second joint angle versus time; (d) Centre o mass trajectory in x direction; (e) Centre o mass trajectory in y direction; () Centre o mass trajectory o space vehicle. (a) (b) Figure 6. (a) Error in X trajectory with respect to time; (b) Error in Y trajectory with respect to time.

9 0 A. KUMAR ET AL. o the reduced model based controller is that it does not require the link dynamic parameters and the trajectory tracking is very good. However the model has limitation as compared to analytical model as it is inite approximation o continuous system. The uture work will be ocus on developing a reduced model based controller or three dimensional lexible space manipulator. 7. Reerences [] Y. Murotsu, S. Tsujio, K. Senda and M. Hayashi, Trajectory Control o Flexible Manipulators on a Free Flying Space Robot, IEEE Control Systems, Vol., No. 3, 99, pp. 55. doi:0.09/ [] Y. Murotsu, K. Senda and M. Hayashi, Some Trajectory Control Schemes or Flexible Manipulators on a Free Flying Space Robot, Proceedings o International Conerence on Intelligent Robots and Systems, Raleigh, 99. [3] B. Samanta and S. Devasia, Modeling and Control o Flexible Manipulators Using Distributed Actuators: A Bond Graph Approach, Proceedings o the IEEE International Workshop on Intelligent Robots and Systems, 988. doi:0.09/iros [4] B. Fardanesh and J. Rastegar, A New Model Based Tracking Controller or Robot Manipulators Using the Trajectory Pattern Inverse Dynamics, Proceedings o the 99 IEEE International Conerence on Robotics and Automation, Nice, 99. doi:0.09/robot [5] F. Y. Wang and Y. Gao, Advanced Studies o Flexible Robotic Manipulators: Modeling, Design, Control and Application: Series in Intelligent Control and Intelligent Automation, Word Scientiic, New York, 003. [6] M. B. Leahy Jr., D. E. Bossert and P. V. Whalen, Robust ModelBased Control: An Experimental Case Study, Proceedings o the 990 IEEE International Conerence on Robotics and Automation, Cincinnati, Vol. 3, 990, pp [7] K. Fawaz, R. Merzouki and B. OuldBouamama, Model Based Real Time Monitoring or Collision Detection o an Industrial Robot, Mechatronics, Vol. 9, No. 5, 009, pp doi:0.06/j.mechatronics [8] S. K. Tso, L. Law, Y. Xu and H. Y. Shum, Implementing ModelBased VariableStructure Controllers or Robot Manipulators with Actuator Modeling, Proceedings o the 993 IEEE/IRSJ International Conerence on Intelligent Robots and Systems, Yokohama, 993. doi:0.09/iros [9] H. A. Zhu, C. L. Teo and G. S. Hong, A Robust ModelBased Controller or Robot Manipulators, Proceedings o the IEEE International Symposium on Industrial Electronics, Vol., 99, pp [0] Z. Qu, J. F. Dorsey, X. Zhang and D. M. Dawson, Ro Law, bust Control o Robots by the Computed Torque Systems and Control Letters, Vol. 6, No., 99, pp.53. doi:0.06/06769(9)9005a [] P. K. Khosla and T. Kanade, RealTime Implementation and Evaluation o ComputedTorque Schemes, IEEE Transactions on Robotics and Automation, Vol. 5, No., 989, pp doi:0.09/ [] P. M. Pathak, R. Prasanth Kumar, A. Mukherjee and A. Dasgupta, A Scheme or Robust Trajectory Control o Space Robots, Simulation Modeling Practice and Theory, Vol. 6, 008, pp doi:0.06/j.simpat [3] R. M. Berger, H. A. EIMaraghy and W. H. EIMaraghy, The Analysis o Simple Robots Using Bond Graphs, Journal o Manuacturing Systems, Vol. 9 No., 003, pp. 39, 003. [4] A. K. Samantaray and A. Mukherjee, Users Manual o SYMBOLS Shakti, HighTech Consultants, S.T.E.P., Indian Institute o Technology, Kharagpur, 006.