Imece2014‑37602

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Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014 November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-37602

FIVE BAR PLANAR MANIPULATOR SIMULATION AND ANALYSIS BY BOND GRAPH

Shengqi Jian Cheng Yin Faculty of Engineering and Applied Science, Faculty of Engineering and Applied Science, Memorial University of Newfoundland Memorial University of Newfoundland St. John’s, Newfoundland and Labrador, Canada St. John’s, Newfoundland and Labrador, Canada

Luc Rolland Lesley James Faculty of Engineering and Applied Science, Faculty of Engineering and Applied Science, Memorial University of Newfoundland Memorial University of Newfoundland St. John’s, Newfoundland and Labrador, Canada St. John’s, Newfoundland and Labrador, Canada

ABSTRACT entertainment devices in cinemas. James. E. Gwinnett firstly This work focuses on the bond graph modelling method designed and patented a spherical parallel mechanism. In 1954, and its application on multi-body system, especially on the Dr. Eric Gough designed and built the first octahedral hexapod five-bar parallel robot. Five-bar parallel robot is comprised of mechanism. It was used for tire testing and this mechanism was four arms, two revolute actuators and five revolute joints. This duplicated until present. Klaus Cappel later revised parallel paper adopts five-bar parallel robot in symmetric configuration mechanism as flight motion simulator [1]. Parallel mechanisms as simulation object. As it will be used as a pickup and placing compensate serial mechanisms' insufficient. It compromised a machine, its workspace is fixed on Cartesian coordinate. The more rigid structure with many advantages that serial robots do relationship between the two rotating angles and end effector's not have. Parallel robots own high stiffness, relatively large desire position is built by inverse kinematics. Bond graph is payload capacity and low inertia. However, it also has some used to describe moment, torque, velocity, angle relationships. drawbacks, such as limited workspace and nonlinear dynamics In this project, the dynamic performances between arms, leading to control hardships. motors at robot basement and end effector will be researched. Five bar planar manipulator is a relatively simple In this paper, an investigation about how to use bond graph to mechanism and its kinematics is explicit. However, its model DC (direct current) servo motor and an integrated characteristics are high speed, high accuracy, low inertia and motion control system is carried out. During a typical end high stiffness. It has 2 degree-of-freedom (DOF). For these effector point-point displacement, the torque change between reasons, it draws a lot of researchers’ attention. Some arms is plotted. Finally, 3-D animation experiment is prototypes and commercial products were made, such as conducted. Experiment results show that bond graph can 'double SCARA' RP-AH series offered by Mitsbishi Electrics, simulate robot dynamics performance without having to make a and DexTAR, five bar planar manipulator designed by ETS. large number of equations. It is able to simulate and solve five- With price decreasing, it has the potential to be widely used in bar kinematics problem in the process. production line for picking and placing tasks and elements assembly tasks. 1. INTRODUCTION In reference [2], It studies five bar manipulator bond graph The need for high speed and high accuracy has justified. model. While it is composed of four revolute joints, one The application of parallel robots in areas such as aerospace, prismatic joint and two actuators, main motor is driving crank precise machining, assembly line tends to be controllable close and another motor is driving linear power screw. In reference chain mechanism. These mechanisms attract more attention [3], Dr. Karnopp and Dr. Margolis studies five bar mechanism from researchers. bond graph model, however, it only researched building Parallel robot was introduced and researched decades ago. mechanism model without control or kinematics. This paper They were firstly applied as air plane simulators and simulation object is an implementary parallel robot system,

which is engineering related. Its workspace is Cartesian labeled as L3 and L4. Two active joints are connected with two coordinate. fixed actuators, labeled as A1 and A2. Three passive joints are Nowadays, mechanism dynamic analysis mainly adopts the labeled as B1, B2 and P. It contains two DOFs on XY plane. closed vector polygon method, the complex number method, or End effector is settled at P joint. matrix method to build mechanism dynamic description. Then Inverse kinematics is used to derive the actuator angles use methods of vector mechanics or analytical mechanics, such from end effector position. Forward kinematics is used to as Lagrange–d'Alembert principle, Hamilton principle, calculate end effector position from actuator angles. Wittenburg method or Kane method, to solve the dynamic problems. Adopting these methods, the whole system dynamic functions must be built. As for complex rigid coupling mechanism, especially multi-loop, multi-bar mechanisms, whole system dynamic functions are difficult to be built. Bond graph is a method to describe system power exchange, transmission, storage and dissipation. By introducing real parameters and variables, it can objectively show the relationship between all variables and get explicit state equations. This method was firstly raised by Dr. Paynter [4] in 1960. Then Dr. Karnopp and Dr. Rosenberg used bond graph simulating different mechanisms, electrical devices and hydraulic mechanism. As systems have power transmission, they are qualified to have bond graph model [5].

Fig. 2: Sketch of five-bar robot (in - + configuration)

As figure 3 shown, if one point is defined as the desired point on the XY plane, usually five-bar parallel robot can have four configurations to reach that point, as + +,+ -, - +,- - [9]. In this work, the - +configuration is chosen. Fig. 1: DexTAR robot, ETS, Montreal [6]

Figure 1 is a typical five bar planar robot, DexTAR (photo courtesy of Dr. Bonev). It is designed by École de technologie supérieure (ÉTS), Montreal [6]. Its main task is to proceed with object palletization. So how to derive equation between end effector desired point and desired motor angle, how to design controller to send motors to set points, and how to build relationship between arms and joints by bond graph, these questions are very critical to this project and all waiting to be solved. (a) - + configuration (b) - - configuration

2. INVERSE KINEMATICS As figure 2 showed, the planar five-bar parallel mechanism compromises two actuators, four arms and five revolute joints. The workspace, parallel singularities and serial singularities are all influenced by four arm lengths ratio and distance offset between two actuators [7]. In order to at best avoid singularity points, symmetric configuration is chosen with two equal long forearms and two equal short arms. Symmetric structure is chosen in proposed design approach also due to the benefits of workspace symmetry and simpler kinematics equations [8]. (c) + + configuration (d) + - configuration Proximal arms are labeled as L1 and L2. Distal arms are Fig. 3: Five-bar four configurations

In this section, hereafter kinematic derivations obey ground Using inverse tangent value to calculate five-bar reference frame shown in figure 2. The end effector's position is inverse kinematics, however, this approach is not accurate defined as enough. During the calculation, some points on XY plane can Px ,y (1) lead to equaling to infinity. So here is chosen instead of , it will not turn to infinity resulting program In the X-Y base reference coordinate, and collapse. positions are shown in (5)(6). and positions are shown Parallel robot’s inverse kinematics problem is easier to be as following (7)(8). , is the length of proximal arms and solved than forward kinematics problem. Multiple solutions they all equal to . , is the length of distal and they all might occur. Because reaching the same point on X-Y plane equal to . is motor one’s angle. is motor two's angle. leads to different configurations. In this case, four is half distance between two motors and it is set as . configurations are caused by multiple solutions. r |AB| |AB| L L (2) 3. BOND GRAPH MODELING r |BP| |BP| L L (3) Planar mechanism is simulated using bond graph by other | | r (4) researchers. In Karnopp and Margolis paper, they described planar mechanism referring to bond graph [3]. In Ashraf’s paper, although he focused on simulate joint effects on A r,0 (5) mechanism, there are still many methods to solve bond graph A r,0 (6) problems in it [10]. A bond graph model is particularly useful to describe systems in which a variety of elements in different B L cos θ , L sin θ (7) energy domains interact [13]. Robot system is constituted by mechanical, electrical, magnetic components. B L cos θ , L sin θ (8) Model of Actuators (9) r r 250 As figure 2 shows, it is the sketch of a DC motor. It

r 25 (10) comprises inductance , resistance , motor constant n, rotation inertia J, bearing friction b and gear header constant m. and are unknown variables. Supposing guide lines and , and was devided into two angles, and , by . Same as , it is devided into and by . From geometry knowledge, and are motor desired angles. Point P position is known variable. Supposing guide lines and , The lengths of and are provided in 1112. The cosine values of Fig. 4: DC motor sketch , are shown in 1314. |PA| x r y (11) For DC motor, reference [11] derived the following transformation function. |PA| x r y (12) DC motor relative parameters are shown table 1. Parameters Amount Units || cos θ (13) || n 253 N. m/A L 121 μH cos θ (14) J 98.6 gcm || R 0.346 Ω b 310 Ns/m So from figure 2, motor one’s desired angle can be derived, m 0.00641 (1:156) Tab. 1: DC motor specifications θ cos θ cos θ (15)

As a symmetric mechanism, and cosine values are derived in 1617. Motor two's desired angle value is in equation 18. || cos θ (16) ||

cos θ (17) || Fig. 5: DC motor model θ cos θ cos θ (18) Bond Graph of Five-Bar Robot

Karnopp and Rosenberg made multiple rigid body model. In this arm one (proximal arm) sub-model, It gives out Eulerian Junction Structure (EJS) model [12]. In AG BG = 125mm (20) this work, robot arm is a rigid body. This model is built based on body fixed frame and center mass. In this paper, as five-bar V V AGsinθ ω 0 (21) workspace is XY coordinate, gravity influence is less than XZ or YZ coordinate. So its center mass is assumed as 100 grams. V V AGcosθ ω 0 (22) Through frame transformation matrix , variables in body V V BGsinθ ω (23) frame can be transformed to ground frame. ω ω (19) V V BGcosθ ω (24) Every arm is an EJS model. Its rotation referring ground frame and it can be shown in matrix. Bond graph is easy to show the relationship between velocity and angular velocity. It is a suitable method to model planar manipulator. In this project, all variables are referred to the same ground coordinate. The bond graph sub-model of arm one is shown in figure 6. In this model, J_G1 is an continuous power storage element and represents arm rotation inertia referring to Z axis. Another two M1x and M1y stand for beam mass. Since this mechanism workspace is XOY Cartesian coordinate, 100 grams are manually valued to this two elements. Same as other arms. Where is the x direction velocity of arm one on point , and is the y direction velocity of arm one on point . Where is the x direction velocity of arm one on point , and is the y direction velocity of arm one on point . is the x direction velocity of arm one on mass center G. is the y direction velocity of arm one on mass center G. For the adoption of symmetrical configuration, arm two

's model can be built same method. Point and point Fig. 7: Bond graph of arm three BP are fixed on ground, thus equations (19)(20) both equal to zero. Arm three (distal arm) is EJS model and it is shown in figure 7. In this sub-model, Where is the x direction velocity of arm one on point , and is the y direction velocity of arm one on point . Where is the x direction velocity of arm one on point , and is the y direction velocity of arm one on point . is the x direction velocity of arm one on mass center G. is the y direction velocity of arm one on mass center G. All these velocity variables are referring to ground frame. For the adoption of symmetrical configuration, arm four 's model can be built same method. PG BG 125mm (25)

V V BGsinθ ω (26)

V V BGcosθ ω (27)

V V PGsinθ ω (28)

V V PGcosθ ω (29) Parasitic Element It is relatively common to encounter derivative causality in the mechanical part of the system due to the assumption that Fig. 6: Bond graph of arm one AB inertia elements such as rigid bodies are connected rigidly [13].

To eliminate derivative causality, parasitic element is built to Arm Principle Units isolate inertial components from the rest system. Figure 8 is the Moments of inertia 2 ports parasitic element sub-model. It can be treated as a set of #1 (Proximal) 0.013562 ∙ parallel spring and damper with large values. In this model #2 (Proximal) 0.012068 ∙ and , which stand for damping #3 (Distal) 0.013556 ∙ #4 (Distal) 0.012881 and compliance. ∙ Tab. 2: Principle moments of inertia Controller

Fig. 8: Bond graph isolator

4. SIMULATION RESULTS In this section, simulation results are shown. As a whole robotic system, controllers is studied. Dynamic results will be shown in graphs. As figure 9, Integral bond graph model is built in 20-sim. Parameters such as arms rotation inertia in 20-sim are created in Solidworks Fig. 10: PID controller The properties of this five-bar robot are needed. Referred In 20-sim, PID controller structure is shown in figure 10. to Z axis, four bars has four principle moments of inertia are This controller has four parameters including proportional gain shown in table 2. Kp, derivative time constant Td, integral time constant Ti, and tameness constant beta. In this model, two PID controllers are implemented. They are used to control 24V DC motors. So in integral model, after PID controller there are two limits ranging from -24 to 24. Trial and error method is implemented to gain robust performance. PID Tuned parameters are offered in table 3. Angle and position control results are in figure 11 and table 4. Parameters Amount Units Kp 2050 second TauD 1970 second Beta 0.95 TauI 2150 second Tab. 3: PID Tuned Parameters (Same On Two Controllers)

Theta1 & Desired Theta1 X Position & Desired X Position

1.55 Desired_Theta_1 0.25 Desired_X_Position {meter} Theta_1 {rad} X_Position {meter} 1.5 0.2

1.45 0.15

1.4 0.1

1.35 0.05

1.3 0 0 10203040506070 0 1020304050607080 time {s} time {s} Theta2 & Desired Theta2 Y Position & Desired Y Position 1.5 0.5 Desired_Theta_2 Desired_Y_Position {meter} 1 Theta_2 0.4 Y_Position {meter}

0.5 0.3 0 0.2 -0.5 0.1 -1 0 -1.5 0 1020304050607080 0 10203040506070 time {s} time {s} Fig. 9: Bond graph of five-bar parallel robot Fig. 11: Angles and positions controlled results

Theta 1 Theta 2 X Y Arms Maximum Torque Minimum (rad) (rad) Position Position (N.m) Torque (N.m) (meter) (meter) #1 (Proximal Arm) -0.38278 0.39580 Initial 1.57080 1.57080 0 0.498675 #2 (Proximal Arm) 1.18674 -1.16695 Final 1.31766 -1.42768 0.09960 -0.000475 #3 (Distal Arm) 0.00800 0.14354 Desired 1.31812 -1.42023 0.1 0 #4 (Distal Arm) 0.31577 -0.33532 Difference 0.00046 0.00745 0.00040 0.000475 Tab. 6: Motors max. and min. torque and angular velocity

Errors 0.182% 0.249% 0.4% 0.095% Arm1 Force Curve (Proximal) Arm2 Force Curve (Proximal) Tab. 4: Angles and positions controlled results 1 Arm_1_Force_x {N} 1 Arm_2_Force_x {N} Typical point to point process, it needs two inputs, desired Arm_1_Force_y {N} Arm_2_Force_y {N} X position and desired Y position (if this desired position is 0.5 0.5 reachable in - + configuration workspace). Then the algorithm 0 0 will automatically calculate out all parameters and variables. -0.5 Motors will drive this system to push end effector to set point. -0.5 -1 Dynamics -1

DC Motor 1 & DC Motor 2 Torque Curves DC Motor 1 & DC Motor 2 Angular Velocity Curves 0 1020304050607080 10 20 30 40 50 60 70 80 0 time {s} time {s} DCMotor_1_Torque {N.m} DCMotor_1_Angular_Velocity {rad/s} DCMotor_2_Torque {N.m} DCMotor_2_Angular_Velocity {rad/s} 1 Arm3 Force Curve (Distal) Arm4 Force Curve (Distal) 2 -0.02 1 Arm_3_Force_x {N} Arm_4_Force_x {N} Arm_3_Force_y {N} 1.5 Arm_4_Force_y {N} 0.5 1 0.5 0.5 -0.04 0 0 0 -0.5

-0.06 -0.5 -1

-0.5 -1.5 -1 -2 0 1020304050607080 0 102030405060708090 -0.08 time {s} time {s} -1 Fig. 14: Force applied on proximal and distal arms -0.1 02468100 102030405060708090100 time {s} time {s} Arm Max Max Min Min Max Min Fig. 12: Motors torque and angular velocity Force Force Force Force Force Force X (N) Y (N) X (N) Y (N) (absol (absol DC Maximum Minimum Maximum Minimum ute) ute) Motor Torque Torque Angular Angular #1 1.19210 0.30232 -1.22862 -0.20836 1.2298 1.2462 (N.m) (N.m) Velocity Velocity #2 1.34251 1.34251 -1.31690 -1.31690 1.8986 1.8624 (rad/s) (rad/s) #3 1.17007 -0.36887 -1.10840 0.28881 1.2268 1.1454 #1 0.03772 -0.24814 0 -0.01691 #4 2.10682 0.02677 -2.10074 0.13606 2.1070 2.1051 #2 1.24112 -1.36744 0 -0.09695 Tab. 7: Force applied on proximal and distal arms Tab. 5: Motors max. and min. torque and angular velocity

Arm1 & Arm2 Torque Curves (Proximal) Arm3 & Arm4 Torque Curves (Distal) In figure 12 and table 5, motors torque and angular

Arm_1_Torque {N.m} 0.3 Arm_3_Torque {N.m} velocity output are important criteria for robot design. Peak Arm_2_Torque {N.m} Arm_4_Torque {N.m} 1 torque is no more than 2 N.m, which is available.

0.2 In figure 13 and table 6, the torque changing during the

0.5 process are provided. In reality application, these plots can be 0.1 used to analyze the material properties and to select optimal configuration. 0 0 In figure 14 and table 7, the force changing during the point to point process is shown. The absolute max. and min. -0.1

-0.5 values are calculated. These results can be used to optimize

-0.2 robot arm strength. In order to optimize model objectiveness, 3D animation -1 -0.3 was made to testify this whole process. The end effector can

02468100246810 successfully reach the setting point. Since the setting point is time {s} time {s} within - + configuration workspace, during the whole moving Fig. 13: Toque applied on arms

process, five-bar manipulator has not changed its original ACKNOWLEDGMENT configuration. The Z in figure 15 is Z axis. Thanks to Dr. Rideout's course ENGI 9496 about bond graph and his selfless help.

REFERENCES [1] Bonev I.A. The true origins of parallel robots. January, 24, 2003, http://www.parallemic.org/Reviews/Review007.html. [2] Zhang K, Huang T, Wang C. Kinematics and dynamics analysis of a planar hybrid five bar actuator[C]//Control, Automation, Robotics and Vision, 2006. ICARCV'06. 9th International Conference on. IEEE, 2006: 1-6. [3] Karnopp D, Margolis D. Analysis and simulation of planar mechanism (a) (b) systems using bond graphs. J Mech Des, Trans ASME. 1979;101(2):187- 191. [4] Paynter, Henry M., Analysis and design of engineering systems, The M.I.T. Press, Boston, 1961 ISBN 0-262-16004-8. [5] Karnopp, Dean C., Margolis, Donald L., Rosenberg, Ronald C., 1990: System dynamics: a unified approach, Wiley, ISBN 0-471-62171-4. [6] Campos L, Bourbonnais F, Bonev IA, Bigras P. Development of a five- bar parallel robot with large workspace. . 2010;2:917-922. [7] Jesus Cervantes-Sanchez J, Cesar Hernandez-Rodriguez J, Angeles J. On the kinematic design of the 5R planar, symmetric manipulator.Mechanism and Machine Theory. 2001;36(11-12):1301- 1313. http://dx.doi.org/10.1016/S0094-114X(01)00053- (c) (d) 2.doi:10.1016/S0094-114X(01)00053-2. Fig. 15: 3D animation [8] Sun S, Cheung JWF, Lou Y. A study on five-bar manipulators for semiconductor packaging applications. . 2007:1811-1816. [9] MacHo E, Altuzarra O, Pinto C, Hernandez A. Workspaces associated to CONCLUSION assembly modes of the 5R planar parallel manipulator. Robotica. This work shows bond graph is a good modeling tool to 2008;26(3):395-403. simulate closed loop mechanisms. Velocity, force, torque and [10] Zeid A. Bond graph modeling of planar mechanisms with realistic joint effects. Journal of Dynamic Systems, Measurement and Control, angle velocity are all can be presented by bond graph model. Transactions of the ASME. 1989;111(1):15-23. This work also demonstrates that bond graph model is not only [11] Karnopp D. Energetically consistent bond graph models in able to solve dynamic problems but also kinematic problems of electromechanical energy conversion. Journal of the Franklin Institute. planar manipulator. 1990;327(5):677-686. http://dx.doi.org/10.1016/0016-0032(90)90076-U. doi: 10.1016/0016-0032(90)90076-U. As for the model application, it can be used to evaluate planar [12] D. Karnopp and R. C. Rosenberg, Analysis and Simulation of Multiport Systems - The Bond Graph Approach to Physical System Dynamics. MIT manipulator's workspace, to check material's durability, to Press, Cambridge, MA, 1968. testify actuator's performance, to testify the feasibility of new [13] Karnopp D. Approach to derivative causality in bond graph models of algorithm. mechanical systems. Journal of the Franklin Institute. 1992;329(1):65- 75.http://dx.doi.org/10.1016/0016-0032(92)90096-Y. doi: 10.1016/0016- 0032(92)90096-Y.