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- 1. Proceedings of ESDA2002: 6th Biennial Conference on Engineering Systems Design and Analysis Istanbul, Turkey, July 8-11, 2002 ESDA2002/APM-131 A CYLINDRICAL ROBOT SIMULATION SOFTWARE FOR ROBOTICS EDUCATION Cenk Yuksel Osman S. Turkay Department of Mechanical Engineering Department of Mechanical Engineering Bogazici University Bogazici University Bebek 80815, Istanbul, TURKEY Bebek 80815, Istanbul, TURKEY Email: cenky@hotmail.com Email: turkay@boun.edu.tr ABSTRACT 2.1 DIRECT KINEMAT ICS In this paper the simulation software of a cylindrical robot The aim of direct kinematics is to determine the end effector arm developed for educational purpose that can be used in an position and orientation (outputs) as a function of the joint academic environment, in seminars or for continuous education variables (inputs) (Sciavicco L. and Siciliano, B., 2000). purpose is presented. The software developed in Matlab® A general manipulator may be thought of as a set of links consists of direct kinematics, inverse kinematics, trajectory connected in a chain by joints. For the purpose of manipulator planning and PID-computed torque control modules. Visual kinematics, a link is considered as rigid body, which defines the animations have been incorporated into simulations for better relationship between two adjacent joint axes on the manipulator. perception of the learners. PID-computed torque controller is The coordinate frames and the link parameters are shown in Fig. developed using Matlab -Simulink block diagram approach. The 1. The parameters are summarized below: impressions of the Robotics course students at Bogazici a i-1 = The normal distance from z i-1 to zi measured along xi-1 University have been positive in understanding the αi-1 = The angle between zi-1 and zi measured about xi-1 fundamental concepts of robot mechanics. d i = The distance from xi-1 to xi measured along zi θ i = The angle between xi-1 to xi measured about zi 1 INTRODUCTION These parameters are defined according to Denavit- The main advantages of a virtual educational robot over the Hartenberg convention [Craig, J.J., 1989) and are generally physical one are that it is easier for demonstration, cheaper, accepted as a standard notation. They are used to define the portable and convenient for multi-users. relative positions of two consecutive links so that the The developed package having visual animation is very mechanics of the whole is system consisting of a chain of links suitable as an educational tool in an introductory Robotics can be undertaken. course in undergraduate or graduate level. 2 MODELING Robot arm kinematics deals with the geometry of robot arm motion with respect to a fixed-reference coordinate system as a function of time without considering forces/moments that cause the motion. Thus, it deals with the spatial configuration of the robot as a function of time, in particular the relations between the joint-variable space and the position and orientation of a robot arm. The kinematics problem usually consist of two sub- problems; the direct and inverse kinematics problems. 1 Copyright © 2002 by ASME
- 2. The link variables associated to the frames in Fig. 2, which are known as the Denavit-Hartenberg parameters, are shown in Table 1. (Yuksel, C., 2002). Table 1. Denavit - Hartenberg parameters Link i αi ai di θi 1 0 0 0 θ1 2 0 0 d2 0 3 π/ 2 100 d3 0 4 0 0 100 0 Fig. 1. Link relationships (Craig,J.J. 1989) The corresponding homogeneous link transformations are For a revolute joint, θ i is the joint variable and the other obtained by successively inserting the Denavit-Hartenberg parameters from Table 1 into Eqn. (1). To describe the position three are constants. For a prismatic joint, di is the joint variable and orientation of the manipulator’s tip with respect to the base and θi, a i-1 and αi-1 are constants, respectively. 0’th frame the following transformation chain is used: Coordinates Transformations. It can be shown that the coordinates of a vector between consecutive frames can be 0 T = 0 T 1T 2T 3T (2) 4 1 2 3 4 transformed using the following transformation matrix (Craig, J.J., 1989): The result is; c θi − sθi 0 α i −1 c θ1 0 sθ1 100cθ1 + (100 + d 3 )sθ1 sθ cα cθ i cα i −1 − sα i −1 − sαi −1d i sθ1 0 cθ1 100sθ1 − (100 + d3 ) cθ1 (3) i i −1 iT = 4T = i −1 0 0 sθi sα i −1 sθ i sα i −1 cα i −1 c αi −1 d i 1 0 d2 0 0 0 1 0 0 0 1 (1) 2.2 INVERSE KINEMATICS The inverse kinematics problem consists of the where s and c denote sine and cosine, respectively. determination of the joint variables necessary to bring the end- Figure 2 shows the three degrees of freedom of cylindrical effector to a desired posture. manipulator structure and the coordinate frame attachments The inverse kinematics computation can be carried out step used in this work where the frames and parameters are assigned by step as follows (Riley, D.L., 1986), as described previously. 100 z2 0 4 T = 0 T 1T 2T 3T 1 2 3 4 (4) x2 O2 y 3 y4 x3 x4 0 1T −1 0T = 1T 2T 3T 4 2 3 4 (5) 100 z4,3 − cθ1 nx + sθ1 n y cθ1 ox + s θ1 o y cθ1 a x + s θ1a y cθ1 px + sθ1 p y mm d3 O32 O4 − sθ n + cθ n − s θ1o x + cθ1 oy − s θ1 a x+ + cθ1 a y − sθ1 p x + cθ1 p y 1 x 1 y nz oz az pz d2 0 0 0 1 z0,1 x0 1 0 0 100 O0, O1 0 0 − 1 − 100 − d3 (6) x1 = 0 1 0 d2 θ1 0 0 0 1 Fig. 2. Coordinate frame attachment of cylindrical manipulator It is seen from Eqn. (6) that the position vector column, i.e., the fourth column can be used to solve for θ 1, d 2 and d 3. 2 Copyright © 2002 by ASME
- 3. ± p 2 + p 2 − (100) 2 J + m2 d 3 2 0 0 θ 2 d 3m2 θd 3 && && 0 n1 θ1 = arctan 2 x y + arctan 2 p y (7) && 100 px 0 ( m1 + m2 ) 0 d 2 + 0 + ( m1 + m2 ) g = f 2 0 0 m2 d 3 − m2 d 3θ 2 && & 0 f3 d 2 = pz (8) (14) d 3 = −100 + p 2 + p 2 − ( 100) 2 x y (9) 2.4 TRAJECTORY PLANNING The goal of trajectory planning is to generate the reference 2.3 DYNAMICS inputs to the motion control system that ensures the The goal of this section is to obtain the dynamic model of manipulator to execute the planned trajectories. The user the cylindrical manipulator that describes the relationship typically specifies a number of parameters to describe the between the joint actuator torques and the motion of the desired trajectory. Planning consists of generating a time structure. The Lagrange-Euler formulation is used for this sequence of values obtained by polynomial function derivation. The Lagrangian is defined as: interpolating the desired trajectory (Sciavicco L. and B. Siciliano, 2000). L=K– P (10) In this study trapezoidal velocity profile generation technique will be used. where K and P are respectively the total kinetic energy and The case examined is one where only the initial and final potential energy of the system. points on the path and the traveling time are specified. This The Lagrange’s equations of motion for a conservative method is called as point-to-point motion case. The manipulator system are obtained by: has to move from an initial to a final joint configuration in a given time tf. In this case, the actual path is of no concern. The d ∂L ∂L (11) algorithm should generate a trajectory, which, in respect to the − =τ dt ∂q ∂q & above general requirements, is also capable to optimize joint variables and their velocities and accelerations when the joint is The dynamic model of cylindrical manipulator is illustrated moved from one position to another. in Figure 3. This model shows link masses, inertial moments and In the case of a trapezoidal velocity profile, the assigned motion capabilities of the cylindrical manipulator. trajectory imposes a constant acceleration in the start phase, a cruise velocity, and a constant deceleration in the arrival phase. d3 Th following equations are the mathematical descriptions of m1 m2 trapezoidal velocity profile, and are used in the simulation code. 1 qi + q c t 2 && 0 ≤ t ≤ tc g 2 (15) d2 J q (t ) = q i + q c tc (t − tc / 2 ) && tc < t ≤ t f − t c 1 t f − tc < t ≤ t f q f − q c (t f − t ) 2 && θ 2 2.5 COMPUTED TORQUE CONTROL METHOD Figure 3. Dynamic model The problem of manipulator control is to find the time behavior of the forces and torques to be delivered by the joint Using this model, we obtain the following Lagrangian actuators so as to ensure the execution of the reference equality: trajectories. PID-computed torque control, which is a well- known control scheme, is considered for the simulation (Lewis, Jθ + 2d 3m2θ&& 3 + m2 d 3 θ && d 2 && F. L., et. al., 1993). d ∂L ∂ L && && (12) − = m d + m1 d 2 + (m1 + m2 ) g The general robot arm dynamics is given by the Eqn. (13). dt ∂ q ∂q 2 2 & && & m2 d 3 − m 2d 3θ 2 The PID-computed torque scheme could be derived as, τ = M ( q)( q d + Kv e + K p e + K i ε ) + N ( q , q && & &) (16) Equation (12) can be arranged in matrix form as, M (q )q + V ( q, q) + G(q ) = τ && & (13) The block diagram representation is presented in Fig.4. 3 Copyright © 2002 by ASME
- 4. Figure 4. PID Computed Torque Control 3 SIMULATION SOFTWARE The simulation and visual animation software are developed using the mathematical models of the previous section and the Matlab software version 5.1. Figure 5. Main GUI (Graphical User Interface) 3.1 SOFTWARE STRUCTURE The desired module can be selected for the execution. The software is composed of four main modules and three There is also a mini information window at the bottom of this supplementary functions. The main modules are: figure, which informs the user about the executed module. · Direct Kinematics · Inverse Kinematics 3.2 MAIN MODULES · Trajectory Planning In this section, the structures of the main modules are · PID Computed Torque Control described. The supplementary modules are developed for an effective Direct Kinematics module is based on the mathematical usage. These are Reset, Workspace Bounds, Close Function derivations obtained in Section 2.1. In the GUI of this module, functions. the desired input joint variables (theta1, d2 and d3) are Software is run with the execution of quot;robotedu.mquot; code and requested in order to simulate and animate the resulting motion. then it executes quot;main.mquot; file, which is the main GUI (Graphical This module processes the given data and then passes the User Interface). The main GUI interface is shown in the Fig. 5. results to trajectory matrix generation module. The trajectory generation file quot;TP.mquot; constructs a separate vector for all three joint variables. These vectors show the changes of the joint variables until the motion ends. While constructing these vectors, the software code calculates the position values according to trapezoidal velocity profile equations explained in Section 2.4. The computed joint variables vectors and the transformation matrices of Section 2.1 are used in the animation code to visualize the motion of the manipulator in 3-D space. During this animation, path of the end-effector path is traced and displayed on the screen. Also, the user can see the initial and final end-effector positions by means of the coordinate frame axes attached to the links. At the end of the animation, the user can change the view angle at any perspective view that enables the user to investigate the realized motion, path and manipulator structure within t e h kinematics context. With the execution of the “Inverse Kinematics” module, the related GUI requests the desired end-effector position coordinates px, py and pz , and then it calculates the necessary 4 Copyright © 2002 by ASME
- 5. joint variables using the inverse kinematics equations of Section position, velocity and acceleration matrices for all three joint 2.2. The code works in a similar way to direct kinematics module variables as shown in Fig. 6. Then the code combines these data and the visual animation occurs in the same way. The 3 -D into the position, velocit y and acceleration matrices of qd, qdp, animation code developed in this project is based on the simple qdpp that include 4 columns and 209 rows (see Fig.4). The first version of the 3-D code of reference (Piner, Z. and Hansel, A., columns of the time values are the same for all of the three 1998). matrices. After these matrices are constructed, the code of When the third trajectory -planning module is executed, the quot;contdata.mquot; defines the mass and inertia values for the robot related GUI requests the desired joint variables as in direct arm and also creates Kp, Ki and Kv diagonal PID control kinematics module and also asks for the acceleration and motion matrices. Then the simulation model quot;cont.mdlquot; starts to operate time to describe the trapezoidal velocity profile. The difference and uses these data as the main input for Simulink simulation of this module from the previous ones is simply the request of execution. acceleration and time as interactive input variables whereas; in Figure 7 is a sample error gra ph between the reference the direct and inverse kinematics animation modules a fixed positions and the actual positions of the three links obtained for acceleration value of 5 mm/sec2 and a fixed motion time of 20 the PID computed-torque controller values of: seconds are used. Figure 6 shows a user described sample K v = diag {0 .1} , K p = diag{ } , K i = diag{1} 10 trajectory curve obtained by the execution of trajectory module for an end-effector speed of 5 mm/sec 2 and duration of 20 seconds together with θ 1=180o, d2=200mm and d3=200mm joint variable inputs. The resulting curves of the joint variable positions and their first and second derivatives as a function of time are seen in this figure. q : θ [ degree ] q : d [ mm ] q : d [ mm ] 1 1 2 2 3 3 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 q′ q′ q′ 1 2 3 6 6 6 4 4 4 2 2 2 Figure 7. Sample error graph 0 0 0 0 5 q′′ 10 15 20 0 5 q′′ 10 15 20 0 5 q′′ 10 15 20 CONCLUSIONS 1 2 3 5 5 5 The direct and inverse kinematics equations, the trajectory planning technique and the PID computed-torque control method of a cylindrical type three-link manipulator has been 0 0 0 presented. A special interactive software has been developed in Matlab environment to simulate and animate the robot for -5 -5 -5 education purpose. The software has been used in teaching a 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 graduate level robot mechanics course. The feedback of the students has been encouraging. Figure 6. Sample trajectory module graphics output This software is open to development for other manipulator Upon the execution of the PID-computed control module types. Including the gripper kinematics and animation can from the main GUI in Figure 5, quot;contstart.mquot; code runs and ameliorate the kinematics module. The dynamics and control loads the trajectory information into Matlab workspace. The module can be improved by including the motor dynamics and user can change the default trajectory data using the M file friction effects. Also the motion of the links can be scaled to editor before executing this mo dule. The default trajectory data match the motion of an actual robot. The virtual robot needs includes an end-effector speed of 5 mm/sec2 and duration of 20 some more visual elaboration concerning shape and dimension seconds together with θ1=180o, d2=200mm and d3=200mm joint of links and the attached coordinate frames. variable inputs. The trajectory matrix code creates the desired 5 Copyright © 2002 by ASME
- 6. The use of a virtual robot has proved to be a valuable tool in robotics education for its cost effectiveness, parametric investigation, portability and multi-end user facility. REFERENCES Craig, J. J., “Introduction to Robotics: Mechanics and Control”, Addison-Wesley, Mass, 1989. Lewis, F. L., Abdallah, C. T. and Dawson, D. M., “Control of Robot Manipulators”, MacMillan Pub. Co., New York, 1993. Piner, Z. and Hansel, A., “Robot Animation of Linear Trajectory with Real Time Error Plot”, http://www.mathworks.com/matlabcentral/fileexchange/Files.jspf ileId=156. Riley, D. L., “Design Modeling, Trajectory Calculation and General Calibration of a Three Degrees of Freedom Cylindrical Coordinate System Robot”, M.S. Thesis, Washington State University, 1986. Sciavicco L. and Siciliano, B., “Modeling and Control of Robot Manipulators”, Springer, New York, 2000. Yuksel, C., “Development of a Simulation Software for Robotics Education”, M.S. Thesis, Bogazici University, 2002. 6 Copyright © 2002 by ASME

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