ManDy: Tool for Fast Development of Open Chain Multibody Systems

Transcription

1 ManDy: Tool for Fast Development of Open Chain Multibody ystems W. Weber,. Rothenbücher Fachhochschule Darmstadt University of Applied ciences chöfferstrasse 3, D Darmstadt Phone: , Fax: , Abstract - The tool ManDy (Manipulator Dynamics) is a consistent software system for programming, simulation and visualization of robot arms or other multi-body systems with open kinematic chains. ManDy is written in Matlab, using MEX-Files for time consuming algorithms. A 3D- animation program is automatically generated in VRML (Virtual Reality Modelling Language). Objective of ManDy is the fast development of multi-axes systems like robot arms and tool machines. It enables the user to design easily a solution in a short time for a particular application and to present alternative solutions to the customers. I. INTRODUCTION In the field of robotics many off-line programming systems are offered for simulation of robot motion in 3D space. Engineers who develop robotic systems can study robotic applications without the physical need and expense for a prototype setup. One class of tools are tailored to robots of a particular robotic manufacturer [ABB, 2005, KUKA, 2005] other tools are open for various kinematics of different manufactures [Delmia, 2005, Easy-Rob, 2005]. While the first class is not suitable to design new kinematics, the second class of tools demand mostly CAD models of the robot arms and uses sophisticated programming languages to program the whole robot process in a robot cell. Additionally often reach verification, collision detection, cycle time validation and so forth are realized. But in most cases a possibility to estimate the control errors via modeling of the dynamics and the feedback control system is missing. Those tools are developed to simulate an application close to reality on the base of an existing robot not for a fast and rough design of a robot system. More and more multi-axes machines are used as manipulators, which are well fitted to a class of application and offer a flexible and reasonable solution. Often it is an advantage in development process to design the kinematical structure of such a special manipulator in a rough manner without details of the mechanical design. A tool to define the multi-axes machine in an easy way and in short time is needed. If the development engineer can additional program and visualize the desired motion for a certain application he can present first solutions to a customer or colleagues in an early stage of development. In this way it can be checked whether the proposed kinematical structure is able to fulfill the requirements of the application and alternative solution can be developed in fast time. The next stage in development of a new or modified multi-axes system on the base of such a tool can be the definition of the dynamic parameters like masses of the links, inertias, center point of masses and the relevant parameters of the actuator system. Furthermore the control structure and parameters are defined and now it can be tested whether the actuators are able to achieve the desired accelerations and velocities of the axes and whether the predicted control errors are in the demanded region. The development of ManDy is a first step to achieve the mentioned objectives with a reasonable tool. II. KINEMATIC TRUCTURE, PROGRAMMING AND VIUALIZATION A. Overview about the workflow Fig. 1 shows the workflow of design process. The user defines the kinematical structure of a serial or open chain multi-axes system. Up to ten links and joints can be defined with help of a special menu. Each joint can be a rotational or prismatic one. If the user invokes the programming interface in connection with the name of an already defined kinematical scheme of a manipulator. Definition of the kinematics Automatic generation of VRML files Automatic adaption of programming interfacet Off-line programming Fig. 1. Work-flow of programming and visualization Page 287

2 The programming interface and motion commands are automatically adapted to the actual kinematical structure. Now the user has the possibility to generate motion commands with an easy and menu-based language. Alternatively he can invoke and modify yet available programs. Mandy offers the feature to animate the desired motion. The model and the motion are created automatically on the base of the user defined kinematical structure and the motion program. B. Defining of the kinematical structure The user can define or modify the kinematical structure by choosing robot in the menu list. It can be chosen an existing description for modification or one can create a new kinematical structure. Fig. 2 shows an example for a six joint robot. First the number of joints and the (desired) limits of velocity and acceleration of the robot tool center point as well as the limits of velocity and acceleration to change the orientation of the robot tool in respect to the non moving base frame must be fixed. The kinematical structure of the manipulator arm has to be defined via the well-known Denavit-Hartenberg convention [Weber, 2002], [ciavicco and iciliano, 2000]. Additionally to the four Denavit-Hartenberg parameters,, a and d the limits of joint coordinates, joint velocities and joint accelerations have to be determined. The user has to define whether the joint is a revolute or prismatic one (Fig. 2). If the user intends to use the robot for simulation of dynamics too (see chapter IV), he has to put a tick in the respective button. A set of parameters for a certain robot arm can be given a name and saved. Fig. 2. Menu for definition of the kinematical parameters TABLE I COMMAND OF THE MANDY PROGRAMMING INTERFACE command description PTP Point-To-Point path, absolute target positions PTPREL Point-To-Point path, relative target position PEEDPTP velocity of each joint in percent of maximum ACCELPTP acceleration of each joint in percent of maximum LIN Cartesian straight-line motion, absolute target LINREL Cartesian straight-line motion, relative target CIRC circular path, absolute target CIRCREL circular path, relative target PEEDCP Cartesian velocity in percent of maximum ACCELCP Cartesian acceleration in percent of maximum WAIT waiting time before next command is executed COMMENT comment C. Off-line programming interface The off-line programming environment of ManDy has the following tasks: 1. Creation and editing of motion programs 2. Translation of programs with interpreter 3. Draw of graphs for a reference path The user invokes the off-line programming environment via the corresponding menu point. He must choose a robot arm, which he had described before. Then the interface is fitted to the kinematical structure of this robot arm by Matlab programs of ManDy. The number of parameters that have to be given is also adapted to the actual manipulator. Like this the number of parameters for joint related motion commands are adopted to the number of joints and it will be distinguished whether a joint is a revolute or prismatic one. The programming language consists only of a few important motion commands (see Table 1) that can be selected by a pop-up menu (see Fig. 3). The target location can give in joint coordinates or in Cartesian coordinates of the robot end effector (operational space). For trajectory generation the user can choose between ramp velocity profile (trapezoidal velocity profile) and the smoother sinusoidal velocity profile. The changes in orientation if a command in operational space is used are based on the description in quaternions [Craig, 2005], [Rothenbücher, 2004]. The user can also define the time distance T_IPO for interpolation. Each time distance T_IPO the path-planning algorithm calculates and stores actual joint coordinates and the appropriate velocities and accelerations. If the motion program is complete, the translation by an interpreter can be executed. During the translation process ManDy checks whether a programmed motion is inside the workspace of the robot. This verification is also related to the actual robot. In the case of such an error the translation will be stopped and an error message is given. Results of the translation are the time history of desired position, time history of desired Page 288

3 desired joint variables, joint velocities and joint accelerations condensed in the vectors q (), t q (), t q d d d() t is available. This reference path can be stored and corresponds to the programmed motion. ManDy provides the draw of graphs of all time histories in joint space and operational space as well as a 3D-view and 2D-views of the trajectory in operational space. III. Animation A. Purpose Fig. 3. Programming interface velocity and acceleration for each joint. To map a desired motion from operational space to joint space the inverse kinematics solution must be realized by ManDy for all sorts of admitted kinematical structures from two to ten joints and links. To solve this problem ManDy generates the Jacobian matrix J ( q ) for the actual manipulator and gives a numerical solution for the inverse kinematics in the form: q J 1 ( q) w When for a certain point in time the vector w of coordinates in operational space and the according Vector q of joint coordinates are known, the Jacobian is calculated and one can calculate the displacement in joint coordinates with (1) as function of the displacement in operational space. An actual displacement w is given on the base of the reference path each time distance T_IPO. Equation (1) is an approximation to the solution since J ( q ) is assumed to be constant for the time distance T_IPO. A special method to handle target location in the case of a manipulator with less than six degrees of freedom is the mask-vector m ( mx, my, mz, mc, mb, ma). The path planning-algorithm must know the degrees of freedom in operational space. The user must sign in the mask-vector which degrees of freedom for positioning are available. For example the manipulator can move the end-effector in x-direction of the base frame, he gives a one for m x, otherwise a 0. Corresponding to the positioning degrees the availability of the orientation degrees are defined by mc, mb, ma. A, B, C describe the Euler-Angles in respect to the base frame. For example a manipulator with three rotational axes perpendicular to the x-y-plane of the base frame can positioning the end effector in x- and y- direction and can orientate around the end-effector around the z-axis, as a result the mask-vector must be m (1,1,0,0,0,1). After execution of the translator program that is also written in Matlab a time history of all (1) A three dimensional model of the designed multi-axes system has advantages for several objectives. First it can be recognized whether the kinematical structure is defined in a correct way by the Denavit-Hartenberg parameters. econd the available workspace and the manoeuvrability of the manipulator can be checked. Furthermore a customer can get a first and clear impression of the proposed solution for his problem. B. VRML VRML (Virtual Reality Model Language) is a programming language for virtual visualization of complex 3D-objects [VRML 1997]. The description of the 3D-world is carried out in a usual ACII-file. The ACII-file can be generated and manipulated by any texteditor. The visualization is executed by a web-browser with a help of a VRML-plug-in. This plug-ins can be loaded down without any costs. The XML-based language X3D will replace in future VRML, but the most X3D-browser will continue to support the VRML standard. C. Automatic generation of VRML ACII-File A Matlab function generates the necessary ACII-File for animation (see Fig. 4). Input parameters are the kinematical structure of the manipulator which is defined Fig. 4. Automatic generation of VRML animation by the number and type of joints and the Denavit- Hartenberg parameters,, a and d plus the limits of joint coordinates of each joint (see section II, B). To animate the motion a reference path must be loaded. The Kinematical reference path tructure qd, qd, qd Generation of VRML-Code ACII-file animation Fig. 4. Automatic generation of VRML animation Page 289

4 reference path is described by the time history of the joint coordinates (see section II, C). Within the Matlab function various VRML-Code-blocks are predefined. Dependent on the actual manipulator and the actual reference path the Matlab functions write the necessary parameters to the blocks. Matlab will start the web browser with VRML plug-in and the animation is running. An example for a VRML model is presented in Fig. 5. model of kinematics dynamic parameters reference path q, q, q d d d simulation algorithms control scheme and parameters simulation parameters IV IMULATION OF DYNAMIC A. Overview about the workflow It was mentioned that the tool Mandy could be divided in two stages. o far the kinematical design has been described. The user can define various kinematical structures of a manipulator; he can program characteristic motions and check whether his solution is able to execute a certain task. Now the user must define mass properties (mass, inertia tensor, center of mass) of each link, essential characteristics of chosen actuators and gear train as well as structure and parameters of the feedback control loop. Mandy provides also a menu for this purpose. Furthermore parameters of simulation like sampling time of the control system; loads and external forces/torques can be defined. Fig. 6 shows an overview of the interfaces for dynamic simulation. All parameter sets are stored in files and can be invoked and modified. B. Formulation and solution of Dynamics There are two formulations of dynamics. In the first formulation, a trajectory point (all joint coordinates, joint velocities and joint acceleration in a point in time) is given and the required joint torques respectively joint forces are to be calculated. This problem is named inverse dynamics. The second problem is to calculate how the mechanism will move under application of a set of joint torques or forces [Craig, 2005]. That is, given a vector of joint torques respectively joint forces, calculate the resulting motion described by joint predicted real path qqq,, Fig 6. Interfaces for simulation coordinates, joint velocities and joint accelerations that are condensed to the vectors qqq,,. This is called Direct Dynamics and is useful for simulating a manipulator. For serial manipulators the Direct Dynamics have the form 1 q M( q) bqq (, ) (2) Mq ( ) is the (n n) mass-matrix, while the ( n 1) -vector b is caused by gravitational-, centripetal-, Coriolis- and friction influences. To predict the actual motion of a manipulator it is necessary to include the actuator dynamics and the gear train. If the vector of manipulated variables, which is provided by the feedback control system, is denoted as U, we get Fig. 7. Fig 7 describes the plant of the feedback control system. The servo system includes secondary current feedback control loops. It can be assumed, that these control loops are working ideally: the reference current is approximately proportional to the motor torque. In ManDy the user can decide whether the gear train is assumed as mechanical stiff with no friction losses or the gear train is considered as torsional spring and damping characteristics. In the first case (1) can be extended to a vector differential equation with same structure and order [Weber, 2002]: 1 q M ( q) U b ( q, q ) (3) In the second case the order of the vector differential equation must be doubled and the numerical solution is more time consuming [Rothenbücher, 2004]. For a certain point in time t 0 Mandy can calculates M q( t0 ) and b q( t0), q ( t0) on the base of the recursive Newton- Euler formulation [Walker and Orin, 1982], [Weber, 2003]. U ervos, actuators geartrain robot arm qqq,, Fig. 5. Example for a VRML animation Fig. 7. Plant of model based control Page 290

5 With (3) and given U ( t0 ) the Runge-Kutta 2 or 4 method [Conte and C. DeBoor, 1972] is used by ManDy to solve the dynamic equation for the next sample. Results are the vectors q ( t0 T), q ( t0 T), q(t0 T). Each sample time the controller delivers an updated vector U of manipulated variables. In ManDy only joint space control is realized. Here U is a function of the desired motion (reference path) given by q d, qd, qd and the actual values of motion qq,, q. Fig. 8 shows the general scheme of joint space control. B. Predefined Control schemes In Mandy four predefined control structures are available: 1. single axis cascaded control system with P-PI structure 2. single axis cascaded control system with P- ReDu structure 3. model based cascaded control system with P-PI structure 4. model based cascaded control system with P- Redu structure In Fig. 9 the often-used cascaded control system with P- position controller and velocity precontrol and PI-velocity controller is depicted. Here the designs of the parameters are based to a linear approximation of a single axis. Nonlinearities and couplings between the links are assumed as disturbances. The ReDu-Controller is a linear approach too. The plant of velocity control is assumed as linear system and the ReDu velocity controller forces a user defined P-T 2 - or P-T 1 - performance between input v d and output v of the velocity control loop [Weber, 2001]. The second class of control methods is named modelbased manipulator control [Craig, 2005] or inverse dynamics control [ciavicco and iciliano, 2000]. The mathematical model of the plant is not only used for design purposes but also directly in the control algorithm. These methods lead to a decoupled and linearized system. First we consider (2) in the inverse system form. This is to solve (2) for U : control algorithm q, q, q U M ( q) q b ( q, q ). (4) d d d U manipulator mechanism actuator system gear train Fig. 8. General scheme of joint space control. qqq,, q di, q d,i q i K Vor v d,i 1T K N s L K P T s - - q i velocity controller Fig. 9. Block diagram of P-PI single axis control. For calculation of the vector U of manipulated variables, the vector r is used instead of q. r is given by a control law. ince the model used for control purposes not exactly correspond with the reality or a reduced model is used as a result of the calculation effort [Weber and Anggono, 2003] the sign ~ is used to mark the deviation. U M ( q) rb ( q, q ) If M ( q) M( q) and bqq (, ) b( qq, ) hold and (5) is put in the equation of motion (3), we get N U,i (5) q r (6) Equation (6) is the replacement plant to design the vector r. As we have to control the vector q, this replacement plant consists of a double I-term for each joint. As a result the coupled and nonlinear system (3) is decoupled and linearized by (5). The methods of model-based control can be distinguished between the linear control laws to obtain the vector r. In the simplest case an element r i is given by ri qd, i KP, i ( qd, i qi) KD, i ( q d, i q i). (7) ManDy uses the well known and efficient cascaded control system with velocity control as auxiliary control loop and position control as final control loop [Weber, 2000]. Like in the case of single axis control for velocity controller a PI-structure or ReDu-structure is used. Fig. 10 shows the block diagram. The user can choose a certain structure of a controller and than he can define or modify the parameters via a menu. A special possibility is to define deviations of the inverse model in (5) by variations in M ( q ) und bq (, q ). In this way the robustness of model-based control against model errors can be checked out. ManDy provides all essential graphs of control performance in joint space and operational space. Time histories of reference and simulated values of coordinates, Page 291

6 q d,i K Vor q di, K L - v d,i ri qi velocity controller replacement plant Fig 10. Cascaded model-based control velocities and accelerations as well as manipulated variables are plotted. In Fig. 11 the corresponding menuwith the control performance of joint 2 is depicted. The reference and simulated path in 2D- and 3D-views can also be displayed. Additionally an animation with VRML of performance of the path is available. Reference path and simulated path are shown as traces in space with different colors. V REMARK ABOUT REALIZATION uch complex programs with many mathematical calculations especially on the base of matrices and vectors can be effective realized with Matlab. But tasks with high computational effort can result in a long calculation time. It is not the objective of ManDy to simulate and animate the motion in real time, but a too high response time in simulation is tiresome for the user. That s why Mandy uses Matlab externals (MEX) for the following tasks: direct kinematics, generation of the Jacobian, calculation of M ( q ) and b ( q, q ), inverse dynamics q i q i These parts of Mandy are written in C/C++. The time for execution of a simulation is nearly two hundred times faster compared with the solution in Matlab. REFERENCE [ABB, 2005] Robottudio., [KUKA, 2005] KUKA.im, [Delmia, 2005] Delmia V5 Robotics, [Easy-Rob, 2005] EAY-ROB Version 4.0, [Weber, 2002] W. Weber, Industrieroboter, München/Wien, Hanser [ciavicco and iciliano, 2000] L. ciavicco and B. iciliano: Modelling and Control of Robot Manipulators, London: pringer. [Craig, 2005] J.J. Craig, Introduction to Robotics, Upper addle River, Pearson Prentice Hall. [Rothenbücher, 2004]. Rothenbücher, Flexible Programmier- und imulationsumgebung für Mehrkörpersysteme, submitted for a diploma, Faculty for Electrical Engineering and Information cience Fachhochschule Darmstadt University of Applied ciences. [VRML, 1997] The VRML Consortium Incorporated, The Virtual Reality Modeling Language, International tandard II/IEC, 1997 ( [Weber, 2003] W. Weber Automatic generated real-time models of robot dynamics, Prepr. Int. ymp. on Robot Control (YROCO), Wrocaw, Poland, pp , ept [Walker and Orin, 1982] M.W. Walker and D.E. Orin, Efficient dynamic computer simulation of robotic mechanism, Journal of Dynamic ystems, Measurement and Control, vol. 104, pp , [Conte and DeBoor, 1972]. Conte and C. DeBoor, Elementary numerical analysis: an analytic approach, New York: Mc Graw-Hill. [Weber, 2001] W. Weber, Modifizierter Drehzahlregler für automatischen Entwurf, in Wt Werkstattstechnik online, Volume 91, 2001, Pages [Weber and Anggono, 2003] W. Weber and L. Anggono, tochastic approach to generate approximated robot models, Proc. 4 th IMAC ymposium on mathematical modelling (MATHMOD), Vienna, Vol. 1, Page 203, Vol. 2, Pages , Febr [Weber, 2000] W. Weber, Modellbasierte Gelenkregelung in Kaskadenstruktur mit vorgegebenem Regelungsverhalten, in: Robotik 2000, Düsseldorf, VDI-Berichte 1552, VDI-Verlag, Düsseldorf, 2000, Pages Fig 11. control performance of joint 2 Page 292