Low Cost Mechatronic Design Tools Applied to Robotic Manipulators

Copyright © IFAC Low Cost Automation, Buenos Aires, Argentina, 1995

LOW COST MECHATRONIC DESIGN TOOLS APPLIED TO ROBOTIC MANIPULATORS

Eduardo R. Benzo, Jorge J. Gleizer, M. Lehmann, M. Torrez Contreras

Industrial Design Research Center o/Complex Products and Open Laboratory - Electronics Department. University 0/ Buenos Aires. Buenos Aires, Argentina.

Abstract: The virtual conception of mechatronic systems and their validation through the fulfillment of the mathematic models which describe them, allow to reduce the margin of uncertainty in the fulfillment of the specification. The acquisition of suitable commercial products for the work, represents an investment which exceeds the U$S 50000. It difficults their implementation in small and medium enterprises. An effective process of low cost mechatronic design which uses standard CAD tools combined with dedicated software of dynamic kinematic simulation, is presented. Keywords: CAD models, Digital Simulation, Interdisciplinary design, Robotics, Mechanical Manipulators.

reducing the research costs.

l.INTRODUCTION The virtual conception of mechatronic systems and their validation through the fulfillment of the mathematic models which describe them, allow to reduce the margin of uncertainty in the solutions of design proposed.

development

time

and

the

Most commercial products existing for the work need workstation or Micro VAX platforms with prices exceeding U$S 50.000 including soft and hard, becoming an important inversion for any small and mediumsized enterprise .

The evolutive process of the design was directed towards the virtual modelization of the systems during the last years, in the stage of definition of shapes and in the numerical simulation.

The aim of this work was to develop a design methodology which allows to predict the mechatronic product behaviour accordingly with new technologies and achieve a low cost .f:; product.

The utilization of numerical and graphic simulation programs in the stage of design of mechatronic products has increased the evaluation capacities of the system behaviour, allowing to predict its real functioning with a great precision. The suitable use of the software lets the designers teams select the best options before building the physic models,

The methodology presented requires hardware equipment based on PC and DOS operative system, which can be acquired by any small and medium-sized enterprise .

409

Basically we propose to use a standard CAD program (AutoCad 12 and AME module or MicroStation 5.0) combined with numerical simulation software designed for a DOS environment, which allows to analize rigid configurations up to six degrees of freedom .

B-Choice of the electric and mechanic models. C-Three-dimensional product.

F-Numerical simulation of the kinematic and dynamic behaviour. G-Generation of production.

In this sense, the method will be applied to an example consisting of the design of a simple manipulator of three degrees of freedom , which can be built with the didactic kits LEGO DACTA.

the

documentation

for

For the fulfillment of the stages C to G, an equipment which includes expensive hardware and software is necessary. Some important software commercial packages can be acquired for U$S 140.000 to PLACE (McDowell Douglas), ROBCAD (ROBCAD Ltd.) whose cost is about U$S 130.000, and IGRIP (Dened Robotics Inc.) whose cost is U$S 60.000.

2. INTERDISCIPLINARY DESIGN The term Mechatronics is related to the whole of technics, proceedings and bases for the service, production and development of machines, elements and devices futureoriented. The term "Mecha" is related with the total development from the technician point of view, and "tronics" is related to intelligent control.

4.PROPOSED METHODOLOGY The presented methodology respects the stages described in the point 3, but it needs hardware equipment based on PC and DOS operative system, which can be acquired by any small and medium-sized enterprise.

Therefore, the Mechatronics is a discipline technically interdisciplinary which bounds the classic mechanic, the industrial design and the software and electronic engineerings.

For the stages C, D and G it's proposed the use of a standard CAD program (AutoCad 12 and AME module or MicroStation 5.0) which modelizes the product three-dimensionally, supporting on definitive materials associated to each piece and compromising the relationship between complexity of shapes and precision of calculations.

of affordable From the appearance computational systems and microelectronics, this approach is being used for the development of new products increasingly.

In this way, the dynamic parameters of the mechanic structure are calculated with error margins strictly limited.

So, a robotic manipulator arises as a mechatronic product; that is why its design must be carried out from an interdisciplinary point of view.

Specifically the following parameters are calculated: mass, centroid, inertia matrix (inertia about axis and crossed products) and main moments (module and direction) . This information is essential for the F stage of numerical simulation.

3. EXISTENT METHODOLOGY An judgement accepted in the area of engineering for the design process of a product implies the concretion of, at least, the following stages: the

the

E-Selection of the satisfactory control method.

Moreover, the methodology presented allows the utilization of these tools with an educative purpose.

of

of

D-Obtention of the dynamic parameters of the structure.

The development and validation of the proposed method came out as a consequence of the design of a robotic manipulator of six degrees of freedom.

A-Initial specification characteristics.

modelization

For the F stage, an user friendly application based on the simulation package RODIS for DOS was created. It allows to analize rigid configurations of six degrees of freedom and needs as data the kinematic parameters of the structure (defined during the stages A and B) ,

desired

410

dynamic parameters of the structure (calculated by the CAD software), dynamic parameters of the actuators (they are data of the respective manufacturers), the gravity vector (because the direction and sense determine how it will influence on the structure) and the algorythms of control and generation of trajectory wished.

neccesary dynamical mechatronics.

properties

needed

in

The algorithms neccesary for the creation of solids required an upgrade of eXlstmg hardware and software, this fact generated a migration to workstation level systems, unavailable for small and medium enterprises.

From the results of numerical simulation, it can be determined the necessity to modify an earlier stage, so it begins an iterative process which ends at the moment when the initial specification is reached.

There are two basic solid-generation structures, one called Constructive Solid Geometry (CSG) and used by AutoCAD AME module (by AutoDesk Inc.) and the other one known as Boundary Representation (B-rep) .

To finish the G stage, it is used the CAD tool previously adopted to obtain all the technical data necessary for the mechanical construction of the designed product (Drafting process).

Table 1. Example of the results of the mass properties calculated by AutoCAD's AME module. Ray projection along X axis, level of sulxlivision: 8. Mass: 2.39534 kg Volume: 673156 cu mm (Err: 301.8221)

4.1 Solid representation systems

The evolution of computational graphics systems dedicated to the representation of solids received an important development (see Tensg, 1989), requiring each time more resources which can be translated in terms of processing capacity and memory space.

Bounding box:

Centroid:

The simplest geometric modelization, which defined the first step in the generation of CAD systems, was the wire-frame representation. This kind of modelization showed big limitations such as not being able to assign physical properties to the represented solids, in this way it is not possible to calculate centroids, volumes and the inertia moments of the objects.

X: -435.0001 -- 35.00007 mm Y: -35.00001 -- 36.50007 mm Z: -64.50003 -- 175.5 mm

X: -277.5391 mm (Err: 0.4354172) Y: 3.888128 mm (Err: 0.06783473) Z: 65.83989 mm (Err: 0.182621)

Moments of inertia: X: 18912.41 kg sq mm (Err: 18.24722) Y: 247048.5 kg sqmm (Err: 192.3111) Z: 229646.3 kg sq mm (Err: 193.8078) Products of inertia: XY: -1883.341 kg sq mm (Err: 21.87754) YZ: 879.0575 kg sq mm (Err: 5.132088) ZX: -40537.37 kg sq mm (Err: 68.7156)

This problem limits the use of CAD systems only as a visualization tool that can not interact with the kinematic and dynamical simulations .

Radii of gyration: X : 88.85664 mm Y: 321.1497 mm Z: 309.6322 mm

The Polygon and Sculptured Surface representations introduce curve-smoothing algorithms for the curves that represent the surfaces of the objects through the use of mathematic or para metrical functions and splines, improving the visualization of real models but leaving unsolved the problem of assigning physical parameters to the analized entities.

Principal moments(kg sq mm) and X-Y-Z directions about centroid: I: 8197.483 along [0.99604 0.01642290.087376] J: 52174.81 along [0.01373 -0.99941 0.0312895] K: 45379.12 along [0.087839 -0.02996 -0.99568] In the constructive geometry of solids, a set of pnmltlve forms are related by boolean operators. Solids are generated by means of a tree-like structure. A compact representation of the drawing database is obtained, which

With Solid Modelling, the road to the study of virtual solids was opened, with the capacity of establishing physical properties to the represented objects. Through graphical representation, it is now possible to obtain the 411

contains the historical information about the creation process of the object.

desired trajectories using linear interpolation in the joint or cartessian space. It's also possible to simulate the kinematic and dynamical behavior using PD or Computed Torque control strategies. The simulator analyzes rigid configurations of six degrees of freedom of open chain and requires as input data the kinematic and dynamic parameters of the structure and the dynamic parameters of the actuators and the gravity vector (the chosen direction and heading will determine how the structure will be influenced) .

Modelers that use boundary representation generate geometric and topological information about the involved objects boundaries. If CSG modelers are used, boundaries details should be calculated if they are required for any purpose. The AutoCAD software package and its AME modeling extension mantain a good ratio between costs and profit. It runs on IBM-PC or compatibles and there are DOS and Windows versions. Its modeling capabilities make possible the calculation of mass properties. Table 1 shows the output format of AME solid evaluation results.

It calculates for each link the desired and real values of: angular position, angular speed, angular acceleration and torques . Error ratios in all magnitudes and final positioning are also obtained.

The diferential equations resolution were achieved using a fourth order Runge-Kutta's algorithm, while the Walker and Orin's method (see Walker and Orin, 1982) solved the direct dynamic problem.

4.2 Kinematic and Dynamical Simulation

To complete this stage, a simulator adequated to the kind of product developed should be built or bought.

For this application it was added to the package a graphics presentation module, in the joint space as well as in the cartesian space and a user-friendly general interface module.The documentation correspondent to the package modules and libraries was also generated.At last the original package was adapted to make possible the simulation of structures with less than six degrees of freedom and its application to the example shown in point 7.

Two kinds of software exist: the first one consists of software developments based on a structurated language (i .e. Pascal, C, C++) . The second one consists on rutines for mathematical software such as MatLab or Mathematica. The first option was chosen by Anigstein (1987) who developed the RODIS software package based on Pascal language. The second option was followed by Honey and Jamshidi (1992) who developed the ROBO_ SIM MatLab toolbox.

Graphics presentation module: To obtain a quick display of the simulation results in all the links and all the variables under study, all the curves corresponding to angular positions, angular speeds, angular accelerations and torques vs time are shown in the modules first screen. Figure 2 and subsequents show such screen images. Each graphics scale are maximized to use as much as possible the available space of the computer monitor.

The main modules of the robotic simulators consist of differential equations and inverse dynamics resolution algorithms. 5. ROBOTIC SIMULATOR Having got the software package developed by Anigstein (1987), this alternative was chosen. The package libraries can calculate position, velocity, acceleration and forces for each element of the structure.(see Anigstein, M. and Ramos, R., 1992)

An enlarged version of each curve can be displayed selecting it with the cursor. In this image the corresponding scale values are shown. A second screen presents the results of the last frame in the cartesian space.

This software runs on DOS under IBM PC-AT or compatibles platforms, and it doesn't have any special hardware requirements.

This application for being friendly has a very interesting use as a demonstrative and didactic tool for learning purposes in introductory as well as advanced robotics courses .

With this software it's possible to input and modify the structure data and generate the 412

Figure 1 represents a renderized image of the prototype developed.

6 A SIX DEGREE OF FREEDOM ROBOTIC MANIPULATOR DESIGN APPLICATION The presented mechatronic design methodology was validated while it was being applied to the design of the first six degree robotic manipulator prototype. The following paragraphs are a resume about how the diferent design stages (mentioned before) were carried out. Stage A: The initial specifications of the first prototype were:

-Minimum Load Capacity: 1.5 kgf. -Maximum Positioning Error: 10 mm. -Manipulator Length (Fully Extended) : 800 mm. -Minimize manufacture and manteinance costs. -Maximize confiability, efficiency, easiness in assembly and setup. Stage B: The mechanic and electric models adopted are :

-Denavit and Hartemberg's kinematic structure representation model (Denavit, et aI., 1955). -Mechanic configuration of six rotational joints like PUMA (see Lee and Arbor, 1982). -Simplified models were adopted for the reducer-motor-encoder sets used.

Figure 1. Image of the manipulator developed. It was rendered from the three dimensional model designed with AutoCAD .

The definitive kinematic parameters are shown in table 2.

The modular design concept allows the assembling and the maintainance operations just as fitting it quickly towards new constructive alternatives during the product evolutive process.

Table 2 Kinematic parameters of PUMA configuration (Denavit - Hartemberg's convention ). Link 1 2 3 4

5 6

Length [m] 0 0.40 0 0 0 0

Stage D: For each link, the associated frames are defined and the dynamic parameters of the structure (essential for the latter simulation) are calculated, using the numerical algorhytms of the AutoCAD AME package .

Distance [m] 0 0.18 0 0.30 0 0.10

The mass and centroid values can be found in table 3 and the inertia moments are presented in table 4. Analyzing the results, we can find a strong symmetry condition which approximates the origins and orientations of the::~elected frames to their autovalues becoming, therefore, the values of the crossed elements (centrifugal moments) of the inertia matrix which are very small. The data obtained are consistent with the MKS system.

Stage C: As CAD software, the AutoCAD v. 12 and its AME module were adopted.

To fulfil the initial specifications low costs, standard pieces which the local market, were used for materials, the actuators and elements.

related to the can be got at the structure the sensor

413

Table 3. Dynamic Qarameters. Mass and centroid for each link. Link 1 2 3 4 5 6

Mass

Centroid

[K~]

(x,~,z)[m]

3.94 2.22 1.69 1.93 0.01 1.51

-0.004 -0.082 -0.006 -0.282 0.002 0.066 0.001 0 0 0 0.136 -0.002 0 0 0 0 0 0.010

specifications to check if it satisfies the torque requirements to the correspondent speed conditions. Figure 2a shows the graphical display of results in the joint space for a simulation that starts at the following angles (45°, 45°, 90°, 0°, 0°, 0°) until the angles (0°, _5°, 90°, 0°, 0°,0°) .

Table 4. Dynamic Qarameters. Inertia moments for each link. Link 1

2

3

4

56.8E-3 1. 15E-3 236E-6 17.3E-3 -807E-6 -38 .3E-3 3.6E-3 -7.6E-6 -45E-6 45.8E-3 149E-6 -348E-6

I [K~m:Z] 1.15E-3 4.56E-3 126E-6 -807E-6 0.23344 510E-6 -7.6E-6 3.65E-3 48E-6 149E-6 2.98E-3 -741E-6

263E-6 126E-6 54.9E-3 -38 .3E-3 51OE-6 0.21723 -45E-6 48E-6 982E-6 -348E-6 -741E-6 46.1E-3

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Fig.2a. Graphical outputs in the joints space. The described trajectory starts at the following angles (45°,45°,90°,0°,0°,0°) until the angles (0°,_5°,90°,0°,0°,0°). Note: The graphic scales are maximized for each case. Figure 2b shows the enlarged curve of torque applied on the 4th link joint for the defined trajectory and the table 6 shows the motors characteristics available for this joint.

Stage E: Initially a PD (proportionalderivative) control strategy is probed. The correspondent constants are shown in the table 5.

Table 5. Feedback constants according to the PD control strategy.

KE Kv

1 150 30

2 500 100

3 300 18

4 50 5

5 40 10

6 9 2

Fig. 2b. Torque vs time graphic for the 4th link in the most exigent trajectory.

Stage F: Two aims are tried to be fulfilled:

Table 6. Available motors characteristics for the 4th link.

I-Select the motors in the suitable way. 2-Check that the margin of position exactness is fulfilled.

Type nom. vel nom. torque max torque n weight [gr] [rad] [N.m] [N.m] 314 675 21 0,74 3 MR5 12 0,51 2,54 500 185 MR2

Motors selection : The first step consists of simulating the movement through the most exigent trajectories, using the software package. The necessary torques for each link are calculated to satisfy the following of the predetermined trajectory. The second step implies the comparison between the results of the simulation and the manufacturer's

The utility of a numerical simulation previous to the prototype construction is shown in this

414

case, comparing the results of figure 2 with the data of the table 5.

Electronics Department. The described methodology was applied to the design of a simple didactical manipulator with three degrees of freedom which can be subsequently built using LEGO DACT A building kits.

Initially, a MR5 motor was selected, while comparing the results it was concluded that it was overdimensionated diminishing the load capacity of the manipulator. When a smaller motor of the same line was adopted (type MR2), the link mass was reduced in 490 gr., making it possible to increase the load capacity of the manipulator in the same proportion.

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Verifying of the margin of position: From the reading of the results of angular errors and the final traslation error graphic, it was concluded that the selected control algorythm doesn't satisfy the specifications, having to adopt a computed torque strategy (see Vubratovik, 1989), which solves the conflict. Figures 3 and 4 show the traslation errors for both mentioned control strategies.

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Fig. 5. CAD software display image showing the design of the manipulator based on LEGO DACT A with three degrees of freedom.

Table 7. Kinematical Parameters of the three degrees of freedom manipulator. Fig. 3. Traslation error vs time graphic according to a PD control strategy.

Number of active links: 3 1st. link length: 0.11150 m 2nd. link length: 0.11260 m 3rd. link length: 0.11950 m

·------·------·------·----------:r:::::f:t:\:~.

-:------+ ------i-------:-------:-----I------~------ .. ------~------- - ---·--··---· I ---\~.

1st. link distance: 0.00000 m 2nd. link distance: 0.00000 m 3rd. link distance: O.OOOOOm Fig.4 Traslation error vs time graphic according to a computed torque control strategy.

1st. link torsion: 0.00000 rad 2nd. link torsion : 0.00000 rad 3rd. link torsion: 0.00000 rad

Stage G: In this design stage, all the necessary documentation for the product manufacture is generated. It includes: sets, subsets and pieces drawings, materials and specifications manufacture schedule.

The presented methodology implies the product virtual conception before any attempt of first prototype construction.

One application of the method resides in the educative environment of the Mechatronic area .

To follow the adopted criterion, AutoCAD was used in the prototype design and its frame definition . The AME module was used for mass properties calculations. Figure 5 shows the design and Tables 7 and 8 re~ectively show the kinematic and dynamical parameters obtained with the AME module.

In this sense, it was developed an experience with mechanic and electronical engineering students in the Open Laboratory of the

Through the use of the software package the motor torque requirements were defined. They were satisfied by the own motors of the kit.

7 DIDACTICAL APPLICATION

415

Figures 6, 7 and 8 show the graphics obtained with the simulation.

Table 8 Dynamical parameters of the three degrees of freedom manipulator. Mass (Kg) I: 0 .0366148 2: 0.0126006 3: 0 .0080146 Centroid (m) 1: (-0 .0706092, 0.0005649,-0.0033129) 2: (-0.0830570,-0.0005038,-0.0001515) 3: (-0.0798175,0.0005203,-0.0026669) Inertia (Kgm2) 2.02E-0005 3.32E-0008 3.50E-0006 1: 3.32E-0008 5.81E-0005 1.41E-0007 3.50E-0006 1.41E-0007 3.98E-0005

2:

3:

Fig. 7. Enlarged curve shows the real and desired angular position of the first link for the desired trajectory shown in Figure 6.

Fig. 8. Torque enlarged curves of the first link for the trajectory shown in figure 6 .

1.68E-0005 -5 .23E-0008 -3 .03E-0008 -5 .23E-0008 1. 76E-0006 2.45E-0006 -3.03E-0008 2.24E-0006 1.55E-0005 1.29E-0005 1.1lE-0008 6 .69E-0008

inertia influence: It was interesting to compare simulations that included the inertia of the structure and the actuator with those that didn't. Figures 9, 10 and 11 show the diferent curves obtained in the first case only the link masses were considerated, meanwhile in the succesive figures the structure and the actuators inertia are succesively added.

1.1lE-0008 6.69E-0008 1. 26E-0005 8.93E-0007 8.93E-0007 5.31E-0007

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They were done in a open loop simulation and starting at an initial angular position (80°,0°,0°) .

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Fig. 6 . The image shows the graphic data output of the simulator. The curves show the desired and real values of angular position, angular speed, angular acceleration and torque for a trajectory starting at (80°,0°,0°) until (90° ,0°,0°) . The feedback constants are: Kp=(6, 5, 5) and Kv=(4 , 3, 3).

Fig. 9. Enlarged curve of real and desired angular position of the first link obtained with open loop simulation considering only link masses and starting at (80°,0°,0°) . It can be appreciated the small difference between the simulations shown in figures 9 and 10, this is reasonable due to the small values of the inertia matrixes.

With the help of the simulator, the inertia influence of the structure and actuator was analyzed and then the resonance conditions of the mechanical structure were studied.

416

Fig. 10. Enlarged curve of real and desired angular position of the first link obtained with open loop simulation considering the link masses and inertia.

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Fig. 13 . Response for Kp=(10,6 ,6) and Kv=(6,4,4) constants. The trajectory is the same of figure 6. A strong resonance condition can be observed.

Three dimensional animation: Later a three dimensional animation of the manipulator describing a circular trajectory was generated using 3D-Studio software package (by Autodesk Inc .) which used the files generated with AutoCAD. Figure 14 shows an image that belongs to the animation. It serves as an approximation to the virtual construction of the product and allows the visualization of the product movement characteristics without the need of prototype construction.

Fig. 11 . Enlarged curve of real and desired angular position of the first link obtained with open loop simulation considering the link masses and iner~ia and the motor inertia (0, 1095 Kgm ). From the comparison between figures 10 and 11 it can be appreciated a great difference because the actuator inertia is greater than the structure and therefore determines the characteristical response.

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Fig . 12. Response for Kp=(5.5 ,5) and Kv=(5 ,4,4) constants . The trajectory is the same of figure 6. Mechanical resonance analysi s: Modifying the feedback constants for a PD type control loop it can be observed how the mechanical resonance condition is reached, this fact is shown in figures 12 and 13 and belongs to the same trajectory fixed in figure 6.

Fig . 14. Image that belongs to the animation, which shows the manipulator describing a circle in a vertical plane.

417

Fig. 15. Photography of the manipulator built with the LEGO-DACT A kit.

Fig. 17. Renderized image from the same viewpoint of the photography shown in figure 15 .

Fig . 16. Enlarged photography of the manipulator.

Fig . 18. Renderized image from the same viewpoint of the photography shown in figure 16.

Finally the manipulator was built and it resembled the virtually conceived very closely .

8 CONCLUSIONS The application of this method in the design of mechatronic products results in a equipment investment of less than U$S 3.000 although the additional investments in simulation software packages or its development must be taken into account.

Figures 15 and 16 show its final aspect. A later programming of the motors made possible the verifying of the motion previously observed in the animation. It's proposed to compare figures 15 and 16 with figures 17 and 18 respectively.

For the medium and small enterprise the use of this method makes possible the development of highly competitive products with an elevated added value.

The similitude between the virtually conceived product (fig. 17 and 18) and the finally built one (fig. 15 and 16) can be observed.

The manipulator of six degrees of freedom is currently undergoing an assembly proccess and was designed as an aid tool for handicapped persons. It can also be applied in education and 418

training fields. It has characteristics that would make possible its use and application in some industries.

Vubratovik, M. (1989). Applied Control of Manipulation Robots. Analysis, Synthesis and Exercises. Springer-Verlag. Berlin.

In reference to the method application in the educative field, it was really auspicious, not only for the motivation and interest that it arised but also for the learning that the participants acquired in the integration of an interdisciplinary work team. This experience should be incorporated in the contents of Mechatronic related to careers and study programs of technical high school.

Walker, M. and Orin, D.(1982) Efficient Dynamic Computer Simulation of Robotic Mechanisms. ASME Journal of Dynamic Systems, Measurement, and Control. 104, pp. 69-76

ACKNOWLEDGEMENTS To LD. Mario Marino, Director of the Industrial Design Research Center (CIDIF ADU) and Project Director for his constant support and encouragement.

REFERENCES Anigstein, M. (1987) Software para robots. V Congreso Nacional de lnformatica y Telecomunicaciones . Buenos Aires. Junio.

To Dr. Ricardo Sanchez Pena, Codirector of the Project for his unconditional support.

Anigstein, M. and Ramos, R. (1992). Analisis dinamico de robots mediante simulacion. 1 Congreso lberoamericano de ingenieria mecanica. Univ. Politecnica de Madrid. Madrid. Septiembre.

To Prof. Mauricio Anigstein, Principal of the Robotics Cathedra (Applied Mechanics Department, FIUBA), for the availability of his Pascal written RODIS software package. To the collaborators : Pablo Barros Martinez, Hugo Nahuys, Mario Valverde, Martin Cassino, Lucas Diodatti e Irene Scarso.

AutoCad 12. User Manual . AutoDesk AME for AutoCad 12 . User Manual. AutoDesk.

To LEGO-DACT A and PMK S.R.L. for the donation of the kits employed in the educative application of the method .

Craig, J (1986). Introduction to Robotics, Mechanics & Control. Addison Wesley. .Massachusetts. Denavit, J. and Hartemberg, R. , Evanston, L (1955) A kinematic Notation for Lower Pair Mechanisms Based on Matrices. J. Applied Mech. 77, pp.215-221. Honey, Wand Jamshidi, M. (1992) ROBO SIM: A robotics simulation environment on personal computers. Robotics and Autonomous Systems 9 pp. 305-317. Lee, G. and Arbon, A. (1982) Robot Arm Kinematics, Dynamics, and Control. Computer. December. (1981) Robot Manipulators: Programming and Control. The MIT Press. Massachusetts.

Paul,

R.

~Mathematics,

Tensg, A.(1989) Software for Robotic Simulation. Adv. Eng. Software . Vol. 11, N° I, pp. 26-36.

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