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Third International Conference on Production Research – Americas’ Region 2006 (ICPR-AM06) IFPR – ABEPRO - PUCPR - PPGEPS

Luís Paulo Laus UTFPR - Department of Mechanical Engineering (DAMEC) Av. 7 de Setembro, 3165 - Curitiba, PR, Brazil - 80.230-901 laus@cefetpr.br Alfranci Freitas Santo UTFPR – freitas@cefetpr.br Luiz Carlos de Abreu Rodrigues UTFPR– lcar@cefetpr.br João Antonio Palma Setti UTFPR – setti@cefetpr.br Leandro Magatão UTFPR – magatao@cefetpr.br Abstract: Robots' kinematic models are the base for the study of robotics and the construction of the dynamic models used for robots’ control and simulation. Lack of a well established procedure for obtaining the kinematic model may lead to a trial and error process, resulting in waste oftime and effort. With a didactic perspective in mind, a simple and intuitive modeling method is presented. This method facilitates learning and reduces the occurrence of mistakes in the construction of the forward kinematic model. The presented method, in opposition to the available ones, does not require that the robot be in any special pose. Any pose can be used, what is also an importantpractical advantage. Keywords: robotics, robotic manipulator, robot kinematics, kinematic modeling, kinematic modeling method.

Third International Conference on Production Research – Americas’ Region 2006 (ICPR-AM06) IFPR – ABEPRO - PUCPR - PPGEPS



Kinematics describes – in robotics – the relations among the movements of joints and links of a robot. The joint variables expressthe position of joints. These positions are given as an angular measure for revolute joints and as a linear measure for prismatic joints. Through kinematics, a mapping among joint variables and the position and orientation is generated for each robot link. The mappings are static, with no concerns about speed (Craig, 1986). Position and orientation of robot’s links are defined attaching acoordinates system – also called frame – to each link. Therefore, the problem evolves to describing the relative position and orientation of the frames. This problem is solved using homogeneous transformations matrices, as indicated by Craig (1986). These matrices have many useful properties being the concatenation the most important one. Concatenation allows describing the position and orientation of anysegment, specially the last one where the tool is attached to, in respect to any coordinated system of interest. Usually, the coordinated system of interest is attached to the robot unmovable link (the base). The most convenient places to locate the frames on a link are the joints, making one of the coordinated axes to match the joint axis. Since an internal link has two joints, it is possible tochoose the “closer joint” or the “distant joint”, when the base of the robot is taken as a reference. Craig (1986) prefers the closer joint but many authors, Gonzalez

et al. (1987), Niku (2001), Paul (1981), Sciavicco & Siciliano (1996) and Stone (1987),
prefer the distant joint.


The Modified Denavit-Hartenberg Notation (MDH)

The original notation proposed by Denavit & Hartenberg(1955) describes a link by attaching a frame on the distant joint. Later on, Craig (1986) began to use the closer joint. However the choosing seems to be arbitrary. There is a possibility that this modification leads to a simpler inverse kinematic. Moreover, it was observed that models that use closer joints are easier to calibrate (Laus, 1998). So, the used notation in this work is called“Modified Denavit-Hartenberg Notation (MDH)”. In order to reduce the number of parameters, Denavit & Hartenberg (1955) have restricted the relative position and orientation of two adjacent frames making one of the

Third International Conference on Production Research – Americas’ Region 2006 (ICPR-AM06) IFPR – ABEPRO - PUCPR - PPGEPS

axes (the X axis) falls on the mutual perspective of the joint...
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