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Kinematical and Dynamical Models of KR 6 KUKA Robot, including the kinematic control in a parallel processing platform

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Kinematical and Dynamical Models of KR 6 KUKA Robot, including the kinematic control in a parallel processing platform
Universidad Industrial de Santander. UIS Colombia

John Faber Archila Díaz

Max Suell Dutra and Fernando Augusto de Noronha Castro PintoUniversidade Federal do Rio de Janeiro, UFRJ Brazil

1. Introduction
This chapter presents the study and modelling of KR 6 KUKA Robot, of the Robotics Laboratory, Federal University of Rio de Janeiro, see fig 1. The chapter shows the CAD model (Computer Aided Design), the direct kinematics, the inverse kinematics and the inverse dynamical model. The direct kinematic is based in the use ofhomogeneous matrix. The inverse kinematics uses the quadratic equations model. The dynamical model is based on the use of Euler-Lagrange equations, using the D-H (Denavit-Hartenberg) algorithm and taking into account the inertia tensor, which was found with help of CAE tools (Computer Aided Engineering), On the other hand the Jacobian of robot manipulator is present, it‘s necessary for the kinematiccontrol. The chapter finishes with the implementation of the inverse kinematic in one parallel processing platform and analyzes its performance.

Fig. 1. KR 6 KUKA Robot, Robotics Laboratory, Federal University of Rio de Janeiro UFRJ.

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Robot Manipulators, New Achievements

Space movement representation For the representation of space movements there are several methods such asrotation matrix, vectors, quaternions, roll pitch and yaw, Euler angles, homogenous matrix, among others (Barrientos, 1997). The selected method used for the developing of the direct kinematic model in this work is the homogeneous matrix. The basic concepts for mathematical models formulations are: Homogeneous Matrix Homogeneous matrices are 4X4 matrixes, which can represent rotations, translations,scales and perspectives (Ollero, 2001). In general, the homogeneous matrices represent linear transformations. The general form is presented in equation (1)
  R  3  3   A     P  1  3   T  3  1       E  1  1    

(1)

R  3  3  Corresponds to a matrix of three rows by three columns, representing rotations. T  3  1  Corresponds to an array of three rows bya column that represents translation. P  3  1  Represents a vector of a row of three columns representing the perspective. E  1  1  Corresponds to a scalar that represents the scale of the transformation.   For this case P  0 and E  1 Principal homogeneous matrix Rotation around the Z axis, figure 2.
Cos( ) Sin( )  Sin( ) Cos( )   0 0  0 0  0 0 1 0 0 0  0  1

Fig.2. Rotation around to axis Z. Translation Px, Py, Pz, figure 3. Y Py W Pz Z Fig. 3.Translation The movements in the space are represented by a series of rotations and translations, these rotations and translations, can be represented as a homogeneous matrix multiplication.
P

V U Px X
1 0  0  0 0 0 Px  1 0 Py   0 1 Pz   0 0 1

Kinematical and Dynamical Models of KR 6 KUKA Robot,including the kinematic control in a parallel processing platform

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Direct Kinematics The direct kinematics is the robot kinematic model. In this model, the movements of the robot (coordinates of degrees of freedom) are given and the final positions are found. See Figure 4.

Fig. 4. Direct Kinematics To find the direct kinematic model, using the homogeneous matrix method, is necessary tomake the moves of coordinated system from the fixed base until the last link. For each movement, homogeneous matrices are obtained and the final result is the product of these matrices. Inverse kinematics The inverse kinematics seeks the coordinates of each degree of freedom based on the final position of the robot. Figure 4.

Fig. 5. Inverse Kinematics The methods used are: the geometric...