Master Thesis - Hochschule Bonn-Rhein-Sieg

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5. Algorithms Master Thesis Björn Ostermann page 88 of 126 Both, the rotation and the translation between the coordinate systems, are defined by the setup of the workplace. They have to be calculated, if the setup of the workplace was changed. To automatically acquire the transformation matrix, the workplace was outfitted with three highly reflective balls, used as reference points, whose coordinates can be determined in the robot’s coordinate frame, as well as in the camera’s coordinate frame, as described in chapter 4.3.5. Figure 63 graphically shows the determination of the shift in origin and rotation. a) b) P 1 P P3 0 P 2 0‘ P 3‘ P 1‘ P 2‘ c) d) P 1 P 3 0 P 2 0‘ P 3‘ P 1‘ P 2‘ e) f) P 1 P 3 0 �a � � d T � �g � �0 P 2 b e h 0 c f i 0 ? � ? � � ? � � 1� 0‘ P 3‘ P 1‘ P 2‘ P 1 P 1 P 1 P 3 0 P 3 0 P 3 0 P 2 P 2 �a � � d T � �g � �0 Figure 63: Gaining the transposition matrix from one coordinate frame to another P1 to P3 mark the reference points measured by the camera 0 marks the origin of the camera’s coordinate system P1’ to P3’ and 0’ mark the respective points in the robot’s coordinate system P 2 b e h 0 c f i 0 0‘ P 3 ‘ P 1‘ P 2‘ 0‘ P 3‘ P 1‘ P 2‘ k � l � � m� � 1 � 0‘ P 3‘ P 1‘ P 2 ‘ 0 � k � � � � l � � � �m �
5. Algorithms Master Thesis Björn Ostermann page 89 of 126 Figure 63a to f show the camera’s coordinate system to the left and the robot’s coordinate system to the right. The first step in Figure 63a is the determination of the coordinates of the three reference points in both coordinate systems. In case of the robot’s coordinate system these points have been measured by moving the robot’s tool centre point above those points and noting the coordinates in the developed program. Since the robot as well as the reference points have a fixed location in this thesis, these measurements do not have to be repeated and can thus have been hardcoded in the program. An approach on the measurement in different workplaces is described in chapter 7.2. For measuring the points with the camera, a subprogram was created, that allows the user to visually target the reference points in the GUI. Since the balls are covered in highly reflective material, they show as white pixels in the reflection image (see Figure 29 in chapter 4.3.5). To compare the rotations of both coordinate systems, the measurement of these systems in Figure 63a would be sufficient, if all coordinates were acquired without any error. Since the camera’s measurement as well as the manual robot measurement both induce small errors, those errors have to be compensated. The chosen way to achieve this is to create coordinate systems from the measured points that are equal in their dimensions. This is achieved from Figure 63b to d. The second step of the algorithm in Figure 63b calculates the vectors from point one to point two and three in both coordinate systems. From those two vectors a third vector (Figure 63c) is created, that is orthogonally to both previous vectors. This is achieved by using the cross product (see Equation 10). Since the points are numbered, both coordinate systems’ third vector creates a right handed coordinate system. � p � � p � � p 1x 1y 1z � � p � � � � � p � � � � p 2x 2 y 2z � � p � � � � � p � � � � p 1y 1z 1x � p � p � p 2z 2x 2 y � p � p � p 1z 1x 1y � p � p � p Equation 10: Cross product of two three-dimensional vectors [61] Those three vectors build a three dimensional coordinate system in the camera’s respectively robot’s system. By comparing the rotations of each new system to their own original system, the rotation of the original systems towards each other could be determined. To compensate for the described errors in measurement, an additional step was included (Figure 63d) that creates a second orthogonal vector from the vector between point one and two and the previously calculated orthogonal vector. This results in three vectors that are orthogonal towards each other. After reducing their length to one, both coordinate systems are identical in all but their rotation. Thus the numerical errors from measuring are eliminated in further calculation. 2 y 2z 2x � � � � �
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Master Thesis ”Industrial jointed
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Statutory Declaration Master Thesis
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Contents Master Thesis Björn Oster
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1. Introduction Master Thesis Björ
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2. Overview on human-robot Master T
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3. Hard- and software Master Thesis
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Master Thesis - Hochschule Bonn-Rhein-Sieg

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