Creating the reach dome of the printer

The reach of the robotic-arm is shown in figure1.8, yet the printing range is smaller due to the factors given in subsection1.4.3, these were the following factors:

1. The shape of the robot head;

2. Singularities;

3. Breakpoint of the 5th axis.

The reach of the printer is calculated with help of the forward kinematic model given in section 2.3, equations 2.7 to 2.12. The exact range of the printer will be used for a pre-check. By placing the to be printed object in the exact range, all pixels will be checked and a pre-check is executed. By performing this check, the tool is able to exclude the set of pixels that are located deep within the range of the printer, where no errors can occur. After this check, only the pixels that are located on the edge or outside of the range, are checked by the inverse kinematic model of subsection2.4.1. The operator will check these pixels in the current situation by doing a physical dry test. This approach eliminates the biggest proportion of the pixels in the object and reduces the calculation time.

3.1.1 Forward kinematic model

By plotting every combination of θ2 and θ3 where θ5 is being determined by the constraints all the possible points within the reach are determined. Each point represents a combination of axes 2, 3 and 5. All combinations need to fit the constraints stated in section2.3. This creates a point cloud that is the section of the reach, this mesh is shown in figure3.1. Every point that is colored blue in figure3.1is reachable by the robotic-arm.

Figure 3.1: The point cloud that describes the reach of the printer.

The point cloud is loaded into Rhino. The outline can be traced to describe the exact reach of the printer. The section can be revolved around the Z-axis to create the 3D dome of the reach, this dome is visible in figure 3.2. This dome can be used by the D2P engineers to check if the objects they design fit within the reach. The object cannot intersect the dome to ensure that the reach is sufficient. The D2P engineer has to keep in mind that if the printer is standing in its lowest position it cannot print lower than the ground, unless the ground is in fact lower at the print position.

Figure 3.2: The 3D dome that describes the reach of the printer.

A test is done to check if a pixel fits within the reach of the robot by checking if the pixel is between the inner point and outer point of the dome. Figure3.3shows the outline, where the inner points are coloured red and the outer points are blue. The inner and outer points are calculated with the angle from the X-axis and the exact length of the vector from 0 to the point. The set of angles are put in order. A delta is taken from the set, which the shortest and longest vector are

saved. These vectors represent the inner and outer point of the dome for the taken delta. Doing this for each delta in set of points will result in the outline of the dome.

An offset of the outline can be made with the same method as the outline, by subtracting an offset length from each outerpoint-vector and adding the offset length to the innerpoint-vector.

Afterwards a check is needed to confirm if the points are within the reach. The inner offset has to be shorter compared to the outer offset vector length and vice versa. This offset can be used to create a border zone between the safe space within the reach and the space that needs to be checked by the inverse kinematic model. The pixels can be checked in a similar way as the outline and offset are created:

1. By calculating the vector length and angle of the pixel relative to the robot base.

2. By comparing the vector length of the pixel to the vector length of the section that fits within the delta angle in the set.

If the angle is not available in the set, the pixel is either above or below outside the reach. If the length of the vector is longer than the outerpoint or shorter than the innerpoint, then the pixel is outside the reach as well. A similar comparison is applicable to the offset points to check if the pixel is within the safe zone or in the ”danger” zone. The set that is outside the safe zone (the points in the danger zone and outside the reach) will be sent to the inverse pixel checker for further inspection. The set which fits in the safe area can be neglected.

Figure 3.3: Outline of the section of figure3.2.

3.1.2 Differences between the printer and the robotic arm

The differences between the printer reach and the robotic-arm reach are shown in figure3.4. There are two zones in the reach that are outside the printer range. Zone 1 due to the singularity and break point of the 5th-axis. The red line that separates zone 1 and the printer range is a line where singularities occur. The area indicated by 1 is only reachable if the 5th-axis is in a negative configuration.

Zone 2 - including the area behind the robot - can not be reached due to the break point of axis 5. The 3rd-axis has to go over its break point as well in order to gain access to that area. So printing in that area is not possible in any way.

The small reach limitation of the 5th-axis due to the printer head is not visible in figure 3.4.

This reach problem has a small effect on the top blue line next to number 2. The robot is able to extend this line a little upward with the full reach of the 5th-axis and due to the singularity problem in the bottom, there is no limitation there. The reach limitation is not visible because it is relatively small. Figure3.4validates the cross-section of the reach by placing the cross-section from figure3.3over the full reach of the robot from figure1.8. The outer blue line is exact on the edge of the outside reach, and the inner red line is exact on the edge of the inner reach.

Figure 3.4: The outline of figure3.3 in the reach of the robot of figure1.8, where sections 1 and 2 are not reachable during printing, dimensions are in mm.

3.1.3 Standardized sizes of objects

With the dome created in Rhino standardized objects sizes can be made. These object sizes give the D2P engineers the possibility to create objects that are not too big or barely fit within the range of the printer. This also helps the clients of CyBe, who have difficulties finding a printer with the correct range for the product they want to make. If an object fits within the standardized size it will automatically fit within the range, so no further inspection is required to ensure printability.

The standardized sizes are located in table3.1to3.3. These objects all have rectangular shapes.

Width [mm] Depth [mm] Height [mm]

1000 300 3450

1000 500 3050

1000 1000 2600

1000 1500 2250

1000 1750 1900

Table 3.1: Standardized sizes of boxes that fit within the range with a width of 1000 mm.

Width [mm] Depth [mm] Height [mm]

Table 3.2: Standardized sizes of boxes that fit within the range with a width of 2000 mm.

Width [mm] Depth [mm] Height [mm]

Table 3.3: Standardized sizes of boxes that fit within the range with a width of 3000 mm.

Width [mm] Depth [mm] Height [mm]

Table 3.4: Standardized sizes of boxes that fit within the range with a width of 4000 mm.