one man

Alexey Ivanov

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A way of cutting holes on an implicit plot. This is from the field of numerical parameterization of surfaces. On the example of the surface  x3 = 0.01*exp (x1) / (0.01 + x1^4 + x2^4 + x3^4)  consider the approach to producing holes. The surface is locally parameterized in some suitable way and the place for the hole and its size are selected. In the first example, the parametrization is performed on the basis of the section of the initial surface by perpendicular planes. In the second example, "round"  parametrization. It is made on the basis of the cylinder and the planes passing through its axis. Holes can be of any size and any shape. In the figures, the cut out surface sections are colored green and are located above their own holes at an equidistant to the original surface.

It can be considered as continuation of these topics: “Determination of the angles of the manipulator with the help of its mathematical model. Inverse problem.”  and  “The use of manipulators as multi-axis CNC machines”.
There is a simple two-link manipulator with three degrees of freedom. We change its last link with a fixed length to a variable length link. By this action we added one degree of freedom to our manipulator. Now we have a two-link manipulator with 4 degrees of freedom.
In a particular mathematical model the variable length of the link is related to the x7, and in the figure the total length of the last link is displayed as the result of subtracting the fixed part of the link (L2) from x7 (equation f2). At the same time, the value of the variable x7 depends on the inclination of the first link (equation f4).  x1, x2, x3 are the coordinates of the moving point of the first link, x4, x5, x6 are the coordinates of the operating point.  The  equation f2 defines the type of connection between the links, the equations f5 and f6  are the trajectory of the working point.

On the convex part of the surface we place a curve (not necessarily flat, as in this case). We divide this curve into segments of equal length (in the text Ls [i]) and divide the path that our surface will roll (in the text L [i]) into segments of the same length as segments of curve. Take the next segment of the trajectory L [i] and the corresponding segment on the curve Ls [i], calculate the angles between them. After that, we perform well-known transformations that place the curve in the space so that the segment Ls [i] coincides with the segment L [i]. At the same time, we perform exactly the same transformations with the equation of surface.

For example, the ellipsoid rolls on the oX1 axis, and each position of the ellipsoid in space corresponds to the equation in the figure.


And similar examples:

We produce only one rotation of the cube by the angle Pi/2, and then repeat it at the following points, changing the colors of the faces in turn. And so the illusion is created that the cube is rolling along a straight line without slipping.
(Just a picture without any sense.)

It post can be called a continuation of the theme “Determination of the angles of the manipulator with the help of its mathematical model. Inverse  problem”.
Consider  the use of manipulators as multi-axis CNC machines.
Three-link manipulator with 5 degrees of freedom. In these examples  one of the restrictions on the movement of the manipulator links is that the position of the last link coincides with the normal to the surface along the entire trajectory of the working point movement.
That is, we, as it were, mathematically transform a system with many degrees of freedom to an analog of a lever mechanism with one degree of freedom, so that we can do the necessary work in a convenient to us way.
It seems that this approach is fully applicable directly to multi-axis CNC machines.

(In the texts of the programs, the normalization is carried out with respect to the coordinates of the last point, in order that the lengths of the integration interval coincide with the path length.)

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