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Alexey Ivanov

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12 years, 53 days

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Controlled platform with 6 degrees of freedom. It has three rotary-inclined racks of variable length:

and an example of movement parallel to the base:

Perhaps the Stewart platform may not reproduce such trajectories, but that is not the point. There is a way to select a design for those specific functions that our platform will perform. That is, first we consider the required trajectories of the platform movement, and only then we select a driving device that can reproduce them. For example, we can fix the extreme positions of the actuators during the movement of the platform and compare them with the capabilities of existing designs, or simulate your own devices.
In this case, the program consists of three parts. (The text of the program directly for the first figure : In the first part, we select the starting point for the movement of a rigid body with six degrees of freedom. Here three equations f6, f7, f8 are responsible for the six degrees of freedom. The equations f1, f2, f3, f4, f5 define a trajectory of motion of a rigid body. The coordinates of the starting point are transmitted via disk E for the second part of the program. In the second part of the program, the trajectory of a rigid body is calculated using the Draghilev method. Then the trajectory data is transferred via the disk E for the third part of the program.
In the third part of the program, the visualization is executed and the platform motion drive device is modeled.
It is like a sketch of a possible way to create controlled platforms with six degrees of freedom. Any device that can provide the desired trajectory can be inserted into the third part. At the same time, it is obvious that the geometric parameters of the movement of this device with the control of possible emergency positions and the solution of the inverse kinematics problem can be obtained automatically if we add the appropriate code to the program text.
Equations can be of any kind and can be combined with each other, and they must be continuously differentiable. But first, the equations must be reduced to uniform variables in order to apply the Draghilev method.
(These examples use implicit equations for the coordinates of the vertices of the triangle.)

As an addition to the post.
Non-orientable surface in the sequence of orientable surfaces. In the picture we see the equations corresponding to the current surface plot.
Just entertainment.

One way to find the equation of an ellipse circumscribed around a triangle. In this case, we solve a linear system of equations, which is obtained after fixing the values of two variables ( t1 and t2). These are five equations: three equations of the second-order curve at three vertices of the triangle and two equations of a linear combination of the coordinates of the gradient of the curve equation.
The solving of system takes place in the ELS procedure. When solving, hyperboles appear, so the program has a filter. The filter passes the equations of ellipses based on by checking the values of the invariants of the second-order curves.
FOR_ELL_Т  ( Fixed comments in the text  01, 08, 2020)

An attempt to find the equation of an ellipse inscribed in a given triangle. 
The program works on the basis of the ELS procedure.  After the procedure works, the  solutions are filtered.
ELS procedure solves the system of equations f1, f2, f3, f4, f5 for the coefficients of the second-order curve.
The equation f1 corresponds to the condition that the side of the triangle intersects t a curve of the second order at one point.
The equation f2 corresponds to the condition that the point x1,x2  belongs to a curve of the second order.
Equation f3 corresponds to the condition that the side of the triangle is tangent to the second order curve at the point x1,x2.
The equation f4 is similar to the equation f2, and the equation f5 is similar to the equation f3.
For example

One of the forums asked a question: what is the maximum area of a triangle inscribed in a given ellipse x^2/16 + y^2/3 - 1 = 0? It turned out to be 9, but there are infinitely many such triangles. There was a desire to show them in one of the possible ways. This is a complete (as far as possible) set of such triangles.
(This is not an example of Maple programming; it is just an implementation of a Maple-based algorithm and the work of the Optimization package).

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