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Alexey Ivanov

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@Kitonum  Гениально, Юрий Николаевич. 

@vv  the numerical solution of an ODE cannot claim to be accurate when finding solutions to nonlinear equations. Each such task must be approached individually. 

@vv  @mmcdara 

This is a collaboration of the Draghilev method with NextZero. I ask you not to pay attention to the quality of the text.
A new variable v is introduced into the original equation; at its zero value, we get a solution to the original equation.
x1*(1+1.000100000*sin(x1^2)) - 2.129280519*v = 0;
In the program, we track when this variable takes a value of 0 and fix the solution to the original equation.
The starting point of the original variable in this case is 1.1, and the initial value of v is always 1 (this is like in the method of continuation of the solution by parameter).
For example, for the integration interval from -50 to 0 and for the starting point 1.1, 3614 solutions are found in the positive direction (printed in reverse order due to the integration interval).

@mmcdara  Come on, it's okay. The main thing is that we figured out the reason.

@vv  Thank you for your so kind feedback. 

@mmcdara  Look where the function changes sign

x := (1/2)*(4.155071365+4.158473414); f := x*(1+1.0001*sin(x^2));
 x := 4.155071365; f := x*(1+1.0001*sin(x^2));
 x := 4.158473414; f := x*(1+1.0001*sin(x^2));

All solutions are additionally verified by Draghilev's method. If interested, this method didn't miss any solutions.

@Carl Love  Think this coefficient is not directly related to Draghilev's method. I don’t know how to get a closed form solution for this autonomous differential equation system (and Maple does not show the progress of the solution). The expression will probably look better if there is no such factor for s. But I don't know how to do it. I only have experience with parameterizing the arc length of a curve for a numerical solution.

@Carl Love  I just have no words. Thank you. You and vv show me what Maple is.  


It seems that Draghilev's method works more "neatly". This can be seen in the example of a line segment with ends
[0.5, 1, 1.5], [-1, 0, 1]. If we weaken the attention to the solution based on Gröbner bases and do not additionally divide the segment into parts, then formally possible, but in reality hardly admissible discreteness may arise: 
This, as we can see, does not happen when using Draghilev's method:

An example of exactly the same movement as in the first example  PLAT 1. To do this, replace the third part of the program with this text:
Platform design options are endless. 

@Ronan  Thank you.
If “dynamic systems” means autonomous ODE systems, then Draghilev’s method is based on them. In turn, this method can be used to find a set of bifurcations of autonomous systems, both discrete and continuous. This is where my knowledge on this issue ends.
As for the Stewart platform, in the 3rd part of the program, you can easily remove my 3 rotary-inclined racks of variable length and replace them with your 6 variable-length racks.  My racks are controlled by rotation, tilt and length, while your racks will be adjustable in length. The number of Cij points in the text (where i is the number of the point, and j is its coordinate) you need to increase from 3 to 6 and connect to the platform (solid) in any way convenient for you. Then you will automatically receive a solution to the inverse kinematic problem for your new device, because at any time you will know the lengths of all the racks and, of course, the coordinates of all the necessary points. At the same time, we are not talking about possible emergency situations such as: crossing of racks, exceeding the permissible length, etc., however, all this can be controlled and, if necessary, make changes to its design.
As a matter of fact, the third part of the program is designed to use preliminary calculations of the first two parts and to save time when obtaining a solution to the inverse kinematic problem for any models that can reproduce the given trajectory of the platform movement.

My help also says: "If T is a triangle, and A a vertex of T, FindAngle (A, T) returns the internal angle of T at A".

Yes, thanks, checked. I used to think that VectorAngle calculates the angle in a similar way, that is, no more than Pi / 2.

Fixed comments in the text of the program, 

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