@roman_pearce 

Thank you too. I believe that the Gröbner bases is a great achievement in mathematics as a generalization of the Gauss method. I have a lot of examples of polynomial systems for finding a finite number of solutions to find then infinite solutions. However, the systems of equations occur transcendental, and then have to work without the Gröbner bases ..

@acer  Thank you.

@vv  It seems to me that so clear to understand.
Maple not always uses complex numbers by default.
For example, I do not know how to get all the solutions of a polynomial system by Maple, if not to make the proposed replacement (x+Iy). See RootFinding [Isolate]
Unless, of course, we do not use evalf (solve (...))

restart;
z := x+I*y;
f1 := evalc(Re(exp(z)+1));
f2 := evalc(Im(exp(z)+1));
fsolve({f1, f2});
solve({f1, f2});
solve({f1, f2}, allsolutions);

@Ronan If it is necessary a list
LS := convert(`union`(convert(L, set), convert(S, set)), list);
(or  L:=...)

У Вас есть ещё какие-нибудь открытия?

   @Bendesarts  Animation and text of the program

   Benjamin.mw 

        About  the idea creating a CAD lever mechanisms. We can consider as degrees of freedom levers length. For example, in the animation for the selection of the desired trajectory of the Q, we change the length of AB (green), keeping unchanged all the other values and fix the position of point A.

@Bendesarts 
Thank you for your feedback on this work. And a special thank you for your interest in our method.
I cannot accurately determine the timing of writing the work in English. While try using a translator to parse the publication of this section  http://www.mapleprimes.com/posts/202821-Calculating-Link-Mechanisms 
http://www.mapleprimes.com/view.aspx?sf=201279_Comment/Method_Mechan_PDF.pdf
I suggest you ask me specific questions on the text of the program. Literally you can ask for each line of text you selected.
Still, as happens, I will say on general provisions of my ideas calculation linkages. The idea is simple: we make a mathematical model of the mechanism in the form of underdetermined systems of nonlinear equations. It is such a system of equations, in which the number of variables is greater than the number of equations. The difference between the number of variables and the number of equations corresponding to the number of degrees of freedom of the mechanism. As the variables we take the coordinates of the moving points of the mechanism, but the variables can be any other convenient values. And all the mechanism is described by the system of equations (relations). The solution to this system of equations is a solution of the Cauchy problem for the ODE system and complies with all the possible provisions of the  mechanism for a particular assembly.
We make no distinction between 2d and 3d mechanisms.
According to the text of the program we with your possible questions can make out the work of a particular mechanism.
(There is an idea and an offer for all wishing to establish on this basis CAD lever mechanisms.)

@Markiyan Hirnyk Answer, what impression about you after you have removed my correspondence with Kitonum and kept silent?
And more, why do you climb into topics where your expertise leaves much to be desired?

@Markiyan Hirnyk Start with http://www.mapleprimes.com/posts/202821-Calculating-Link-Mechanisms, then read carefully my previous message, there http://www.mapleprimes.com/posts/200663-Spatial-Fourbar-Linkages-
And do not forget to share your successes in mastering MapleSim.

@Thomas Richard 

Thank you. It is known that for flat linkages there are many methods of calculation. I promote a universal method for all kinds of linkages and plan to write a paper in English and put on Maplesoft
I can show some the calculations that are unlikely to get be done in MapleSim, for example:
http://www.mapleprimes.com/posts/200663-Spatial-Fourbar-Linkages- 

Variant of animation nearer to a drawing

@vv  So are you talking about ... Well, turn to the desired angle. This is all done by hand

@vv  What kind of animation you say? I gave the main text of the program to the first animation (the program works on  Maple13). The only difference is in the color reproduction.
Another text to the first animation I have not.