I won’t say that I understood what the text of the message is about, but as for solving the inverse problem of the kinematics of manipulators, there is one simple and unambiguous way to solve the inverse problem – a way to reduce this problem to the kinematics of lever mechanisms. This is achieved by mathematically reducing the number of degrees of freedom to 1 (that is, by the introduction of additional restrictions using mathematical equations).
Maple examples:
https://www.mapleprimes.com/posts/214084-One-More-Way-Of-Inverse-Kinematics
https://www.mapleprimes.com/posts/215233-Another-Example-Of-A-Real-Manipulator-Model
https://www.mapleprimes.com/posts/213631-Something-About-One-Degree-Of-Freedom
https://www.mapleprimes.com/posts/213534-A-Little-About-Controlled-Platforms-parallel-Manipulators
 

Why an optimization package? The zeros of the derivative of the Gamma function indicate its extreme values. Having found the zeros of the derivative in the desired range, you will select the amplitudes you need. 
Or did I not understand something?

@vv 
I wish I could see all these pictures at school. In due time, of course.
 I wanted to see what the area of polynomial roots looks like and how to graphically interpret the presence of complex and real roots in the same area. Well, with a reasonable dimension, I think it is possible to somehow understand what's what, using various examples. 
By the way, going around the circle, you can find all the branches leading to solutions, and go to them. (As an alternative solution.)

Have you tried using the nops procedure?
 For example:
SN:=(solve(..));
nops(SN);

@mmcdara   It's okay. This is done intentionally - this can be seen from the text of the program: look at the display, where graphs of different types are located in one direction and in the opposite direction.
(This is if I understood you correctly, because I use an online translator.)

@Pepini 
Apparently I couldn't figure out what the question was. 
The spatial method (Draghilev's method) finds the same:
[-1.675392259, 0.108903871]

@Pepini  
Basically, you have one equation and one variable. If you want to view your expressions in 3d, then for example
 

restart; with(plots):
a:= 10; 
smin:= -3; smax:= 2; 
f:= exp(t)+exp(-t^2-2*t); 
g:= exp(-t^2)+exp(-t^2-2*t); 
spacecurve([f, g, t], t = smin .. smax, thickness = 3, view = [-a .. a, -a .. a, -a .. a], axes = normal, color = red)

(Formally, f, g, t can act as independent variables, and you will consider their original expression as a system of two equations with three variables.)

@mmcdara Thanks a lot for your additional work. I will try to adopt your technique, but, most likely, as usual for me, in the form of a ready-made block. Of course, to some extent I understand your text, but I still never can work at such high level. Maple is something fantastic, and that's why I manage to somehow show the some algorithms that were once interesting to me. I do not hide, I am not a professional.

@tomleslie  Oh, I get it, thanks.

@mmcdara  Yes, thanks, everything works now. (But hard for me to understand the text, because I work with Maple at a very amateur level.)
 

@hamideh  For this polynomial equation, you can express one variable in terms of two others. This is the same as my previous message, only with the formulas.

restart:
 f1 := -(1/400)*Pi^2+(.816196912*x1-.408547794*x2-.408547794*x3)^2+(.707106781*x2-.707106781*x3)^2; 
 solve(f1, [x3]);

@hamideh  For example, start a double loop. In the outer loop, change the value of x1, and in the inner loop, change the value of x2 relative to x1, and  solve (f1, [x3]). The solve (f1, [x3]) procedure finds all solutions x3, and you choose the ones you need.

f1 := -(1/400)*Pi^2+(.816196912*x1-.408547794*x2-.408547794*x3)^2+(.707106781*x2-.707106781*x3)^2;

(This is a fairly accurate polynomial approximation to your equation.) Do the same for all periodic solutions that fall within the required range.
If you are not satisfied with the accuracy of the solution, then you can always supplement the loop with the fsolve procedure, but this time relative to your original equation, indicating the corresponding range for the fsolve procedure.
You will succeed.

@Kitonum 
 Yes, it was necessary to start much more to the left. Then everything fits together.

                               1
                         -47.53726862
                               2
                          -46.64054105
                               3
                          -29.30913089
                               4
                          -28.07302137

@dharr  please show your 27 solutions. In my version of Maple, your program doesn't work. I counted 23 on the graph, NextZero and Draghilev's method also gave 23
 

                                                           -8
   1, [HFloat(-4.3095768833401635)], 9.06687679380624446 10  
                                                           -7
   2, [HFloat(-1.1023789158944424)], 1.50646519592179118 10  
                                                          -9
   3, [HFloat(13.407730779581346)], 7.79979625331606030 10  
                                                          -8
   4, [HFloat(18.113661217228824)], 1.66432153059226096 10  
                                                          -8
   5, [HFloat(31.783308581635165)], 2.72685562741070698 10  
                                                          -7
    6, [HFloat(37.07999963464075)], 1.09762171685012788 10  
                                                          -9
    7, [HFloat(39.84837784271576)], 3.93514354435353653 10  
                                                          -8
    8, [HFloat(43.82587731863776)], 6.71648974492899953 10  
                                                          -8
    9, [HFloat(50.25908342435418)], 2.93121961577290369 10  
                                                           -7
   10, [HFloat(55.575394679681445)], 1.35574310347608140 10  
                                                           -7
   11, [HFloat(55.882759086494076)], 1.18089220589590038 10  
                                                          -8
   12, [HFloat(63.09487085630448)], 6.04421363792351229 10  
                                                          -8
    13, [HFloat(68.2966429866009)], 8.61217160919025559 10  
                                                          -9
   14, [HFloat(71.81338112721974)], 8.58604581877031592 10  
                                                          -8
    15, [HFloat(74.6968525451681)], 4.96600724708695652 10  
                                                          -8
   16, [HFloat(82.69383468107193)], 1.58464290578308464 10  
                                                          -8
   17, [HFloat(84.61651574863721)], 2.03942768228770888 10  
                                                          -8
   18, [HFloat(89.47047663737627)], 6.66622110845338512 10  
                                                          -8
   19, [HFloat(93.33508470604818)], 8.73743196683207657 10  
                                                           -7
   20, [HFloat(108.15523037467786)], 1.70414072853120047 10  
                                                           -7
   21, [HFloat(111.66954228728133)], 1.16034193475833548 10  
                                                           -8
   22, [HFloat(127.67295282695558)], 2.57006522730307552 10  
                                                           -8
   23, [HFloat(128.95061538591767)], 1.56371757764617314 10  

@hamideh 
For example, just print all the points that are animated and you will find out how many there are and see their numerical values...
Do you know how to get the equation of a circle by three  points that do not lie on one straight line? Also, are you familiar with the angles of rotation around the coordinate axes? This is all plus the selection of the values ​​of the periods, and you get algebraic equations instead of trigonometric ones, but with a certain accuracy. There are 5 parameters in total.
I have given the values ​​of these parameters rather roughly, but they can be refined using Maple optimization procedures.