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I'm trying to carry out the forward displacement analysis of a parallel mechanism. This question is simplified as follows. Let $Q_1$, $Q_2$, $Q_3$, $A_1$, $A_2$, $A_3$ be six unit vectors that pass the origin. $Q_1$, $Q_2$, $Q_3$ are variables, and $A_1$, $A_2$, $A_3$ are constants. Assume $Q_1$ equals to $(x_1, y_1, z_1)$, $Q_2$ equals to $(x_2, y_2, z_2)$, $Q_3$ equals to $(x_3, y_3, z_3)$, $A_1$ equals to $(a_1, b_1, c_1)$, $A_2$ equals to $(a_2, b_2, c_2)$, $A_3$ equals to $(a_3, b_3, c_3)$. Besides the unit vector condition, the cosine value between every two vectors among $Q_1$, $Q_2$, $Q_3$ are constant, the cosine value between $Q_i$ and $A_i$, $i = 1, 2, 3$, are constant. Thus, the system of equations is described as follows: $$ \begin{cases} |Q_1| = 1, \\ |Q_2| = 1, \\ |Q_3| = 1, \\ x_1·x_2 + y_1·y_2 + z_1·z_2 = k_1, \\ x_1·x_3 + y_1·y_3 + z_1·z_3 = k_2, \\ x_3·x_2 + y_3·y_2 + z_3·z_2 = k_3, \\ x_1·a_1 + y_1·b_1 + z_1·c_1 = k_4, \\ x_2·a_2 + y_2·b_2 + z_2·c_2 = k_5, \\ x_3·a_3 + y_3·b_3 + z_3·c_3 = k_6, \\ \end{cases} $$ where

  • $k_1, k_2, k_3, k_4, k_5, k_6$ are constants, and
  • $Q_1, Q_2, Q_3$ are the unknown variables to be calculated.

I would appreciate it if someone could help me.

Jinping Zhang
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    MSE is a better fit for this question since MO is focused on questions related to current research in mathematics; see https://mathoverflow.net/help –  Jun 08 '22 at 12:40
  • Welcome to MO. The question will be, probably, moved to MSE (which shouldn't bother you in the slightest: it is pretty much the same set of people). However I want to ask you what exactly you mean by "solve": do you want an explicit analytic expression (which seems sort of hopeless because the corresponding polynomial is of high degree unless you want it in some perturbative regime when one can linearize everything) or just a numeric algorithm that produces a solution when it exists? – fedja Jun 08 '22 at 12:40
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    Potential solvers would likewise appreciate it if you format this nicely. –  Jun 08 '22 at 12:46
  • In the fifth equation, do you really mean $y_2$ or it must be $y_3$ instead? – მამუკა ჯიბლაძე Jun 08 '22 at 14:11
  • You say verbally that each $Q_k,\ A_k$ are unit vectors. But you only explicitly mention the $Q_k$ in the system of equations. I would think either you would mention both or neither of $Q_k,\ A_k$ in the system. They need not be since you already say verbally they are unit vectors. – coffeemath Jun 09 '22 at 10:38
  • Geometrically this is two triples of points on the unit sphere where the angular distance between the points of one triple are given by $k_1,k_2,k_3$ and the the angular distance between corresponding points in the two triples are given by $k_4,k_5,k_6$. You can use this interpretation to simplify the problem. There may not be a solution for some values of the $k$ constants. – Somos Jun 09 '22 at 10:49

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