Solving quadratic equations involving Quaternion variables

Questions
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1

I am solving a system of quadratic equations that comes from the 3D rotational kinematics problem expressed by Quaternion variables after simplifying the expressions:

Solve[{-2/dT (-qbr1 qbrp0 + qbr0 qbrp1 - qbr3 qbrp2 + qbr2 qbrp3) + 
     2 constC (-qb1 qbr0 + qb0 qbr1 - qb3 qbr2 + qb2 qbr3) (qb0 qbr0 +
         qb1 qbr1 + qb2 qbr2 + qb3 qbr3) == 
    const1 && -2/
      dT (-qbr2 qbrp0 + qbr3 qbrp1 + qbr0 qbrp2 - qbr1 qbrp3) + 
     2 constC (-qb2 qbr0 + qb3 qbr1 + qb0 qbr2 - qb1 qbr3) (qb0 qbr0 +
         qb1 qbr1 + qb2 qbr2 + qb3 qbr3) == 
    const2 && -2/
      dT (-qbr3 qbrp0 - qbr2 qbrp1 + qbr1 qbrp2 + qbr0 qbrp3) - 
     2 constC (qb3 qbr0 + qb2 qbr1 - qb1 qbr2 - qb0 qbr3) (qb0 qbr0 + 
        qb1 qbr1 + qb2 qbr2 + qb3 qbr3) == const3 && 
   qbr0*qbr0 + qbr1*qbr1 + qbr2*qbr2 + qbr3*qbr3 == 1}, {qbr0, qbr1, 
  qbr2, qbr3}, Cubics -> False, Quartics -> False]

It keeps running but, so far for 5 hours, cannot get the solution yet. This seems possible to solve and not too difficult. Did I make something wrong, or should I solve using other methods or functions? Please advise me.

2

We can calculate very quickly using a numerical algorithm using NSolve:

dT = 1;
constC = 2;
qbrp0 = 3;
qbrp1 = 4;
qbrp2 = 5;
qbrp3 = 6;
qb0 = 7;
qb1 = 8;
qb2 = 9;
qb3 = 10;
const1 = 11; 
const2 = 12;
const3 = 13;

EQ = {-2/dT *(-qbr1*qbrp0 + qbr0* qbrp1 - qbr3*qbrp2 + qbr2*qbrp3) + 
    2*constC*(-qb1*qbr0 + qb0 *qbr1 - qb3*qbr2 + 
       qb2*qbr3)*(qb0 *qbr0 + qb1* qbr1 + qb2*qbr2 + qb3 *qbr3) == 
   const1, -2/
      dT *(-qbr2 *qbrp0 + qbr3* qbrp1 + qbr0* qbrp2 - qbr1*qbrp3) + 
    2*constC *(-qb2* qbr0 + qb3 *qbr1 + qb0*qbr2 - 
       qb1*qbr3)* (qb0* qbr0 + qb1* qbr1 + qb2* qbr2 + qb3*qbr3) == 
   const2, -2/
      dT* (-qbr3*qbrp0 - qbr2 *qbrp1 + qbr1* qbrp2 + qbr0* qbrp3) - 
    2* constC* (qb3*qbr0 + qb2*qbr1 - qb1*qbr2 - 
       qb0*qbr3)* (qb0* qbr0 + qb1* qbr1 + qb2 *qbr2 + qb3* qbr3) == 
   const3, qbr0*qbr0 + qbr1*qbr1 + qbr2*qbr2 + qbr3*qbr3 == 
   1};
 NSolve[EQ, {qbr0, qbr1, qbr2, qbr3}]
 (*{{qbr0 -> 0.300808 - 0.0888247 I, qbr1 -> 6.31248 - 12.6133 I, 
   qbr2 -> -13.5724 - 0.470986 I, 
   qbr3 -> 6.92771 + 10.5743 I}, {qbr0 -> 0.300808 + 0.0888247 I, 
   qbr1 -> 6.31248 + 12.6133 I, qbr2 -> -13.5724 + 0.470986 I, 
   qbr3 -> 6.92771 - 10.5743 I}, {qbr0 -> -0.393081, qbr1 -> -0.471046,
    qbr2 -> -0.528483, qbr3 -> -0.586778}, {qbr0 -> 0.39098, 
   qbr1 -> 0.469899, qbr2 -> 0.528865, 
   qbr3 -> 0.588754}, {qbr0 -> 0.897994, qbr1 -> -0.21354, 
   qbr2 -> -0.285171, qbr3 -> -0.258236}, {qbr0 -> -0.909058, 
   qbr1 -> 0.243877, qbr2 -> 0.169177, qbr3 -> 0.292432}}*)

 NSolve[EQ, {qbr0, qbr1, qbr2, qbr3}, Reals]
 (* {{qbr0 -> -0.393081, qbr1 -> -0.471046, qbr2 -> -0.528483, 
   qbr3 -> -0.586778}, {qbr0 -> 0.39098, qbr1 -> 0.469899, 
   qbr2 -> 0.528865, qbr3 -> 0.588754}, {qbr0 -> 0.897994, 
   qbr1 -> -0.21354, qbr2 -> -0.285171, 
   qbr3 -> -0.258236}, {qbr0 -> -0.909058, qbr1 -> 0.243877, 
   qbr2 -> 0.169177, qbr3 -> 0.292432}}*)

Regards M.I.

3

Thanks, M.I. But that’s numerical solution is not what I really look for. I’d like the closed form solution, or the algorithm, so that I can implement in code and solve in real-time.

4

The numeric solving takes just over 0.1 seconds on my slow laptop. This might be faster than plugging numerical values into a huge complicated set of parametrized Root objects. And given the way the parameters appear in the equations, my guess is the symbolic result will be huge.

5

Finally, the Solve quit silently with no return or abort. I still believe that there is a way to solve this nice-structured and physically meaningful algebraic system. Maybe, we need some trick to solve this kind of math Olympiad-like problem.