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$$ R_2+R_3-R_4-R_0=0 $$
$$ R_3-R_4 = R_0 - R_2 $$
$$ 7&(cos(\theta_3)+jsin(\theta_3)) - 6&(cos(\theta_4)+jsin(\theta_4)) = 10.5 - 2.5&(cos(\theta_2) + j sin(\theta_2)) $$
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\begin{arraycases}{rcl} y&cos 7cos(\theta_3) - 6&cos6cos(\theta_4) = 10.5 - 2.5&cos5cos(\theta_2) \\ f(x,y,z) & = & x + y + z \end{array} 7jsin(\theta_3) - 6jsin(\theta_4) = - 2.5j sin(\theta_2) \end{cases} |
This is a non-linear system of equations. There are an infinite number of solutions for theta_3 and theta_4. Even when we limit our range to [0,2π], there are 2 unique solutions, because the mechanism has a toggle position. To keep the solution consistent with the real world, all non-linear equations are solved numerically with initial conditions based on the measurements for the chair. In this case, the initial conditions and solutions are
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$$ \theta_{3,guess} = 45 \hspace{1ex} degrees ; \theta_{4,guess} = 120 \hspace{1ex} degrees ; \theta_{2,t=0} = 165 \hspace{1ex} degrees $$
\begin{cases} \theta_{3,t=0} = 2.6 \hspace{1ex} degrees \\ \theta_{4,t=0} = 170.8 \hspace{1ex} degrees \end{cases} |