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Algebraic methods

The algebraic solution of equation (2) exists only for a restricted class of cases. The joint angles could be expressed using the end effector position. The number of nonlinear equations increases with the DOFs ($n$ DOFs mean $n$ equations). Each joint angle - one by one - could be solved by the system of $n$ equations [Chi96]:

\begin{displaymath}\prod\limits^n_{i=2} A^i_{i-1}(\theta_i) = A^{-1}_1(\theta_1) \cdot A_n \eqno (3)\end{displaymath}

The $m+1^{st}$ joint angle is expressed using the $m$ previous angles. The last joint angle is expressed only by the end effector position and orientation.

The forward kinematics solution for the end effector position $X_{EE}(x,y)$ in 2 DOF structure could be expressed as follows:

\begin{displaymath}X_{EE} = \left( l_1 \cos \theta_1 + l_2 \cos (\theta_1 + \the... \theta_1 + l_2 \sin (\theta_1 + \theta_2 \right) ) \eqno (4)\end{displaymath}

Thus, by applying elementary trigonometry the inverse solution is:

\theta_2 = \cos^{-1} {x^2 + y^2 - l_1^2 - l_2^2 \over 2 l_1 ...
...r (l_2 \sin \theta_2)y + (l_1 + l_2 \cos \theta_2)x}
\eqno (5)

As the DOFs for most of cases are higher, the state vector is not analytically expressible using such a trivial way. Therefore more sophisticated approaches are necessary.

Lukas Barinka 2002-03-21