In many cases it is more important to be able to derive the joint angles given the end-of-arm position in world space. The typical situation is where the robot’s controller must compute the joint angles required to move its end-of-arm to a point in space defined by the point’s coordinates.

For the two-link manipulator we have developed, there are two possible configurations for reaching the point (*x,y*), as shown in fig.

This is so because the relation between the joint angles and the end effector coordinates involve ‘sine’ and ‘cosine’ terms. Hence we can get two solutions when we solve the two equations as given in the section above.

Some strategy must be developed to select the appropriate configuration. One approach is that employed in the control system of the Unimate PUMA robot.

In the PUMA’s control language, VAL, there is a set of commands called ABOVE and BELOW that determines whether the elbow is to make an angle θ_{2} that is greater than or less than zero, as illustrated in fig.

For our example, let us assume the θ_{2} is positive as shown in fig. Using the trigonometric identities,

cos (A + B) = cos A cos B – sin A sin R

sin (A + B) = sin A cos B + sin B cos A

we can rewrite Eqs

*x* = L_{1} cos θ_{1} + L_{2} cos θ_{1} cos θ_{2} – L_{2} sin θ_{1} sin θ_{2}

*y* = L_{1} sin θ_{1} + L_{2} sin θ_{1} cos θ_{2} – L_{2} cos θ_{1} sin θ_{2} _{ }

Squaring both sides and adding the two equations yields

\theta_{2}=\frac{x^{2}+y^{2}-L_{1}^{2}-L_{2}^{2}}{2 L_{1} L_{2}}Defining α and β as in fig we get

\begin{gathered} \tan \alpha=\frac{L_{2} \sin \theta_{2}}{L_{2} \cos \mid \theta_{2}+L_{1}} \\ \quad \tan \beta=\frac{y}{x} \end{gathered}Using the trigonometric identity

\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}we get

\tan \theta_{1}=\frac{\left[y\left(L_{1}+L_{2} \cos \theta_{2}\right)-x L_{2} \sin \theta_{2}\right]}{\left[x\left(L_{1}+L_{2} \cos \theta_{2}\right)+y L_{2} \sin \theta_{2}\right]}Knowing the link lengths L_{1} and L_{2} we are now able to calculate the required joint angles to place the arm at a position (*x,y*) in world space.

- See More : Robot sensors and actuators
- See More : Types of robot control
- See More : Configuration of robot controller
- See More : Robot applications