The arm we have been modeling is very simple; a two-jointed robot arm has little practical value except for very simple tasks. Let us add to the manipulator a modest capability for orienting as well as positioning a part or tool.

Accordingly, we will incorporate a third degree of freedom into the previous configuration to develop the *RR:R* manipulator shown in fig.

This third degree of freedom will represent a wrist joint. The world space coordinates for the wrist end would be

*x* = L_{1} cos θ_{1} + L_{2} cos (θ_{1} + θ_{2}) + L_{3} cos (θ_{1} + θ_{2} +θ_{3})

*y* = L_{1} sin θ_{1} + L_{2} sin (θ_{1} + θ_{2}) + L_{3} sin (θ_{1} + θ_{2} +θ_{3})

Ψ = (θ_{1} + θ_{2} +θ_{3})

We can use the results that we have already obtained for the 2-degree of freedom manipulator to do the reverse transformation for the 3 -degree of freedom arm.

When defining the position of the end of the arm we will use *x,y*, and Ψ. The angle Ψ is the orientation angle for the wrist. Given these three values, we can solve for the joint angles (θ_{1}, θ_{2}, and θ_{3}) using

*x*_{3} = *x* – L_{3} cos Ψ

*y*_{3} = *y* – L_{3} sin Ψ

Having determined the position of joint 3, the problem of determining θ_{1} and

θ_{1} reduces to the case of the 2-degree of freedom manipulator previously analyzed.