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 = L₁ cos θ₁ + L₂ cos (θ₁ + θ₂) + L₃ cos (θ₁ + θ₂ +θ₃)
y = L₁ sin θ₁ + L₂ sin (θ₁ + θ₂) + L₃ sin (θ₁ + θ₂ +θ₃)
Ψ = (θ₁ + θ₂ +θ₃)

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 (θ₁, θ₂, and θ₃) using

x₃ = x – L₃ cos Ψ
y₃ = y – L₃ sin Ψ

Having determined the position of joint 3, the problem of determining θ₁ and

θ₁ reduces to the case of the 2-degree of freedom manipulator previously analyzed.