Inverse Kinematics Example

To give you more insight into the solution process, particularly for the first three joint variables, we present an example of an RRP manipulator.

The point z_c could be considered to be the wrist center of a spherical wrist. We will assume that the Cartesian coordinates of z_c have already been calculated. To find 𝜃₁, we need to project z_c onto the ground plane - a trivial task since it only requires setting the z-coordinate = 0! Thus,

\mathbf{\theta_{1} = atan2(y_{c},x_{c})}

To solve for 𝜃₂, imagine 𝜃₁ = 0 and project links 2 and 3 onto the the x-z plane. Again, we use the atan2 function,

\mathbf{\theta_{2} = atan2(s,r)}

where,

\mathbf{r^{2} = x_{c}^{2} + y_{c}^{2}}

and,

\mathbf{s = z_{c} - d_{1}}

The final joint variable, d3, represents the length of the prismatic joint. From the figure, it is clear that d₃ ,is the hypotenuse of a right triangle with sides “r” and “s”. Solving for d₃,

\mathbf{d_{3} = \sqrt{r^{2}+s^{2}} =\sqrt{x_{c}^{2}+y_{c}^{2} + (z_{c}-d_{1})^{2}}}

And that’s it! The first three joint variables are now described explicitly in closed-form.