forward and inverse kinematics of arm

Question

I'm trying to figure out to a problem with a planar manipulator. The manipulator has a base effector and L1 and L2 with 3 joints.

1. Question: INVERSE KINEMATICS

Given the posit_ion(x, y, z) of the tool I calculate the values of the joint variables ($\theta_1$, $\theta_2$, $\theta_3$ and $\gamma$) as follow:

From the position(x, y, z) I calculate:

1. $\gamma= atan2(z, x)$
2. $x'= \cos(\gamma)*x + \sin(\gamma)*z$
3. $y'= y$

I used $x'$ and $y'$ to calculate (through) the angles of joints like this:

is it correct?

2. Question FORWARD KINEMATICS:

Given the joints variables ($\theta_1$, $\theta_2$, $\theta_3$ and $\gamma$) I need to calculate the position(x, y, z) of the tool. In this case I don't know how to continue.

Anyone can help me?

Answer 1

From now, I will solely answer your second question about forward kinematics, which is usually easier to solve than inverse kinematics.

First you should sketch your robot in a plan using textbook representations for revolute and prismatic joints, this will make it clearer for you and the others. I needed to look at the picture to see that the 4 rotations are not along parallel axis.

There are two approaches based on your degree of confidence;

1. you consider yourself at ease and you derive it from the position of the base by propagating in plans, this easily doable here are 3 joints remain in the same plan thus you only need to compute position in the plan, plus you don't have spherical joint or other complex mechanism

2. you fell less at ease and you use a systematic way, for example one of Denawit-Hartemberg parametrization. This option is well documented in any textbook or in DeLuca's slides, freely available online. This way is systematic, and less error-prone.

Both methods boil down to the use of transformation matrices so are pretty efficient to implement nowadays.

An hint about your inverse kinematics you have 4 degrees of freedom and want to find the inverse kinematic of a 3D position, there is potentially an infinite number of solution for such a problem.