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I have a 4 degree of freedom robot arm with 3 revolute joints and one prismatic joint as shown in the image:enter image description here

I can give just Target Point position to the Inverse kinematic chain and find a solution for the 4 DOF robot. Which moves 4 joints to touch the Target Point (Red Dot). However I also want certain condition for the rotation of the last Joint. I don't want to find a solution where the orientation of the last joint rotate along Z axis. In other words orientation of the end effector is important for me.

I am given a H.M to the inverse kinematic chain for the robot show above. My H.M is something like this (Target_x, Target_y, Target_z represents the red dot position):

[X X X Target_x]
[X X X Target_y]
[X X X Target_z]
[0 0 0 1       ]

As you can see Rotation in H.M, not set (I wrote Xs) and solution is found only for the position. But I would like to set a orientation, so that end-effector would not rotate over its own Z axis. In other words rotation in Yaw should be zero. What kind of a rotation matrix should I set inside the H.M. to achieve this?

                  [X_? X_? X_?]
   Rotation =     [X_? X_? X_?]
                  [X_? X_? X_?]

P.S: I am using IKPY inverse kinematic python library which allows me to find inverse kinematic solution by Position, Orientation, or Position and Orientation together. I don't think it is directly related with the question it self, but I wanted to point out.

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  • $\begingroup$ Is this a planar arm? What IK library are you using? Are you only trying to get the end-effector to reach the target point or do you want to achieve a target pose? In other words, does the orientation of the end-effector matter? $\endgroup$
    – Ben
    Nov 18 '21 at 17:09
  • $\begingroup$ @Ben This is UR5 robot. I added a prismatic joint to end of it and simulate it over Gazebo, I gave gave constraints to the first three joint of the UR5. In other words my robot becomes last three joints of UR5 robot, and a prismatic joint. I am using ikpy python library for inverse kinematic solutions. Lastly yes, orientation of the end-effector is important. $\endgroup$ Nov 18 '21 at 18:27
  • $\begingroup$ @Ben I edited the question, I simplified the the problem not to confuse you. You can ignore my comment and re-read the question. $\endgroup$ Nov 18 '21 at 18:34
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    $\begingroup$ ikpy tutorial for orientation shows aligning an eef axis with a target axis. Is this not what you want? Just choose the axis that gets you aligned the way you want? $\endgroup$ Nov 18 '21 at 18:51
  • $\begingroup$ @andymcevoy Problem is not aligning the axis with target axis, problem is giving constraint to the orientation of the I.K solution. $\endgroup$ Nov 19 '21 at 6:48
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Honestly, I would just treat this arm as a 2-dof arm whose end effector approach angle and distance from desired 2d point lives on a circle with a radius that matches the length of the last link (the prismatic joint), and not actually on the EE tip. Then, the approach angle is calculated given the fact that the location of joint 3 and the EE tip are known.

In other words, assume the length of the prismatic joint is "0" and solve the arm.

This isn't a super difficult arm to solve IK for, it is just that there are 4 parameters that you can play with on a planar arm, but there are a maximum of 3 dof to describer the EE tip frame. You have to apply an extra constraint somewhere, and I would do that at the prismatic joint. I say that because the longer the link, the more torque exerted on the joints across the entire arm.

Does this make sense?

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