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I'm trying to calculate angular velocity of each axis by movement of end-effector,

If S1 and S2 is finite screw, and S2 had infinitesimal movement from S1. Also, let S1_ be (-)array of S1 and instantaneous screw of S1 is St1

So If I triangle product S2 and S1_ (S2△S1_=St1), It becomes almost instantaneous screw of S1 (I believe)

What I want to calculate is, if St1 is instantaneous screw, than can I calculate the angular velocity of each axis by using inverse jacobian with [ (J^-1)*St1 = answer ]? (jacobian is from S1, if S1=S1_6△S1_5△S1_4△S1_3△S1_2△S1_1, (the robot has 6 axis), jacobian matrix 'J' = [Su1_1, Su1_2, Su1_3, Su1_4, Su1_5, Su1_6], Su is for 'unit twist')

ZAE
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    I have a MSME in Robotics and wrote my thesis on Screw theory. I still don't understand what you are asking. If this is indeed a programming question and not a [Mathematics.SE] or [Physics.SE] question then **a lot more details** are needed to effectively answer. You need to explain the kinematics framework used, any conventions you follow (what is the triangle product here) and what you want to calculate. – John Alexiou Jan 09 '22 at 23:52
  • Maybe ask at https://robotics.stackexchange.com/ – John Alexiou Jan 10 '22 at 19:00

1 Answers1

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When you look at the kinematics recursively from the base to the end effector you have

vi = vi-1 + si ui

where vi is the velocity screw of each link, vi-1 is the velocity screw of the previous link, si is the unit screw of the joint axis, and ui the joint speed.

So the end effector has a final velocity screw of

v6 = s1 u1 + s2 u2 + s3 u3 + s4 u4 + s5 u5 + s6 u6

and I think you are asking on how to find the vector of joint speeds u = (u1, u2, u3, u4, u5, u6)

So you compose the 6×6 jacobian matrix J, by combining the individual joint axis unit screws in columns

J = [ s1 s2 s3 s4 s5 s6]

and invert the kinematics

v6 = J * u ⇒ u = J-1 v6

John Alexiou
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  • Thank you for your kind comment. I'm trying to figure out the numerical inverse kinematics which using Jacobian. For example, I made a model about PUMA type 6-axis articulated robot. and by triangle product of screws, I can easily calculate screw of end-effector but what I want to know is, If I give the deviation to end-effector made by adding a minimal angular displacement to each axes, and I knew the screw of end-effector after deviation, if I let the screw of before deviation as "S_b", screw after deviation as "S_a" and the vector of minimal angular displacement to each axes as "q" – ZAE Jan 10 '22 at 14:53
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    Can I calculate the minimal angular displacement "q" by -S_b△S_a = Jq ? The J is jacobian matrix as you mentioned above. If not, is there any numerical inverse kinematics method such as null space projection method or configuration control method? + Sorry for my bad english, and right now, I'm reading about your graduate thesis(Projective Articulated Dynamics) which seems very helpful for me. If there are any information to help me, or any recommending citation, please let me know if you're not bothered about it. – ZAE Jan 10 '22 at 14:54
  • Please edit your question and explain the Δ operator and maybe an example usage. – John Alexiou Jan 10 '22 at 15:41