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Inverse Kinematics of Standford Manipulator
using Pieper’s Theorem
Presented By: Mian Waqar ul Mulk
Given
Solution
As We Know That
is Given which is H. we will first find T03 i.e. inverse position and then inverse orientation T36.
To find Ɵ1, Ɵ2 and d3, We Multiply first Three Matrics as below and Compare Last columns of H and T03.
𝐶1𝐶2 −𝑆1 𝐶1𝑆2 𝐶1𝑆2𝑑3 − 𝑆1𝑑2
𝑆1𝐶2 𝐶1 𝑆1𝑆2 𝐶1𝑑2 + 𝑆1𝑆2𝑑3
−𝑆2 0 𝐶2 𝐶2𝑑3
0 0 0 1
Three Equations and Three Unknowns, By Solving these we get:
𝑇6
3
= 𝑇3
0 𝑇
. T
R.H.S =
−𝑆2 𝐶1𝐶2 𝑆1𝐶2
0 −𝑆1 𝐶1
𝐶2 𝐶1𝑆2 𝑆1𝑆2
Comparing 3rd columns of Both sides, we get
𝐶5 = 𝑆1. 𝑆2 = 1
Considering only 3×3 Matrix Portion of L.H.S and R.H.S
In this case
𝑀𝑒𝑎𝑛𝑠 Ɵ5 = 0
Putting Values from 𝑇3
0 𝑇
. T, we get:
𝜃4 = 0, 𝑎𝑛𝑑 𝜃6 = 0

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Inverse Kinematics of Standford Manipulator using Pieper's Theorem.pptx

  • 1. Inverse Kinematics of Standford Manipulator using Pieper’s Theorem Presented By: Mian Waqar ul Mulk
  • 4. As We Know That is Given which is H. we will first find T03 i.e. inverse position and then inverse orientation T36. To find Ɵ1, Ɵ2 and d3, We Multiply first Three Matrics as below and Compare Last columns of H and T03. 𝐶1𝐶2 −𝑆1 𝐶1𝑆2 𝐶1𝑆2𝑑3 − 𝑆1𝑑2 𝑆1𝐶2 𝐶1 𝑆1𝑆2 𝐶1𝑑2 + 𝑆1𝑆2𝑑3 −𝑆2 0 𝐶2 𝐶2𝑑3 0 0 0 1 Three Equations and Three Unknowns, By Solving these we get:
  • 5. 𝑇6 3 = 𝑇3 0 𝑇 . T R.H.S = −𝑆2 𝐶1𝐶2 𝑆1𝐶2 0 −𝑆1 𝐶1 𝐶2 𝐶1𝑆2 𝑆1𝑆2 Comparing 3rd columns of Both sides, we get 𝐶5 = 𝑆1. 𝑆2 = 1 Considering only 3×3 Matrix Portion of L.H.S and R.H.S
  • 7. Putting Values from 𝑇3 0 𝑇 . T, we get: 𝜃4 = 0, 𝑎𝑛𝑑 𝜃6 = 0