Lecture 13:inverse kinematics I

Prof. Katie Driggs-CampbellCo-Teaching with Prof. Belabbas

October 14, 2021

Administrivia

• Project Update 1 is due Saturday 10/16 at midnight• All informaBon about the project updates (including due dates) are online

• HW 4 is due next Friday 10/22 at 8pm• Next Thursday (10/21) will be project catch-up day

What is going on in this class?• Rigid Body Mo*on• Forward Kinema*cs

• Calculate the posi.on of the end-effector of an open chain given joint angles𝑥 𝑡 = 𝑓 𝜃 𝑡

• Velocity Kinema*cs• Calculate the velocity (twist!) of the end-

effector!!"𝑥 𝑡 = #$

#%𝜃 ⋅ �̇�

• Inverse Kinema*cs• Computes the possible joint angles from the

pose of the end-effector• Given 𝑇(𝜃), find solu.ons 𝜃 that sa.sfy 𝑇 𝜃 = 𝑋

Where is forward kinematics used?

Inverse Kinematics in Animation

Image Credit: Rive

Inverse Kinematics• Forward Kinematics: computes the end-effector position from joint angles:

• Inverse Kinematics: computes the possible joint angles from the pose of the end-effector

• Given 𝑇(𝜃), find solutions 𝜃 that satisfy 𝑇 𝜃 = 𝑋

Inverse Kinema6cs (IK)

6R PUMA arm • Joint 1 axis is aligned in the �̂� direc7on• Joint 2 axis is aligned in the - $𝑦 direc7on• Joint 3 axis lies on the x-y plane and is parallel

with joint 2 axis

6R PUMA arm – orientation

Numerical Inverse Kinematics• Forward Kinematics: 𝜃 → T 𝜃

• Inverse Kinematics: 𝑋 → 𝜃

• When the analytic solution is hard or impossible to come by, we numerically solve:

Newton-Raphson Method (1)

Newton-Raphson Method (2)

Summary

• Reviewed analytic inverse kinematics, which helps us find the joint angles that will produce a desired end-effector pose• We can do this with the help of atan2 and ZYX Euler Angles• However, this is only works for relatively simple robot chains

• Introduced the Newton-Raphson method, which can be modified to solve inverse kinematics numerically