CSCE 452 Intro to Robotics Inverse Kinematics 1

CSCE 452 Intro to Robotics

Forward/Direct Kinematics:

CSCE 452 Intro to Robotics X Given x  a unique q

CSCE 452 Intro to Robotics

Workspace Reachable Workspace Dexterous Workspace

CSCE 452 Intro to Robotics

X Given X Find q=(  1,  2,  3 )

CSCE 452 Intro to Robotics Algebraic Solution The kinematics of the example seen before are: Assume goal point is specified by 3 numbers:

CSCE 452 Intro to Robotics Algebraic Solution (cont.) By comparison, we get the four equations: Summing the square of the last 2 equations: From here we get an expression for c 2

CSCE 452 Intro to Robotics Algebraic Solution (III) When does a solution exist? What is the physical meaning if no solution exists? Two solutions for  2 are possible. Why? Using c 12 =c 1 c 2 -s 1 s 2 and s 12 = c 1 s 2 -c 2 s 1 :

CSCE 452 Intro to Robotics Algebraic Solution (IV) k1k1 k2k2 22 l1l1 l2l2 φ Then: k 1 =r cos φ, k 1 =r sin φ, and we can write: x/r= cos φ cos  1 - sin φ sin  1 y/r= cos φ cos  1 - sin φ sin  1 or: cos(φ +  1 ) = x/r, sin(φ +  1 ) =y/r

CSCE 452 Intro to Robotics Algebraic Solution (IV)

CSCE 452 Intro to Robotics Geometric Solution IDEA: Decompose spatial geometry into several plane geometry problems x y L1L1 L2L2 Applying the “ law of cosines. ”

CSCE 452 Intro to Robotics