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Inverse Kinematics

Inverse Kinematics. Forward/Direct Kinematics:. X. Given x  a unique q. Workspace. Reachable Workspace Dexterous Workspace. X. Given X Find q=(  1 ,  2 ,  3 ). Algebraic Solution. The kinematics of the example seen before are:. Assume goal point is specified by 3 numbers:.

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Inverse Kinematics

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  1. Inverse Kinematics

  2. Forward/Direct Kinematics:

  3. X Given x a unique q

  4. Workspace • Reachable Workspace • Dexterous Workspace

  5. X Given X Find q=(1, 2, 3)

  6. Algebraic Solution The kinematics of the example seen before are: Assume goal point is specified by 3 numbers:

  7. 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 c2

  8. 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 c12=c1c2-s1s2 and s12= c1s2-c2s1: where k1=l1+l2c2and k2=l2s2. To solve these eqs, set and φ

  9. k1 l2 k2 2 φ l1 Algebraic Solution (IV) Then: k1=r cosφ, k1=r sin φ,and we can write: x/r= cosφcos 1 - sin φsin 1 y/r= cosφcos 1 - sin φsin 1 or: cos(φ +1) = x/r, sin(φ +1) =y/r

  10. Algebraic Solution (IV) Therefore: φ+1 = Atan2(y/r,x/r) = Atan2(y,x) and so: 1 = Atan2(y,x) - Atan2(k2,k1) Finally, 3 can be solved from: 1+2+3 =

  11. y L2 L1 x Geometric Solution IDEA: Decompose spatial geometry into several plane geometry problems Applying the “law of cosines.”

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