ME469 HW #6 Solution

\(<< LinearAlgebra`MatrixManipulation`; 

    T[α_, a_, d_, 
        θ_] := {{Cos[θ], \(-Sin[θ]\), 0, a}, {
          Sin[θ]*Cos[α], Cos[θ]*Cos[α], 
          \(-Sin[α]\), \(-Sin[α]\)*d}, {
          Sin[θ]*Sin[α], Cos[θ]*Sin[α], 
          Cos[α], Cos[α]*d}, {0, 0, 0, 1}}; \)

#  Prob 1

\(T04 = 
      T[0, 0, 0, θ\_1] . T[Pi/2, 0, 0, θ\_2] . 
        T[0, a2, 0, θ\_3] . \ T[0, a3, 0, 0]; 

    MatrixForm[T04 // Simplify]\)

\(j11 = D[\(T04[\([1]\)]\)[\([4]\)], θ\_1]; 
    j12 = D[\(T04[\([1]\)]\)[\([4]\)], θ\_2]; 
    j13 = D[\(T04[\([1]\)]\)[\([4]\)], θ\_3]; 

    j21 = D[\(T04[\([2]\)]\)[\([4]\)], θ\_1]; 
    j22 = D[\(T04[\([2]\)]\)[\([4]\)], θ\_2]; 
    j23 = D[\(T04[\([2]\)]\)[\([4]\)], θ\_3]; 

    j31 = D[\(T04[\([3]\)]\)[\([4]\)], θ\_1]; 
    j32 = D[\(T04[\([3]\)]\)[\([4]\)], θ\_2]; 
    j33 = D[\(T04[\([3]\)]\)[\([4]\)], θ\_3]; 

    Jacobian1 = 
      Simplify[{{j11, j12, j13}, {j21, j22, j23}, {j31, j32, j33}}]; 

    MatrixForm[Jacobian1]\)

\(Jacobian1 = 
      N[Jacobian1 /. {a2 -> 1. , a3 -> 1. , θ\_1 -> Pi/6. , 
            θ\_2 -> Pi/6. , θ\_3 -> Pi/6. }]; 

    Eigensystem[Jacobian1\  . Transpose[Jacobian1]]\)

The smallest eigenvalue is the 3 rd one , 
        thus the 3 rd eigenvalue is the direction of maximum mechanical advantage; 

The biggest eigenvalue is the 1 st one, thus the 1 st eigenvalue is the direction of maximum velocity amplitude;

\(\(

#  Prob\ 2\)\)

\(T04 = 
      T[0, 0, d\_1, 0] . T[0, a\_1, 0, θ\_2] . 
          T[0, a2, 0, θ\_3] . \ T[0, a3, 0, 0]\  // Simplify; 

    MatrixForm[T04 // Simplify]\)

\(j11 = D[\(T04[\([1]\)]\)[\([4]\)], θ\_1]; 
    j12 = D[\(T04[\([1]\)]\)[\([4]\)], θ\_2]; 
    j13 = D[\(T04[\([1]\)]\)[\([4]\)], θ\_3]; 

    j21 = D[\(T04[\([2]\)]\)[\([4]\)], θ\_1]; 
    j22 = D[\(T04[\([2]\)]\)[\([4]\)], θ\_2]; 
    j23 = D[\(T04[\([2]\)]\)[\([4]\)], θ\_3]; 

    j31 = D[\(T04[\([3]\)]\)[\([4]\)], θ\_1]; 
    j32 = D[\(T04[\([3]\)]\)[\([4]\)], θ\_2]; 
    j33 = D[\(T04[\([3]\)]\)[\([4]\)], θ\_3]; 

    Jacobian2 = 
      Simplify[{{j11, j12, j13}, {j21, j22, j23}, {j31, j32, j33}}]; 

    MatrixForm[Jacobian2]\)

\(Jacobian2 = 
      N[Jacobian2 /. {a2 -> 1. , a3 -> 1. , θ\_2 -> Pi/6. , 
            θ\_3 -> Pi/6. }]; 

    Eigensystem[Jacobian2 . Transpose[Jacobian2]]\)

The smallest eigenvalue is the 3 rd one , 
        thus the 3 rd eigenvalue is the direction of worst velocity amplitude; 

The biggest eigenvalue is the 1 st one, thus the 1 st eigenvalue is the direction of worst mechanical advantage;

\(\(

#\ Prob\ 3\)\)

\(T04 = 
      T[0, 0, d\_1, 0] . T[\(-Pi\)/2, 0, d\_2, \(-Pi\)/2] . 
          T[\(-Pi\)/2, 0, d\_3, 0] . \ T[0, d\_4, 0, 0]\  // Simplify; 

    MatrixForm[T04 // Simplify]\)

\(j11 = D[\(T04[\([1]\)]\)[\([4]\)], d\_1]; 
    j12 = D[\(T04[\([1]\)]\)[\([4]\)], d\_2]; 
    j13 = D[\(T04[\([1]\)]\)[\([4]\)], d\_3]; 

    j21 = D[\(T04[\([2]\)]\)[\([4]\)], d\_1]; 
    j22 = D[\(T04[\([2]\)]\)[\([4]\)], d\_2]; 
    j23 = D[\(T04[\([2]\)]\)[\([4]\)], d\_3]; 

    j31 = D[\(T04[\([3]\)]\)[\([4]\)], d\_1]; 
    j32 = D[\(T04[\([3]\)]\)[\([4]\)], d\_2]; 
    j33 = D[\(T04[\([3]\)]\)[\([4]\)], d\_3]; 

    Jacobian3 = 
      Simplify[{{j11, j12, j13}, {j21, j22, j23}, {j31, j32, j33}}]; 

    MatrixForm[Jacobian3]\)

Eigensystem[Jacobian3 . Transpose[Jacobian3]]

\(
The\ eigenvalues\ are\ the\ same . This\ indicates\ that\ there\ is\ 
      no\ preferred\ \)direction of Maximum force fidelity and maximum velocity 
      fidelity .