Jacobian computation

From velocity recursion

j1 = Jacobian1[DH, 0, 3] //Simplify ; j1 // TF

( -L1 sin(θ1) - L2 sin(θ1 + θ2) -L2 sin(θ1 + θ2) ... 0 1 1

From differentiation of forward kinematics

j2 = Jacobian2[DH, 0, 3] ; j2 //  TF

( ... θ2) + sin (θ1) cos (θ2) + cos (θ1) sin (θ2) + sin (θ1) sin (θ2)

Are they the same?

j1 // TF

( -L1 sin(θ1) - L2 sin(θ1 + θ2) -L2 sin(θ1 + θ2) ... 0 1 1

j2b = j2 // Simplify ; j2b // TF

( -L1 sin(θ1) - L2 sin(θ1 + θ2) -L2 sin(θ1 + θ2) ... 0 1 1

j1  j2b

True

Step-by-step derivation of Jacobian from forward kinematics

Homogeneous transform

T03 = T[DH, 0, 3] // FullSimplify ; T03 // TF

( cos(θ1 + θ2) -sin(θ1 + θ2) ... 0 0 1

Position vector from homogeneous transform

P03 = PFromT[T03] ; P03 //Nice

( L1 cos(θ1) + L2 cos(θ1 + θ2) ) L1 sin(θ1) + L2 sin(θ1 + θ2) 0

Differentiation of position vector ( 3x2 linear velocity submatrix of Jacobian)

vj2 = Outer[D, P03, {θ1, θ2}] ; vj2 // TF

( -L1 sin(θ1) - L2 sin(θ1 + θ2) -L2 sin(θ1 + θ2) ... θ2) L2 cos(θ1 + θ2) 0 0

Rotation matrix from homogeneous transform

R03 = RFromT[T03] ; R03 // TF

( cos(θ1 + θ2) -sin(θ1 + θ2) 0  ... 2) cos(θ1 + θ2) 0 0 0 1

S(Ω) = Overscript[R, .] R^T(3x3 angular velocity, skew-symmetric matrix)

SΩ = Dt[R03, t] . Transpose[R03] ; SΩ // TF

( θ1 θ2 ... 0

S(Ω) = Overscript[R, .] R^T(simplified)

SΩ = SΩ // FullSimplify ; SΩ // TF

( θ1 θ2 ... 0 0 0 0

Corresponding angular acceleration vector

Ω = {SΩ[[3, 2]], SΩ[[1, 3]], SΩ[[2, 1]]} ; Ω // Nice

( ) 0 0 & ... 63308;θ2 --------------- + --------------- t t

3x2 angular velocity submatrix of Jacobian

ωj2 = Transpose[Coefficient[Ω, Dt[#, t]] & /@ {θ1, θ2}] ; ωj2 // TF

( 0 0 ) 0 0 1 1

Put linear and angular velocity Jacobian together

j2b = Join[vj2, ωj2] ; j2b // TF

( -L1 sin(θ1) - L2 sin(θ1 + θ2) -L2 sin(θ1 + θ2) ... 0 1 1

Compare to previous result

j1 // TF

( -L1 sin(θ1) - L2 sin(θ1 + θ2) -L2 sin(θ1 + θ2) ... 0 1 1

Created by Mathematica  (October 1, 2003)