12. inv_Kinematics

# I i 1 ai 1 di i 0 0 0 0 0 1 0 a0 0 1 2 cos1

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Unformatted text preview: sin θ ⎢ ⎢0 ⎣ − sin θ cos θ 0 ⎡ cos θ Ry = ⎢ 0 ⎢ ⎢ − sin θ ⎣ ⎡1 R z = ⎢0 ⎢ ⎢0 ⎣ 0 cos θ sin θ 0 1 0 0⎤ 0⎥ ⎥ 1⎥ ⎦ sin θ ⎤ 0⎥ ⎥ cos θ ⎥ ⎦ 0⎤ − sin θ ⎥ ⎥ cos θ ⎥ ⎦ Rotation around the ZAxis Rotation around the YAxis Rotation around the XAxis Y2 i Z0 X0 Y0 Y1 a0 V X 0 Y0 Z 0 di θi 0 0 0 θ0 0 a0 0 θ1 2 X1 a(i-1) 1 X2 α(i-1) 0 Z1 -90 a1 d2 θ2 d2 a1 ⎡V X2 ⎤ ⎢ Y2 ⎥ ⎢V ⎥ =T ⎢V Z2 ⎥ ⎢ ⎥ ⎢1⎥ ⎣ ⎦ T =( 0T)( 01T)( 12T) Note: T is the D-H matrix with (i-1) = 0 and i = 1. i α(i-1) a(i-1) di θi 0 0 0 0 θ0 1 0 a0 0 θ1 2 ⎡cosθ1 ⎢ sinθ 1 0 T=⎢ 1 ⎢0 ⎢ ⎣0 -90 a1 − sinθ1 cosθ1 0 0 d2 θ2 0 a0 ⎤ 0 0⎥ ⎥ 0 0⎥ ⎥ 0 1⎦ This is a translation by a0 followed by a rotation around the Z1 axis ⎡cosθ0 − sinθ0 ⎢sinθ cosθ0 0 ⎢ 0T = ⎢0 0 ⎢ 0 ⎣0 0 0⎤ 0 0⎥ ⎥ 1 0⎥ ⎥ 0 1⎦ This is just a rotation around the Z0 axis ⎡ cosθ 2 ⎢0 1 ⎢ 2T = ⎢− sinθ 2 ⎢ ⎣0 − sinθ 2 0 − cosθ 2 0 0 a1 ⎤ 1 d2 ⎥ ⎥ 0 0⎥ ⎥ 0 1⎦ This is a translation by a1 and then d2 followed by a rotation around the X2 and Z2 axis T =( 0T)( 01T)( 12T) INVERSE OF DH – Revolute Joints ⎡ Cθ i ⎢− Cα Sθ i i Aii−1 = ⎢ ⎢ Sα i Sθ i ⎢ 0 ⎣ Sθ i Cα i Cθ i 0 Sα i − Sα i Cθ i Cα i 0 0 − ai ⎤ − ai Sα i ⎥ ⎥ − d i Cα i ⎥ ⎥ 1⎦ 44 DH matrix – Prismatic joint i i −1 T = T ( zi −1 , d i ) R( zi −1 , θ i )T ( xi , ai ) R( xi , α i ) ⎡Cθ i ⎢ Sθ i −1 ⎢i Ai = ⎢0 ⎢ ⎣0 − Cα i Sθ i Sα i Sθ i Cα i Cθ i Sα i 0 − Sα i Cθ i Cα i 0 0⎤ ⎥ 0⎥ di ⎥ ⎥ 1⎦ 45 Inverse DH matrix – Prismatic joint ⎡ Cθ i ⎢− Cα Sθ i i i ⎢ Ai −1 = ⎢ Sα i Sθ i ⎢ 0 ⎣ Sθ i 0 Cα i Cθ i − Sα i Cθ i Sα i Cα i 0 0 ⎤ ⎥ − d i Sα i ⎥ − d i Cα i ⎥ ⎥ 1⎦ 0 46 Transform graph INVERSE KINEMATICS - FINDING FINDING EQUATIONS FOR THE JOINT VARIABLES IN TERMS OF THE CARTESIAN CARTESIAN SPACE DES...
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