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eqsummary - ∂x ∂q i ∂y ∂q i ∂z ∂q i ⇒ ∂x...

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ME/ECE 739 Equation Summary - Spring 2006 - Professor Nicola J. Ferrier 1 Notation/Definitions Assuming original DH conventions: i - 1 i T = i - i i i i a i i i i i - i i a i i 0 i i d i 0 0 0 1 d dt [ i - 1 i T ( q i )] = d dq i [ i - 1 i T ( q i )] dq i dt = Q i i - 1 i T ( q i ) ˙ q i where Q i = 0 - 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 joint i revolute 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 joint i prismatic D i = 0 i - 1 TQ i 0 i - 1 T - 1 W N = N i =1 D i ˙ q i α N = N i =1 [( W i - 1 D i - D i W i - 1 ) ˙ q i + D i ¨ q i ] dD i dt = [ W i - 1 D i - D i W i - 1 ] d W N dt = N i =1 dD i dt ˙ q i + D i ¨ q i = N i =1 [[ W i - 1 D i - D i W i - 1 ] ˙ q i + D i ¨ q i ] d dt { 0 N T } = W N 0 N T d dt { 0 N T - 1 } = - 0 N T - 1 W N Kinematics Assuming point, P is fixed in frame { N } then 0 P = 0 N T N P 0 P = 0 1 T ( q 1 ) 1 2 T ( q 2 ) · · · N - 1 N T ( q N ) N P 0 V P = d dt 0 P = N i =1 D i ˙ q i 0 N T N P = W N 0 P 0 A P = d dt 0 V P = [ α N + W N W N ] 0 P Building The Jacobian We build each column of J using expressions com- puted for velocity (pull out terms from D i and D i 0 N T ): D i = 0 i - 1 TQ i 0 i - 1 T - 1 = 0 - ∂θ z ∂q i ∂θ y ∂q i * ∂θ z ∂q i 0 - ∂θ x ∂q i * - ∂θ y ∂q i ∂θ x ∂q i 0 * 0 0 0 0 ∂θ x ∂q i ∂θ y ∂q i ∂θ z ∂q i D i 0 N T =
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Unformatted text preview: ∂x ∂q i * * * ∂y ∂q i * * * ∂z ∂q i ⇒ ∂x ∂q i ∂y ∂q i ∂z ∂q i Stack these elements into columns of J : J = ∂x ∂q 1 ∂x ∂q 2 ··· ··· ∂x ∂q N ∂y ∂q 1 ∂y ∂q 2 ··· ··· ∂x ∂q N ∂z ∂q 1 . . . . . . ∂θ x ∂q 1 ∂θ y ∂q 1 . . . . . . ∂θ z ∂q 1 ··· ··· ∂θ z ∂q N...
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