haptic control of hydraulic machinery using proportional valves

⎡ ⎤
−c 1 a c24 ê x
P 41 = P t0 + P 02 =
⎢ −s 1 a c24 ê y ⎥
⎣ ⎦
−a s24 ê z
⎡ ⎤
−c 1 a c34 ê x
P 42 = P t0 + P 03 =
⎢ −s 1 a c34 ê y ⎥
⎣ ⎦
−a s34 ê z
⎡ ⎤
−c 1 a c4 ê x
P 43 = P t0 + P 04 =
⎢ −s 1 a c4 ê y ⎥
⎣ ⎦
−a s4 ê z
(3.14)
(3.15)
(3.16)
Taking the cross products.
⎡ ⎤
⎡
0
s 1
ê z1 =
⎢ 0
⎥ 2
= ê z3 = ê z4 =
⎢ −c 1
⎣ ⎣
1
1
⎡ ⎤
−s 1 a c14
P 40 × ê z1 =
⎢ c 1 a c14 ⎥
⎣ ⎦
0
⎡ ⎤
−c 1 a s24
P 41 × ê z2 =
⎢ −s 1 a s24 ⎥
⎣ ⎦
⎤
⎥
⎦
(3.17)
(3.18)
(3.19)
⎡
P 42 × ê z3 =
⎢
⎣
a c24
⎤
−c 1 a s34
−s 1 a 34 ⎥
⎦
a c23 + a c24
(3.20)
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