haptic control of hydraulic machinery using proportional valves

⎡
p cyl
= 1 d d ⎢
⎣
J −1
⎤
−d d /a c13 0 0 0
0 −a c3 −a s3 0
0 a c23 a s23 0 ⎥
⎦
0 −a c2 −a s2 −a s3 a 2
+ a s2 a c3
(3.37)
Note that given θ 1 , θ 2 , θ 3 and θ 4 , the position vectors, Jacobian and Jacobian
inverse can be evaluated with only eight trigonometric evaluations: c 1 , c 2 , c 23 , c 24 , s 1 ,
s 2 , s 23 , s 24 for either the cylindrical coordinate or Cartesian coordinate case. Another
important fact that will later be exploited is that θ is decoupled from r, z and φ in
the cylindrical coordinates Jacobians 3.31.
3.2.3 Inverse-Kinematics
The inverse kinematics can also be derived using cylindrical coordinates. It will be
assumed that the radius will be constrained by a minimal value to assure that the
point being used for coordinated motion control does not pass through the vertical z 0
axis. This is the only singularity that a backhoe can pass through due to joint angle
limitations. This software constraint will be discussed later after the unconstrained
inverse kinematic solution is derived. Given r, θ, z, φ, it is possible to calculate θ 1−4 .
First off θ 1 = θ.
For distal bucket control:
⎛
P 03 =
⎜
⎝
⎞ ⎛
r − a 4 cos(φ)ê r
0
⎟
⎠ = ⎜
⎝
z − a 4 sin(φ)ê z
⎞
r 3 ê r
0
⎟
⎠
z 3 ê z
(3.38)
For proximal bucket control:
⎛
P 03 =
⎜
⎝
⎞ ⎛
rê r
0
⎟
⎠ = ⎜
⎝
zê z
⎞
r 3 ê r
0
⎟
⎠
z 3 ê z
(3.39)
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