haptic control of hydraulic machinery using proportional valves

the closed-form inverse kinematic solution. Geometrically different regions, denoted
by R ij , map to the workspace, or a point or curve on the edge of the workspace. Note
that this can be done with out loss of generality for the case of both proximal and
distal bucket control since both of these cases rely on calculating θ 2 and θ 3 based on
P 03 in the r − z plane.
The downfall of this method is that it gives priority to the position of the bucket
over its orientation. Another method would be to optimize a cost function, L, that
includes a bucket angle error term (see equation 3.47).
L = 1 2 W r∆r 2 + 1 2 W z∆z 2 + 1 2 W φ∆φ 2 (3.47)
where ∆r = r d − a c1 4, ∆z = z d − a s24 and ∆φ = φ d + π − θ 2 − θ 3 − θ 4 represent
the difference between the desired and commanded position. W r , W z and W φ are
weighting matrices. Since r and z are both measure distance, W r is typically equal
to W z and W φ is different. Minimizing this error can be done numerically using the
steepest descent method to find the optimal vector of joint angles θ 2−4 that minimize
L.
Ξ k+1 = Ξ k − µ∇ Ξ L T (3.48)
where is the vector of joint angle that is being optimized to minimize L.
⎡ ⎤
θ 2 (k)
Ξ k =
⎢ θ 3 (k)
⎥
⎣ ⎦
θ 4 (k)
(3.49)
and ∇ Ξ L T is the transpose of the gradient of L with respect to Ξ.
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