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v = c f = c ( f D + f τ )
.
(4.8)
Let us now introduce a new compliance c τ
,
1] that attenuates the non-preferred
component of the force input. With this we arrive at
[0
v = c ( f D + c τ f τ )
= c ([ D ]+ c τ
D
) f
.
(4.9)
Thus, the final control law is in the general form of an admittance control with a
time-varying gain matrix determined by D ( t ). By choosing c
we control the overall
compliance of the system. Choosing c τ low imposes the additional constraint that
the robot is stiffer in the non-preferred directions of motion. As noted above, we
refer to the case of c τ = 0asa hard virtual fixture , since it is not possible to move in
any direction other than the preferred direction. All other cases will be referred to as
soft virtual fixtures. In the case c τ = 1
,
,
we have an isotropic compliance as before.
1 and create a virtual fixture where it is easier to
move in non-preferred directions than preferred. In this case, the natural approach
would be to switch the role of the preferred and non-preferred directions.
It is also possible to choose c τ >
4.2.3
Choosing the Preferred Direction
The development to this point directly supports the following types of guidance:
motion in a subspace: suppose we are supplied with a time-varying, continuous
function D = D ( t )
.
Then applying (4.9) yields a motion constraint within that
subspace; and
motion to a target pose x t
SE (3): suppose that we have a control law u =
f ( x
,
x t ) such that by setting v = u
,
lim
t
x = x t .
Then by choosing D = u and applying (4.9), we create a virtual fixture that guides
the user to the given target pose.
These two tasks are, in some sense, at the extremes of guidance. In one case, there
is no specific objective to attain; we are merely constraining motion. In the second,
the pose of the manipulator is completely constrained by the objective. What of
tasks that fall between these two extremes?
To study this problem, let us take a simple yet illustrative case: the case of main-
taining the tool tip within a plane through the origin. For the moment, let us neglect
manipulator orientation and consider the problem when controlling just the spatial
position of the endpoint. We define the surface as P ( p )= n
·
p = 0where n is a unit
vector expressed in robot base coordinates.
Based on our previous observations, if the goal was to allow motion parallel to
this plane, then, noting that n is a non-preferred direction in this case, we would
define D =
n
and apply (4.9). However, if the tool tip is not in the plane, then it is
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