IEEE Robotics & Automation Magazine - March 2022 - 13

()Lmax and minimum ()Lmin
lengths, respectively, were computed,
as presented in Figure 3. For the malleable link, with y
in the vertical direction and x in the horizontal, the involute
curve is defined as
xr cossintt at
(( )( )( )),
=-1
+=-1
((
() () ())
yr sincostt at=- -
and the cycloid curve is defined as
xr (())cos t
yr sintt Lmin
=- + (( ))
r
,
(1)
(2)
(3)
(4)
where a is the maximum bending angle of the malleable link,
t is the variable curvature of the malleable link, and r is the
radius of curvature. The values of
min = 110 , Lmax = 700 ,
L
min = 550 , and a 5= rad were used.
The total area captured within these curves represents the
distal joint at the end of the malleable-link (/ ,P34 shown in
Figure 3) reconfiguration space. Then, using the length of the
distal link, it was possible to compute the resulting end-effector
reconfiguration workspace based on the reconfiguration
workspace of /.P34 Then, by sweeping this 2D area around
the vertical base joint, a 3D theoretical reconfiguration workspace
of the end effector was generated.
To validate this, a reconfiguration space-exploration
experiment was carried out, where the position of P /34 and
P5 were tracked while the robot was moved throughout its
reconfiguration space. These 3D results were then rotated
around the vertical base axis, removing the z component, to
form a 2D plot for increased readability in comparison
against the theoretical reconfiguration space. These results
can be seen in Figure 4. The robot's mounting platform was
also added, highlighting why negative y values cannot be
seen. From these results, we can see that the theoretical
reconfiguration space is a good representation of the actual
reconfiguration space of the robot, assuming that the theoretical
space is adjusted to remove all negative y values. A
3D model of the theoretical reconfiguration workspace of
the end effector, with this adjustment
made, was then produced for use in
the AR application.
With the generated reconfiguration
workspace, several design decisions
were made with regard to how
it will interact with the other holographic
objects. Initially, we considered
utilizing the reconfiguration
workspace purely to provide feedback
to the user, without it appearing
in the user's vision, by constantly
checking for collisions among the
objects and changing the color of the
holograms once it collides with the
reconfiguration workspace. However,
following preliminary user feedback, it
was recognized that having a visible workspace can not
only speed up the reconfiguration, as the user can easily
anticipate which positions and orientations are achievable,
it can also visualize the range of robot motions a priori to
confirm that it will not coincide with objects in the physical
space. Therefore, the availability to toggle this workspace
on and off was provided in the developed system,
and an on-demand feasibility-checking system (initiated
using a button press) was adopted.
Malleable Robot Optimal Solution Visualization
For the developed AR application, it was also necessary to
guide the user in reconfiguring the robot once an ideal
configuration was determined. From the topology computation
equations, four distances (D13, D14, D23, and
D24, shown in Figure 2) are returned that represent the
topology of the robot based on the interpoint distances
between P1 at the robot base origin, P2 directly above the
origin, and P3 and P4, which define either side of the distal
joint.
Visualizing solely interpoint distances is quite difficult. In
our previous work on end-effector alignment, a non-AR,
tracking-based alignment strategy was proposed [7]. An
OptiTrack system with six cameras was set up around the
malleable robot. The user was given real-time feedback on the
error between the expected and current positions during
IEEE Robotics & Automation Magazine - March 2022

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