IEEE Robotics & Automation Magazine - September 2022 - 97

1.5
1
Cable 3
Cable 3
0.5
-0.5
-1
-1.5
-1.5 -1 -0.5 0 0.5 1 1.5
X
(a)
MP
(b)
Figure 9. (a) Hypocycloid curves were used for trajectory planning with zero velocity at the target points [95]. (b) A 6-3 spatial RRPM
equivalent to a 3-3 purely translational IRPM using a parallelogram. (Source: [99]; used with permission.)
the camera. In [99] and [100], a parallelogram configuration
of cables was used to change the feasible dynamic motions
of a 6-6 IRPM [Figure 9(b)]. Each parallelogram consists of
two parallel cables sharing the same length. The robot can
move like a 3-3 purely translational CDPR, while the MP orientation
remains approximately the same.
Since fewer cables are used in IRPMs than CRPM or
RRPM, the payload can have additional DoF and exhibit
swaying motions or oscillations. A trajectory planner was
implemented for a 4-6 IRPM to eliminate unwanted oscillations
using a zero-vibration input-shaping scheme [101].
Motion in 3D space was mirrored to two vertical planes perpendicular
to each other, and afterward, the natural frequency
was calculated for planar IRPMs with two cables.
In terms of considering the mass and elasticity of the
cables, a trajectory-planning approach was presented in [102]
for a 6-6 IRPM considering a virtual
equivalent spring model for the cables.
The trajectory-planning technique
permitted the MP to move beyond its
static workspace in a controlled, predictable
manner.
Regarding the trajectory planning
O1
of CRPMs and RRPMs, a robot was
designed in [103] for simulating underwater
forces, such as the buoyancy
for a walking humanoid robot. The
simulator robot followed the desired
trajectories, ensuring a smooth path
while avoiding any perturbation in the
cables. In [104], the trajectory planning
of a 4-3 CRPM for sensing and
mapping of an aquatic environment
was presented. Trajectory planning for
minimum energy consumption when the robot is maneuvering
periodic trajectories was determined. This was achieved
by minimizing the second norm of the cable tension vector.
A rapid calibration method was developed to reduce the deployment
time using a laser rangefinder and plumb lines suspended
from each cable origin. In this case, the horizontal
distance between the plumb lines can be measured quickly
and precisely.
In a cluttered environment, the cables can wrap around
obstacles or tangle with each other [33]. Therefore, optimized
collision-free path planning is studied in the literature to tackle
this problem. In [33], a homotropy-signature augmented graph
[Figure 10(a)] was used for path planning of an IRPM in an area
with polygonal obstacles. Two different trajectories connecting
the same start and end points are homotopic if one can continuously
deform into the other without intersecting any obstacle.
τ1
Xg
-τ2
O2
τ2
Xs
O3
(a)
(b)
Figure 10. (a) 1x and 2x are homotopic, but 3x is not homotopic to 1x and 2x [33].
(b) A CDPR with a reconfigurable MP for collision-free path planning. (Source: [34]; used
with permission.)
SEPTEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
97
τ3
Reconfigurable MP
Cable 6
Cab
Cable 5
Cable 5
Cable 1
le 1
Cable 2
l
Cable 4
Cable 6
Y
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