IEEE Robotics & Automation Magazine - September 2022 - 116

where (, )xy is in the following space:
{,|}cos
xy xy L
max
JP ,Lr
()k 1
new
22 22 i+= ^h .
(12)
where Pobs
In this space, the extreme value of the fitness function is
" +=
xy xy
22
consistent with spherical crown M. Finally, the undetermined
parameters are (, ),xy and the search space is ,|
22 imax ,
L sco () .
Fitness Function
The fitness function must weigh the obstacle avoidance and
joint movement. For the obstacle avoidance, it can directly set
the fitness to infinity when a collision is detected. However, to
measure the amount of joint movement, there are different
methods.
For the limited root joint, we expect that, by being closer
to the root joint, there will be a smaller change in the joint
position during forward planning. According to [18], these
requirements can be met when the joint is as close as possible
to the extension line of the last position of the link. Therefore,
in forward planning, fitness is defined as the angle a between
the vectors () ()
JJ and
kk 1+
JJ .
() ()k
new In the case of reverse plank
ning,
the end joint is not as limited as the root joint, but it is
still expected to be as close as possible to the last position.
Therefore, the fitness is defined as
d JJ
() ()k
k
=
Finally, the fitness function is
Q = *d
fit
3
a
forward
reverse
collision
tor JP .
()k 1
new obs
+
.
(13)
Collison Detection
Considering that each point needs to be judged when
searching for the optimal position, this study uses the following
methods to simplify the calculation of the collision
detection. For example, when planning the position of the
kth link, obstacles that are located outside the sphere that
crown M is on will definitely not collide with the kth link
and can be eliminated in advance. They satisfy the following
formula:
αobs
βobs
γ > βobs
With this technique, before the search starts, some obstacles
can be eliminated, and obsb for each obstacle can be obtained.
Then, obsb for the discriminant can be used to quickly detect
collisions during the search.
In summary, by analyzing the constraints of the
mechanical structure, we converted the link planning into
an optimization search and determined the suitable undetermined
parameters and search space. Then, the angle or
distance was used to quantify the amount of link movement
while planning in different directions, and the obstacleavoidance
function was realized by the collisiondetection
and fitness functions.
γ < βobs
Pobs
J(k+1)
new
J(k+2)
new
(a)
c .
J(k+1)
new
J(k+2)
new
(b)
Figure 12. The collision detection for (a) angle obsb and (b) two
cases of angle
116 * IEEE ROBOTICS & AUTOMATION MAGAZINE * SEPTEMBER 2022
Root Planning and Root-Free Planning
The manipulator that is described in this study often needs
to be used with a propulsion platform. Here, we used a linear
sliding table. Because the cable-driven mechanism is
fixed on the sliding table, the root of link 1, that is, joint 1,
is always located on a line segment determined by the sliding
table, which is called the slider segment. Its length is the
effective stroke of the sliding table, and its direction is parallel
to the direction of movement of the sliding table.
Then, according to the constraints of the joint mechanism,
it is easy to see that joint 2 should be located in a bulletshaped
area with the slider segment as the axis, as shown
in Figure 13.
When deciding the position of link 2, the optimal position
on spherical crown M should meet the following three
new
.
cb1 obs
where c is the angle between the link axis JJ
() ()
newnew
+
kk1
(17)
and vecis
the center position of the obstacle, robs
is the
radius of the obstacle, and L is the length of the kth link.
Then, some obstacles that are inside the sphere but still
outside the cone will not collide with the kth link, and they
can also be eliminated. They satisfy the following formula:
ab $ i
-
obsobs
where obsa
kk
JJ obsb is the angle between the kth link and the
obstacle when the edge of the link is assumed to be tangential
to the obstacle, as shown in Figure 12. It can be calculated
as follows:
() ()
newnew
++
21
.
b = cm+
+
obs
where Rlink
arcsin Rr
P
J()k 1
new obs
linkobs
is the radius of the kth link.
Next, for the remaining obstacles, it is necessary to evaluate
whether it collides with the kth link according to J()k
new
when searching for the optimal position. Specifically, when
the following formula is satisfied, a collision occurs:
,
(16)
ma ,x
is the angle between the vectors JP
()k 1
new obs
+
(15)
and
+ -+2obsobs
(14)
IEEE Robotics & Automation Magazine - September 2022

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