IEEE Robotics & Automation Magazine - September 2023 - 100

Other continuum robots, in addition to active control over
their whole shape, employ rotary and linear stages to achieve
follow-the-lead motion [10], [12]. This is, however, achieved
by moving the entire actuation pack for the tendon-driven
section with these base stages [8], [9], requiring overengineered
or bulky motors. Conversely, COBRA's design is
unique in " dragging " the robot throughout its linear and
rotary stage while still allowing a static actuation pack, thus
significantly reducing the overall system size. This feature
also enables the separate deployment of the two actuation
units when both kinds of controlled motion (active and passive
segments) are not needed together for a given operation
(e.g., acting as a teleoperated borescope with a twist-andfeed
mechanism only or deploying the robot manually while
actuating the active segment at the tip).
CONTROL
In this section, the control system for COBRA is explained
with its four main elements: first, the active segment is
modeled with a piecewise constant curvature model; then,
tendon actuation is discussed with a focus on tension feedback
as well as backlash and stretch compensation; a novel
efficient algorithm for tip-following motion is presented;
and, finally, a human-machine interface is developed with a
user-friendly mapping of the robot's motion variable onto a
joystick controller.
CONTINUUM ROBOT MODELING
The active segments of COBRA have been modeled with a
piecewise constant curvature model, which describes a continuum
robot as n serially connected sections that can be independently
bent in any direction with constant curvature in
space [7]. This kinematic model is simple and efficient,
enabling real-time control for a smooth response to joystick
commands. Key assumptions include a continuous differential
along the backbone curve, with consecutive sections sharing
the same backbone tangent at their common point, and a rigid
{S1}
/1
z0ˆ
{S0}
x0ˆ
y0ˆ ϑ1
ϕ1
(a)
αij
xiˆ
yiˆ
(b)
FIGURE 5. The active segment modeling. (a) A piecewise constant
curvature model for a 6-DoF, three-section active segment. (b) A
model of a vertebra with geometrical parameters for tendon routing.
100
IEEE ROBOTICS & AUTOMATION MAGAZINE SEPTEMBER 2023
{Si}
ϕi
x1ˆ
ϕ2
z1ˆ
ϑ2
y2ˆ
/2
x2ˆ
{S2}
ϑ3
ϕ3
z2ˆ
/3
x3ˆ
{S3} z3ˆ
y3ˆ
ziˆ
put relationship between tendon input ll ,TTi=1R=
motion parameters ^^ ^hhh can be obtained.
ij
j{
ii
tt ,
Two additional active DoF of the robot are controlled by
the twist-and-feed mechanism: twist angle {0 t ,^h which
rotates the base frame of around the backbone axis, and the
exposed length of the robot
, t ,^h defined as the backbone
length inserted in the environment. As such, the example prototype
can actively control 8 DoF. However, different combinations
of active and passive sections can result in a higher or
lower mobility.
By repeating (4) for all tendons and sections, an input-outn
ij
i and
cross section at any backbone length, always normal to the
backbone tangent. A fixed frame S0 " , is defined at the base
of the active segment, with n further frames Si " , at the end of
each of the n sections [Figure 5(a)]. A homogeneous transformation
TSE 3
i
i
-1 ! ^h from Si 1- , to Si " , can be written as
"
T -1 = = --1G
1
i
i
where RSO 3
i
i
Rt
01
i
i
13
i
i
##
11
-1 ! ^h is the rotation, and t Ri
i
1
"
- ! 3 is the translation
along the section circle arc. This transformation is conventionally
parameterized by the direction of bending { and
bending angle .j Thus, the rotation from Si 1,
to Si " , is
Rrot^^ ^hh h
i
i
1 zi yi zi$$
{j {
rotrot
-=- (2)
and the corresponding translation is expressed as
t -1 = ,i
i
i
j >
i
coscos
sin
{j
{j
j
ii
ii
i
^
^
1
1
-
-
sin
cos
h
h
H
"
(3)
where i, is the length of the backbone from Si 1- , to " S .i,
Once the shape of the backbone is defined through these
equations, the length of each tendon is required to actuate
the robot. The length of each of the 12 tendons in the
reported example is defined as
l ,ij where = " ,,
sents
j SE NW= "
i 12 3, reprethe
sect ion at which the tendon ends, and
,, ,
, represents the routing position of the tendon
in the section [south (S), east (E), north (N), and west
(W)], working in antagonistic S-N and E-W pairs. The pose
of each tendon in a vertebra is defined by the angle ija [Figure
5(b)] around the z-axis of the local frame. As a single
tendon is routed in each hole of a vertebra, the corresponding
pairs for consecutive sections are shifted by a 30° angle
[(Figure 2(c)].
Assuming that each tendon is routed at a constant distance
from the central backbone of the robot, defined by
radius
r ,0 its change in length along the ith section lt,ij iT ^h
^ ^ h
j{
ii
tt ,
i
when moving from a straight backbone position 00ij = h
to a bent configuration defined by ^^ ^hhh can be
computed as
Tlt lt rthh hhh .
ij,, ,^^ ^^ ^ t
iiji ii ij
cos
=- ja {=- (4)
(1)
IEEE Robotics & Automation Magazine - September 2023

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