IEEE Robotics & Automation Magazine - December 2021 - 47

contact frame are given by the nonzero pose components as
follows (Figure 3, red arrows):
3D surface:
3D edge:
FPtS () (, ,; ,, ),
FPt xz0
S () (, ,; ,, ).ab c
= 00 z ab 0
=
(2)
(3)
Components that cannot be predicted from tactile measurements
are set to zero (Figure 3, blue arrows). These motions
leave the object feature invariant (such as along an edge).
In practice, these sensory invariances are idealizations
that hold only for infinite straight edges or planar surfaces.
Here, we consider them to hold approximately on local
regions of a curved edge or surface. In principle, this
approximation could be improved with prior knowledge of
the edge or surface geometry, which is an interesting topic
for further research.
Reference Pose
The reference pose used to define the pose error [(1)]
depends on the geometry of the object feature, with components
in the contact frame as follows:
3D surface:
3D edge:
FP (, ,; ,, ),
FP (, ,; ,, ).rr rr rab c
R = rr rr r rab c
r
xy z
R xzy
=
(4)
(5)
The blue components of the reference set how the object is
explored, and the red components set the desired contact.
By comparing the predicted sensor pose in (2) and (3) with
the reference in (4) and (5), we see two types of pose error
components that differ in their actions on the sensor pose:
●
those where both the reference and predicted sensor
pose can be nonzero, which change how the tactile sensor
contacts the object feature [colored red in Figure 3
and (2)-(5)]
●
those where just the reference can be nonzero, which are
invariances that do not change how the sensor contacts the
object feature [colored blue in Figure 3 and (2)-(5)].
These blue reference components move the sensor in a
way that the contact frame appears identical to the sensor.
We use these motions for exploration steps along the edge
or surface.
In practice, these displacements and invariances hold
approximately on local regions of an object. As the sensor
moves over a curved edge or surface, the sensor pose is disturbed
in the contact frame on a local part of the object
(which itself moves relative to the fixed base frame of the
robot when the object is nonuniform). The disturbance produces
a pose error, which drives the controller to move the
sensor toward the reference in the new contact frame. Repeating
this procedure servos the tactile sensor over a curved edge
or surface.
Note that the geometry of the object feature determines
how a tactile sensor can explore that feature extended over the
object. This is different from the approach of Li et al. [5] (see
the " Tactile Servo Control " section), which composed an
external motion signal with the controlled motion. In
principle, the reference components for the invariances could
be set to zero [(4) and (5), blue terms] and the control composed
with an external signal. However, an external motion
signal can disturb the pose components that are being controlled.
Therefore, we unify the two motions within a single
reference, keeping complete freedom to explore the object
without disturbing the servo control.
Controller
The control law that aims to drive the pose error in (1) to zero
remains to be specified. In this work, we control the sensor
pose using a proportional integral controller in discrete time
,, ,
(t 01 2 f=
=
the sensor pose,
R () (),
). The controller computes a correction to
SUt UtSl = () (), from the pose error,
SEt Et using a parameterized representation in Cartesian
coordinates and Euler angles:
()
Uu uu uu uab c
=
xy z
and KI
Ut KE tKfE
t
=+ x
x=
PI ;/ E() .
()
and Ee ee ee eab c
xy z
= (, ,; ,, )
(6)
For the purpose of specifying the controller, these parameterized
SE(3) transformations are treated as 6D vectors
(, ,; ,, )
in
the coordinates of the sensor frame. The 66# gain matrices
KP
with an antiwindup function []f $
between a minimum and maximum.
In our system, the position control of the robot is specified
in its base frame B, which is set at the start of the
experiment along with an initial sensor pose ().P 0B
S
the control signal
base frame:
SUtSl
B () () () ().
Pt 1 Pt Pt Ut
S += =ll
B
S
Implementation
All poses (, ,; ,, )
B
S
S
S
Then,
() updates the sensor pose in the
(7)
xy z ab c ! SE(3) are represented in Cartesian
coordinates and Euler angles (in the extrinsic-xyz convention).
The sensor pose components in the contact frame
[(4) and (5)]) are assumed to lie within the following ranges:
x ! [−5, 5] mm horizontally centered on the edge,
z ! [−5, −1] mm vertically into the surface or top of the
edge,
ab , ! [−15°, 15°] roll and pitch around the surface/
edge normal, and !c [−45°, 45°] yaw centered on the edge
(Table S1 in " Training Data, Control, and PoseNet Parameters " ).
These ranges determine the span of data used to train
the neural network for predicting the sensor pose. A typical
reference pose is set in the center of these ranges, representing
a −3-mm contact depth with the sensor oriented normal to
the surface or edge.
The control gain matrices are diagonal and have values
that give good overall performance while not being specific
to any of our experiments (Table S2 in " Training Data,
Control, and PoseNet Parameters " ). The proportional
gains are 0.5 for components of the displacement (red) and
one for invariances (blue); the integral gains are 0.3 for
translational and 0.1 for rotational displacements and zero
DECEMBER 2021 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
47
act on these 6D vectors and are supplemented
to bound the integral error
IEEE Robotics & Automation Magazine - December 2021

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