1120 Auton Robot (2018) 42:1117–1132
derives the feasible joint trajectories that satisfy whole-arm
collision avoidance, robot’s joint limits and target visibility
constraints. We present experimental scenarios and robust-
ness results for our algorithm in Sect. 4, and then conclude
in Sect. 5.
2 Robot teach-by-demonstration
Robot teach-by-demonstration helps to define safe regions
for the robot motion without the need of expensive mapping
(especially in case of having a single eye-in-hand vision sen-
sor). In the following paragraphs we briefly summarize the
steps taken to generate the feasible motion used in our work.
Further details are reported in Chan et al. (2013) and Shen
et al. (2013).
Step 1 User demonstrations. To start, a reference image
of the target is taken by the user as a prerequisite for robot
teach-by-demonstration. Next, the user moves the robot arm
towards different target locations, so that we can extract sta-
tistical information about the demonstrated joint trajectories.
Variations among the demonstrated joint trajectories approx-
imate how closely the robot should track a given trajectory.
Specifically, when the robot is in close proximity to obstacles,
we expect this set of joint trajectories to have little variation.
When the workspace is relatively free of obstacles, we expect
the demonstrated trajectories to result in larger variations.
Step 2 Canonical time warping (CTW). We assume that
the trajectory variations follow a Gaussian distribution in
time and space. We wish to extract a robust trajectory from
a set of demonstrations. In addition, we wish to quantify any
variation that may exist between the trajectories. To remove
the effect of temporal variations, we use CTW to solve for the
best temporal alignment between pairs of trajectories (Zhou
and Torre 2016) while adhering to temporal precedence and
continuity constraints.
Step 3 Gaussian mixture model (GMM). The robot’s
workspace is represented using a multivariate GMM of M-
components with dimensionality N + 1 (for a robot with N
DoF and the time index t):
P(q(t)) =
M
m=1
π
m
N (q(t); μ
m
,
m
) (1)
where π
m
is the prior probability on the Gaussian compo-
nent m and N (q (t ); μ
m
;
m
) is the (N + 1)-dimensional
Gaussian density of component m, with μ
m
and
m
as the
mean and covariance matrix, respectively. These parameters
are estimated using the expectation maximization (EM) algo-
rithm. For analysis, μ
m
and
m
can be separated into their
spatial and temporal constituents:
μ
m
=[μ
t
m
,μ
q
m
],
m
=
t
m
tq
m
qt
m
q
m
. (2)
Step 4 Gaussian mixture regression (GMR). We perform
GMR along the time index to reconstruct the average joint tra-
jectory
¯
q(t) and its time-dependent covariance matrix
q
(t):
¯
q(t) =
M
m=1
β
m
(t)
¯
q
m
(t),
q
(t) =
M
m=1
β
m
(t)
q
m
(3)
where:
β
m
(t) =
π
m
N (t; μ
t
m
;
t
m
)
M
j=1
π
j
N (t; μ
t
j
;
t
j
)
, (4)
¯
q
m
(t) = μ
q
m
+
qt
m
(
t
m
)
−1
(t − μ
t
m
). (5)
The cost function
(
q(t) −
¯
q(t)
)
W(t)
(
q(t) −
¯
q(t)
)
, (6)
where W(t) = (
q
(t))
−1
gives a measure of the relative
importance of each joint, evaluates the covariance-weighted
distance between the average joint trajectory
¯
q(t) and its can-
didate servoing trajectory q(t). We sample the average joint
trajectory
¯
q(t) and its time-dependent weighting matrix W(t)
and denote them as
¯
q
i
and W
i
, i = 1,..., p for subsequent
optimization. The cost function for the sampled data is:
(
q
i
−
¯
q
i
)
W
i
(
q
i
−
¯
q
i
)
, i = 1,..., p, (7)
where q
i
is the sampled candidate servoing trajectory that
minimizes the above quadratic function.
Minimization of (7) defines a safe region in joint space;
however, the visibility constraint can be violated by just
following the intended joint trajectory or by applying the
controller presented in Chan et al. (2011, 2013). Herein, we
propose to transform the intended trajectory and the safe
region into robot’s workspace, where we explicitly model
the camera trajectory and apply constrained optimization to
find a satisfactory camera path that also satisfies the target
visibility.
3 Constrained optimization
There are two main constraints considered while optimizing
a modeled camera trajectory. One is the safe region trans-
formed from joint space, which is expressed in a tube form,
and the other is continuous target visibility. Beginning with
the safe region defined by the intended joint trajectory
¯
q
i
and its weighting matrices W
i
, we obtain the corresponding
camera trajectory
¯
g
i
and its time-dependent weighting matri-
ces for camera translation M
i
, the minimum eigenvalues of
which serve as conservative variances for the three transla-
tional coordinates in
¯
g
i
. The intended camera trajectory
¯
g
i
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