IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 6

Monocular Visual Autonomous Landing System for Quadcopter Drones Using Software in the Loop
KALMAN FILTER
The Kalman filter is an estimator that infers hidden states
from indirect, inaccurate, and uncertain observations. It is
possible to use the Filter to handle noisy observations
from the detection module and produce a continuous estimate
of the template position at each time step t [6].
We assume that we have a linear dynamic system for a
landing platform such as in (3), where xt is the x coordinate
of a pixel at time index t and Dt is the time between
two consecutive image frames. Similarly, in (4),
x_ t corresponds
to the x velocity component of a pixel in the image
xt ¼ xt1 þ Dt _xt1
x_ t ¼ _xt1:
(3)
(4)
The set of states X 2 R10 is given by (5), with xc;yc
as the position of the centroid of the template in the image
frame. The filter states are the same as the vector of observations
plus their first-order derivatives
X ¼½xc;yc;Ow;Oh; u; x_c; y_c; O_w; O_h; _uT:
(5)
Knowing the transition dynamics and states of the filter,
the motion model of the system is then given by (6).
The matrix A 2 R1010, shown in (7), is the state transition
matrix of the system and w is a white noise random
vector such that wNð0;QQÞ. Q 2 R1010 is defined as
the covariance matrix of the process noise [31]. For the
sake of notation, I represents the identity matrix
Xt ¼ At1Xt1 þ wt1
A ¼
I55 DtI55
055
I55
(6)
:
1010
Kt ¼ PtHT
Likewise, the measurement model of the filter is given
in (8). H ¼ I510 is defined as the observation matrix and
v is a white noise random vector such that vNð0;RRÞ.
As for Q, here R 2 R55 is the covariance matrix of the
observation noise
Yt ¼ HtXt þ vt:
(8)
With the motion and measurement models defined, it is
possible to formulate the pose estimation process of the
platform by giving the Kalman filter equations (9)-(15). In
this set of equations, P is defined as the covariance matrix
of the posterior estimate, Y is the innovation vector, K is
the Kalman gain and S is the covariance matrix ofthe innovation.
Additionally,
X^ the corrected states after a measurement update.
The first two equations (9) and (10) are used in the
prediction step, and give an estimate of the states X at
6
X^t ¼Xt þKtYt
Pt ¼ðI10x10KtHtÞPt:
(14)
(15)
The set of states produced by the Kalman filter can be
X represents the predicted states and
applied in an IBVS module to control the landing procedure
and to obtain the position of the landing platform in
the current image frame, as shown in Figure 4. These
states are computed in the 2D image frame to reduce the
computation carried out by the on-board computer of the
UAV. Furthermore, since we are not estimating the relative
pose between the vehicle and the landing platform,
we are not feeding IMU data to the tracking module; this
allows for the use of a linear Kalman filter and avoids the
calculation of Jacobians at each time-step. Pseudocode for
the vision-based detection and tracking system is given in
Algorithm 1.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
t S1
t
(13)
(7)
Pt ¼ AtPt1AT
t þQt:
(10)
When an observation is obtained by the detection
module, the correction phase (11)-(15) is computed
immediately after the prediction step. This step aims to
correct the error in the estimations using an observation of
the template in the current image frame at time t
Yt ¼ ZtHtXt
St ¼ HtPtHT
t þRt
(11)
(12)
each time step regardless of whether an observation was
obtained
Xt ¼ AtXt1
(9)
Figure 4.
Estimation of the platform position using the Kalman filter vector
of states X.
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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