A typical mission of a RUAV is split in several phases. The helicopter takes-off from its base, it flies to an area for mission execution usually using waypoint navigation, and once the task is completed the RUAV starts the approach to the landing area. Finally, once the helicopter is over the landing location, it starts the descent until it lands. The majority of the navigation strategies employed in autopilots are based on fusing the GNSS information with the navigation solution calculated using the accelerometers and gyros of the inertial measurement unit. In fact, this is the most common strategy for taking-off, and the waypoint navigation phases, where the positioning in absolute coordinates provides enough accuracy for performing the different maneuvers. However, if the landing phase has to be performed on moving platforms, a more robust strategy is required based on more accurate sensors. Moreover, the use of a GNSS-free landing system increases the robustness of the system by providing more sources of positioning, especially in non-GNSS friendly environments.
The work and the experiments presented in this article focused on the landing maneuver of the RUAV. The different steps that compound this phase are summarized below:
GNSS-Free Navigation Algorithm
One of the main contributions of this work is the relative navigation strategy by using a tether for the landing phase of the rotary-wing UAV. To obtain useful information from the tether, a specific device was developed. It consists of two-axis coupled cardan joints that allow estimating the angles between the tether and the helicopter frame in terms of the two successive rotations and a load sensor to measure the tension level of the tether.
Three different reference frames were considered in this study (see
Figure 2):
Body frame (B): The body frame is a non-inertial coordinate system associated with the vehicle with the origin at its center of gravity. The x-axis points in the forward direction, the z-axis down through the vehicle and the y-axis completes the right-hand coordinate system. This frame is denoted by the superscript b.
Local Navigation Tangent Plane frame (N): This is an inertial coordinate system determined by fitting a tangent plane to the geodetic reference ellipsoid at a fixed point. This point is taken as the origin of the coordinate system. The x-axis points to the true north, the y-axis points to the west and the z-axis points up. This frame is denoted by the superscript n.
The tether frame (T): It is a non-inertial coordinate system associated with a cardan joint mechanism. It has its origin in the point where the tether is connected to the helicopter. The x- and y-axes rotate with respect to the fuselage of the helicopter and the z-axis is always pointing towards the landing point. This frame is denoted with superscript t.
Figure 2 shows a scheme with the different elements that play a role in the landing procedure presented along this work.
The precision obtained by usual algorithms based on fusing a GNSS sensor with an Inertial Navigation System (INS) is not enough to perform a safe approach and landing, especially in moving platforms. Therefore, a technique to estimate the position in real time with high accuracy is needed in order to successfully accomplish the autonomous landing safely. In our study, a relative estimator was developed and implemented in an autonomous helicopter in order to be used during the landing phase on static or mobile platforms.
Figure 4 shows the architecture of the relative estimation module. As can be seen, the inputs to this module are the data provided by the tether system (angles
and
, and tension T), the altitude of the altimeter (
), the accelerations and angular velocities of the INS (
and
) and the magnetic field measurements of the magnetometer (
). This scheme is composed by:
Attitude and Heading Reference System (AHRS) block is the module in charge of calculating the attitude of the UAV (roll , pitch and yaw ). The attitude is defined as the inclination of its body-axes reference frame to the navigation reference frame. In addition, in this block, the accelerations are rotated to the navigation axes().
Tether Conversion block performs all the geometric and rotation operations needed to translate the tether information to a relative position vector ().
Sensor Fusion block fuses all the information obtained from the AHRS and the Tether conversion block and estimates the relative state vector that is used by the controller of the RUAV.
In our work, a crucial requirement for the estimation module is to obtain a precise tracking of the relative position and velocity between the helicopter and the landing platform. Most of the relative kinematics works are based on the fact that both vehicles have an external positioning source (generally GNSS) and they can share their own information through a communication link. This is very common for example in leader–slave architectures for formation flights [
22] where the system model uses information of the state vector received from the other vehicles and the relative position measurements are obtained by using a Differential GPS architecture. However, in this work, we did not rely on communication links or external very accurate positioning systems. In this way, as the true behavior of the vehicles was not known, the control input of the relative state vector was modeled as a random process with certain properties.
To build the model for the relative estimation in the approach maneuver, we had to take into account that the filter does not have any information about the dynamic of the landing platform. In this case, we chose to use a stochastic dynamic model of the relative vector between the vehicles, where a random variable represents an unknown time-varying quantity. In particular, our system model is a modified version of a Singer acceleration model. The Singer acceleration model [
23] is a popular model [
24,
25] for target maneuvers that characterizes the unknown target acceleration as a time-correlated stochastic process. It is an a priori model since it does not use online information about the target maneuver, although it can be made adaptive through an adaptation of its parameters. In this case, the acceleration is modeled as a zero-mean first-order stationary Markov process with an autocorrelation function [
26]:
where
is the variance process noise and
is the reciprocal of a maneuver time constant
that depends on how long the maneuver lasts, for instance, in a slow turn of an aircraft
, ∼60 s, and, in an evasive maneuver
, ∼10–20 s [
23]. In a Markov process, its value at a given time depends on values at other times only through its nearest neighbors. To provide values for these parameters, some typical simplifying assumptions for the ship model were taken [
27,
28]: the landing platform follows a straight trajectory with (nearly) constant velocity. Regarding the helicopter, it was assumed that the autopilot is capable of following the ship in a soft way during the maneuver (this last assumption was proved by the tests that are presented in
Section 4). One of the shortcomings of the Singer model is that the acceleration has a zero mean at any moment [
26]. However, we could use information from the inertial sensors on-board the RUAV, so some modifications can be done in the model in order to overcome this limitation. In the landing scenario, as the ship was assumed to have a slow dynamic, most of the changes in the relative velocity between both vehicles are due to the accelerations of the helicopter. These accelerations are not zero and can be measured by the accelerometers on-board. Hence, the Singer model can be modified to have a non-zero mean of the acceleration. This approach it is potentially more effective than the Singer model because it includes in the model most of the dynamics of the relative vector that is associated with the helicopter accelerations. In this way, the acceleration model satisfies
where
is the mean acceleration of the helicopter in the navigation frame, which was assumed to be constant during the sampling period. On the other hand,
is the zero-mean Markov process of the Singer model, with the autocorrelation function shown in Equation (
1), and it satisfies
In Equation (
3),
w(
t) is modeled as a zero-mean white noise. If
is expressed in Equation (
2) and plugged into Equation (
3), it is possible to obtain
If we note from Equation (
2) that
and it was assumed that the acceleration of the helicopter
is constant over a sampling interval
, we obtain
Through Equation (
5), it is possible to write the complete stochastic differential equation as:
In Equation (
6),
is the relative state vector [pr(t),vr(t),ar(t)] where pr(t), vr(t) and ar(t) are the relative position, velocity and acceleration, respectively. By applying a standard discretization step in Equation (
6), it is possible to obtain the discrete state equation
where
is the state transition matrix,
is the discrete time input matrix and
is a white noise sequence with covariance matrix
. The values of these matrices are given by Equations (
8)–(
10).
where the terms of the process noise covariance matrix (a similar methodology can be found in [
29,
30], where a velocity model is derived) are given by
In Equation (
10),
is a conditional density modeled as a Rayleigh distribution variance of the acceleration for
and its equation is:
where
is the current predicted acceleration and
and
are design parameters that correspond to the maximum and minimum accelerations, respectively. In this equation, if the absolute values of the design parameters are small, the accuracy of the estimation will tend to be high; however, the filter will have a slow response in cases where the changes of the relative motion are very aggressive. On the other hand, if the maximum and minimum values are larger, the model allows a quick response to the dynamics changes but the tracking accuracy becomes lower. In this work, the helicopter can follow the mobile platform in a soft manner, thus it was preferred to model these acceleration parameters using a low profile in order to obtain more accurate estimations. By generalizing the one-dimensional Equation (
6), the equations of the model for three dimensions are given by
Once the discrete time dynamic equations have been modeled, the measurement equation of the discrete-time system is presented as:
where
is the measurement vector that contains the relative positions between the rotary-wing UAV and the landing point calculated using the tether system,
is the measurement matrix and
is the noise in the measurements. The measurement matrix is given by
Figure 2 presents the different elements that are used by the Tether Calculation Block shown in
Figure 4 for the computation of the vector
. The first step for obtaining the relative positioning measurements consists on calculating the altitude to the landing point from the contact point. Because the altimeter is not installed in the same place, to have as much accuracy as possible, it is necessary to correct the lever arm according to the equations
where
is the rotation matrix from the body to the navigation frame calculated in the AHRS block and “
c” and “
s” denotes cosine and sine, respectively. The cardan joint is rigidly attached to the helicopter and perfectly aligned with the
and
body axes of the vehicle. This device rotates
and
angles with respect to the helicopter fuselage around its
and
axes, respectively, thus the rotation matrix from the tether to the body frame is
Once the altitude h
CP has been calculated, it is necessary to compute the relative position in navigation axes from the CP to the landing point as
where
and
are the positions vectors of the contact point in the navigation and the tether frame, respectively. In the tether frame, the horizontal XY coordinates of the landing point are 0 so the position vector is expressed as [0, 0,
zt ] and Equation (
19) can be written as:
By using Equation (
20), it is possible to obtain the relative coordinates of the contact point in the navigation frame as
The last step is to translate this relative position to the center of mass (CG) of the vehicle by applying another lever arm correction:
Once the relative position vector has been computed from the sensors outputs, it can be used as the measurement vector
of Equation (
14). Because this vector has been calculated through rotations and translations to work in the navigation frame, the components of the measurement noise covariance become correlated. To calculate the terms of the covariance matrix, it is necessary to calculate the error propagation in a multi-input multi-output system. The measurement noise covariance matrix has the form:
where the terms of the diagonal of the covariance matrix can be calculated as:
where
f is the measurement function presented in Equation (
24),
denotes the derivate of the function f with respect to the
ith sensor,
represents the variance of the
ith sensor and
is the covariance between the
ith and
jth sensors. In this case, the sensors are independent so the covariance
disappears and the resulting variance is
For the correlated terms of the covariance matrix, it is possible to calculate the terms solving the equation
where the second term can be eliminated because the sensors are independent. Once Equations (
27) and (
28) are solved, the covariance matrix become extremely complex and it changes every time step. This model involves a very high computational load because the functions that compound each term of the covariance matrix involve many trigonometric functions. After a simulation stage and a post-processing of real data captured during the preliminary experimental phase, it was obtained that this complexity was not necessary, so a simplification was made and the noise covariance matrix was modeled as
where the values of
and
are given taking into account the accuracy of the relative measurements calculated during the laboratory tests and the current value of the tether tension (high tension is related to a better accuracy). In fact, using these values showed better performance of the filter than using the covariance matrix calculated with Equations (
27) and (
28). Hence, the measurement noise covariance matrix chosen for the final tests was Equation (
29).
Finally, for obtaining the solution of the estimation problem, a linear Kalman filter [
31,
32] with Equations (
6) and (
14) is used. This filter will have as output an accurate estimation of the relative state vector at 100 Hz that will be used as the inputs of the control module of the helicopter autopilot.