4.1. Polynomial Interpolation for Quadrotor Navigation
Bézier curves have been used widely and properly for path smoothing in robot navigation [
31] and in motion control schemes for electric motors [
32] and mechanical systems [
33]. In the former, curves are expressed, such as parametric equations, where the time
t is used to determine the values of coordinate pairs of
points graphed on the plane. In this work, a cubic Bézier curve is used and is defined by end points:
and
, and control points:
and
such illustrated in
Figure 4. In the second case, Bézier interpolation polynomials are suitably configured as position or velocity trajectory reference profiles, in order to soft the transition between two operation points for electromechanical and mechanical systems.
It is worthwhile to note that, due to its structure and after a proper selection of endpoints and control points, Bézier curves can be successfully implemented in a quadrotor to online computing the navigation path in cluttered environments, in order to ensure adequate obstacle avoidance manoeuvres while accomplishing a specific mission. On the other hand, it should be noted that derivatives of the trajectory references are not available in advance, and, in consequence, the proposed approach in this paper can be effectively implemented for this experiment.
During the first experiment, the quadrotor is tasked to perform the following: soft take-off to a height of 3 m; navigation through specific operation points in the space; and finally, soft landing, all of them by means of Bézier curves. It is worthwhile to note that the use of these curves is a viable strategy for solving properly the navigation and obstacle avoidance problems. Thus, in order to obtain smooth transitions between initial and final vertical operation points, the following motion scheme is adopted for take-off and landing tasks:
where
and
, given in meters, stand for the desired initial and maximum vertical positions. The time values given in seconds are as follows:
and
. In addition,
is a Bézier polynomial [
32] defined as
with
and
as the initial and final transition times. Moreover,
, and
.
Subsequently, after the take-off, the rotorcraft is carry to desired positions in the horizontal plane, where the third order parametric equations used for navigation are defined as follows:
Here, the values of the endpoints and control points are selected for performing a continuous navigation according to the parameters summarized in
Table 2. Observe that four Bézier curves are used to define the whole navigation path and which is segmented for purposes of mathematical description.
On the other hand, external vibrating disturbance forces have been included after 12 s for robustness assessment purposes of the introduced motion control scheme, and are given by
with
,
,
, and
.
In
Figure 5, it is presented the quadrotor flight performance by implementing the proposed controller, where a proper path following is exhibited. Throughout the manuscript, the use of solid and dashed lines for representing real and desired trajectories is adopted, respectively. As observed in
Figure 6, the Bézier curves are successfully implemented for navigation between operation positions, and as a consequence of the proposed controller, a proper trajectory tracking of the planned references is achieved. Moreover, according to this figure, angular tracking of the online computed references
and
is achieved in spite of there is not information about the derivatives of these references since a properly integration of integral reconstructors and neural networks within the robust motion control approach is achieved.
Furthermore, it is evident the satisfactory performance of the quadrotor tracking motion control scheme even though the quadrotor is subjected to undesired harmonic forces. Notice that regulation around
rad is performed in this experiment. Additionally,
Figure 7 portrays the controlled vertical quadrotor dynamics, the height control, and yaw motion regulation. From this figure, the utility of the Bézier polynomial curve, where a soft take-off and landing are achieved thanks to the mathematical framework introduced by Equations (30) and (52) is appreciated. In the next section, the ground effect is included within the analysis in order to assess the control scheme robustness for controlling the quadrotor vertical motion.
For this experiment the following desired Hurwitz polynomial has been selected,
where, in order to ensure close-loop stability and the properly tracking of the planned trajectory, the control gains in (23) should match the following
where
, for
, is the unique online computed control parameter. To improve and ease the parameter selection process in this experiment, each of these control parameters are suitably derived by the adaptive framework introduced in
Figure 2, where the output of each individual neural network is the value for the control parameter
. As it is presented in
Figure 8, dynamical updating, as well as a successful parameter computation of the control gains, is achieved by using the adaptive B-spline artificial neural networks.
In
Figure 9, it has been included results considering both perturbed and unperturbed cases in order to contrast the compensation action of the adaptive robust control scheme. It is worthwhile to note, from
Figure 9b, that it is possible to track, satisfactorily, the references, as well as being demonstrated in
Figure 9a. Nevertheless, the vibrating disturbance compensation is not present in the unperturbed case. By analyzing
Figure 9b, it is evident the reachability of the control commands which benefits the non-saturation of the actuators. It is also important to mention that similarly as the oscillations due to the control compensation action, in
Figure 6 it is appreciated the compensation of the vibrating disturbance forces affecting translational dynamics since are related with the rotational trajectory tracking trough the under-actuation property.
According to the results, the proposed control method is robust and able to efficiently reduce induced oscillations. Additionally, it is demonstrated that Bézier polynomial interpolation can be widely and satisfactorily exploited in quadrotor motion control systems: path and trajectory tracking. The experiment presented in this section illustrates that the complex quadrotor non-linear system is motion controlled in an acceptable way. As no information is required about derivatives of the trajectory references and from the external disturbances the control process is simplified significantly.
4.2. Improved Robust Quadrotor Autonomous Landing
One of the most essential requirements for a VTOL vehicle is to ensuring a safe landing flight phase. Rotorcraft are subjected to significant variations in motion control during take-off and landing stages due to the increase in lift force when they are close to the ground. Such phenomena are known as the ground effect [
34]. The aim of this experiment is to assess the capabilities of the proposed controller for dealing with the ground effect in simulation. Therefore, the Cheeseman and Bennett modified ground effect model, proposed for quadrotors by authors in [
35], are used, which state the following:
where the ratio
is equal to one outside of the ground-effect. In addition,
r is the propeller radius,
represents the distance from the rotor to the ground,
u and
is the input thrust commanded and the generated real thrust, respectively. Notice, the third expression of equations set (4) is affected by the introduced model representation of the ground effect phenomenon, where it is evident that
and referring to the above equation and using the real generated input thrust in the nominal mathematical model it yields the following
or
Thereafter, without loss of generality
with
where
should be compensated by the adaptive robust motion control approach. In addition, the following data have been used during the simulation:
,
m, and
m.
On the other hand, in
Figure 10 the quadrotor landing is illustrated. Here, it is used two different values for the learning rate
ℓ and for the weighting vector for vertical motion
, in order to illustrate two cases where the effect of increasing or decreasing the parameter values within the adaptive framework defines the quadrotor operation. Moreover, it is observed that a better tracking performance of the closed-loop system is achieved when a suitably selection of the parameters is done. In
Table 3 are showcased the respective values for the aforementioned parameters in each case.
It is relevant to mention that in this experiment it is adopted the same set up outlined by expressions (33) and (34) defined in the previous section. Thus, as corroborated by the dynamic behavior of
in
Figure 10, online computation of the control parameters is accomplished dynamically by the adaptive framework. From the same figure, it is also appreciated that the magnitude of the control effort is modified in function of the disturbance force exerted as a consequence of the ground effect. Nevertheless, a significant deviation of the actual motion from the planned reference is observed in the first case. In contrast, in the second case, acceptable attenuation levels of induced oscillations is attained by a proper selection of the parameters presented in
Table 3.
The key for a successfully performance of the adaptive scheme depends on a properly selection of the adaptive parameters during the design process. Note that the selection of the initial weights within the offline training procedure, different operational conditions can be take into account for improving the initial system response, and, in this way, leading the quadrotor non-linear system to stable scenarios. In the next section, a different setup is introduced for selection of the control parameters: a desired Hurwitz polynomial where three parameters will be computed and a optimized selection by means of particle swarm theory.
4.3. Bs-ANN Offline Training by Particle Swarm Optimization
Inspired by the social behavior observed in fish schools and bird flocks, particle swarm optimization (PSO) has been proposed as an effective solution for solving a wide range of optimization problems [
36]. The use of intelligent agents, called particles, allows this algorithm to iteratively find the best solution on a defined space of searching. For this reason, potentials of PSO has been properly exploited in different engineering and researching applications, such as tuning of automatic controllers [
37] and artificial neural networks training [
38]. In the second experiment, the PSO is used for the offline training of the BsNN (selection of the initial weights). The training process is performed while the system is commanded to reach a step reference for vertical translational motion, where the closed-loop response information is used for designing the objective function
to be minimized.
Figure 11 portrays the closed-loop response of a second order system. Here, it can be observed that there exist several parameters can be used in the design of the objective function in order to minimize the tracking error and the control efforts:
,
,
, and
stand for the rise time, settling time, maximum peak, or overshoot and peak time, respectively.
In this study, only the overshoot data are used as design parameter of the following objective function
where the coefficients
and
penalize the error and the magnitude of the control inputs, respectively. On the other hand, the integral time absolute error (ITAE) index is computed as follows
here
is the tracking vertical error and
t is the time variable. Additionally, the integral squared control input (ISCI) term is introduced in Equation (43).
In contrast with the previous experiments, it has been selected the following Hurwitz polynomial:
here
, are the controller adjustment parameters. Therefore, concerning Equation (22), the control gains can be selected as follows for ensuring closed-loop stability and the properly tracking of the planned trajectory
For the third experiment, the quadrotor take-off stage is analyzed. In order to improve and ease the tuning process, the control parameters are properly computed online by using artificial neural networks which trained offline by a PSO framework.
In the Algorithm 1, it is presented the pseudocode for the training process, where a simulation time of 10 seconds is adopted.
Algorithm 1: Evaluation of the objective function . |
|
The MATLAB optimization toolbox is used for the execution of the PSO algorithm. It is worthwhile to note that the procedure in Algorithm 1 is evaluated in each iteration of the optimization process, in order to determine the best set of control parameters who minimizes the objective function, which has been designed in function to the vertical motion tracking error, as well as the control input effort. Moreover, for this simulation experiment the PSO algorithm is configured with the dimensions of the search space defined by the low and upper boundaries and , respectively, and a swarm size of 50 particles.
Additionally, in order to highlight the performance of the introduced novel adaptive robust control strategy, in this section it is illustrated the applicability of offline training of Bs-ANN neural networks by description of two relevant scenarios: in the first the offline training is carried out for determining initial values of control parameters
and
without using the online learning. On the other hand, online training is considered for computation of the parameters values throughout second scenario. Henceforth, we identified the scenarios, respectively, as fixed and adaptive. The yielded results are portrayed in
Figure 12 and
Figure 13.
It is worth to mention that from
Figure 12 it is observed that the performance for both scenarios looks similar. Nevertheless, the control signal efforts and the error are significantly decreased by using the adaptive strategy. The ISCI and the ITAE indexes are used also for a quantitative comparison and is summarized in
Table 4 for both cases in experiment 3.
It is important to point out that in first scenario it is also achieved an acceptable performance of the introduced control approach. The tuning procedure of the control gains in automatic control systems is not always an easy tasks since it depends on the designer experience for selecting the control gains. Thus, after a properly setup of the PSO scheme, it is possible to ease the tuning process where several control gains or parameters need to be selected: five gains in the present study. Moreover, in
Figure 12, we highlighted the useful of the offline and online training process in the quadrotor motion control. Here, large overshoot and oscillation is avoided from the the closed-loop response by an efficient implementation of the adaptive robust motion control strategy.
On the other hand, in
Figure 13, it can be appreciated the effects for using the online training of the Bs-ANN in contrast with the fixed case utilized in the first scenario of third experiment. According to the information presented in this figure, it is corroborated that by using the full adaptive scheme it is possible to improve the closed-loop response of the quadrotor system by suitably adjusting the control parameters. Notice that the introduced control scheme, it is able to perform efficiently regulation and trajectory tracking tasks even though there is not full knowledge of the non-linear quadrotor mathematical model, as well as the external vibrating disturbances.
4.4. Quadrotor Subjected to Wind Gust Disturbances
In this section, a Dryden wind gust model is used for the assessment of control robustness. From the set of Equation (4), it is evident that in presence of induced disturbance torques, the angular, as well as the translational trajectory tracking, will be deteriorated. Therefore, in this experiment, the quadrotor is disturbed while it is hovering and path following in order to simulate different scenarios which it would usually face within a wide range of applications. Consider the wind gust mathematical model [
39] given by
for
. Expression (46) considers that the disturbance caused by wind field is proportional to the wind speed [
39], which is described as a family of time-varying excitations. On the other hand,
and
are randomly selected frequencies and phase shifts, respectively;
n is the number of sinusoids,
is the amplitude, and
is a static term for wind disturbance. Thus, the mathematical expression in (46) can be integrated in (5) for this simulation experiment as torque disturbances
,
and
, with
for
and
, and
for
. Disturbance parameters are summarized in
Table 5.
Consider the following planned references for lateral and longitudinal quadrotor motion in this experiment,
with
, and the Bézier based motion profile for vertical motion defined by Equations (30) and (52) with the following data:
,
,
and
. Additionally, the yaw motion is regulated about a constant angle
rad. Soft transition between initial condition and the regulation point is accomplished by a Bézier polynomial.
Figure 14 and
Figure 15 describes the effective performance of the adaptive robust motion control scheme (20), which compensates the disturbance forces induced by the wind gust model introduced in (46). Moreover, it is evident excellent levels of oscillations attenuation by using our control approach.
From
Figure 16, it is observed that the quadrotor is able to efficiently perform trajectory tracking tasks in spite of there is not previous information of the disturbance torques while tracking the planned references introduced in (47). On the other hand, the control input forces and torques generated by the proposed controller are presented in
Figure 17. Here, it is appreciated a properly compensation of the disturbance effects which by the computed control inputs. The closed-loop system response for rotational dynamics is plotted in
Figure 18, where it is corroborated an efficient performance of the introduced adaptive robust control approach, as done in previous experiments. It is important to mention that during experiment 4 it is adopted the same process for the computation of the control gains in the first experiment.
Finally, in
Figure 18, it can be seen the control parameters for rotational motions, which are computed online by means of the adaptive BS-ANN scheme. In addition, it is corroborated that even though there is not available information from derivatives of the angular references, the under-actuation problem is properly solved by the use of the neural networks and the integral reconstructors, thereby a good tracking of the online computed references
and
is achieved.
4.5. Robustness against Uncertainty of Quadrotor Mass
Another important issue for controlling a quadrotor is the variations of the nominal mass. Notice that the online computed references in (8) which define a proper motion on the plane depends on the nominal mass. Therefore, the quadrotor is supposed to follow the references considering the nominal mass value. Thus, during this experiment, it is probed if the vehicle flight may be deteriorate significantly when an extra mass is added.
Consider the following mass variation for this experiment
where
stands for the nominal quadrotor mass in kg, and
is an abrupt change of the mass quadrotor described by a modified impulse function given by
with
,
,
,
, and
. During the experiment it is adopted a spiral shape planned reference given by the next parametric equations
where
and
. Inspecting
Figure 19, it is appreciated that the abrupt variation in the quadrotor mass does not affect significantly the following of the planned reference. In
Figure 20, it is observed a slightly deviation of the quadrotor angular tracking in contrast with the nominal references
and
, computed with the nominal mass. Moreover, it is evident that after a brief period of time the quadrotor is able to recover from the perturbation and perform a proper tracking of the desired references thanks to the robustness of the proposed control scheme.
In
Figure 21a, it is presented the compensation to the mass variation at
s by the control input
u.
Figure 21b portrays the vertical motion which is performed before the path following, and is given by the following Bézier polynomial
where
and
, given in meters, stand for the desired initial and maximum vertical positions. The time values given in seconds are as follows:
. In addition,
is the Bézier polynomial introduced in (30) with
and
as the initial and final transition times. Moreover,
, and
.
Future studies will address other methodologies for trajectory generation. Interested readers are referred to [
40,
41] and references therein for further information on trajectory generation. Moreover, algebraic estimators [
42] will be explored for determining variation in the quadrotor nominal mass due to unknown payload and damage or failure in the quadrotor frame.