Adaptive Control of Quadrotors in Uncertain Environments

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Adaptive Control of Quadrotors in Uncertain Environments

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Submitted:

11 March 2024

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11 March 2024

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Abstract

The problem addressed in this article consists of the motion control of a quadrotor, comprising model disturbances and uncertainties. In order to tackle model uncertainty, adaptive control based on reinforcement learning is used. The distinctive feature of this article, in comparison with other works on quadrotor control using reinforcement learning is the exploration of the underlying optimal control problem in which a quadratic cost and a linear dynamics allow an algorithm that runs in real time. Instead of identifying a plant model, adaptation is obtained by approximating the performance index given by the Q-function using directional forgetting recursive least squares that rely on a linear regressor build from quadratic functions of input/output data. The adaptive algorithm proposed is tested in simulation in a cascade control structure that drives a quadrotor. Simulations show the improvement in performance that results when the proposed algorithm is turned on.

Keywords:

Subject: Engineering - Control and Systems Engineering

1. Introduction

1.1. Motivation

Unmanned Air Vehicles are currently becoming an important player in areas such as inspection and maintenance, photography, surveillance, mapping, agriculture, military warfare, delivery, and more [1,2]. Fully automated vehicles without human intervention have been receiving increased attention [3], as advanced controller architectures are [1,2] progressing towards full autonomy, an example of which being adaptive controllers that can tackle wind perturbations or uncertainty in the model parameters.

Reinforcement learning (RL) is a technique in which the control action to apply to the manipulated variable of the plant is selected such as to maximize a reward signal [4]. Some RL algorithms can provide adaptation in unique ways, for example, without the need for any model knowledge about the system upon which it acts. This opens a gateway for a new class of adaptive controllers, that can attain some degree of control robustness, even when faced with unmodeled dynamics. The topic of drone control using RL has been the subject of many recent works. In addition to the works reviewed in [2], other references include [5,6,7], as well as [8,9], that address real-time RL.

Although general RL based controllers yield remarkable performances for a wide variety of robotic and aircraft plants [10], the fact that they rely on deep neural networks imply long training periods, with a massive computational load, that may only be executed off-line. Nevertheless, for a class of problems in which the plant dynamics is linear and the cost to optimize is quadratic, the computational cost is greatly reduced, being compatible with on-line adaptive control.

This work has the objective of developing an adaptive controller for a quadrotor based on such a specific RL model-free algorithm, according to the lines presented in [11]. It is stressed that, opposite to the works reported in [2], the algorithm proposed in this article does not rely on off-line training using massive amounts of data but, instead, provides real time adaptation of a vector of state feedback gains, without knowing or estimating the parameters of any plant model. By real time adaptation it is meant that the algorithm is fast enough to be run as the drone operates. This procedure is possible because we restrain the class of problems addressed to be of linear quadratic type, i. e., defined by an underlying plant dynamics to be linear but unknown and a quadratic performance cost to be minimized.

Assuming the underlying plant dynamics to be linear is justified by the type of application envisaged;: the algorithm, developed is embedded in the control part of the quadrotor GNC (guidance, navigation and control) system and has the task of regulating the quadrotor state around a reference trajectory generated by the guidance system. Hence, the controller is required to tackle only incremental dynamics that may be approximated by a linear model.

With respect to other works described in the literature for motion control of quadrotors based on reinforcement learning [2], the approach proposed here has the drawback of not allowing large amplitude maneuvers, in which the nonlinear character of the dynamics dominates. In compensation, the control algorithm proposed provides a model free adaptive controller that operates in real time with little a priori plant knowledge. The definition and use of such a controller for drone motion is the focus of this article.

1.2. Problem Definition, Objectives and Contribution

A common control architecture for quadrotors consists of two controllers interconnected in a cascade structure: an outer control loop, responsible for keeping the quadrotor in a desired position, and an inner control loop, that tracks the quadrotor attitude demanded by the outer controller.

The attitude controller works on the premise that the quadrotor is not performing extreme maneuvers, so that it is valid to assume that the drone dynamics is approximately linear. In order to tackle uncertainty in dynamics, the inner loop, that regulates the attitude, relies on an adaptive controller that embeds a reinforcement learning (RL) algorithm to learn the gains that optimize a quadrati cost.

The outer loop consists of a linear quadratic (LQ) controller with constant gains, designed using a nominal model and is such that it has a time scale that is much slower than the one of the inner loop.

The contribution of this article consists on the design of the above cascade adaptive control architecture and its demonstrations in simulations for a quadrotor model. The main innovative feature is the embedding of a reinforcement learning algorithm in the inner loop to yield a model free adaptive controller. The use of a directional forgetting least-squares estimation algorithm to avoid identifiability problems in the Q-function approximator is also is also a contribution.

The article is organised as follows: section 2 describes the drone model used in the simulations; section 3 presents the control architecture; section 4 explains the class of RL based contriol algorithms used; section 5 peresents simulation results; finally, section 6 draws conclusions.

2. Dynamics Modeling

Although the proposed controller does not rely on a model, the dynamics of the drone considered is described in this section for the purposes of simulation and interpretation of the results presented later-on.

The model dynamics follows the rigid body equations of motion [12]. For this sake, an inertial reference frame

{I}

attached to the ground and a fixed body frame

{B}

, attached to the vehicle, are defined as represented in Figure 1.

The rigid body state comprises a position in the inertial frame p, a velocity in the body frame v, an angular velocity in the body frame

ω

, and a rotation matrix R that represents the transformation between the body fixed frame

{B}

and the inertial frame

{I}

orientations. The full kinematics equations are

\{\begin{matrix} \dot{p} = R v \\ \dot{λ} = Q (λ) ω \end{matrix},

(1)

where the quadrotor attitude

λ

is expressed in Euler angles, a vector with the elements

ϕ

as the angle of rotation about the x axis,

θ

as the angle of rotation about the y axis and

ψ

as the angle of rotation about the z axis. The matrix Q is defined as

Q (λ) = [\begin{matrix} 1 & sin ϕ tan θ & cos ϕ tan θ \\ 0 & cos ϕ & - sin ϕ \\ 0 & sin ϕ sec θ & cos ϕ sec θ \end{matrix}] .

(2)

The dynamics equations are

\{\begin{matrix} m \ddot{p} = T R e_{3} - m g e_{3} + w \\ J \dot{ω} = - S (ω) J ω + n \end{matrix},

(3)

where J represents the moment of inertia matrix, that is assumed to be diagonal. The application

S (.)

corresponds to a skew-symmetric matrix. The inputs T and n represent the thrust force and torque respectively.

3. Controlling the Quadrotor

3.1. Controller Structure

The relationship between both sets of equations (1) and (3) allows the development of a cascade control structure that has an inner loop for controlling the attitude of the quadrotor, and an outer loop for controlling the position and that generates as manipulated variable the desired orientation to the inner (attitude) controller. This type of controller is commonly used in UAVs and its block diagram is shown in Figure 2.

The trajectory generator block creates the references

p_{d}

{\dot{p}}_{d}

and

{\ddot{p}}_{d}

for the position controller, and the reference

ψ_{d}

for the attitude controller. These signals represent the desired position and its derivatives, and the desired

ψ

angle, respectively. Besides generating an input thrust T, the position controller also generates a desired upwards orientation

r_{3 d}

, corresponding to the third column of R, that is combined with

ψ_{d}

in the attitude controller to track the desired orientation angles.

3.2. Position Controller

Neglecting the rotational dynamics, it is assumed that the desired orientation

r_{3 d}

, can be forced upon the system. Considering that

r_{3} = R e_{3}

, the virtual input

u_{T} = \frac{T}{m} r_{3 d}

is defined, and transforms the first line of (3) into

\ddot{p} = u_{T} - g e_{3},

(4)

which is now a linear system, that can be controlled with a linear controller.

For the sake of reference trajectory tracking, the error variable is defined as

e = p - p_{d},

(5)

where

p_{d}

represents the reference position. Taking the second derivative of (5) and resorting to the system defined in (4), the error dynamics

\ddot{e} = \ddot{p} - {\ddot{p}}_{d} = u_{T} - g e_{3} - {\ddot{p}}_{d}

(6)

is obtained. Selecting

u_{T}

to be

u_{T} = - (k_{1} e + k_{2} \dot{e}) + g e_{3} + {\ddot{p}}_{d},

(7)

introduces linear state feedback, where each state has a proportional gain. There is also a correction term in order to eliminate the second derivative of the error present in the dynamics. Applying the control law transforms (6) into

\ddot{e} = - k_{1} e - k_{2} \dot{e},

(8)

where the gains

k_{1}

and

k_{2}

are selected such as to the error e to zero, enabling reference tracking given a desired trajectory

p_{d}

, and its derivatives

\dot{p_{d}}

and

\ddot{p_{d}}

The virtual control input, which is a three-dimensional vector, is translated into the thrust T input and desired orientation

r_{3 d}

(forced upon the system) values. Via equation (7) the thrust T is calculated with

T = m | | u_{T} | |,

(9)

and the desired orientation

r_{3 d}

is obtained through

r_{3 d} = \frac{u_{T}}{| | u_{T} | |} .

(10)

The thrust input is fed directly into the system, as opposed to

r_{3 d}

, which is passed to the attitude controller so that the orientation errors can also be regulated.

3.3. Attitude Controller

Neglecting the translational dynamics, a desired orientation

r_{3 d}

is fed to the attitude controller, that produces an input torque for the system.

To track the desired orientation value, the desired

ψ

angle,

ψ_{d}

is combined with

r_{3 d}

to obtain the remaining desired angles

ϕ_{d}

and

θ_{d}

Through decomposition of the desired R matrix, the equation

R_{z} (- ψ_{d}) r_{3 d} = [\begin{matrix} cos ϕ_{d} sin θ_{d} \\ - sin ϕ_{d} \\ cos ϕ_{d} cos θ_{d} \end{matrix}]

(11)

allows the computation of the desired attitude

λ_{d}

. To track this value, a possible strategy is to linearize the second equations of systems 1 and 3 around the hover condition, a valid approach for smooth trajectories where aggressive maneuvers are not required, resulting in

\{\begin{matrix} δ \dot{λ} = δ ω \\ J δ \dot{ω} = δ n \end{matrix},

(12)

given that the matrix J is diagonal. The linearized system obtained is also a double integrator. The input n is chosen to account for the angle and angular velocity error, resulting in

n = - k_{1} (λ - λ_{d}) - k_{2} ω .

(13)

The inner loop that comprises the attitude controller must react faster than the outer loop containing the position controller. This time scale separation is forced to ensure that the system quickly corrects angle displacements, that are detrimental to effectively tracking the desired position. To guarantee correctness, the gains of each controller must be chosen such that the inner loop poles contain a real part that is ten times (or more) larger than the real part of the outer loop poles.

3.4. Underlying LQ control

The adaptive RL controller proposed in this article consists of a state feedback in which the gains converge to the ones of a LQ controller. Hereafter, the model and the quadratic cost that define this underlying controller are defined. Assuming that the inputs are piece-wise constant over the sampling period h, the discrete-time equivalent [13] of the double integrator dynamics takes the form

x_{k + 1} = Φ x_{k} + Γ u_{k}

, where

Φ = [\begin{matrix} 1 & h \\ 0 & 1 \end{matrix}], Γ = [\begin{matrix} h^{2} / (2 J_{i, i}) \\ h / J_{i, i} \end{matrix}],

(14)

where

J_{i, i}

is the diagonal element from the inertial moment matrix J associated with the angle to be controlled, for the rotational dynamics. This value is unitary for the translational dynamics.

Having both

Φ

and

Γ

, the discrete controller gains for the control laws in (7) and (13) can be calculated resorting to the LQR algorithm, that aims at minimizing the quadratic cost

J_{c} = \sum_{k = 0}^{\infty} (x_{k}^{T} Q_{c} x_{k} + R_{c} u_{k}^{2}) .

(15)

A careful choice of the

Q_{c}

and

R_{c}

weight matrices is necessary for a good performance.

The sampling period selected throughout all experiences is

h = 0.01

, equivalent to a frequency of 100 Hz. The quadrotor state variables are assumed to be fully observable. However, a sensor noise effect is reproduced by adding white noise after the A/D sampling process. A standard deviation of

σ = 3.16 \times 10^{- 5}

(with zero mean) is used in the simulation.

4. Reinforcement Learning

4.1. Introduction

Reinforcement learning is an area of Machine Learning that aims to learn what is the best action an agent can execute as it interacts with the environment on which it acts upon [4]. From a control perspective, we can think of the environment as the system and the action as the output produced by the controller, which is the agent [14].

In this setting, the action is now the control action

u_{k}

and the state is

x_{k}

. The control policy determines the value of

u_{k}

and it is defined as

u_{k} = h (x_{k}),

(16)

consisting of a map

h (.)

from the state space to the control space. Since this map depends only on the current state

x_{k}

, it defines a state feedback controller. This definition needs to be coherent with the definition of the reward signal, as detailed further ahead.

Let the system be described by

x_{k + 1} = A x_{k} + b u_{k},

(17)

where A and b are matrices that define the linearized quadrotor dynamics.

4.2. Dynamic Programming and Q-Function

The control policy is to be designed such as to minimize

V_{h} (x_{k}) = \sum_{i = k}^{\infty} γ^{i - k} r (x_{i}, u_{i}),

(18)

often called the cost-to-go or return. The discount factor

γ

factor, takes values in the interval

[0, 1]

For the reward signals, this article considers the quadratic function

r (x_{k}, u_{k}) = x_{k}^{⊺} Q_{c} x_{k} + u_{k}^{⊺} R_{c} u_{k} .

(19)

In order to obtain model-free controllers, define the Q-function

Q_{h} (x_{k}, u_{k}) = r (x_{k}, u_{k}) + γ V_{h} (x_{k + 1}),

(20)

The Q-function verifies a Bellman like equation given by

Q_{h} (x_{k}, u_{k}) = r (x_{k}, u_{k}) + γ Q_{h} (x_{k + 1}, u_{k + 1}) .

(21)

The optimal control policy is then calculated through

h^{*} (x_{k}) = \underset{u}{arg min} (Q_{h} (x_{k}, u_{k})) .

(22)

In the absence of constraints, the optimal control policy can then be computed by solving

\frac{\partial Q^{*}}{\partial u} = 0 .

(23)

4.3. Approximate Dynamic Programming

To determine the optimal policy at time k, it is necessary to know the optimal policy at time

k + 1

. This minimization process is done backwards in time, and it is only feasible as an off-line planning method, using iterative algorithms. The resulting Dynamic Programming methods are not enough to obtain a controller which can learn in real time. Approximate Dynamic Programming solves this problem and introduces two other concepts that will make the operation forwards in time possible.

The first concept is that of Temporal Difference (TD). Taking advantage of equation (21), it is possible to define a residual error

e_{k}

given by

e_{k} = r (x_{k}, u_{k}) + γ Q_{h} (x_{k + 1}, u_{k + 1}) - Q_{h} (x_{k}, u_{k}) .

(24)

If the Bellman Equation is satisfied, then the error is zero. This error can be seen as the difference between observed and predicted performances as a previously calculated action is applied. The next important tool is the Value Function Approximation (VFA), that approximates the Q-function by a parametric approximator. For linear plants and quadratic costs, it can be shown [11] that the Q-function can be rewritten as

Q_{h} (x_{k}, u_{k}) = W^{⊺} ϕ (x_{k}, u_{k}),

(25)

where W is a vector of coefficients and

ϕ (x_{k}, u_{k})

is a vector containing the basis functions with the square terms and cross terms for all elements of

x_{k}

and

u_{k}

Combining the equations (25) and (24), the TD error becomes

e_{k} = r (x_{k}, u_{k}) + W^{⊺} (γ ϕ (x_{k + 1}, u_{k + 1}) - ϕ (x_{k}, u_{k})),

(26)

that defines a linear regression model with parameters W. Approximating

Q_{h}

by a linear combination of basis functions, enables the use of estimation techniques to obtain the coefficients which will then define the

Q_{h}

closest approximation [11]. Equations (21), (22) and (25) allow building an on-line forward-time learning algorithm for state feedback control.

4.4. Q Learning Policy Iteration

In this article, the Policy Iteration adapted for Q Learning is chosen to be applied in the controller.

The Policy Evaluation step, where the VFA coefficients are estimated, requires determining a least squares solution to find the estimate of the parameter vector W in the linear regression model (26). This can be done by resorting to Recursive Least Squares (RLS) or batch least squares. In this article Directional Forgetting RLS is used.

A Gaussian dither noise is added to the observed control input signal

u_{k}

, derived from the current policy

h_{j}

obtained from (23). The reason behind this procedure is the necessity to ensure a persistency of excitation condition. The absence of this effect can cause the algorithm to completely fail [11].

For estimation purposes, Recursive Least Squares with Directional Exponential Forgetting (RLS-DF) is used [15], where the estimated Q-function gains

W_{j + 1}

are updated at every time step k every time step, and the policy

h (x_{k})

is updated after a fixed number of time steps. This approach divides the computational effort of estimation across every time step, whereas a batch least-squares update concentrates all estimation in one run. The Q-Learning algorithm for the versions using RLS-DF is detailed in Algorithm 1. Preprints 101090 i001

For a two-dimensional state system, the VFA is defined as (25) with the basis functions given by

ϕ (x_{k}, u_{k}) = (x_{k, 1}^{2}, x_{k, 2}^{2}, x_{k, 1} x_{k, 2}, x_{k, 1} u_{k}, x_{k, 2} u_{k}, u_{k}^{2}),

(27)

which means that the policy improvement step at the update

j + 1

, as defined in Algorithm 1, takes the form

h_{j + 1} (x_{k}) = - \frac{w_{j + 1}^{(4)}}{2 w_{j + 1}^{(6)}} x_{k, 1} - \frac{w_{j + 1}^{(5)}}{2 w_{j + 1}^{(6)}} x_{k, 2} .

(28)

The control action takes the form of a linear state feedback

h_{j + 1} (x_{k}) = K x_{k}

, with a K gain of

K = [\begin{matrix} \frac{w_{j + 1}^{(4)}}{2 w_{j + 1}^{(6)}} & \frac{w_{j + 1}^{(5)}}{2 w_{j + 1}^{(6)}} \end{matrix}],

(29)

where

w^{(i)}

represents the estimate of the ith element of W.

The importance of the use of DF-RLS stems from the fact that, although there are only 2 controller gains, the Q-function is approximated by 6 parameters, a fact that may cause identifiability problems.

5. Simulation Stusdy

In this section, simulations are shown to illustrate the results obtained when applying the above algorithm to control the motion of the quadrotor model described in Section 2. The simulation experiments show that, with the proposed controller, the algorithm is able to track a reference in which there are curved sections followed by segments that are approximately straight. Most important, the simulations illustrate adaptation. In the experiments, there is an initial period in which a constant vector of, a priori chosen, controller gains is used. During this period the algorithm is learning the optimal gains but these ones are not used in feedback. After this initial period, the gains learned by the RL algorithm are used. There are two sets of experiments. In the first set, the initial gains are far from the optimum. In the second set of experiments, the initial set of gains are not optimal, but they are closer to optimum than the gains in the first set. Two relevant results are shown: one is the capacity of the algorithm to improve the performance when the gains obtained by RL are used; furthermore, the dither noise can be reduced when the initial gains are closer to the optimum (that is to say, when more a priori information is available). The evaluation and comparison of the different situations is done using an objective index (the “score”) defined below.

The wind force w is characterized by its x, y, and z components, that determine its direction and magnitude. White noise is applied to a low-pass filter and added to w to produce more realistic deviations from the average magnitude. The average value of the disturbance is assumed to be measurable and is compensated for, but the model is still affected by the white noise effect, that has a standard deviation of

σ = 3.16 \times 10^{- 5}

. The wind disturbance value for the i component is described as

w_{i} = v_{i} + o_{i},

(30)

where

w_{i}

represents the generated wind,

v_{i}

represents the average value, and

o_{i}

represents the filtered white noise. The dynamics of the motor are neglected, being considered much faster than the remaining quadrotor dynamics. The aerodynamic drag effect is also neglected, since most of the quadrotor operations are maintained in a near hover condition.

The model considered has the following parameters:

$m = 1.35$ Kg
$l = 0.18$ m
$J_{1, 1} = J_{2, 2} = 0.01$ Kg.m²
$J_{3, 3} = 0.01$ Kg.m²

For simulation purposes, the constant wind disturbance is compensated, assuming that it is possible to measure its average value. The noisy oscillations around the average value still affect the system.

To gain a better insight on how well the controller performs before and after the learning process, a performance metric is defined as

score = \frac{\sum_{i = 1}^{N} | | p_{i}^{*} - p {| |}^{2}}{N},

(31)

where

p^{*}

represents the desired position and p the actual position. The metric calculates the average value of the distance between the moving point in the reference trajectory and the actual position of the quadrotor. The lower this value, the better the controller can track the given reference trajectory.

5. The control algorithm parameters are selected by trial and error in simulations. The main parameters to adjust are the weights in the quadratic cost considered and the dither noise variance, added to the control variable. The dither noise variance must be chosen according to a trade-off between not disturbing optimality (meaning that the variance must be small) and providing enough excitation to identify the parameters of the quadratic function that approximates the Q-function (which requires increasing the dither variance). The

R_{c}

weight adjusts the controller bandwidth and must be selected to ensure that the inner loop is much faster than the outer loop.

In order to avoid singularities on the model, the references are such that the attitude angle deviates only by a maximum value with respect to the vertical.

5.1. Experiment 1

The first test attempts to improve on a controller that is tuned with the following weights in relation to the quadratic cost defined above:

Position controller: $Q_{c}$ = diag(200,1), $R_{c}$ = 100;
Attitude controller (except $ϕ$ ): $Q_{c}$ = diag(100,1), $R_{c}$ = 10;
Attitude controller ( $ϕ$ ): $Q_{c}$ = diag(10,5), $R_{c}$ = 10;

This calibration affects the attitude control of the angle

ϕ

, where a poor selection of weights is chosen so it can be improved on.

The algorithm has the parameters shown in Table 1.

The learning process (during which a priori chosen controller gains are used) lasts 400 seconds, and the trajectory in the form of a lemniscate has cycles of 20 seconds, meaning that each learning cycle of 10 seconds comprises half a curve. The starting point is (2,0,0) at rest. The lemniscate has been selected as a test reference since it combines approximated straight and curved stretches.

Simulation results for the trajectory tracking improvement are presented in Figure 3. To test the performance of both the initial and learned gains, a single cycle of the lemniscate curve is used.

For a starting point at (2,0,3), that starts right at the beginning of the lemniscate curve, the controller after learning a better set of gains has a score (as defined in equation (31)) of 0.0177, whereas the untuned controller produced a score of 0.1203. These results show a clear improvement in the trajectory tracking performance. The corresponding gain evolution in time is presented in Figure 4.

Despite some oscillations, the convergence is quick. This happens due to the selection of high variance for the dither noise. However, this procedure can be a problem, given that in real life quadrotors have limitations on the actuators that might render such high values impossible. The magnitudes of the input during the learning stage can induce big oscillations in the quadrotor, since the angular velocity and the angle get excited by the effect of the dither noise.

In order to be able to reduce the dither noise power and still obtain satisfactory results, an alternative is to increase the number of steps required for an update. However, this approach slows down the learning process. Another possibility is to increase the values of the covariance matrix of the LS estimator at each update, so that the updates are less influenced by previous updates. This comes at the price of bigger oscillations. The new parameters are shown in Table 2.

With these updates, a new simulation rendered the results in Figure 5 that shows the evolution of the gains with time. Even though the learning process slows down as expected, improvements are still achieved. Figure 6 demonstrates that good trajectory tracking was obtained.

For a starting point at (2,0,3), that coincides with the beginning of the lemniscate curve, the controller performs reference tracking with a score (as defined in equation (31) of 0.0175 after the learning process, whereas the untuned controller produced a score of 0.1204, with the input torque

n_{x}

acquiring smaller values as a direct consequence of the reduction of the dither noise.

5.2. Experiment 2

The previous two tests aim to improve the controller gains starting from a non-optimum scenario. The following three tests try to improve on already optimized results. This might pose a better challenge for the algorithm, since there will be no need for big amounts of excitation to learn an improved set of gains.

The starting point for the controller gains corresponds to the following configuration of weights:

Position controller: $Q_{c}$ = diag(200,1), $R_{c}$ = 100;
Attitude controller: $Q_{c}$ = diag(100,1), $R_{c}$ = 10;

The difference between experiments is the value of the moment of inertia

J_{1, 1}

, which is assumed in each experiment to be:

Third experiment $J_{1, 1} = 0.04$ Kg.m²
Fourth experiment $J_{1, 1} = 0.005$ Kg.m²
Fifth experiment $J_{1, 1} = 0.01$ Kg.m²

The real value of

J_{1, 1}

is 0.01 Kg.m². This means that each experience has its

ϕ

attitude controller gains with different values, upon the application of the LQR algorithm.

The algorithm has the same parameters throughout all three experiences, presented in Table 3.

The dither is now significantly reduced, as less excitation is required. The small expected improvements allow for some reduction of the confidence matrix P values, to dampen more some oscillations that might appear.

The third experiment rendered the results that follow in Figure 7 and Figure 8.

A closer look at the trajectories reveals that improvements do occur, with the improved gains producing a better tracking performance. The score for the unoptimized version is 0.0202, whereas the version with learned gains scored 0.0175, reflecting a small improvement over the original configuration.

The simulation of the fourth experiment rendered the results in Figure 9 and Figure 10.

The score for the performance metric before RL is 0.01781, and after applying the algorithm, the score becomes 0.01765. Even if there was an improvement, in this case, it is almost negligible. This can be seen in Figure 10 where both trajectories pratically overlap each other, ending up having the same distance to the reference lemniscate curve.

The simulation of the fifth experiment rendered the results in Figure 11 and Figure 12

The score metric for the original set of gains is 0.01811, and after applying the algorithm, the score becomes 0.01765. In this last experience, a small improvement was achieved, even with the real value of

J_{1, 1}

being known and used to calculate the controller gains that are used before applying RL. A good tuning of the algorithm allowed for a small performance improvement over a very optimized controller.

6. Conclusions

The development of a reinforcement learning based adaptive controller for a quadrotor, that includes an adapted version of the Q-learning policy iteration algorithm for linear-quadratic problems, was performed. The particular class of RL based controllers considered is such that it allows adaptation in real time.

Adjustments to the baseline algorithm were necessary to make it robust to external perturbations, something achieved through thoughtful choices of the parameters.

Disturbances affecting the system input and output have a big effect on the correct functioning of the algorithm. A careful choice of algorithm parameters and balance between the estimation algorithm parameters is the solution to this problem. However, when big disturbances are present, it is only possible to make the gains converge close to optimality. Increasing the influence of prior estimations allows for a greater degree of robustness, with the drawback of deviating the convergence process to nearby values of the original optimal gains. Nonetheless, that is necessary to prevent harsh oscillations in the learned gains, which are still present in the quadrotor tests, most likely due to the non-linear nature of the rotational dynamics and other unmodelled dynamics, besides the perturbations.

Still, the algorithm produced good results provided that the drone is kept working within the near-linear zone of operation, that is, where safe maneuvers with the quadrotor close to the hovering position are prevalent.

Author Contributions

D. Leitão: research, implementation, writing; R. Cunha: research, methodology, writing; J. M. Lemos: research, methodology, writing

Funding

Part of this work was supported by INESC-ID under project UIDB/50021/2020 and by LARSyS under project UIDB/50009/2020, both financed by Fundação para a Ciência e a Tecnologia (Portugal).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data are not publicly available. The data files are stored in corresponding instruments in personal computers.

Acknowledgments

No acknowledgments.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

RL	Reinforcement Learning
LQR	Linear Quadratic Regulator
VFA	Value Function Approximation
TD	Temporal Difference
RLS	Recursive Least-Squares
MDP	Markov Decision Process
ZOH	Zero-Order Hold

References

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Figure 1. Quadrotor with inertial reference frame and fixed body frame defined.

Figure 2. Cascade controller for the quadrotor motion.

Figure 3. Trajectory before and after RL; first case.

Figure 4. Evolution of the learned attitude controller gains with time (1st case: dither covariance

σ_{D N} = 0.5

); the blue line represents the first gain and the red the second gain.

Figure 4. Evolution of the learned attitude controller gains with time (1st case: dither covariance

σ_{D N} = 0.5

); the blue line represents the first gain and the red the second gain.

Figure 5. Evolution of the learned

ϕ

attitude controller gains with time (2nd case: dither covariance

σ_{D N} = 0.1

Figure 5. Evolution of the learned

ϕ

attitude controller gains with time (2nd case: dither covariance

σ_{D N} = 0.1

Figure 6. Trajectory before and after RL; second case.

Figure 7. Evolution of the learned attitude controller gains with time (3rd case: dither covariance

σ_{D N} = 0.05, P_{r e s e t} = 500

Figure 7. Evolution of the learned attitude controller gains with time (3rd case: dither covariance

σ_{D N} = 0.05, P_{r e s e t} = 500

Figure 8. Detail of the trajectory before and after RL; third case.

Figure 9. Evolution of the learned

ϕ

attitude controller gains with time; fourth case.

Figure 9. Evolution of the learned

ϕ

attitude controller gains with time; fourth case.

Figure 10. Detail of the trajectory before and after RL; fourth case.

Figure 11. Evolution of the learned

ϕ

attitude controller gains with time; fifth case.

Figure 11. Evolution of the learned

ϕ

attitude controller gains with time; fifth case.

Figure 12. Detail of the trajectory before and after RL; fourth case.

Table 1. Algorithm parameters for the first test.

Weight matrices	$Q_{c} = [\begin{matrix} 100 & 0 \\ 0 & 1 \end{matrix}], R_{c} = 10$
$γ$	0.99
Update step	1000 (10 seconds)
$λ_{f}$	0.99
$σ_{D N}$	0.5
w	${[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{⊺}$
$P_{0}$	$10^{7}$
$P_{reset}$	$10^{4}$

Table 2. Algorithm parameters for the second test.

Weight matrices	$Q_{c} = [\begin{matrix} 100 & 0 \\ 0 & 1 \end{matrix}], R_{c} = 10$
$γ$	0.99
Update step	1000 (10 seconds)
$λ_{f}$	0.99
$σ_{D N}$	0.1
w	${[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{⊺}$
$P_{0}$	$10^{7}$
$P_{reset}$	$10^{6}$

Table 3. Algorithm parameters for the third, fourth and fifth tests.

Weight matrices	$Q_{c} = [\begin{matrix} 100 & 0 \\ 0 & 1 \end{matrix}], R_{c} = 10$
$γ$	0.99
Update step	1000 (10 seconds)
$λ_{f}$	0.99
$σ_{D N}$	0.05
w	${[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{⊺}$
$P_{0}$	$10^{7}$
$P_{reset}$	500

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Adaptive Control of Quadrotors in Uncertain Environments

Abstract

1. Introduction

1.1. Motivation

1.2. Problem Definition, Objectives and Contribution

2. Dynamics Modeling

3. Controlling the Quadrotor

3.1. Controller Structure

3.2. Position Controller

3.3. Attitude Controller

3.4. Underlying LQ control

4. Reinforcement Learning

4.1. Introduction

4.2. Dynamic Programming and Q-Function

4.3. Approximate Dynamic Programming

4.4. Q Learning Policy Iteration

5. Simulation Stusdy

5.1. Experiment 1

5.2. Experiment 2

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

References

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