Boundary conditions.

## Abstract

Quadrotor control is an exciting research area. Despite last years developments, some aspects demand a deeper analysis: How a quadrotor operates in challenging trajectories, how to define trajectory limits, or how changing physical characteristics of the device affects the performance. A visual interface development platform is a valuable tool to support this effort, and one of these tools is briefly described in this Chapter. The quadrotor model uses Newton-Euler equations with Euler angles, and considers the effect of air drag and propellers’ speed dynamics, as well as measurement noise and limits for propeller speeds. The tool is able to test any device just by setting a few parameters. A three-dimensional optimal trajectory defined by a set of waypoints and corresponding times, is calculated with the help of a Minimum Snap Trajectory planning algorithm. Small Angle Control, Desired Thrust Vector (DTV) Control and Geometric Tracking Control are the available strategies in the tool for quadrotor attitude and trajectory following control. The control gains are calculated using Particle Swarm Optimization. Root Mean Square (RMS) error and Basin of Attraction are employed for validation. The tool allows to choose the control strategy by visual evaluation on a graphical user interface (GUI), or analyzing the numerical results. The tool is modular and open to other control strategies, and is available in GitHub.

### Keywords

- Quadrotor
- Trajectory Planning
- Trajectory Tracking

## 1. Introduction

Quadrotors are a special type of unmanned aerial vehicles (UAVs), increasingly employed last years for mapping, surveillance, searching and tracking operations, in rescue missions, agriculture, traffic management, landscape film making, and others [1, 2, 3].

When quadrotors applications demand large angle attitude control and obstacle avoidance, the following areas still need to be strengthened: Aggressive maneuvering control, visual-based control, localization in indoor environments, optimizing the computational cost for complex algorithms, and fault-tolerant disturbance rejection [4]. Several controllers have been proposed for these tasks: classic techniques as PD [5] and PID [6], optimal techniques as LQR [7] and LQG [8], non-linear techniques such as Lyapunov [9] and * Backstepping*[5, 9] and intelligent and adaptive control techniques as Fuzzy200 [10] and Reinforcement Learning [11].

Research in quadrotor demand sophisticated equipment and costly laboratory. However, it can be optimized employing low cost virtual development tools. In autonomous control the path planning, path tracking and joint operation with other UAVs, can be supported by optimization techniques with support of software tools [1, 2, 12]. Published works relate the use of software such as Visual Basic, MatLab, Panda3D, Gazebo, or more open applications developed in languages as Python and C++ [2]. In [1], a UAV 3D flight environment programmed in Python and developed in Panda3D is presented. A GUI developed in LabVIEW was published in [2], and other developed on MatLab-Simulink employs quadratic linear regulator (LQR) control [13].

Gazebo is an important virtual environment for robotics. Its integration with ROS provides a powerful testbed to analyze control algorithms. In [12] works that employ Gazebo and ROS for developing simulation of UAVs are described.

Most of the cited simulation tools have a difficult start for users with little programming experience. Even open-source models can be tricky. An interesting tool depicted in [14, 15] gives support for quadrotor control development, analyzing and comparing various control strategies on challenging trajectories. It may be used by beginning users with not much knowledge in ROS, Gazebo or in programming languages such as Python or C ++. The tool easies the understanding of quadrotor dynamics and related equations, as well as the development of control strategies. The present Chapter describes this user-friendly framework, the employed techniques are described in [14, 15, 16, 17].

Section 2 presents a description of the employed model. Section 3 explains the optimal trajectory planning method. Section 4 describes the controllers available in the tool. Section 5 presents the graphical user interface developed by the tool. Section 6 presents the graphical and numerical results. In the end of the Chapter, the Section 7 emphasizes the main aspects and critical issues related.

## 2. Quadrotor model

The equations are described in the rigid body model of the quadrotor, and its displacement is related to an inertial * frame*, fixed on the Earth surface [18]. Three main

*are considered for a quadrotor model: Inertial (*frames

The quadrotor owns 4 propellers in cross configuration. Each pair of propellers (1,3) and (2,4) rotates in opposite directions (Figure 1). Setting different rotor speeds to each pair (

Newton-Euler equations describe quadrotor dynamics and kinematics [9, 18]. The Rotation Matrix represents the rotation around the three axes in the sequence

The angle * roll angle*, the angle

*, and the*pitch angle

*.*yaw angle

The orientation vector is defined by ** I**by

**(**B

The derivative of the angles * Frame*of the Rigid Body are not the same.

*. The relationship with the angular rates*Frame

frame

Considering that the body is symmetrical with respect to the * Frame*can be described using Newton’s Second Law. Air drag forces are represented by a matrix in (5), and the values of

Besides the force each rotor produces a moment in the rigid body perpendicular to the plane of rotation of the propeller (* L*is the size of the quadrotor arm (Figure 1) and

Due to the forces produced by the rotors, moments are produced in the rigid body (

For simulating the measurement noise, the tool allows to add a random signal to position and orientation variables and their derivatives. A first order delay between the comands

## 3. Trajectory planning for a set of waypoints

Due to the low inertia of quadrotor it is necessary to calculate a smooth trajectory to minimize the risk of collapse. Euler–Lagrange equations are used to find the minimum

Time | Position | Velocity | Acceleration | Jerk |
---|---|---|---|---|

0 | 0 | 0 | 0 | |

0 | 0 | 0 |

With this boundary conditions the equations’ coefficients for the two-points optimal desired trajectory are calculated for each coordinate of position (

Desired angles for

This procedure yields a minimum

The tool calculates the complete optimized trajectory for any set of waypoints. Desired trajectory

## 4. Attitude and trajectory following control strategies

Some of the challenges to be overcome in quadrotor operation are: attitude stability for large angles, trajectory following, collision avoidance through aggressive maneuvers, monitoring, and others [2, 4].

The control architecture employed in the following control strategies uses two cascaded loops. The inner loop (attitude control) runs in a fast time-scale and is assumed exponentially stable. The outer loop (position control) runs in a slow time-scale, with a higher bandwidth [4]. All of them employ a feed-forward with proportional plus derivative structure (FF-PD). The tool is open to easily add more control strategies.

### 4.1 Small angle control

Small Angle control assumes an operation not too far from the hovering condition. A simple heuristic method with FF-PD control calculates the required accelerations for the desired trajectory (9).

It is assumed small deviations from zero in

After linear simplifications the equations for attitude control are defined in (11) and (12) [16].

A PD control law is used for attitude control and to calculate

### 4.2 Desired thrust vector control

At high speeds and

The axis

As we know

With the Rodrigues formula [20], the axis of rotation

A PD control law is used to calculated

Basin of attraction

### 4.3 Geometric tracking control

Geometric Tracking Control exhibits almost global exponential attractiveness to the zero equilibrium of tracking errors [17].

With

(27) calculates the desired attitude for the quadrotor given

Attitude tracking error and angular velocity error are calculated from (29, 30). The control vector

### 4.4 Particle swarm optimization for control gains tuning

A PSO algorithm is employed to tune the control gains. Some adjustments were performed to reduce the processing [14, 15], s. Each particle

Using a predefined set of waypoints chosen by the user, the code calculates the desired trajectory and tests each particle, calculating the RMS error. The PSO algorithm evolves in the direction of the best validated solution in each iteration, until achieving a minimum error tolerance. For faster convergence, every time a particle is evaluated, the evaluation is interrupted in the middle of the trajectory if the error increases above a predefined limit.

## 5. The graphical user interface

A user-friendly 3D animated GUI was developed in MatLab. It is able to evaluate and compare the performance of various quadrotor control strategies for any user-chosen trajectory. A few of quadrotor parameters are easily set in this interface [15].

A red vertical line indicates the upper side of the device, and a red circle the front rotor. Waypoints are represented by red markers, the desired trajectory is on dashed blue line, and the 3D interface axis limits are automatically adjusted from the waypoints (Figure 2).

Waypoints, quadrotor parameters and simulation comands are set in th buttons on the left side of the GUI.

The user may test its own control strategy just by creating the corresponding code like the following example:

The code

## 6. Results

The tool developed in MatLab and is available in GitHub [21]. The values employed for these results are in Table 2 [3].

For simulating the measurement noise, a random signal was added to position (

Two sets of waypoints of challenging trajectories were tested, each one was simulated in three different initial conditions: Case 1 starting with an orientation angle of

### 6.1 Elliptical helix trajectory

The first trajectory waypoints (Table 3) describe an elliptical helix in a forward trajectory along with the

Parameter | Value | Units |
---|---|---|

0.00 | −0.40 | 0.00 | 2.00 | 0.00 |

1.20 | 0.00 | 0.00 | 2.00 | 0.00 |

1.80 | 0.20 | 0.00 | 2.60 | 0.00 |

2.40 | 0.40 | 0.40 | 2.00 | 1.57 |

3.00 | 0.60 | 0.00 | 1.40 | 3.14 |

3.60 | 0.80 | −0.40 | 2.00 | 4.71 |

4.20 | 1.00 | 0.00 | 2.60 | 6.28 |

4.80 | 1.20 | 0.40 | 2.00 | 7.85 |

5.40 | 1.40 | 0.00 | 1.40 | 9.42 |

6.00 | 1.60 | −0.40 | 2.00 | 11.00 |

6.60 | 1.80 | 0.00 | 2.60 | 12.57 |

7.20 | 1.80 | 0.00 | 2.60 | 12.57 |

With this set it was calculated the desired optimal trajectory. Using this trajectory the PSO algorithm calculates the optimal gains for each of the three evaluated control algorithms.

With this gains the simulation was performed and validation errors in each of the four coordinates were registered. Tables 4–6 present the gains used in each case, and corresponding validation errors.

Control Strategy | ||||||||
---|---|---|---|---|---|---|---|---|

Small Angle | 17.96 | 7.11 | 0.231 | 0.16 | 1.373 | 7.763 | 6.301 | 1.0300 |

DTV | 18.91 | 10.77 | 44.920 | 23.43 | 2.112 | 4.907 | 5.300 | 0.6455 |

Geometric Tracking | 7.34 | 7.35 | 90.590 | 20.75 | 3.172 | 8.807 | 26.680 | 0.3973 |

Control Strategy | ||||||||
---|---|---|---|---|---|---|---|---|

Small Angle | 22.76 | 2.00 | 0.765 | 0.197 | 2.01 | 32.69 | 26.22 | 0.6344 |

DTV | 19.70 | 4.85 | 155.400 | 26.710 | 1.58 | 22.03 | 27.65 | 0.4949 |

Geometric Tracking | 8.24 | 5.62 | 220.200 | 27.670 | 3.17 | 35.94 | 51.57 | 0.4027 |

Control Strategy | ||||||||
---|---|---|---|---|---|---|---|---|

Small Angle | 8.180 | 2.349 | 2.45 | 0.4371 | 5.988 | 64.98 | 146.1 | 1.223 |

DTV | 7.268 | 3.434 | 236.40 | 31.4300 | 4.709 | 33.71 | 105.4 | 1.351 |

Geometric Tracking | — | — | — | — | — | — | — | — |

When calculating the gains using PSO it was observed that the processing time increases substantially with the initial orientation angle (

Small angle control is more sensible to variations in rotational positions as observed in the results. Optimal small angle control gains are bigger than rotational (Tables 4–6). The opposite occurs for DTV and geometric tracking.

Case 3 is the most challenging since the quadrotor starts almost upside down. Geometric tracking controller is more sensitive to noise error, PSO did not succeed in finding a set of gains in Case 3.

A visual validation is possible using the tool. Graphics in Figure 3 show the performance of small angle control in Case 3. The quadrotor drops and recoverers the vertical position, and continues along with the desired trajectory. Graphics on the right supports a visual validation of position and orientation variables.

In the graphics of propellers’ speeds of Figure 4, dashed red lines indicate the speed extremes. In extreme situations, the controller attains the propellers’ limits.

Figure 5 presents the performance of DTV Control on Case 3. It is observed a faster reaction and a more accurate trajectory following.

Near the third point the trajectory leads the quadrotor to its limits. Propellers’ speeds in Figure 6 show a more stable performance than the previous strategy.

Compared with small angle, this strategy seems to be more robust and accurate.

Figure 7 presents the performance of geometric tracking control in Case 2. This strategy reacts faster to challenging situations, since it recovers in a shorter time its hovering orientation. This fast reaction makes it more sensitive to measuring noise.

Propellers’ speeds in Figure 8 confirm the faster command reaction of the controller compared with Figures 4 and 6. This strategy is also sensitive to nonlinear behaviour. When it reaches the propeller limits, it is highly prone to instability.

The gains calculated for the more challenging Cases were later tested on Cases 1 and 2. Despite being less optimal they displayed a stable behaviour. The performance is presented in Table 7.

Control Strategy | |||||
---|---|---|---|---|---|

Small Angle | 0° | 1.373 | 7.763 | 6.301 | 1.0300 |

Small Angle | 0° | 3.111 | 21.960 | 28.390 | 0.2430 |

Small Angle | 88° | 2.010 | 32.690 | 26.220 | 0.6344 |

Small Angle | 88° | 3.242 | 51.760 | 49.580 | 0.3378 |

DTV | 0° | 2.112 | 4.907 | 5.300 | 0.6455 |

DTV | 0° | 4.625 | 15.650 | 13.310 | 0.2350 |

DTV | 88° | 1.584 | 22.030 | 27.650 | 0.4949 |

DTV | 88° | 4.886 | 41.580 | 44.530 | 0.3509 |

Geometric Tracking | 0° | 3.172 | 8.807 | 26.680 | 0.3973 |

Geometric Tracking | 0° | 3.338 | 5.917 | 28.660 | 0.2615 |

Graphical results of small angle control performance on Case 1, when using the optimal gains compared with the performance with the gains calculated for Case 3. The system holds the attitude performance. The same comparison was performed for DTV control. The decrease in performance is lower than with the small angle controller [15].

### 6.2 Lemniscate Shape Trajectory

A second set of waypoints presented in Table 8, depicts a lemniscate shape trajectory varying the position in

0.00 | 0.00 | 0.00 | 2.00 | −1.57 |

0.40 | 0.12 | −0.28 | 1.58 | −0.79 |

0.80 | 0.40 | −0.40 | 1.40 | 0.00 |

1.20 | 0.68 | −0.28 | 1.58 | 0.79 |

1.60 | 0.97 | 0.00 | 2.00 | 0.79 |

2.00 | 1.25 | 0.28 | 2.42 | 0.79 |

2.40 | 1.53 | 0.40 | 2.60 | 0.00 |

2.80 | 1.81 | 0.28 | 2.42 | −0.79 |

3.20 | 1.93 | 0.00 | 2.00 | −1.57 |

3.60 | 1.81 | −0.28 | 1.58 | −2.36 |

4.00 | 1.53 | −0.40 | 1.40 | −3.14 |

4.40 | 1.25 | −0.28 | 1.58 | −3.93 |

4.80 | 0.97 | 0.00 | 2.00 | −3.93 |

5.20 | 0.68 | 0.28 | 2.42 | −3.93 |

6.60 | 0.40 | 0.40 | 2.60 | −3.14 |

6.00 | 0.12 | 0.28 | 2.42 | −2.36 |

6.40 | 0.00 | 0.00 | 2.00 | −1.57 |

Figure 9 depicts the second trajectory.

As with the to previous trajectory, the small angle control compensates its limitations with lower gains to guarantee stability, gains are bigger for position control than for attitude.

DTV Control is again the best validated strategy. Geometric tracking presents a lower performance due to its sensitivity to noise and nonlinear operation conditions. The DTV control had presented the best validation even in the most challenging conditions.

Detailed results and analysis of this trajectory are available in [15].

Using the tool the graphical and numerical analysis was easier. Not much programming knowledge was necessary for using and configuration, just basic MatLab programming. Access to quadrotor parameters of the analyzed device, analyzing the influence of measurements noise, and comparing and simulating different control strategies is easier than with other visual platforms. Graphical and numerical simulation results are easily available with the interface buttons.

## 7. Summary

Quadrotor control is a fascinating research area, but the equations involved and programming skills requirements can be arduous for initiating students. It is a worth to develop motivational appliances for beginners. This was the motivation to present a beginner-friendly visual interface tool for the development of quadrotor control strategies. It is easy to understand, device characteristics are simple to configure, and control algorithms can be implemented and analyzed with not much effort. It is not necessary to have a deep knowledge in programming languages, and may be an introduction to this field of research.

This tool uses RMS and basin of attraction for numerical validation, and the GUI may help to evaluate stability, robustness, and accuracy. It integrates these criteria in a unique interface and helps to measure and visualize details and requirements that may not be so clear using other visual tools.

Baseline controllers are offered so students may compare performance, and is open to easily introduce other strategies for comparison. A trajectory planning based on minimum

The tool makes easier and faster to realize critical quadrotor requirements and limitations for challenging applications. These requirements may be related with the complexity of a defined trajectory, the weakness of a control strategy, or the improvements that may be carried out in the quadrotor (size, weight, propeller power, etc.) to accomplish the desired results.

Other controllers can be studied and compared using this tool, such as * Backstepping*and intelligent strategies. In the trajectory planning stage, applications with obstacles may be simulated. Support for multiple quadrotors, communication with the controller via Robotics Operating System (ROS) and implementation of obstacles are some of the future improvements planned for this tool.

## Abbreviations

ROS | |

PSO | |

DTV | |

RMS | |

GUI | |

UAV | |

PD | |

PID | |

LQR | |

LQG | |

FF-PD |

## Nomenclature

I | |

V | |

B | |

ωi | |

Mi | |

Fi | |

ϕ | |

θ | |

ψ | |

ϕ̇ | |

θ̇ | |

ψ̇ | |

sϕ,sθ,sψ | Sine of angles ϕ, θ and ψ |

cϕ,cθ,cψ | Cosine of angles ϕ, θ and ψ |

RXY | |

p | |

q | |

r | |

ṗ | |

q̇ | |

ṙ | |

η | |

r | |

r¨ | |

ν | |

T | |

L | |

J | |

x | |

y | |

z | |

er | |

er | |

Kp | |

Kd | |

m | |

g | |

KR | |

Kν | |

t | |

bi | |

I3x3 | |

v | |

β | |

eR | |

eν | |

Ψ | |

α | |

k | |

b | |

τ |