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Abstract

A simple, robust nonlinear controller for quadcopters to avoid collisions, based on the geometry approach and kinematics equation, is proposed. The controller allows the quadcopter to avoid single and multiple obstacles. Once an obstacle with a high possibility of collision is detected, a boundary sphere of the obstacle is generated to determine the collision zone. Afterward, the tracking error angles between the quadcopter’s motion direction and the tangential lines from the quadcopter’s current position to the boundary sphere are computed to steer the direction of the quadcopter for collision avoidance. A guidance law and a velocity control law are obtained from the Lyapunov stability based on the tracking error angles and relative distances between vehicle and obstacles. In addition, a method to drive the quadcopter to the target position after the completion of collision avoidance is introduced. The effectiveness of the proposed collision-avoidance algorithm is demonstrated through the result of a numerical simulation.

Keywords

Collision avoidance geometric approach nonlinear control collision cone quadcopter

Introduction

At present, quadcopters are used in a variety of applications. They are well suited for autonomously performing complicated civilian or military operations. However, because a low altitude is usually required in most of these missions, the issue of collisions between quadcopters and various obstacles is very serious and occurs frequently. Therefore, quadcopters must have the autonomous capability to avoid collision for safe and stable operation.

To solve this problem, various solutions have recently been reported in the literature. Many of them are inspired by path-planning algorithms. Chen et al.¹ proposed a path-planning approach for unmanned aerial vehicles (UAVs) with tangent-plus-Lyapunov vector field guidance to avoid obstacles. Richards and How² proposed a method to find optimal trajectories for multiple aircraft to avoid collision with each other. Budiyanto et al.³ introduced a potential field to avoid collisions with obstacles based on optimal path planning. Lin and Saripalli⁴ proposed a real-time path-planning method for UAVs to avoid collision with other aircraft or obstacles through motion uncertainty. In this method, collision prediction for UAVs is performed using reachable sets, and collision-avoidance path planning is generated through a sampling-based method. Several other studies related to the collision-avoidance ability for autonomous robot system can be found in the works of Dai et al.⁵ and Rubio et al.⁶ However, these methods are based on optimization techniques, which require intensive computation.

Another approach inspired by geometry was proposed in 1998 by Chakravarthy and Ghose,⁷ who introduced a collision cone for collision detection and collision avoidance between two irregularly shaped moving objects with unknown trajectories. The collision cone is effectively used to detect the possibility of collision between a robot and an obstacle (in both static and dynamic environments). Also, recently Wang et al.⁸ proposed a three-dimensional (3D) navigation strategy to reach a target position while avoiding collisions with obstacles based on the enlarged vision cone. Seo et al.⁹ proposed a method to avoid collision for UAVs in formation flight. Zhiyong et al.¹⁰ presented a method to equip UAVs with collision-avoidance capacity based on the minimum angle shift. Another collision-avoidance approach was proposed by Goss et al.,¹¹ in which collision avoidance was realized between two aircraft in a 3D environment using a combination of a collision cone and the geometric approach. Shin et al.^12,13 presented an obstacle-avoidance method for UAVs based on differential geometry. Generally, these collision-avoidance algorithms seem to be simple and facilitate fast response with low computational requirements. However, most of them are limited to guidance laws, and the UAV’s velocity is considered constant throughout the flight time. Therefore, collision may occur if a UAV is in flight with a high velocity and the heading angular rate is limited in the control loop.

A more practical method for the collision avoidance of UAVs based on vision sensors has been developed by Gosiewski and Cieśluk.¹⁴ In this method, the UAV avoids collision with obstacles based on the Lucas–Kanade method for estimating optical flow and gradient, which is implemented on a real-time embedded system. Fasano et al.¹⁵ proposed a fully autonomous multisensor (radar and camera) anti-collision system for UAVs. Choi et al.¹⁶ proposed a real-time mid-air reactive collision-avoidance system for UAVs based on a single vision sensor. Another interesting method is to mimic the human behavior of detecting collision using a monocular camera, as proposed by Al-Kaff et al.¹⁷ Aguilar et al.¹⁸ also proposed an obstacle-avoidance system for UAVs using the monocular onboard camera. These methods are the good solutions for collision-avoidance system. However, most of them are focused on the image processing techniques for detection and size estimation of obstacles.

The present study proposes a simple, robust nonlinear controller for quadcopters to avoid collisions based on the geometry approach and kinematics equation. Once an obstacle is detected, a boundary sphere of the obstacle is defined to determine the collision zone (or collision cone). The radius of boundary sphere is determined considering both the dimension of vehicle and obstacles in order to facilitate more rapid finding of avoidance angle. Afterward, the tracking error angles between the quadcopter’s motion direction and the tangential lines from the quadcopter’s current position to the boundary sphere are computed to steer the quadcopter in the left or right directions for collision avoidance. A guidance law and a velocity control law are obtained from Lyapunov stability based on the tracking error angles and relative distances between vehicle and obstacles. In this manner, both the heading angular rate and velocity of the quadcopter are analyzed and controlled simultaneously. The advantages of this algorithm are the ability to find the shortest path to avoid obstacles and reach the preplanned target position after completion of collision avoidance. In addition, the guidance and velocity control laws are considered simultaneously to ensure safe flights within speed and angular rate limitations.

Furthermore, this study proposes a simple and practical nonlinear controller and compares with other most advanced approaches^19
–21 with considerations of the implementation on a real-time embedded system. The proposed full control scheme consists of multi-loop architecture (i.e. outer loop and inner loop). The proposed algorithms for collision avoidance and approaching to the target position are implemented in the outer loop of position and heading control to generate the set point of velocity and heading angular rate. In the inner loop, the conventional proportional integral derivative (PID) control law for velocity and angular rate control is implemented. To verify the collision-avoidance performance and to demonstrate the effectiveness of the proposed algorithm, simulations with various scenarios were performed.

The remainder of this article is organized as follows. The “System modeling” section presents the dynamic model and kinematic equation of the quadcopter. The definitions and assumptions for collision avoidance based on the geometric approach are presented in the “Collision-avoidance algorithm” section, and the guidance and velocity control laws are analyzed using Lyapunov stability theory in this section. The results of numerical simulations are presented in the “Simulation results and discussion” section. Finally, concluding remarks are given in the “Conclusions” section.

System modeling

Brief mathematical model of quadcopter

In many previous studies, the dynamic model of the quadcopter was clearly explained and proved through many simulations and experimental results.^{22

–30} The dynamic model of the quadcopter is configured in this study by the inertial frame E and body frame B, as shown in Figure 1. Let the vector $[x, y, z]'$ represent the position of the center of the quadcopter; I_x, I_y, and I_z denote the moment of inertia of the quadcopter; m denotes the total mass; and J_r denotes the rotor inertia. Let the vector ${[ϕ, θ, ψ]}^{'}$ denote three Euler angles representing the roll, pitch, and yaw angles, respectively, where $ϕ \in (- π / 2, π / 2), θ \in (- π / 2, π / 2), and ψ \in (- π, π)$ . The mathematical model of the quadcopter is described by the following equations²³

{\begin{cases} \ddot{x} = (cos ϕ sin θ cos ψ + sin ϕ sin ψ) \frac{1}{m} U_{1} \\ \ddot{y} = (cos ϕ sin θ sin ψ - sin ϕ cos ψ) \frac{1}{m} U_{1} \\ \ddot{z} = - g + (cos ϕ cos θ) \frac{1}{m} U_{1} \\ \ddot{ϕ} = \dot{θ} \dot{ψ} (\frac{I_{y} - I_{z}}{I_{x}}) - \frac{J_{r}}{I_{x}} \dot{θ} Ω + \frac{l}{I_{x}} U_{2} \\ \ddot{θ} = \dot{ϕ} \dot{ψ} (\frac{I_{z} - I_{x}}{I_{y}}) + \frac{J_{r}}{I_{y}} \dot{ϕ} Ω + \frac{l}{I_{y}} U_{3} \\ \ddot{ψ} = \dot{ϕ} \dot{θ} (\frac{I_{x} - I_{y}}{I_{z}}) + \frac{1}{I_{z}} U_{4} \end{cases}

Figure 1.

Quadcopter configuration, inertial frame E, and body frame B.

The first three equations of expression (1) describe the linear velocity vector, and the next three equations describe the angular velocity vector.

The relation among force, moment, and velocity of the rotor is described as follows²³

{\begin{cases} U_{1} = b (Ω_{1}^{2} + Ω_{2}^{2} + Ω_{3}^{2} + Ω_{4}^{2}) \\ U_{2} = b (- Ω_{2}^{2} + Ω_{4}^{2}) \\ U_{3} = b (- Ω_{1}^{2} + Ω_{3}^{2}) \\ U_{4} = d (- Ω_{1}^{2} + Ω_{2}^{2} - Ω_{3}^{2} + Ω_{4}^{2}) \\ Ω = - Ω_{1} + Ω_{2} - Ω_{3} + Ω_{4} \end{cases}

Equation (2) represents the system’s input variables of movement; Ω_i denotes the speed of the propeller i; $F_{i} = b Ω_{i}$ denotes the thrust generated by rotor i; and b and d denote the thrust and drag coefficients, respectively.

Kinematic equation

Let R_mc denotes the radius of the quadcopter (including propellers), and $M (x_{m}, y_{m})$ represents its current position in Cartesian coordinates. ψ_m denotes the heading angle, and v_mc and ω denote the velocity and heading angular rate of the quadcopter, respectively. The kinematic equation of the quadcopter and the relation among its coordinates are shown in Figure 2. The equation is expressed in two dimensions as follows

[\begin{array}{l} {\dot{x}}_{m} \\ {\dot{y}}_{m} \\ {\dot{ψ}}_{m} \end{array}] = [\begin{matrix} cos ψ_{m} & 0 \\ \begin{array}{l} sin ψ_{m} \\ 0 \end{array} & \begin{array}{l} 0 \\ 1 \end{array} \end{matrix}] [\begin{array}{l} v_{m c} \\ ω \end{array}]

Figure 2.

Geometry of the quadcopter and an obstacle.

Collision-avoidance algorithm

Fundamental definitions

This section presents the definitions of important parameters of the quadcopter for laying the foundation of the proposed collision-avoidance algorithm. In order to be easy to compute and design a controller, the obstacle is considered as a circumsphere of real obstacle shape. These definitions are based on the geometry, as shown in Figure 2.

Definition 1: Boundary sphere of obstacle

The radius of boundary sphere of an obstacle is defined as follows

R = R_{m c} + R_{o b}

where R_ob denotes the radius of the obstacle and $R_{o b_min} \leq R_{o b} \leq R_{o b_max}$ , $R_{o b_min}, R_{o b_max}$ denote the minimum and maximum obstacle radius, respectively. The position of the obstacle is stationary at $O (x_{o}, y_{o})$ .

Definition 2: Obstacle detection

The quadcopter detects an obstacle when the relative distance Dt satisfies the following relation

‖ D t ‖ \leq D_{detect}

where $‖ D t ‖ = ‖ \vec{M O} ‖$ denotes the relative distance between the quadcopter and an obstacle. $D_{detect}$ denotes the detection radius; it is also the detection range of the sensor.

Definition 3: Collision

The collision of the quadcopter with an obstacle occurs when the relative distance Dt satisfies the following relation

‖ D t ‖ < R

Definition 4: Collision detection and avoidance direction

Let β⁺ and β⁻ $(0 < β^{+} < π / 2, - π / 2 < β^{-} < 0)$ denote the angles of tangent line vectors $\vec{M N}$ and $\vec{M P}$ with respect to the vector Dt, respectively.

Collision detection: Collision will occur in the future if the velocity vector v_mc of quadcopter is inside the collision cone NMP or the heading angle ψ_m satisfies the following relation

∠ P M x^{'} < ψ_{m} < ∠ N M x^{'} \Leftrightarrow ψ_{m o} - | β^{-} | < ψ_{m} < ψ_{m o} + β^{+}

Avoidance direction. The quadcopter has two options to change the direction to avoid collision based on the heading angle. Let α denote collision detection angle and $α = (v_{m c}, D t) = ψ_{m} - ψ_{m o}$ (as shown in Figure 2).

option 1: $α \geq 0 \Leftrightarrow ψ_{m o} \leq ψ_{m} < ψ_{m o} + β^{+}$ , or the velocity vector v_mc is in ∠NMO. The best direction of quadcopter for collision avoidance is to turn left. Therefore, α tracks β⁺.

option 2: $α < 0 \Leftrightarrow ψ_{m o} - | β^{-} | < ψ_{m} < ψ_{m o}$ , or the velocity vector v_mc is in ∠PMO. The best direction of quadcopter for collision avoidance is to turn right. Therefore, α tracks β⁻.

Definition 5: Completion of collision avoidance

The collision avoidance is successfully completed if the projection of vector $D t (D t \neq 0)$ onto vector $v_{m c} (v_{m c} \neq 0)$ is opposite to the direction of vector v_mc, as shown in Figure 3

\begin{array}{l} {(Proj)}_{v_{m c}} D t = σ v_{m c} = \frac{‖ D t ‖ ‖ v_{m c} ‖ cos α}{{‖ v_{m c} ‖}^{2}} v_{m c} \end{array}

where σ is scalar.

Figure 3.

Completion of collision avoidance.

According to equation (8), if $cos α < 0,$ $(α \in (π / 2, π) or α \in (- π, - π / 2))$ then the collision avoidance is completed. That is, at the time the quadcopter has just successfully avoided an obstacle, the velocity vector v_mc is perpendicular to vector Dt, and the collision detection angle α reaches π/2. From this statement, the collision-avoidance algorithm is activated if α satisfies the following relation

[\begin{array}{l} 0 < α < β^{+} < π / 2 \\ - π / 2 < β^{-} < α < 0 \end{array}

Definition 6: Collision algorithm activation

Let $D_{active} (0 < D_{active} \leq D_{detect})$ denote the relative distance between the quadcopter and an obstacle at which the collision-avoidance algorithm activated. The avoidance action will work when the relative distance Dt satisfies the following relation

‖ D t ‖ \leq D_{active}

Expressions (9) and (10) are the conditions to activate the collision-avoidance algorithm.

Controller design

In this section, the guidance and velocity control laws are designed to avoid obstacles. The objective is to design a controller such that the collision detection angle α tracks the angle β^± until $α \geq β^{+}$ or $α \leq β^{-}$ . Let e be the tracking error

e = β^{\pm} - α

From geometry constraints, β^± and α are computed as follows

β^{\pm} = \pm {sin}^{- 1} (\frac{R}{‖ D t ‖})

The relative distance ‖Dt‖ is computed as follows

‖ D t ‖ = \sqrt{{(x_{o} - x_{m})}^{2} + {(y_{o} - y_{m})}^{2}}

The derivative of ‖Dt‖ is computed as follows

‖ \dot{D} t ‖ = - \frac{{\dot{x}}_{m} (x_{o} - x_{m}) + {\dot{y}}_{m} (y_{o} - y_{m})}{‖ D t ‖}

From equation (12), the derivative of β⁺ and β⁻ iscomputed as follows

{\dot{β}}^{\pm} = \frac{\pm 1}{\sqrt{1 - {(\frac{R}{‖ D t ‖})}^{2}}} (\frac{- R ‖ \dot{D} t ‖}{{‖ D t ‖}^{2}}) \Leftrightarrow {\dot{β}}^{\pm} = \frac{\mp R}{‖ D t ‖ \sqrt{{‖ D t ‖}^{2} - R^{2}}} ‖ \dot{D} t ‖

Substituting equations (3) and (14) into equation (15), we obtain

{\dot{β}}^{\pm} = \frac{\pm R v_{m c}}{‖ D t ‖ \sqrt{{‖ D t ‖}^{2} - R^{2}}} (cos ψ_{m} \frac{(x_{o} - x_{m})}{‖ D t ‖} + sin ψ_{m} \frac{(y_{o} - y_{m})}{‖ D t ‖})

The collision detection angle α is obtained as follows

α = ψ_{m} - ψ_{m o}

where

ψ_{m o} = {tan}^{- 1} (\frac{y_{o} - y_{m}}{x_{o} - x_{m}})

From equations (17) and (18), the derivative of α is computed as follows

\begin{array}{l} \dot{α} = {\dot{ψ}}_{m} - {\dot{ψ}}_{m o} \\ = {\dot{ψ}}_{m} - [\frac{1}{{(\frac{y_{o} - y_{m}}{x_{o} - x_{m}})}^{2} + 1} (\frac{- {\dot{y}}_{m} (x_{o} - x_{m}) + {\dot{x}}_{m} (y_{o} - y_{m})}{{(x_{o} - x_{m})}^{2}})] \\ = {\dot{ψ}}_{m} - \frac{{\dot{x}}_{m} (y_{o} - y_{m}) - {\dot{y}}_{m} (x_{o} - x_{m})}{{(x_{o} - x_{m})}^{2} + {(y_{o} - y_{m})}^{2}} \end{array}

\Leftrightarrow \dot{α} = {\dot{ψ}}_{m} - \frac{{\dot{x}}_{m} (y_{o} - y_{m}) - {\dot{y}}_{m} (x_{o} - x_{m})}{{‖ D t ‖}^{2}}

Substituting equation (3) into equation (19), we obtain

\dot{α} = ω - \frac{1}{‖ D t ‖} v_{m c} (cos ψ_{m} \frac{(y_{o} - y_{m})}{‖ D t ‖} - sin ψ_{m} \frac{(x_{o} - x_{m})}{‖ D t ‖})

$\dot{e}$ is computed from equation (11) as follows

\dot{e} = {\dot{β}}^{\pm} - \dot{α}

Substituting equations (11), (16), (17), and (20) into equation (21), we obtain

\dot{e} = [\frac{sin e}{‖ D t ‖ cos β^{\pm}} - 1] [\begin{array}{l} v_{m c} \\ ω \end{array}]

(Equation (22) is proved in Appendix 1.)

Consider that the candidate Lyapunov function and its derivative are given as follows

V_{0} = \frac{1}{2} e^{2}

\begin{array}{l} {\dot{V}}_{0} = e \dot{e} = e (\frac{sin e}{‖ D t ‖ cos β^{\pm}} v_{m c} - ω) \end{array}

A method to achieve a negative value of ${\dot{V}}_{0}$ is to choose $(v_{m c}, ω)$ as follows

{\begin{cases} v_{m c} = k_{1} ‖ D t ‖ cos β^{\pm} \\ ω = k_{1} sin e + k_{2} e \end{cases}

where k₁ and k₂ are positive values.

Controller gain turning

We assume that the heading angle of the quadcopter satisfies option 1 in definition 4. Let $v_{m c_max} and ω_{max}$ $(v_{m c_max} > 0, ω_{max} > 0)$ denote the limited velocity and heading angular rate of the quadcopter, respectively

{\begin{cases} 0 \leq v_{m c} \leq v_{m c_max} \\ 0 \leq ω \leq ω_{max} \end{cases}

Let $β_{min} (β_{min} > 0)$ denote the minimum value of angle β⁺.

According to definition 1

{sin}^{- 1} (\frac{R_{m c} + R_{o b_min}}{D_{active}}) \leq β_{active} \leq {sin}^{- 1} (\frac{R_{m c} + R_{o b_max}}{D_{active}})

where $β_{active} = {sin}^{- 1} (R / D_{active})$ is the value of angle β⁺ at the time the collision-avoidance algorithm is activated.

From equation (27), β_min is obtained as follows

β_{min} = Min (β_{active}) = {sin}^{- 1} (\frac{R_{m c} + R_{o b_min}}{D_{active}})

Let $e_{max} (e_{max} > 0)$ denote the maximum value of tracking error e. At the time that the collision-avoidance algorithm is activated, e = e_max.

From equation (27), e_max is obtained as follows

e_{max} = Max (β_{active}) = {sin}^{- 1} (\frac{R_{m c} + R_{o b_max}}{D_{active}})

According to equations (25) and (26)

0 < k_{1} \leq \frac{v_{m c_max}}{D_{active} cos (β_{min})}

0 < k_{2} \leq \frac{ω_{max} - k_{1} sin (e_{max})}{e_{max}}

Expression (31) is satisfied if

k_{1} < \frac{ω_{max}}{sin (e_{max})}

The controller gains k₁ and k₂ are obtained from equations (30) to (32).

In option 2 (definition 4), the controller gains k₁ and k₂ can also be determined by the same method as for option 1. Expressions (30) to (32) are also effective for option 2.

Collision avoidance with multiple obstacles

In this section, the collision-avoidance algorithm is extended for multiple obstacles. To avoid collisions with multiple obstacles, a new avoidance direction is recalculated based on the number of obstacles and geometry constraints. Let us assume that the quadcopter can detect all obstacles within the range of sensor. Typical cases of multiple obstacles are shown in Figure 4(a) and (b). Two obstacles are considered to define the collision zone; the relative position between the quadcopter and multiple obstacles is considered as follows.

Figure 4.

(a) Relation between quadcopter and two obstacles O₁ and O₂ when collision will occur. (b) Relation between quadcopter and two obstacles O₁ and O₂ when collision will not occur.

In Figure 4(a), the collision may occur between the quadcopter and two obstacles (obstacles O₁ and O₂) because the velocity of the quadcopter v_mc is in the collision zone of obstacle O₁ $(∠ N_{1} {MP}_{1})$ (as in definition 4). For multiple obstacles, two consecutive obstacles must satisfy the following relation

∠ P_{1} M x < ∠ N_{2} M x < ∠ N_{1} M x

The collision zone is defined as follows in Figure 4(a)

∠ P_{2} M x < ψ_{m} < ∠ N_{1} M x

Collision may occur if the relative position of the quadcopter and multiple obstacles satisfies both expressions (33) and (34).

Conversely, collision will not occur between the quadcopter and two obstacles O₁ and O₂ in Figure 4(b) because the relative position of the quadcopter and two obstacles does not satisfy expression (33). In this case, the velocity v_mc is not in the collision zone of obstacle O₁ or O₂, and collision-avoidance action is not required. If the velocity v_mc is in the collision zone of obstacle O₁ or O₂, this case becomes the single-obstacle problem.

In Figure 4(a) and (b), two obstacles are considered. However, additional obstacles can be included with the same method. The proposed algorithm is extended to n obstacles. Again, we assume that the quadcopter is able to detect n obstacles within the range of the sensor; the relative position between two consecutive obstacles and quadcopter satisfies expression (33); and vector v_mc is in the zone of collision $∠ N_{1} M P_{n}$ , as shown in Figure 5.

Figure 5.

Relation between quadcopter and multiple obstacles: extended method.

The collision detection angles of n obstacles are calculated as follows

{\begin{cases} α_{1} = ψ_{m} - ψ_{m o (1)} \\ ⋮ \\ α_{i} = ψ_{m} - ψ_{m o (i)} \\ α_{i + 1} = ψ_{m} - ψ_{m o (i + 1)} \\ ⋮ \\ α_{n} = ψ_{m} - ψ_{m o (n)} \end{cases}

where $i = 1, n - 1$ .

From Figure 5, it is easy to recognize that the quadcopter has two directions to avoid obstacles: The first direction is to turn left of the obstacle O₁ and the second direction is to turn right of the obstacle O_n. The collision-avoidance strategy is divided into three cases:

case 1: $α_{1}, ..., α_{n} \geq 0$

Let $α_{turnleft} = Min (α_{1}, ..., α_{n})$ $\Rightarrow α_{1} = α_{turnleft} \geq 0$ ; therefore, the current velocity vector v_mc is in $∠ N_{1} {MO}_{1}$ of obstacle O₁. This case becomes the single-obstacle problem (option 1 of definition 4 for obstacle O₁) with $α = α_{1}$ tracks $β_{1}^{+}$ ; the quadcopter turns left for collision avoidance. Therefore, the proposed controller at equation (25) based on the Lyapunov stability is also effective in this case.

case 2: $α_{1}, ..., α_{n} \leq 0$

Let $α_{turnright} = Max (α_{1}, ..., α_{n})$ $\Rightarrow α_{n} = α_{turnright} \leq 0$ ; therefore, the current velocity vector v_mc is in $∠ P_{n} {MO}_{n}$ of obstacle O_n. This case becomes the single-obstacle problem (option 2 of definition 4 for obstacle O_n) with α = α_n tracks $β_{n}^{-}$ ; the quadcopter turns right for collision avoidance. Therefore, the proposed controller at equation (25) based on the Lyapunov stability is also effective.

case 3

Let

{\begin{cases} α_{1}, ..., α_{i} < 0 \\ α_{i + 1}, ..., α_{n} > 0 \end{cases}

Therefore, the current velocity v_mc is in $∠ O_{1} {MO}_{n}$ .

From expression (36), the quadcopter can choose from two directions to avoid the series of obstacles: left or right.

The avoidance direction is determined as follows

α_{turnleft} = Min (α_{1}, ..., α_{i}) = α_{1}

α_{turnright} = Max (α_{i + 1}, ..., α_{n}) = α_{n}

The tracking error angles of two avoidance directions are obtained as follows

e_{turnleft} = β_{1}^{+} - α_{turnleft}

e_{turnright} = β_{n}^{-} - α_{turnright}

From equations (39) and (40),

If $| e_{turnleft} | > | e_{turnright} |$ , the quadcopter turns right with

{\begin{cases} α = α_{turnright} = α_{n} \\ e = e_{turnright} = β_{n}^{-} - α \end{cases}

and α tracks $β_{n}^{-}$ .

If $| e_{turnleft} | \leq | e_{turnright} |$ , the quadcopter turns left with

{\begin{cases} α = α_{turnleft} = α_{1} \\ e = e_{turnleft} = β_{1}^{+} - α \end{cases}

and α tracks $β_{1}^{+}$ .

The proposed controller at equation (25) based on the Lyapunov stability is also effective in this case (The proposed controller at equation (25) based on the Lyapunov stability is also effective in case 3. It is proved in Appendix 2.).

Target position approach

This section introduces the algorithm to steer the quadcopter to the target position after the completion of collision avoidance.

In option 1, let $S (x_{S}, y_{S})$ and $T (x_{T}, y_{T})$ denote the start and target positions of the quadcopter, respectively. We assume that M₁ is the position at which collision avoidance is completed (definition 5). However, the quadcopter cannot directly move to point T from M₁ along a straight line M₁T (Figure 6). Instead of M₁T, the curve $M_{1} {M^{'}}_{1} T$ is a good path candidate. This path is divided into two areas.

Figure 6.

The path of quadcopter to reach the target position after obstacle-avoidance completion.

Area 1

The current position of the quadcopter $M \in ∠ M_{1} T I \Leftrightarrow ψ_{m t} > ψ_{I}$ . In this area, the normal heading controller (P controller) is applied to control the heading angle to steer the quadcopter from position M₁ to a position outside the angle $∠ M_{1} T I$ with a constant velocity of $v_{m c_0}$ m/s. Therefore, ψ_m tracks ψ_mt with error e_t. The heading angular rate set point is generated from the outer control loop as follows (Figure 7)

ω = e_{t} K p_{yaw} = K p_{yaw} (ψ_{m t} - ψ_{m})

where $K p_{yaw} > 0$

ψ_{m t} = {tan}^{- 1} (\frac{y_{T} - y_{m}}{x_{T} - x_{m}})

ψ_{I} = {tan}^{- 1} (\frac{y_{T} - y_{o}}{x_{T} - x_{o}}) - {sin}^{- 1} (\frac{R}{\sqrt{{(x_{T} - x_{o})}^{2} + {(y_{T} - y_{o})}^{2}}})

Figure 7.

General block diagram of proposed algorithm.

Area 2

Once the position of the quadcopter M is outside the angle $∠ M_{1} T I (ψ_{m t} \leq ψ_{I})$ , the quadcopter can directly move to the target position along a straight line. The normal position and heading controller of the quadcopter (P controller) are used in this case, as shown in Figure 7. To make the quadcopter move toward the target point with a constant velocity $v_{m c_0}$ and stop at this point, the errors $e_{x t} and e_{y t}$ of the position controller are calculated as follows

e_{x t} = \frac{x_{T} - x_{m}}{d} (\frac{v_{m c_0}}{K p_{pos}})

e_{y t} = \frac{y_{T} - y_{m}}{d} (\frac{v_{m c_0}}{K p_{pos}})

where $d = \sqrt{{(x_{T} - x_{m})}^{2} + {(y_{T} - y_{m})}^{2}}$ , $K p_{pos} > 0$ .

When the quadcopter closely approaches target point $(d \leq 1 m)$ , the velocity must be decreased to stop at the destination. Therefore, the error is calculated as follows

e_{x t} = x_{T} - x_{m}

e_{y t} = y_{T} - y_{m}

The method to reach the target point in option 2 (definition 4) is identical to that in option 1.

Simulation results and discussion

Simulation conditions

Numerical simulations were conducted to test the performance of the proposed method. Several conditions and assumptions were considered for the simulations. First, the case of a stationary obstacle is considered. The position and velocity of the quadcopter are determined from the embedded global positioning system (GPS)/inertial navigation system (INS) systems. The obstacles are detected by a light detection and ranging (LiDAR) or vision sensor system.

Second, the simulation assumed that the quadcopter is in flight at a constant altitude h = 10 m, with the start position S(1, 1) and a constant velocity of 2 m/s. An obstacle exists at position $O (8, 15)$ (for the single-obstacle problem). The radius of the quadcopter is $R_{m c} = 0.35 m$ (including propellers). The sensor of the quadcopter can detect an object within 8 m. The other parameters are listed in Table 1.

Table 1.

Numerical parameters and initial values for simulation.

Parameters	Values	Unit	Note
R_mc	0.35	m	Including propellers
m	1.12	kg
I_x	0.0119	kg m²
I_y	0.0119	kg m²
I_z	0.0223	kg m²
g	9.81	N/s²
R_ob	2.0	m
D _detect	8.0	m
D _active	6.0	m
h	10.0	m
$v_{m c_0}$	2.0	m/s	Constant velocity
ω ₀	0.0	rad/s	Initial angular rate
$(x_{S}, y_{S})$	(1,1)	(m, m)	Start position
$(x_{o}, y_{o})$	(8,15)	(m, m)	Obstacle position
$v_{m c_max}$	5.0	m/s
ω _max	100	deg/s
$R_{o b_max}$	3.0	m
$R_{o b_min}$	0.1	m
k ₁	0.3
k ₂	0.4

Simulation results

In this section, the results of the simulation are presented to demonstrate the effectiveness of the proposed collision-avoidance algorithm.

Single obstacle

The first simulation was conducted for option 1 in definition 4. This simulation was performed with the target position of the quadcopter set to T(11, 23). Hence, the velocity v_mc is in the left-half collision cone NMO (Figure 2). The simulation results are shown in Figures 8 to 15. All the figures are divided into three areas by the time of collision-avoidance algorithm activation and the time at which α ≥ β⁺ (as shown in Figures 10 and 12).

Figure 8.

Path performance in three dimensions with target position $T (11, 23)$ .

Figure 9.

Path performance in two dimensions with target position $T (11, 23)$ .

Figure 10.

Relative distance between the quadcopter and obstacle with target position $T (11, 23)$ .

Figure 11.

Heading angular rate performance of the quadcopter with target position $T (11, 23)$ .

Figure 12.

Angle α tracking β⁺ with target position $T (11, 23)$ .

Figure 13.

Tracking error e with target position $T (11, 23)$ .

Figure 14.

Velocity performance of the quadcopter with target position $T (11, 23)$ .

Figure 15.

Attitude angles and heading angle of the quadcopter with target position $T (11, 23)$ .

In area 1, the collision-avoidance algorithm is not activated because the relative distance is greater than active distance $(‖ D t ‖ > D_{active})$ . Hence, the quadcopter moves toward the obstacle with a constant velocity of 2 m/s (Figure 14), the heading angular rate is zero (Figure 11), and the heading angle does not change, as shown in Figure 15. In this area, the distance ‖Dt‖ linearly decreases, and the quadcopter approached the obstacle R (Figure 10).

At time t = 3.9 s, the obstacle is detected $(‖ D t ‖ \leq D_{detect})$ , and at t = 4.9 s, the relative distance ‖Dt‖ is less than the active distance $(‖ D t ‖ = 5.9 m < D_{active})$ (Figure 10). Therefore, the collision-avoidance algorithm is activated to control the behavior of the quadcopter for avoiding collision with the obstacle, as shown in area 2 of Figure 10. In this area, the velocity v_mc and heading angular rate ω are controlled simultaneously to ensure that collision does not occur.

As shown in Figure 14 (in area 2), the velocity gradually decreases as the quadcopter comes closer to the obstacle. In the meantime, the heading angular rate ω starts to increase and then gradually decreases when the quadcopter approaches the obstacle (Figure 11). Hence, α tracks β⁺ (Figure 12), and tracking error converges to zero (Figure 13). The heading angle ψ_m gradually increases to turn the quadcopter left for collision avoidance, as shown in Figures 8, 9, and 15. During this process, the relative distance is always greater than the boundary radius $(‖ D t ‖ \geq R)$ (Figure 10).

Area 3 begins at time $t = 13.6 s, α = 82.6 °$ . Here, α becomes greater than β⁺ (Figure 12). Thus, the velocity vector of the quadcopter is outside the collision zone. Therefore, the algorithm to avoid obstacles stops working. However, α and β⁺ continuously approach 90°. From this point onward, collision will not occur, and the velocity of the quadcopter starts to increase to a constant velocity of 2 m/s (Figure 14). The angular rate ω becomes zero (Figure 11), and the heading angle ψ_m becomes a constant value (Figure 15).

At time t = 14.07 s, α ≥ 90° ≥ β⁺ (Figure 12), the collision avoidance is successful, and the target position approach algorithm begins to work. Therefore, the magnitude of the heading angular rate ω starts to increase (toward the negative direction) to drive the quadcopter toward the target position with a constant velocity of 2 m/s (Figures 11 and 14). Once the quadcopter is close to the target position, the velocity decreases so that the quadcopter stops at the destination. In addition, the movement trajectory of the quadcopter for collision avoidance is shown in 3D and two-dimensional (2D) in Figures 8 and 9, respectively.

The second simulation was performed for option 2 in definition 4, with the target position of the quadcopter set to T(12, 22). Hence, the velocity v_mc is in the right-half collision cone $PMO$ (Figure 2). The results of the simulation are shown in Figures 16 to 23. Similar to option 1, the quadcopter moves toward the obstacle with a constant velocity of 2 m/s, the angular rate is zero, and the heading angle does not change in area 1. The collision-avoidance algorithm is not activated because the relative distance ‖Dt‖ is greater than D_active (Figure 18).

Figure 16.

Path performance in three dimensions with target position T(12, 22).

Figure 17.

Path performance in two dimensions with target position T(12, 22).

Figure 18.

Relative distance between the quadcopter and obstacle with target position T(12, 22).

Figure 19.

Heading angular rate performance of the quadcopter with target position T(12, 22).

Figure 20.

Angle α tracking β⁻ with target position T(12, 22).

Figure 21.

Tracking error e with target position T(12, 22).

Figure 22.

Velocity performance of the quadcopter with target position T(12, 22).

Figure 23.

Attitude angles and heading angle of the quadcopter with target position T(12, 22).

At time t = 4.9 s, the relative distance $‖ D t ‖ = 6 m \leq D_{active}$ , and the collision-avoidance algorithm is activated. Therefore, the velocity begins to decrease (Figure 22). At the same time, the magnitude of the heading angular rate ω starts increasing (toward the negative direction) and then gradually decreases when the quadcopter comes closer to the obstacle (Figure 19). α tracks β⁻ (Figure 20). Therefore, the heading angle ψ_m gradually decreases to turn the quadcopter right for collision avoidance (Figures 16, 17, and 23). During this process, the relative distance is always greater than the boundary radius of obstacle $(‖ D t ‖ \geq R)$ (Figure 18).

At time t = 13.67 s, α becomes less than β⁻ (Figure 20). Thus, the velocity vector of the quadcopter is outside the collision zone. Therefore, the algorithm to avoid obstacles stops working. However, α and β⁻ are continuously approaching −90°. From this point onward, there is no possibility of collision, and the velocity of the quadcopter starts to increase to a constant velocity of 2 m/s (Figure 22). The angular rate ω is zero (Figure 19), and heading angle ψ_m becomes a constant value (Figure 23).

At time $t = 14.12 s, α \leq - 90 ° \leq β^{-}$ (Figure 20), the collision avoidance is successful, and the target position approach algorithm begins to work. Therefore, the heading angular rate ω starts to increase to drive the quadcopter toward the target position with a constant velocity of 2 m/s (Figures 19 and 22). Once the quadcopter is close to the target point, the velocity decreases so that the quadcopter stops at the destination. In addition, the movement trajectory of the quadcopter for collision avoidance is shown in 3D and 2D in Figures 16 and 17, respectively.

Multiple obstacles

The last simulation was performed with multiple obstacles. Three obstacles were considered in this case, and their positions are O₁(8, 15), O₂(12, 14), and O₃(13, 10). The other conditions are listed in Table 1. Simulations were performed with the target positions of the quadcopter set to T(18, 24) and T(20, 22).

For T(18, 24)

The quadcopter moves toward the multiple obstacles $(O_{1}, O_{2}, and O_{3})$ with a constant velocity of 2 m/s, initial heading angular rate of zero, and initial heading angle $ψ_{m 0} = 53.5$ °. Therefore, v_mc is in the collision cone $∠ O_{1} M O_{2}$ (Figure 25). The results of the simulation are shown in Figures 24 to 31.

Figure 24.

Path performance in three dimensions with multiple obstacles and $T (18, 24)$ .

Figure 25.

Path performance in two dimensions with multiple obstacles and $T (18, 24)$ .

As shown in Figure 26, once $‖ D t ‖ \leq D_{active}$ , the collision-avoidance algorithm is activated. Hence, α = α₁ tracks $β_{1}^{+}$ (Figure 28), and the tracking error e = e₁ converges to zero (Figure 29). The heading angle ψ_m increases to turn the quadcopter left to avoid obstacles (Figures 24, 25, and 27). During this collision-avoidance process, the relative distances are always greater than the radius of obstacles $(‖ D t_{1} ‖, ‖ D t_{2} ‖, ‖ D t_{3} ‖ \geq R)$ . After the completion of collision avoidance, the quadcopter starts to move to the destination. The heading angle of the quadcopter decreases as it moves toward the destination with a constant velocity of 2 m/s (Figures 27 and 30). Figures 24 and 25 show the performance of the collision-avoidance algorithm in 3D and 2D, respectively.

Figure 26.

Relative distance between the quadcopter and multiple obstacles with T(18, 24).

Figure 27.

Attitude and heading angle of the quadcopter with multiple obstacles and T(18, 24).

Figure 28.

Angle α = α₁ tracking $β_{1}^{+}$ with multiple obstacles and T(18, 24).

Figure 29.

Tracking error e with multiple obstacles and T(18, 24).

Figure 30.

Velocity performance of the quadcopter with T(18, 24).

Figure 31.

Heading angular rate performance of the quadcopter with T(18, 24).

For T(20, 22)

The next simulation was conducted with the target position of the quadcopter set to T(20, 22); the other conditions are listed in Table 1. The results of the simulation are shown in Figures 32 to 39. In this case, the velocity v_mc is in the collision cone $∠ O_{2} M O_{3}$ (Figure 33). Hence, α = α₃ tracks $β_{3}^{-}$ (Figure 36), and the tracking error e = e₃ converges to zero (Figure 37). The heading angle ψ_m decreases to turn the quadcopter right to avoid obstacles (Figures 32, 33, and 35). Subsequently, the quadcopter starts to move to the destination. During the collision-avoidance process, the relative distances are always greater than the radius of boundary of obstacles (Figure 34). The performance of the collision-avoidance algorithm in 3D and 2D is shown in Figures 32 and 33, respectively.

Figure 32.

Path performance in three dimensions with multiple obstacles and T(20, 22).

Figure 33.

Path performance in two dimensions with multiple obstacles and T(20, 22).

Figure 34.

Relative distance between the quadcopter and multiple obstacles with T(20, 22).

Figure 35.

Attitude angles and heading angle of the quadcopter with multiple obstacles and T(20, 22).

Figure 36.

Angle α = α₃ tracking $β_{3}^{-}$ with multiple obstacles and T(20, 22).

Figure 37.

Tracking error e with multiple obstacles and T(20, 22).

Figure 38.

Velocity performance of the quadcopter with T(20, 22).

Figure 39.

Heading angular rate performance of the quadcopter with T(20, 22).

The simulation results demonstrate the effectiveness of proposed method. The advantages of this algorithm are simplicity, low computational requirements, and ability to find the shortest path to avoid obstacles and reach the target position after completion of collision avoidance. Both heading angular rate and velocity of quadcopter are considered simultaneously to ensure that collisions do not occur when the vehicle travels at high speed. This method overcomes the disadvantages of previous studies that only concern guidance law.^9,13,16 This algorithm is also effective to multiple obstacles. Finally, the proposed simple method can be used to avoid collisions for quadcopter or multicopter in a static environment.

Conclusions

In this article, a new approach to control quadcopters to avoid obstacles was introduced. This method can be applied to avoid single or multiple obstacles. To design the guidance and velocity control laws, a tracking error was defined to determine the avoidance direction. The controllers are obtained from Lyapunov stability. Furthermore, the controller gain can be found easily with the presented turning method. In addition, a target position approach method to steer the quadcopter to reach the destination was presented.

Numerical simulations for different scenarios were performed, and the results demonstrated the good performance of the proposed method. However, the study did not consider collision avoidance for moving objects and external disturbances. Therefore, the extension of this method to dynamic environments and external disturbances will be studied in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Korea Institute for Advancement of Technology (KIAT) grant funded by the Korean government (MOTIE: Ministry of Trade, Industry, and Energy) (no. N0002431).

Appendix 1

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Simple nonlinear control of quadcopter for collision avoidance based on geometric approach in static environment

Abstract

Keywords

Introduction

System modeling

Brief mathematical model of quadcopter

Kinematic equation

Collision-avoidance algorithm

Fundamental definitions

Definition 1: Boundary sphere of obstacle

Definition 2: Obstacle detection

Definition 3: Collision

Definition 4: Collision detection and avoidance direction

Definition 5: Completion of collision avoidance

Definition 6: Collision algorithm activation

Controller design

Controller gain turning

Collision avoidance with multiple obstacles

Target position approach

Area 1

Area 2

Simulation results and discussion

Simulation conditions

Simulation results

Single obstacle

Multiple obstacles

For T(18, 24)

For T(20, 22)

Conclusions

Footnotes

Declaration of conflicting interests

Funding

Appendix 1

Appendix 2

References