Event-driven PID control of autonomous quadrotor helicopters

Abstract

This paper proposes an event-driven PID control mechanism for autonomous quadrotor helicopters that will reduce the usage of communication channels. Compared to traditional PID controller, the event-driven PID controller can maintain the satisfactory stabilization effect with the ability of reducing the number of transmissions significantly. The simplified dynamics model of quadrotor helicopter is also established. Finally, the improvements realized by the developed method are verified in the computer simulations.

Introduction

In recent years, with the development of microelectronics, micro mechanical technology and computer technology, control of small aircrafts can be realized. Due to the advantages with low cost, convenient control manipulation, flexibility, low noise, small volume and light weight, and quadrotor helicopters have become increasingly popular and have drawn considerable attention in recent years. The small quadrotor helicopter is an under actuated system with six degrees of freedom and four inputs. In addition to vertical landing and taking off, landing and hovering, and other flight characteristics, compared with the conventional helicopters, it has more concise structure and simpler control mode achieving a variety of flight control by changing the speeds of four rotors. However, the quadrotor helicopters are also unstable systems with multivariable, nonlinear, strong coupling, and other characteristics. Many results, which report progresses regarding the modeling, analysis, stability, and control of quadrotor helicopter systems, have appeared in the literature (see, e.g., literature^1–8). In Chen et al., ² the authors designed an optimal reconfiguration control method for a quadcopter, which encounters actuator faults and applied adaptive control and combined multiple models with consideration of optimal performance index and the minimum cost to reach a good tracking performance. Mian et al. ⁵ constructed a nonlinear model of a six-DOF quadcopter based on the Newton–Euler formalism and presented a control strategy with coupled feedback linearization. By applying the quantum information mechanism, Chen et al. ⁶ developed an adaptive compensation control method for a four-rotor helicopter, which can deal with the attitude control problems in the case of external disturbance. Hoffmann et al. ⁷ described the vehicle dynamics of the STARMAC quadrotor testbed and the control system design with flight test results for each control loop culminating in multiple waypoint trajectory tracking demonstrations with sub-meter accuracy. Sumantri et al. ⁸ proposed a sliding mode controller in order to track trajectory robustly under wind disturbance and analyzed electricity consumption with verification in experiment. While the control goals in the above contributions were achieved via continuous communication, it is more realistic to consider intermittent control signals at discrete instants. ⁹

Recently, in contrast to such periodically sampled control schemes, which may yield a conservative solution when the communication resources, event-driven control has become hot research topics and drawn considerable attention,^10–13 because it may allow to significantly reduce the usage of the communication channel. For instance, the distributed event-driven control scheme has been presented for cooperative control of multi-agent systems over networks with limited resources.^14–16 By comparison with time driven control, event-based control has the often cited advantage on communication reduction. ¹⁰ In spite of these advances, it has not received much attention until now with few results appearing on event-driven control of quadrotor helicopters.

Motivated by the use of event-driven control methods for multi-agent systems, the main result of this paper is a methodology for constructing event-driven PID control strategy guaranteeing the considered quadrotor helicopter stop moving and remains stable. The simplified mathematical model of quadrotor helicopter is firstly presented. Then, we introduce the PID controller for the dynamics to follow a designated trajectory and develop an event-driven control strategy for the attitude stabilization of the quadrotor helicopter.

The rest of this paper is organized as follows: In “Model of quadrotor helicopter” section, we give the mathematical framework for quadrotor helicopters. “PID controller” section presents the PID controller and event-driven controller for achieving the attitude stabilization. In “Simulation” section, numerical simulations are provided to show the effectiveness of the proposed method. Finally, some conclusions are drawn in the last section.

Model of quadrotor helicopter

A quadrotor helicopter is usually equipped with four equal-spaced rotors, which are arranged at the corners. A free body diagram of quadrotor helicopter is depicted in Figure 1. Before delving into the physics of quadcopter motion, let us formalize the kinematics in the body and inertial frames. In the inertial frame, denote $\overline{x}={(x,y,z)}^{\mathrm{\top }}$ and $\dot{\overline{x}}={(\dot{x},\dot{y},\dot{z})}^{\mathrm{\top }}$ as the position and velocity, respectively. In the body frame, $\phi$ is denoted as the roll angle and then $\dot{\phi }$ represents the angular velocity; θ is denoted as the pitch angle and then $\dot{\theta }$ represents the corresponding angular velocity; ψ is denoted as the yaw angle and then $\dot{\psi }$ represents the corresponding angular velocity. Denote the angular velocity vector in the body frame ¹⁷ as

$$\omega =\left[\begin{array}{ccc}1& 0& -s_{\theta }\\ 0& c_{\phi }& c_{\theta }{s}_{\phi }\\ 0& -s_{\phi }& c_{\theta }{c}_{\phi }\end{array}\right]\stackrel{.}{\overline{\theta }}$$

(1)

where $\overline{\theta }={(\phi ,\theta ,\psi )}^{\mathrm{\top }}$ , $\stackrel{.}{\overline{\theta }}={(\dot{\phi },\dot{\theta },\dot{\psi })}^{\mathrm{\top }}$ . ω is applied to convert the angular velocities.

Figure 1.

Flight simulation of a simulated quadrotor helicopter.

Forces and torques

Under reasonable assumptions on Brushless motors of quadrotor helicopter, the power of each motor is calculated as

$$P\hspace{0.17em}\approx \hspace{0.17em}\frac{K_v}{K_t}\tau \omega$$

where τ represent the motor torque, K_t denotes the torque proportionality constant, K_v is a proportionality constant, ¹⁸ and ω is the angular velocity of the motor.

It follows from Hoffmann et al. ¹⁹ that in the general case, $\tau =\overrightarrow{r}\times \overrightarrow{F}$ and the torque τ is related to the thrust T and a positive ratio $K_{\tau }$ , which is determined by the blade configuration, i.e. $\tau =K_{\tau }T$ . Meanwhile, the power P equals to the product of the air velocity v_h (as it is used to keep the quadcopter aloft) and the thrust T, i.e.

$$P=T{v}_h$$

where $v_h=\sqrt{\frac{T}{2\rho A}}$ , A denotes the area swept out by the rotor, and ρ represents the density of the surrounding air.  Therefore, we have

$$P=\frac{K_v}{K_t}\tau \omega =\frac{K_v{K}_{\tau }}{K_t}T\omega =\frac{T^{\frac{3}{2}}}{\sqrt{2\rho A}}$$

With mathematical operation to get the thrust magnitude T, it is obtained that the thrust can be represented as the product of a ratio k and the square of angular velocity of the motor

$$T={\left(\frac{K_v{K}_{\tau }\sqrt{2\rho A}}{K_t}\omega \right)}^2=k{\omega }^2$$

where $k=2\rho A{K}_v^2{K}_{\tau }^2/K_t^2$ is some appropriately dimensioned constant.

As there are four equally spaced rotors in a quadrotor helicopter, considering all the motors yields the whole thrust on the quadrotor helicopter (in the body frame) below

$$T_B={\displaystyle \sum _{i\hspace{0.17em}=\hspace{0.17em}1}^4{T}_i}=k\left[\begin{array}{c}0\\ 0\\ {\displaystyle {\sum }_{i\hspace{0.17em}=\hspace{0.17em}1}^4{\omega }_i^2}\end{array}\right]$$

(2)

The friction is simply modeled as a force which is regarded as a proportional term with the linear velocities and such a modeling choice is sufficient from the view point of modeling and simulation. Then, the drag forces are modeled as

$$F_D=\left[\begin{array}{c}-k_d\dot{x}\\ -k_d\dot{y}\\ -k_d\dot{z}\end{array}\right]$$

(3)

where the constant k_d can be replaced by three different friction constants if additional precision is desired.

The torque τ_B in the body frame consisting of the roll torque ${\tau }_{\phi }$ , the pitch torque ${\tau }_{\theta }$ , and the yaw torque ${\tau }_{\psi }$ , is given by

$${\tau }_B=\left[\begin{array}{c}Lk({\omega }_1^2-{\omega }_3^2)\\ Lk({\omega }_2^2-{\omega }_4^2)\\ b\left({\omega }_1^2-{\omega }_2^2+{\omega }_3^2-{\omega }_4^2\right)\end{array}\right]$$

where L is the distance from the center of the quadrotor helicopter to any of the propellers and b is the drag coefficient.

Equations of motion

With the aid of rotation matrix, the linear motion in the inertial frame can be obtained as

$$m\ddot{\overline{x}}=\left[\begin{array}{c}0\\ 0\\ -mg\end{array}\right]+R{T}_B+F_D$$

(4)

where $\overline{x}$ and m denote the position and mass of the quadrotor helicopter, respectively. g is the acceleration due to gravity, T_B and F_D are defined in (2) and (3), respectively. The rotation matrix R is given by

$$R=\left[\begin{array}{ccc}{c}_{\phi }{c}_{\psi }-c_{\theta }{s}_{\phi }{s}_{\psi }& -c_{\psi }{s}_{\phi }-c_{\phi }{c}_{\theta }{s}_{\psi }& s_{\theta }{s}_{\psi }\\ c_{\theta }{c}_{\psi }{s}_{\phi }+c_{\phi }{s}_{\psi }& c_{\phi }{c}_{\theta }{c}_{\psi }-s_{\phi }{s}_{\psi }& -c_{\psi }{s}_{\theta }\\ s_{\phi }{s}_{\theta }& c_{\phi }{s}_{\theta }& c_{\theta }\end{array}\right]$$

which is applied to transform the thrust vector between the body frame and the inertial frame.

For rigid body dynamics, it follows from Euler’s equations that the rotational motion can be modeled as

$$\dot{\omega }=\left[\begin{array}{c}{\dot{\omega }}_x\\ {\dot{\omega }}_y\\ {\dot{\omega }}_z\end{array}\right]=\left[\begin{array}{c}{\tau }_{\phi }{I}_x{x}^{-1}\\ {\tau }_{\theta }{I}_y{y}^{-1}\\ {\tau }_{\psi }{I}_z{z}^{-1}\end{array}\right]-\left[\begin{array}{c}\frac{I_{yy}-I_{zz}}{I_{xx}}{\omega }_y{\omega }_z\\ \frac{I_{zz}-I_{xx}}{I_{yy}}{\omega }_x{\omega }_z\\ \frac{I_{xx}-I_{yy}}{I_{zz}}{\omega }_x{\omega }_y\end{array}\right]$$

(5)

where ${\tau }_{\phi },{\tau }_{\theta }$ and ${\tau }_{\psi }$ denote the roll torque, the pitch torque, and the yaw torque, respectively; ω represents the angular velocity and I is the inertia matrix of the form

$$I=\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& I_{yy}& 0\\ 0& 0& I_{zz}\end{array}\right]$$

Denote $x_1=\overline{x},x_2=\dot{\overline{x}},x_3=\stackrel{.}{\overline{\theta }}$ and $x_4=\omega$ . Then, combining (1), (4), and (5) yields

$$\{\begin{array}{l}{\dot{x}}_1=x_2\\ \dot{x_2}=\left[\begin{array}{c}0\\ 0\\ -g\end{array}\right]+\frac{1}{m}R{T}_B+\frac{1}{m}{F}_D\\ \dot{x_3}={\left[\begin{array}{ccc}1& 0& -s_{\theta }\\ 0& c_{\phi }& c_{\theta }{s}_{\phi }\\ 0& -s_{\phi }& c_{\theta }{c}_{\phi }\end{array}\right]}^{-1}{x}_4\\ \dot{x_4}=\left[\begin{array}{c}{\tau }_{\phi }{I}_x{x}^{-1}\\ {\tau }_{\theta }{I}_y{y}^{-1}\\ {\tau }_{\psi }{I}_z{z}^{-1}\end{array}\right]-\left[\begin{array}{c}\frac{I_{yy}-I_{zz}}{I_{xx}}{\omega }_y{\omega }_z\\ \frac{I_{zz}-I_{xx}}{I_{yy}}{\omega }_x{\omega }_z\\ \frac{I_{xx}-I_{yy}}{I_{zz}}{\omega }_x{\omega }_y\end{array}\right]\end{array}$$

(6)

where ${[x_1,x_2,x_3,x_4]}^{\mathrm{\top }}\in R^{12}$ is the state of the system (6).

PID controller

To achieve the stabilization of attitude of the quadrotor helicopter, the usual PID controller is employed, with a component proportional term to the error, an integral term and a derivative term of the error. The control input $u(t)=-{[u_{\phi }{u}_{\theta }{u}_{\psi }]}^{\mathrm{\top }}$ is designed as

$$\begin{array}{l}{u}_{\phi }=K_p{e}_{\phi }(t)+K_i{\displaystyle {\int }_0^{\mathrm{\top }}{e}_{\phi }}(t)\text{d}t+K_d{\dot{e}}_{\phi }(t)\\ u_{\theta }=K_p{e}_{\theta }(t)+K_i{\displaystyle {\int }_0^{\mathrm{\top }}{e}_{\theta }}(t)\text{d}t+K_d{\dot{e}}_{\theta }(t)\\ u_{\psi }=K_p{e}_{\psi }(t)+K_i{\displaystyle {\int }_0^{\mathrm{\top }}{e}_{\psi }}(t)\text{d}t+K_d{\dot{e}}_{\psi }(t)\end{array}$$

where $e_{\phi }(t)={\int }_0^{\mathrm{\top }}\dot{\phi }\text{d}t$ , $e_{\theta }(t)={\int }_0^{\mathrm{\top }}\dot{\theta }\text{d}t$ , $e_{\psi }(t)={\int }_0^{\mathrm{\top }}\dot{\psi }\text{d}t$ , and K_p, K_i, and K_d are corresponding coefficients.

Compared to PD controllers despite of their ease of implementation, the PID controller will not lead to steady state error, especially in the case of having noise and disturbances. Note that the control input u(t) will be not used in (6) directly due to the fact that the controlling variables are the voltages across the motors relating to the angular velocities ω_i ( $i=1,2,3,4$ ). However, with the relationship ${\tau }_B=Iu(t)$ and the constraint keeping the quadrotor helicopter aloft, the control can be imposed through values for τ_B and the angular velocities ω_i ( $i=1,2,3,4$ ).

The complete specification for the PID controller is provided and simulations under this controller can be performed. To avoid integral windup, we apply a switched strategy, that is, disable the integral term until the error is close to the steady state and activate the integral term near the desired steady state so as to push the system towards a low steady-state error.

Event-driven PID controller

In this section, we introduce an event-driven PID control scheme, where the event detector monitors the event-driven condition and determines when to transmit the newest errors to the controller. More specifically, when an event happens with $t_k<t_{k+1}$ , we denote the time instants by ${\left\{t_k\right\}}_{k\hspace{0.17em}=\hspace{0.17em}0}^{\infty }$ . With the errors $e_{\phi }(t_k),e_{\theta }(t_k)$ and $e_{\psi }(t_k)$ sampled at time t_k, the next sampling instant $t_{k+1}$ can be determined by

$$t_{k+1}=\inf \{t>t_k:|e_j(t)-e_j(t_k)|\hspace{0.17em}\mathrm{⩾}\hspace{0.17em}\overline{e}\}$$

(7)

for all $k\in \{1,2,\cdots \}$ and $j=\{\phi ,\theta ,\psi \}$ , where $\overline{e}$ is the event threshold. Then, the event-driven controller is designed based on the sampled errors

$$u(t)=-{[u_{\phi }(t_k)u_{\theta }(t_k)u_{\psi }(t_k)]}^{\mathrm{\top }}, t\in [t_k,t_{k+1})$$

(8)

for $k\in \{1,2,\cdots \}.$ Note that the control input (8) is only updated at the sampling instants. Hence, a zero-order holder (ZOH) will be embedded to keep a continuous control signal.

In order to smooth the error characteristic, we also consider another event-driven strategy with a different event-driven condition. Then, the new condition is given by

$$t_{k+1}=\inf \{t>t_k:|e_j(t)-e_j(t_k)|\hspace{0.17em}\mathrm{⩾}\hspace{0.17em}\gamma (t)\}$$

(9)

for all $k\in \{1,2,\cdots \}$ and $j=\{\phi ,\theta ,\psi \}$ , where $\gamma (t)=\beta \sqrt{{\mathrm{ϵ}}^{-\alpha t}+{\mathrm{ϵ}}_0}$ is the event threshold determined by the time t and parameters $0<\beta <1,\mathrm{ϵ}>1,$ $\alpha >1$ and ${\mathrm{ϵ}}_0\hspace{0.17em}\mathrm{⩾}\hspace{0.17em}0$ .

Note that the event threshold decreases exponentially. However, the event monitoring process remains unchanged, that is, the event detector monitors the event-driven condition and determines when to transmit the updated state estimates.

It is worth mentioning that the continuous-time event detector case is considered to determine the transmission of updated state estimates. However, in some cases, it is necessary to reduce sustained supervision times. Then, one can adopt a discrete-time detector which works in a discrete manner and is not difficult to be deduced.

Simulation

Having obtained the event-driven PID controller, we evaluate its performance with simulated realizations of the dynamics of the quadrotor helicopter, equation (6). In contrast to the literature for tuning PID parameters, the intent of this section is not to modify the dynamics of the quadrotor helicopter nor is the intent to design an algorithm to output the “optimal” PID gains. Instead, the comparisons between the PID controller and the event-driven PID controller are provided to find a tradeoff between control accuracy and reduced usage of the communication channel.

This event-driven method relies on detecting the upper bound of the real and sampled errors related the roll, pitch, and yaw angles; the value of the upper bound is informally referred to here as the guard of event condition. In simulations, we simply take the event threshold $\overline{e}=0.001$ , and the PID gain parameters $K_p=3,$ $K_i=5,$ $K_d=4,$ which can be tuned by hand, or by applying some methods such as extremum seeking method. ²⁰ Figure 1 gives the transient motion of the simulated quadrotor helicopter. Figure 2 shows the evolutions of the angular velocities $\dot{\phi },\dot{\theta },\dot{\psi }$ under the PID controller. Figures 3 and 4 show the evolutions of the angular velocities $\dot{\phi },\dot{\theta },\dot{\psi }$ under event-driven PID controllers with different event-driven conditions. It is observed that the angular velocities for both cases can be stabilized and the PID control scheme provides the trajectories smoothly within expectation due to the continuous control actions. When applying the event-driven PID controller, altitude control is provided by an event-driven control on the angle errors as well as the usual PID terms. For the event-driven PID controller with condition (9), taking $\beta =0.01,$ ϵ = 2, $\alpha =-1.5$ , and ${\mathrm{ϵ}}_0=0.001,$ it is seen from Figures 3 and 4, the transient behavior performs better due to the exponentially decreasing event threshold.

Figure 2.

Evolutions of the angular velocities $\dot{\phi },\dot{\theta },\dot{\psi }$ under the PID controller.

Figure 3.

Evolutions of the angular velocities $\dot{\phi },\dot{\theta },\dot{\psi }$ under the event-driven PID controller with condition (7).

Figure 4.

Evolutions of the angular velocities $\dot{\phi },\dot{\theta },\dot{\psi }$ under the event-driven PID controller with condition (9).

Compared to the PID control, the event-driven controllers can retain a satisfactory closed-loop behavior and can significantly reduce the number of transmissions, but we pay the price in allowing the quadrotor helicopter’s angular velocities varying within small scopes. However, this alteration is still tolerable and affordable.

Conclusions

This paper has addressed the event-driven PID control problem for autonomous quadrotor helicopters. Simplified dynamics model of quadrotor helicopters in hovering flight mode has been established. Then, an event-driven control strategy with the usual PID terms for the attitude stabilization of the quadrotor helicopter has been provided. Simulation results have been proposed to corroborate the theoretical results and have shown the comparisons with the effects under the PID controller.