Design and experiment of adaptive modified super-twisting control with a nonlinear sliding surface for a quadrotor helicopter

Abstract

Quadrotor unmanned aerial vehicle is a nonlinear system of 6-degree-of-freedom motion. In order to handle the nonlinearity that causes undesirable behavior, robustness of flight control has been studied. In this work, we consider the combination of higher order sliding mode control and nonlinear time-varying sliding surface for robustness and accuracy in tracking. An adaptive super-twisting control, a second-order sliding mode control, is utilized to compensate for the uncertainty and perturbation of a quadrotor system. A time-varying sliding surface is designed with a nonlinear function to provide varying properties of closed-loop dynamics and to improve control performance with energy consumption reduction. The proposed control system performance including energy consumption was compared among nonlinear adaptive super-twisting control algorithm, linear adaptive super-twisting control algorithm, and linear super twisting controllers, without and under wind disturbance. The robustness and effectiveness of the proposed control system are demonstrated by several times simulation and experiment using a quadrotor helicopter test bed.

Keywords

Sliding mode control quadrotor helicopter adaptive super-twisting control nonlinear sliding surface

Introduction

Quadrotor helicopter or quadcopter unmanned aerial vehicles (UAVs) have attracted research interest due to its wide range of applications such as navigation task (surveillance, mapping, rescuing, etc.)^1–3 and recently physical interaction (environment and manipulation of object).^4,5 However, the quadcopter requires robust control performance under aerodynamic forces, gyroscopic effect, variation of altitude, and wind payload and its resources and also the limited operational time due to power supply capacity. In reality, the characteristic of quadcopter dynamics is a nonlinear system and coupled; therefore, it requires a robust controller to compensate the uncertainties and external disturbances and also consider energy consumption. Most studies on reducing energy consumption for a quadcopter discuss the design of the quadcopter’s platform or mechanical parts. Another approach to reduced energy consumption is by changing only the control algorithm on the existing hardware, which is of low cost and shows higher efficiency. Therefore, the effect of a robust control algorithm for energy saving is addressed in this study.

A quadcopter has highly nonlinear and time-varying behavior because it is always influenced by unpredictable disturbances. In the conventional control approaches, feedback linearization⁶ is developed which obtains the linear decoupled closed-loop system after a change of coordinates in the state–space. The requirement to measure all the states for this feedback controller is the most important limitation and it is vulnerable to parameter uncertainties. Sliding mode control (SMC) is used as a solution to deal with model uncertainties⁷ because it provides perturbation rejection by designing boundary layer on sliding mode surface in the range of uncertainties/perturbation.⁸ However, the boundary is difficult to estimate in most practical cases, therefore often leading to overestimation control, and also aggravates the chattering phenomenon which may decrease the tracking performance and increase energy loss. To overcome this weakness, an adaptive SMC was introduced for stabilizing the control system in altitude and attitude motion against the boundedness of uncertainty.² The idea of this adaptation algorithm is by increasing/decreasing the gain in order to adapt its magnitude to the perturbations and uncertainties in the sliding mode condition. Furthermore, it will force to increase the control gain to continue the adaptation process when the condition is not in the sliding mode. A backstepping and sliding mode controller was developed in simulation to control the position and attitude with an observer to know the bound fault of the system.⁹

Another technique known as second-order sliding mode control (SOSMC) was introduced for chattering reduction.^7,10 SOSMC has the main advantages of SMC and improves control performance with respect to chattering effects. SOSMC generates continuous control that drives the sliding variable and its derivative to zero in finite time. Among SOSMC, the super-twisting algorithm (STA) is a well-known controller which ensures robustness with respect to modeling errors and external disturbances because it contains a discontinuous function under the integral term and can reduce the chattering effect.¹ However, in practice, it is not easy to choose appropriate parameters of the STA for good control performance because the classical STA gain requires the bound of the disturbance as well as its time derivatives. To improve this weakness, adaptive super-twisting control algorithm (ASTA) was introduced without knowing bound of the disturbance and unmodeled dynamics,¹¹ and a modified STA was proposed by adding a linear stabilizing term to provide faster convergence to the sliding mode condition and robustness from any initial condition.¹² Multivariable STA was applied with neurofuzzy observer to estimate the bound of the uncertainty in SMC attitude of quadcopter in simulation.¹³ However, all these SMC methods were conducted not in real-time application and not considering energy consumption.

The closed-loop dynamics in SMC¹ depends on the design of the stable sliding surface. To improve the performance in SMC, a fuzzy system strategy was utilized to update parameters of a sliding surface,¹⁴ a time-varying sliding surface was considered,¹⁵ and a variable damping ratio of closed-loop dynamics was applied to reduce high overshoot.¹⁶ However, these methods will increase the computation time.

In addition, recent studies on disturbance rejection for a quadcopter have been developed for outdoor environment against wind gust. A controller based on active disturbance rejection was introduced to observe the presence of disturbance and maintain the position at a fixed point.¹⁷ Another approach is using an adaptive observer with incremental nonlinear dynamic inversion, to counter the wind gust disturbance using a small quadcopter.¹⁸ However, these methods are only verified in hovering motion and do not consider the energy consumption.

In this study, we propose a real-time robust control algorithm and a nonlinear sliding surface (NSS)¹⁹ for a quadcopter in order to obtain robust tracking control and energy saving. The robust control algorithm is inspired from the ASTA²⁰ and is based on adaptive SOSMC and a modified STA that have a robust behavior to modeling errors, uncertainties, and perturbations with unknown bounds while having controller gain dynamically adapted. The control algorithm consists of the equivalent control part with a nonlinear function and an ASTA. The NSS is a function of tracking error, and it is designed to reduce the energy consumption of quadcopter during the flight. We verify the effectiveness and control performance of a quadrotor by simulation and experiment. In addition, we show the control signals in order to verify the chattering reduction that relates to reduce energy consumption. In order to verify the repeatability of the proposed controller’s performance and ensure the fair comparison with conventional controllers, we conducted repeated experiments in an indoor environment with an artificial wind disturbance. This article is an extension to the work in a conference paper.²¹ Differences include the following: (1) an indoor experiment based on a rigid link quadcopter to demonstrate the performance without and under wind disturbance, (2) the stability proof of the designed controller using the Lyapunov stability theory, (3) demonstration with desired trajectory x-y-z motion simultaneously with disturbances in the simulation, and (4) energy consumption evaluation.

Dynamics model of quadrotor helicopter

A quadcopter consists of two pairs of contrarotating rotors, one pair (M2 and M4) rotates clockwise and the other (M1 and M3) counterclockwise, as shown in Figure 1. The changing altitude of the quadcopter is managed by rotating all rotors. The rotation body along z-axis (yaw motion) is generated by managing the differences of clockwise and counterclockwise rotor speeds simultaneously. The rotation body along x- and y-axes (pitch and roll motion) is achieved by controlling speed of clockwise rotor and counterclockwise rotor that will generate forward, backward, left, and right motions.

Figure 1.

Quadrotor model.

A rigid 6-degree-of-freedom (DOF) body of quadcopter will operate considering in two model frames as shown in Figure 1. A body frame {B} is fixed at the quadcopter’s center of gravity, in which the rotor axes point to positive z-axis and the two arms point to x- and y-axes. An earth/inertial frame {E} is a rigid body with respect to the ground. In {E}, we describe the quadcopter’s position and orientation as $X = [x, y, z]^{T}$ and $Θ = [ϕ, θ, ψ]^{T}$ (represents pitch, roll, and yaw angles) and also linear and angular velocities as $\overset{\cdot}{X} = [\overset{\cdot}{x}, \overset{\cdot}{y}, \overset{\cdot}{z}]^{T}$ and $\overset{\cdot}{Θ} = [\overset{\cdot}{ϕ}, \overset{\cdot}{θ}, \overset{\cdot}{ψ}]^{T}$ . The position and orientation in {E} are measured from sensors attached on the body in {B}. Here, the linear and angular velocities in {B} are denoted by $ν$ and $ω$ . Then, the relation linear and angular velocities in {E} and {B} are defined as follows

\begin{matrix} \overset{\cdot}{X} = R ν \\ H \overset{\cdot}{Θ} = ω \end{matrix}

where

\begin{matrix} R = [\begin{matrix} - s ψ s ϕ s θ + c ψ c θ & - s ψ c ϕ & s ψ s ϕ c θ + c ψ s θ \\ c ψ s ϕ s θ + s ψ c θ & c ψ c ϕ & - c ψ s ϕ c θ + s ψ s θ \\ - c ϕ s θ & s ϕ & c ϕ c θ \end{matrix}] \\ H = [\begin{matrix} c θ & 0 & - c ψ s θ \\ 0 & 1 & s ψ \\ s θ & 0 & c ψ c θ \end{matrix}] \end{matrix}

R is a rotational matrix derived from ZXY Euler angle of yaw, pitch, and roll; H is a transformation matrix from angular velocity {B} to {E}; and s and c represent sine and cosine, respectively.

The Newton–Euler approach in {B} is utilized to obtain the system dynamics of quadcopter for translational and rotational motion. For controller design, the dynamics will be represented in {E}. The dynamics of quadcopter is described as follows

{\overset{\cdot\cdot}{ξ}}_{1} = f + U

(1)

where

\begin{matrix} ξ_{1} = [X^{T}, Θ^{T}]^{T} \\ f = {[\begin{matrix} m I_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & J \end{matrix}]}^{- 1} [0, 0, - mg, K_{1}, K_{2}, K_{3}]^{T} \\ U = {[\begin{matrix} m I_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & J \end{matrix}]}^{- 1} [\begin{matrix} (s ϕ c θ s ψ + s θ c ψ) u_{1} \\ (- s ϕ c θ c ψ + s θ s ψ) u_{1} \\ c ϕ c θ u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}] \\ J = [\begin{matrix} I_{x} c θ & 0 & - I_{x} c ϕ s θ \\ 0 & I_{y} & I_{y} s ϕ \\ I_{z} s θ & 0 & I_{z} c ϕ c θ \end{matrix}] \\ \begin{matrix} K_{1} = (I_{x} + I_{y} - I_{z}) \overset{\cdot}{ϕ} \overset{\cdot}{θ} s θ + (I_{y} - I_{z}) {\overset{\cdot}{ψ}}^{2} s ϕ c ϕ c θ \\ + (- I_{x} + I_{y} - I_{z}) \overset{\cdot}{ϕ} \overset{\cdot}{ψ} s ϕ s θ \\ + (I_{x} + I_{y} - I_{z}) \overset{\cdot}{θ} \overset{\cdot}{ψ} c ϕ c θ \end{matrix} \\ \begin{matrix} K_{2} = (- I_{y} + (I_{z} - I_{x}) c 2 θ) \overset{\cdot}{ϕ} \overset{\cdot}{ψ} c ϕ \\ + (I_{z} - I_{x}) ({\overset{\cdot}{ϕ}}^{2} - {\overset{\cdot}{ψ}}^{2} c^{2} ϕ) s θ c θ \end{matrix} \\ \begin{matrix} K_{3} = (- I_{z} + I_{x} - I_{y}) \overset{\cdot}{ϕ} \overset{\cdot}{θ} c θ - (I_{x} - I_{y}) {\overset{\cdot}{ψ}}^{2} s ϕ c ϕ s θ \\ + (I_{z} + I_{x} - I_{y}) \overset{\cdot}{ϕ} \overset{\cdot}{ψ} s ϕ c θ \\ + (I_{z} - I_{x} + I_{y}) \overset{\cdot}{θ} \overset{\cdot}{ψ} c ϕ s θ . \end{matrix} \end{matrix}

$I_{x}, I_{y}, and I_{z}$ are the quadcopter’s moment of inertia around the x-, y-, and z-axes in {B}. m is the total mass of quadcopter, g is the gravitational acceleration, $I_{3 \times 3}$ is the identity matrix, and $0_{3 \times 3}$ is the $3 \times 3$ null matrix. We define control input $u_{j} (j = 1, \dots, 4)$ as follows

[\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}] = [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & - L & 0 & L \\ L & 0 & - L & 0 \\ - d & d & - d & d \end{matrix}] [\begin{matrix} f_{1} \\ f_{2} \\ f_{3} \\ f_{4} \end{matrix}]

Input $u_{1}$ is a thrust force from all rotor forces (Figure 1); $u_{2}$ , $u_{3}$ , and $u_{4}$ are pitch, roll, and yaw moments of the torque differences around the axes in {B} between clockwise rotor, counterclockwise rotor, and clockwise and counterclockwise rotors. L is the distance between the center of gravity and rotor and d is a drag coefficient relating force to moment. This dynamics model has been studied by Sumantri et al.^19,22

The gyroscopic and aerodynamic effects are not included in quadrotor model but it is considered as a disturbance term. In addition, four rotors have limits for speed and torque, and thus the control inputs have minimum and maximum values in operation.

Control system design

In this section, we present the control scheme for the quadcopter which is a 6-DOF underactuated system with four control inputs. The system dynamics is changed into a decoupled system by transforming four control inputs, u, into six new synthetic inputs, v, in order to obtain a linear form. Then, the new input v is transformed into original control input u, desired attitude $ϕ_{d}$ , and $θ_{d}$ . To derive the system dynamics, we choose a combination of robust controller named the modified super-twisting control algorithm (STA)²⁰ with adaptation gain¹¹ and NSS, to attenuate chattering phenomenon, increase accuracy tracking, and avoid gain overestimation. The STA is based on a second-order sliding mode, and we design the sliding surface with the nonlinear function.

In addition, the control of pitch and roll angles is quite sensitive and the discontinuous control law could bring unstable behavior of the system. It can reduce the interest to use robust discontinuous control laws like the first-order SMC. The second-order sliding mode controller is an attractive approach to avoid discontinuity. In addition, the proposed controller adaptively limits the magnitude of the control input and then the chattering. The STA also limits the essential discontinuous behavior of the control input which does not use the time derivative of the sliding variable as the standard second-order SMC.

Control system structure

To simplify controller design, we transform the system dynamics in equation (1) into a decoupled system like in the work by Sumantri et al.,²² and we consider the virtual input

v = f + U

(2)

Hence, the decoupled system in equation (1) considering equation (2) is rewritten into a linear state form as follows

\begin{matrix} {\overset{\cdot}{ξ}}_{1} = ξ_{2} \\ {\overset{\cdot}{ξ}}_{2} = v + ρ_{d} \end{matrix}

(3)

where $ξ_{1} = [x, y, z, ϕ, θ, ψ]^{T}$ is the positional vector, $ξ_{2}$ is the velocity vector, $v = [v_{x}, v_{y}, v_{z}, v_{ϕ}, v_{θ}, v_{ψ}]^{T}$ is a new virtual control input vector, and $ρ_{d}$ is aerodynamics and gyroscopic effects together with wind effects and uncertainty as a disturbance vector. Therefore, the virtual input v enables us to consider a fully actuated system. The upper bound of $| | ρ_{d} | |$ is generally unknown, so adaptive is considered in this study. Furthermore, it is necessary to design robust controller to compensate for the effect of uncertainties/perturbations. For this reason, the adaptive STA is applied as presented in the next section.

From the new definition virtual input vector in equation (3), considering equation (2), we obtain two sets of equations: translation and rotational dynamics. For translational dynamics, we have

\begin{matrix} v_{x} = (\sin θ \cos ψ + \sin ϕ \cos θ \sin ψ) \frac{u_{1}}{m} \\ v_{y} = (\sin θ \sin ψ - \sin ϕ \cos θ \cos ψ) \frac{u_{1}}{m} \\ v_{z} = \cos ϕ \cos θ \frac{u_{1}}{m} - g \end{matrix}

(4)

For rotational dynamics, we have

v_{ϕ}, v_{θ}, v_{ψ}]^{T} = J^{- 1} ({[u_{2}, u_{3}, u_{4}]}^{T} + {[K_{1}, K_{2}, K_{3}]}^{T})

(5)

Substituting the attitude variables into desired attitude variables in equation (4), we can calculate desired pitch and roll angles

\begin{array}{l} 1 ϕ_{d} = \arctan (\frac{v_{x} \sin ψ_{d} - v_{y} \cos ψ_{d}}{v_{z} + g}) \\ θ_{d} = \arctan (\frac{v_{x} \cos ψ_{d} + v_{y} \sin ψ_{d}}{\sqrt{{(v_{x} \sin ψ_{d} - v_{y} \cos ψ_{d})}^{2} + {(v_{z} + g)}^{2}}}) \end{array}

(6)

To obtain the original inputs $u_{1}, u_{2}, u_{3}, and u_{4}$ in equation (1), we utilize the method presented in the work by Sumantri et al.,²² using a least square method. The input $u_{1}$ is obtained from equations (3) and (4) as follows

u_{1} = m \sqrt{v_{x}^{2} + v_{y}^{2} + {(v_{z} + g)}^{2}}

(7)

The inputs $u_{2}, u_{3}, and u_{4}$ are obtained from equation (5); by utilizing a simple dynamic inversion, we have

[u_{2}, u_{3}, u_{4}]^{T} = J [v_{ϕ}, v_{θ}, v_{ψ}]^{T} - [K_{1}, K_{2}, K_{3}]^{T}

(8)

The above control strategy is summarized in Figure 2 and described as follows:

Step 1. Desired trajectory of $X_{d} = [x_{d}, y_{d}, z_{d}]^{T}$ and yaw angle $ψ_{d}$ are assigned to the position controller. The position controller explained in the following section computes the virtual control input $[v_{x}, v_{z}, v_{y}]^{T}$ from tracking errors between current positional states X and their desired trajectories $X_{d}$ . This step provides the desired trajectory of roll angle $ϕ_{d}$ and pitch angle $θ_{d}$ , while $ψ_{d}$ remains as assigned.

Step 2. The attitude controller which is explained in the following section handles the attitude tracking error between current angular states $Θ$ and their desired trajectories $Θ_{d}$ and then provides virtual control inputs $[v_{ϕ}, v_{θ}, v_{ψ}]^{T}$ .

Step 3. The virtual input vector $v = [v_{x}, v_{y}, v_{z}, v_{ϕ}, v_{θ}, v_{ψ}]^{T}$ is transformed into the real vector $u = [u_{1}, u_{2}, u_{3}, u_{4}]^{T}$ with equations (7) and (8).

In the following section, the calculation of v is explained as one vector. However, all the calculations are decoupled into each DOF so that the sequential calculations in Figure 2 are possible.

Figure 2.

Control system structure.

Design of nonlinear sliding surface

We apply SMC for the calculation of virtual input v for achieving the robust control performance. Using an NSS function, the damping ratio of the closed loop system is changed from its initial low value to a final high value as the output approaches the desired trajectory. The nonlinear sliding surface equation for the dynamics in equation (3) is designed as follows¹⁶

s_{l} = [F - Ψ I] [e_{1} e_{2}]^{T}

(9)

where $s_{l} = [s_{l 1}, \dots, s_{l 6}]^{T}$ is the sliding surface vector in all DOF, $F = diag {F_{i}}$ and $Ψ = diag {Ψ_{i}}$ with $i = 1, 2, \dots, 6$ are diagonal matrices with positive and non-positive function diagonal elements, I is a $6 \times 6$ identity matrix, $e_{1} = ξ_{1} - ξ_{1 d}$ and $e_{2} = ξ_{2} - ξ_{2 d}$ are tracking error vectors with the desired trajectory vectors $ξ_{1 d} = [x_{d}, y_{d}, z_{d}, ϕ_{d}, θ_{d}, ψ_{d}]^{T}$ and $ξ_{2 d} = [{\overset{\cdot}{x}}_{d}, {\overset{\cdot}{y}}_{d}, {\overset{\cdot}{z}}_{d}, {\overset{\cdot}{ϕ}}_{d}, {\overset{\cdot}{θ}}_{d}, {\overset{\cdot}{ψ}}_{d}]^{T}$ , respectively. The matrix $Ψ$ consists of nonlinear function as follows¹⁹

Ψ_{i} = - β_{i} (\frac{1 - \exp (- 1)}{\exp (ε_{i}^{2}) - \exp (- 1)}), ε_{i} \in e_{1}

(10)

where $β_{i} > 0$ satisfies $Ψ_{i} < 0$ . The stability of using the NSS has been discussed and proved by Sumantri et al.²³ by illustration for $β_{i} ⩽ 3 F_{i} (F_{i} > 0)$ . Appendix 1 briefly shows the stability analysis.

The nonlinear function, $Ψ_{i}$ , in equation (10) generates a varying time constant for the NSS as a function of error. It changes the time constant from a minimum value to a high value if the error magnitude changes from zero to a high value. In addition, it is common that a system has a faster response with a lower time constant than that with a higher time constant, but the lower time constant requires more energy in a control system. Therefore, in order to reduce the energy consumption, a relatively fast response is considered by varying time constant.²³

To derive the SMC law achieving $s_{l} = {\overset{\cdot}{s}}_{l} = 0$ , we have the first derivative of sliding surface equation from equation (9)

{\overset{\cdot}{s}}_{l} = {\overset{\cdot}{e}}_{2} + (F - Ψ) {\overset{\cdot}{e}}_{1} - \overset{\cdot}{Ψ} e_{1}

(11)

Equations (3) and (11) lead to

{\overset{\cdot}{s}}_{l} = v - {\overset{\cdot}{ξ}}_{2 d} + (F - Ψ) {\overset{\cdot}{e}}_{1} - \overset{\cdot}{Ψ} e_{1} + ρ_{d}

(12)

Then, we choose a control law v which consists of equivalent control $v_{eq}$ and the STA part $v_{st}$

v = v_{eq} + v_{st}

(13)

where

v_{eq} = - (F - Ψ) {\overset{\cdot}{e}}_{1} + \overset{\cdot}{Ψ} e_{1} + {\overset{\cdot}{ξ}}_{2 d}

(14)

Here, we assume that the desired accelerations of pitch and roll motion are zero to avoid second derivative calculation of equation (6) in implementation.

Design of adaptive super-twisting control

In this study, we consider the STA that is an extension of previous control algorithm¹⁰ with a constant plus proportional and power rate reaching law to include an extra linear correction term, which ensures finite-time convergence of sliding variable for broader class of uncertainties. It provides stronger robustness and faster convergence speed.²⁴ It also has an important feature which is directly applicable to systems with relative degree 1 and without measuring derivative of sliding variable ${\overset{\cdot}{s}}_{li}$ . The controller in equation (13) is designed as follows

v = v_{eq} + v_{st} = v_{eq} - k_{a} Φ_{1} - k_{b} \int Φ_{2} dt

(15)

where

\begin{matrix} Φ_{1} = s_{l} + k_{c} {| s_{l} |}^{\frac{1}{2}} sign (s_{l}) \\ Φ_{2} = s_{l} + \frac{3}{2} k_{c} {| s_{l} |}^{\frac{1}{2}} sign (s_{l}) + \frac{1}{2} k_{c}^{2} sign (s_{l}) \end{matrix}

(16)

are the nonlinear stabilizing terms and $| s_{l} |^{1 / 2} = diag {| s_{li} |^{1 / 2}}, sign (s_{l})$ is the vector signum function, and $k_{a} = diag {k_{ai}}, k_{b} = diag {k_{bi}},$ and $k_{c} = diag {k_{ci}}$ (i = 1, 2, …, 6) are diagonal matrices with positive diagonal elements.

Generally, gains of $k_{a}$ and $k_{b}$ are selected with sufficiently high values considering the upper bound of perturbation. Since the upper bound on $ρ_{di}$ in equation (3) is not known, we consider gain adaptive law which is utilized in the work by Shtessel et al.¹¹ for each DOF as follows

\begin{matrix} {\overset{\cdot}{k}}_{ai} = {\begin{matrix} {\bar{k}}_{ai} \sqrt{\frac{γ_{ai}}{2}} sign (| s_{li} | - μ_{i}) & if k_{ai} > k_{mi} \\ η_{i} & if k_{ai} ⩽ k_{mi} \end{matrix} \\ k_{bi} = 2 k_{di} k_{ai} \end{matrix}

(17)

where ${\bar{k}}_{ai}, γ_{ai}, η_{i}, k_{di}, and k_{mi}$ are all positive constants, $μ_{i}$ is a positive constant that relates to the sliding mode bound. Under a few assumptions in the work by Shtessel et al.,¹¹ the gain adaptive law in equation (17) achieves the finite-time convergence of sliding variable to a real two-sliding mode when $| s_{li} | ⩽ ϖ_{1 i}$ and $| {\overset{\cdot}{s}}_{li} | ⩽ ϖ_{2 i}$ , with $ϖ_{1 i} ⩾ μ_{i}$ and $ϖ_{2 i} > 0$ . The idea of ASTA is to dynamically increase or decrease the control gains $k_{ai}$ and $k_{bi}$ until a second-order sliding mode with respect to $\overset{\cdot}{s}$ is established. Then, the gains start reducing. This gain becomes increasing again to force the adaptation process again when the condition on $\overset{\cdot}{s}$ is lost (because of a very small gain). When $| s_{li} | ⩽ μ_{i}$ , the ASTA will reduce the control gains until the two-sliding mode establishes. The gain will be increased when the sliding variable starts deviating from the two-sliding mode. Appendix 1 partially shows the stability analysis.

The second line of adaptation law is used to ensure that the gains are always positive. We may consider to tune $k_{mi}$ as small as possible when only small uncertainties need to be considered.

The overall closed-loop dynamics is obtained by substituting equation (13) into equation (3) and considering equation (9) as follows

\begin{array}{l} {\overset{\cdot\cdot}{e}}_{1} + (F - Ψ + k_{a}) {\overset{\cdot}{e}}_{1} + (k_{a} (F - Ψ) - \overset{\cdot}{Ψ}) e_{1} \\ - ρ_{d} + k_{a} k_{c} {| s_{l} |}^{\frac{1}{2}} s i g n (s_{l}) + {k_{b}}_{0}^{t} {{\overset{\cdot}{e}}_{1} + (F - Ψ) e_{1} \\ + \frac{1}{2} k_{c} {| s_{l} |}^{\frac{1}{2}} s i g n (s_{l}) + \frac{3}{2} k_{c}^{2} s i g n (s_{l})} d t = 0 \end{array}

(18)

Implementation and evaluation

In this section, we examine the effectiveness and robustness of the proposed controller design, by comparative results with STA and ASTA controllers in numerical and experimental tests without and under disturbance. In the experimental test, we use a quadrotor experimental test bed as shown in Figure 3. This test bed was built to verify the repeatability by maintaining the same experimental condition. The parameters in equation (1) for these numerical and experimental tests are obtained from previous work¹⁹ as follows: $m = 0.285 kg$ , $L = 0.212 m$ , $g = 9.807 m / s^{2}$ , $d = 1 m$ , $I_{x} = I_{y} = 5.136 \times 10^{- 3} kg m^{2}$ , and $I_{z} = 1.016 \times$ $10^{- 2} kg m^{2}$ . For the comparison purpose, we assign the linear sliding surface $s_{l} = λ e_{1} + e_{2}$ and then the control input $v_{eq}$ in equation (13) is designed as follows

v_{eq} = - λ {\overset{\cdot}{e}}_{1} + {\overset{\cdot}{ξ}}_{2 d}

(19)

$λ$ is decomposed into F and $β$ , where $λ = F + β$ and $β = diag {β_{1}, \dots, β_{6}}$ in equation (10). The designed parameters for ASTA and NSS are important. From the existing literature, to the best of the authors’ knowledge, there is no systematic method to choose the values of the parameters. In this study, all values are selected by the extensive trials. We choose the STA and adaptive gains start from low to high value in order to bound the disturbance. The nonlinear function is adjusted after obtaining the minimum of linear sliding surface. In simulation, we employed the same trajectory with our previous work²¹ to verify the effectiveness of the proposed new controller over the previous one, and the further effectiveness to disturbance rejection was confirmed. Because the trajectory in simulation consisted of constant velocity motion, we considered a more practical trajectory with varying velocity motion in experiment to verify the further effectiveness.

Figure 3.

Experimental system configuration.

Simulation results

Simulation results of the proposed controller for trajectory tracking are illustrated in this section in order to evaluate the robustness of the control performance with sampling time 0.005 s. The desired trajectory in Figure 4 consists of the following four different motions (total 60 s): (1) take-off motion from A to B (0–25 s) with the height of 0.1 m, (2) maneuvering in the $x - y$ plane from B to C (25–40 s) with the height of 0.35 m, (3) performing a yaw motion from A to C (0–40 s), and (4) landing (40–60 s) from C to D. The designed trajectory will perform x-y-z motion simultaneously from 25 to 40 s and the robustness of the controller design may be evaluated under disturbances during this motion. We assume that there are fix disturbances such as wind in x and y-axes around 4 m/s and small error measurement in x-y-z.

Figure 4.

The 3D desired trajectory in simulation.

In this study, we demonstrate the simulation with different parameters from the previous work²¹ by considering disturbances. The same tuning parameters are chosen for fair comparison including NSS parameters and initial parameters for adaptation (units are omitted): $k_{a} = diag {5, 5, 20, 8, 8, 15}$ , $k_{b} = diag {1, 1, 10, 81, 8}$ , and $k_{c} = diag {0.1, 0.1, 0.1, 0.1, 0.1, 0.1}$ . The NSS parameters are $F = diag {5, 10, 5, 5, 55}$ and $β = diag {10, 10, 10, 10, 10, 10}$ . The tuning parameters for the ASTA are chosen as follows: $\bar{k} = [2, 2, 10, 2, 2, 2]$ , $γ_{i} = 2$ , $μ_{i} = 0.01$ , $k_{mi} = 0.1$ , $k_{d} = [1, 1, 3, 1, 1, 1]$ , $i = 1, 2, \dots, 6$ , and $η = k_{m}$ . In order to demonstrate the performance of the system which is like in a realistic environment, we put static wind disturbance at 20–30 s and the time-varying disturbance which is given as follows: $ρ_{d} = [0.2, 0.2, 3, 1, 1, 1]^{T} \times \cos (t)$ .

The ASTA with NSS (ASTAN) achieves smaller tracking position error in Figure 5, which reduces the mean and variance of the tracking position error by 33% without disturbance and 26.8% under disturbance on the average. Figure 6 shows the control performances of these controllers. The control input $u_{1}$ is generated a relative high total force value especially at time 0.2 s to perform take-off motion and it shows that ASTAN performs a faster initial take-off motion and has a reduced overshoot under disturbance. Figure 7 shows that the control gains change dynamically according to the behavior of tracking error without and under disturbance. Gains for pitch and roll motions have different results after 20 s under disturbance because there occurs significant tracking error changes in x and y positions that influence directly to $ϕ_{d}$ and $θ_{d}$ , and both adaptive methods can detect the significant error and try to compensate for these errors well.

Figure 5.

Tracking error results in simulation.

Figure 6.

Control input profiles in simulation.

Figure 7.

Results of adaptive gain $k_{a}$ in simulation.

The root-mean-square error (RMSE) and the standard deviation (SD) for all methods are summarized in Table 1, which shows that the ASTAN performs better than ASTA. In addition, it is verified that the adaptive mechanism is also effective for ASTAN case without and under disturbance. The variance of control input for these methods is summarized in Table 2, which shows that the ASTAN is also effective to reduce the chattering of the system very well.

Table 1.

Tracking error comparison results in simulation.

Condition	Controller	RMSE		SD
Condition	Controller	(cm)	(°)	(cm)	(°)
None	ASTAN	1.2 × 10⁻¹	6.74 × 10⁻²	1.2 × 10⁻¹	6.74 × 10⁻²
	ASTA	1.8 × 10⁻¹	5.85 × 10⁻²	1.8 × 10⁻¹	5.01 × 10⁻²
	STA	5.6 × 10⁻¹	7.8 × 10⁻²	5.5 × 10⁻¹	7.8 × 10⁻²
Dist.	ASTAN	7.1 × 10⁻¹	8.14 × 10⁻¹	7	8.14 × 10⁻¹
	ASTA	9.9 × 10⁻¹	5.7 × 10⁻¹	9.7	5.7 × 10⁻¹
	STA	3.6	1	35.9	1

ASTA: adaptive super-twisting control algorithm; ASTAN: ASTA with nonlinear sliding surface; RMSE: root-mean-square error; SD: standard deviation; STA: super-twisting algorithm.

Table 2.

Control input variance comparison results in simulation.

Controller	Variance
Controller	None	Disturbance
STA	4.48 × 10⁻²	8.20 × 10⁻²
ASTA	3.52 × 10⁻²	5.87 × 10⁻²
ASTAN	2.85 × 10⁻²	5.32 × 10⁻²

ASTA: adaptive super-twisting control algorithm; ASTAN: ASTA with nonlinear sliding surface; STA: super-twisting algorithm.

Experimental results

Experimental results of the proposed controller for trajectory tracking and stabilization are illustrated in this section. Figure 8 is a desired trajectory in experiment, which consists of six motions during 60 s as follows: (1) take-off motion from A to B (0–10 s), (2) perform yaw motion and maneuvering in the x–y plane from B to C (10–15 s), (3) hovering at C (15–30 s), (4) maneuvering from C to B (30–34 s), (5) hovering at B (34–50 s), and (6) landing from B to A (50–60 s). We applied wind disturbance from an electric fan (power: 57 W) to verify the robustness of the proposed method with distance 0.5–1.5 m, as shown in Figure 3. The disturbance magnitude is proportional to the distance between the quadcopter and the fan. The variable disturbance conditions are considered along the trajectory from B to C and C to B, where it takes the maximum at C. With the position and direction of the fan are fixed, it generates different air flow directions to the motion of quadcopter from B to C and C to B. Therefore, we may evaluate the effectiveness of the proposed controller design under variable magnitudes and directions of the wind disturbance. The environment in experimental test bed includes dynamics uncertainties and perturbation different from simulation. We obtain the controller parameters by several experiments as follows (units are omitted): $k_{a} = diag {1, 1, 4, 8, 8, 15}$ , $k_{b} = diag {0.5, 0.5, 0.5, 1, 1, 8}$ , and $k_{c} = d i a g {0.1, 0.1, 0.1, 0.1, 0.1, 0.1}$ . The NSS parameters are $F = diag {5, 5, 5, 5, 5, 50}$ and $β = diag {15, 15, 15, 10, 10, 150}$ . The values of $β$ should be designed properly because if we choose a relatively small value, the chattering still occurs, while a relatively large $β$ will reduce the robustness according to the previous work.¹⁹ The tuning parameters for the ASTA are chosen as follows: $\bar{k} = [0.01, 0.01, 0.01, 0.05, 0.05, 0.04]$ , $γ_{i} = 2$ , $μ_{i} = [2, 2, 1, 1, 1, 1]$ , $k_{mi} = 0.1$ , $k_{d} = [0.07, 0.1, 0.1, 0.06, 0.06, 0.6]$ , $i = 1, 2, \dots, 6$ , and $η = k_{m}$ .

Figure 8.

The 3D desired trajectory in experiment.

Control performance evaluation

The tracking error resulting from three control strategies in the experiments without/under wind disturbance is shown in Figure 9. It shows that both control strategies provide robustness in experiments without/under wind disturbance. Generally, the larger error occurs in hovering motion after maneuvering motion. Under wind disturbance, all states have large magnitude error than without wind disturbance. During the hovering motion, the pitch motion exhibits larger error results than the other state motions because it is largely affected by wind disturbance. From Table 3, we conclude that the ASTAN provides the smallest tracking errors than the others in all conditions. The repeatability of these controllers is also evaluated by performing five-times experiments and the results are illustrated in Figures 10 and 11. On average, the ASTAN provides around 2% and 3.5% smaller mean value than that by ASTA and STA in both conditions without/under wind disturbance and has around 1.5% and 5% smaller variance in the condition without wind disturbance. Figure 12 shows the control performance in each trajectory condition in Figure 8. On average, the ASTAN provides more effectiveness for maneuvering and hovering without wind disturbance.

Figure 9.

Tracking error profiles in experiment.

Table 3.

Tracking error comparison results in experiment.

Disturbance	Controller	RMSE		SD
Disturbance	Controller	(cm)	(°)	(cm)	(°)
None	STA	1.99	1.675	1.95	1.675
	ASTA	1.96	1.135	1.88	1.134
	ASTAN	1.92	1.134	1.85	1.133
Wind	STA	2.30	1.589	2.14	1.588
	ASTA	2.17	1.332	2.11	1.33
	ASTAN	2.14	1.313	2.08	1.311

ASTA: adaptive super-twisting control algorithm; ASTAN: ASTA with nonlinear sliding surface; RMSE: root-mean-square error; SD: standard deviation; STA: super-twisting algorithm.

Figure 10.

Comparison of root mean square and variance of tracking errors without wind diturbance.

Figure 11.

Comparison of root mean square and variance of tracking errors under wind disturbance.

Figure 12.

Comparison of root mean square and variance of tracking errors in six trajectory conditions.

Figure 13 shows control input profiles. It is seen that the control input $u_{1}$ produces great total thrust to lift the quadcopter minimum value around 2 N. For initial take-off, it requires total force about 3.5–4 N. It is increasing in under wind disturbance, especially in the maximum disturbance near to position C (30–35 s) which is near to the fan. The variances of control inputs are summarized in Figures 14 and 15 and Table 4. Figure 15 shows that take-off motion produces larger magnitude in chattering, and the chattering increases in hovering motion close to the fan. The ASTAN provides low variance in control input and the effectiveness of the chattering reduction is verified in all the conditions.

Figure 13.

Control input profiles.

Figure 14.

Control input variance in five-times experiment.

Figure 15.

Control input variance in six trajectory conditions.

Table 4.

Control input variance comparison results in experiment.

Controller	Variance
Controller		None	Wind
STA	0.1472	0.1566
ASTA	0.1484	0.1544
ASTAN	0.1458	0.146

ASTA: adaptive super-twisting control algorithm; ASTAN: ASTA with nonlinear sliding surface; STA: super-twisting algorithm.

We use the voltage formula and power measurement for each motor which are obtained from the work by Sumantri et al.²² including how to calculate power of each motor from control inputs $u_{1}$ , $u_{2}$ , $u_{3}$ , and $u_{4}$ . The DC motors are assumed to have identical resistance loads to estimate power consumption. Chattering of the control input causes corresponding chattering in the electric power of each motor.

The total energy consumed in each actuator is calculated by integrating its electric power during motion. The effect of chattering evaluation on energy consumption for five-times experiments in the conditions without/under wind disturbance is shown in Figure 16 and Table 5. It can be seen that the ASTAN is more effective to reduce energy consumption in both conditions. The ASTAN can reduce energy about 3% and 5% than with ASTA and STA without disturbance and also 1.4% and 3.3% under wind disturbance.

Figure 16.

Total energy consumed in actuators in five-times experiment.

Table 5.

Total energy comparison results in experiment.

Dist.	None	Wind
Cont.	E (mWh)	E (mWh)
STA	302.5	299.5
ASTA	297.0	293.4
ASTAN	288.3	289.8

ASTA: adaptive super-twisting control algorithm; ASTAN: ASTA with nonlinear sliding surface; STA: super-twisting algorithm.

In addition, we extend the evaluation in order to obtain information of energy consumption in six trials during operational flight, which is shown in Figures 17 and 18. Figure 17 describes average energy consumption for each motion in which the hovering motion consumed less energy than others. Figure 18 describes energy consumption for several experiment in each motion, and ASTAN has less energy consumption significantly for a case without disturbance. However, under wind disturbance, while there is no significant difference energy consumption, ASTAN has less energy consumption on average during hovering motion.

Figure 17.

Total energy consumed in actuators for each trajectory condition.

Figure 18.

Total energy consumed in actuators for six trajectory conditions ( $E_{Ti}$ : energy consumption in the $i th$ trajectory condition).

Conclusion

In this article, we design an ASTAN for trajectory tracking of a quadrotor system. Because the parameter adjustment of NSS is sensitive, adaptation mechanism is included based on the Lyapunov stability theory. The designed ASTAN has been verified by comparative simulation and experiments with conventional STA and ASTA. It maintains stability to external disturbances and provides smaller tracking errors. In experiments, the proposed ASTAN provides better performance compared to ASTA and STA in which the average mean of tracking error is reduced by 2% and 3.5% in both conditions (cm), and 1% and 32% without disturbance (°) and 1.5% and 17.3% under wind disturbance (°); the variance of tracking error is reduced by 1.5% and 5% without wind disturbance (cm) and 1.5% and 2.8% under wind disturbance (cm), and 1% and 30% without disturbance (°) and 1.5% and 17.5% under wind disturbance, and is also effective to reduce the energy consumption by 3% and 5% without wind disturbance and 1.4% and 3.3% under wind disturbance, respectively. In future work, we will include unmatched disturbance rejection and conduct outdoor experiment with the proposed control method.

Footnotes

Appendix 1

Handling Editor: Parvathy Rajendran

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Reesa Akbar

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