1 Introduction

Unmanned aerial vehicles (UAVs) have seen a boost in popularity and generated great interest for both military and civil applications [1,2,3]. Among various types of UAVs, quadcopters are adopted in many fields due to their maneuverability and simplicity [4–7]. However, the mobility and maneuverability of the quadcopter in standard configuration are limited since all propeller force vectors are limited in a single plane. Leading to only the cartesian position and the yaw angle can be controlled independently due to this under-actuated version [8]. With the increasing requirements in tasks, the abilities to interact with narrow and cluttered environments have attracted considerable attention, such as full-actuated, fault-tolerant and maneuverable version [9]. This indicates that exploring fully actuated quadcopter is preferred to greatly improve their performance in tasks [10].

Motivated by these requirements, some actuation strategies have constantly evolved in recent years. Considering an intuitive way by adding actuators in other directions, Hugo in [11] propose a new configuration with additional propellers in horizontal directions, where the translational and rotational movements can be controlled by two sets of rotors. Inspired by the tilt-wing in [12,11,14], Ryll first proposes a novel actuation concept in [15], where the propellers are allowed to tilt with respect to (w.r.t.) their axes. Due to the introduction of the tilting mechanism, the tilting quadcopter has the ability to generate arbitrary direction force or torque, which contributes to a fully-actuated version. And flight tests, although preliminary, clearly show its superior capability compared with the common quadcopter in [16]. A flight transition of the tilting quadcopter from \(0\) to \(90^\circ\) is then discussed since the maximum torque decreases as the pitch angle increasing [17]. To further improve the control bandwidth, a dual axis tilting quadcopter is proposed in [18] offering a wider range of control torques, which makes the vehicle be a more agile vision. Moreover, compared with the conventional quadcopter, the dual-axis version can still complete the task despite of half of actuators damaged [19, 20]. However, once a rotor failure of a common one, we will lose its control in yaw where the reliability and the survivability cannot be guaranteed [21]. As discussed, this tilting configuration can not only achieve full controllability with 6 DOF, but also fault-tolerant ability is dramatically improved [22].

Recently, the control design for the tilting quadcopter is receiving increased attention [23]. To address nonlinearities, an input-output linearization is applied to track arbitrary trajectories and desired orientations simultaneously in [24]. Considering external disturbances, a mixed sensitivity \(H_\infty\) optimal control is synthesized for the tilting quadcopter to compensate these effects [25]. The dynamics of tilting quadcopter with disturbances and uncertainties are studied in [26], where a second-order sliding mode is constructed to avoid the chattering phenomenon. In [27], the stability and PD control of tilting quadcopter are developed upon the failure of propeller during flight, where fully known dynamics are assumed. Moreover, a control scheme for the tilting quadcopter with H-configuration used for transporting unknown sling loads is proposed in [13]. Based on barrier Lyapunov functions, robust model reference adaptive control is employed to guarantee prior constraints on both tracking errors and control inputs despite of poor knowledge about vehicles and payloads. Various control designs have exploded over the years, such as backstepping control [28], sliding mode control [26], model predict control [29], robust control [30] and input output linearization [31]. Although the reliability and the flexibility are improved, the tilting mechanism makes the system into a complex nonlinear system. The additional unknown nonlinearities from the tilting mechanisms, aerodynamic damping and external environment should be fully considered in the control design to improve the robustness of the control scheme.

Adaptive control has been studied for decades with characterizing the ability of compensating uncertainties [32,30,34]. A well-known assumption in many existing literature is that uncertainties should follow the linear-in-parameter form or growth condition as supposed in [35,33,34,35,36,37,38,39,40,41,45]. However, this assumption is stringent in practice since many systems contain complex nonlinearities. To relax this condition and handle nonlinear uncertainties, function approximators are incorporated into the adaptive control, e.g., NNs [46–49] and fuzzy logic systems (FLSs) [50,47,52]. Gradient descent adaptive algorithm is developed in [32] to update the NNs weights, which may encounter a bursting phenomenon. To remedy this issue, several modified parameter estimations are proposed, e.g., e-modification [53] and \(\sigma\)-modification [54]. Due to additional damping terms in these algorithms, the closed-loop system can be guaranteed uniformly ultimately bounded [55]. However, most adaptive laws designed in [56,53,54,55,60] only concern about the overall system stability. Few considerations have been given to the estimated parameters convergence since the estimation errors is immeasurable or unknown generally. According to the certainty equivalence principle in [33], the control system performance can be greatly improved once the estimated parameters in adaptive laws converge to their optimal values. A well-recognized is noted that adaptive laws prefer containing some information on the parameter estimation error to guarantee its convergence. In [61], a novel composite-adaptation-based adaptive robust control method is proposed to achieve high tracking performance and accurate parameter estimation. This approach capturing the parameter error in the adaptation law can obtain better parameter estimation results than the traditional adaptive robust control. In [62], another parameter estimation scheme is proposed which features the ability to reconstruct the unknown parameters within finite time provided the PE condition being satisfied. And this idea is incorporated into the control design in [63] to obtain exponential convergence. Note that the above approaches need to online test the invertibility of a regressor-based matrix. And the designed auxiliary matrix and vector in [61,58,63] with unstable integrator inevitably increase all the time or even to infinite. Motivated by these requirements, to explore a control scheme, incorporated with a novel adaptive law with the ability of guaranteeing the system stability and the estimated parameters convergence, can pave the way for controlled systems with better tracking performance. Although these existing controllers are stable, the tracking errors converge to equilibrium points as time to infinity [64]. Then, an extension is paid to the fractal and fractional derivatives, where analysis and solution have been discussed for the linear and nonlinear fractional differential equations [65,62,63,64,65,66,71]. A finite-time stability theory has attracted considerable attention, which provides fast convergence, high-precision and better robustness to uncertainties [72]. Up to now, homogeneous [73], terminal sliding mode (TSM) [74] and nonsingular terminal sliding mode (NTSM) [75] are widely developed, where tracking errors converge to the equilibrium in finite-time. However, the existing TSM delivers a slower convergence when the system state is far away from the equilibrium, although its convergence rate lies in exponential one when near the equilibrium [76,73,78]. Thus, a fast finite-time theorem is proposed in [9, 79, 80], which can perform a fast convergence during the whole tracking process.

Motivated by the above discussion, we in this paper propose a novel tilting quadcopter conception with the translational and rotational movements controlled independently. To compensate for uncertainties in systems, an adaptive finite-time neural control is proposed, where a novel estimation error-based adaptive scheme is introduced. Compared to the existing work, the main contributions of this paper include:

  1. (1)

    A novel fully actuated tilting quadcopter conception is proposed, where the push mechanism can achieve large tilting range. Compared with [13, 15,14,15,16,17,20, 24,22,23,24,25,26,27,31, 81,78,83], we establish the tilting quadcopter dynamics simultaneously considering uncertainties. The nonlinearities in systems do not need to follow the linear-in-parameter for or growth condition as assumed in [35,33,34,35,36,37,38,39,40,41,45], which can pave the way for the design of more general conditions.

  2. (2)

    We propose a new auxiliary filter, where parameter estimation errors can be obtained explicitly and then be used as new leakage terms. By introducing these terms in the adaptive laws, compared with [56,53,54,55,60], high estimation performance can be achieved since weights of identifier NNs can converge to their optimal values in a bounded sense. Moreover, we provide a simple online approach to verify the parameters convergence, a relaxed alternative to the conventional PE-condition in [61,58,63].

  3. (3)

    An adaptive finite-time neural control is proposed for the tilting quadcopter to compensate for the uncertainties in systems. By incorporating the new adaptive law into the control design, the tracking errors can fast converge to a small neighborhood around the equilibrium within finite time as well as the estimated errors.

The rest of this paper is organized as follows. Section 2 presents the design of a novel tilting quadcopter and develops its complete dynamics with uncertainties. The main results are given in Sect. 3, where an ANC with fast finite-time convergence is proposed for tilting quadcopter to achieve tracking and precise estimation simultaneously. In Sect. 4, simulation results are presented. Finally, we draw a conclusion in Sect. 5.

Notations: Throughout this paper, let \({\mathbb {R}}\) denote the set of real numbers and \({\mathbb {R}}^{n\times m}\) denote a \(n\times m\)-dimensional matrix. Let \({K}_i=\text {diag}[k_{ij}]\in {\mathbb {R}}^{3\times 3}\) and \({\bar{K}}_i=\text {diag}[{\bar{k}}_{ij}]\in {\mathbb {R}}^{3\times 3}\) denote positive-definite matrices with \(i=1,2,\cdots ,6,\;j=1,2,3\). Let \(z^k=[|z_1|^k\text {sgn}(z_1),\cdots ,|z_n|^k\text {sgn}(z_n)]^{\text {T}}\), where \({z}=[z_1,\cdots ,z_n]^{\text {T}}\) and \(\text {sgn}(\cdot )\) is the sign function.

2 Dynamic modeling

In this section, we propose a novel conception of the tilting quadcopter, which consists of four main rigid parts: the Body, Servo, Push and Propeller as shown in Fig. 1. Servo and Push mechanisms can be actuated to tilt about the corresponding axes w.r.t. the main body, which suggests a fully actuated vehicle can be obtained. Compared with [18] and [19], the tilting mechanisms of this novel conception can be indeed arbitrary within \([-\frac{\pi }{2},\frac{\pi }{2}]\) to provide more agile and reliable abilities.

Fig. 1
figure 1

The tilting quadcopter vehicle

Full size image

2.1 Preliminary definitions

Let \(R_{W}:\{o_{w},x_{w},y_{w},z_{w}\}\) and \(R_{B}:\{o_{b},x_{b},y_{b},z_{b}\}\) denote the World reference frame and the Body frame. Let \(R_{S_{i}}:\{o_{s_{i}},x_{s_{i}},y_{s_{i}},z_{s_{i}}\}\) and \(R_{P_{i}}:\{o_{p_{i}},x_{p_{i}},y_{p_{i}},z_{p_{i}}\},i=1\cdots 4\) denote the Servo and Push frames associated with the i-th propeller as shown in Fig. 2.

Fig. 2
figure 2

The reference frame of 3-th propeller group

Full size image

In this paper, we select Euler angles to describe the attitude, and the rotation matrix \(R_{E2B}\) which represents the orientation of \(R_{W}\) w.r.t. \(R_{B}\) is given

$$\begin{aligned} {{R_{E2B}}{\mathrm{= }}\left[ {\begin{array}{*{20}{c}} {\mathrm{c}\theta \mathrm{c}\psi }&{}\quad {\mathrm{c}\theta \mathrm{s}\psi }&{}\quad { - \mathrm{s}\theta }\\ { - \mathrm{c}\phi \mathrm{s}\psi + \mathrm{s}\phi \mathrm{s}\theta \mathrm{c}\psi }&{}\quad {\mathrm{c}\phi \mathrm{c}\psi +\mathrm{s}\phi \mathrm{s}\theta \mathrm{s}\psi }&{}\quad {\mathrm{s}\phi \mathrm{c}\theta }\\ {\mathrm{s}\phi \mathrm{s}\psi + \mathrm{c}\phi \mathrm{s}\theta \mathrm{c}\psi }&{}\quad { - \mathrm{s}\phi \mathrm{c}\psi + \mathrm{c}\phi \mathrm{s}\theta \mathrm{s}\psi }&{}\quad {\mathrm{c}\phi \mathrm{c}\theta } \end{array}} \right] }, \end{aligned}$$
(1)

with the denotations \(\text {c}\cdot =\cos (\cdot )\) and \(\text {s}\cdot =\sin (\cdot )\) for clarity and conciseness. \(\phi\), \(\theta\) and \(\psi\) are roll, pitch and yaw, respectively. The rotational equations take the form

$$\begin{aligned} \left[ {\begin{array}{*{20}{l}} {{{\dot{\phi }}} }\\ {{{\dot{\theta }}} }\\ {{{\dot{\psi }}} } \end{array}} \right] {\mathrm{= }}\left[ {\begin{array}{*{20}{c}} 1&{}\quad {\mathrm{s}\phi \mathrm{t}\theta }&{}\quad {\mathrm{c}\phi \mathrm{t}\theta }\\ 0&{}\quad {\mathrm{c}\phi }&{}\quad { - \mathrm{s}\phi }\\ 0&{}\quad {\dfrac{{\mathrm{s}\phi }}{{\mathrm{c}\theta }}}&{}\quad {\dfrac{{\mathrm{c}\phi }}{{\mathrm{c}\theta }}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} p_v\\ q_v\\ r_v \end{array}} \right] = {R_1}^{-1}\omega _b, \end{aligned}$$
(2)

where \(\omega _b=[p_v,q_v,r_v]^{T}\) is the angle velocity projected into \(R_B\). Let \(\eta =[\phi ,\theta ,\psi ]^{T}\) be the attitude vector. From (2), we have \(R_1{\dot{\eta }}=\omega _b\) where the inverse of matrix \(R_1\) exists with the assumption of \(\theta \in (-\frac{\pi }{2},\frac{\pi }{2})\).

2.2 Dynamic modeling of tilting quadcopter

According to the Euler–Lagrange equations, the dynamics of the tilting quadcopter can be described as

$$\begin{aligned} J(\eta )\ddot{\eta }+C(\eta ,{\dot{\eta }}){\dot{\eta }}=\tau +\tau _f\nonumber \\ m_v\ddot{\xi }=R_{E2B}^\text {T}u+F_g+F_f, \end{aligned}$$
(3)

where \(J(\eta )\in {\mathbb {R}}^{3\times 3}\) is the effective inertial matrix, \(C(\eta ,{\dot{\eta }})\in {\mathbb {R}}^{3\times 3}\) is the Coriolis and Centrifugal force matrix, \(m_v\) is the mass of vehicle, \(\xi =[x_v,y_v,z_v]^{\text {T}}\) and \(F_g=[0,0,m_vg]^{\text {T}}\) denote the position and the gravity vector in \(R_W\); \(\tau _f=[-g_1|p_v|p_v,-g_2|q_v|q_v,-g_3|r_v|r_v]^{\text {T}}\) and \(F_f=[-d_1|{\dot{x}}_v|{\dot{x}}_v,-d_2|{\dot{y}}_v|{\dot{y}}_v,\) \(-d_3|{\dot{z}}_v|{\dot{z}}_v]^\text {T}\) are aerodynamic damping with positive coefficients \(g_i\) and \(d_i\), \(\tau \in {\mathbb {R}}^3\) and \(u\in {\mathbb {R}}^3\) are control inputs expressed in \(R_B\). From (2), we have \(J=R_1^TI_vR_1\) where \(I_v=\text {diag}(I_{xx},I_{yy},I_{zz})\) is the moment inertia of the whole vehicle. If readers are interested in \(J(\eta )\) and \(C(\eta ,{\dot{\eta }})\), more details can be found in Appendix A.

When outdoor flights, the aerodynamic damping may dramatically deteriorate the tracking performance. The coefficients like \(g_i\) and \(d_i\) in \(\tau _f\) and \(F_f\) are always obtained based on the specific operation conditions. However, these coefficients may not cover all the realistic environment which becomes more complex than the specific conditions. It is impossible to capture these explicit parameters to achieve a perfect compensation. However, such a novel tilting conception makes the vehicle decouple the translational and rotational movements. Note that the skeleton of tilting quadcopter is varied while mechanisms tilting. This means that inherent parameters like the inertia of moment \(I_v\) would be different from the normal value but bounded by \([{\underline{I}}_v,{\bar{I}}_v]\) with positive constants \({\underline{I}}_v\) and \({\bar{I}}_v\). Motivated by the above discussions, the dynamics of tilting quadcopter is inevitably subjected to parametric uncertainties. For our control design, the following properties of the dynamics (3) are proposed.

Property 1 (see [36]): There exist positive constants \({\underline{j}}\) and \({\bar{j}}\) such that for any attitude vector \(\eta\) and vector \(\vartheta =[\vartheta _1,\vartheta _2,\vartheta _3]^T\) with \(\vartheta _i,(i=1,2,3)\) being the real constants:

  1. 1.

    The effective inertial matrix \(J(\eta )\) is symmetric, positive definite, and is bounded from below and above, i.e., \({\underline{j}}\le \Vert J(\eta )\Vert \le {\bar{j}}\).

  2. 2.

    The matrix \({\dot{J}}(\eta )-2C(\eta ,{\dot{\eta }})\) is skew-symmetric such that \(\vartheta ^\text {T}({\dot{J}}(\eta )-2C(\eta ,{\dot{\eta }}))\vartheta\) \(=0\)

3 Adaptive finite time neural control

3.1 Preliminaries

Lemma 1

(see [84]): Suppose a continuous positive definite function \(V(\zeta )\), and scalars \(a>0\), \(b>0\), \(0<\beta <1\) and \(\varrho >0\) such that

$$\begin{aligned} {\dot{V}}(\zeta )\le -aV(\zeta )-bV^{\beta }(\zeta )+\varrho . \end{aligned}$$
(4)

Then for any given time \(t_0\), the function \(V(\zeta )\) is bounded by a small region \(\varOmega _\zeta =\{\zeta |V^\beta (\zeta )\le \frac{\varrho }{b(1-c)}\}\) with \(0<c<1\) and the settling time \(t_s\)

$$\begin{aligned} t_s\le t_0+\frac{1}{a(1-\beta )}\ln \left( \frac{aV^{1-\beta }(\zeta _0)+bc}{bc}\right) . \end{aligned}$$
(5)

where \(\zeta _0\) and \(t_0\) are initial values of \(\zeta\) and initial time, respectively.

Lemma 2

(see [85]): For \(x_i\in {\mathbb {R}},\,i=1,2,\cdots ,q,\,0<k\le 1\), the following inequalities hold

$$\begin{aligned} \left( \sum _{i=1}^q|x_i|\right) ^{k} \le \sum _{i=1}^q|x_i|^{k}\le q^{1-k}\left( \sum _{i=1}^q|x_i|\right) ^{k}. \end{aligned}$$
(6)

Lemma 3

(see [86]): The Radial Basis Function (RBF) networks are employed to emulate any continuous function \(g(X): {\mathbb {R}}^{r}\rightarrow {\mathbb {R}}\):

$$\begin{aligned} g_{nn}(X)=W^\text {T}S(X), \end{aligned}$$
(7)

where the input vector \(X\in {\mathbb {R}}^{r}\), the weight vector \(W\in {\mathbb {R}}^m\), the weight number \(m>1\) and the basis function vector \(S(X)=[s_1(X),s_2(X),\cdots ,s_m(X)]^{\text {T}}\) with

$$\begin{aligned} s_i(X)=\text {exp}\left[ \frac{-(X-\mu _i)^\text {T}(X-\mu _i)}{\sigma _i^2} \right] , \end{aligned}$$
(8)

where \(\mu _i=[\mu _{i1},\mu _{i2},\cdots ,\mu _{ir}]^{\text {T}}\) is the center of the receptive field and \(\sigma _i\) is the width of the Gaussian function. RBF NNs can approximate any smooth function over a compact set with convergent errors

$$\begin{aligned} g(X)=W^{*\text {T}}S(X)+\epsilon , \end{aligned}$$
(9)

where \(W^*\) is the ideal weight vector, and \(\epsilon\) is the convergent error, satisfying \(|\epsilon |\le \epsilon _n\) with a positive constant \(\epsilon _n\).

Lemma 4

(see [87]): For any real variables \(y_1\), \(y_2\), and any positive constants c and d, the following inequality holds

$$\begin{aligned} |y_1|^c|y_2|^d\le \frac{c}{c+d}|y_1|^{c+d}+\frac{d}{c+d}|y_2|^{c+d}. \end{aligned}$$
(10)

3.2 Estimation error-based adaptive scheme design

Following the function approximation given in (9), we can always develop our system into a form as:

$$\begin{aligned} {\dot{y}}={W}^{\text {T}}{S}+{\epsilon }, \end{aligned}$$
(11)

where \(y\in {\mathbb {R}}^{3\times 1}\) can be system states; \({W}=\left[{\bar{W}}_1,{\bar{W}}_2,{\bar{W}}_3\right]\in {\mathbb {R}}^{h\times 3}\) and \({S}\in {\mathbb {R}}^{h\times 1}\) are augmented weight matrix and regressor, respectively; \({\epsilon }=[{\epsilon }_1,{\epsilon }_2,{\epsilon }_3]^{\text {T}}\in {\mathbb {R}}^{3\times 1}\) is the residual error. Note that \({\dot{y}}\) always represents acceleration information which may be sensitive to noises. A stable filter is thus adopted

$$\begin{aligned} \begin{array}{ll} l{\dot{y}}_f=-y_f+y, &{} y_f(0)=0,\\ l\dot{{S}}_f=-{S}_f+{S}, &{} {S}_f(0)=0,\\ l\dot{{\epsilon }}_f=-{\epsilon }_f+{\epsilon }, &{} \epsilon _f(0)=0, \end{array} \end{aligned}$$
(12)

where \(y_f\), \({S}_f\) and \({\epsilon }_f\) are the filtered variables, l is a positive constant. We then introduce the following auxiliary matrices P and vector Q

$$\begin{aligned}&{\dot{P}}=-\delta P+{S}_f{S}_f^{\text {T}}, \end{aligned}$$
(13)
$$\begin{aligned}&{\dot{Q}}_{i}=-\delta Q_{i}+{S}_f\left( \frac{y_i-y_{fi}}{l}\right) ^{\text {T}},\,i=1,2,3, \end{aligned}$$
(14)

where \(\delta\) is any positive constant. Based on matrices P and \(Q_i\), a new leakage term \(L_i\) can be constructed as

$$\begin{aligned} L_i=P\hat{{\bar{W}}}_{i}-Q_{i}, \end{aligned}$$
(15)

where \(\hat{{\bar{W}}}_i\) is the estimated vector of \({\bar{W}}_i\). One can obtain a solution of the differential Eq. (13) as,

$$\begin{aligned}&P=\int _0^te^{-\delta (t-r)}{S}_f{S}_f^{\text {T}}\text {dr} \end{aligned}$$
(16)
$$\begin{aligned}&Q_{i}=\int _0^te^{-\delta (t-r)}{S}_f\left( \frac{y_i-y_{fi}}{l}\right) ^{\text {T}}\text {dr}, \end{aligned}$$
(17)

From (15) and (16), \(L_i\) can be rewritten as

$$\begin{aligned} L_i=P\hat{{\bar{W}}}_{i}-Q_{i}=P\hat{{\bar{W}}}_i-P{\bar{W}}_i+C_i=-P\tilde{{\bar{W}}}_i+C_i, \end{aligned}$$
(18)

where \(C_i=-\int _0^te^{-\delta (t-r)}{S}_f{\epsilon }_{fi}^{\text {T}}\text {dr}\) and \(\tilde{{\bar{W}}}_i={\bar{W}}_i-\hat{{\bar{W}}}_i\) is the estimation error. Since the NNs error \({\epsilon }\) and the regressor S are both bounded, then the filtered terms \({\epsilon }_{fi}\) and \({S}_f\) are bounded. Thus, the term \(C_i\) is bounded by \(\epsilon _{ci}\).

As the introduction of the leakage term \(L_i\), the unknown estimation error \(\tilde{{\bar{W}}}_i\) can be abstracted, which can be used to drive the adaptive law to improve the estimation convergence. And we first give the following lemma.

Lemma 1

If the regressor vector S satisfies the persistently excited (PE) condition, then the matrix \(P_f\) in (13) is positive-definite, i.e., its minimum eigenvalue fulfills \(\lambda _{\min }(P_f)>\delta _{P_f}>0\) for a positive constant \(\delta _{P_f}\). Moreover, if the matrix P is positive-definite, then S is PE.

Proof

The first part is omitted here for clarity and conciseness, and we provide the proof for the second part in Appendix B. \(\square\)

Remark 1

Lemma 1 indicates that the required excitation \(\lambda _{\min }(P_f)>\delta _{P_f}>0\) can be obtained under the standard PE condition. According to [33], PE condition is necessary for the parameter estimation convergence. However, it is difficult to test the PE condition on-line. Lemma 1 provides an intuitive way to validate the PE condition, i.e., whether the minimum eigenvalue of \(\lambda _{\min }(P_f)>0\) holds. Namely, we provide a simple online approach to verify the parameters convergence, a relaxed alternative to the conventional PE-condition in [61,58,63].

Remark 2

The gradient-based adaptive law in [88] has not fully considered the convergence of parameter estimation. Although the e-modified adaptive law in [50] can eliminate the potential bursting phenomenon incurred by the gradient scheme, it only contains the estimated parameter \(\hat{{\bar{W}}}_i\). And this scheme cannot achieve the estimated parameters converging to their optimal values. It should be noted that if we adopt the leakage term \(L_i\) in the adaptive law, with the help of the estimation error \(\tilde{{\bar{W}}}_i\), the estimated parameter \(\hat{{\bar{W}}}_i\) characters the ability to converge to a small neighborhood around its optimal value. The performance of the overall control system can be dramatically improved. Please refer to the following control design for more details.

3.3 Rotational subsystem control

This section contributes to develop an estimation error-based AFTNC with for the rotational subsystem of (3). Let \(\eta _d\) be the desired attitude and we define \(z_1=\eta -\eta _d\) as attitude errors. Then, a virtual control \(\alpha _1\) is given as

$$\begin{aligned} \alpha _1={\dot{\eta }}_d-K_1z_1-K_2z_1^{2\beta _1-1}, \end{aligned}$$
(19)

with \(0<\beta _1<1\). We then introduce an integral term

$$\begin{aligned} \chi _1=\int _0^tz_2(s)\text {d}s, \end{aligned}$$
(20)

where \(z_2={\dot{\eta }}-\alpha _1\). This integral term can dramatically reduce the sensitivity to parametric uncertainties [89]. Differentiating \(z_1\), we have

$$\begin{aligned} {\dot{z}}_1&={\dot{\eta }}-{\dot{\eta }}_d=z_2+\alpha _1-{\dot{\eta }}_d. \end{aligned}$$
(21)

Inspired by the backstepping idea, the convergence of \(z_1\) is completely depended on the convergence of \(z_2\). This recursive methodology provides systematic steps on designing the control input to guarantee \(z_2\) to converge. Differentiating \(z_2\) w.r.t. time leads to

$$\begin{aligned} J{\dot{z}}_2+Cz_2&=J\ddot{\eta }-J{\dot{\alpha }}_1+Cz_2\nonumber \\&=\tau +\tau _f-C\alpha _1-J{\dot{\alpha }}_1, \end{aligned}$$
(22)

where J and C are the abbreviations of \(J(\eta )\) and \(C(\eta ,{\dot{\eta }})\). Then the above equation can be rewritten as

$$\begin{aligned} {\dot{F}}_{11}(z)+F_{12}(z)=\tau +F_{13}(z), \end{aligned}$$
(23)

where \(F_{11}=Jz_2\), \(F_{12}=Cz_2-{\dot{J}}z_2\), \(F_{13}=\tau _f-C\alpha _1-J{\dot{\alpha }}_1\) and \(z=\left[\eta ^{\text {T}},{\dot{\eta }}^{\text {T}},\alpha _1^{\text {T}},{\dot{\alpha }}_1^{\text {T}}\right]^{\text {T}}\). Thus, the control input \(\tau\), which can be used to develop an estimation error-based adaptive scheme, can indirectly reflect all uncertainties in system. According to Lemma 3, we apply NNs to approximate the unknown dynamics functions \(F_{11}\), \(F_{12}\) and \(F_{13}\)

$$\begin{aligned}&F_{11}(z)=W_{11}^{*\text {T}}S_{11}+\epsilon _{11},\nonumber \\&F_{12}(z)=W_{12}^{*\text {T}}S_{12}+\epsilon _{12},\nonumber \\&F_{13}(z)=W_{13}^{*\text {T}}S_{13}+\epsilon _{13}, \end{aligned}$$
(24)

where \(W_{1i}^{*}\in {\mathbb {R}}^{m_1\times 3}\) is the optimal weight matrix; \(S_{1i}(z)\in {\mathbb {R}}^{m_1\times 1}\) is the corresponding regressor, \(\epsilon _{1i}\in {\mathbb {R}}^{3\times 1}\) is the estimation error bounded by \(\Vert \epsilon _{1i}\Vert \le \epsilon _{1in}\) with positive constants \(\epsilon _{1in}\), \(m_1\) is the number of NNs node. We can rewrite (23) as

$$\begin{aligned} W_1^{*\text {T}}S_1+{\bar{\epsilon }}_1=\tau , \end{aligned}$$
(25)

where \(W_1^*\in {\mathbb {R}}^{3m_1\times 3}\), \(S_1=\left[{\dot{S}}_{11}^{\text {T}},S_{12}^{\text {T}},-S_{13}^{\text {T}}\right]^{\text {T}}\in {\mathbb {R}}^{3m_1\times 1}\) and \({\bar{\epsilon }}_1={\dot{\epsilon }}_{11}+\epsilon _{12}-\epsilon _{13}\) with \(\Vert {\bar{\epsilon }}_1\Vert \le {\bar{\epsilon }}_{1n}\) where \({\bar{\epsilon }}_{1n}\) is a positive constant. We define the form \(W_1^*\) as

$$\begin{aligned} W_1^*=\left[ \begin{array}{c} W_{11}^* \\ W_{12}^* \\ W_{13}^* \\ \end{array} \right] =\left[ W_{c1}^{*},W_{c2}^{*},W_{c3}^{*} \right] , \end{aligned}$$
(26)

where \(W_{ci}^{*}\in {\mathbb {R}}^{3m_1\times 1}\). According to the estimation error-based adaptive scheme, we first construct the following stable filters

$$\begin{aligned} \begin{array}{ll} l_1{\dot{\tau }}_{fi}=-\tau _{fi}+\tau _i,&{}\tau _{fi}(0)=0,\\ l_1{\dot{S}}_{1f}=-S_{1f}+S_1,&{}S_{1f}(0)=0,\\ l_1\dot{{\bar{\epsilon }}}_{1fi}=-{\bar{\epsilon }}_{1fi}+{\bar{\epsilon }}_{1i},&{}{\bar{\epsilon }}_{1fi}(0)=0,\\ \end{array} \end{aligned}$$
(27)

where \(\tau _{fi}\), \(S_{1f}\) and \({\bar{\epsilon }}_{1fi}\) are the filtered states. Then the following auxiliary matrices are established

$$\begin{aligned}&{\dot{P}}_1=-\delta _1P_1+S_{1f}S_{1f}^{\text {T}},\nonumber \\&{\dot{Q}}_{1i}=-\delta _1Q_{1i}+S_{1f}\tau _{fi},\nonumber \\&{\mathcal {L}}_{1i}=P_1{\hat{W}}_{ci}-Q_{1i}, \end{aligned}$$
(28)

where \(\delta _1\) is a positive constant and \({\hat{W}}_{ci}\) are the estimated vector of \({W}_{ci}\). From (18), we have

$$\begin{aligned} {\mathcal {L}}_{1i}=-P_1{\tilde{W}}_{ci}+C_{1i}, \end{aligned}$$
(29)

where \(C_{1i}=-\int _0^te^{-\delta _1(t-r)}S_{1f}{\bar{\epsilon }}_{1fi}\text {dr}\) with \(\Vert C_{1i}\Vert \le {\bar{\epsilon }}_{1ci}\), \({\bar{\epsilon }}_{1ci}\) is a positive constant and \({\tilde{W}}_{ci}=W_{ci}^*-{\hat{W}}_{ci}\) is the estimated error. Considering the dynamics (21) and (22), we propose the following estimation error-based AFTNC

$$\begin{aligned}&\tau =\tau _{c1}+\tau _{c2}+\tau _{c3},\nonumber \\&\tau _{c1}=-K_3z_2-K_5z_2\chi _1^{\text {T}}\chi _1,\nonumber \\&\tau _{c2}=-K_4z_2^{2\beta _1-1}-K_6z_2\chi _1^{\text {T}}\chi _1^{2\beta _1-1},\nonumber \\&\tau _{c3}=-z_1-\chi _1-{\hat{W}}_{1}^{\text {T}}{\bar{S}}_1,\nonumber \\&\dot{{\hat{W}}}_{ci}=\varGamma _{1i}\left( {\bar{S}}_1 z_{2i}-\gamma _{1i}\frac{P_1^{\text {T}}{\mathcal {L}}_i}{\Vert {\mathcal {L}}_i\Vert }\right) ,\,i=1,2,3, \end{aligned}$$
(30)

where \({\bar{S}}_1=[0,0,S_{13}^{\text {T}}]^{\text {T}}\in {\mathbb {R}}^{3m_1\times 1}\) in which \(S_{13}\in {\mathbb {R}}^{m_1\times 1}\), \(\gamma _{1i}\) is a positive constant and \(\varGamma _{1i}\) is positive-definite matrix.

Theorem 1

Consider systems (21) and (22) with AFTNC and learning algorithm (30). If matrix \(P_1\) is a positive-definite matrix, it can be derived that tracking errors \(z_1\), \(z_2\)and estimated error \({\tilde{W}}_{ci}\) converge to a small neighborhood around the origin in finite-time. Moreover, the estimated weight \({\hat{W}}_{ci}\) converges to the region around its optimal values, simultaneously.

Proof

We first adopt the following Lyapunov function candidate

$$\begin{aligned} V_1=\frac{1}{2}z_1^Tz_1. \end{aligned}$$
(31)

Taking its derivative, we have

$$\begin{aligned} {\dot{V}}_1&= z_1^Tz_2-z_1^TK_1z_1-z_1^TK_2z_1^{2\beta _1-1}\nonumber \\&\le z_1^{\text {T}}z_2-{\underline{k}}_1\sum _{i=1}^3z_{1i}^2-{\underline{k}}_2\sum _{i=1}^3\left( z_{1i}^2\right) ^{\beta _1}, \end{aligned}$$
(32)

where \({\underline{k}}_1=\text {min}(k_{1j_1}),\;{\underline{k}}_2=\text {min}\left( k_{2j_1}\right) ,\;j_1=1,2,3\). From the above inequality, the tracking error \(z_1\) can achieve finite-time stability once \(z_2\) converges. Then we consider a Lyapunov function

$$\begin{aligned} V_2=V_1+\frac{1}{2}\chi _1^{\text {T}}\chi _1+\frac{1}{2}z_2^{\text {T}}Jz_2+\frac{1}{2}\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}}\varGamma _{1i}^{-1}{\tilde{W}}_{ci}. \end{aligned}$$
(33)

Differentiating \(V_2\) along (22) yields

$$\begin{aligned} {\dot{V}}_2&=z_1^{\text {T}}z_2-z_1^TK_1z_1-z_1^TK_2z_1^{2\beta _1-1}\nonumber \\&\quad +\chi _1^{\text {T}}z_2+\frac{1}{2}z_2^{\text {T}}{\dot{J}}z_2+z_2^{\text {T}}\Big (\tau \nonumber \\&\quad +\tau _f-C{\dot{\eta }}-J{\dot{\alpha }}_1\Big )-\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}}\varGamma _{1i}^{-1}\dot{{\hat{W}}}_{ci}\nonumber \\&=z_1^{\text {T}}z_2-z_1^TK_1z_1-z_1^TK_2z_1^{2\beta _1-1}\nonumber \\&\quad +\chi _1^{\text {T}}z_2+\frac{1}{2}z_2^{\text {T}}\left( {\dot{J}}-2C\right) z_2\nonumber \\&\quad +z_2^{\text {T}}(\tau +F_{13})-\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}}\varGamma _{1i}^{-1}\dot{{\hat{W}}}_{ci}. \end{aligned}$$
(34)

From the Property (2), the matrix \({\dot{J}}-2C\) is skew-symmetric where \(z_2^{\text {T}}({\dot{J}}-2C)z_2=0\) holds. Substituting (24) and (30) into (34), we have

$$\begin{aligned} {\dot{V}}_2&=z_1^{\text {T}}z_2-z_1^TK_1z_1-z_1^TK_2z_1^{2\beta _1-1}\nonumber \\&\quad +\chi _1^{\text {T}}z_2+z_2^{\text {T}}\Big (\tau +W_{13}^{*\text {T}}S_{13}\nonumber \\&\quad +{\epsilon }_{13}\Big )-\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}} \left( {\bar{S}}_1 z_{2i}-\gamma _{1i}\frac{P_1^{\text {T}}{\mathcal {L}}_i}{\Vert {\mathcal {L}}_i\Vert }\right) \nonumber \\&=-z_1^TK_1z_1-z_2^\text {T}K_3z_2-z_2^\text {T}K_5z_2\chi _1^{\text {T}}\chi _1\nonumber \\&\quad -z_1^TK_2z_1^{2\beta _1-1}-z_2^\text {T}K_4z_2^{2\beta _1-1}\nonumber \\&\quad -z_2^\text {T}K_6z_2\chi _1^{\text {T}}\chi _1^{2\beta _1-1}+z_2^{\text {T}}\left( {\tilde{W}}_{13}^{\text {T}}S_{13}+{\epsilon }_{13}\right) -\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}}{\bar{S}}_1 z_{2i}\nonumber \\&\quad +\sum _{i=1}^3\gamma _{1i}{\tilde{W}}_{ci}^{\text {T}}\frac{P_1^{\text {T}}{\mathcal {L}}_i}{\Vert {\mathcal {L}}_i\Vert }\nonumber \\&=-z_1^TK_1z_1-z_2^\text {T}K_3z_2-z_2^\text {T}K_5z_2\chi _1^{\text {T}}\chi _1\nonumber \\&\quad -z_1^TK_2z_1^{2\beta _1-1}-z_2^\text {T}K_4z_2^{2\beta _1-1}\nonumber \\&\quad -z_2^\text {T}K_6z_2\chi _1^{\text {T}}\chi _1^{2\beta _1-1}+z_2^{\text {T}}\epsilon _{13}-\sum _{i=1}^3\gamma _{1i}\frac{{\tilde{W}}_{ci}^{\text {T}}P_1^{\text {T}}P_1{\tilde{W}}_{ci}}{\Vert {\mathcal {L}}_i\Vert }\nonumber \\&\quad +\sum _{i=1}^3\gamma _{1i}\frac{{\tilde{W}}_{ci}^{\text {T}}P_1^{\text {T}}C_{1i}}{\Vert {\mathcal {L}}_i\Vert }. \end{aligned}$$
(35)

Note that \(P_1\) is a symmetrical positive-definite matrix such that \(\frac{{\tilde{W}}_{ci}^{\text {T}}P_1^{\text {T}}P_1{\tilde{W}}_{ci}}{\Vert {\mathcal {L}}_i\Vert }\ge \lambda _1{\tilde{W}}_{ci}^{\text {T}}{\tilde{W}}_{ci}\) with \(\lambda _1>0\) being the minimum eigenvalue of matrix \(\frac{P_1^{\text {T}}P_1}{\Vert {\mathcal {L}}_i\Vert }\). And \(\sum _{i=1}^3\gamma _{1i}\frac{{\tilde{W}}_{ci}^{\text {T}}P_1^{\text {T}}C_{1i}}{\Vert {\mathcal {L}}_i\Vert }\) is bounded by a positive constant \(\epsilon _{1c}\). Since \(K_5\) and \(K_6\) are both positive-definite matrices, we can always find positive constants \({\underline{k}}_5\) and \({\underline{k}}_6\) such that \(z_2^\text {T}K_5z_2\ge {\underline{k}}_5,\) and \(z_2^\text {T}K_6z_2\ge {\underline{k}}_6,\;\forall z_2\ne 0\). Then, we have

$$\begin{aligned}&{\dot{V}}_2 \le -{\underline{k}}_1\sum _{i=1}^3z_{1i}^2 -{\underline{k}}_2\sum _{i=1}^3\left( z_{1i}^2\right) ^{\beta _1}\nonumber \\&\qquad - \left( {\underline{k}}_3-\frac{1}{2}\right) \sum _{i=1}^3z_{2i}^2-{\underline{k}}_4\sum _{i=1}^3\left( z_{2i}^2\right) ^{\beta _1}\nonumber \\&\qquad -{\underline{k}}_5\sum _{i=1}^3\chi _{1i}^2-{\underline{k}}_6 \sum _{i=1}^3\left( \chi _{1i}^2\right) ^{\beta _1}\nonumber \\&\qquad -\lambda _1\sum _{i=1}^{3}\sum _{j=1}^{3m_1}\gamma _{1i}{\tilde{W}}_{cij}^{2}+\frac{1}{2}\epsilon _{13n}^2+\epsilon _{1c}, \end{aligned}$$
(36)

where \({\underline{k}}_3=\text {min}(k_{3j_1}),\;{\underline{k}}_4=\text {min}(k_{4j_1}),j_1=1,2,3\), and a suitable value of \({\underline{k}}_3\) with \({\underline{k}}_3>\frac{1}{2}\). For term \(-\sum _{j=1}^{3m_1}\gamma _{1i}{\tilde{W}}_{cij}^{2}\) according to Lemma 4 yields

$$\begin{aligned}&-\gamma _{1i}\sum _{j=1}^{3m_1}{\tilde{W}}_{cij}^{2}&\nonumber \\&\quad =-\gamma _{1i}\sum _{j=1}^{3m_1}{\tilde{W}}_{cij}^{2}-\gamma _{1i}\sum _{j=1}^{3m_1}\left( {\tilde{W}}_{cij}^{2}\right) ^{\beta _1}+\gamma _{1i}\sum _{j=1}^{3m_1}\left( {\tilde{W}}_{cij}^{2}\right) ^{\beta _1}\nonumber \\&\quad \le -\gamma _{1i}\sum _{j=1}^{3m_1}{\tilde{W}}_{cij}^{2}-\gamma _{1i}\sum _{j=1}^{3m_1}\left( {\tilde{W}}_{cij}^{2}\right) ^{\beta _1}\gamma _{1i}\sum _{j=1}^{3m_1}\left( \beta _1{\tilde{W}}_{cij}^{2}+\left( 1-\beta _1\right) \right) \nonumber \\&\quad \le -\!\gamma _{1i}\sum _{j=1}^{3m_1}\left( 1\!-\!\beta _1\right) {\tilde{W}}_{cij}^{2}\!-\!\gamma _{1i}\sum _{j=1}^{3m_1}\left( {\tilde{W}}_{cij}^{2}\right) ^{\beta _1}\!+\!\sum _{j=1}^{3m_1}\gamma _{1i}\left( 1\!-\!\beta _1\right) . \end{aligned}$$
(37)

Substituting (37) into (36), we can get

$$\begin{aligned} {\dot{V}}_2&\le -{\underline{k}}_1\sum _{i=1}^3z_{1i}^2 -\left( {\underline{k}}_3-\frac{1}{2}\right) \sum _{i=1}^3z_{2i}^2-{\underline{k}}_5\sum _{i=1}^3\chi _{1i}^2-{k}_7\sum _{i=1}^3\sum _{j=1}^{3m_1}{\tilde{W}}_{cij}^{2}\nonumber \\&\quad -{\underline{k}}_2\left( \sum _{i=1}^3z_{1i}^2\right) ^{\beta _1} -{\underline{k}}_4\left( \sum _{i=1}^3z_{2i}^2\right) ^{\beta _1}-{\underline{k}}_6 \left( \sum _{i=1}^3\chi _{1i}^2\right) ^{\beta _1}+\epsilon _{1c}\nonumber \\&\quad -{k}_8\sum _{i=1}^3\left( \sum _{j=1}^{3m_1}{\tilde{W}}_{cij}^{2}\right) ^{\beta _1} +\lambda _1\sum _{i=1}^3\sum _{j=1}^{3m_1}\gamma _{1i}\left( 1-\beta _1\right) +\frac{1}{2}\epsilon _{13n}^2, \end{aligned}$$
(38)

where \({k}_7=\text {min}\left( \lambda _1\gamma _{1i}(1-\beta _1)\right)\) and \({k}_8=\text {min}(\lambda _1\gamma _{1i})\). The effective inertial matrix J satisfies Property (1), and thus we have,

$$\begin{aligned}&-\sum _{i=1}^3z_{2i}^2=-\frac{1}{{\bar{j}}}\sum _{i=1}^3{\bar{j}}z_{2i}^2\le -\frac{1}{{\bar{j}}}z_2^{\text {T}}Jz_2\nonumber \\&-{k}_7\sum _{j=1}^{3m_1}{\tilde{W}}_{cij}^{2}=-\frac{{k}_7}{\lambda _2}\sum _{j=1}^{3m_1}{\lambda _2}{\tilde{W}}_{cij}^{2}\le {\underline{k}}_7{\tilde{W}}_{ci}^{\text {T}}\varGamma _{1i}^{-1}{\tilde{W}}_{ci}, \end{aligned}$$
(39)

where \(\lambda _2\) is the minimum eigenvalue of positive-definite matrices \(\varGamma _{1i}\). Substituting (39) into (38) and applying Lemma 2 yields

$$\begin{aligned} {\dot{V}}_2&\le -a_1\left( \frac{1}{2}\sum _{i=1}^3z_{1i}^2+\frac{1}{2}z_{2}^{\text {T}}Jz_2\right. \nonumber \\&\quad\left. +\frac{1}{2}\sum _{i=1}^3\chi _{1i}^2+\frac{1}{2}\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}}\varGamma _i^{-1}{\tilde{W}}_{1i}\right) \nonumber \\&\quad -b_1\left( \frac{1}{2}\sum _{i=1}^3z_{1i}^2+\frac{1}{2}z_{2}^{\text {T}}Jz_2 +\frac{1}{2}\sum _{i=1}^3\chi _{1i}^2\right. \nonumber \\&\quad \left. +\frac{1}{2}\sum _{i=1}^3{\tilde{W}}_{ci}^{\text {T}}\varGamma _i^{-1}{\tilde{W}}_{ci}\right) ^{\beta _1}\nonumber \\&\quad +\frac{1}{2}\epsilon _{13n}^2+\epsilon _{1c}+\lambda _1\sum _{i=1}^3\sum _{j=1}^{3m_1}\gamma _{1i}(1-\beta _1), \end{aligned}$$
(40)

where \(a_1=\text {min}\left( 2{\underline{k}}_1,\frac{2{\underline{k}}_3-1}{{\bar{j}}},2{\underline{k}}_5,2{\underline{k}}_7\right)\), \(b_1=\text {min}\left( 2^{\beta _1}{\underline{k}}_2,2^{\beta _1}{\underline{k}}_4,2^{\beta _1}{\underline{k}}_6,2^{\beta _1}{\underline{k}}_8\right)\) where \({\underline{k}}_8=\frac{k_8}{\lambda _2^{\beta _1}}\). Referring to Lemma 1 yields,

$$\begin{aligned} {\dot{V}}_2\le -a_1V_2-b_1V_2^{\beta _1}+\varrho _1, \end{aligned}$$
(41)

where \(\varrho _1=\frac{1}{2}\epsilon _{13n}^2+\epsilon _{1c}+\lambda _1\sum _{i=1}^3\sum _{j=1}^{3m_1}\gamma _{1i}(1-\beta _1)\). According to the Lemma 1, the function \(V_2\) is bounded by a small region \(\varOmega =\left\{ V_2^\beta \le \frac{\varrho _1}{b_1(1-c_1)}\right\}\) with \(0<c_1<1\). This suggests that \(z_1\), \(z_2\) and \({\tilde{W}}_{ci}\) converge to a small neighborhood around the equilibrium within finite-time and this completes the proof. Note that \(\varrho _1=\frac{1}{2}\epsilon _{13n}^2+\epsilon _{1c}+\lambda _1\sum _{i=1}^3\sum _{j=1}^{3m_1}\gamma _{1i}(1-\beta _1)\) where \(\gamma _{1i}\) and \(\beta _1\) are control scheme parameters, \(\epsilon _{13n}\) represents the boundary of residual error, and \(\epsilon _{1c}\) is related to \(C_{1i}\) as described in (35). If the number of NN nodes is sufficiently large, the residual error \(\epsilon _{1i},(i=1,2,3)\) can be zero in the ideal case, which means \(\epsilon _{13n}=0\) holds. Due to the stable filters in (27) and the leakage term \(L_i\) in (29), we can also derive \(C_{1i}\) to be zero in this case and lead to \(\epsilon _{1c}=0\). As the above condition, the item \(\varrho _1\) related to the ultimate convergence region \(\varOmega\) becomes \(\varrho _1=\lambda _1\sum _{i=1}^3\sum _{j=1}^{3m_1}\gamma _{1i}(1-\beta _1)\). If we appropriately select the control parameters, the final region can be arbitrary small. Moreover, although the number of NN nodes cannot be infinity, \(\epsilon _{13n}\) and \(\epsilon _{1c}\) are only small residual errors, and little effect on the final region \(\varOmega\). \(\square\)

Remark 3

For the existing gradient law or the modified adaptive law used in [88] and [50], the adaptive laws are established to guarantee the tracking errors convergence. However, the convergence of the estimated parameters to their optimal values may be difficult due to the uncertainties and damping effect in adaptive laws. This paper proposes a novel way to guarantee this convergence by introducing a new leakage item \(L_i\) in the adaptive law, where the estimation errors can be obtained explicitly to achieve the convergence of the parameter estimation.

Remark 4

The adaptive scheme (30) contains the estimated error \(\tilde{{\bar{W}}}_i\) in the new leakage term \(L_i\), which contributes to the convergence of the estimated parameter to its optimal value. In order to clearly illustrate the superiority of the adaptive law in (30), we adopt the e-modified adaptive law design scheme and let the adaptive law be \(\dot{{\hat{W}}}_{ci}=\varGamma _{1i}({\bar{S}}_1 z_{2i}-\gamma _{1i}{\hat{W}}_{ci}),(i=1,2,3)\). Although finite-time stability can be achieved, due to the introduction of the damping term, the ultimate convergence region \(\varOmega\) depends not only on the residual error \(\epsilon _{1i}\), but also the optimal value \(W_{ci}^*\) of the unknown NN weights. For its ultimate convergence region proof, we omit here for clarity and conciseness. As discussed above, the ultimate region under e-modified adaptive law is much larger than \(\varOmega\) under the estimation error-based adaptive law (30). Thus the estimated parameters of e-modified cannot achieve converging to their optimal values. In other words, the estimation error-based adaptive scheme (30) can get better results in estimating, and the overall control system can be greatly improved.

3.4 Translational subsystem control

Incorporated with the estimation error-based adaptive scheme, an ANC is then developed for the translational subsystem. We first define \(e_1=\xi -\xi _d\) where \(\xi _d\) is the desired position. A virtual control input is developed as,

$$\begin{aligned} \alpha _2={\dot{\xi }}_d-{\bar{K}}_1e_1-{\bar{K}}_2e_1^{2\beta _2-1}, \end{aligned}$$
(42)

where \(0<\beta _2<1\). Similar to the rotational control, an integral term \(\chi _2=\int _0^te_2(s)\text {ds}\) is developed where \(e_2={\dot{\xi }}-\alpha _2\). Differentiating \(e_1\) and \(e_2\) yields,

$$\begin{aligned}&{\dot{e}}_1=e_2+\alpha _2-{\dot{\xi }}_d\nonumber \\&{\dot{e}}_2=\ddot{\xi }-{\dot{\alpha }}_2=f_{2}(\eta ,u)+F_{2}(e), \end{aligned}$$
(43)

where \(f_{2}(\eta ,u)=\frac{1}{m_v}\left( R_{E2B}^{\text {T}}u+F_g\right)\), \(F_{2}(e)=\frac{1}{m_v}F_f-{\dot{\alpha }}_2\) with \(e=\left[ |{\dot{\xi }}|{\dot{\xi }},{\dot{\alpha }}_2\right] ^{\text {T}}\). And we employ NNs to estimate the uncertain term \(F_{2}(e)\) as,

$$\begin{aligned} F_{2}(e)=W_2^{*\text {T}}S_2+{\bar{\epsilon }}_2, \end{aligned}$$
(44)

where \(W_2^*=\left[ W_{b1}^*,W_{b2}^*,W_{b3}^*\right] \in {\mathbb {R}}^{m_2\times 3}\) is the optimal weight matrix, \({\bar{\epsilon }}_2\in {\mathbb {R}}^{3\times 1}\) is bounded by a positive constant \({\bar{\epsilon }}_{2n}\) and \(S_2\in {\mathbb {R}}^{m_2\times 1}\) is regressor. Then stable filters are given as,

$$\begin{aligned} \begin{array}{ll} l_2{\dot{e}}_{2fi}=-e_{2fi}+e_{2i}, &{} e_{2fi}(0)=0, \\ l_2{\dot{f}}_{2fi}=-f_{2fi}+f_{2i}, &{} f_{2fi}(0)=0, \\ l_2{\dot{S}}_{2f}=-S_{2f}+S_2, &{} S_{2f}(0)=0, \\ l_2\dot{{\bar{\epsilon }}}_{2fi}=-{\bar{\epsilon }}_{2fi}+{\bar{\epsilon }}_{2i}, &{} {\bar{\epsilon }}_{2fi}(0)=0, \\ \end{array} \end{aligned}$$
(45)

where \(l_2\) is a positive constant. Employing the estimation error-based adaptive scheme, the following auxiliary matrices are developed,

$$\begin{aligned}&{\dot{P}}_2=-\delta _2P_2+S_{2f}S_{2f}^{\text {T}}\nonumber \\&{\dot{Q}}_{2i}=-\delta _2Q_{2i}+S_{2f}\left( \frac{e_{2i}-e_{2fi}}{l_2}-f_{2fi}\right) ^{\text {T}}\nonumber \\&{\mathcal {L}}_{2i}=P_2{\hat{W}}_{bi}-Q_{2i}, \end{aligned}$$
(46)

where \(\delta _2\) is a positive constant. According to (18), we have,

$$\begin{aligned} {\mathcal {L}}_{2i}=-P_2{\tilde{W}}_{bi}+C_{2i}, \end{aligned}$$
(47)

where \({\tilde{W}}_{bi}=W_{bi}^*-{\hat{W}}_{bi}\) are estimated errors and \(C_{2i}=-\int _0^te^{-\delta _2(t-r)}S_{2f}{\bar{\epsilon }}_{2fi}\text {dr}\). Then, we propose an estimation error-based ANC with fast finite-time convergence as,

$$\begin{aligned}&u=u_{1c}+u_{2c}+u_{3c}\nonumber \\&u_{1c}=H_2\left( -{\bar{K}}_3e_2-{\bar{K}}_5e_2\chi _2^{\text {T}}\chi _2\right) \nonumber \\&u_{2c}=H_2\left( -{\bar{K}}_4e_2^{2\beta _2-1}-{\bar{K}}_6e_2\chi _2^{\text {T}}\chi _2^{2\beta _2-1}\right) \nonumber \\&u_{3c}=H_2\left( -e_1-\chi _2-\frac{1}{m_v}F_g-{\hat{W}}_2^{\text {T}}S_2\right) \nonumber \\&\dot{{\hat{W}}}_{bi}=\varGamma _{2i}\left( S_2 e_{2i}-\gamma _{2i}\frac{P_2^{\text {T}}{\mathcal {L}}_{2i}}{\Vert {\mathcal {L}}_{2i}\Vert }\right) ,\,i=1,2,3, \end{aligned}$$
(48)

where \(\varGamma _{2i}\) and \(\gamma _{2i}\) are positive-definite matrix and constant respectively; and \(H_2=m_vR_{E2B}\). Similar to the rotational subsystem, the following theorem can be obtained.

Theorem 2

Consider the translational subsystem (43) with the estimation error-based AFTNC (48). If the matrix \(P_2\) is positive-definite, tracking errors \(e_1\), \(e_2\) and estimated errors \({\tilde{W}}_{bi}\) converge to a small region around zero in finite-time. Moreover, estimated weights of RBF NNs can converge to the region around their optimal values simultaneously.

The proof of Theorem 2 is similar to Theorem 1, we omitted its proof process for clarity and conciseness. If readers are interested in proof, please refer to the stability analysis in the rotational subsystem.

4 Simulation

In this section, we conduct comparative simulations to illustrate the effectiveness of our proposed control scheme for the tilting quadcopter with unknown nonlinearities. The vehicle is commanded to track the translational and rotational trajectories simultaneously. More details about the simulation designs are elaborated as follows.

4.1 Rotational simulation results

The initial attitude states of the tilting quadcopter are set to zero. The value selections of the inherent parameters, adaptive laws and control parameters are given in Table 1. To verify the robustness to the uncertainties, note that the moment inertia and damping parameters exist unknown nonlinearities as shown in Table 1. Moreover, the vehicle is commanded to track the desired rotational trajectories as \(\eta _d(t)=[\cos (t-1),\cos t,\cos t+0.5]^{\text {T}}\). To illustrate the superiority of our scheme, we adopt the following Adaptive Neural Control (ANC) without finite-time convergence and estimation error-based adaptive law as the comparative method

$$\begin{aligned}&\alpha _1={\dot{\eta }}_d-K_1z_1\nonumber \\&\tau =-K_3z_2-z_1-{\hat{W}}_{1}^{\text {T}}{\bar{S}}_1\nonumber \\&\dot{{\hat{W}}}_{ci}=\varGamma _{1i}\left( {\bar{S}}_1 z_{2i}-\gamma _{1i}{{\hat{W}}}_{ci}\right) ,~i=1,2,3, \end{aligned}$$
(49)

where the above control parameters are the same to Table 1 to guarantee the reasonable results. The comparative simulation results are shown as follows.

Table 1 Parameters for Rotational Simulations
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Fig. 3
figure 3

The tracking performance of attitude

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Fig. 4
figure 4

The estimation performance of \(F_{13}\)

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Fig. 5
figure 5

The weight \(W_c\) with error-based adaptive scheme

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Fig. 6
figure 6

The profile of auxiliary vector \(L_1\)

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The tracking performance is exhibited in Figs. 3, 4, 5 and 6. We use the abbreviation AFTNC as the proposed control scheme (30) and ANC as the results of (49). Figure 3 depicts the rotational tracking trajectories. AFTNC can not only enjoy better transient performance with fast convergence to the desired trajectories within finite-time, but also robustness rejection ability to the uncertainties. ANC can only achieve infinite-time stabilization, which leads to a long transient response. The estimation performance of neural networks with estimation error-based adaptive laws is shown in Fig. 4. Compared with ANC results, AFTNC can fast and well estimate the unknown nonlinearities \(F_{13}\), where RBF NNs can converge to a small range around their real values. Let \({\hat{W}}_c=\left[ {\hat{W}}_{c1}^{\text {T}},{\hat{W}}_{c2}^{\text {T}},{\hat{W}}_{c3}^{\text {T}}\right] ^{\text {T}}\), and the norm value of \(\hat{W_c}\) is shown in Fig. 5. The estimated weights of AFTNC can converge to a region around their real values in finite-time. Additionally, according to the estimation error-based adaptive scheme (29), the term \({\mathcal {L}}_{1i}\) contains the information of estimated error \({\tilde{W}}_{ci}\). Thus, \({\mathcal {L}}_{1i}\) should converge to a small neighborhood around zero, which depends on the estimation error \({\bar{\epsilon }}_1\). Let \(L_1=\left[ {\mathcal {L}}_{11}^{\text {T}},{\mathcal {L}}_{12}^{\text {T}},{\mathcal {L}}_{13}^{\text {T}}\right] ^{\text {T}}\) and the norm value of \(L_1\) is shown in Fig. 6. Note that \(L_1\) converges to a small range around zero which further represents the estimated weights can converge to their optimal values. All simulation results have illustrated the effectiveness and superior performance of AFTNC with estimation error-based adaptive law.

Table 2 Parameters for Translational Simulations
Full size table

4.2 Translational simulation results

The initial position \(\xi (0)\) is set to zero and the vehicle is commanded to track the trajectories \(\xi _d=[\cos (t-0.8),\cos (t),\cos (t)+0.3]^{\text {T}}\). All simulation parameters used in dynamics and control are listed in Table 2. As the rotational simulation design, we adopt the aerodynamic damping coefficient \(d_i\) with uncertainties as shown in Table 2 to demonstrate the robustness of the proposed control scheme. Moreover, the comparative ANC is designed as

$$\begin{aligned}&\alpha _2={\dot{\xi }}_d-{\bar{K}}_1e_1,\nonumber \\&u=H_2-{\bar{K}}_3e_2+H_2\left( -e_1-\frac{1}{m_v}F_g-{\hat{W}}_2^{\text {T}}S_2\right) ,\nonumber \\&\dot{{\hat{W}}}_{bi}=\varGamma _{2i}\left( S_2 e_{2i}-\gamma _{2i}{{\hat{W}}}_{bi}\right) ,~i=1,2,3, \end{aligned}$$
(50)

where all the above control parameters are set the same to Table 2 to guarantee reasonable simulation results. The translational tracking results are shown as follows.

Fig. 7
figure 7

The tracking performance of position

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Fig. 8
figure 8

The estimation of \(F_{12}(e)\)

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Fig. 9
figure 9

The weight \(W_b\) with error-based adaptive scheme

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Fig. 10
figure 10

The profile of auxiliary vector \(L_2\)

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The tracking performance of comparative results is exhibited in Figs. 7, 8, 9 and 10. We define AFTNC as the performance of the proposed control scheme and ANC as (50). Fig. 7 shows the translational tracking results. ANC can only achieve infinite-time convergence, and there exists obvious over-shoot leading to a long transient response. However, AFTNC can fast converge to the desired commands within finite-time and achieve well tracking performance to uncertainties. The estimations of unknown nonlinearities \(F_{2,i}\) are shown in Fig. 8. With the help of estimation error-based adaptive law, it can be demonstrate that AFTNC can well approximate the smooth nonlinearities in high precision. Denote \(W_b=\left[ W_{b1}^{\text {T}},W_{b2}^{\text {T}},W_{b3}^{\text {T}}\right] ^{\text {T}}\). And compared with ANC, NNs weights of AFTNC can fast converge to a smaller neighborhood around their optimal values as shown in Fig.  9, which can support the superior estimation results in Fig. 8. Moreover, let \(L_2=\left[ {\mathcal {L}}_{2i}^{\text {T}},{\mathcal {L}}_{2i}^{\text {T}},{\mathcal {L}}_{2i}^{\text {T}}\right] ^{\text {T}}\), which contains the estimated error information \({\tilde{W}}_{bi}\). According to the rigorous theoretical analysis, \(L_2\) should converge around zero which suggests that \({\hat{W}}_{bi}\) can estimate their optimal value \(W_{bi}^*\). And Fig. 10 can demonstrate the above convergence of \({\hat{W}}_{bi}\), where \(L_{2}\) converges to a small region around the equilibrium within finite-time.

According to the above simulation results, the tilting quadcopter can provide a fully actuation version with rotational and translational movements controlled independently. In comparison with the existing ANC, the proposed AFTNC can not only perform robustness ability to the uncertainties, but also guarantee finite-time convergence of tracking errors leading to a fast transient response. Furthermore, with the help of estimation error-based adaptive law, AFTNC enable NNs weights \(W_{ci}\) and \(W_{bi}\) converge to a smaller neighborhood around their optimal values in finite-time, which contributes to achieve better tracking performance.

5 Conclusion

In this paper, we first propose a novel conception of a tilting quadcopter with fully-actuated version, which suggests that the translational and rotational movements can be controlled independently. The dynamics of the tilting quadcopter is developed based on Euler-Lagrange equations and NNs are utilized to approximate the unknown nonlinearities in systems. An estimation error-based AFTNC is then proposed, where a new auxiliary filter is constructed to obtain the estimation errors explicitly as new leakage terms into the adaptive law. This indicates that not only the tracking and estimation errors converge to a small neighborhood around zeros, but also the estimated weights of NNs can converge to their optimal values simultaneously. Finally, the simulation results have been provided to demonstrate the effectiveness of our proposed ANC. In our future work, we will extend our approach to practical situations, such as states constrains, external disturbance and fault-tolerant control.