We assume that the reference trajectory, generated by the motion planning algorithm, fulfils the following model:

(1) [x˙ry˙rθ˙r]=[cos⁡θr0sin⁡θr001][vrwr]

where xr , yr and θr represent the desired (x,y) position and orientation of the UGV, vr and wr are the desired linear and angular velocities respectively.

It is obvious that the real controls v and w rely on the state measurements x,y and θ (Fig. 1). Due to measurement noise and modeling uncertainties, here we consider input uncertainties for both v and w [39]. Thus, the real equation of the robot trajectory fulfills the following model:

(2) [x˙y˙θ˙]=[cos⁡θ0sin⁡θ001][v+δvw+δw]

where the linear and angular velocities, v and w , are defined as follows:

(3) v=r2(φ˙d+φ˙l)

(4) w=r2L(φ˙d−φ˙l)

parameter r represents the radius of the wheels; 2L is the distance between the two driving wheels; φ˙l and φ˙d represent the angular velocity of two wheels left and right-hand side respectively; δv and δw are the uncertainties on v and w .

The objective of trajectory tracking is to asymptotically stabilize the tracking errors ex=xr−x , ey=yr−y and eθ=θr−θ to zero.

Transforming the tracking errors expressed in the inertial frame to the robot frame, the error coordinates can be denoted as follows:

(5) [e1e2e3]=[cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001][exeyeθ]

Thus, the tracking-error model is represented by the following equation:

(6) e˙=f1(e)+f2(e)(U+δ)

where

(7) {e=[e1,e2,e3]Tf1(e)=[vrcos⁡e3,vrsin⁡e3,wr]Tf2(e)=[−1e20−e10−1]U=[v,w]Tandδ=[δv,δw]T

Sliding Mode Control (SMC) is widely used in the control of different type of systems, such as wheeled mobile robot [40], five DOF redundant robot [41], PH process in stirred tanks [42], tunnel bow thrusters [43] or for other purpose such as multi-sensors data fusion [44]. In this part, we propose to enhance the SMC with the integral action, to improve its disturbances rejection. This controller is then applied to our UGV system.

For system Eq. (5), the control law is defined as follows:

(8) U=U0+U1

U0 is the nominal control and U1 represents the ISMC part which is designed to be discontinuous in order to reject the disturbance.

The first part of the control design is to find a saturated control law U0 so that the nominal system e˙=f1(e)+f2(e)U0 is globally asymptotically stable (see Jiang et al. [45] for more details). The nominal control input is chosen as follows:

(9) U0=[v0=vrcos⁡e3+λ3tanh⁡e1w0=wr+λ1vre2sin⁡e3e3(1+e12+e22)+λ2tanh⁡e3]

The positive parameters λ1 , λ2 and λ3 can be designed so that the bounds of the controls are complied with. This can be represented as follows:

(10) |v0|≤vmax+λ3,|w0|≤wmax+λ1vmax2+λ2

For the ISMC part U1 , the sliding variable ‘ s ’ is defined as follows:

(11) s=[s1,s2]T=s0(e)+z

where

(12) {s0(e)=[−e1,−e3]Tz˙=−∂s0∂e(f1(e)+f2(e)U0)z(0)=[e1(0),e3(0)]T

The variable z includes the integral term and provides one more degree of freedom in the construction of the sliding variable. According to Defoort et al. [46], the sliding mode is established as the initial moment and the phase of convergence is eliminated. Then, the control law is given by the following equation:

(13) U1=[−K1sign(s1)−K2sign(−e2s1+s2)]

with K1>δv+μ and K2>δw+μ such as: μ>0

In order to reduce the chattering phenomena, the sign function is replaced by: f(x)=2πtanh⁡(ηx) , with η being a positive constant [47]. Following from this, Eq. (13) can be written as follows:

(14) U1=[−2K1πtanh⁡(η1s1)−2K2πtanh⁡(η2(−e2s1+s2))]

Remark 1: The trajectory evolves on the manifold s=0 from t=0 and remains there in presence of the disturbances. The time derivative of the sliding variable is s˙=∂s0∂e(e˙−f1(e)−f2(e)U0) . Therefore, the motion equation in sliding mode is e˙=f1(e)+f2(e)U0 which is globally asymptotically stable (see Defoort et al. [39] for more details).

Let us study the effect of the approximation of the sign function by tanh on global stability.

Lemma 1. [48] For every given scalar x and positive scalar η , the following inequality holds:

xtanh⁡(ηx)=|xtanh⁡(ηx)|=|x||tanh⁡(ηx)|≥0

Proof of Lemma 1. According to the definition of tanh function, we have:

xtanh⁡(ηx)=xeηx−e−ηxeηx+e−ηx=1e2ηx+1x(e2ηx−1)

Since

{e2ηx−1≥0ifx≥0e2ηx−<0ifx<0Thenx(e2ηx−1)≥0

Therefore

xtanh⁡(ηx)=1e2ηx+1x(e2ηx−1)≥0

And

xtanh⁡(ηx)=|xtanh⁡(ηx)|=|x||tanh⁡(ηx)|≥0

Proof of stability. Consider the following Lyapunov function candidate:

V=12sTs

The discontinuous control term must satisfy the condition V˙≤0 guaranteeing the global asymptotic stability.

V˙=sTs˙=sT(s˙0(e)+z˙)

V˙=sT[∂s0∂ee˙−∂s0∂e(f1(e)+f2(e))U0]

V˙=sT∂s0∂ef2(e)(U1+δ)

V˙=(s1−e2s1+s2)(U1+δ)

According to Lemma 1, the above condition is satisfied if:

U1=[−2K1πtanh⁡(η1s1)−2K2πtanh⁡(η2(−e2s1+s2))]

with 2K1π>δv+μ and 2K2π>δw+μ ( μ>0 )

Therefore, according to LaSalle’s theorem, the control system with the smoothed ISMC is asymptotically stable in the sense of Lyapunov.

Ground targets (UGVs) were always implicitly assumed to be non-manoeuvring and the noise statistics involved in the dynamics model/target observation (matrices Q(k) and R(k) ) were assumed to be known. In practice, it goes without saying that these parameters are never well known and may vary over time depending on the manoeuvring capacity of the ground targets. For this reason, we have opted for the use of a discrete kinematic model with quasi-constant acceleration [49]. We can model the equations of the UGV as a linear system in following representation:

(15) Xk+1=AXk+Γwk

where:

(16) A=[1T12T200001T0000010000001T12T200001T000001]

(17) Γ=[12T20T010012T20T01]

and wk is the random process noise. Xk is the state vector describing the motion of the UGV (its position, velocity and acceleration):

Xk=[xkx˙kx¨kyky˙ky¨k]T

and T is the sampling period. The measurements of the mobile robot position p at time k are described as follows:

(18) Zk=Hkpk+vk

where vk is a random measurement noise. We assume that the process and measurement noises wk and vk are white, zero-mean.

As the measurement vector is the UGV position (xk,yk) , the matrix Hk in Eq. (18) (called measurement matrix) is given by:

Hk=(100000000100)

The predicting and update equations for the Kalman filter are presented as follows:

(19) {X^k+1|k=AkX^k|kPk+1|k=AkPk|kAkT+Qky~k+1=Zk−HkXk+1|kSk+1=Hk+1Pk+1|kHk+1T+Rk+1Kk+1=Pk+1|kHk+1TSk+1−1X^k+1|k+1=X^k+1|k+Kky~k+1Pk+1|k+1=(I−Kk+1Hk+1)Pk+1|k

Let us describe the Kalman filter parameters ( X0,Rk,Qk and P0 ) selection:

For the initialization ( X0 , P0 ), several simulations were performed by considering initialization accuracy up to 70% the true value of the UGV position. We have noticed that the results of the estimation and the convergence speed of the filter are highly dependent on initialization. The suitable values of ( X0 , P0 ) are estimated after many simulations.

Rk=(σx200σy2) ; where σx and σy are the standard deviations of the position of x and y, respectively. In our case, the sensor used to measure the UGV position is assumed to be a camera embedded on the UAV. Thus, in our simulations the matrix Rk is selected based on real camera characteristics.

The covariance matrix of the process noise ( Qk ) is estimated based on the odometer model of the mobile robot given by Eq. (2), in our simulation a suitable values of the matrix Qk are determined from a real robot (Pioneer 3-AT).

Our interest in such type of UAV (Fig. 2) is its hovering capability and high manoeuverability.

The dynamic models of the quadrotor are well studied. The details of the following Newton-Euler equations can be found in Refs. [50–53].

(20) {ϕ¨=Jy−JzJxθ˙ψ˙−JrJxΩ¯θ˙−KfaxJxϕ˙2+1Jxu2θ¨=Jz−JxJyϕ˙ψ˙+JrJyΩ¯ϕ˙−KfayJyθ˙2+1Jyu3ψ¨=Jx−JyJzθ˙ϕ˙−KfazJzψ˙2+1Jzu4x¨=−Kftxmx˙+1muxu1y¨=−Kftymy˙+1muyu1z¨=−Kftzmz˙−g+1m(cos⁡ϕcos⁡θ)u1

The control inputs of the UAV are defined as follows:

(21) {u1=b(w12+w22+w32+w42)u2=lb(w42−w22)u3=lb(w32−w12)u4=d(w12−w22+w32−w42)ux=(cos⁡ϕcos⁡ψsin⁡θ+sin⁡ϕsin⁡ψ)uy=(cos⁡ϕsin⁡ψsin⁡θ−sin⁡ϕcos⁡ψ)Ω¯=(w1−w2+w3−w4)

The description of the parameters used in this model is given in Tab. 1.

In order to control the UAV a backstepping control scheme is used. The inner controller stabilizes the orientation angles in order to achieve a stable flight while the outer controller is responsible for the control of the position of UAV. Further the state vector of the UAV is defined as follows:

X=[ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,z,z˙,x,x˙,y,y˙]T=[x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12]T

Thus we obtain the following equations:

(22) {x˙1=x2x˙2=a1x4x6+a2x22+a3Ω¯x4+b1u2x˙3=x4x˙4=a4x2x6+a5x42+a6Ω¯x2+b2u3x˙5=x6x˙6=a7x2x4+a8x62+b3u4x˙7=x8x˙8=a9x8−g+1m(cos⁡x1cos⁡x3)u1x˙9=x10x˙10=a10x10+1muxu1x˙11=x12x˙12=a11x12+1muyu1

with

{a1=Jy−JzJx,a2=−KfaxJx,a3=−JrJx,a4=Jz−JxJy;a5=−KfayJy,a6=JrJy,a7=−Jx−JyJz,a8=−KfazJz;a9=−Kftzm,a10=−Kftxm,a11=−Kftym;b1=1Jx,b2=1Jy,b3=1Jz

The following Lyapunov functions are used:

(23) Vi={12zi2/i∈{1,3,5,7,9,11}12(Vi−1+zi2)/i∈{2,4,6,8,10,12}

with

zi={xid−xi/i∈{1,3,5,7,9,11}xi−x˙(i−1)d−α(i−1)z(i−1)/i∈{2,4,6,8,10,12}

The application of the backstepping technique [50] and [54] on the quadrotor state model give the following control inputs:

(24) {u1=mcos⁡x1cos⁡x3(z7−a9x8+g−α7(z8+α7z7)−α8z8+x¨7d)u2=1b1(z1−a1x4x6−a2x22−a3Ω¯x4−α1(z2+α1z1)−α2z2+x¨1d)u3=1b2(z3−a4x2x6−a5x42−a6Ω¯x2−α3(z4+α3z3)−α4z4+x¨3d)u4=1b3(z5−a7x2x4−a8x62−α5(z6+α5z5)−α6z6+x¨5d)ux=mu1(z9−a10x10−α9(z10+α9z9)−α10z10+x¨9d)uy=mu1(z11−a11x12−α11(z12+α11z11)−α12z12+x¨11d)

with cos⁡x1cos⁡x3≠0,αi>0 and ∀i∈{1,2…,12} . Variables ux and uy are virtual control inputs which will be used to find desired Euler angles as follows:

(25) {ϕd=x1d=arcsin⁡(uxsin⁡x5d−uycos⁡x5d)θd=x3d=arcsin⁡(uxcos⁡x5d+uysin⁡x5dcos⁡x1d)

The simulation results are obtained based on the following realistic parameters of quadrotor in Tab. 2 and characteristics of Pioneer 3-AT mobile robot in Tab. 3.

In this first simulations set, we aim to evaluate the ISMC controller of the ground agent (UGV), thus the first tracking approach based on the Kalman filter and the backstepping control of the quadrotor.

In this context of the ground agent tracking by a UAV, and in order to get closer to reality, several scenarios are considered as follows.

The initial conditions of the UGV: x0=0.5y0=−0.5θ0=π4

(26) {0≤u1≤4bΩmax2|u2|≤lbΩmax2|u3|≤lbΩmax2|u4|≤2dΩmax2

The initial parameters of the Kalman estimator are given as follows:

(27) {X0=[0.400−0.400]Rk=diag([0.0520.032])Qk=Γ(5.10−3)2ΓTP0=zeros(6,6)

Scenario 1: The quadrotor may be affected by external disturbances such as wind. Several models of wind are proposed in Gawronski [55]. In our work, we assume that the wind has caused the same acceleration intensity on all axes x , y and z [56], as shown in Fig. 3. The mathematical model of wind is given by the following equation:

(28) a(t)={0when0s<t≤30s0.7sin⁡(π(t−30)31)+0.4sin⁡(π(t−30)7)⋯⋯+0.08sin⁡(π(t−30)2)+0.056sin⁡(24π(t−30)11)when30s<t≤41s0when41s<t≤45s3.35sin⁡(π(t−45)21)+0.5sin⁡(π(t−45)2)⋯⋯+0.45sin⁡(π(t−45)5)+0.205sin⁡(24π(t−45)11)when45s<t≤65s0when60s<t≤70s

Scenario 2: This scenario is designed to evaluate the robustness of the control system with respect to modelling errors and measurement noises. In terms of parametric uncertainty, we assume that the elements of the inertia matrix Jx , Jy and Jz are underestimated to 60%, the coefficients b and d are also underestimated, whereas the values used in the control are only 80% of the actual values.

(29) {J~x=0.6JxJ~y=0.6JyJ~z=0.6Jzb~=0.8bd~=0.8d

As for the measurement noises, an additive Gaussian white noise with the density 150 μg/Hz was considered.

In the presence of wind gusts (Fig. 3) and as shown in Fig. 4, the quadrotor tracks the UGV that traveled Eight trajectory using the ISMC controller. By looking at Figs. 5 and 6, it can be noticed that the ISMC allows the robot to follow the set point correctly according to the two axes x,y and accounts for the saturation constraints with a very low tracking error on the trajectory realized by the UGV. The two sliding surfaces (s1,s2) tend to zero at the initial moment. Figs. 7 and 8 show that the wind affects the performances of the quadrotor during its mission, resulting in maximum absolute tracking errors of 0.27 m along z , 0.18 m along x and y .

As for scenario 1, even with the existence of parametric uncertainties and measurement noises, the quadrotor succeeds in tracking the mobile robot with small fluctuations in its tracking trajectory (Fig. 9). The tracking error along x axis takes a value of approximately 0.35 m, which shows that these disturbances have significant effects on the control by backstepping as indicated in Figs. 10 and 11.

One of the major challenges for the tracking algorithm is the uncertainty in the motion of UGV. This uncertainty refers to the fact that a precise dynamic model of the movement is not available at the level of the tracking algorithm [57,58]. However, the Kalman filter (KF) can only achieve a good performance (optimal solution) under the assumption that the complete and exact information of the process model and the noise distribution are to be known as a prior. In practice, state and observation models are often poorly known, or contain uncertain parameters, and the statistical properties of noise (state and observation) are also poorly known, coming to the optimality of the solution obtained. Therefore, to improve our tracking algorithm, and to overcome these limitations, we propose to use a new filter or estimator called Smooth Variable Structure Filter (SVSF) to process the tracking problem of a UGV [59,60].

The Smooth Variable Structure Filter is a relatively new estimating strategy proposed by Habibi in 2007 [61]. This strategy is based on the concepts of sliding mode control and the theory of systems with variable structure, outcome and similar design to variable structure filter (VSF) [62]. This filter is formulated in the predictive-correction format, and can be used for linear or non-linear systems. It uses a correction gain simpler than the one used by the VSF. The SVSF is introduced to provide more stability and robustness to the estimation process. This technique is generally used for the estimation of states and parameters of dynamic systems [63], the prediction and diagnosis of defects in systems [64] and targets tracking problems [59,60,65].

To formulate the tracking problem, we use the same model that has been described in detail by the two Eqs. (15) and (18).

The SVSF estimation method is described by the following series of equations. Note that this formulation includes state error covariance equations as presented in Gadsden et al. [66], which was not originally presented in the standard SVSF form [61]. The prediction stage is similar to the KF; its steps are as follows:

- Initialization

(30) {X^0|0=X0E^0|0=E0

- Prediction

(31) X^k+1|k=AkX^k|k

(32) Pk+1|k=AkPk|kAkT+Qk

where X^k|k is the state estimated at time k of the state Xk|k .

(33) Y^k+1|k=HX^k+1|k

then, pretest measurement error is calculated by the following equation:

(34) Ek+1|k=Yk+1−Y^k+1|k

- Update

For the state estimate, the SVSF correction gain is calculated by MacArthur et al. [22] and Phan et al. [31]:

(35) Kk+1|k=H+Diag[(|Ek+1|k|abs+γ|Ek|k|abs)∘Sat(Ψ¯−1Ek+1|k)][Diag(Ek+1|k)]−1

where ∘ signifies Schur (or element-by-element) multiplication, the superscript + refers to the pseudo inverse of a matrix and Ψ¯−1 is a diagonal matrix constructed from the smoothing boundary layer vector Ψ , defined as follows:

(36) Ψ¯−1=[Diag(Ψ)]−1=[1Ψ1000⋱0001Ψm]

The form of saturation used in Eq. (35) is defined as follows:

(37) Sat(Ψ¯−1Ek+1|k)={Ek+1|kiΨi≥1−1<Ek+1|kiΨi<1Ek+1|kiΨi≤−1

The gain is used to update the predicted state as follows:

(38) X^k+1|k+1=X^k+1|k+Kk+1|kEk+1|k

The covariance associated with the state updates is then calculated as follows:

(39) Pk+1|k+1=[I−Kk+1|kHK]Pk+1|k[I−Kk+1|kHK]T+Kk+1|kRk+1Kk+1|kT

Thus, the estimated measurement and the corresponding empirical measurement error are calculated as follows:

(40) {Y^k+1|k+1=HX^k+1|k+1Ek+1|k+1=Yk+1−Y^k+1|k+1

Two critical variables in this process are the pretest and empirical measurements (output) error estimates, defined by Eqs. (34) and (40). It shall be noted that Eq. (40) is the empirical measurement error estimates from the previous time step, and is used only in the gain calculation.

The selection of the smoothing boundary layer width vector Ψ reflects the level of uncertainties in the filter and the disturbances (i.e., system and measurement noise, and uncertain parameters).

The backstepping control cannot ensure the favorable tracking performance of the quadrotor if unpredictable disturbances from the unknown external disturbance, modeling errors, as well as measurement noise occur. In order to improve these performances and consequently the robustness, we propose to combine the conventional PID with the backstepping control. This will allow for integral backstepping. However, this control technique has been proposed in several research studies [50,52,67,68,69], which demonstrated that the integral backstepping controller allows rejection of external disturbances and is robust to parametric uncertainties.

The application of the integral backstepping control on the quadrotor state model gives the following control inputs:

(41) {u1=mcos⁡x1cos⁡x3(z7−a9x8+g+x¨7d−α7(α7z7+z8+λ4χ4)+λ4z7−α8z8)u2=1b1(z1−a1x4x6−a2x22−a3Ω¯x4+x¨1d−α1(α1z1+z2+λ1χ1)+λ1z1−α2z2)u3=1b2(z3−a4x2x6−a5x42−a6Ω¯x2+x¨3d−α3(α3z3+z4+λ2χ2)+λ2z3−α4z4)u4=1b3(z5−a7x2x4−a8x62+x¨5d−α5(α5z5+z6+λ3χ3)+λ3z5−α6z6)ux=mu1(z9−a10x10−α9(z10+α9z9+λ5χ5)+λ5z9−α10z10+x¨9d)uy=mu1(z11−a11x12−α11(z12+α11z11+λ6χ6)+λ6z11−α12z12+x¨11d)