# Robust tracking of an unknown trajectory with a multi-rotor UAV: A high-gain observer approach

We study a trajectory tracking problem for a multi-rotor in the presence of modeling error and external disturbances. The desired trajectory is unknown and generated from a reference system with unknown or partially known dynamics. We assume that only position and orientation measurements for the multi-rotor and position measurements for the reference system can be accessed. We adopt an extended high-gain observer (EHGO) estimation framework to estimate the feed-forward term required for trajectory tracking, the multi-rotor states, as well as modeling error and external disturbances. We design an output feedback controller for trajectory tracking that comprises a feedback linearizing controller and the EHGO. We rigorously analyze the proposed controller and establish its stability properties. Finally, we numerically illustrate our theoretical results using the example of a multi-rotor landing on a ground vehicle.

## Authors

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## I Introduction

We envision a future in which aerial vehicles provide services such as parcel delivery, remote monitoring, and maintenance. In such scenarios, a ground-based vehicle may provide charging services for multi-rotors or act as a staging area for storing parcels. In these cases, the multi-rotors will be required to make multiple landings on the ground vehicle, which need not be stationary.

One of the principle challenges in achieving a landing on a moving vehicle is the generation of a trajectory. The multi-rotor and ground vehicle may be operated by different service-providers, which may prohibit communication between the vehicles due to security reasons, or if the two have incompatible communication equipment. In this case, the trajectory of the ground vehicle will need to be inferred by the multi-rotor.

The multi-rotor may only be able to measure the position of the ground vehicle in real-time. However, to achieve efficient control performance, higher-order derivatives of the trajectory are required. Furthermore, the modeling error and external disturbances—such as, wind gusts, the ground effect, aerodynamic drag, parcel size and weight, or items shifting inside a parcel—may also deteriorate the multi-rotor’s tracking performance resulting in a poor landing. In this work, we study this problem and design an output feedback controller that addresses these challenges.

From control design to path planning and disturbance rejection, much work has been devoted to studying multi-rotor UAV control design; see [1, 2] for a survey. The problem of autonomous landing of a multi-rotor on a mobile platform has also received some attention [3, 4]. Many control methodologies have been applied to landing on a mobile platform, including model predictive control [5, 6], PI control [7, 8, 9], and feedback linearizing control [10].

State estimators such as Kalman filter have been used to estimate the dynamic state of the mobile platform

[9, 11] under the assumptions that the dynamic model of the mobile platform is known and it travels with unknown constant velocity. Through our EHGO design, these assumptions are relaxed, requiring no information about the mobile platform’s dynamics or input.

An alternative approach to estimate the state of the mobile platform uses optical flow data [7, 8], or visual cue data [12] in which a dynamic model of the mobile platform is not required. In these cases the relative velocity is estimated through the optical flow algorithms and is minimized in the control to ensure tracking.

Many of the approaches in the literature either do not consider modeling error and external disturbances, or consider them to be constant or slowly time-varying [7, 8]. In contrast, our approach only requires that the disturbance be bounded and continuously differentiable.

In this paper, we study a trajectory tracking problem for a multi-rotor in the presence of modeling error and external disturbances. The desired trajectory is unknown and generated from a reference system with unknown or partially known dynamics. We assume that only position and orientation measurements for the multi-rotor and position measurements for the reference system can be accessed. We design an output feedback controller that robustly tracks such unknown desired trajectories. The contributions of this work are as follows:

• We design and rigorously analyze an EHGO to estimate modeling error and external disturbances, feed-forward control for trajectory tracking, and multi-rotor states for output feedback control.

• We design and analyze a robust feedback linearizing controller that mitigates modeling errors and external disturbances using their estimates.

• We rigorously characterize the stability of the overall output feedback system.

• We illustrate the effectiveness of our output feedback controller through simulation using the example of a multi-rotor landing on a mobile platform.

The remainder of the paper is organized as follows. The system dynamics are introduced in Section II. The control is presented in Section III, with rigorous stability analysis in Section IV, and validation of the proposed controller through simulation in Section V. Conclusions are presented in Section VI.

## Ii System Dynamics

A multi-rotor UAV is an underactuated mechanical system. While there can be

rotors, only four degrees of freedom can be controlled in the classic configuration where all rotors are co-planar. To handle the underactuation, as discussed below, the rotational dynamics are controlled to create a virtual control input for the translational dynamics.

### Ii-a Rotational Dynamics

The rotational dynamics of the multi-rotor are [13]

 τ=J˙Ω+Ω×JΩ, (1)

where is the inertia matrix, is the torque applied to the multi-rotor and is the angular velocity, each expressed in the body-fixed frame.

Consider the orientation of the multi-rotor expressed in terms on Euler angles . The angular velocity is related to the Euler angle rates in the inertial frame by

 θ2=ΨΩ,Ψ=⎡⎢ ⎢⎣1sϕtθcϕtθ0cϕ−sϕ0sϕ/cθcϕ/cθ⎤⎥ ⎥⎦,Ω=Ψ−1θ2,

where denote , , , respectively. The rotational dynamics (1) can be equivalently written in terms of Euler angles

 ˙θ1=θ2,˙θ2=˙ΨΨ−1θ2−ΨJ−1(Ψ−1θ2×JΨ−1θ2)+ΨJ−1τ+σξ, (2)

where is an added term to represent the lumped rotational disturbance.

Let be the rotational reference signal. Define rotational tracking error variables

 ξ1=θ1−θr,ξ2=˙ξ1=θ2−˙θr,ξ=[ξ⊤1ξ⊤2]⊤.

The rotational dynamics (2) can be written in terms of tracking error

 ˙ξ1=ξ2,˙ξ2=f(ξ,θ1,˙θr)+G(θ1)τ+σξ−¨θr, (3)

where

 f(ξ,θ1,˙θr)=˙ΨΨ−1(ξ2+˙θr)−ΨJ−1(Ψ−1(ξ2+˙θr)×JΨ−1(ξ2+˙θr)),G(θ1)=ΨJ−1.

### Ii-B Translational Dynamics

The translational dynamics of the multi-rotor are [13]

 ˙p1=p2,˙p2=−ufmR3(θ1)+gez+σρ. (4)

Here, is the position of the center of mass of the aerial platform in the inertial frame, is the cumulative thrust force, is the force generated by the -th rotor, is the mass of the aerial platform, is the gravitational constant, , and is the lumped translational disturbance. is the third column of the rotation matrix describing the multi-rotor body frame with respect to the inertial frame,

 R3(θ1)=⎡⎢ ⎢⎣cϕsθcψ+sϕsψcϕsθsψ−sϕcψcϕcθ⎤⎥ ⎥⎦.

Let be the translational reference signal. Define translational error variables

 ρ1=p1−pr,ρ2=˙ρ1=p2−˙pr,ρ=[ρ⊤1ρ⊤2]⊤.

The translational dynamics (4) can be written in terms of tracking error

 ˙ρ1=ρ2,˙ρ2=−ufmR3(θ1)+gez+σρ−¨pr. (5)

### Ii-C Reference System Dynamics

We assume that the reference trajectory that the multi-rotor UAV will track is generated by the system

 ˙xc1=xc2,˙xc2=fc(xc,uc), (6)

where is the position of the reference system, is the system state, is the unknown system input, and is some unknown function. We take system input and let . We assume that is twice continuously differentiable and and its second partial derivatives with respect to are bounded on compact sets of for all .

## Iii Control Design

In this section, we first design a trajectory tracking feedback linearizing controller for rotational system and subsequently use the rotational trajectory to design a trajectory tracking controller for the translational system.

### Iii-a Rotational Control

The rotational control feedback-linearizes the rotational tracking error dynamics (3) as follows

 τd=G−1(θ1)[fr−f(ξ,θ1,˙θr)], (7)

for constants where . This results in the following closed loop rotational tracking error system

 ˙ξ1=ξ2,˙ξ2=−β1ξ1−β2ξ2. (8)

### Iii-B Translational Control

The translational control uses the total thrust, as the direct control input and the desired roll and pitch trajectories, and , as virtual control inputs. The translational control will be designed in view of the potential tracking errors in roll and pitch trajectories, leading to the following modification of the translational error dynamics (5)

 ˙ρ1=ρ2,˙ρ2=−ufmR3(θr+ξ1)+gez+σρ−¨pr. (9)

The translational subsystem dynamics can be redefined in terms of the nominal translational model with the addition of an error term as follows

 ˙ρ1=ρ2,˙ρ2=−ufmR3(θr)+gez+σρ−¨pr+eθ(ξ1), (10)

where

 eθ(ξ1)=−ufm(R3(θr+ξ1)−R3(θr)),

and it can be verified that is a Lipschitz function. Assuming perfect rotational tracking, i.e., , the system (10) can be feedback linearized using the following direct and virtual control inputs 111During normal operation, , ensuring differentiability of .

 ϕr=tan−1⎛⎜ ⎜⎝−fysign(fz−g)√f2x+(fz−g)2⎞⎟ ⎟⎠,θr=tan−1(fxfz−g),ψr=0,uf=−m(fz−g)cϕrcθr, (11)

where is the forcing function, defined by

 ft=−γ1ρ1−γ2ρ2−σρ+¨pr, (12)

for constants . This leads to the following closed-loop translational subsystem with the inclusion of tracking error from the rotational subsystem

 ˙ρ1=ρ2,˙ρ2=−γ1ρ1−γ2ρ2+eθ(ξ1). (13)

Note that the controller (7) requires , which can be computed analytically from (11)

 ˙ϕr=fy(˙fxfx+˙fz(fz−g))−˙fy(f2x+(fz−g)2)(f2x(fz−g)2+1)1/2(fz−g)(f2x+f2y+(fz−g)2),˙θr=˙fx(fz−g)−fx˙fzf2x+(fz−g)2,˙ψr=0,

where and

 ˙ft=−γ1ρ2−γ2[−ufmR3(θr)+gez+σρ−¨pr]−˙σρ+p(3)r.

### Iii-C Extended High-Gain Observer Design

A multi-input multi-output EHGO is designed similar to [14, 15] to estimate higher-order states of the error dynamic systems (3) and (5), uncertainties arising from modeling error and external disturbances, as well as the reference trajectory based on the reference system dynamics (6). It is shown in [16] that it is necessary to include actuator dynamics in the multi-rotor model for EHGO design. The actuator dynamics reside in the same time-scale as the EHGO, and therefore can not be ignored in the EHGO dynamics.

The actuators used on multi-rotor UAVs are Brushless DC (BLDC) motors, which require electronic speed controllers. These controllers introduce dynamic delays [17] of the following form

 τm˙ω=(ωdes−ω), (14)

where is the time constant of the actuator system,

is a vector of angular rates of the rotors and

is a vector of rotor angular rate control inputs. Since feedback of the rotor angular rate is not available, it can be simulated by the following system

 τm˙^ω=(ωdes−^ω), (15)

where is a vector of simulated rotor angular rates. The translational dynamic model (5) takes a cumulative thrust force, , as an input and the rotational dynamic model (3) takes three body-fixed torques, , as inputs. The thrust and torques are generated by applying different forces with each actuator, which is a function of the rotor angular rate as follows

 ¯fi=bω2i,for i={1,…,n}, (16)

where is a constant relating angular rate to force and is the -th rotor angular rate. These individual actuator forces are then mapped through a matrix, , based on the geometry of the multi-rotor aerial platform, allowing the squared rotor angular rates to be taken as the control input to the model

 [ufτ]=bMωs,ωs=[ω21…ω2n]T. (17)

The inverse operation is used to generate the desired rotor speeds from the feedback linearizing control signals and . For , the inverse of (17) is an over-determined system which admits infinitely many solutions. In this case, we focus on the minimum energy solution , where is the pseudo-inverse of .

The rotational and translational tracking error dynamics and the reference system dynamics (6) can be combined into one set of equations for the observer. The state space will be extended to estimate disturbance terms and the control input for the reference system dynamics. Since the third derivative of the reference trajectory is required in the rotational control (7), the reference system’s dynamics will be extended to also include the third derivative of its position. The full system dynamics become

 ˙ρ1=ρ2,˙ρ2=−ufmR3(θ1)+gez+σρ−¨pr,˙σρ=φρ(t,ρ),˙ξ1=ξ2,˙ξ2=f(ξ,θ1,˙θr)+G(θ1)τ+ςξ,˙ςξ=φξ(t,ξ),˙xc1=xc2,˙xc2=xc3,˙xc3=σxc,˙σxc=φxc(t,xc), (18)

where . Since the reference system dynamics may not be known, they have been replaced by the disturbance term in their entirety. The estimated reference system states will be taken as the reference trajectory.

Define which is a vector of unknown functions describing the disturbance, where is continuous and bounded on the set defined in Lemma 1. Note that the second order derivative of the reference trajectory, , is lumped into the disturbance . Since the disturbance term must be first order differentiable, must be differentiable, therefore requiring the translational reference signal to be fifth order differentiable. The observer system with extended states can be written compactly as

 ˙^χ=A^χ+B[¯f(^χ,θ1,˙θr)+¯G(θ1)^ωs]+H^χe,^χe=C(χ−^χ),

where

 ^χ=[^ρT1 ^ρT2 ^σTρ ^ξT1 ^ξT2 ^ςTξ ^xTc1 ^xTc2 ^xTc3 ^σTxc]T,
 ¯f(^χ,θ1,˙θr)=[gez−¨prf(^ξ,θ1,˙θr)03×1],
 ¯G(θ1)=b[−R3(θ1)m0303×1G(θ1)03×103]M,

where is designed by choosing such that

 sϱi+αi1sϱi−1+⋯+αiϱi−1s+αϱi, (19)

is Hurwitz and .

### Iii-D Output Feedback Control

The state feedback controllers (7) and (11) are rewritten as the following output feedback controllers using the EHGO estimates. The rotational control becomes

 ^τd=G−1(θ1)[^fr−f(^ξ,θ1,˙θr)], (20)

where . The output feedback translational controller becomes

 ^ϕr=tan−1⎛⎜ ⎜⎝−^fysign(^fz−g)√^f2x+(^fz−g)2⎞⎟ ⎟⎠,^θr=tan−1(^fx^fz−g),^ψr=0,^uf=−m(^fz−g)c^ϕrc^θr, (21)

where the forcing function, , is now defined using tracking error estimates as follows

 ^ft=−γ1^ρ1−γ2^ρ2−^σρ+^xc3. (22)

These control signals result in the following output feedback controller

 u(^χ)=[^uf^τd]. (23)

The output feedback controller must be saturated to overcome the peaking phenomenon, see Remark 1. The following saturation function is used to saturate each estimate individually

 ^χis=Mχisat(χiMχi),  sat(y)={y, if% |y|≤1,sign(y), if |y|>1,

for , where the saturation bounds are chosen such that the saturation functions will not be invoked under state feedback. The feedback linearizing control then becomes .

## Iv Stability Analysis

The domain of operation will now be restricted and the stability of the state feedback control, observer estimates, and output feedback control will now be proven.

### Iv-a Restricting Domain of Operation

In order to ensure that the rotational feedback linearizing control law remains well defined, the domain of operation must be restricted leading to the following assumption

###### Assumption 1

The rotational reference signals remain in the set , where .

We will now establish that for sufficiently small initial tracking error, , the tracking error for all . Consequently the system will operate away from any Euler angle singularities. A Lyapunov function in the rotational error dynamics is taken as

 Vξ=ξTPξξ,where PξAξ+ATξPξ=−I6, (24)
 Aξ=[03I3−β1I3−β2I3],

where . A Lyapunov function in the translational error dynamics, assuming perfect rotational tracking, is taken as

 Vρ=ρTPρρ,where PρAρ+ATρPρ=−I6, (25)
 Aρ=[03I3−γ1I3−γ2I3],

where , and define the following two positive constants . Let the positive constant be chosen such that , and define , leading to the following lemma.

###### Lemma 1 (Restricting Domain of Operation)

For the feedback linearized rotational error dynamics (8) with initial conditions in the set the system state remains in the set for all . Similarly, the feedback linearized translational error dynamics (13) with initial conditions in the set the system state remains in the set for all .

Over the set , the Lyapunov function (24) is bounded by

 Vξ≤β21δ2+β1b2ξ+β1δ2+(β2δ+bξ)22β1β2, (26)

leading to the following bound on in

 Vξ≤cξ⇒β1ξT1ξ12β2≤cξ<β1δ22β2⇒∥ξ1∥<δ.

The Lyapunov function (24) also satisfies the following inequalities

 λmin(Pξ)∥ξ∥2≤Vξ≤λmax(Pξ)∥ξ∥2,˙Vξ≤−∥ξ∥2,

showing that is positively invariant, which ensures the reference signals of the rotational subsystem remain in the set of Assumption 1.

The translational error dynamics Lyapunov function, , satisfies the following inequalities by design

 λmin(Pρ)∥ρ∥2≤Vρ≤λmax(Pρ)∥ρ∥2,˙Vρ≤−∥ρ∥2.

By these inequalities, is compact and positively invariant. The domain of operation is now defined as the set .

###### Corollary 1

By Lemma 1 and Assumption 1, the rotational states remain in the set , where is some positive constant. Thereby ensuring singularities in the Euler angles are avoided and the feedback linearizing controller (7) remains well defined.

### Iv-B Stability Under State Feedback

###### Theorem 1 (Stability Under State Feedback)

For the closed-loop state feedback rotational and translational subsystems, (8) and (13), respectively, with initial conditions , the system states and will remain in for all and the states will exponentially converge to the origin.

The translational and rotational closed-loop systems can be written as a cascaded system in the following form

 (27)

Taking the Lyapunov functions for the rotational and translational subsystems, (24) and (25), a composite Lyapunov function can be written

 Vsf=d1Vξ+Vρ,d1>0. (28)

It can be shown following the generalized proof in the Appendix that for small enough, the entire closed-loop state feedback system converges exponentially to the origin, and the set is convex and positively invariant.

### Iv-C Convergence of Observer Estimates

The scaled error dynamics of the EHGO are written by making the following change of variables

 ηij=(χij−^χij)ϵϱi−j,~ωs=ωs−^ωs, (29)

for and . In the new variables, the scaled EHGO estimation error dynamics become

 (30)

where and (30) is an perturbation of the system

 ϵ˙ηi=Fiηi+Bi1[Δf+¯G(θ1)~ωs], (31)

where

 Fi=⎡⎢ ⎢ ⎢⎣−αi1I3I3⋯03⋮⋱⋮−αiϱi−1I303⋯I3−αiϱiI303⋯03⎤⎥ ⎥ ⎥⎦,Bi1=⎡⎣03⋮I303⎤⎦,
 Bi2=[03 ⋯ 03 I3]T,ηi=[ηi1 ⋯ ηiϱi]T.

The actuator error dynamics in terms of the error in squared rotor angular rate, , and rotor angular rate error, can be written as

 τm˙~ωs=−2~ωs+2Wdes~ω,τm˙~ω=−~ω, (32)

where , and is time-varying.

###### Lemma 2 (Stability of Actuator Dynamics)

For bounded input , i.e., , where is some positive constant, the actuator error dynamics (32) exponentially converge to the origin.

The actuator error dynamics are a cascaded system with the Lyapunov functions

 V~ωs=~ωTs~ωs,V~ω=~ωT~ω, (33)

and the composite Lyapunov function

 Vv=d2V~ωs+V~ω,d2>0. (34)

Using the general result for cascaded systems in the Appendix, it can be shown that the origin is exponentially stable.

Combining these states as , the entire actuator error dynamic system can be written as

 ˙v=Aωv,where Aω=[−2In2Wdes0n−In]. (35)

The EHGO estimation error system can now be written as a cascade connection of the actuator error system with the observer error dynamic system as follows

 ϵ˙ηi=Fiηi+Bi1[Δf+[¯G(θ1) 06×n]v],τm˙v=Aωv. (36)
###### Theorem 2 (Convergence of EHGO Estimates)

The estimates of the EHGOs, , in (31) converge exponentially to the states, , i.e., converges exponentially to the origin for any for sufficiently small .

Taking the cascaded system (36), cascade analysis is performed to arrive at a composite Lyapunov function for the cascaded system. The Lyapunov function for the actuator error system is (34). A Lyapunov function for the EHGO error system with the input, , set to zero is written as

 ϵV3=3∑i=1(ηi)TPi3ηi,Pi3Fi+FTiPi3=−I, (37)

 ϵ˙V3=3∑i=1−(ηi)Tηi+2(ηi)TPi3Bi1Δf.

The function is Lipschitz and can be bounded by

 ∥Δf∥≤Lη∥∥χ22−^χ22∥∥,
 ⇒∥Δf∥≤ϵLη∥∥η22∥∥≤ϵLη∥∥η2∥∥≤ϵLη∥η∥,

leading to the following bound on the derivative of the Lyapunov function

 ϵ˙V3≤3∑i=1(−∥∥ηi∥∥2+2ϵLη∥∥ηi∥∥2∥∥Pi3Bi1∥∥),=−∥η∥2+2ϵLη∥N∥∥η∥2,

where the elements of the diagonal matrix are . Since are tunable and is a design parameter, pick such that resulting in the following inequality

 ϵ˙V3≤−12∥η∥2. (38)

Taking the composite Lyapunov function as

 Vη=d3V3+Vv,d3>0, (39)

and following the Appendix, the origin of (31) is exponentially stable.

The complete scaled observer error system (30) is (31) with an added perturbation. The perturbation is bounded by for some and is continuous and bounded, therefore it can be treated as a nonvanishing perturbation. Following Lemma 9.2 in [18] and Theorem 2, the estimation error of the EHGO converges exponentially to an neighborhood of the origin.

###### Remark 1 (Peaking Phenomenon)

The EHGO estimation error can be bounded by

 |~χi|≤bϵϱ−1∥~χ(0)∥e−at/ϵ, (40)

for some positive constants and , by Theorem 2.1 in [19]. Initially, the estimation error can be very large, i.e., , but will decay rapidly. To prevent the peaking of the estimates from entering the plant during the initial transient, the output feedback controller needs to be saturated outside a compact set of interest. This is done by saturating the individual estimates and results in the output feedback controller .

There is some set that the estimation error will enter after some short time, , where . Since the initial state of resides on the interior of , choosing small enough will ensure that will not leave the set during the interval . This establishes the boundedness of all states.

### Iv-D Stability Under Output Feedback

The system under output feedback is a singularly perturbed system which can be split into two time-scales. The multi-rotor dynamics and control reside in the slow time-scale while the observer and actuator dynamics reside in the fast time-scale.

###### Theorem 3 (Stability Under Output Feedback)

The closed-loop system under output feedback, with initial conditions on the interior of will exponentially converge to the origin, when is chosen small enough.

The entire output feedback closed-loop system can now be written in singularly perturbed form

 ˙χ=Aχ+B[¯f(^ξ,θ1,˙θr)+¯G(θ1)u(^χs)]+¯Bφ(t,χ), (41a) ϵ˙ηi=Fiηi+Bi1[Δf+G(θ1)~ωs],