I Introduction
We envision a future in which aerial vehicles provide services such as parcel delivery, remote monitoring, and maintenance. In such scenarios, a groundbased vehicle may provide charging services for multirotors or act as a staging area for storing parcels. In these cases, the multirotors 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 multirotor and ground vehicle may be operated by different serviceproviders, 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 multirotor.
The multirotor may only be able to measure the position of the ground vehicle in realtime. However, to achieve efficient control performance, higherorder 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 multirotor’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 multirotor UAV control design; see [1, 2] for a survey. The problem of autonomous landing of a multirotor 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 timevarying [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 multirotor 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 multirotor 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, feedforward control for trajectory tracking, and multirotor 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 multirotor 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 multirotor 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 coplanar. To handle the underactuation, as discussed below, the rotational dynamics are controlled to create a virtual control input for the translational dynamics.
Iia Rotational Dynamics
The rotational dynamics of the multirotor are [13]
(1) 
where is the inertia matrix, is the torque applied to the multirotor and is the angular velocity, each expressed in the bodyfixed frame.
Consider the orientation of the multirotor expressed in terms on Euler angles . The angular velocity is related to the Euler angle rates in the inertial frame by
where denote , , , respectively. The rotational dynamics (1) can be equivalently written in terms of Euler angles
(2) 
where is an added term to represent the lumped rotational disturbance.
Let be the rotational reference signal. Define rotational tracking error variables
The rotational dynamics (2) can be written in terms of tracking error
(3) 
where
IiB Translational Dynamics
The translational dynamics of the multirotor are [13]
(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 multirotor body frame with respect to the inertial frame,
Let be the translational reference signal. Define translational error variables
The translational dynamics (4) can be written in terms of tracking error
(5) 
IiC Reference System Dynamics
We assume that the reference trajectory that the multirotor UAV will track is generated by the system
(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.
Iiia Rotational Control
The rotational control feedbacklinearizes the rotational tracking error dynamics (3) as follows
(7) 
for constants where . This results in the following closed loop rotational tracking error system
(8) 
IiiB 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)
(9) 
The translational subsystem dynamics can be redefined in terms of the nominal translational model with the addition of an error term as follows
(10) 
where
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 ^{1}^{1}1During normal operation, , ensuring differentiability of .
IiiC Extended HighGain Observer Design
A multiinput multioutput EHGO is designed similar to [14, 15] to estimate higherorder 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 multirotor model for EHGO design. The actuator dynamics reside in the same timescale as the EHGO, and therefore can not be ignored in the EHGO dynamics.
The actuators used on multirotor UAVs are Brushless DC (BLDC) motors, which require electronic speed controllers. These controllers introduce dynamic delays [17] of the following form
(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(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 bodyfixed 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
(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 multirotor aerial platform, allowing the squared rotor angular rates to be taken as the control input to the model
(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 overdetermined system which admits infinitely many solutions. In this case, we focus on the minimum energy solution , where is the pseudoinverse 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
(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
where
where is designed by choosing such that
(19) 
is Hurwitz and .
IiiD 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
(20) 
where . The output feedback translational controller becomes
(21) 
where the forcing function, , is now defined using tracking error estimates as follows
(22) 
These control signals result in the following output feedback controller
(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
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.
Iva 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
(24) 
where . A Lyapunov function in the translational error dynamics, assuming perfect rotational tracking, is taken as
(25) 
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)
Over the set , the Lyapunov function (24) is bounded by
(26) 
leading to the following bound on in
The Lyapunov function (24) also satisfies the following inequalities
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
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.
IvB Stability Under State Feedback
Theorem 1 (Stability Under State Feedback)
The translational and rotational closedloop 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
(28) 
It can be shown following the generalized proof in the Appendix that for small enough, the entire closedloop state feedback system converges exponentially to the origin, and the set is convex and positively invariant.
IvC Convergence of Observer Estimates
The scaled error dynamics of the EHGO are written by making the following change of variables
(29) 
for and . In the new variables, the scaled EHGO estimation error dynamics become
(30) 
where and (30) is an perturbation of the system
(31) 
where
The actuator error dynamics in terms of the error in squared rotor angular rate, , and rotor angular rate error, can be written as
(32) 
where , and is timevarying.
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
(33) 
and the composite Lyapunov function
(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
(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
(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
(37) 
leading to
The function is Lipschitz and can be bounded by
leading to the following bound on the derivative of the Lyapunov function
where the elements of the diagonal matrix are . Since are tunable and is a design parameter, pick such that resulting in the following inequality
(38) 
Taking the composite Lyapunov function as
(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
(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.
IvD Stability Under Output Feedback
The system under output feedback is a singularly perturbed system which can be split into two timescales. The multirotor dynamics and control reside in the slow timescale while the observer and actuator dynamics reside in the fast timescale.
Theorem 3 (Stability Under Output Feedback)
The closedloop 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 closedloop system can now be written in singularly perturbed form
(41a)  
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