## 1 Introduction

The problem of slung load transportation arises in a variety of essential tasks, such as transportation of containers in harbors (1), aerial delivery of supplies in search-and-rescue missions (2), and landmine detection (3). The suspended load is usually connected to the mobile platform by means of a rope, considerably changing its dynamic behavior and adding unactuated degrees of freedom to the whole system. Moreover, the rope is a non-rigid body and is not always taut, which increases the task challenge. Several studies can be found in the literature, concerning different modeling approaches and control strategies for load transportation using overhead cranes (4), robotic manipulators (5), and aerial vehicles (6; 7).

An important issue in slung load transportation is the recurrent necessity of knowing the load position to accomplish the task, mainly when precise positioning of the load is required. Since available sensors are often embedded in the mobile platform, information on the load position may not be directly obtained. The problem of estimating the load position then arises, being commonly addressed through visual systems and Bayesian state estimators. The Kalman filter is employed in

(8) for state estimation of a quadrotor unmanned aerial vehicle (UAV) with suspended load, in which measurements are provided by external cameras and sensors embedded at the aircraft. Considering a helicopter with suspended load platform, (9) designs a data fusion algorithm based on the unscented Kalman filter (UKF) to estimate the load’s position and velocity with measurements from an inertia measurement unit (IMU) and a vision system, both located at the helicopter. In (10), algorithms based on the UKF are proposed for estimation of the full state vector of a helicopter with suspended load, with measurements provided by a Global Positioning System (GPS), a magnetometer, a camera, an IMU on the helicopter and another one on the load. Kalman filtering algorithms require knowledge on statistical properties of existing process and measurement disturbances, which may not be easily obtained. In view of the exposed, the present work pursues set-membership estimation approaches, which require knowledge only on bounds of existing disturbances. These techniques are based on the construction of sets that include, with guarantee, the system states consistent with available measurements

(11; 12). This work extends the zonotopic state estimation strategy proposed in (11) to receive measurements provided by sensors with different sampling times.The versatility and autonomous operation of UAVs are useful advantages in aerial load transportation. The main control design objectives in the literature include path tracking of the UAV with load swing attenuation (6; 13; 14; 15; 16; 17; 18; 19), obstacle avoidance (20; 21), transportation by multiple aircrafts (22; 23), and trajectory tracking of the suspended load (24; 25; 26)

. This paper focuses on the latter, which is the appropriate goal in tasks requiring precise maneuvering of the load. In contrast to the swing attenuation problem, the knowledge on the load position is usually required for such purpose. A model-free, open-loop approach based on trajectory generation by machine learning is proposed in

(24) for path tracking of a suspended load using a quadrotor UAV. However, the lack of a feedback structure prevents compensation of external disturbances affecting the load. A nonlinear cascade control strategy is designed in (25), based on model decoupling, for trajectory tracking of a suspended load using a quadrotor UAV. Nevertheless, compensation of unmodelled dynamics and external disturbances is not addressed in the proposed strategy, and convergence issues are well known for cascade control systems. Assuming the aircraft as a system actuated by total thrust and orientation, (26) proposes another nonlinear solution to the problem of suspended load path tracking using a quadrotor UAV. Nevertheless, such assumption is valid for a very limited repertory of mechanical systems, which do not comprise the convertible UAV configuration addressed by this work.Most of the unmanned aerial vehicles used in load transportation tasks are in helicopter and quadrotor configurations. These rotary-wing UAVs have vertical take-off and landing (VTOL) and hovering capabilities, and achieve high maneuverability in low velocities. However, due to their limited flight envelope, such UAVs are not appropriate for missions that require long distance traveling, such as deployment of supplies to risky zones. To overcome such constraint, researches are looking into the design of small-scale convertible aircrafts, being the tilt-rotor configuration among the most popular ones (27; 28; 29). Provided with both fixed and rotary wings, tilt-rotor UAVs achieve an enlarged flight envelope by switching between helicopter and airplane flight-modes through tilting of the thrusters. However, such advantages come with several design and control challenges, since these aircrafts are complex, underactuated mechanical systems with highly coupled dynamics. Additionally, when these UAVs are connected to a payload through a rope, the dynamic behavior of the system varies due to the load’s swing, which can destabilize the whole system if it is not well attenuated. A model predictive control (MPC) strategy is designed in (16) for path tracking of a tilt-rotor UAV with suspended load, in which the aircraft tracked a desired trajectory, while the load remained stable. A cascade strategy composed of three levels of feedback linearization controllers is proposed in (14), for trajectory tracking of a tilt-rotor UAV with load swing attenuation. The problem of path tracking of a suspended load using a tilt-rotor UAV is solved in (30), in which a model predictive controller is designed, taking into account time-varying load’s mass and rope’s length, and estimating the load’s position and orientation by means of an unscented Kalman filter. However, the state estimation is not guaranteed, and nothing can be said about the transient response of the closed-loop system. The present work addresses the problem of trajectory tracking of a suspended load using a tilt-rotor UAV as mobile platform, with guaranteed time-response properties, compensation of unmodelled dynamics and external disturbances, and state estimation of the load’s position and orientation, based on the set-membership approach to perform the task.

This paper is an extended, consolidated version of the previous work presented in (31). To solve the aforementioned challenges, this work develops the whole-body dynamic equations of a tilt-rotor UAV with suspended load, from the perspective of the load. The position and orientation of the latter are chosen as degrees of freedom of the system, yielding a nonlinear state-space representation with these variables among the system states. As shown in (31; 30), this choice allows state-feedback control strategies to steer the trajectory of the load with respect to an inertial reference frame. In contrast to previous works, a reduced number of assumptions is made with respect to the physical system. This work designs a discrete-time state-feedback mixed control strategy with an enlarged domain of attraction for path tracking of the suspended load with disturbance rejection and guaranteed time-response properties, taking into account the desired accelerations of the load in the control design through an uncertain linear parameter-varying framework. In addition, this work proposes a zonotopic state estimation strategy to estimate the load’s position and orientation when available measurements are provided by sensors with different sampling times. To demonstrate and compare the performance of the proposed state estimator, a Kalman filter is also designed. The performance of the proposed strategies are demonstrated through numerical experiments, performed in a platform based on the Gazebo simulator and on a Computer Aided Design (CAD) model of the system. The contributions of this work can be summarized as: (i) a detailed modeling from the load’s point of view that comprises the dynamic coupling of the load and the tilt-rotor UAV, with few assumptions on the system, leading to an input-affine state-space representation with the load’s position and orientation as state variables; (ii) a set-membership state estimation strategy based on zonotopes to provide the load’s position and orientation, formulated for measurements with different sampling times and unknown-but-bounded disturbances; (iii) a single-loop state-feedback control strategy for trajectory tracking of the suspended load, robust to unmodeled dynamics, parametric uncertainties and external disturbances, with enlarged domain of attraction; and (iv) formulation of pole placement constraints in discrete-time for overshoot requirements.

This paper is organized as follows: the dynamic equations of the tilt-rotor UAV with suspended load are developed in Section 2, from the perspective of the load; Section 3 proposes the zonotopic state estimation strategy to provide the entire state vector, formulated for sensors with different sampling times, and also the derivation of a Kalman filter is presented for comparison purposes; Section 4 presents the design of the state-feedback mixed control strategy with constraints in pole placement for path tracking of the suspended load, with feedback from estimated states; Section 5 presents results from numerical experiments to demonstrate and compare the performance of the zonotopic state estimator along with the designed controller; and Section 6 concludes the work.

## 2 System modeling from the perspective of the load

This section develops the equations of motion of the tilt-rotor UAV with suspended load, formulated from the perspective of the load. The system is regarded as a multi-body mechanical system, and its dynamic equations are obtained through the Euler-Lagrange formulation. The dynamic coupling between the aircraft and the load is taken into account naturally. By choosing the latter’s position and orientation as degrees of freedom, nonlinear state-space equations are obtained with these coordinates represented by state variables. The aircraft’s position and orientation are described only with respect to the load.

### 2.1 System description

The tilt-rotor UAV with suspended load is regarded as a multi-body mechanical system composed of four rigid bodies: (i) the aircraft’s main body, composed of Acrylonitrile Butadiene Styrene (ABS) structure, landing skids, batteries, instrumentation and electronics; (ii) the right thruster group, composed of the right thruster and a tilting mechanism (a revolute joint); (iii) the left thruster group, composed of the left thruster and a tilting mechanism; and (iv) the suspended load group, composed of the load and the rope. The actuators of the system are the aircraft’s thrusters and servomotors. The Computer Aided Design (CAD) model of the tilt-rotor UAV with suspended load is shown in Figure 1.

For control purposes, the rope is assumed to be rigid and massless. Moreover, the aircraft’s center of mass is displaced from its geometric center in order to improve pitch moment and to yield non-null equilibria for the angular positions of the tilting mechanisms and pitch angle. This mechanical feature improves the controllability of the aircraft in hover flight, yielding horizontal projections of the thrust forces without tilting the thrusters.

### 2.2 Kinematics from the perspective of the load

The approach presented in this paper consists in formulating the forward kinematics of the system considering the load as a free rigid body, while the tilt-rotor UAV as a multi-link system rigidly coupled to it. For such objective, six reference frames are defined, shown in Figure 2: (i) the inertial reference frame, ; (ii) the suspended load group center of mass frame, ; (iii) the aircraft’s geometric center frame, ; (iv) the main body center of mass frame, ; (v) the right thruster group center of mass frame, ; and (vi) the left thruster group center of mass frame, . Three auxiliary frames are also defined: (i) a reference frame attached to the point of connection of the rope to the aircraft, ; (ii) a reference frame attached to the tilting axis of the right servomotor, ; and (iii) a reference frame attached to the tilting axis of the left servomotor, .

The position of the load with respect to the inertial frame is denoted by . The displacement vector from to corresponds to the rope, and is defined in by , where is the rope’s length. The displacement vectors from to , from to , from to , and from to are model parameters of the tilt-rotor UAV, denoted by , , , , respectively, expressed in the respective previous frames, with .

The orientation of the load with respect to is parametrized by Euler angles, , using the convention about local axes. The associated rotation matrix is defined by

(1) |

Since the rope is assumed rigid, it cannot twist. Thus, the orientation of frame with respect to , corresponding to the orientation of the UAV with respect to the rope, is parametrized by two angles, , such that

(2) |

The orientations of the thrusters’ groups with respect to are defined by

(3) |

where and are the tilting angles of the right and left servomotors, respectively, and is a fixed inclination angle of the thrusters towards the aircraft geometric center, designed to improve the aircraft controllability (32). The reference frames , , and are parallel to each other and attached to the same rigid body, thus the relative orientation is null, i.e., ^{2}^{2}2In this work,

is an identity matrix with dimension

, denotes an by matrix of zeros, and is an by matrix of ones.. Similarly, we have that . Then, , and , for .Given that , where denotes the angular velocity of with respect to , expressed in , and

denotes an operator that maps a vector to a skew-symmetric matrix

(33), we have . Similarly, the remainder angular velocities of the system are given by , , , and , with(4) |

and . Moreover, , , and .

The defined rigid transformations yield the forward kinematics of points that belong to each rigid body, given by

(5) | |||

(6) | |||

(7) |

where and , is the position of a point that belongs to the suspended load body, and belongs to the rigid body with attached frame .

The generalized coordinates of the system are chosen according to the defined rigid transformations. Note that, since the load’s position and orientation are defined with respect to , such variables are independent of each other. Therefore, these are included in the generalized coordinates vector, which is chosen as

(8) |

Due to the chosen perspective, the position and orientation of the aircraft with respect to are not degrees of freedom of the system, being not included in (8). Consequently, their time evolution will not be described explicitly by the obtained equations of motion.

### 2.3 Equations of motion

In order to derive the equations of motion through the Euler-Lagrange formulation, on one hand the kinetic and potential energies of each body of the mechanical system must be obtained. These energies can be computed for the -th rigid body through the volume integrals (33)

(9) | |||

(10) |

respectively, where denotes its density, its volume, its mass, corresponds to the gravitational acceleration vector expressed in , and is the position vector obtained from the forward kinematics of the origin of . The quadratic terms , , and are computed using the time derivatives of (5)–(7

), respectively. Moreover, by defining the inertia tensors

and , taking into account the parallel axis theorem (34), yields , , , , and , .The total kinetic energy of the system is given by , in which the kinetic energy of each rigid body is obtained using (9). Then, by defining , , , and , and using several properties of skew-symmetric matrices (33), writing the total kinetic energy as leads to the inertia matrix ,

(11) |

where denotes elements that are deduced by symmetry, and

(12) | ||||

(13) | ||||

(14) | ||||

(15) | ||||

(16) | ||||

(17) | ||||

(18) |

with , , , , , and . Note from the inertia matrix that the dynamics of the four rigid bodies are coupled, allowing one to consider the existing interactions in the control design without the need of cascade control structures.

The Coriolis and centripetal forces matrix, , is obtained through Christoffel symbols of the first kind (33). The element of its -th row and -th column is given by

(19) |

where , with being an element of the inertia matrix (11).

The forward kinematics of each body’s center of mass is obtained using (5)–(7), with the potential energies of the load and of each body of the aircraft then computed using (10). The total potential energy of the system is given by , yielding

(20) |

The gravitational force vector is then obtained through

(21) |

On the other hand, this work assumes that the system is also subject to generalized forces from the aircraft’s actuators, viscous friction, and external disturbances affecting the load. Therefore, let and denote non-conservative force and torque vectors, respectively, actuating on the mechanical system. Furthermore, let denote the point of application of , and be a reference frame rigidly attached to the body to which is applied. The contributions of and to the generalized forces can be computed through (35)

(22) | |||

(23) |

where , and .

The input forces and torques of the system are the right and left thrust forces, denoted by and , and right and left servomotor torques, denoted by and , respectively. They are expressed in their respective frames by , , , and , where (see Figure 2). Therefore, in the inertial reference frame, we have

(24) | |||

(25) | |||

(26) | |||

(27) |

Besides, this work assumes that the thrust forces are applied to the centers of mass of the respective thrusters’ groups, i.e, the origins of and . To obtain the corresponding mappings to generalized forces, it is necessary to compute and . Firstly, making in the time derivative of (7) leads to

(28) | ||||

Analogously, for the left thrust force, we have

(30) |

The servomotor torques are applied to the respective thrusters’ bodies, and opposite torques due to reaction are applied to the aircraft’s main body. These torques are mapped to generalized forces through (23). From the addition of angular velocities (33), we have

(31) | |||

(32) | |||

(33) |

Recalling that , and , leads to

(34) |

(35) |

This work also takes into account drag torques generated by the propellers, which are reaction torques applied to the thrusters’ bodies, due to the blades’ acceleration and drag (36). Assuming steady-state for the angular velocity of the blades, the drag torques are given in the thrusters’ reference frames by

(36) |

where and are parameters obtained experimentally, and and are given according to the direction of rotation of the corresponding propeller: if counter-clockwise, ; if clockwise, . In the inertial reference frame, we then have

(37) | |||

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