Adrian Manzanilla
email: amanzanilla@cinvestav.mx

Sergio Reyes

Miguel Garcia
email: miguelgarciar@cinvestav.mx

Diego Alberto Mercado
email: diego.mercado@cimat.mx

Rogelio Lozano

Autonomous Navigation for Unmanned Underwater Vehicles: Real-Time Experiments Using Computer Vision

Keywords: Marine robotics, autonomous vehicle navigation, AUVs, trajectory tracking, visual-based navigation

This letter studies the problem of autonomous naviga-tion for unmanned underwater vehicles, using computer vision for localization. Parallel tracking and mapping is employed to localize the vehicle with respect to a visual map, using a single camera, whereas an extended Kalman filter (EKF) is used to fuse the vi-sual information with data from an inertial measurement unit, in order to recover the scale of the map and improve the pose estima-tion. A proportional integral derivative controller controller with compensation of the restoring forces is proposed to accomplish trajectory tracking, where a pressure sensor and a magnetometer provide feedback for depth control and yaw, respectively, while the remaining states are provided by the EKF. Real-time experiments are presented to validate the navigation strategy, using a commer-cial remotely operated vehicle (ROV), the BlueROV2, which was adapted to perform as an autonomous underwater vehicle with the help of the robot operative system.

I. INTRODUCTION

In the last decades, the scientific and tecnological developments in underwaters robotics have played a very important role in many civilian and military applications, such as the attention of disasters, environmental inspection and industrial tasks. Within the Unmanned Underwater Vehicles (UUV) there are several classifications, among which stand out the Remotely Operated Vehicles (ROVs) and Autonomous Underwater Vehicles (AUVs), which can be used for coast patrolling, surveillance, bridge inspection, marine cartography, 3D reconstruction, mapping and tracking [START_REF] Melo | Survey on advances on terrain based navigation for autonomous underwater vehicles[END_REF], etc.

In order to increase the level of autonomy in navigation, localization arises as one of the main challenges. The use of the mathematical model to improve the measurement in some navigation techniques is a common approach, such is the case of dead reckoning, which also makes use of inertial measurements to integrate accelerations and velocities to estimate the position, although they present unbounded error accumulation (drift), and can not be thrust in long term missions. Alternatively, the Doppler Velocity Logs (DVLs) are used to estimate the velocity of the vehicle using acoustic signals directed to the bottom of the ocean and observing the Doppler effect. Nevertheless, integration is required to recover position, resulting in error drift.

Due to these issues, most of the localization methods are based on acoustic sensors, which are mostly heavy, large, expensive and with some other deficiencies such as: a small bandwidth, high latency in water, reduced battery life as they increase the power used to transmit the ultra sound signals, variable sound speed due to temperature and salinity. Some systems track pings from underwater units, measuring the Time-Of-Arrival (TOAs). Different solutions enter this classification depending on the number of transducers and receivers, and the locations of them, for example, using a receive-only Ultra-Short Baseline (USBL) array, to determine the location of the objects and then transmit it to the vehicle using acoustic modem technology [START_REF] Megdal | Underwater acoustic navigation systems and methods[END_REF]. Other related alternatives are the Long-Baseline (LBL) systems, USBL systems and the related Super-Short-Baseline (SSBL). Global Positioning Systems (GPS) Intelligent Buoys (GIB) work as an inverted LBL device where the transducers are replaced by floating buoys, self-positioned by GPS. The tracked position is calculated in real-time at the surface from the TOAs of the acoustic signals sent by the underwater vehicle, and acquired by the buoys [3]- [START_REF] Paull | AUV navigation and localization: A review[END_REF].

Vision-based localization is a powerful alternative for autonomous navigation. In some works, the use of visual Simultaneous Localization and Mapping (vSLAM) [START_REF] Taketomi | Visual slam algorithms: A survey from 2010 to 2016[END_REF] has been proposed, which can be implemented through the use of conventional cameras, providing a significant amount of information at low cost, such as color, edges, size, texture, shape, etc. This data allows a vehicle to be located in a certain way, using a sequence of captured images that estimates the movement of the vehicle in a mostly static scene. The implementation of SLAM in real underwater environments can be considered an open problem of great interest in robotics [START_REF] Guth | Underwater slam: Challenges, state of the art, algorithms and a new biologicallyinspired approach[END_REF].

Within the literature of vSLAM applied to underwater vehicles, we can find in [START_REF] Concha | Real-time localization and dense mapping in underwater environments from a monocular sequence[END_REF] the use of a direct mapping method that estimates in real time a dense reconstruction in 3D of an underwater scene and the position of the vehicle, using a monocular camera. In [START_REF] Negre | Cluster-based loop closing detection for underwater slam in feature-poor regions[END_REF], a technique for loop closures grouping the key points is presented, with the purpose of validating it in an AUV, in marine environments with low textures. Based on insights from [START_REF] Garcia | Detection of interest points in turbid underwater images[END_REF], they use Speeded-Up Robust Features (SURF) for feature detection, showing that the algorithm is capable of localizing an AUV in different environments. However, this is not a purely camera driven SLAM solution since they incorporate a DVL and sonar into their framework. The authors in [START_REF] Silveira | An open-source bio-inspired solution to underwater slam[END_REF] propose an underwater SLAM called DolphinSlam, which claims to be able to improve the perception at low resolution, ambiguity of perception and long-term tasks for a 2D plane, using a neural network.

The application of these techniques depends mainly on the computational cost, light and image conditions, i.e., illumination, shadows, blurriness, turbidity, scale, rotation, reflectance, among some other environmental factors. These are negative effects that affect the features detection and alter the estimation of the pixel shift between image frames, leading to errors in the estimation [START_REF] Aqel | Review of visual odometry: Types, approaches, challenges, and applications[END_REF].

A qualitative study of visual navigation algorithms, using SLAM in the underwater scenario, is presented in [START_REF] Kahlefendt | Implementation and evaluation of monocular SLAM for an underwater robot[END_REF], where several SLAM algorithms are tested off-line in data sets captured by ground, aerial and marine vehicles. The authors encounter that the best results for underwater applications are achieved by PTAM (Parallel Tracking and Mapping), ORB-SLAM, COLMAP, g2o and Ceres algorithms. The document further studies the ORB-SLAM algorithm and implements it on the BlueROV2.

In this work, a conventional low-cost monocular camera is used for underwater location, based on the pose estimation obtained from the PTAM algorithm [START_REF] Klein | Parallel tracking and mapping for small AR workspaces[END_REF], fused with data provided by the Inertial Measurements Unit (IMU) using an Extended Kalman Filter (EKF). This method, originally conceived for augmented reality, is used to estimate the scaleless pose of the camera in an unknown scene. The algorithm can be divided into two separate tasks, one in charge of tracking and the other in mapping, processed in parallel threads in a computer. This algorithm can be employed even in a phone camera [START_REF] Klein | Parallel tracking and mapping on a camera phone[END_REF]. PTAM has been successfully implemented for localization of aerial vehicles, fusing it with inertial measurements to recover the scale and improve the estimation [START_REF] Rodríguez-Canosa | A real-time method to detect and track moving objects (DATMO) from unmanned aerial vehicles (UAVS) using a single camera[END_REF]. To our knowledge, this technique has not been applied to underwater vehicles before. Estimation from the EKF is complemented with a pressure sensor and a magnetometer to measure depth and yaw respectively. Once a suitable feedback of the pose of the vehicle is available, a Proportional Integral Derivative controller (PID) with compensation of the restoring forces is used to accomplish trajectory tracking in fully autonomous operation. Realtime experiments are provided to validate the proposed strategy, in a customized BlueROV2 vehicle (see Fig. 1), adapted to perform autonomously. The proposed scheme has potential application in inspection missions for flat structures such as hulls, damns and bridges, as well as coast patrolling, among others.

The outline of the document is as follows: in Section II the general dynamic model of the vehicle is presented. In Section III the implementation of the vision-based localization and the data fusion are described, while in Section IV the control strategy is provided. In Section V the characteristics and modifications of the prototype are discussed. In Section VI, the experimental results of trajectory tracking for fully autonomous operation using visual-based navigation are presented. Finally, a discussion about this work and future developments is detailed in Section VII.

II. DYNAMIC MODEL

The dynamic model for an UUV must consider a body reference frame O B and an inertial one O I . This model considers the hydrodynamic effects, generalized inertial forces, gravity, buoyancy and force given by the thrusters, as shown in Figure 2. The dynamics of the vehicle can be represented based on the matrix equations proposed by Fossen [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF], which are based on the general notation for submarine vehicles populated by [START_REF] Society | Nomenclature for treating the motion of a submerged body through a fluid[END_REF], and can be written as follows:

M ν + C(ν)ν + D(ν)ν + g(η) = τ + ω η = J(η)ν (1)

A. The BlueROV2 Underwater Vehicle

For the particular case of the BlueROV2, it is composed of six thrusters from which 1 to 4 are used for the rotational movement in yaw, as well as the forward-reverse and lateral movements, while to raise and descend, the thrusters 5 and 6 are used. The configuration of the thrusters allows the vehicle to move freely within these four degrees of freedom, as illustrated in Figure 3. By design the submarine robot has symmetry in two of its planes and is stable mechanically in the roll and pitch angles, thanks to a careful selection of the center of buoyancy, i.e., φ, θ ≈ 0, which implies that the submarine has the ability to return to steady state after a disturbance (movement generated by water flow), therefore K, M, p, q, ṗ, q ≈ 0. From now on, we consider only the reduced state vector η = [x, y, z, ψ] T and the reduced force vector τ = [X, Y, Z, N ] T .

From equation (1), using the short notation sin(a) = s(a) and cos(a) = c(a), following the methodology as in [START_REF] Manzanilla | Design and control of an autonomous underwater vehicle (AUV-UMI)[END_REF] and considering only the reduced system, the translational and rotational dynamics particularly for this robot can be expressed as follows

X = (m -X u ) u + X v v -(my g -X ṙ ) ṙ + (Y v v + mw)r -(X u )u + f B s(θ) (2) Y = (m + Y v ) v + (mx g -Y q ) ṙ -(X u u -(mu))r -(Y v )u -f B c(θ)s(φ) (3) Z = (m -Z ẇ ) -(Z w )w -f B c(θ)c(φ) (4) N = (-my g -X ṙ ) u + (mx g -Y ṙ ) v + (I z z -N ṙ ) ṙ -(Y v v -mv)u + (X u u + mu)v -(N r )r (5)
where X u , Y v , Z w , N r are the linear damping coefficients; Taking into account the physical properties of the vehicle and that only slow motions below 2m/s are considered, moreover, the origin of the body fixed frame O B coincides with the center of gravity C G , and is colinear with the main axes of the inertial reference frame O I , the reduced vector of forces and moments τ that acts in the vehicle can be expressed as:

X k , Y k , Z k , K k , M k , N k with k : u, v
τ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -F 1 c(ϕ 1 ) -F 2 c(ϕ 2 ) -F 3 c(ϕ 3 ) -F 4 c(ϕ 4 ) F 1 s(ϕ 1 ) -F 2 s(ϕ 2 ) + F 3 s(ϕ 3 ) -F 4 s(ϕ 4 ) F 5 + F 6 ξ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (6) with ξ = l x (F 1 c(ϕ 1 ) -F 2 c(ϕ 2 ) -F 3 c(ϕ 3 ) + F 4 c(ϕ 4 )) + l y (F 1 s(ϕ 1 ) + F 2 s(ϕ 2 ) -F 3 s(ϕ 3 ) -F 4 s(ϕ 4 )
). F i:1,...6 are the forces generated by each thruster; ϕ i:1,...4 are the angles between the x B axis and the force F i:1,...4 applied in the AUV; l x and l y are the distances between the horizontal and vertical thrusters with the center of gravity C G (see Figure 3). Note that all propellers are in the same plane with the C G .

III. VISION-BASED LOCALIZATION

In order to accomplish fully autonomous navigation underwater, real-time localization arises as one of the main challenges. Given that acoustic sensors, which are the preferred solution, are normally heavy, large and expensive, computer vision presents an interesting alternative.

PTAM is an algorithm developed to estimate the pose of a camera in an unknown environment and was originally conceived for augmented reality (AR), the main advantages of this algorithm are that it does not require special markers, previously known maps or structured scenarios, and it only requires a single camera. The operation basically consists of separating the tracking and mapping into two separate tasks, processed in parallel, generating a 3D map of the observed characteristic points [START_REF] Klein | Parallel tracking and mapping for small AR workspaces[END_REF]. The generated map estimates a dominant visual plane from the feature points while refining the map, having a fast, accurate and robust tracking.

The monocular vision algorithm provides a scaleless estimation of the pose of the camera, without drift over time, at a slow rate. With the purpose of recovering the scale of the position and to improve the estimation, the PTAM algorithm is fused with inertial measurements from the IMU (accelerometers and gyroscopes) using an EKF. This allows as well to increment the estimation rate, taking advantage of the faster IMU measurements. In order to do so, the sensor fusion libraries from [START_REF] Weiss | Real-time metric state estimation for modular vision-inertial systems[END_REF], [START_REF] Weiss | Versatile distributed pose estimation and sensor self-calibration for autonomous MAVs[END_REF] are used. This libraries where developed for aerial vehicles and are also available as a ROS package. The PTAM estimation is used as the update sensor, while the accelerations and angular speeds are used directly in the prediction step, running at a faster rate. Another interesting advantage of this approach is that it auto-calibrates itself since it does not only attempt to estimate the pose and velocity, it also estimates the missing scale and the intrinsic rotations between the different coordinate frames, ensuring long-term operations without error drift (only spacial drift may be present from the visual algorithm).

IV. CONTROL STRATEGY

Provided that a suitable state feedback is available from the IMU, pressure sensor and vision algorithm with the EKF, the control objective is to track a time varying trajectory for monitoring and inspection applications. It is common to use sensor measurements referred to the Euler angles (for orientation) and the inertial frame (for the position). Then, the reduced dynamic model of an AUV is expressed in the frame fixed to the ground using the kinematic transformations proposed by Fossen [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF], and is represented by the following equation:

M η (η) η + C η (ν, η) η + D η (ν, η) η + g η (η) = τη (7)
where the subindex η represents coordinates in the inertial frame. In this section, the implementation of a PID controller is approached for two cases, depth and yaw stabilization, and for trajectory tracking in x and y position. The PID controller with compensation of the restoring forces is implemented in the underwater robot for regulation, and is given by:

τη = g η (η) -J T (η) K p e + K d ė + K i e ( 8
)
where e = η -ηd defines the error and the feedback gains K p , K d and K i are diagonal positive definite matrices such that K j :{p,d,i} = diag(k j x , k j y , k j z , k j ψ ). The proposed controller renders the system asymptotically stable, as can be probed by Lyapunov's second method of stability and LaSalle's invariance principle (see [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF], chapter 12).

V. EXPERIMENTAL PLATFORM

The robot used for this work is the BlueROV2, by Blue Robotics, presented in Figure 4. It is a small-size ROV of approximately 45.71 × 33.81 × 22.1cm [START_REF]Bluerov2[END_REF]. The electronic components are installed in a hermetic acrylic tube, for the battery there is another acrylic tube which has a direct connection to the electronic tube. The prototype has the capacity of sinking up to 100 m depth according to the manufacturer. It is equipped with six Blue Robotics T200 thrusters configured as shown in Figure 3. The main physical parameters of the prototype are presented in Table I, where SI base units are adopted.

The BlueROV2 uses the open source software ArduSub installed on a PixHawk autopilot, furthermore, it runs ROS in a Raspberry Pi on-board computer, while the MAVLink protocol is used to communicate the on-board computer and the autopilot, to obtain the sensors information. Via ROS the vehicle is connected to a ground station using a Fathom-X tether interface trough 100 m of wire providing communication with a bandwidth of 80 Mb/s. Video from a frontal camera, with a resolution of 640 × 480, is transmitted to the ground station at a frame rate of 15fps. The vision algorithm, along with the EKF and the control strategy are computed in the ground station and return the Pulse-Width Modulation (PWM) value to the vehicle's thrusters.

Figure 5 displays the main ROS nodes running in the ground station, where the BlueROV2 node provides the sensor measurements from the vehicle, including the video stream which is fed to the PTAM node. The sensor fusion node reads the pose estimation from PTAM and merges it with inertial measurements from the embedded IMU. The control node was developed for this work and uses the pose estimation from the EKF, along with the pressure and magnetometer measurement to accomplish trajectory tracking using a PID with compensation of the restoring forces, and delivers the PWM signals to the ROV's thrusters. Real-time visualization of the states is possible thanks to the rqt_plot node, while online parameter tuning is available through a Graphical User Interface (GUI).

VI. REAL-TIME EXPERIMENTS

In this section, the validity of the vision algorithm for underwater applications is demonstrated. The main objective is to autonomously track a trajectory, using vision-based localization and applying the control law given by [START_REF] Concha | Real-time localization and dense mapping in underwater environments from a monocular sequence[END_REF].

The test was performed in an olympic pool whose dimensions are (25 × 50 × 1.6m). It consisted in stabilizing yaw and depth at a constant reference ψ d = 110 • , z d = -35cm, respectively, and performing trajectory tracking in (x, y). For the trajectory tracking of the longitudinal movements the desired signal is given by: x d = 0.7 * cos(2πt/25) and y d = 0.5 * sin(2πt/25). Several perturbations were applied during the test to further test the robustness of the proposed algorithms, at seconds 310, 400 and 475. The controller parameters were heuristically adjusted and are presented in Table II, using SI base units. The covariance matrix for the EKF update step is provided by the PTAM algorithm, while for the prediction step the noise variances for the accelerometers σ respectively. For more details about the EKF implementation see [START_REF] Weiss | Real-time metric state estimation for modular vision-inertial systems[END_REF], [START_REF] Weiss | Versatile distributed pose estimation and sensor self-calibration for autonomous MAVs[END_REF].

In Fig. 6, we can observe the x, y and z position of the vehicle during the path tracking, as measured by the PTAM (pink dotted) and estimated by the EKF (dashed blue) against the desired trajectory (solid red line). We can appreciate that the proposed strategy is good enough to accomplish this task, with bounded error, in spite of unexpected disturbances at time 310s, 400s and 475s. The orientation of the AUV is depicted in Fig. 7. There we can appreciate the mechanical stability property of the vehicle in roll φ and pitch θ angles, achieved through a suitable selection of thr center of buoyancy. An aerial view of the position estimation and the behavior of the controller before the perturbations can be seen in Fig. 8. We can note the action of the EKF (dashed blue) in order to recover the missing scale in the PTAM algorithm (dotted pink), and improve the estimation. Finally, the control inputs are shown in Figure 9.

The behavior of the AUV during this experiment can be appreciated in the following video: https://www. youtube.com/watch?v=v0Uj-IuztIs. From there, we can observe the system performance and robustness while facing unexpected disturbances such as important rotations and displacements, or camera occlusions, causing the vehicle to deviate from the trajectory. As expected, the use of an EKF with data from the IMU helps to recover the map.

VII. CONCLUSIONS AND FUTURE WORK

In this work, fully autonomous trajectory tracking was successfully achieved using PTAM and IMU data merged via an EKF. The PID controller with compensation of restoring forces showed a good performance, even when unexpected disturbances were present. The implementation of different control laws is contemplated as future work to improve the system's behavior, for example, adaptive control may be used to compensate unknown dynamics, or predictive control to take care of potential communication delays in a tetherless vehicle.

The use of PTAM in an aquatic environment was validated in real time experiments, exposing potential use for underwater applications in real world scenarios such as bridge, dams and hull inspection; coast patrolling; monitoring in shallow waters; etc. Field experiments are envisioned to demonstrate the limits of the algorithms under different real conditions.

The main contribution of this work is then, the implementation of the PTAM algorithm to perform autonomous navigation of an underwater vehicle, and the validation on real-time experiments. We are currently working on the implementation of the algorithms fully embedded in the vehicle, with this we intend to avoid the use of a long tether, increasing the level of autonomy and the coverage of the vehicle.

Fig. 1 .

 1 Fig. 1. A) AUV in an autonomous mission using vision based localization. B) A frontal camera in the vehicle is used to obtain the feature points for localization. C) Map generated by the PTAM algorithm for localization.

Fig. 2 .

 2 Fig. 2. Coordinate frames. A body fixed frame O B moves with respect to an inertial frame fixed to the earth O I .

Fig. 3 .

 3 Fig. 3. The forces acting on the submarine are due to 6 thrusters. Thrusters 1 to 4 are used for the rotation movement in yaw and the advance-recoil, while thrusters 5 and 6 are used to raise and descend. Roll and pitch are stable by design. Thrusters 1, 4 and 5 use clockwise propellers, while 2, 3, and 6 are counter-clockwise, cancelling the reactive moments.

  , ẇ, ṗ, q, ṙ represent the hydrodynamic added mass coefficients; m defines the mass of the vehicle. In addition, r g = [x g , y g , z g ] T form the distance vector from the origin of the body fixed frame O B to the center of gravity of the robot C G ; f B = W -B stands for the restoring forces vector resulting from the difference between the vehicle's weight W and the buoyancy force B. Finally, I z z is the inertia moment in z.

Fig. 4 .

 4 Fig. 4. Underwater vehicle BlueROV2 during a mission. The BlueROV2 was modified in this work to accomplish fully autonomous operation as an AUV instead of remotely operation as a ROV.

Fig. 5 .

 5 Fig. 5. Main ROS nodes used in the implementation. The BlueROV2 node provides information from the embedded sensors in the vehicle. The PTAM node takes the video stream from the BlueROV2 and estimates the scaleless pose of the vehicle w.r.t. the visual map. The sensor fusion node merges together the PTAM estimation along with inertial measurements from the IMU to improve the pose estimation. The control node implements a PID controller to accomplish autonomous trajectory tracking. Finally, the rqt_reconfigure and rqt_plot nodes are used respectively for on-line parameter tuning and real-time visualization.

2 a ; gyroscopes σ 2 ω and their bias σ 2 ba respectively σ 2 2 s and σ 2 bω = 0.0013 rad 2 s 3

 2223 bω are taken into account, i.e., σ 2 a = 0.25 m 2 s 3 , σ 2 ba = 0.025 m 2 s 5 , σ 2 ω = 0.013 rad

Fig. 6 .

 6 Fig. 6. Autonomous trajectory tracking using vision-based localization: position. Desired (solid red line); PTAM measurement (pink dotted line) and EKF estimation (dashed blue line) position; x coordinate response on top, y coordinate on the center and z coordinate on bottom. Important perturbations were applied at time 310s, 400s and 475s.

Fig. 7 .

 7 Fig. 7. Autonomous trajectory tracking using vision-based localization: orientation. Desired (solid red line) vs IMU measurements (dashed blue line). The vehicle is mechanically stabilized by design in roll φ and pitch θ angles, thanks to a careful selection of the center of buoyancy.

Fig. 8 .

 8 Fig. 8. Autonomous trajectory tracking using vision-based localization: aerial view before perturbations. Desired trajectory (solid red line); PTAM measurement (dotted pink line) and EKF estimation (dashed blue line).

Fig. 9 .

 9 Fig.9. Autonomous trajectory tracking using vision-based localization: control inputs. The motors operate in a PWM range from 1100 to 1500 in one direction and from 1500 to 1900 in the other.

TABLE I PHYSICAL

 I PARAMETERS FOR THE VEHICLE

TABLE II CONTROL GAINS

 IIGAINS

ACKNOWLEDGMENT

The author would like to thank the Dirección de Desarrollo y Fomento Deportivo del IPN, Mexico, for facilitating the use of the swimming pool.

Science and Technology CONACyT, Mexico.


Jesús Irving Rosales

Ricardo López Gutiérrez

Angel Zamora

Sergio Salazar

Antonio Osorio-Cordero

Hipolito Aguilar

Rogelio Lozano

Comparison of Control Techniques in a Weight Lifting Exoskeleton

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,


Rosa Adela Mejia-Nava
email: rosa-adela.mejia-nava@utc.fr

Adnan Ibrahimbegovic
email: adnan.ibrahimbegovic@utc.fr

Rogelio Lozano-Leal

Instability phenomena and their control in statics and dynamics: Application to deep and shallow truss and frame structures

Keywords: instability, control, viscous damping, friction damping Corresponding

In this paper we study the control for nonlinear geometric instability problem of a deep or a shallow truss or yet a frame structure. All the structural models are built with geometrically exact truss and beam finite elements. The proposed models can successfully handle large overall motion under static or dynamic conservative load. The control strategy considers adding a damping from either friction device or viscous damper. This kind of control belong to well-known concept of passivity. Different examples are presented in order to illustrate the proposed theoretical developments.

Introduction

The design of a structural systems involves many parameters that influence the response of the structure under conservative static or dynamic loads. For the optimal result for the design close to reality, it is very important to know how to detect the presence of instabilities, and to solve any such problem for preventing the structure collapse. Most of the instability problems such as Euler buckling have been studied under a conservative load and small pre-buckling displacement. Much less is known on nonlinear instability that is characterized by large overall motion, and in particular very large pre-buckling displacements. Gaining a better insight in non-linear instability the goal of this paper is the study of instability of structures in dynamic framework, which has not been studied sufficiently especially on how to control instability-induced vibrating. More precisely to avoid instability of structure or to be able to control it under dynamic loads, can be achieved by adding a damping provided by some kind of external damping mechanism. One can use a viscous damper, a friction device or a smart shape memory alloy [START_REF] Oliveira | Study of geometric nonlinear instability of 2D frame structures[END_REF][START_REF] Machado | Lyapunov exponents estimation for hysteretic systems[END_REF].

Any control of this kind is equivalent to the well-known concept of passivity (e.g., see [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]. This paper presents an application of the finite element method to solving such instability problem with large overall displacements of elastic truss and frame structures, subject to a conservative force. Different ways to achieve the passive control are discussed for nonlinear instability problem for a shallow two-dimensional elastic truss and frame structure.

Fig. 1 Energy criteria of instability

The outline of the paper is as follows. Different criteria for detecting instability are presented in Section 2. In Section 3 we discuss different types of instability phenomena, illustrated on a geometrically nonlinear truss structure. In section 4 we introduce geometrically exact beam model, with both linear and nonlinear instability problems. The dynamic framework with different ways of instability control, including viscous damping or friction devices, are presented in Section 5. Several illustrative numerical examples are presented in Section 6. In section 7, we give some conclusive remarks.

Instability phenomena: linear versus nonlinear buckling

The instability phenomena imply in general that a small perturbation of loading can lead to a disproportional amplification of the response. The instability depends on many factors, such as the nature of load, the spatial geometry, the material properties, boundary conditions like connections and supports, as well the mass (see [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics[END_REF][START_REF] Argyris | Nonlinear finite element analysis of elastic systems under nonconservative loading-Natural formulation. Part I. Quasistatic problems[END_REF][START_REF] Simitses | Fundamentals of Structural Stability[END_REF][START_REF] Xu | Aerodynamic stability analysis of geometrically nonlinear orthotropic membrane structure with hyperbolic paraboloid in sag direction[END_REF].

Detection criteria for instability in statics

Contrary to linear instability (see Fig. 1) the pre-buckling displacement for nonlinear instability is very large. Hence, such problem solution requires the full power of the nonlinear formulation and solution method.

Energy criteria

For the class of conservative problems, the instability of an equilibrium state can be evaluated by the second variation of total the potential energy, which can be written as

δΠ(𝑢 + 𝛼𝑤) = Π(𝑢) + δΠ(𝑢; 𝑤) + 1 2 δ 2 Π(𝑢; 𝑤) + 𝑂(‖δu‖ 2 ) (1)
For illustration, we look at a simple conservative system of the body in the gravity field with three different positions (see Fig. 1). For positon B, if the body is disturbed for a small disturbance, it will simply oscillate around the static equilibrium position, and this position is called stable. The second derivative of the total potential energy for this position remains positive. If the body is disturbed at point A, it will tend to move away from the static equilibrium position; and such an equilibrium position is called unstable. In this case, the second derivative of the total potential energy is negative. Finally, if the body is disturbed at point C, it will tend to remain in disturbed position and the value of the second derivative of the total potential energy is equal to zero; such equilibrium point regarding from state to unstable equilibrium is referred to as critical (see Ibrahimbegovic 2009, Simitses and[START_REF] Simitses | Fundamentals of Structural Stability[END_REF].

Criteria based on singularity of stiffness Matrix

The energy criterion can also be restated in terms of the stiffness matrix. Namely, when the tangent stiffness matrix becomes singular, its determinant takes zero value at the critical state of equilibrium.

𝛿 2 𝜋(𝛿𝑢; 𝛿𝑢) = 𝛿𝑢 𝑇 𝐾𝛿𝑢 = 0 ⟹ 𝑑𝑒𝑡[𝐾 ̂(𝑑 𝑐𝑟 )] = 0 (2)
Due to excessively high computing cost of the determinant of the stiffness matrix and rapid increase/decrease in values of the determinant around zero can lead to significant convergence difficulties of Newton's iterative method for computing the corresponding displacement. Thus the determinant-based criterion for detection of the critical equilibrium state is not very practical (see [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics[END_REF].

Criteria based on zero eigenvalue

Criterion based on zero eigenvalue reveals instability in terms of the corresponding zero eigenvalue. Such criterion not only can be used to verify if the tangent stiffness is a singular matrix, but also with this detection criterion we can indicate the type of instability mode by computing the eigenvector of the tangent stiffness at the critical equilibrium point associated with zero eigenvalue. The transition of the structure from stability to instability occurs with a zero-eigenvalue (see [START_REF] Argyris | Nonlinear finite element analysis of elastic systems under nonconservative loading-Natural formulation. Part I. Quasistatic problems[END_REF][START_REF] Argyris | Nonlinear finite element analysis of elastic systems under nonconservative loading-Natural formulation. Part I. Quasistatic problems[END_REF]see Ibrahimbegovic 2009) [𝐾 -𝜆 𝑐𝑟 𝐼]𝜓 𝑐𝑟 = 0 ⟹ 𝐾𝜓 𝑐𝑟 = 0

(3)

Instability computation for truss structures

Criteria based on zero eigenvalue for linear instability

For the special case, of small pre-buckling displacements the tangent stiffness can be expressed in a simplified form, where the geometric part of the tangent stiffness is a linear function of applied loading [START_REF] Ibrahimbegovic | Linear instability or buckling problems for mechanical and coupled thermomechanical extreme conditions[END_REF]. For example, for a simple structure that consists of 2 truss-bar elements, (see Fig. 3) the present case of geometric instability, the critical values can be obtained as

𝑑𝑒𝑡[𝐾 𝑚 + 𝜆 𝑐𝑟 𝐾 𝑔 ] = 0 ⟹ 2𝐸𝐴 𝑙 3 ℎ 3 𝑏 2 (𝑏 2 + ℎ𝑢 𝑐𝑟 + 1 2 𝑢 𝑐𝑟 2 ) = 0 (4)
where the material and geometric part of the stiffness matrix can be written as

𝐾 𝑚 𝑒 = 𝐸𝐴 𝑙 𝑒 [ 1 0 -1 0 0 0 0 0 -1 0 1 0 0 0 0 0 ] ; 𝐾 𝑔 𝑒 = 𝑁 𝑙 𝑒 [ 0 0 0 0 0 1 0 -1 0 0 0 0 0 -1 0 1 ] (5)
By improving the boundary conditions and solving the resulting eigenvalue problem, we can finally obtain, the values of the critical load as We can express the critical displacement as function of the geometric characteristics of the truss structure as

𝜆 𝑐𝑟1 = 𝐸2𝐴 𝑙 𝑒 𝑏 2 ℎ ; 𝜆 𝑐𝑟2 = 𝐸2𝐴 𝑙 𝑒 ℎ 3 𝑏 2 (6)
𝑢 𝑐𝑟 = -ℎ ± √ℎ 2 -2𝑏 2 (7)
We note that these values of the critical displacement are only possible for deep truss (b<<h, see Fig. 2), and more importantly they lead to small pre-buckling displacement

Nonlinear instability

In the class of problems referred to as nonlinear instability, all the displacements, rotations and strains are all large in general before arriving at the critical equilibrium point. For that reason, it is often impossible to obtain the analytic solution, as the one available for linear instability problem of Euler´s buckling (see [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics[END_REF]. One exception is a general problem in nonlinear instability for a shallow truss composed of two truss-bar elements with 2-nodes, which is loaded by a vertical force at the apex. We can note that maintaining the stability of equilibrium in the system will require zero value of vertical external force for the displacement values v=h and v=2h. These two equilibrium states correspond, respectively, to the truss equilibrium state where both bars are horizontal and the equilibrium state where bars have been moved through to the opposite side so that the deformed length of each bar remains the same as in the initial configuration, which implies zero internal force.

The natural deformation measure for such truss is the stretch, which can be computed as the ratio of the deformed versus initial length of the bar

𝜆 𝑡 = 𝑙 𝑡 𝜑 𝑙 ⁄ (8)

Instability computation for frame structure

For the computational purpose, it is better to use the Green-Lagrange strain, which can be easily computed for a 2-node truss bar element as

𝐸 11 ℎ | 𝑙 = 1 (𝑙) 2 𝑥 𝑒𝑇 𝐻 𝑒 𝑑 𝑒 + 1 2(𝑙) 2 𝑑 𝑒𝑇 𝐻 𝑒 𝑑 𝑒 (9)
Where 𝑥 𝑒𝑇 are nodal coordinates 𝑑 𝑒 nodal displacements and 𝐻 𝑒 has block unit matrix structure that means

𝐻 𝑒 = [ 𝐼 -𝐼 -𝐼 𝐼 ] (10)
When the truss constitutive behaviour is linear elastic, which means it is governed by the Saint-Venant-Kirchhoff material law, we can easily obtain the corresponding finite element approximation of the second Piola-Kirchhoff stress as

𝑆 11 ℎ | 𝑙 = 𝐸 ( 1 (𝑙) 2 𝑥 𝑒𝑇 𝐻𝑑 𝑒 + 1 2(𝑙) 2 𝑑 𝑒 𝐻𝑑 𝑒 ) ( 11
)
Where E is Young's module.

We can also obtain the corresponding finite element approximation of the virtual Green-Lagrange strain

Γ 11 = 1 (𝑙) 2 (𝑥 𝑒𝑇 + 𝑑 𝑒𝑇 𝐻𝑤 𝑒 ) ; 𝑤 𝑒 = ( 𝑤 1,1 𝑒 𝑤 1,2 𝑒 𝑤 2,1 𝑒 𝑤 2,2 𝑒 ) (12)
Where 𝑤 𝑒 are nodal values of virtual displacements. Finally, the contribution to the discrete approximation allow us to define the internal virtual work according to

𝐺 𝑖𝑛𝑡,𝑒 = 𝑤 𝑇 𝑓 𝑖𝑛𝑡,𝑒 ; 𝑓 𝑖𝑛𝑡,𝑒 = 𝐻 𝑇 1 𝑙 (𝑥 𝑒 + 𝑑 𝑒 )𝑆 11 ℎ (13)
where we denote

𝑥 𝑒 = ( 𝑥 1 𝑦 2 𝑥 2 𝑦 2 ) ; 𝑑 𝑒 = ( 𝑑 1,1 𝑒 𝑑 1,2 𝑒 𝑑 2,1 𝑒 𝑑 2,2 𝑒 ) ; 𝐻 𝑒 = ( 1 0 -1 0 0 1 0 -1 -1 0 1 0 0 -1 0 1 ) (14)
Introducing the consistent linearization of this week form, we obtain the tangent stiffness matrix, with the material and geometric part that can be written as

𝐾 𝑒 = 𝐾 𝑚 𝑒 + 𝐾 𝑔 𝑒 (15)
By assuming the motion symmetry, the weak form of the equilibrium equations for this shallow truss structure can be rewritten as

𝐺 𝑒𝑥𝑡 = 𝐺 𝑖𝑛𝑡 ⇔ 𝑤𝑓 = 𝑤2𝐴𝑆 11 1 𝑙 (ℎ + 𝑣) ; 𝑆 11 = 𝐸 1 𝑙 2 (ℎ𝑣 + 1 2 𝑣 2 ) (16)
which allows us to define the explicit form of internal force as a function of (the only non-zero component)of vertical displacement

𝑣 𝑓 = 2𝐸𝐴 𝑙 3 (ℎ + 𝑣) (ℎ𝑣 + 1 2 𝑣 2 ) (17)
The corresponding tangent stiffness matrix also reduces to a single entry. This, it is easy to confirm the presence of two critical equilibrium states, which can be computed from zero value of 𝐾 The classical example of linear geometric instability is Euler's buckling (see Fig. 4), which assumes small pre-buckling displacements. Namely we consider an ideal column assumed initially to be perfectly straight and submitted at the top to the compressive axial load P. A small transverse perturbation is applied producing small pre-buckling displacement in terms of a small deflection. If the load P is smaller than the critical value, no buckling will happen and the column will remain straight once the perturbation is removed. We define the critical load as the axial load sufficient to keep the bar in a slightly bent form, and producing disproportionality large transverse displacement for any further, even small increase.

0 = 𝐾 ∶= 𝑑𝑓 𝑖𝑛𝑡 𝑑𝑣 = 2𝐸𝐴 𝑙 3 (ℎ + 𝑣) 2 + 2𝐸𝐴 𝑙 3 (ℎ𝑣 + 1 2 𝑣 2 ) (18)
The critical equilibrium point of this kind is referred to as bifurcation (see Ibrahimbegovic 2009), [START_REF] Simitses | Fundamentals of Structural Stability[END_REF]Hodges 2006, Chen and[START_REF] Chen | Structural Stability[END_REF]. The critical force for Euler's beam can be calculated by solving differential equation that corresponds to beam equilibrium. The Euler solution method enabled us to reduce the problem of instability to a simpler problem of finding the minimum characteristic value of the corresponding boundary-value problems [START_REF] Simitses | Fundamentals of Structural Stability[END_REF].

𝑑 2 𝑦(𝑥) 𝑑𝑥 2 + 𝑝 𝐸𝐼 𝑦(𝑥) = 0 (19)
The general solution of this ordinary equation can be written as

𝐹 𝑐𝑟 = 𝜋 2 𝐸𝐼 𝑐𝑙 2 ; 𝑦 = 𝐴𝑐𝑜𝑠𝑘𝑥 + 𝐵𝑠𝑖𝑛𝑘𝑥 ( 20
)
where c is the constant that depend, on boundary conditions, whereas A and B are constants defining the instability mode. If we solve this equations for each particular boundary condition, we will be able to find the critical load (see Fig. 4 for different cases).

𝐹 𝑐𝑟 = 𝑛 2 𝜋 2 𝐸𝐼 𝑙 2 (21) 𝐹 𝑐𝑟 = 𝜋 2 𝐸𝐼 4𝑙 2 (22) 𝐹 𝑐𝑟 = 2,05𝜋 2 𝐸𝐼 𝑙 2 (23)

Nonlinear instability of geometric exact beam model: large pre-bucling displacements and rotations

As possible enhancement of the truss model, we can use geometrically exact beam instead of the truss (see [START_REF] Ibrahimbegovic | On the choice of finite rotation parameters[END_REF][START_REF] Ibrahimbegovic | Finite element analysis of linear and elastic initially curved beams non-linear planar deformations[END_REF][START_REF] Imamovic | Coupled testing-modeling approach to ulitmate state computation of steel structures with connections for statics and dynamics[END_REF]. This time, the instability phenomena can be present in both large displacement and large rotations setting of geometrically nonlinear problems. When a deformable beam is submitted to large displacements, large rotations and large deformations, the initial configuration at the beginning of a given loading program and the deformed configuration at the end are quite different, such as in the case of a shallow beam after to be submitted to the critical load.

The generalized strain measures proposed by Reissner (e.g., [START_REF] Ibrahimbegovic | On finite element implementation of geometrically nonlinear Reissner´s beam theory: three-dimensional curved beam elements[END_REF] are expressed as function of the rotation matrix Λ, position vector 𝜑 and unit vector 𝑒 1 as

( 𝐸 Γ ) = Λ 𝑇 𝜑 ´-𝑒 1 ; 𝐾 = 𝑑𝜓 𝑑𝑥 (24)
For 2d case, the rotation matrix and position vector are written as

Λ = [ cos 𝜓 -sin 𝜓 sin 𝜓 cos 𝜓 ] ; 𝜑 ´= 𝑑 𝑑𝑥 ( 𝑥 + 𝑢 𝑣 ) (25)
Therefore, the generalized strain measures can be expressed as

𝐾 = 𝑑𝜓 𝑑𝑥

The virtual work can be expressed in terms of non-symmetric Piola-Kirchhoff stress P and its energy conjugate strain measure as the deformation gradient F. All the instability criteria already by the truss model are also valid for this geometrically exact beam model, for the case of conservative loading where we can define the potential energy.

G(𝑢

Dynamic framework and time-integration schemes

For the case where total potential energy cannot be defined (for a e.g., non-conservative loading, follower force), we can switch to dynamic frame work for instability [START_REF] Hemat | Dynamic instability response in nanocomposite pipes conveying pulsating ferrofluid flow considering structural damping effects[END_REF][START_REF] Mohammad | Dynamic instability region analysis of sandwich piezoelectric nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal strain gradient theory[END_REF]. In general case, the dynamics equations are solved by using time integration schemes.

The instability criteria already presented by the truss model are all valid for this geometrically exact beam model, for the case of conservative loading where we can define the total potential energy. The solution of the differential equation of motion can be obtained through numerical methods with step-by-step time integration scheme. The solution depend on initial conditions, and has to include the equations of motion at time 𝑡 𝑛+1 .

𝑀𝑑 ̈𝑛+1 + 𝐶𝑑 ̇𝑛+1 + 𝐾𝑑 𝑛+1 = 𝐹 𝑛+1 𝑒𝑥𝑡 (29)
The best solution can be obtained by the second-order time-integration schemes, such as the central difference or the Newmark method. In this work, we can work with either the first or the second method, where the velocity and displacement can be expressed as

𝑑 𝑛+1 = 𝑑 𝑛 + ℎ𝑑 ̇𝑛 + ℎ 2 [( 1 2 -𝛽) 𝑎 𝑛 + 𝛽𝑎 𝑛+1 ] 𝑑 ̇𝑛+1 = 𝑑 ̇𝑛 + ℎ[(1 -𝛾)𝑑 ̈𝑛 + 𝛾𝑑 ̈𝑛+1 ] (30)
Here, the factor 𝛾 affects, the acceleration in the velocity equation, whereas 𝛽 affects displacements. With values 𝛽 equal to 0 and 𝛾 equal 1/2 we obtain the central difference and with values 𝛽=1/4 and 𝛾=1/2 correspond to the trapezoidal rule, which assumes that acceleration will remain constant (see [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics[END_REF][START_REF] Clough | Dynamics of Structures[END_REF][START_REF] Chapra | Numerical Methods for Engineers[END_REF].

The dynamic response of a simple dynamic system can oscillate indefinitely if it does not have any force that can dissipate energy and reduce motion to zero. However, in real life the oscillation will reduce gradually due to damping of the system. Here, we present two types of damping, which both fit into the same framework.

𝑓 𝑖𝑛𝑡 (𝑑) + 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 = 𝑓 𝑒𝑥𝑡

(31)

Viscous damping

This model is used successfully to model the exponential vibration amplitude decay in a variety of mechanical systems. One example of viscous damper can be a piston fit into a cylinder filled with oil. Here the damping force is proportional to the velocity of the piston, in direction opposite of the motion. Such a damping force illustrated in Fig. 5 has the form 𝑓 𝑐 = 𝑐𝑑(𝑡) ̇(32)

Frictional damping

This kind of damping with sliding friction is well known Coulomb damping, which is characterized by One way to illustrate this kind of damping in a system shown in Fig. 6, It is equivalent to perfect plasticity model where the sliding resistance is equal to elasticity limit 𝜎 𝑦 (see [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics[END_REF][START_REF] Inman | Engineering Vibration[END_REF].

Finally, such damping modifies the equation of motion as expressed in the following way

𝑀𝑑 ̈+ 𝑓 𝑐 𝑠𝑔𝑛(𝑑 ̇) + 𝑓 𝑖𝑛𝑡 (𝑑) = 𝑓 𝑒𝑥𝑡 (34)
In short, adding different damping mechanism correspond to passive control of vibration (see Lozano andBrogliato 1992, Lozano et al. 2008) Passive control of vibrations was used successfully in many control applications, for example, the stabilization of unmanned aerial vehicle using unit quaternions (see [START_REF] Guerrero-Sanchez | Passivity-based control for a micro air vehicle using unit quaternions[END_REF] where the authors reduced the problem to the longitudinal plane and the potential energy V=mgz, as a conservative load by the product of the mass m, gravity g, and the position in the vertical axis in the internal frame z:

Hence, to design structures or machines and have a desired transient and steady state response to some extent the designer can play with mass, stiffness or damping. Adding mass and changing stiffness values are also methods of passive control. However, it is not always possible to choose all the parameters to play with them due to design constraints that have to be satisfied. Hence, in order to avoid vibrations, it may be necessary to add an external damper or some mass could be added to a given structure to lower its natural frequency. Damping treatments increase the rate of decay of vibrations as the nonlinear control technique based on passivity. Passivity is a fundamental property exhibited by many physical systems which involves energy dissipation and transformation (see [START_REF] Inman | Vibration with Control[END_REF]).

Numerical examples

In this section, we present several illustrative numerical examples of instability computations and control. First example is a simple cantilever beam in free vibration represented, with a truss element, where we control the beam vibrations by adding two different kinds of damping in order to dissipate the oscillations, viscous damping and equivalent friction damping, in order to dissipate the oscillations. Then we consider the case of a nonlinear instability case with the Williams toggle frame which is modelled with either two truss elements for 8 beam elements. In the first case, we will present arc-length method results and after that the results, we obtain by adding viscous damping and equivalent friction damping.

Simple cantilever with truss element

We propose control of axial oscillations of a simple cantilever modeled by truss in order two different a simple cases of vibration control and show how the structure will behave. We take a cantilever first subjected to a triangular pulse load, which will continue to oscillate in free vibration (see Fig. 7). The chosen material and geometric characteristics are EI=2.66×10 5 Ncm 2 , EA=8.251×10 6 N, ρ=7.79×10 -3 kg/cm 3 .

For dissipating these oscillations, we can use two kinds of damping, which is modelled with special elements placed at the end of the bar. The first one is viscous damping, which depends on the viscosity coefficient, and the second is a friction damping, which depends on frictional coefficient.

Cantilever truss element with viscous damping

In the case when we add a viscous damping, the cantilever oscillation will decrease, with decrement that depends on the value of the viscosity coefficient. If we further increase this value, the amplitude of oscillations decay time will be shorter (see Fig. 8).

Cantilever truss element with friction damping

For the second option of vibration control we take the same cantilever truss, and we add a friction element at free end (see Fig. 9), we propose an equivalent friction damping. The first control model is a truss element with perfect plasticity, and the second one is plasticity with kinematic hardening. We note that these models are not exactly pure friction. However, they can still store dissipate energy, since the rheological model of plasticity consists of elastic plus friction part.

Williams toggle frame

In this section we present a vibration control example of William toggle frame one, with truss elements and other with beams (see Fig. 12). This example considers the nonlinear instability. This frame consists of two legs each supported on one end and on the other loaded by a vertical force at the apex. The chosen material characteristics are EI=2.66×10 5 Ncm 2 , EA=8.251×10 6 N, ρ=7.79×10 -3 kg/cm 3 . A numerical solution is presented in [START_REF] Mamouri | Study of geometric nonlinear instability of 2D frame structures[END_REF].

When increasing the values of applied load, the frame displacement increases proportionally until the load reaches the critical value. At critical load, the sudden increase of displacement will occur. In computing this first part of response, we perform the static analysis. Subsequently we continue with the dynamic analysis, where we apply the Newmark time integration scheme (see Figs. 13 and14). We present two different models of this structure: first with two truss elements At the end of instability induced sudden jump in dynamics we will see some oscillations (see Fig. 13(b) and 14 (b)), and the goal will be to damp out these oscillations. Thus we propose to first add a viscus damping and then to add an equivalent friction damping like in the case of cantilever beam.

Williams toggle frame model with truss elements

In this part we model the frame with two truss elements chosen material with the properties and geometry shown in Fig. 12. First we add viscous damping and then an equivalent friction damping, with a perfect plasticity. We check their ability to control oscillations after the critical load.

In the first case when we add a viscous damping the behavior is similar to the cantilever truss. Namely, the decay of vibration amplitude directly depends on viscosity coefficient. The oscillations will decrease faster when we increase this value. For the case of perfect plasticity, we change area values. The oscillations are changing directly with changing this value. The values in this analysis were chosen as .4125, 4.125 and 41.25 of the cross section of the damping device. We can see the results in Fig. 16. With the value of .425, the amplitude of oscillations decreases but is not meaningful. Here the value that dissipates energy is 41.125. We change this area in the function of the area of the frame element. With these elements, the behavior is similar to the previous two when we add viscous damping, as the viscosity coefficient increases the dissipation also increases until one point where there are no oscillations. On the other hand, for plastic damping the key factor is the chosen value of the yield stress. The oscillations are changing directly with changing this value. The values in this analysis were chosen as 0.3, 0.4 and 1. We can see the results in Fig. 18.

Conclusions

We have presented different criteria for nonlinear instability problem, and corresponding illustrative examples. A shallow beam under a static load in the apex will be unstable after the critical load. If after the critical load we continue the analysis in dynamics the structure will start to oscillate. One way to control this oscillation is by adding either viscous damping or friction damping. In nonlinear instability under dynamic load, we implement two different kinds of damping mechanism (viscous damping and friction damping) for verifying the dissipative behavior.

When we add viscous damping to the truss structure and William toggle frame the reduction of oscillations depends of the viscous damping coefficient, if we increase this value we will obtain a reduction of the oscillations. Equivalent friction damping with perfect plasticity behavior is more particular for each case. For example, in the case of the truss in cantilever dissipation depends on the yield stress, as well as in the in the case of beam William Toggle frame. However, for the William toggle frame with truss elements dissipation depend on area of the devise, greater area greater dissipation.

In future developments we plan to generalize the proposed approach to the case of nonconservative loading, such as those produced by fluids pressure brought by wind loads (e.g., [START_REF] Ibrahimbegovic | Long-term simulation of wind turbine structure for distributed loading describing long-term wind loads for preliminary design[END_REF]; the latter requires rather special treatment [START_REF] Bolotin | Fundamentals of Structural Stability[END_REF] within the proposed dynamics framework. We also plan to generalize to using this approach for practical application, such as planned destruction of an existing building (e.g., El Houcine et al. 2018).

Fig. 2

 2 Fig. 2 Deep truss structure

Fig. 4

 4 Fig. 4 Euler's buckling. (a) Simple supported (b) One fixed-one free end (c) Both end fixed beam

  can be restated in terms of stress resultants as follow (see[START_REF] Ibrahimbegovic | Finite element analysis of linear and elastic initially curved beams non-linear planar deformations[END_REF] G(𝑢; 𝑤) = ∫ 〈𝑑 𝑇 (𝑤)Λ + 𝜓 ̂ℎ𝑇 (𝑎)

Fig

  Fig. 5 Viscous damping

Fig. 7

 7 Fig. 7 Oscillations Cantilever truss

  Fig. 9 Cantilever truss, oscillations displacement-time of free vibration and friction damping

Fig. 12

 12 Fig. 12 Geometry and load conditions of toggle frame

  Fig. 15 Williams toggle frame modelled with truss elements with viscous damping

  Fig. 17 Williams toggle frame modelled with beam elements with viscous damping


Miguel Angel

Garcia Rangel

Adrian Manzanilla

Angel Eduardo Zamora Suarez

Filiberto Muñoz

Sergio Salazar

Rogelio Lozano

Adaptive Non-singular Terminal Sliding Mode Control for an Unmanned Underwater Vehicle: Real-time Experiments

Keywords: Non-singular terminal sliding mode controller, real-time experiments, unmanned underwater vehicle

This research work focuses on the design of a robust-adaptive control algorithm for a 4DOF Unmanned Underwater Vehicle (UUV). The proposed strategy is based in a Non-Singular Terminal Sliding Mode Control (NTSMC) with adaptive gains, where the proposed adaptation mechanism ensures that the gains remain bounded. In this control strategy a non-singular terminal sliding surface is proposed to obtain a faster convergence of the tracking errors. The NTSMC ensures Practical Finite-Time Stability for the closed-loop system as well as exhibits a chattering reduction. In order to demonstrate the satisfactory performance of the proposed controller, a set of experiments was performed with a Non-Singular Terminal Sliding Mode Controller and an Adaptive Non-Singular Terminal Sliding Mode Control (ANTSMC) in real time for trajectory tracking in the X-Y plane, the graphs showed that the ANTSMC converges faster to a smaller region and reduces oscillations.

INTRODUCTION

In recent years, Unmanned Underwater Vehicles (UUV's) have been used in different activities, such as: civilian, military and research tasks [START_REF] Naeem | A review of guidance laws applicable to unmanned underwater vehi-cles[END_REF][START_REF] Veres | Autonomous vehicle control systems-a review of decision making[END_REF]. Among their main applications are inspection of coastal structures, oceanographic monitoring, hull 3D reconstruction, under-water structure mapping, geological sampling and deep sea archaeology [START_REF] Melo | Survey on advances on terrain based navigation for autonomous underwater vehicles[END_REF]. Moreover, the underwater vehicles are used to obtain relevant data from the underwater environment, such as temperature, dissolved oxygen, pH and salinity [START_REF] Partan | A survey of practical issues in underwater networks[END_REF]. During the execution of the after mentioned undersea missions, underwater vehicles are expected to conduct these tasks in an automated way without the interaction of a human operator. Therefore, UUV's repre-sents an extensive field of study for many research groups around the world, due to the remarkable modelling dif-ficulties, such as highly non-linear terms, in addition to coupled dynamics, uncertainties and external disturbances in the UUV.

For a successful performance of the UUV's, it is nec-essary to have a control strategy that allows the vehicle to follow the desired trajectory [START_REF] Stutters | Navigation technologies for autonomous underwater vehicles[END_REF], even under external disturbances. In this sense, the motion control problem for UUV's has received considerable attention due to its theoretical challenges. For example, classical controllers like: Proportional Integral Derivative (PID) [START_REF] Lee | Clonal selection algorithms for 6-dof PID control of autonomous underwater vehicles[END_REF], hierarchi-cal Proportional Derivative (PD) [START_REF] Bak | Positioning control of an underwater robot with tilting thrusters via decomposition of thrust vector[END_REF], robust PD [START_REF] Zamora | Depth control of an underwater vehicle using robust pd controller: real-time experiments[END_REF]. As well as nonlinear and intelligent controllers, such as: back-stepping [START_REF] Pang | Three-dimensional leader-follower formation control of multiple autonomous underwater vehicles based on line-of-sight measurements using the backstepping method[END_REF], sliding mode [START_REF] Borlaug | Trajectory tracking for an articulated intervention AUV using a super-twisting algorithm in 6 dof[END_REF], backstepping with slid-ing modes [START_REF] Santos | Backstepping sliding mode control with functional tuning based on an instantaneous power approach applied to an underwater vehicle[END_REF], adaptive model predictive control [START_REF] Anderlini | Control of a rov carrying an object[END_REF], radial bases neural networks [START_REF] Li | Neural network-based sliding mode variable structure control for mars entry[END_REF], adaptive fuzzy sliding mode [START_REF] Chu | Adaptive fuzzy sliding mode diving control for autonomous underwater vehicle with input constraint[END_REF], to name a few. However, the control of UUV's presents key challenges associated with complex dynam-ics, system uncertainties and environmental disturbances. These challenges are of notable interest and serve as a mo-tivation for this research work.

To cope with these challenges, sliding modes is a ro-bust control technique which provides robustness against unmodeled dynamics, insensitivity to parameter variations and disturbance rejection [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF]. The main drawback of the sliding mode technique is the well-known chattering ef-fect [START_REF] Levant | Chattering analysis[END_REF]. However, there exist several ways to reduce or mitigate the chattering effect: using High Order Sliding Modes (HOSM) [START_REF] Levant | Chattering analysis[END_REF] or using a fast Terminal Sliding Mode (TSM)-type reaching law [START_REF] Yu | Continuous finite-time control for robotic manipulators with terminal sliding mode[END_REF].

Miguel Angel Garcia Rangel, Adrian Manzanilla, Angel Eduardo Zamora Suarez, Filiberto Muñoz, Sergio Salazar, and Rogelio Lozano are with the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV), Laboratory LAFMIA-UMI, Mexico (e-mails: {miguelgarciar, amanzanilla, angel.zamora, filiberto.munoz}@cinvestav.mx, sergio.salazar.cruz@gmail.com, rlozano@hds.utc.fr). Filiberto Muñoz is also with the National Council of Science and Technology (CONACYT), Mexico. Rogelio Lozano is also with Université de Technologie de Compiegne, Compiegne, France.

The Super-Twisting (ST) is a HOSM algorithm that is commonly used for ro-bust control and observation, due to it reduces the chat-tering effect, respect to conventional sliding modes [START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF]. Moreover, for mechanical systems, the ST algorithm en-sures convergence to the sliding surface in finite-time, however the tracking errors converges to zero only exponentially [START_REF] Derafa | Super twisting control algorithm for the attitude tracking of a four rotors uav[END_REF][START_REF] Chalanga | Implementation of super-twisting control: Super-twisting and higher order sliding-mode observer-based approaches[END_REF][START_REF] Mobayen | Design of an adaptive tracker for n-link rigid robotic manipulators based on super-twisting global nonlinear sliding mode control[END_REF].

Furthermore, to force the tracking errors to zero or a bounded set in finite-time, TSM and Non-Singular Terminal Sliding Mode controllers have been developed [START_REF] Yu | Continuous finite-time control for robotic manipulators with terminal sliding mode[END_REF][START_REF] Yang | Nonsingular fast terminal slidingmode control for nonlinear dynamical systems[END_REF][START_REF] Sheng | Nonsingular finite-time second order sliding mode attitude control for reentry vehicle[END_REF][START_REF] Wang | Continuous fast nonsingular terminal sliding mode control of automotive electronic throttle systems using finite-time exact observer[END_REF]. These control strategies have been applied to control marine vehicles. For example, a depth control for a remotely operated underwater vehicle using a NTSMC was developed in [START_REF] Wang | Depth control of remotely operated vehicles using nonsingular fast terminal sliding mode control method[END_REF]. In [START_REF] Yan | Globally finite-time stable tracking control of underactuated UUVs[END_REF], the trajectory tracking problem of an under-actuated unmanned underwater vehicle was solved using a finite-time control strategy. In this approach, the tracking errors globally converge to a neighborhood of the origin in finite-time. In [START_REF] Elmokadem | Terminal sliding mode control for the trajectory tracking of underactuated autonomous underwater vehicles[END_REF] is developed a robust control strategy for the lateral motion of underactuated AUV, a TSM controller guarantees the finitetime convergence of the tracking errors to zero. Moreover, a Non-Singular Terminal Sliding Mode (NSTSM) controller was developed in [START_REF] Londhe | Non-singular terminal sliding mode control for robust trajectory tracking control of an autonomous underwater vehicle[END_REF] for robust trajectory tracking of an AUV.

A well known issue with ST and NSTSM controllers is that the bounds of uncertainties and disturbances need to be know in advance. In order to overcome this drawback, adaptive strategies for sliding mode controllers have been developed [START_REF] Shtessel | Super-twisting adaptive sliding mode control: A lyapunov design[END_REF]. Moreover, adaptive control automatically adjust the controller's gains in real-time in order to guarantees a successful performance when the parameters of the dynamic model are unknown. An adaptive sliding mode control is developed in [START_REF] González | A practical approach to adaptive sliding mode control[END_REF], where adaptive sliding surfaces and adaptive control gains are designed. Adaptive versions of the super-twisting algorithm are developed in [START_REF] Shtessel | A novel adaptivegain supertwisting sliding mode controller: Methodology and application[END_REF][START_REF] Guerrero | Trajectory tracking for autonomous underwater vehicle: An adaptive approach[END_REF][START_REF] Jeong | Tracking error constrained super-twisting sliding mode control for robotic systems[END_REF]. These versions reduce the chaterring effect, however only ensure asymptotic convergence of the tracking errors. An adaptive global terminal sliding mode controller is proposed in [START_REF] Mobayen | Adaptive global terminal sliding mode control scheme with improved dynamic surface for uncertain nonlinear systems[END_REF], where two adaptive laws are employed for the upper bound of uncertainties and disturbances.

In order to successfully control the dynamics of marine vehicles, adaptive sliding modes approaches have been proposed. In [START_REF] Guerrero | Trajectory tracking for autonomous underwater vehicle: An adaptive approach[END_REF] is developed an adaptive gain generalized super-twisting algorithm for trajectory tracking of an AUV. In this manuscript experimental results are conducted for yaw and depth dynamics. In [START_REF] Yan | Robust adaptive sliding mode control of underactuated autonomous underwater vehicles with uncertain dynamics[END_REF] an adaptive integral sliding mode for a class of UAVs is developed. To cope with parametric uncertainties and external disturbances, a neural neural network controller is added. An integral terminal sliding mode controller with adaptive compensation of uncertainties and disturbances is developed in [START_REF] Qiao | Double-loop integral terminal sliding mode tracking control for uuvs with adaptive dynamic compensation of uncertainties and disturbances[END_REF] for an autonomous underwater vehicles.

In [START_REF] Qiao | Adaptive second-order fast nonsingular terminal sliding mode tracking control for fully actuated autonomous underwater vehicles[END_REF] an adaptive second-order fast non-singular terminal sliding mode is developed for tracking control of AUVs. The controller designed yields local exponential convergence of the position and attitude tracking errors to zero. An adaptive non-singular integral terminal sliding mode controller is developed in [START_REF] Qiao | Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles[END_REF] for an AUV. The control strategy ensures that the velocity tracking errors locally converge to zero in finite t ime a nd a fter t hat the position tracking errors locally converge to zero exponentially.

The main contribution of this research work is the development of a Robust Adaptive Gain Non Singular Terminal Sliding Mode Controller. The proposed approach guarantees that the tracking errors converge to a bounded set in finite-time, even under unmodeled dynamics and external disturbances. In addition, a set of real-time experiments were conducted in X, Y, Z and yaw dynamic to demonstrate the performance of the proposed method.

From the comparison performed with [START_REF] Londhe | Non-singular terminal sliding mode control for robust trajectory tracking control of an autonomous underwater vehicle[END_REF][START_REF] Qiao | Double-loop integral terminal sliding mode tracking control for uuvs with adaptive dynamic compensation of uncertainties and disturbances[END_REF][START_REF] Qiao | Adaptive second-order fast nonsingular terminal sliding mode tracking control for fully actuated autonomous underwater vehicles[END_REF] and [START_REF] Qiao | Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles[END_REF], we can deduce that the controller proposed in this article has advantages, such as: a low computational cost since it does not depend of estimation of the dynamic parameters, presenting a rapid convergence to a small region [START_REF] Mobayen | Adaptive global terminal sliding mode control scheme with improved dynamic surface for uncertain nonlinear systems[END_REF] due to the adaptive gains of the controller, it does not present oscillations when the error is small. Moreover, the efficiency o f t he c ontroller i s v alidated b y r eal-time experiments and obtain Root Mean Square Error, in contrast to the comparative articles where only simulation results are presented, the advantages found among the articles are summarized in the Table 1.

The remainder of the document is organized as follows: Section 2 describes the development of the dynamic model for the UUV. Section 3 presents the main contribution of this paper, which is the derivation of an adaptivegain non-singular sliding mode control strategy for the UUV, as well as the stability analysis based on Lyapunov theory. Next, Section 4 demonstrates the controller performance through a series of experimental results for trajectory tracking tasks. Finally, Section 5 provides concluding remarks, as well as future research directions.

DYNAMIC MODEL OF UNDERWATER VEHICLES

The dynamic model for an underwater vehicle involves two coordinate frames: a body reference frame O B and an inertial frame O I , see Fig. 1. For this analysis, some of the main hydrodynamic effects of this type of vehicles are considered, in addition to considering restorative forces such as gravity and buoyancy. Considering the SNAME notation for underwater vehicles [START_REF] Sname | Nomenclature for treating the motion of a submerged body through a fluid[END_REF], the dynamic model in matrix form is written as [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF]:

M ν +C(ν)ν + D(ν)ν + g(η) = τ + ω, η = J(η)ν, (1)
Table 1. Comparative study between our research efforts and the existing literature closely related to our paper. In convergence speed of is fast and is very fast.

Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles [START_REF] Qiao | Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles[END_REF] Adaptive second-order fast nonsingular terminal sliding mode tracking control for fully actuated autonomous underwater vehicles [START_REF] Qiao | Adaptive second-order fast nonsingular terminal sliding mode tracking control for fully actuated autonomous underwater vehicles[END_REF] Non-singular terminal sliding mode control for robust trajectory tracking control of an autonomous underwater vehicle [START_REF] Londhe | Non-singular terminal sliding mode control for robust trajectory tracking control of an autonomous underwater vehicle[END_REF] Double-loop integral termal sliding mode tracking control for UUV's with adaptive dynamic compensation of uncertainties and disturbances [START_REF] Qiao | Double-loop integral terminal sliding mode tracking control for uuvs with adaptive dynamic compensation of uncertainties and disturbances[END_REF] Our research.

Speed convergence of sliding surface: High computational cost:

M = diag [m -X u, m -Y v, m -Z ẇ, I zz -N ṙ] , (2)
C( ν) =     0 0 0 -(m -Y v)v 0 0 0 (m -X u)u 0 0 0 0 (m -Y v)v -(m -X u)u 0 0     , (3)
D( ν) = -     X u + X u |u| |u| 0 0 0 Y v +Y v |v| |v| 0 0 0 Z W + Z w |w| |w| 0 0 0 0 0 0 N r + N r |r| |r|     , (4)
g( η) = [0, 0, -(W -B), 0] T , (5)
where m, W and B denote the mass, weight and buoyancy of the underwater vehicle, respectively. I zz denotes the moment of inertia in the z-axis and N r ,

N r |r| , Z W , Z w |w| , Y v , Y v |v| , X u , X u |u| , X u, Y v, Z
ẇ and N ṙ are the hydrodynamic parameters due to aggregate mass and vehicle shape.

The transformation between the body and inertial coordinate frames is given by η

= J( η) J( η) -1 η + J( η) ν, (6)
with J( η) defined as

J( η) =     c ψ -s ψ 0 0 s ψ c ψ 0 0 0 0 1 0 0 0 0 1     , (7)
where c * = cos( * ) and s * = sin( * ). From ( 6), the dynamic model of the underwater vehicle, expressed in the inertial frame O I , is as follows:

η = Cη ( η, η) η + Dη ( η, η) η + ḡη ( η) + τη , (8)
where matrix Cη ( η, η), Dη ( η, η), ḡη ( η), τη . A detailed derivation of the dynamic model ( 8) is presented in Appendix A.

ADAPTIVE NON-SINGULAR SLIDING MODE CONTROL

In this section, the proposed control strategy for trajectory tracking of UUV's is presented. A non-singular terminal sliding mode controller is developed, where an adaptive mechanism for the controller's gain is introduced. The strategy proposed ensure that the tracking errors converge to a bounded region in finite-time. Some useful preliminaries are introduced below.

Definition 1: Assume a nonlinear system ζ = f (ζ , u), where ζ and u are the state and control signal. The solution is defines as Practical Finite-Time Stable (PFS) if for all ζ 0 , there exist δ > 0 and T (δ , ζ 0 ) such that ζ (t) < δ for all t ≥ t o + T . It is worth mentioning that PFS means finite-time boundedness.

Lemma 1 [START_REF] Zhu | Attitude stabilization of rigid spacecraft with finite-time convergence[END_REF]: Consider the nonlinear system ζ = f (ζ , u). Assume that exists a continuous function V (ζ ) and parameters λ > 0, 0 < β < 1 and 0 < ω < ∞, such that

V (ζ ) ≤ -λV β (ζ ) + ω. (9)
Then, the system ζ = f (ζ , u) is PFS. Moreover, the system trajectory is bounded as

lim a→a 0 ζ ∈ V β (ζ ) ≤ ω (1 -a 0 ) λ , (10)
in finite time t ≤ T given by

T ≤ V 1-β (ζ 0 ) λ a 0 (1 -β ) , 0 < a 0 < 1. (11)

Control problem formulation

Consider a second-order dynamical system

χ1 = χ 2 , χ2 = f (χ 1 , χ 2 ) + g (χ 1 , χ 2 ) u + ξ (χ 1 , χ 2 ,t), (12)
where χ 1 , χ 2 are the state variables, f (•), g (•) are nonlinear functions with g (•) -1 = 0, u is the control input and [START_REF] Chalanga | Implementation of super-twisting control: Super-twisting and higher order sliding-mode observer-based approaches[END_REF][START_REF] Guerrero | Trajectory tracking for autonomous underwater vehicle: An adaptive approach[END_REF]. In this approach the surface is designed as

s = e 2 + ce, c > 0. ( 13
)
In comparison with the conventional linear sliding surface, the control strategy developed in this research work has a faster convergence, enhanced with finite-time properties.

Control scheme design

Based on [START_REF] Yu | Continuous finite-time control for robotic manipulators with terminal sliding mode[END_REF], an adaptive version for a Non-singular sliding mode controller for the trajectory tracking problem of an AUV is developed. The sliding surface employed is given by [START_REF] Yang | Nonsingular fast terminal slidingmode control for nonlinear dynamical systems[END_REF]

s = e 1 + β 1 sig γ 1 e 1 + β 2 sig γ 2 e 2 , (14)
where

β 1 > 0, β 2 > 0, 1 < γ 2 < 2, and γ 1 > γ 2 .
The notation sig γ a denotes sig γ a := |a| γ sign (a), where sign (•) represents the sign function.

Theorem 1: Consider the dynamic system given by ( 12) and the non-linear sliding surface defined in [START_REF] Chu | Adaptive fuzzy sliding mode diving control for autonomous underwater vehicle with input constraint[END_REF]. If the adaptive non-singular sliding mode control strategy is designed as

u = - 1 g (χ 1 , χ 2 ) 1 β 2 γ 2 sig 2-γ 2 e 2 • 1 + β 1 γ 1 |e 1 | γ 1 -1 + f (χ 1 , χ 2 ) + k1 sig κ s + k2 s -χ2 d , (15)
whit 0 < κ < 1, and the adaptive gains k1 and k2 are updated [START_REF] Wang | Adaptive super-twisting fractional-order nonsingular terminal sliding mode control of cable-driven manipulators[END_REF] as

k1 =            -σ 1 , if k1 > k 1 max -σ 2 , if k 1 min < k1 < k 1 max and |s| < ∆ s σ 1 , if k1 ≤ k 1 min or k 1 min < k1 < k 1 max , and |s| > ∆ s , (16)
k2 = σ 3 k1 , (17)
where σ 1 , σ 2 , σ 3 , ∆ s , and k 1 max > k 1 min are positive constants. Then, there exists a finite-time t F > 0 such that the system trajectory will converge to the set defined by

|s| ≤ Ω = min{Ω 1 , Ω 2 }, (18)
Ω 1 = ω 1 (1 -θ ) α 01 , (19
)

Ω 2 = ω 2 (1 -θ ) α 02 1/2 , ( 20
)
where 0 < θ < 1, and parameters ω 1 , ω 2 , α 01 and α 02 are defined below. Moreover, the tracking error e 1 and e 2 converge to the following regions

|e 1 | ≤ 2Ω, |e 2 | ≤ β -1 2 Ω 1/γ 2 , (21)
in finite-time.

Proof: Consider the Lyapunov function candidate as

V = 1 2 s 2 + 1 2 k2 1 + 1 2 k2 2 , (22)
where k2 1 = k1k 1 and k2 2 = k2k 2 . Notice that the variables k 1 and k 2 are arbitrary constants used for stability analysis purposes. That is to say, k 1 and k 2 will not be employed in the control strategy. The time derivative of [START_REF] Yang | Nonsingular fast terminal slidingmode control for nonlinear dynamical systems[END_REF] along system ( 12) is given as

V =s e 2 + β 1 γ 1 |e 1 | γ 1 -1 e 2 + β 2 γ 2 |e 2 | γ 2 -1 f (χ 1 , χ 2 ) + g (χ 1 , χ 2 ) u + ξ (χ 1 , χ 2 ,t) -χ2 d + k1 k1 + k2 k2 . (23)
Substituting the control law [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF], we obtain

V = -ρ k1 |s| κ+1 + k2 |s| 2 + sξ (χ 1 , χ 2 ,t) + k1 k1 + k2 k2 , (24)
where ρ = β 2 γ 2 |e 2 | γ 2 -1 . From the fact that the disturbances is bounded and 0 < κ < 1, we obtain

V ≤ -ρ k1 |s| -ρ k2 |s| 2 + ρξ + |s| + k1 k1 + k2 k2 . ( 25
)
The adaptive gain k1 is contained in the set

[k 1 min , k 1 max ].
Then from ( 25) we obtain

V ≤ -ρ|s| k 1 min -ξ + -ρ k2 |s| 2 + k1 k1 + k2 k2 . ( 26
)
If k 1 min is chosen such that k 1 min > ξ + , and k1 = k

1 min -ξ + , then V ≤ -ρ k1 |s| -ρ k2 |s| 2 + k1 k1 + k2 k2 . (27)
From [START_REF] Elmokadem | Terminal sliding mode control for the trajectory tracking of underactuated autonomous underwater vehicles[END_REF], the following two inequalities will hold

V ≤ -ρ k1 |s| + k1 k1 + k2 k2 , (28)
V ≤ -ρ k2 |s| 2 + k1 k1 + k2 k2 .

From inequality [START_REF] Londhe | Non-singular terminal sliding mode control for robust trajectory tracking control of an autonomous underwater vehicle[END_REF], we can obtain

V ≤ - α 11 √ 2 |s| - α 21 √ 2 | k1 | - α 31 √ 2 | k2 | + k1 k1 + k2 k2 + α 21 √ 2 | k1 | + α 31 √ 2 | k2 |, (30)
where

α 11 = √ 2ρ k1 , α 21 > 0 and α 31 > 0. In view of b 2 1 + b 2 2 + b 2 3 ≤ |b 1 | + |b 2 | + |b 3 | is fulfilled for arbitrary real numbers b 1 , b 2 , b 3 ,
the above inequality can be expressed as

V ≤ -α 01 V 1/2 + ω 1 , (31)
where α 01 = min (α 11 , α 21 , α 3 1 ) and

ω 1 = k1 k1 + k2 k2 + α 21 √ 2 | k1 | + α 31 √ 2 | k2 |.
Thus, from Lemma 1 the system trajectory will converge in finite-time to the neighborhood of sliding surface s = 0 as

|s| ≤ ω 1 (1 -θ ) α 01 . ( 32
)
Applying a similar procedure to inequality (29) we obtain

|s| ≤ ω 2 (1 -θ ) α 02 1/2 , ( 33
)
where 14), we can observe that the condition e 2 = 0 may inhibit the reachability to the region [START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF]. To prove that this situation will not occur we proceed as follow. The control law [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF] is replaced in the dynamic model [START_REF] Anderlini | Control of a rov carrying an object[END_REF] and setting e 2 = 0, we arrive to ė2 = -k1 sig κ s + k2 sξ (χ 1 , χ 2 ,t) .

ω 2 = k1 k1 + k2 k2 + α 22 √ 2 | k1 | 2 + α 32 √ 2 | k2 | 2 with α 22 > 0, α 32 > 0; α 02 = min (α 12 , α 22 , α 32 ) with α 12 = √ 2ρ k2 . From (
(

) 34
When the sliding surface s is outside of the region Ω (s / ∈ Ω), we can obtain

ė2 = -k1 -ξ (χ 1 , χ 2 ,t)sig -κ s sig κ s + k2 s = 0, ( 35
) ė2 = -k1 sig κ s + k2 -ξ (χ 1 , χ 2 ,t)s -1 s = 0. ( 36
)
Notice that from previous equations we get e 2 = 0, which means that the system trajectory will no remain in the region e 2 = 0, s / ∈ Ω. Therefore, the finite-time reachability of the sliding surface ( 14) is still guaranteed.

Combining ( 14) and (18) we obtain

s = e 2 + β 1 sig γ 1 e 1 + β 2 sig γ 2 e 2 <= Ω, (37)
which can be rewritten as

e 2 + β 1 sig γ 1 e 1 + β 2 -s • sig 1/γ 2 e 2 sig γ 2 e 2 = 0. ( 38
)
If β 2s • sig 1/γ 2 e 2 > 0 is fulfill, inequality [START_REF] Qiao | Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles[END_REF] remains as a TSM. Thus, the velocity tracking error will converge to the following region

|e 2 | ≤ |β 1 2 s| 1/γ 2 ≤ |β 1 2 Ω| 1/γ 2 . ( 39
)
Combining ( 38) and ( 39) we obtain that the position tracking error will converge to the following bound

|e| = 2Ω. ( 40
)
It is worth mentioning that the control signal (15) doesn't involve negative fractional power, thus it is non-singular.

Control strategy for the 4DOF autonomous underwater vehicle

The control strategy for the UUV in surge, sway, heave and heading is based on the robust non-singular sliding mode controller developed in Subsection 3.2. From control signal (15), we have the following control signals

• Surge X = - 1 g x 1 β 2 x γ 2 x sig 2-γ 2x e 2 x • 1 + β 1 x γ 1 x |e 1 x | γ 1x -1 (41)
+ f x + k1 x sig κ x s x + k2 x s x -x 2 d . (42)
• Sway

Y = - 1 g y 1 β 2 y γ 2 y sig 2-γ 2y e 2 y • 1 + β 1 y γ 1 y |e 1 y | γ 1y -1
+ f y + k1 y sig κ y s y + k2 y s yy 2 d .

(43)

• Heave Z = - 1 g z 1 β 2 z γ 2 z sig 2-γ 2z e 2 z • 1 + β 1 z γ 1 z |e 1 z | γ 1z -1 + f z + k1 z sig κ z s z + k2 z s z -z 2 d . (44)
• Heading

N = - 1 g ψ 1 β 2 ψ γ 2 ψ sig 2-γ 2ψ e 2 ψ • 1 + β 1 ψ γ 1 ψ |e 1 ψ | γ 1ψ -1 + f ψ + k1 ψ sig κ ψ s ψ + k2 ψ s ψ -ψ 2 d . ( 45
)
The functions f x , f y , f z , f ψ , g x , g y , g z , g ψ are given in Appendix A. The sliding surfaces s x , s y , s z and s ψ are defined accord to (14).

REAL-TIME EXPERIMENTS

In this section, a brief description of the UUV used to implement the control strategy developed is presented. As well as the description of the real-time experiments realized in surge, sway, heave and heading.

Description of the underwater vehicle

The BlueROV2 is the underwater vehicle used for the realization of real-time experiments, see Fig. 2. This vehicle is connected to a ground station through metaoperating system ROS (Robot Operating System), the power supply is by means of Li-Po batteries, which are in the vehicle. The ground terminal acquires the data of the robot's sensors such that the controller calculates the torque for the thrusters. The control algorithms are executed with a sampling period of 50 milliseconds. The vehicle's dimensions are 45.71 × 33.81 × 22.1 cm with a weight of 11.5 Kg. This vehicle has the capacity to submerge up to 100 m. The known vehicle parameters are presented in Table 2.

The electronic structure of the BlueROV2 is illustrated in Fig. 3, where you can see that it uses a Pixhawk autopilot, which is connected to a Raspberry pi 3, which has a camera that can transmit video in real time through a Fathom-X Tether, which connects to the ground station.

The estimation of the current position and orientation of the UUV it is transmitted of the vehicle is by water linked API and BlueROV2, the submarine has a hydrophone that sends an acoustic pulse to four receiving beacons. The received acoustic signal is processed in a positioning com-Fig. 2. This figure shows the system used to locate the BlueROV2, which consists of four hydrophones, a transmitter mounted on the vehicle and a ground station (master). puter that calculates the time of arrival of the signal to each receiving beacon and using the data fusion of sensors, the robot position is triangulated. The range of the underwater GPS system is 200 × 200 × 100 meters, its update rate is 4 Hz.

It is important to mention that the accuracy of the location system is affected by acoustic reflections, therefore, a second order low pass filter is implemented to attenuate background noise, for a better understanding you can see Fig. 4. The ground station is the Lenovo Thinkpad T430 notebook, equipped with Intel Core i5-3210M (3.1 GHz frequency), 8GB RAM memory, SSD hard drive under Linux Xenial 16.04 LTS operating system and ROS Kinetic. Such a computer was proposed for the autonomous control of the BlueROV2, to conduct all necessary calculations at the ground station, including: data (measurements) processing, autonomous control, etc.

Real-time experiments

In this section, the control strategy [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF] developed in this research work was applied to control the surge, sway, depth and heading dynamics of the UUV. In order to show the outperformance of the strategy proposed, we conducted a comparative study, in the surge and sway dynamic, with the NSTSM controller presented in [START_REF] Londhe | Non-singular terminal sliding mode control for robust trajectory tracking control of an autonomous underwater vehicle[END_REF].

The parameters of the controller and the adaptive law Get the BlueROV2 orientations and positions from the IMU and the underwater GPS, respectively. Send the command to the thrusters of the vehicle based on the previous control laws. 9: end while 10: Close all communications. Table 3. Controller gains and adaptive law parameters.

Parameters of the controller and adaptive laws

β x 1 = 4 β x 2 = 5 γ x 1 = 1.3 γ x 2 = 1.1 ∆ x = 3.5 k 1xmax = 15 k 1 x min = 12.5 σ x 1 = 0.2 σ x 2 = 2 β y 1 = 8 β y 2 = 7 γ y 1 = 1.3 γ y 2 = 1.1 ∆ y = 3.5 k 1ymax = 13 k 1 y min = 12.5 σ y 1 = 0.2 σ y 2 = 2 β z 1 = 150 β z 2 = 50 γ z 1 = 1.2 γ z 2 = 1.1 ∆ z = 3.5 k 1zmax = 15 k 1 z min = 13 σ z 1 = 1 σ z 2 = 2 β ψ 1 = 5 β ψ 2 = 6 γ ψ 1 = 1.5 γ ψ 2 = 1.05 ∆ ψ = 10 k 1ψ max = 1.35 k 1 ψ min = 1.1 σ ψ 1 = 0.25 σ ψ 2 = 2
were adjusted heuristically and are summarized in Table 3.

Experimental tests are performed in a pool of dimensions 25 × 20 × 3 m, four test scenarios are proposed. In three scenarios the same trajectory is used, in surge, sway and yaw angle. Algorithm 1 presents the steps to implement the algorithm to real-time experiments. The desired route is carried out using the parametric equations given by: where χ f is the final position and t f is the final route time.

χ 1d (t) = χ f 10 t t f 3 -15 t t f 4 + 6 t t f 5 , χ 2d (t) = χ f 30 t 2 t f 3 -60 t 3 t f 4 + 30 t 4 t f 5 , (46)
In the first scenario, the comparative tests the trajectory tracking of the submarine vehicle on the x-axis is given by (46) with χ f ± 2.5 and t f = 30. The first comparative test is performed with the trajectory tracking on the x-axis, the results of UUV can be seen in Fig. 5. We can observed that the reference is reached in both cases, however, it is visible that the trajectory tracking error is smaller with the proposed controller.

The graphs of the evolution of the control input and the sliding surface for ANTSMC are shown in Fig. 6, note that there is an attenuation in the chattering effect and that the surface is maintained in a convergence region close to zero.

Moreover, in Fig. 7 we can see the response of the control input and the sliding surface for NSTSM controller, notice that the sliding surface doesn't converge in a region close to zero. Finally, for the first scenario the evolution of the adaptation of the parameters k1x and k2x are presented in Fig. 8.

In the second scenario, trajectory tracking is performed on the y-axis using the reference proposed by (46) with the same conditions as scenario 1. As shown in Fig. 9, the behavior of the trajectory tracking performed is reported. We can appreciate that it is close to the reference in both cases but the trajectory tracking error on Sway with NSTSM controller is bigger that with the ANTSMC. We can observe that the reference is reached and it is visible that when it reaches the final position there is an overshoot using the NSTSM controller. Fig. 10 shows that there was also attenuation in the chattering effect and that the control input is very small approximately only 10% of the force provided by the thrusters. Moreover, it can be seen that sliding surface being maintained in a region. On the other hand, in Fig. 11 the response of the control input and the evolution of the sliding surface of the NSTSM controller are displayed. We can observe that the sliding surface presents oscillations when it converges to a close region zero. We can observe in Fig. 12 the temporal evolution of the adaptation of gains k1y and k2y , it can be noted that the adaptive gains remain bounded. In the Table 4, we obtain the RMSE for each controller and conclude that the controller proposed in [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF] is more efficient in both scenarios, i.e, the proposed controller has a very small tracking error.

The third scenario was followed by a sinusoidal reference in depth given by equations presented. Note that the reference signal is reached in a short time and that the tracking is quite satisfactory with almost no tracking error. Also, we can be seen in Fig. 14 the response of the control signal and the behavior of sliding surface from stay in a region very close to zero. The evolution of the adaptive parameters k 1z and k 2z are shown in Fig. 15, note that both parameters remain at the lower limit because the sliding surface converges to a region near zero. Finally, in the fourth scenario the yaw trajectory tracking given by (46) was carried out with χ f = 100 and t f = 30. The results of this experimental test are presented in Fig. 16, we can see that the desired orientation is reached along trajectory tracking.

z 1d = -1.4 + 1.
The efficiency of the UUV trajectory tracking during the four scenarios can be seen in the following video: https://youtu.be/2DrcwiENBHI. We can observe the behavior of the robot during experimental tests and the ac- Remark 2: The adaptive controller's gains increasing and decreasing in a bounded region, which implies that the closed loop system remains stable. For the robustadaptive control strategies developed in [START_REF] Shtessel | Super-twisting adaptive sliding mode control: A lyapunov design[END_REF][START_REF] Guerrero | Trajectory tracking for autonomous underwater vehicle: An adaptive approach[END_REF], the controller gains keep increasing. If the initial tracking errors are large or in presence of external disturbances, these sitwould give rise to large control signals, actuator saturation and even failures in the system.

CONCLUSION

In this research work, we developed and implemented an ANTSMC for the trajectory tracking of an UUV, moreover to perform a practical comparison with an NSTSM controller, in order of know the advantages that the proposed controller offers. The real-time experimental results obtained demonstrate the behavior of the closed loop system and the effectiveness of the proposed controller strategy. We conclude, based on the results obtained and the calculation of RMSE, that the strategy proposed is more efficient that NSTSM controller. In addition, the graphics of the behavior of adaptive parameters show the evolution of gains by decreasing or increasing its value in a bounded region, according to the magnitude of the sliding surface. Also, a significant reduction of the chattering effect in the control signal is observed. To further our research we will develop a disturbance observer that enable us to estimate external unknown disturbances. Furthermore, we will conduct sea-trial in more aggressive environments.

APPENDIX A

From ( 1) and ( 6), the dynamic model of the underwater vehicle, expressed in the inertial frame O I , is given as The BlueROV2 consists of 6 thrusters which allows the vehicle to move freely in four degrees of freedom, the thrusters of 1 to 4 generate lateral movements, forward and reverse, in addition to generating the rotational movement in yaw. The thrusters 5 and 6 is are used to generate immersion and emersion. The configuration of the thrusters is shown in Fig. 17. The vector of forces and torques τ is given as follows:

τ =     -∑ 4 i=1 F i c ϕ i ∑ 4 i=1 (-1) i F i s ϕ i -F 5 -F 6 ∑ 4 i=1 (l x F i s ϕ i + l y F i c ϕ i )sign(i -2.5)     , (A.7)
where l y , l x is the distance between the center of gravity of the vehicle to the thrusters on the axis y and x, respectively. In order to simplify the notation, we rewrite equation

Fig. 1 .

 1 Fig. 1. Reference frames. Body fixed-frame O B and earthfixed reference frame O I .

Fig. 3 .Fig. 4 .

 34 Fig. 3. BlueROV2 electronic elements diagram.

Algorithm 1 :

 1 Procedure for the implementation of ANTSMC Controller by using BlueROV2, ROS and Water Linked GPS API 1: Requiere: Install mavros, bluerov-ros-pkg, Water Linked Underwater GPS API and all necessary dependencies. 2: Initialize the communication with the BlueROV2 vehicle by modifying the network configuration and selecting a valid IP. 3: Initialize the Water Linked GPS through the API. 4: Define a desired trajectory for the position and orientation of the BlueROV2 vehicle. 5: while No error do 6:

7 :

 7 Calculate the control laws[START_REF] Zhu | Attitude stabilization of rigid spacecraft with finite-time convergence[END_REF][START_REF] Wang | Adaptive super-twisting fractional-order nonsingular terminal sliding mode control of cable-driven manipulators[END_REF](43)(44) for the BlueROV2 to reach trajectory in position (Surge, Sway and Heave) and orientation (Yaw).

Fig. 5 .Fig. 6 .

 56 Fig. 5. Trajectory tracking in position x using acoustic localization. x d desired (dashed red line), x trajectory with ANTSMC (solid blue line) and x n trajectory with NSTSM controller (dashed point magenta line).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Control input and sliding surface in surge controller using NSTSM controller.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Trajectory tracking in position y using acoustic localization. y d desired (dashed red line), y trajectory with ANTSMC (solid blue line) and y n trajectory with NSTSM controller (dashed point magenta line).

Fig. 13 ,Fig. 11 .Fig. 12 .

 131112 Fig.13, the results of the depth trajectory are

Fig. 13 .Fig. 14 .Fig. 15 .

 131415 Fig.[START_REF] Li | Neural network-based sliding mode variable structure control for mars entry[END_REF]. Autonomous trajectory tracking in depth using the ANTSMC.

Fig. 16 .

 16 Fig. 16. Angular position, ψ is the yaw angle and ψ d is the desired yaw angle, used NSTSM controller.

3 )D 11 =

 311 follows: η = -M-1 ( η) C( η, η) η -M-1 ( η) D( η, η) η -M-1 ( η) ḡ( η) + M-1 ( η) τ, (A.1)whereM( η) = M J-1 ( η), D( η, η) = D( J( η) -1 η) J( η) -1 , C( η, η) = -M J-1 ( η) J( η) J-1 ( η) +C( J( η) -1 η) J( η) -1 . (A.2)The matrices that represent the dynamic model in the inertial framework are defined asM( η) =     (m-X u)c ψ (m-X u)s ψ 0 0 -(m-Y v)s ψ (m-Y u)c ψ 0Calculating the C( η, ˙η) matrix in terms of η and the vehicle's conditions we obtainC( η, η) -X u) ψs ψ C 12 0 -C 14 -(m -Y v) ψs ψ c ψ -C 22 0 C 12 = (m -X u) ψc ψ ; C 14 = (m -Y v)( ẏc ψẋs ψ ); C 22 = (m -Y v) ψs ψ ; C 24 = (m -X u)( ẋc ψ + ẏs ψ ); C 41 = c ψ (m-Y v)( ẏc ψẋs ψ )+s ψ (m-X u)( ẋc ψ + ẏs ψ ) and C 42 = s ψ (m -Y v)( ẏc ψẋs ψ )c ψ (m -X u)( ẋc ψ + ẏs ψ ).Rewriting the damping matrix in the inertial system, we have to -c ψ (X u + X u |u| | ẋc ψ + ẏs ψ |),d 12 = -s ψ (X u + X u |u| | ẋc ψ + ẏs ψ |), d 21 = s ψ (Y v +Y v |v| | ẏc ψẋs ψ |), d 22 = -c ψ (Y v +Y v |v| | ẏc ψẋs ψ |).(A.6)

Fig. 17

 17 Fig. 17. The thrusters 5 and 6 are used to submerge and emerge. Thrusters 1 to 4 are used for the advancerecoil motion and the rotation movement in yaw.

  Fig. 17. The thrusters 5 and 6 are used to submerge and emerge. Thrusters 1 to 4 are used for the advancerecoil motion and the rotation movement in yaw.

  (A.1) as follows: η = Cη ( η, η) η + Dη ( η, η) η + ḡη ( η) + τη , (A.8)whereCη ( η, η) = -M-1 ( η) C( η, η), Dη ( η, η) = -M-1 ( η) D( η, η), ḡη ( η) = -M-1 ( η) ḡ( η), τη = M-1 ( η) τ.(A.9)From (A.8), we obtain the following expressions for the dynamics of the UUVẍ = s ψ Y -(Y ν +Y ν | ν| | ẏc ψẋs ψ |)( ẏc ψẋs ψ ) Y νm + s ψ ψ(X um)( ẋc ψ + ẏs ψ ) Y νm ψ(s ψ + c ψ )( ẋc ψ + ẏs ψ ) + c ψ (X u + X u |u| | ẋc ψ + ẏs ψ |)( ẋc ψ + ẏs ψ ) -X X um + c ψ ψ(Y νm)( ẏc ψẋs ψ ) X um , ÿ = ψc ψ ( ẋc ψ + ẏs ψ ) -ψs ψ ( ẏc ψẋs ψ ) c ψ Y -(Y ν +Y ν | ν| | ẏc ψẋs ψ |)( ẏc ψẋs ψ ) Y νm

  ξ (•) is a bounded lumped disturbance. Moreover, disturbance ξ (•) satisfies |ξ (•)| ≤ ξ + , with ξ + a positive constant. Let us define the tracking errors e 1 = χ 1χ 1 d and e 2 = χ 2χ 2 d , where χ 1 d and χ 2 d denote the reference signals. The problem is to design a sliding mode controller that the tracking errors converges to a bound region in finite-time despite the presence of lumped disturbances.Remark 1: Classical linear sliding surface has been used for a long time. Nowadays, it is widely used in Super Twisting sliding mode controllers

Table 2 .

 2 Physical and dynamic parameters of the BlueROV2.

	Param.	Value.	Param. Value. Param. Value.
	W	[0, 0, 176] T	X u	5.5	Y v	12.7
	r b	[0, 0, -0.1] T	Z ẇ	14.57	I xx	0.16
	B	[0, 0, 9.81m f ] T	I yy	0.16	I zz	0.16
	K ṗ	0.002	M q	0.002	N ṙ	0.12
	X u	3.03	Y v	3.21	Z w	8.18
	K p	2	M q	0.1	N r	0.07
	X uc	4.03	Y vc	6.21	Z wc	5.18
	K pc	0.07	M qc	0.07	N rc	0.07

Table 4 .

 4 Root Mean Square Error criterion for surge and sway controllers.

	RMSE (m)
	NSTSM controller	x	0.2424
		y	0.1932
	ANTSMC	x	0.1288
		y	0.0686

In this work, we assume that the angle ψ is first controlled at zero (ψ = 0, ψ = 0). Under this assumption, (A.10) is rewritten as follows:

Finally, for the development of the control law, the following terms of (A.11) were defined as

(A.12)


Jesús Ricardo López Gutiérrez

A Efraim

Rodriguez Mata

Sergio Salazar

Ivan Gonzalez

Rogelio Lozano

Robust Quadrotor Control: Attitude and Altitude Real-Time Result

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