MATEC Web of Conferences 8303001 , (2016) DOI: 10.1051/matecconf/20168303001 CSNDD 2016

Modeling and position control of tethered octocopters

Davi Ferreira de Castroa, Igor Afonso Acampora Pradob, and Mateus de Freitas Virg´ılio Pereira c, Davi Antonioˆ dos Santosd and Jose´ Manoel Balthazare ∗ Aeronautics Institute of Technology, ITA, Brazil.

Abstract. This work presents the modeling and control of a multirotor aerial vehicle with tethered configura- tion. It is considered an octocopter with a saturated proportional-plus-derivative position control. A viscoelastic model is considered for the tether, which has a tension control. Numerical simulations are carried out to compare the performance of the tethred configuration with the vehicle in free flight.

1 Introduction 2 Tethered configuration model

Multirotor Aerial Vehicles (MAVs) have been the subject This Section presents the multicopter model and tether model, of many academic studies and have attracted a lot of at- which together describe the dynamics of the tethered mul- tention from industry in recent years. MAVs have flight ticopter configuration. capabilities such as hovering, Vertical Take-Off and Land- ing (VTOL) and agile maneuvering capability, which can- not be achieved by conventional fixed wing aircraft. Some 2.1 Multicopter model drawbacks of multicopter vehicles are the reduced endurance, which hardly exceeds some minutes and limited payload. Consider a octocopter and three Cartesian coordinate sys- Tethered MAVs have been introduced to the aerospace sec- tems (CCS) illustrated in Figure 1. tor as a possible alternative. The tether, even though limit- ing the vehicle flight range, can be used for data transmis- ZR ZB sion and power supply during operation, which improves YR

the vehicle endurance. YB In academia, research works on tethered multicopters Horizontal plane are still rare by the present day and there are few compa- XB nies and institutions that have invested in the development CM X ZI R TC of such vehicles. The first references to consider the teth- YI ered configuration are found in [7] and [4]. Some applica- tions of this kind configuration are found in aerostats [1] c Cable and robot satellites [2]. The work in [3] presents a naviga- O

tion strategy in partially mapped environments. This strat- XI egy is applied to the collaboration between an UAV 1 and an UGV 2, in which power is provided to the former by a Fig. 1. Cartesian coordinate systems. cable connecting the two vehicles. The work [5] presents a strategy to control a tethered helicopter with a combination The CCS of the body S B , {XB, YB, ZB} is fixed to the of classical PID control laws. vehicle structure and centered on the center of mass (CM). This paper presents a modeling and position control The inertial CCS S , {X , Y , Z } is fixed to the earth at strategy for tethered MAV. A viscoelastic model is con- I I I I the point O. The CCS S , {X , Y , Z } is parallel to sidered for the cable, in order to reproduce its dynamic R R R R S I and centered in CM. TC is tether tension and c defined behavior. The controller is based on a saturated state feed- tether line vector. back control, thus simplifying the controller. This paper is The differential equations that describe the dynamic organized as follows: in Section 2 we present the tethered behavior of the multirotor can be obtained by the Newton- configuration model, in Section 3 a control system is pro- Euler formalism. This work chooses to represent the at- posed, in Section 4 we present the simulation results and titude by the quaternion of rotation q ∈ R4, however for Section 5 presents the conclusions. visualization is adopted Euler angles (φ, θ and ψ). a e-mail: [email protected] The kinematic attitude model for the quaternion of ro- b e-mail: [email protected] tation is given by the following differential equation [8]: c [email protected] e-mail: q˙ = Ωq (1) d e-mail: [email protected] e e-mail: [email protected] where " # 1 Unmanned Aerial Vehicle (UAV) 1 0 −ωT Ω = , (2) 2 Unmanned Ground Vehicle (UGV) 2 ω −[ω×] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

MATEC Web of Conferences 8303001 , (2016) DOI: 10.1051/matecconf/20168303001 CSNDD 2016

T h i 3 being ω , ωx ωy ωz ∈ R the angular velocity of the CM vehicle represented in S R and [ω×] is given by: CM x2 TC  0 −ω ω   z y  k ×  ω 0 −ω  k b [ω ] =  z x . (3) b   x1 −ωy ωx 0 q9 L da k d k Rewriting the equation (1) explicitly, one obtains: a a a       c c  q˙1   0 −ωx −ωy −ωz  q1        O O  q˙2  1 ωx 0 ωz −ωy  q2    =     . (4) (a) Unstressed tether (b) Elongated tether  q˙  ω −ω 0 ω   q   3  2  y z x   3  q˙4 ωz ωy −ωx 0 q4 Fig. 2. Viscoelastic model of the tether, adapted from [5]. Applying Newton’s second law for rotational motion and neglecting disturbance torques, one can obtain: is reeled out by the ground device and q9 is deployed tether τ = h˙ + ω × h, (5) segment. It is important to note that in the model adopted the mass of the cable is considered negligible in relation T 3 to the vehicle mass. The equations and parameters are pro- where τ , [τx τy τz] ∈ R is the resulting propulsion 3 posed by Sandino et al [5]. The variables that describe the torque in the vehicle represented in S B and h , I ω ∈ R tether state are shown in Figure 2. The transfer function is the angular momentum of the body represented in S B, being I ∈ R3x3 the matrix of inertia of the vehicle repre- associated tether model is proposed by Sandino et al [5] is given by: sented in S B. It is assumed that the vehicle has symmetrical structure with respect to the coordinate axes. The matrix I ! ka resulting in a diagonal matrix defined by: dakb + s T (s) da   G(s) = C = (11) Ix 0 0 U (s) s(k + k + d s)   C a b a I = 0 Iy 0 . (6)   where UC(s) represents the tether velocity of frequency do- 0 0 Iz main. Regarding model parameters da, ka and kb, they are Rewriting the equation (5) explicitly, one can obtain: usually expressed as:

(Iy − Iz) τx q.E1 q.E2 q.η ω˙ x = ωyωz + . (7) ka = , kb = , da = (12) Ix Ix L L L

(Iz − Ix) τy Parameter values that have been used in this study for ω˙ y = ωxωz + . (8) simulation purposes are listed in Table 1. Iy Iy

(Ix − Iy) τz ω˙ z = ωxωy + . (9) Iz Iz Table 1. Parameters of the tether model [5]. Applying the second Newton’s law of motion and ne- Parameter Value Units −3 glecting any perturbation, we obtain the following model Diameter dC 10e m !2 of translational movement in S I: d Cross section q π C m2 2  0  1   Modulus of elasticity E1 0.194 GPa r¨ = f n +  0  , (10) m   E1 0.543 GPa −g Viscosity η 0.096 GPa.s T 3 Initial tether length L 10 m where r , [rx ry rz] ∈ R is the three-dimensional posi- T 3 tion of the CM represented in S I, n , [nx ny nz] ∈ R is the normal unit vector perpendicular to the rotor plane represented in S I, f is the total thrust, g is the acceleration of gravity and m is the mass of the vehicle. 3 Control system

2.2 Tether model 3.1 Multicopter control

A viscoelastic model for the cable is considered in order The multicopter control strategy presented in this section is to reproduce its behavior, as illustrated in Figure 2. More described by Santos et al [6]. Let us define the total thrust precisely, the model consists of a damper with coefficient vector ∈ R3 of the octocopter to be: da and a spring with stiffness ka in parallel, both connected in series with a spring of different stiffness kb. f , f n. (13) Here x1 is an unobservable variable which describes the inner friction, x2 represents the elongation of that seg- Let us define the inclination angle ϕ ∈ R of the octo- ment, L denotes the unstressed tether segment lenght that copter to be:

2

MATEC Web of Conferences 8303001 , (2016) DOI: 10.1051/matecconf/20168303001 CSNDD 2016

height rz responds to the height desired rz, as if it were gov- −1 ϕ , cos nz. (14) erned by the following second order linear time-invariant (LTI) system: Note that ϕ consists of the angle between n and ZR (see Figure 3). r¨z + k2r˙z + k1rz = k1rz, (19)

where k1 > 0 ∈ R e k2 > 0 ∈ R are controller coefficients. n ZR 3.1.2 Horizontal position control

휑 T 2 T 2 Let rxy , [rx ry] ∈ R and nxy , [nx ny] ∈ R denote the horizontal projections of r and n, respectively. From equation (10), one can write the horizon position dynamic X R CM model as f r¨xy = nxy. (20) nxy m

f Consider the system modeled by equation (20) and the YR e r xy= nxy maximum allowable value ϕmax > 0 of the inclination an- m gle ϕ. Assume that m and f , 0 are exactly known. The feedback control law is given by: Fig. 3. Relationship between the normal unit vector n and the ( horizontal acceleration r¨ xy. αxy, kαxyk ≤ sin ϕmax nxy = αxy , (21) sin ϕmax , kαxyk > sin ϕmax kαxyk Let r [r r r ]T ∈ R3 denote the position desired for with , x y z m h   i the octocopter and define the corresponding tracking error αxy , − kz rxy − rxy + k4r˙xy , (22) r˜(i) ∈ R3 by f r˜ , r − r. (15) respecting the inclination angle constraint ϕ ≤ ϕmax and is such that if kαxyk ≤ sin ϕmax, then rxy responds to the hori- Let ϕ ∈ R be the maximum allowable values of the max zontal position desired r , [r r ]T as if it were governed ∈ R ∈ R xy x y inclination angle ϕ, fmin and fmax be, respec- by the following pair of second order LTI systems: tively, the minimum and the maximum allowable values of the total thrust magnitude f . The solution to the problem r¨xy + k4r˙xy + k3rxy = k3rxy, (23) of position control is to find a feedback control law for f, which makes r → r or r˜ → 0, while respecting the con- where k3 > 0 ∈ R e k4 > 0 ∈ R are controller coefficients. straints fmin ≤ f ≤ fmax and ϕ ≤ ϕmax. The control position In practice, the attitude desired for the internal attitude is partitioned into control height and horizontal position control loops need to be computed from the normal unit control. vectors n. In order to specify a unique attitude, it is nec- essary to select a heading. For example, one can choose a zero heading just by taking into consideration the attitude 3.1.1 Height control represented by the principal Euler angle/axis (ϕ, e), where ϕ is the angle of inclination own and e ∈ R3 is a unit vector From Equation (10), the dynamic model of the height of (see Figure 3) is given by: the vehicle rz is given by n × nxy e = (24) nz k × k r¨ = f − g. (16) n nxy z m From (ϕ, e) one can thus represent the attitude of S B Consider the system modeled by Equation (16), and the with respect to S R, using , for example, Euler angles pa- minimum fmin > 0 and maximum fmax > fmin allowable rameterization that permits obtains φ, θ and ψ. values of the command total thrust magnitude f . Assume that m, g and nz , 0 are exactly known. The feedback con- trol law is given by: 3.2 Tether tension control   fmin, αz < fmin The tether tension is controlled by a proportional control  ∈ f =  αz, αz [ fmin, fmax] (17) configuration. Equation 25 and Figure 4 show the architec-  fmax, αz > fmax ture of the tether control, which is described by Sandino et with al[5]. m αz , (g − k1 (rz − rz) − k2r˙z) , (18) ? ? nz UC = kp(TC − TC) (25) ? ? respecting the total thrust magnitude constraint fmin ≤ f ≤ where UC is the desired velocity tether, TC is the desired fmax and is such that if fmin ≤ αz ≤ fmax, then the vehicle tether tension and kp is proportional gain. 3

MATEC Web of Conferences 8303001 , (2016) DOI: 10.1051/matecconf/20168303001 CSNDD 2016

10 TC 9 Tethered MAV TC UC = dL/dt TC kp G(s) Free-Flight MAV 8

Fig. 4. Control system for simplified tether model given by G(s) 7 as proposed by Sandino et al [5]. 6

5 3.3 Tethered multicopter control 4 Amplitude

In order to control the height, a term TC was added in Equa- 3

tion 18. Therefore, Equation 26 represents the proposed 2 controller. 1 m αz , (g − k1 (rz − rz) − k2r˙z) + TC (26) 0 nz 0 10 20 30 40 50 60 70 Time [s] Fig. 5. Position error 4 Simulation results

The multicopter dynamics is integrated by using the fourth Acknowledgements order Runge-Kutta method with time step of T = 0.002 s. The Reference trajectory follows the way-points : w(1) = We would like to thanks Fundac¸ao˜ de Amparo a` Pesquisa T T [0 0 0] m, w(2) = [4.24 4.24 8] m, w(3) = [4.24 − do Amazonas (FAPEAM), Conselho Nacional de Desen- T T 4.24 8] m, w(4) = [−4.24 −4.24 8] m, w(5) = [−4.24 4.24 volvimento Cient´ıfico e Tecnologico´ (CNPq) and Coordenac¸ao˜ T T 8] m and w(6) = [4.24 4.24 8] m. The mass of vehicle is de Aperfeic¸oamento de Pessoal de N´ıvel Superior (CAPES) m = 2.132 kg and the gravity acceleration is g = 9.796 for the grants awarded. 2 m/s . The control parameters of vehicle, k1, k2, k3 and k4, are chosen based on equations (19) and (23). The max- ref imum overshoot of Mp = 0.02 m, and a peak time of References ref tp = 2 s, were selected, which resulted in k1 = 6.29 and ref k2 = 3.91. Similarly, the maximum overshoot of Mp = 1. S. Khaleefa, S. Alsamhi, and N. Rajput. Tethered bal- ref 0.02 m and a peak time of tp = 3 s were selected and the loon technology for telecommunication, coverage and k3 = 2.78 and k4 = 2.61 gains were obtained. The propor- path loss. IEEE Students Conference on Electrical, tional gain tether control selected was kp = 0.01. The sim- Electronics and Computer Science, 2014. ulation was ran for 70 seconds, and the following bounds 2. M. Nohmi. Mission design of a tethered robot satellite ◦ were selected: ϕmax = 30 , fmax = 40 N e fmin = 2 N. A stars for orbital experiment. 18th IEEE International sinusoidal disturbance input was added in axes X and Z at Conference on Control Applications, 2009. t = 15s with amplitude of 5N . The simulations provide 3. C. Papachristos and A. Tzes. The power tethered uav comparisons between free and tethered flight. The control ugv team: A collaborative strategy for navigation in par- error was quantified by the root mean square error, given tially mapped environments. 22nd Mediterranean Con- by: ference on Control and Automation (MED), 2014. 4. D. Rye. Longitudinal stability of a hovering tethered ro- r 1 torcraft. Journal of Guindance, Control and Dynamics, ERMS = [(error x)2 + (error y)2 + (error z)2] (27) 3 8:743–752, 1985. 5. L. A. Sandino, M. Bejar, K. Kondak, and A. Ollero. Ad- Figure 5 presents the position tracking error of the way- vances in Modeling and Control of Tethered Unmanned points. We can note that the disturbance results in a lower Helicopters to Enhance Hovering Performance. Journal amplitude in the position error for tethered vehicle when of Intelligent and Robotic Systems, 73:3–18, 2014. compared with the MAV in free flight. 6. D. A. Santos, O. Saotome, and A. Cela. Position For- mation Control of Multirotors using Thrust Vector Con- straints. IEEE Transactions on Aerospace and Elec- 5 Conclusion tronic Systems, 2013. 7. G. Schmidt and R. Swik. Automatic hover control of an This work has presented the modeling and control of a teth- unmanned tethered rotoplatform. Automatica, 10:393– ered MAV configuration. Numerical simulations showed 403, 1974. that, when compared with the traditional vehicle in free 8. J. R. Wertz. Spacecraft Attitude Determination and flight, the tethered configuration has a better performance control. Kluwer Academic Publishers, The Nether- under external disturbances. The tether reduces the posi- lands, 1978. tion error amplitude, improving the robustness of the vehi- cle. For future work, we consider a lumped mass model for the tether, as well as a pertubation model that corresponds to actual wind profiles in outdoor environments.

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