Extended Simulations for the Observer-based Control of Position and Tension for an Aerial Robot Tethered to a Moving Platform

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1 Extended Simuations for the Observer-based Contro of Position and Tension for an eria obot Tethered to a Moving Patform Marco Tognon, ntonio Franchi To cite this version: Marco Tognon, ntonio Franchi. Extended Simuations for the Observer-based Contro of Position and Tension for an eria obot Tethered to a Moving Patform: Technica ttachment to: M. Tognon, S. S. Dash, and. Franchi Observer-based Contro of Position and Tension for an eria obot Tethered to a Moving Patform pubished in the IEEE obotics and utomation Letters and additionay presented at 6 IEEE IC conference. [esearch eport] LS-CNS. 6. <ha-65> HL Id: ha-65 Submitted on 5 Jan 6 HL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 Extended Simuations for the Observer-based Contro of Position and Tension for an eria obot Tethered to a Moving Patform Technica ttachment to: M. Tognon, S. S. Dash, and. Franchi Observer-based Contro of Position and Tension for an eria obot Tethered to a Moving Patform pubished in the IEEE obotics and utomation Letters and additionay presented at 6 IEEE IC conference Marco Tognon, and ntonio Franchi, This document is a technica attachment to [] as an extension of the simuation s section. I. MODELING WITH LGNGIN FOMLISM To derive the dynamic equations of q using the Lagrangian approach, we first compute the kinetic and potentia energies, which are K = m ṗ W ṗ W + J W ϑ W = m ṗ W ṗ W + J W V = m gp W e, respectivey, where p W = pw C + Cd C and ṗ W = C (ṗc C + Ω C d C + d C + ḋ C), () Ω C = [ω C ], J W = J W /r W, e = [ ] T, and g 9.8 is the gravitationa constant. Finay, the generaized forces that perform work on q are [ ] Q = pw T q e e u t = G(q,, C )u t, where e = [ ] T, e = [ ] T, u t = [ f τ W ] T and τ W = τ W /r W. ppying the Euer-Lagrange equation [] to K and V we obtain M q + c + g + n = Gu t, () where M(q) is the positive definite inertia matrix, c(q, q,ṗ C C, ω C ) contains a the centrifuga/coriois terms, g(q, C ) contains a the gravity terms and n(q, p C C, ω C ) contains the terms depending on the acceeration of the moving patform. In equation (6) of Tabe I we report the fu expression of a the terms in (). LS-CNS, 7 venue du Coone oche, F- Tououse, France. mtognon@aas.fr, antonio.franchi@aas.fr Univ de Tououse, LS, F- Tououse, France This work has been partiay funded by the European Union s Horizon research and innovation programme under grant agreement No 67 EOMS. II. EXTENDED SIMULTIONS In this section we provide severa additiona detaied simuation resuts in order to test the vaidity of the proposed method. We consider an aeria vehice with nomina mass m = [Kg] and inertia J = diag(.5,.5,.5)[kg m ]. The nomina winch radius and inertia are r W =. and J W =.5[Kg m ], respectivey. We set k i and k j such that the error dynamics ξ i and ξ j have poes in (,,, ) and (, ) respectivey. Whie for the observer we choose ε =. and (α,α ) such that s + α s + α has roots (, ). Those vaues guarantees the stabiity and ensure a sufficienty fast exponentia tracking. In the simuation we reproduce a possibe rea appication scenario in which the task consists to et the moving patform and the aeria vehice cooperativey patro an area. The patform foows a certain smooth trajectory in the D space with x C tangent to the curve and y C perpendicuar to z W, as if, e.g., a ground vehice is foowing a mountain road. We require the aeria vehice at time t to takeoff from the moving patform, at time t circ to circe above the patform at a certain atitude, and at time t and to and at the same takeoff point on the moving patform. Moreover we ask that the yaw ange of the aeria vehice foows the one of the patform, in this way the two vehices aways head to the same direction. Finay, during the takeoff the desired stress must go from a sma initia tension of.5 to a steady-state vaue of that is kept for the whoe circing phase. Finay the tension has to go back to the initia vaue during the anding. In order to fuy vaidate our contro strategy we tested it on severa different non idea conditions, reported in the foowing: Case ) initia position and estimation errors, Case B) parametric variations, Case C) partia measurements of the moving patform trajectory, Case D) noisy sensor measurements, Case E) non zero offset between the tether and the center of gravity of the aeria vehice, Case F) non diagona inertia matrix J, Case G) saturation of the inputs,

3 J W + m M = m cos δ m c = [c c c ] T c = m ( δ + ω Cx + ω Cz + ẋ C ω Cz sinδ ż C ω Cx sinδ + ϕ cos δ + ω Cy cos δ ω Cz cos δ δω Cz cosϕ + δω Cx sinϕ + ẏ C ω Cz cosδ cosϕ ż C ω Cy cosδ cosϕ ϕω Cy cos δ + ẋ C ω Cy cosδ sinϕ ẏ C ω Cx cosδ sinϕ ω Cx cos δ cos ϕ + ω Cz cos δ cos ϕ ϕω Cx cosδ cosϕ sinδ + ω Cx ω Cy cosδ cosϕ sinδ ϕω Cz cosδ sinδ sinϕ + ω Cy ω Cz cosδ sinδ sinϕ ω Cx ω Cz cos δ cosϕ sinϕ) c = m cosδ(ż C ω Cy sinϕ ẏ C ω Cz sinϕ ϕ cosδ + ω Cy cosδ + ẋ C ω Cy cosϕ ẏ C ω Cx cosϕ + δ ϕ sinδ + ω Cx ω Cz cosδ δω Cy sinδ + ω Cx cosϕ sinδ + ω Cz sinδ sinϕ + ω Cx cosδ cosϕ sinϕ ω Cz cosδ cosϕ sinϕ + δω Cx cosδ cosϕ + δω Cz cosδ sinϕ + ω Cy ω Cz cosϕ sinδ ω Cx ω Cy sinδ sinϕ ω Cx ω Cz cosδ cos ϕ) c = m ( δ + ω Cx sinϕ + ( ϕ sin(δ))/ + (ω Cy sin(δ))/ (ω Cz sin(δ))/ ẋ C ω Cz cosδ + ż C ω Cx cosδ ω Cz cosϕ + ω Cx ω Cy cosϕ + ω Cy ω Cz sinϕ + ẏ C ω Cz cosϕ sinδ ż C ω Cy cosϕ sinδ ϕω Cy sin(δ) + ẋ C ω Cy sinδ sinϕ ẏ C ω Cx sinδ sinϕ ω Cx cosδ cos ϕ sinδ + ω Cz cosδ cos ϕ sinδ + ϕω Cx cos δ cosϕ ω Cx ω Cy cos δ cosϕ + ϕω Cz cos δ sinϕ ω Cy ω Cz cos δ sinϕ ω Cx ω Cz cosδ cosϕ sinδ sinϕ) (6) g = m g( C p W q )T e n = [n n n ] T n = m (ẍ C cosδ cosϕ ÿ C sinδ + z C cosδ sinϕ) n = m cosδ(ẍ C sinϕ z C cosϕ + ω Cy cosδ + ω Cx cosϕ sinδ + ω Cz sinδ sinϕ) n = m (ÿ C cosδ ω Cx sinϕ + ẍ C cosϕ sinδ + z C sinδ sinϕ + ω Cz cosϕ) TBLE I: Fu expression of the terms in (), where p C = [x C y C z C ] T and ω C = [ω Cx ω Cy ω Cz ]. Case H) motor time constant, Case I) comparison with a controer based on standard hierarchica techniques. For each case we show the contro performances potting the tracking of each output of interest, the goba tracking error ξ track computed as the sum of each errors, and the inputs. Concerning f and τ W we aso show the nomina input coming from the fatness, f n and τw n, that shoud be appied to obtain the desired output tracking in the nomina case. We aso show the observer performances comparing the estimated state and the actua one. Th error is simpy cacuated as the sum of th error for each entry of the state. Finay we dispay the trajectories of the aeria vehice and of the moving patform in the word frame and with respect to F C. In the D pots the position of the moving patform and of the aeria vehice in some particuar instants are represented with a triange and a square respectivey.

4 z. Initia errors In this section we want to show the cosed oop stabiity of the system in dynamic condition even with some initiaization error. The system starts with an error on of., on ϕ and δ of [ ] and on f L of.5. Simiary the initiaization of the observer is done with an error of. on ˆ, of 5[ ] on ˆϕ and ˆδ, whie their veocity are initiaized to zero. In Fig. one can see that after the convergence of the observer, that takes ess than one second, the controer exponentiay steers the outputs aong the desired trajectories, whie the moving patform is foowing its own dynamic trajectory. d f d L f L f n f. -. d (a) Controer performances. 5 ^ 8 ' ^' _ ^_ [s] _ ^_ ^_ ^_ [s] ^ ^ We show here ony the first second because after the estimated state foows the actua one with high fideity for a the remaining simuation. p W t = t and p W C t = t = p C.5 t = t = zc.5 t = t = - - t = t circ y C x C t = t = x y (c) Trajectories visuaization. Fig. : Simuation: initia errors.

5 z B. Parametric variations The purpose of the next sections is to investigate the robustness of the proposed method. In particuar in this one, we consider some parameter variation between the rea mode and controer/observer. Indeed in a rea scenario we can not know exacty each parameter of the system, thus the controer and observer woud be based on some parameter vaue different from the rea one. Fig. dispays the resuts of the simuation with a parametric variation of the 5% for each entry, i.e., m, J, J W and r W. In order to partiay compensate the effects of the uncertainties we added in the controer an integra term with gain k I =. We can notice that due to the uncertainty of the mode we have some nonzero errors in the tracking and in the estimation of the state. Nevertheess the error system remains stabe and thanks to the integrator terms, during the anding maneuver we obtained a decreasing tracking error that aows a correct anding of the aeria vehice. We performed additiona extensive simuations in which we observed that the system remains stabe up to a parametric variation of the %, after this vaue the system resuts unstabe. However notice that in reaity those parameters are very we measurabe with sma errors, certainy ower than the %. d. -. f d L f n f L f 6 t circ t[s] and ^ d (a) Controer performances. 6 t circ t[s] and ' ^' _ ^_ ^ ^ _ ^_ ^_ ^_ 6 t circ t[s] and t circ t[s] and p W t = t and p W C t = t = p C zc t = t = t = t = - - t = t y C circ x C - t = t = - - x y (c) Trajectories visuaization. Fig. : Simuation: parametric variations.

6 z [m=s ] [rad=s ] [m=s ] [rad=s ] [m=s ] [rad=s ] [rad=s] x C() C.. -. _x C C _y C C _z C C Bx C C By C C Bz C C y C() C z C() C ! xc! yc! zc _! xc _! yc _! zc B! xc B! yc B! zc d f d L f L f n f. d x C() C y C() C z C() C 5 5! () xc! () yc! () zc (a) Controer performances Fig. : Pot of xc i, for i =,,,. In Simuation C a the variabes in the ast five pots are considered zero by the controer and the observer. C. Limited knowedge of p W C (t) In the Sections III and IV of [] we saw that the knowedge of XC is needed in order to compute the contro action. In other words, to obtain a perfect tracking one has to know the derivative of p C C (t) up to the fourth order and of ω C (t) up to the third order. though those variabes have to be known to obtain zero tracking error, actuay, without a posteriori knowedge of the trajectory or the mode and contro inputs of the system, it is difficut to measure the higher-order derivatives. Nevertheess, in this section we want to show that even with ony a partia measurement of XC the system stays stabe and the tracking error remains bounded. In particuar, Fig. shows the resuts obtained considering measured ony ω C (t), and p C C (t) up to its second derivative, i.e., assuming w = XC where x C = x C = (, ). Indeed, for a rea moving patform, a standard onboard sensoria configuration, such as optica fow, IMU and magnetometer, is sufficient to obtain ω C (t) and p C C (t) up to its second derivative. In Fig. b we can observe that th error is amost constanty zero even if ω C is assumed zero. Whie in Fig. a one can notice that the outputs osciates around the desired vaue and the tracking error does not go to zero but remains bounded under a reasonabe threshod. Nevertheess, with a more aggressive patform trajectory the negative effects woud be more significants. In Fig. the entries of xc i for i =,,, are potted. The ast five entries are assumed zero by the observer and the controer ^ p W t = t and _ ^_ ^_ ^_ ' ^' _ ^_ p W C t = x t = t = ^ ^ t = p C t = t circ - zc (c) Trajectories visuaization. - y t = t = x C t = t = - y C Fig. : Simuation: imited measurements of the moving patform trajectory.

7 z Type Measurement Noise variance w abs. encoder ϑ W.8[rad] w abs. encoder ϕ.8[rad] w abs. encoder δ.8[rad] w 5 acceerometer ( p W + ge ) - w 6 gyroscope ω.[rad/s] w 7 magnetometer h W - W compementary fiter. TBLE II: List of sensors. D. Noise on the measurements In this ast section we investigate the robustness of the proposed method with the presence of noise in the measurements. Tab. II gathers the variance magnitude set for each measurement. For the encoder and the gyroscope we set some reasonabe vaue found in the iterature []. On the other hand, instead of adding noise on w 5 and w 7 we preferred inserting the noise directy in the measure of the rotationa matrix, i.e., in W. This is done because the direct measure of using the acceerometer and the magnetometer is normay fitered with the gyroscope [], in order to obtain a ess noisy estimation of both and ω. The noise added directy to is comparabe to the one we woud obtain after the fitering. From Fig. 5 we can observe that the estimated state shows some noise but the corresponding error remains imited. Due to the noisy component on the estimated state the outputs presents some osciation as we, especiay on the stress that seams to be the more sensitive output to the noise. Nevertheess the tracking error remains sma and aways bounded. d f d L f n f L f 5 ^ 5-5 d (a) Controer performances. 5 ' ^' _ ^_ -.5 ^ ^ 5-5 _ ^_ ^_ ^_ 5 5 p W p W C t = t and t = t = p C zc.5 t = t =.5 t = t = - - t = t y C circ x C t = t = x y (c) Trajectories visuaization. Fig. 5: Simuation: noisy measurements.

8 z E. Tethered offset Exact attachment of the ink to the center of mass of the aeria vehice is practicay unfeasibe. Therefore there wi aways be a non zero offset, athough sma, between the tether attachment and the center of gravity. This offset makes the transationa and rotationa dynamics of the aeria robot couped and can potentiay ead to the instabiity of the controed system. In this section we want to show the robustness of the proposed method when the distance between the attaching point and the center of gravity of the aeria vehice is non zero. In particuar in this simuation the ink is attached 5 [cm] verticay beow O with respect to F. s expected, the tracking error does not go to zero but however remains bounded, showing good tracking performances. Notice that the error is higher during the circing phase since this part of the goba trajectory is very dynamica and the unmodeed effects due to the offset are arger. However we remark that a good mechanica design coud make the tracking error amost negigibe. We tested the method with even arger offsets and we saw that the system remains stabe up to a vertica offset of [cm], that is an exaggerated vaue for the system considered in the simuation (sma-size quadrotor ike vehice). In fact, note that a arger quadrotor means a arger inertia which actuay reduces the negative effects of the offset. In additiona simuations, which are not reported here for the sake of brevity, we aso tested the robustness of the method with a more genera offset (not ony vertica) noticing that, within some reasonabe bounds, the system remains stabe and with acceptabe tracking performances.. -. d f d L f L f n f 5 d (a) Controer performances. ^ 5 ' ^' 5.5 _ ^_ -.5 ^ ^ 5-5 _ ^_ ^_ ^_ 5 5 p W p W C t = t and t = t = t = p C.5 t = t = zc.5 t = t = - - t = t circ x y C C - - x t = (c) Trajectories visuaization. - y Fig. 6: Simuation: non zero offset between tether attachment and center of gravity of the aeria vehice.

9 z F. Nondiagona inertia matrix In the derivation of the mode and of the controer as we, we assumed a diagona inertia matrix. In this section we check the robustness of the method if the aeria vehice has a non diagona inertia matrix. In particuar, in Fig. 7, we show the resuts for a test in which the rea inertia matrix is J =.5.5.5, whie the controer sti assumes a diagona inertia matrix. One can observe that the tracking error is not exacty zero but is kept imited within a sma bound. For the observer this does not constitute a non ideaity, in fact th error is constanty zero. With further simuations we observed that the system remains stabe up to a vaue of.5 in the off diagona terms (6% of the vaues on the main diagona). With arger vaues the system becomes unstabe.. -. d f d L f L f n f 5 d (a) Controer performances. 5 ^ ' ^' 5.5 _ ^_ _ ^_ ^_ ^_ ^ ^ 5 p W p W C t = t and t = t = t = p C.5 t = t = zc.5 t = t = - - t = t circ y C x C - t = - - x y (c) Trajectories visuaization. Fig. 7: Simuation: non-diagona inertia matrix J.

10 z G. Input saturation For how we panned the desired trajectory, the nomina input needed to track the desired outputs is aways within the imits of the considered system. Indeed, expoiting the fatness, we are abe to a priori check if the inputs exceed the minimum and maximum vaues. Nevertheess, in this section we want to show that the system is sti stabe if the inputs are hardy saturated for some instants. Thus we set some very restrictive imits on the input, i.e., f f and τ τ i τ, where i = x,y,z, f =, τ = and τ =. In order to et the saturation show up during execution we did not re-pan the desired trajectory. In Fig. 8a it can be seen that the inputs are saturated for some time instants during the execution of the task. When the inputs are saturated the tracking error increases, but, as soon as the inputs come back within the imits, the error exponentiay decreases to zero. We stress again the fact that the saturation of the inputs can be avoided expoiting the fatness. Using the fatness one can check if the desired trajectory requires inputs that are too arge. In the worst case one can re-pan the trajectory such that the input imits are respected.. -. f d L f n d f L f 7f 5 d (a) Controer performances. ^ 7= 5 5 = ' ^'.5 _ ^_ _ ^_ ^_ ^_ ^ ^ 5 p W p W C t = t and t = t = t = p C.5 t = t = zc.5 t = t = - - t = t circ y C x C - t = - x - y (c) Trajectories visuaization. Fig. 8: Simuation: saturation of the input.

11 z H. Motor time constant With this simuation we want to further enarge the set of non idea modes considered for the testing of the proposed contro method. Considering an aeria vehice actuated by rotating propeers, in this simuation we add the dynamica mode of the motors described with a first order system characterized by a time constant of τ M =.[s]. In practice the propeer dynamics inserts a frequency dependent phase shift between the commanded contro input and the actuated one, whose ampitude depends on the time constant. In other words the modes acts as a ow pass fiter on the commanded input, cutting its high frequency components. Those effects coud dramaticay decrease the performances or even make the system unstabe. However, from Fig. 9, one can notice that our method is robust to the unmodeed effects of the propeers dynamics. Indeed, in some instant, where the trajectory is more dynamica and requires fast varying inputs, the tracking error increases but it is aways bounded and at steady state converges to zero. We remark that, if needed, one can increase the smoothness of the contro inputs considering an higher order in the dynamic feedback contro. Indeed adding more integrators on the contro channes one can increase the degree of smoothness of the contro input thus guarantying that it is aways beow the cutting frequency proper of the system, and in particuar of the propeers. nother possibe strategy is to expoit the fatness to pan a trajectory that fufis the system imitations. f d L d f L f n f d (a) Controer performances. ^ 5 ' ^' 5.5 _ ^_ _ ^_ ^_ ^_ ^ ^ 5 p W p W C t = t and t = t = t = p C zc.5 t = t =.5 t = t = - t = t - circ y C x C - t = - - x y (c) Trajectories visuaization. Fig. 9: Simuation: system with motors dynamics.

12 I. Hierarchica contro In this section we compare our contro method to a controer based on hierarchica techniques. Simiar techniques were successfuy impemented and tested in, e.g., [5], [6]. In fact, the methods in [5], [6] can not be directy appied to sove our probem because they are designed for different systems, athough simiar. Therefore in the foowing we design the best hierarchica controer we can conceive for controing the position of the aeria vehice and the tension on the ink. The controer is based on the cascaded structure between the transationa and rotationa dynamics. We sha then compare its performances with respect our controer based on dynamic feedback inearization to show that a hierarchica approach performs much worse than our method in terms of both tracking precision and robustness to noise. Given a desired position trajectory p C (t), defined in terms of the generaized coordinates q(t) d we define q = q d + k D q ( q d q) + k P q( q d q), where k P q,k D q +. The vector q coud be seen as the desired acceeration that ets q foow the desired configuration q d using a PD strategy. Then, given a desired trajectory for the interna force of the ink f L (t) d, and inverting the baance of momenta on the winch, we compute the winch torque as τ W = J W f d L. To finay impement q we compute the desired thrust vector inverting the baance force equation on O, f C T e = a x a g fl d d C m J q q. }{{} desired thrust vector From the desired thrust vector we derive the input f as f = a x a g fl d d C m J q q, and the desired z-axis of F, i.e., z = e = C( a x a g f d L dc m J q q ) f. The desired yaw ange ψ d together with z et us define the desired attitude of the vehice described by. In fact, given ψ d we define x = z(ψ d )e where z (ψ d ) is the rotation matrix describing the rotation of ψ d aong z W. The axis x represents the desired heading of the aeria vehice. The desired attitude is computed creating an orthonorma basis using the vectors x and z that is given by = [x y z ] where, y = z x z x, x = y z y z. This concudes the design of the outer oop contro. Given the tracking error it computes the desired winch torque τ W, the desired thrust intensity f and the desired attitude. Now we design the inner oop contro to et the attitude foow the desired one. Let e (the attitude error) be computed as [e ] = ( T T ). In order to steer e to zero we define the desired anguar acceeration based on a PD controer, ω = k D ω ω + k P ωe, where k P ω,k D ω +. Inverting the rotationa dynamics we can finay find the input torque τ, τ = J ω ω + J ω. If the inner oop is sufficienty faster than the outer oop, the asymptotic convergence of q to q d is guaranteed. In Fig. the resuts of the hierarchica controer in idea condition are reported. The desired trajectory and the initia tracking and estimation errors are the same of the ones in Sec. II-. fter a tuning phase we were abe to get some good performances and a sma bounded tracking error, even if the error does not converge exacty to zero. On the other hand, in the same conditions the controer based on dynamic feedback inearization is abe to steer the output aong the desired trajectory with zero error (see Fig. ). However, to obtain good tracking performances with the hierarchica controer we had to set very high gains that make the system more reactive and thus abe to foow the desired trajectory. Nevertheess this requirement has two main drawbacks. The first drawback is that, due to the arge contro gains, the contro effort increases thus possiby requiring an input that is out of the physica imits of the actuators. Indeed with this configuration we reach a maximum thrust and a maximum torque of about 5 and.5 respectivey. This vaues are higher than the nomina inputs required to track the desired trajectory. The second extremey serious issue arises in the presence of noise in the measurements and so in the estimated state. Indeed, the higher the gains, the arger the noise in the commands and the coser the controed system is to instabiity. In fact, simuating the system with the same measurement noise described in Sec. II-D the cosed oop system becomes unstabe. In order to get a stabe behavior we had to significanty ower the gains, an action that, however, ceary degrades the tracking performances. s we can see in Fig. the performances with noise are much worse than the ones obtained using the dynamic feedback inearizing controer in the same noisy condition. Therefore, the hierarchica approach presents a stricty penaizing tradeoff between appicabiity with noise and tracking performances. One cannot obtain both. ttainment of both objectives is instead possibe with our proposed controer. EFEENCES [] M. Tognon, S. S. Dash, and. Franchi, Observer-based contro of position and tension for an aeria robot tethered to a moving patform, IEEE obotics and utomation Letters, 6.

13 z []. M. Murray, Z. Li, and S. S. Sastry, mathematica introduction to robotic manipuation. CC, 99. [] L. Sandino, D. Santamaria, M. Bejar,. Viguria, K. Kondak, and. Oero, Tether-guided anding of unmanned heicopters without GPS sensors, in IEEE Int. Conf. on obotics and utomation, Hong Kong, China, May, pp. 96. []. Mahony, T. Hame, and J.-M. Pfimin, Noninear compementary fiters on the specia orthogona group, IEEE Trans. on utomatic Contro, vo. 5, no. 5, pp. 8, 8. [5] L. Sandino, M. Bejar, K. Kondak, and. Oero, square-root unscented kaman fiter for attitude and reative position estimation of a tethered unmanned heicopter, in Unmanned ircraft Systems (ICUS), 5 Internationa Conference on, June 5, pp [6] S. Lupashin and. D ndrea, Stabiization of a fying vehice on a taut tether using inertia sensing, in IEEE/SJ Int. Conf. on Inteigent obots and Systems, Tokyo, Japan, Nov, pp. 8. d 8 6. f d L f L 6 f n f -. 5 d (a) Controer performances ^ ' ^' _ ^_ _ ^_ ^_ ^_ 5..5 ^ ^ 5 p W p W C t = t and t = t = t = p C.5 t = t = zc.5 t = t = - - t = t circ x y C C - - x t = (c) Trajectories visuaization. - y Fig. : Simuation: Hierarchica contro. Good tracking performances are obtained ony with very high gains in the inner and outer contro oops

14 z d f d L f L d 5 f n f (a) Controer performances. 5 ^ ' ^' _ ^_ _ ^_ ^_ ^_ ^ ^ 5 p W p W C p C t = t and t = t = t = zc t = t circ - t = t = t = t = - y - x C C x t = (c) Trajectories visuaization. - y Fig. : Simuation: Hierarchica contro in the noisy case. To preserve stabiity ower gains have to be used with noise, therefore the performances are significanty degraded. The hierarchica controer presents a stricty penaizing tradeoff between tracking performances and robustness to noise.