A decentralized approach for the aerial manipulator robust trajectory tracking

RESEARCH ARTICLE A decentralized approach for the aerial manipulator robust trajectory tracking Yarai Elizabeth Tlatelpa-OsorioID 1☯ *, Hugo Rodrı ´guez-Corte ´s 1☯ , J. A ´. Acosta 2☯ 1Seccio ´n de Mecatro ´nica, Deparatamento de Ingenierı ´a Ele ´ctrica del Centro de Investigacio ´n y de Estudios Avanzados del Instituto Polite ´cnico Nacional, Ciudad de Me ´xico, Me ´xico, 2Departamento de Ingenierı´a de Sistemas y Automa ´tica, Universidad de Sevilla, Sevilla, Spain ☯These authors contributed equally to this work. *[email protected] Abstract This paper introduces a new decentralized control strategy for an unmanned aerial manipulator (UAM) constrained to the vertical plane. The control strategy comprises two loops: the first compensates for the aerial vehicle’s impact on the manipulator; and the second one implements independent controllers for the aerial vehicle and the manipulator. The controller for the aerial vehicle includes an estimator to compensate for the dynamic influence of the manipulator, even if it is affected by external wind-gust disturbances. The manipulator has two revolute joints; however, it is modeled as an dynamically equivalent manipulator, with one revolute and one prismatic joint. The proposed control strategy’s performance is evaluated using a simulator that includes the vehicle’s aerodynamics and the manipulator’s contact force and moment. 1 Introduction Combining multi-rotor Unmanned Aerial Vehicles (UAVs) with robot manipulators has led to the more versatile and agile devices termed Unmanned Aerial Manipulators (UAMs). The applications of UAMs are vast and diverse since UAMs can take advantage of the manipulator’s dexterity and the UAV’s agility. However, technological and scientific issues must be addressed to exploit their usefulness. Among these problems is the need for lighter materials, better batteries, foolproof safety, and enhanced performance during tracking/positioning manipulation tasks. Synthesizing robust control algorithms can tackle the UAM performance during manipulation tasks. UAMs are classified by the vehicle’s number of rotors, manipulator links, and arms. In particular, the manipulator must have at least one actuated degree of freedom to be considered an UAM. One of the first UAM with a manipulator with 2-DoF revolute joints can be found in [1]. Nowadays, there are many applications including experimental work on UAMs with multiple degrees of freedom. There are two trends in control architectures reported for UAMs: centralized and decentralized. The first considers the UAM as a whole system, and the second one, accounts for separated dynamic systems, the UAV and the robotic manipulator. In the first approach, only one PLOS ONE PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 1 / 32 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Tlatelpa-Osorio YE, Rodrı´guez-Corte ´s H, Acosta JA ´(2024) A decentralized approach for the aerial manipulator robust trajectory tracking. PLoS ONE 19(3): e0299223. https://doi.org/10.1371/ journal.pone.0299223 Editor: Gang Wang, University of Shanghai for Science and Technology, CHINA Received: November 2, 2023 Accepted: February 4, 2024 Published: March 7, 2024 Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https://doi.org/10.1371/journal.pone.0299223 Copyright: ©2024 Tlatelpa-Osorio et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the manuscript. Funding: Y.E. Tlatelpa-Osorio 702178 CONSEJO NACIONAL DE CIENCIA Y TECNOLOGI ´Ahttps:// conahcyt.mx/. mathematical model describes the UAV and robotic manipulator dynamics. Conversely, in the decentralized approach, the quadrotor and the robotic manipulator are considered separate systems; their interaction is viewed as the effect of mutual disturbances. Sometimes, the control architecture is influenced by the dynamic modeling method used for UAMs. Thus, the centralized approach fits better with the Euler-Lagrange formalism, while the Newton-Euler approach follows the decentralized framework. However, under the same conditions both modeling frameworks are equivalent, generating then the same dynamic models. As examples of the application within the centralized control approach, in [2] an UAM with two anthropomorfic manipulators is commanded following the impedance control technique; in [3] the dynamic model of the UAM is linearized around an equilibrium point and a Linear Quadratic Regulator (LQR) is employed. On the other hand, in the decentralized approach, the controllers selected for the UAV and manipulator can have different formulations that may not require knowledge of the whole model. In [4] is employed a kinematic control for the two anthropomorphic UAM’s arms, while a backstepping controller is implemented for the hexacopter that carries the arms. In [5], a general description of the robustness of decentralization is provided with nonlinear controller at the kinematic level. In [6], a PID-based controller with gravity compensation is used. In [7], a PD controller is used for the manipulator, while the UAV uses two level controls, one for translational dynamics and one for attitude dynamics, for an ad-hoc type of UAM satisfying differential flatness, i.e. fully linearizable by feedback. The work in [8] uses a passive nonlinear dynamic controller for the UAV and an integral kinematic controller for the manipulator. Adaptive control has also been implemented in [9]. A review of other centralized and descentralized approaches, summarizing characteristics and differences can be found in [10]. Under the decentralized approach, some authors have pointed out the exogenous nature of the disturbances from the robotic manipulator to the aerial vehicle and vice-versa, see e.g. [1] where the effects on the robotic manipulator of the aerial vehicle displacement of the center of mass and changes in the moments of inertia are characterized through experiments. This crucial observation shows that decentralized approaches are more suitable when external disturbances are present, thus allowing to implement robust control strategies for each system independently. In fact, even though many works, such as those described above, have proposed a variety of controllers, no attention have been paid to the actual nature of the robustness gained with the decentralized approaches. This is the main focus of this work where a detailed analysis of the dynamical interaction between the aerial vehicle and the robotic manipulator and the ad-hoc design of a robust controller are provided. Some efforts have been made to take into account the interaction between the UAV and the robotic manipulator by using equivalent models. For example, in [11], the dynamic model of an n-link manipulator is described by a one-degree-of-freedom (DoF) revolute joint that concentrates the n-link manipulator total mass, assuming that the robot arm reach any commanded reference instantly. Thus, as far as we know, the only work treating theoretically the interaction is in [8], but only at the kinematic level of the robot arm, which means that, in practice, the analysis is only valid for slow movements when the accelerations can be neglected. The present work addresses the problem of controlling the longitudinal UAM dynamics following the decentralized approach taking into account external disturbances. The longitudinal dynamics considered captures the essential nonlinearities of a 3D environment and most practical aerial missions are 2D immersed in a 3D workspace [8]. The robotic manipulator is composed of two links with revolute joints, where the computed-torque methodology is employed to design a trajectory-tracking controller [12]. The robust control implemented in the UAV is based on the PD methodology in combination with PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 2 / 32 Competing interests: The authors have declared that no competing interests exist. the estimator of external forces and moments similarly to [13]. As one of the main contributions of this work and to better illustrate that the manipulator effects over the quadrotor can be tailored as exogenous disturbances, the robotic manipulator dynamics are modeled as an equivalent 1-revolute 1-prismatic robotic manipulator, thus the manipulator model approach proposed in [5] is also extended. The Newton-Euler method and its recursive algorithm [14] are used to obtain the UAM dynamic model. Finally, numerical simulations using the UAM realistic simulator reported in [8] are presented to assess the performances. The main contributions are summarized as follows: 1. The modelization of a 2-revolute links robotic manipulator as an equivalent 1-revolute, 1-prismatic links robotic manipulator; this permits simplifying the stability analysis, by customizing the dynamic effect of the robotic manipulator on the aerial vehicle as an exogenous disturbance. In the previous authors’ works in [8,11,15] a simplified model for the nlink manipulator as a 1-revolute joint manipulator was proposed. However, only kinematics is considered for the robot arm interaction and control. The dynamic model developed in this work allows us to characterize its complete nature and formalize the exogenous nature of the dynamic interaction, i.e. torques and forces. We emphasize that this dynamical analysis is missing in the literature, normally neglected with assumptions such as slow motion, that indeed are not practical because there may be external disturbances such as gusts of wind that cause high accelerations. 2. The model developed paves the way for the design of a decentralized robust PD and computed torque nonlinear controllers together with a exogenous disturbances estimator. A complete stability analysis is provided that ensures that the tracking error is confined to a vicinity of the origin exponentially, which is a well-known desirable robust property. The performances are validated through numerical simulations. The organization of this work is as follows. Section 2 presents the Aerial Manipulator Dynamics model on the plane. In section 3, the control strategy is developed, as well as the stability analysis. In section 4 the numerical simulations results are shown, and conclusions and future work are presented in section 5. 2 UAM dynamic model This work considers a quadrotor with a robotic manipulator at the bottom, i.e an UAM. The robotic manipulator has two degrees of freedom as two revolute joints. The UAM’s dynamic model is obtained under the following considerations in all flight operations: • the aerial vehicle and the robotic manipulator are considered rigid bodies, i.e. links are not flexible; • the relative motions of the propellers to the quadrotor frame are disregarded; • the union between the aerial vehicle and the robotic manipulator is rigid and remains unaltered; • the links move independently only when their actuators generate a moment; Recall that the robotic manipulator can only move on the plane 0x b z b , see Fig 1. To obtain the UAM dynamic model the reference frames 0x i y i z i ,i= 1, 2, 3 on the robotic manipulator’s links are defined, as in Fig 1. PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 3 / 32 2.1 Quadrotor dynamic model From Newton-Euler laws of motion the quadrotor dynamic model is mQ€ X¼mQge3þFi pfi RM J_ O¼  O�JOþMb pMb RM ð1Þ where m Q is the quadrotor mass, gis the gravity acceleration constant, e 3 = [0 0 1] > ,Fi pis the force due to the propulsion system, fi RM is the force due to the robotic manipulator expressed in the inertial frame and X= [x y z] > is the vector of cartesian coordinates. Moreover, J= diag {J xx ,J yy ,J zz } is the quadrotor inertia matrix, O= [p q r] > is the quadrotor velocity in body coordinates, Mb pthe moment due to the propulsion system, and Mb RM the moment due to the robotic manipulator expressed in body coordinates. The propulsion force is given by Fi p¼  TTRe3 where TT¼P4 i¼1Tiis the total thrust produced by the four rotors with T i the thrust produced by rotor r i . Moreover, R2SO(3) is the rotation matrix from body coordinates to inertial coordinates.Where SOð3Þ ¼ nR2R3�3jR>R¼I;detðRÞ ¼ 1owith Ithe identity matrix. Introducing the following notation c σ = cos(σ), s σ = sin(σ) for any angle σ, the propulsion moment is Mb p¼ Mb x Mb y Mb z 2 6 6 43 7 7 5¼ ‘cp=4ðT2þT3T1T4Þ ‘cp=4ðT1þT2T3T4Þ Q1þQ3Q2Q4 2 6 43 7 5 where ℓis the distance between the origin of the body frame and the rotation axis of each rotor, π/4 is the angle between the rotor arm and the body axis 0x b for rotors 1 and 3; and between the body axis 0y b for rotors 2 y 4. Finally, Q i ,i= 1, ���, 4 is the reaction moment of each rotor. Fig 1. Unmanned aerial manipulator. It comprises the aerial vehicle with four rotors n 1 ,n 2 ,n 3 , and n 4 ; and the robotic manipulator with two revolute joints R 1 and R 2 . Body axes 0x b y b z b and inertial axes 0x i y i z i . https://doi.org/10.1371/journal.pone.0299223.g001 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 4 / 32 The force fi RM and the moment Mb RM can be computed from the inward iteration of the Recursive Newton-Euler Algorithm (RNEA) that propagates the forces and moments from the end effector to the robotic manipulator base. The RNEA procedure is completed with the outward iteration to compute links velocities and acceleration. Both iterations allow to obtain the dynamic model of the robotic manipulator [14]. 2.2 Robotic manipulator dynamic model The ideal workspace of the robotic manipulator considered in this work is a semi-circle below the quadrotor. This workspace is achieved through the independent motion of the two revolute joints, 1Rand 2R; see Fig 2.Fig 2 also shows the robotic manipulator’s center of gravity. The robotic manipulator motion can be interpreted as the motion of a fully actuated slung load when it is analyzed from the motion of its center of gravity. This observation gave rise to modeling the 1R2Rrobotic manipulator as an equivalent robotic manipulator composed of one revolute joint 10Rand one prismatic joint 1P, as illustrated in Fig 3. This idea was partially developed in [15] by modeling the 1R2Rrobotic manipulator as an actuated pendulum with constant length. Pursuing this idea, in this work, the second degree of freedom is recovered by considering that the pendulum’s center of mass can move longitudinally using a prismatic joint, unlike there. As a result, the complete robotic manipulator’s workspace can be covered. The reference systems shown in Fig 3 follow the link-frame procedure proposed in [14]. Moreover, Table 1 summarizes the link parameters, also known as the Denavit-Hartenberg in the proximal variant notation [16]. Fig 4 depicts the link parameters. From the link parameters, the following rotation matrices are obtained ð0 1RÞ ¼ cy1sy10 sy1cy10 0 0 1 2 6 6 6 43 7 7 7 5;ð1 2RÞ ¼ 1 0 0 0 0 1 0 1 0 2 6 6 6 43 7 7 7 5 ð2 3RÞ ¼ 1 0 0 0 1 0 0 0 1 2 6 6 6 43 7 7 7 5 ð2Þ the rotation matrices use the notation introduced in [14]. Hence, ( i+1 iR) is the rotation matrix from reference frame ito i+ 1. Moreover, ( i+1 iR) > = ( i i+ 1R) and ðiþ1iþ2RÞðiþ2 iRÞ ¼ ðiþ1 iRÞ. Fig 2. UAM ideal workspace. https://doi.org/10.1371/journal.pone.0299223.g002 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 5 / 32 Also, the distance Pi iþ1between frame iand frame i+ 1 measured from frame iis given by P0 1¼ 0 0 0 2 6 6 6 43 7 7 7 5P1 2¼ 0 0 0 2 6 6 6 43 7 7 7 5P2 3¼ 0 0 L 2 6 6 6 43 7 7 7 5ð3Þ Now, without any loss of generality, the following simplifications are considered to tailor the robotic manipulator dynamic model. The revolute joint 10Ris massless so that the mass of the revolute joints 1Rand 2Ris concentrated at the distal end of equivalent prismatic link P 1 . Hence, m10R¼0;mP¼m1Rþm2R with m 1 0 R the mass of the revolute joint 10R,m P the mass of the revolute joint 1Pand m 1R ,m 2R the mass of the revolute joints 1Rand 2R, respectively. Moreover, the position Pi Cof the center of mass of link iexpressed in the i-th reference frame, is defined by the following vectors P1 C¼ 0 0 0 2 6 43 7 5;P2 C¼ 0 0 L 2 6 43 7 5ð4Þ moreover, the inertia tensors for each point mass are I1 C¼0and I2 C¼0. Fig 3. UAM ideal workspace with revolute and prismatic joints. https://doi.org/10.1371/journal.pone.0299223.g003 Table 1. Link parameters. iα i−1 a i−1 d i θ i 1α 0 = 0 a 0 = 0 d 1 = 0 θ 1 2a1¼p 2a 1 = 0 d 2 = 0 θ 2 = 0 3α 2 = 0 a 2 = 0 d 3 =L θ 3 = 0 With α i the angle from z i to z i+1 measured around x i ,a i is the distance from z i to z i+1 measured along x i ,d i is the distance from x i−1 to x i measured along z i , and θ i is the angle from x i−1 to x i measured around z i . https://doi.org/10.1371/journal.pone.0299223.t001 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 6 / 32 Since the robotic manipulator is attached to a flying base, it is necessary to obtain the rotation matrix between the reference frame 0x 0 y 0 z 0 and the body reference frame 0x b y b z b . Note that both frames are rigidly attached but have different configurations, see Fig 4. The corresponding rotation matrix is the following. ð0 bRÞ ¼ 1 0 0 0 0 1 0 1 0 2 43 5 Thus, the boundary conditions for the flying base are [17–19] O0¼ ð0 bRÞOþ_ y0^ z0 _ O0¼ ð0 bRÞ_ Oþð0 bRÞO�_ y0^ z0þ€ y0^ z0 _ V0¼ ð0 bRÞh_ VbþO�VbþO�Pb 0þO�ðO�Pb 0Þ gR>e3i where O 0 and V 0 are the angular and translational velocities of the frame 0x 0 y 0 z 0 , and Vb¼ R>_ Xis the translational velocity of the quadrotor expressed in body axes. Moreover, Pb 0¼02 R3is the distance from the frame 0x 0 y 0 z 0 to the body frame and θ 0 is the angle of rotation of frame 0x 0 y 0 z 0 with respect to frame 0x b y b z b around the ^ z0axis. The term R > ge 3 is introduced to consider the gravitational acceleration. In the following, O i ,_ Oi,V i and _ Viare the rotational and translational velocities and accelerations of joint iexpressed in the frame 0x i y i z i , respectively. Fig 4. UAM link variables. https://doi.org/10.1371/journal.pone.0299223.g004 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 7 / 32 The frame 0x 0 y 0 z 0 is rigidly attached to the body frame so that y0¼_ y0¼€ y0¼0, thus the boundary conditions reduce to O0¼ ð0 bRÞO _ O0¼ ð0 bRÞ_ O _ V0¼ ð0 bRÞ�_ VbþO�VbgR>e3� The angular velocities are propagated to frames 0x 1 y 1 z 1 and 0x 2 y 2 z 2 following the outward iteration of the RNEA method, see Appendix A, as follows. Therefore, the angular velocity for frame 0x 1 y 1 z 1 attached to a revolute joint with i+ 1 = 1 is given by (44) O1¼ ð1 0RÞO0þ_ y1^ z1 meanwhile, for frame 0x 2 y 2 z 2 attached to the prismatic joint with i+ 1 = 2, it follows that (45) O2¼ ð2 1RÞO1 Once again, from the RNEA outward iteration, the angular and translational accelerations are propagated as follows. For the frames 0x 1 y 1 z 1 and 0x 2 y 2 z 2 with i+ 1 = 1 and i+ 1 = 2, respectively, one obtains (46) and (47), _ O1¼ ð1 0RÞ_ O0þð1 0RÞO0�_ y1^ z1þ€ y1^ z1 _ V1¼ ð1 0RÞh_ O0�P0 1þO0�ðO0�P0 1Þþ _ V0i For the prismatic joint, one has (48) and (49), _ O2¼ ð2 1RÞ_ O1 _ V2¼ ð2 1RÞh_ O1�P1 2þO1�ðO1�P1 2Þþ _ V1iþ2O2�_ L^ z2þ€ L^ z2: Taking into account P0 1and P1 2defined in Eq (3) the accelerations _ V1and _ V2reduce to _ V1¼ ð1 0RÞ_ V0 _ V2¼ ð2 1RÞ_ V1þ2O2�_ L^ z2þ€ L^ z2 The last step of the outward iteration involves the computation of the links’ center of mass acceleration, forces and moments acting on it. The acceleration of the link’s center of mass is computed as follows (50) _ V1 C¼_ O1�P1 CþO1�ðO1�P1 CÞþ _ V1 _ V2 C¼_ O2�P2 CþO2�ðO2�P2 CÞþ _ V2 Considering (4) the acceleration of the revolute link center of mass reduces to _ V1 C¼_ V1 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 8 / 32 The forces acting at the center of mass of each link are (51) F1¼mR0 1 _ V1¼0 F2¼mP1_ V2 C Finally, under the aforementioned considerations, the moments on each link are N 1 =N 2 = 0, and hence the outward iteration is completed. The inward iteration propagates forces and moments acting on the end effector to the robotic manipulator base. The inward algorithm runs from i+ 1 = 3 to i+ 1 = 2. Thus, the force f 2 exerted on link 2, by link 1 and the force f 1 exerted on link 1 by the robotic manipulator’s base are (52) f2¼F2 f1¼ ð1 2RÞf2ð5Þ where it is assumed that f 3 = 0. Additionally, the torque n 2 exerted on link 2 by link 1 and the torque n 1 exerted on link 1 by the robotic manipulator base are given by n2¼P2 C�F2 n1¼ ð1 2RÞn2þP1 2�ð1 2RÞf2ð6Þ where it is assumed that n 3 = 0. Considering (3) and (4), it follows that n1¼ ð1 2RÞn2 Finally, the robotic manipulator dynamic model is described by the following equations tR¼n1>^ z1 fP¼f2>^ z2ð7Þ where τ P is the control moment applied to the revolute link, and f P is the control force applied to the prismatic link. The force f 1 and the moment n 1 can be expressed in the reference frame 0x 0 y 0 0z 0 as follows f0¼ ð0 1RÞf1 n0¼ ð0 1RÞn1ð8Þ thus, fi RM ¼Rð0 bRÞ>f0 nb RM ¼ ð0 bRÞ>n0ð9Þ In the following the angle θ 1 is replaced by θ P measured as shown in Fig 5. Thus, y1¼yPyQ Complex but straightforward computations show that the quadrotor translational dynamic model constrained to the plane 0x i z i becomes � M € x € z " # ¼� Mg 0 1 " #TT syQ cyQ " # � fi RMx � fi RMz " # ð10Þ PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 9 / 32 3.4 UAM closed loop dynamics To sum up everything, the controller and estimator proposed provide the following UAM closed loop dynamics _ e X1¼e X2 _ e X2¼  KPX e X1KDX e X2 u>ryQ mcosðeyQ1ÞcosðeyQ1Þ cosðeyQ1þyQdÞ sinðeyQ1þyQdÞ 2 6 43 7 5 cosðyQd sinðyQdÞ 2 43 5 8 > < > :9 > = > ;þ1 mm1w _ eyQ1¼eyQ2u>S½ðKDX G1Þm1þðKDX IÞm2�w u>u _ eyQ2¼  KPQeyQ1KDQeyQ2Fðu;w;ed4Þ _ eyP1¼eyP2 _ eyP2¼  KPAeyP1KDAeyP2 _ e L1¼e L2 _ e L2¼  KPPe L1KDPe L2 _w¼Awwþm> 3dð3Þ T _ ed4¼  G4ed4þ_ dR ð29Þ where e X1¼e X,e X2¼_ e X,yQ1¼yQyQd,yQ2¼_ yQ_ yQd,yP1¼eyP,yP2¼_ eyP,e L1¼e L,e L2¼_ e L, w1¼ed1,w2¼_ ed1,w3¼€ ed1, and Aw¼ G1I0 0 G2G1I0 G3G2G10 000G4 2 6 6 6 6 4 3 7 7 7 7 5; moreover, Fðu;w;ed4Þ ¼ 1 ðu>uÞ2ðkuk2u>S½ð� KDþG2Þm1þ ðIþG1Þm2þ2m3�w f2u>½ðKDG1Þm1þðKDIÞm2�w u>S½ðKDG1Þm1þðKDIÞm2�wgÞ ed4 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 16 / 32 where m1¼ ½I3�303�303�3� m2¼ ½03�3I3�303�3� m3¼ ½03�303�3I3�3� with I 3×3 and 0 3×3 , the identity and zero matrix in R3�3, respectively. Function Fðu;w;ed4Þaccounts for the estimation error terms that cannot be canceled. Such estimation errors appear because of the control action propagation from the quadrotor rotational dynamics to the quadrotor dynamics along the 0x i axis, this is, through the yQdcomputed as _ yQd ¼u>S_ u kuk2ð30Þ € yQd ¼ðu>S€ uÞu>u2ðu>_ uÞu>S_ u ðu>uÞ2ð31Þ with _ u¼� M�KPe X2þKDe X3�þZ2b2þKDed1_ ed1ð32Þ € u¼� M�� KPe X2� KDe X3�þZ3b3þ� KDed1þ_ ed1þ€ ed1þed3ð33Þ where e X3¼  KPe X1KDe X2,� KP¼KDKPand � KD¼KPþK2 D. 3.5 Stability analysis For the main stability result the following standard assumption for the disturbances is in order. Assumption 1 δ T (t)and dðiÞ TðtÞ,i= 1, 2, 3 and δ R (t)and _ dRðtÞare in L1,t�0. Let us define x≔colðe X1;e X2Þ,eyQ≔colðeyQ1;eyQ2Þ,wa≔colðw;ed4Þand Δ δ ≔col (m> 3dð3Þ T;_ dM). Thus, from (29), the ex,eyQand χ a dynamics in compact form become _ ex¼AxexþCxðeyQÞ�u>ryQ mþDxwa _ eyQ¼AQeyQþCQðuÞ�wa _ wa¼AwawaþDd ð34Þ where C (�) and Δ δ are vector functions and D x a constant matrix of appropriate dimensions, respectively, whose expressions can be easily obtained by direct substitutions in (29). Additionally, Hurwitz matrices are defined as Ax≔0I2 KPKD " #;AQ≔0I2 KPR KDR " #; where K P ,K D ,K PR and K PD are positive definite matrices and let Awa ≔BlockdiagðAw;G4Þ. Thus, the main stability result is stated in the following proposition. PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 17 / 32 Proposition 1 Consider the closed-loop dynamics (29) with all the control-gain matrices positive definite. Then, under Assumption 1, the error is ultimately bounded to a Δ δ -vicinity of the origin. Moreover, if Δ δ (t) = 0 then the error converges exponentially to zero, t �0. Proof 1 First, from (29) is straightforward to see that the error dynamics e Liand eyPi,i= 1, 2, are decoupled from others and hence converge exponentially to zero. Therefore, we only focus on the remaining error dynamics (34).For, let us define the error e≔colðex;eyQ;waÞsuch that (34) can be rewritten in matrix form as _ e¼Aeþ 0 0 Dd 2 6 43 7 5;A≔ AxDx u>ryQ mDx 0AQCQ 0 0 Awa 2 6 6 6 43 7 7 7 5;ð35Þ where we have omitted all the arguments for compactness. Notice that, by design C x (0) = 0 and hence the whole term can be factored by Δ x as in (35).The results follow noting that the block-tri- angular matrix Ais Hurwitz and Δ δ is uniformly bounded under Assumption 1. 4 Numerical simulations Numerical simulations were driven on the UAM realistic simulator reported in [8] and were kindly provided to us by Carlos Rodrı ´guez de Cos from the University of Sevilla. The original simulation platform consists of 4 main blocks: the MANT mathematical model block, the target trajectory block for the UAV and the End effector, a block for the UAV control, and one for the manipulator control. The MANT system is disturbed by a random gust of wind. Note that the proposed controller was designed considering a quadrotor with a manipulator composed of a revolute joint R1 and a prismatic joint P1; however, the simulator considers only revolute joints. Hence, it is mandatory to prove that both manipulators are equivalent in some sense. References [21–23] give definitions for the concept of dynamic systems equivalency; in this work, the equivalence between dynamic systems is addressed based on the following definition, adapted from [23]. Definition 1 It is said that the systems S:_ w¼fðwÞþgðwÞnð36Þ P:_ ew¼efðewÞþegðewÞen;ð37Þ with w;ew2Rk1,n;en2Rk2are equivalent if there exist: i. A diffeomorphism ew¼FðwÞ ð38Þ ii. A static state feedback n¼auðwÞþbuðwÞen;ð39Þ with β u (χ)a nonsingular square matrix, such that the transformation of Sunder (F,α u ,β u ) is equal to P. Fig 7 shows the link variables γ 1 and γ 2 of the R1, R2 manipulator. Hence, for the dynamic systems addressed in this work, Definition 1 can be applied considering that the system S PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 18 / 32 corresponds to the two revolute joints manipulator; thus, w¼ ½g1g2_ g1_ g2�>,ν= [τ 1 τ 2 ] > . At the same time, Pis the revolute-prismatic joints manipulator system; this is ew¼ ½yPL_ yP_ L�>, � n¼ ½�tR� fP�>. It is possible to verify that the diffeomorphism (38) and the static state feedback (39) can be defined as follows yP L _ yP _ L 2 6 6 6 6 4 3 7 7 7 7 5¼Fðg1;g2;_ g1;_ g2Þ ð40Þ t1 t2 " #¼buðg1;g2Þ�tR � fP " # where Fðg1;g2;_ g1;_ g2Þ ¼ 1 2ffiffiffiffi dl p p 2þg1þarcsin l2sinðg2Þ ffiffiffidl p � � l1l2sg2_ g2 ffiffiffidl p _ g1_ g2l2ð2c2 g2l1l2dlcg22l1l2Þ dlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 g2l2 2l2 2þdl p 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 ð41Þ bug1;g2 ð Þ ¼ 1 0 l2 2Lsin p 2þg2a � � l2 2sin ag2 ð Þ " # ð42Þ with a¼arcsin l2sinðg2Þ ffiffiffiffi dl p !and dl¼4l2 1þl2 2þ4l1l2cg2: Fig 7. Angles γ 1 and γ 2 on green. https://doi.org/10.1371/journal.pone.0299223.g007 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 19 / 32 The simulator considers a scenario where the UAM must fly close to target position in the proximity of a virtual object, after 10 seconds the end effector follows a desired trayectory to achive a desired final position. Thus, given a desired end-effector reference position, x EEd ,z EEd , the desired angular positions γ 1d and γ 2d are obtained from the inverse kinematics and using the diffeomorphism (40), one gets yPd Ld _ yPd _ Ld 2 6 6 6 6 4 3 7 7 7 7 5¼Fðg1d;g2d;_ g1d;_ g2dÞ; The physical paramethers are sumarized on Table 2, while the gains values are on Table 3. The diagram in Fig 8 illustrates how the simulations were implemented for the manipulator. The realistic simulator where the simulations were implemented enables disturbances over the system generated through a wind profile. The wind profile can be user-defined or generated from random data. Thus, to evaluate the performance of the proposed control strategy, five simulations were driven, each with a different random wind disturbance profile, as seen in Fig 9. All Figures containing data plots were generated with the tool Professional Plots [24]. Table 4 shows the wind magnitude and direction media values for each simulation. Table 3. UAM controller gains. Gain Value Γ 1 5.5 Γ 2 5.5 Γ 3 10.5 Γ 4 60 K PX diag{4.5, 6.2} K DX diag{5, 6.2} K PQ 125 K DQ 90 K PA 25 K PP 50 K DA 25 K DP 50 https://doi.org/10.1371/journal.pone.0299223.t003 Table 2. UAM physical parameters. Parameter Value M1Kg m 1 0.1 Kg m 2 0.1 Kg l 1 0.4 m l 2 0.4 m Links arm inertia 0.0001 Kg �m 2 UAV inertia 0.048 Kg �m 2 https://doi.org/10.1371/journal.pone.0299223.t002 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 20 / 32 Fig 8. Robot arm simulation implementation. https://doi.org/10.1371/journal.pone.0299223.g008 Fig 9. Wind disturbance profile defined by wind magnitude and direction. https://doi.org/10.1371/journal.pone.0299223.g009 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 21 / 32 Figs 10 and 11 depict the time evolution of the UAV translational axis errors, exand ez, respectively, while Fig 12 depicts the UAV attitude error, eyQ. Figs 13 and 14 show the end effector position error exEE ¼xEE xEEd and ezEE ¼zEE zEEd, respectively. As can be observed, all error signals converge to a zero neighborhood, as the theoretical analysis predicted. The equivalent robotic manipulator errors e Land eyPare shown in Figs 15 and 16, while the original robotic manipulator errors eg1¼g1g1dand eg2¼g2g2dare reported in Figs 17 and 18. Note that the errors on the equivalent robotic manipulator are closer to zero than those from the original. This behavior can be caused by unknown parameters implemented on the realistic simulator. Figs 19 and 20 show the control inputs on the UAM only for the first simulation. The following integral functions were measured for each simulation to understand the controller Table 4. Wind profile media magnitude and direction for simulations 1 to 5. Simulation Wind direction media [rad] Wind magnitude media [m/s] 1 -1.9294 1.3020 2 -1.7974 1.3070 3 -1.0304 1.0674 4 -2.6946 2.2006 5 -0.1799 0.9544 https://doi.org/10.1371/journal.pone.0299223.t004 Fig 10. UAV translational xaxis error. https://doi.org/10.1371/journal.pone.0299223.g010 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 22 / 32 Fig 11. UAV translational zaxis error. https://doi.org/10.1371/journal.pone.0299223.g011 Fig 12. UAV attitude error. https://doi.org/10.1371/journal.pone.0299223.g012 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 23 / 32 performance in all simulations better, F1¼Zt 0ke XðtÞ k dt F2¼Zt 0eyQðtÞ2dt F3¼Zt 0e g1ðtÞ2þe g2ðtÞ2  �dt F4¼Zt 0exEEðtÞ2þezEEðtÞ2  �dt F5¼Zt 0e LðtÞ2þeyPðtÞ2 � �dt ð43Þ Table 5 presents the values for each measurement F i ,i= 1, ���, 5 correspondent to each simulation. From the values in Table 5, it can be concluded that the control performance remains the same for different wind profiles acting on the system as disturbances. Hence, the proposed disturbance estimator performs adequately. Fig 21 presents the disturbance estimated by the proposed estimation strategy for simulation 1. Fig 22 shows the UAM sequence followed during the simulation. From number 1 to number 4, the UAM approaches a reference near the blue dot, the reference for the robotic manipulator, and remains in such a position. In number 5, the UAM is already on its reference so that the robotic arm can also reach its reference. Fig 13. End effector xaxis position error. https://doi.org/10.1371/journal.pone.0299223.g013 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 24 / 32 Fig 14. End effector zaxis position error. https://doi.org/10.1371/journal.pone.0299223.g014 Fig 15. Equivalent manipulator prismatic joint error, e L. https://doi.org/10.1371/journal.pone.0299223.g015 PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 25 / 32 16. Lipkin H. A note on Denavit-Hartenberg notation in robotics. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. vol. 47446; 2005. p. 921–926. 17. Dubowsky S, Papadopoulos E. The kinematics, dynamics, and control of free-flying and free-floating space robotic systems. IEEE Transactions on robotics and automation. 1993; 9(5):531–543. https://doi. org/10.1109/70.258046 18. Antonello A, Valverde A, Tsiotras P. Dynamics and Control of Spacecraft Manipulators with Thrusters and Momentum Exchange Devices. Journal of Guidance, Control, and Dynamics. 2018; 42(1):15–29. https://doi.org/10.2514/1.G003601 19. Chow TL. Classical mechanics. CRC press; 2013. 20. Astolfi A, Ortega R. Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems. Automatic Control, IEEE Transactions on. 2003; 48(4):590–606. https://doi.org/10.1109/ TAC.2003.809820 21. Aranda-Bricaire E, Moog CH. Invariant codistributions and the feedforward form for discrete-time nonlinear systems. Systems & control letters. 2004; 52(2):113–122. https://doi.org/10.1016/j.sysconle. 2003.11.005 22. Aranda-Bricaire E, Califano C, Moog CH. Immersion of Nonlinear Systems into Higher Order Systems. IFAC-PapersOnLine. 2017; 50(1):9480–9484. https://doi.org/10.1016/j.ifacol.2017.08.1581 23. Respondek W, Tall IA. Feedback equivalence of nonlinear control systems: a survey on formal approach. In: Chaos in Automatic Control. CRC Press; 2018. p. 137–262. 24. atharva aalok (2024). Professional Plots (https://www.mathworks.com/matlabcentral/fileexchange/ 100766-professional-plots), MATLAB Central File Exchange. Recuperado January 11, 2024. PLOS ONE A decentralized approach for the aerial manipulator robust trajectory tracking PLOS ONE | https://doi.org/10.1371/journal.pone.0299223 March 7, 2024 32 / 32