^{2} Univ. Grenoble Alpes, GIPSALab, FR38000 Grenoble, France;
^{3} Autonomous University of Puebla (BUAP), Faculty of Electronics, MX 72570 Puebla, Mexico;
^{4} University of Limoges ENSIL  ENSCI, XLIM Laboratory UMR CNRS 7252, Parc Ester Technopole, 16 rue Atlantis, Limoges 87068, France;
^{5} FEMTOST Institute, UMR CNRSUFC/ENSMM/UTBM, Automatic Control and MicroMechatronic Department, FR Besançon, France
Aerial manipulation has been an active area of research in recent times for unmanned aerial vehicles (UAVs) since it increases the application in both the military and civilian sectors. Unlike fixedwing UAV configurations, whose flight profile lacks hovering flight, vertical takeoff and landing (VTOL) rotorcrafts with three, four, six, or eight rotary propellers are wellsuited for aerial manipulation operations. However, numerous scientific challenges, technological and theoretical, remain open. The main issue arises from their limited payload capacity. Thus, alternatively, multiple robots can carry heavier payloads using cables or grippers^{[1]} which feature light and dexterous endeffectors. Furthermore, the dynamics of the robot is significantly altered while shifting and/or carrying payloads, due to the center of gravity shifting and external disturbances. Indeed, this is also an attraction in assembly because aerial robots can use this to sense disturbance forces and moments, as in [2]. Moreover, the performance of the aerial manipulation task relies on effective estimation of dynamic couplings for compensation purposes.
Numerous approaches have been proposed to deal with such problems. In [3], a Lyapunov based model reference adaptive control is used to stabilize a quadrotor with a multi degree of freedom (DoF) manipulator. However, the stability analysis is carried out with a linear approach and only the dynamics of the quadrotor was concerned due to the complexity of the system. JimenezCano et al.^{[4]} present a NewtonEuler approach to model and control a quadrotor through a variable parameter integral backstepping (VPIB) approach. However, the parametrization of the system is made through Euler angles, which present attitude estimation singularities. Kim et al.^{[5]} present aerial manipulation using a quadrotor and a 2DoF robot arm. The dynamic model of the system is obtained by the EulerLagrange formulation. Then, an adaptive sliding mode controller is designed. The effectiveness of the proposed method is showed experimentally by picking up and delivering an object.
In [6], the problem is solved through an autonomous avianinspired grasping method, however the design of the controller is formulated in the vertical plane, which supposes a limitation to the full 3 D space. Sun et al.^{[7]} present an amplitudesaturated nonlinear strategy for underactuated cranes with doublependulum dynamics. Such a strategy is twice validated: theoretically and practically. The experimental platform consists of a car and a doublependulum crane, where the coupling between the two systems (car and pendulum) is taken into account and a dynamic model is obtained. However, compared to an aerial vehicle, the attitude and position dynamics must be taken into account.
Finally, in [8], a new class of aerial manipulator is presented. It consists in a planar vertical takeoff and landing (PVTOL) equipped with parallel manipulator arms attached to the center of mass (CoM) of the aerial vehicle, which is called protocentric. A control law has been proposed for the case of rigid joints and validated through simulations.
The contribution of the present paper is centered on a strategy that combines an alternative torque compensation approach with a nonlinear control design. Specifically, a mathematical model is presented in detail using quaternions and it takes into account the coupling between the two systems, quadcopter and manipulator. Unlike the research previously cited, the design of the attitude control law uses the quaternion parametrization, which avoids the presence of singularities.
With quaternion parametrization, we proposes a constructive control law for the attitude and position stabilization. First, the design of a smooth, almost globally asymptotical control law for attitude stabilization which takes into account the dynamics of the robotic arm, is carried out. After that, a globally asymptotical nonlinear controller for the translational dynamics is proposed. In general, the control law is based in the usage of a sum of nested saturation functions in order to take into account the actuator′s limitation. Realtime experimental results and a statistical study of the obtained results validate the proposed strategy.
The paper is structured as follows. In Section 2, the attitude model of the quadcopter with the manipulator arm is presented. Then, the attitude control design is formulated in Section 3. Section 4 gives a strategy to estimate the angular position of each link in the robot manipulator through a Luenberger observer. Section 5 is devoted to the design of the position control law. Section 6 presents the hardware setup, experimental results, as well as a statistical study. Finally, some conclusions are presented in Section 7.
2 System modeling 2.1 Unit quaternion and attitude kinematicsConsider two orthogonal righthanded coordinate frames: the body coordinate frame,
The cross product between two vectors
The
$\mathcal{U}(\beta, e_v)=I_3+\sin(\beta) [{\rm e}_v^\times] + (1\cos(\beta))[{\rm e}_v^\times]^2.$  (1) 
Hence, a unit quaternion,
$q=\left(\begin{aligned}& \; \cos \frac{\beta}{2} \\& e_v \sin \frac{\beta}{2} \\\end{aligned}\right)=\left(\begin{aligned} q_0 \\ q_v \\\end{aligned}\right) \in \mathbb S^3$  (2) 
where
$\mathcal{R}:=I_3+2q_0 [q_v^\times] + 2[q_v^\times]^2.$  (3) 
Remark 1.
Denoting by
$\left(\begin{aligned} \dot{q}_0 \\ \dot{q}_v \\\end{aligned}\right) = \frac{1}{2}\left(\begin{aligned} & \quad {q}_v^{\rm T} \\ & I_3q_0 + [{q}_v^\times]\end{aligned}\right) \vec{\omega}=\frac{1}{2}\Xi(q)\vec{\omega}.$  (4) 
The attitude error is used to quantify mismatch between two attitudes. If
$ q_e=q_d^{1} \otimes q=(q_{e_{0}} \ {q}_{e_{v}}^{\rm T})^{\rm T}$  (5) 
where
The attitude dynamics and kinematics for the quadcopter have been reported in many works
$\left(\begin{aligned}& \dot{q}_0 \\& \dot{q_v}\end{aligned}\right)=\frac{1}{2}\Xi(q)\vec{\omega}\;\;$  (6) 
$J\dot{\vec{\omega}}=\vec{\omega}^\times J\vec{\omega}+\Gamma_T $  (7) 
where
$\Gamma_T= \Gamma + \Gamma_{arm} + \Gamma_G$  (8) 
where
Taking the robot manipulator as a physical pendulum attached to the fuselage of the aerial vehicle and following a similar process to the one in [13], the manipulator torque can be computed by
$\Gamma_{arm_{dyn}}=m_ag\zeta_{cd}\times \mathcal{R}(q)e_3$  (9) 
where
$\zeta_{cd}=\frac{1}{m_a}\left[\sum_{i=1}^nm_{mi}\varrho_{id}+m_l\varrho_{ld}\right]$  (10) 
where
In this case, let us consider the scheme in Fig. 1, which shows an anthropomorphic arm manipulator. This system has three degrees of freedom and then, the corresponding
$\begin{split} {\varrho _{1d}} = &{[0 \quad 0 \quad  {l_{c1}}]^{\rm T}}\\ {\varrho _{2d}} = & [{l_{c2}}\sin{\theta _{a2}}\cos {\theta _{a1}} \quad {l_{c2}}\sin{\theta _{a2}}\sin {\theta _{a1}}\\& ({l_1} + {l_2})\cos{\theta _{a2}}{]^{\rm T}}\\{\varrho _{3d}} = & [({l_2}\sin{\theta _{a2}} + {l_{c3}}\sin ({\theta _{a2}} + {\theta _{a3}}))\cos{\theta _{a1}} \\& ({l_2}\sin{\theta _{a2}} + {l_{c3}}\sin ({\theta _{a2}} + {\theta _{a3}}))\sin {\theta _{a1}}\\& ({l_1} + {l_2})\cos{\theta _{a2}}  {l_{c3}}\cos({\theta _{a2}} + {\theta _{a3}}){]^{\rm T}}\end{split}$  (11) 
where
$ \theta_{ai_m}=\frac{Ku(t)\dot{\theta}_{ai_m}}{a}$  (12) 
where
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Fig. 1. Manipulator arm with three degrees of freedom 
2.3 Actuator model
The quadrotor under study has four fixedpitch rotors mounted at the four ends of a simple cross frame. The model of the rotors is considered as a simple direct current (DC) motor which may reach high speed values (more than 200 rad/s). The collective input (or throttle input) is the sum of the thrusts of each rotor
$Q_j=ks_j^2 \quad\quad\quad\quad\;$  (13) 
$T = \sum_{j=1}^4 f_j=b\sum_{j=1}^4 s_j^2$  (14) 
where
$\Gamma_G=\sum_{j=1}^4 J_r(\vec{\omega}\times \vec{z}_b)(1)^{j+1}s_j$  (15) 
where
$\Gamma_{1}=d(f_3f_4)=db(s_3^2s_4^2)$  (16) 
$\Gamma_{2}=d(f_1f_2)=db(s_1^2s_2^2)$  (17) 
$\Gamma_{3}=Q_1Q_2+Q_3+Q_4=k(s_1^2s_2^2+s_3^2+s_4^2)$  (18) 
where
The objective is to design a control law which drives the quadcopter to attitude stabilization under the torques and moments exerted to this from the movement of a manipulator arm attached to its lower part. In other words, let
$q\rightarrow [\pm1\;0\;0\;0]^{\rm T}, \;\vec{\omega}\rightarrow 0\;\text{as}\;t\rightarrow\infty.$  (19) 
Furthermore, it is known that actuator saturation reduces the benefits of the feedback. When the controller continuously outputs infeasible control signals that saturate the actuators, system instability may follow. Then, besides the asymptotic stability, the control law also takes into account the physical constraints of the control system, in order to apply only feasible control signals to the actuators.
3.2 Attitude control with manipulator armIn this subsection, a control law that stabilizes the system described by (6) and (7) is proposed. The goal is to design a control torque that is bounded.
Definition 1. Given a positive constant
$ \begin{split}&\sigma_M=s\quad \mathrm{if}\, \, s<M \hbox{} \\&\sigma_M=\mathrm{sgn}(s)M\, \quad \mathrm{elsewhere}. \hbox{} \end{split}$  (20) 
Note that the components of
Theorem 1. Consider a rigid body rotational dynamics described by (6) and (7) with the following bounded control inputs
$\Gamma_i=\sigma_{M_{i2}}\left(\Gamma_{arm_{i}} + \sigma_{M_{i1}}( \lambda_i [\omega_i + \rho_i q_i])\right)$  (21) 
with
The proof of Theorem 1 is given in Appendix.
Remark 2. Note that the stability analysis has been carried out considering the asymptotic condition
$\Gamma_i=\sigma_{M_{i2}}\left(\Gamma_{arm_{i}} + \sigma_{M_{i1}}( \lambda_i [\omega_i + \rho_i q_{e_i}])\right)$  (22) 
where
The objective is to design a strategy for the estimation of the angles on each link in the manipulator arm, combining the data coming from the first order model of the manipulator actuators and the data coming from the end effector position tracked by the VICON system (motion capture system). Since the first order model does not fully describe the behavior of the arm manipulator (nonmodeled dynamics, actuators malfunction, etc.). In order to know the angular positions of the manipulator links with respect to the base body (quadcopter), the inverse kinematics of the manipulator arm is used.
4.2 Manipulator links observer designIn this subsection, a Luenberger observer is designed to estimate the angles on each link in the manipulator. For this, the endeffector position is computed through the arm inverse kinematics to know the angle on each link. The expression that describes a link angle is given by
$ \theta_{aV}=\theta_a+\mu_V$  (23) 
where
$ \dot{\hat{\theta}}_\delta=a\hat{\theta}_\delta+Ku_\delta+L(\theta_{aV}\hat{\theta})$  (24) 
$ \hat{\theta}=\hat{\theta}_V$  (25) 
where
Now, given the expression (24), where
The objective is to design a control law which stabilizes the quadcopter to a desired position, thereby solving the attitude stabilization problem. In other words, once the control law has stabilized the attitude of the system,
The schematic representation of a quadcopter carrying a manipulator arm can be seen in Fig. 2, where the inertial reference frame
${\Sigma _T}:\left\{ {\begin{array}{*{20}{l}}{\dot {\vec p} = \vec v}\\{{m_T}\dot {\vec v} =  {m_T}\vec g + R\left( {\begin{array}{*{20}{c}}0\\0\\u\end{array}} \right)}\end{array}} \right.$  (26) 
${\Sigma _O}:\left\{ {\begin{aligned}& {\dot q = \frac{1}{2}\Xi (q)\vec \omega }\\& {J\dot {\vec \omega} =  {{\vec \omega }^ \times }J\vec \omega + {\Gamma _T}}\end{aligned}} \right.\quad\quad\quad\quad$  (27) 
where
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Fig. 2. Schematic configuration of a quadrotor carrying a manipulator arm 
Note that the rotation matrix
$\begin{split}&R(\phi, \theta, \psi ) = \\&\,\,\,\,\,\,\left( {\begin{array}{*{20}{c}}{{\rm{C}}\psi {\rm{C}}\theta }&{{\rm{S}}\psi {\rm{C}}\theta }&{  {\rm{S}}\theta }\\{{\rm{C}}\psi {\rm{S}}\theta {\rm{S}}\phi  {\rm{S}}\psi {\rm{C}}\theta }&{{\rm{S}}\phi {\rm{S}}\theta {\rm{S}}\psi + {\rm{C}}\psi {\rm{C}}\phi }&{{\rm{C}}\theta {\rm{S}}\phi }\\{{\rm{C}}\psi {\rm{C}}\phi {\rm{S}}\theta + {\rm{S}}\psi {\rm{S}}\phi }&{{\rm{S}}\theta {\rm{S}}\psi {\rm{C}}\phi  {\rm{C}}\psi {\rm{S}}\phi }&{{\rm{C}}\theta {\rm{C}}\phi }\end{array}} \right).\end{split}$  (28) 
Taking into account (26) and (27), this system can be seen as a cascade system, where the translational dynamics (26) depend on the attitude (27), but the attitude dynamics do not depend on the translational one. This property will be used to design the control law. Now, assume that using the control law (21) one can stabilize the yaw dynamics, i.e.,
$\left( {\begin{aligned}{{{\dot p}_x}}\\{{{\dot p}_y}}\\{{{\dot p}_z}}\end{aligned}} \right){\rm{ = }}\left( {\begin{aligned}{{v_x}}\\{{v_y}}\\{{v_z}}\end{aligned}} \right)\quad\quad\quad\quad\quad\quad\quad$  (29) 
$\left( {\begin{aligned}{{{\dot v}_x}}\\{{{\dot v}_y}}\\{{{\dot v}_z}}\end{aligned}} \right) = \left( {\begin{aligned}& \;\;\;\;\;\;{  \frac{u}{{{m_T}}}\sin \theta }\\& \;\;\;{\frac{u}{{{m_T}}}\sin \phi \cos \theta }\\& {\frac{u}{{{m_T}}}\cos \phi \cos \theta  g}\end{aligned}} \right).$  (30) 
With an appropriate choice of target configuration, it will be possible to transform (29) and (30) into three independent linear triple integrators. For this, take
$\begin{split}&\phi_d=\arctan\left(\frac{r_2}{r_3+g}\right) \\&\theta_d=\arcsin\left(\frac{r_1}{\sqrt{r_1^2+r_2^2+(r_3+g)^2}}\right)\end{split}$  (31) 
where
$u=m_T\sqrt{r_1^2+r_2^2+(r_3+g)^2}.$  (32) 
Let the state be
$\begin{aligned}\label{eq:sub1} \,\,\,\,\,\,\,\, \Sigma_x:\left\{ \begin{array}{ll} \dot{p}_1=p_2 & \hbox{} \\ \dot{p}_2=p_3 & \hbox{} \\ \dot{p}_3=r_1 & \hbox{} \end{array} \right. & \!\!\Sigma_y:\left\{ \begin{array}{ll} \dot{p}_4=p_5 & \hbox{} \\ \dot{p}_5=p_6 & \hbox{} \\ \dot{p}_6=r_2 & \hbox{} \end{array} \right. & \!\!\!\!\!\!\!\!\!\Sigma_z:\left\{ \begin{array}{ll} \dot{p}_7=p_8 & \hbox{} \\ \dot{p}_8=p_9 & \hbox{} \\ \dot{p}_9=r_3. & \hbox{} \end{array} \right. \end{aligned}$  (33) 
Note that
Since the chains of integrators given in (33) have the same form, a control law can be proposed as in [14], and can be established by Theorem 2.
Theorem 2. Consider the quadcopter translational dynamics expressed in (29) and (30). Then, the thrust input
$\begin{split}r_1&=\varsigma_1\{a_3\sigma_{M1}[\frac{1}{\varsigma_1}(a_2p_1+p_2+p_3)]+\\& \quad a_2\sigma_{M1}[\frac{1}{\varsigma_1}(a_1p_2+p_3)]+a_1\sigma_{M1}[\frac{1}{\varsigma_1}(p_3)]\} \\r_2&=\varsigma_2\{b_3\sigma_{M1}[\frac{1}{\varsigma_1}(b_2p_4+p_5+p_6)]+\\& \quad b_2\sigma_{M1}[\frac{1}{\varsigma_2}(b_1p_5+p_6)]+b_1\sigma_{M1}[\frac{1}{\varsigma_2}(p_6)]\} \\r_3&=\varsigma_3\{c_3\sigma_{M1}[\frac{1}{\varsigma_1}(c_2p_7+p_8+p_9)]+\\& \quad c_2\sigma_{M1}[\frac{1}{\varsigma_3}(c_1p_8+p_9)]+c_1\sigma_{M1}[\frac{1}{\varsigma_3}(p_9)]\}\end{split}$  (34) 
$\begin{split}&\varsigma_1=\frac{\bar{r}_1}{(a_1+a_2+a_3)}\quad\quad \\&\varsigma_2=\frac{\bar{r}_2}{(b_1+b_2+b_3)}\quad\quad\\&\varsigma_3=\frac{\bar{r}_3}{(c_1+c_2+c_3)}.\quad\quad\end{split}$  (35) 
Due to space constraints, the proof of Theorem 2 is not presented here, but it can be easily derived from the seminal work of [15–17] .
Remark 3. In the above Theorem 2, the stabilization goal is the origin. In the case where the asymptotic condition is different from the origin, the variables
In order to test the effectiveness of the proposed control law, a set of experiments were performed. The aerial system consists of a homemade quadcopter and arm manipulator, see Fig. 3. Both structures were specially designed and built for this project. The characteristics and parameters of each system are described in Table 1. The total weight of the quadcopter with its arm is 315 g and its carrying capacity is about 50 g.
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Fig. 3. Miniquadcopter with its manipulator arm in flight 
The attitude control law (21) for the quadcopter was programmed in a Microwii Copter board, which has gyros, accelerometers and the ATMega32u4 as processor. Then, a ground station estimates the position and attitude of the hexacopter using the Vicon Tracker system and T40s cameras^{[18]}. With this system, it is possible to compute the position and attitude up to 100 Hz. The estimated states are sent to Matlab/Simulink through a user datagram protocol (UDP) frame every 2 ms. The position control algorithm is implemented in realtime at 200 Hz on a computer using number personal computer (xPC) target toolbox^{[19]}. The control variables are finally sent back to the quadcopter on the Microwii, through a Grenoble Image Parole Signal Automatique (GIPSA)Lab′s builtin bridge to DSM2 protocol. For this, a radiofrequency emitter is used. Furthermore, the manipulator arm is controlled by the DS35 Digital Super SubMicro Servo^{[20]}. To transmit the control signals to the manipulator, Spektrum digital spread modulation 2 (DSM2) transmitters are connected to the ground station through another builtin bridge. An overview of the whole hardware architecture is presented in Fig. 4.
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Fig. 4. Block diagram of the system 
6.2 Experimental scenario
A set of experiments is carried out in order to compare the performance of the proposed control with and without accounting for the torque generated by the manipulator, scenarios 1 and 2, respectively. The parameters of thecontrol laws are selected according to the characteristics of the quadrotor actuators and the condition for the manipulator arm. For the attitude control law (21) and the position control law given in (34), where
Fig. 5 shows the general performance of the aerial system under the disturbances coming from the manipulator arm when the arm torque estimation is not taken into account. Fig. 5 shows the angular and linear positions of the quadcopter during the experiment. Note that even when the quaternion parametrization is considered, Euler angles given in (28) are used in order to have a better perspective of the behavior of the system. In this case, attitude stabilization is guaranteed, but the movement of the manipulator causes stability issues for both attitude and linear position.
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Fig. 5. General behavior of the system during scenario 1 
6.4 Experimental results in scenario 2
Fig. 6 shows the angular and linear positions of the quadrotor as well as the computed manipulator torques. The arm perturbations are compensated through the dynamic model and it results in a general stabilization improvement of the quadrotor compared to the precedent approach. In addition, the importance of this approach is that the precision of the angular position knowledge is enhanced, which guarantees a better torque estimation.
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Fig. 6. General behaviour of the system during the experiment using dynamic method estimation torque compensation and the nonlinear observer 
6.5 Results analysis and statistical study
In order to compare the obtained results of the proposed methods and the flight performance of the aerial vehicle under the disturbances coming from the movement of the manipulator arm, a statistical error study is carried out. For this, the experiments described before were performed 8 times.
Figs. 7–9 show the linear position errors, the normal Gaussian distribution errors and the attitude errors for scenarios 1 and 2, respectively.
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Fig. 7. Attitude error and attitude average error value during the different experiments 
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Fig. 8. Position errors during the different experiments 
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Fig. 9. Error Gaussian normal distributions 
In order to calculate the attitude error,
Since the attitude stabilization of the aerial robot is enhanced with the usage of the proposed method, then the linear position stabilization is equally improved, as it is shown in the linear position errors computation.
To have another perspective of the obtained results, the normal Gaussian distribution errors are computed. For this, the linear position errors data were used. Then, from the obtained distributions, it is clear that when the dynamic method estimation for the arm manipulator torque is used, the average error value is closer to 0. Furthermore, the area covered by the distributions when the classical approach is used is bigger, consequently, the probability of error increases.
In general, the stabilization of the system is improved and guaranteed with the use of the dynamic model and the nonlinear observer. Table 2 shows the different average error values for the set of experiments. The first column shows the attitude average value error, the second, third and fourth columns show the average error value for the
7 Conclusions
In this paper, a novel model for a quadcopter carrying a manipulator arm was proposed. In addition, a control law was designed to asymptotically stabilize the attitude and position of the system. Moreover, this work has presented a method for aiding the solution through the design of a feedforward term which allows the estimation of the moments and torques exerted by the manipulator. Since input constraints exist in the actuators, the control law takes into account the actuator′s saturations. Experimental results and a statistical study show the effectiveness of the proposed control law in facing the disturbances coming from the manipulator. In a future work, experimental mass estimation and outdoor picking up and delivery of an object will be pursued.
Appendix Proof of Theorem 1Consider the candidate Lyapunov function
$\begin{split}V&=\frac{1}{2}\vec{\omega}^{\rm T}J\vec{w}+\kappa((1q_0)^2+q_v^{\rm T}q_v)=\\ & \quad \frac{1}{2}\vec{\omega}^{\rm T}J\vec{w}+2\kappa(1q_0)\end{split}$  (36) 
where
$\begin{split}\dot{V}&=\vec{\omega}^{\rm T}J\dot{\vec{\omega}}2\kappa\dot{q}_0=\\& \quad \vec{\omega}^{\rm T}(\vec{\omega}^{\times}J\vec{\omega}+\Gamma + \Gamma_{arm}+\Gamma_G)+\kappa q_v^{\rm T}\vec{\omega}=\\& \quad \underbrace{w_1(\Gamma_1+ \Gamma_{arm_{1}}) +\kappa q_1\omega_1}_{\dot{V}_1}+ \\&\quad\underbrace{w_2(\Gamma_2 + \Gamma_{arm_{2}}) + \kappa q_2\omega_2}_{\dot{V}_2}+\\&\quad\underbrace{w_3(\Gamma_3+\Gamma_{arm_{3}})+ \kappa q_3\omega_3}_{\dot{V}_3}\end{split}$  (37) 
where
$\begin{split}\dot{V}_1&=\omega_1 (\sigma_{M_{12}}\left(\Gamma_{arm_{1}} + \sigma_{M_{11}}( \lambda_1 [\omega_1 + \rho_1 q_1])\right)+\\ & \qquad \Gamma_{arm_{1}} )+\kappa q_1\omega_1\end{split}$  (38) 
if we choose
$\dot{V}_1=\omega_1\sigma_{M_{11}}(\lambda_1[\omega_1+\rho_1q_1])+\kappa q_1\omega_1.$  (39) 
Assume that
$\begin{split}\dot{V}_1&=\omega_1\sigma_{M_{11}}(\lambda_1[\omega_1+\rho_1q_1])+\kappa \omega_1q_1\leq\\& \quad \omega_1\sigma_{M_{11}}(\lambda_1(\rho_1+\epsilon))+\kappa\omega_1.\end{split}$  (40) 
Taking
$\kappa<\min(M_{11}, \lambda_1\rho_1+\epsilon)$  (41) 
one can assure the decrease of
Let
$\kappa<\lambda_1\rho_1+\epsilon.$  (42) 
For
$\lambda_1(\omega_1+\rho_1q_1)\leq3\lambda_1\rho_1\leq M_{11}.$  (43) 
Consequently,
$\Gamma_1=\lambda_1[\omega_1+\rho_1q_1].$  (44) 
As a result, (39) becomes
$\dot{V}_1=\lambda_1\omega_1^2\lambda_1\rho_1\omega_1q_1+\kappa \omega_1q_1.$  (45) 
Choosing
$\dot{V}_1=\lambda_1\omega_1^2\leq0.$  (46) 
The same argument is applied to
$\begin{split}\dot{V}&=\dot{V}_1+\dot{V}_2+\dot{V}_3=\\& \quad (\lambda_1\omega_1^2+\lambda_2\omega_2^2+\lambda_3\omega_3^2)\leq0.\end{split}$  (47) 
In order to complete the proof, the LaSalle invariance principle is invoked. All the trajectories converge to the largest invariant set
This work was supported by CONACYTMexico, LabEx PERSYVALLab (No. ANR11LABX0025) and Equipex ROBOTEX (No. ANR10EQPX4401).
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