A decentralized approach for the aerial manipulator robust trajectory tracking

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 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 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 n-link 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^bz^b, see Fig 1.

Fig 1. Unmanned aerial manipulator.

It comprises the aerial vehicle with four rotors n₁, n₂, n₃, and n₄; and the robotic manipulator with two revolute joints R₁ and R₂. Body axes 0x^by^bz^b and inertial axes 0xⁱyⁱzⁱ.

To obtain the UAM dynamic model the reference frames 0xⁱyⁱzⁱ, i = 1, 2, 3 on the robotic manipulator’s links are defined, as in Fig 1.

2.1 Quadrotor dynamic model

From Newton-Euler laws of motion the quadrotor dynamic model is

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill m_Q\ddot{X}& =& m_{Q}g{e}_3+F_p^i-f_{RM}^i\hfill \\ \hfill J\dot{\mathrm{\Omega }}& =& -\mathrm{\Omega }\times J\mathrm{\Omega }+M_p^b-M_{RM}^b\hfill \end{array}\end{array}$$ (1)

where m_Q is the quadrotor mass, g is the gravity acceleration constant, e₃ = [0 0 1]^⊤, $F_p^i$ is the force due to the propulsion system, $f_{RM}^i$ 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, Ω = [p q r]^⊤ is the quadrotor velocity in body coordinates, $M_p^b$ the moment due to the propulsion system, and $M_{RM}^b$ the moment due to the robotic manipulator expressed in body coordinates.

The propulsion force is given by

$$F_p^i=-T_{T}R{e}_3$$

where $T_T={\sum }_{i=1}^4{T}_i$ is the total thrust produced by the four rotors with T_i the thrust produced by rotor r_i. Moreover, R ∈ SO(3) is the rotation matrix from body coordinates to inertial coordinates.Where $SO\left(3\right)=\{R\in {\mathbb{R}}^{3\times 3}|R^{\top }R=I,\text{det}\left(R\right)=1\}$ with I the identity matrix. Introducing the following notation c_σ = cos(σ), s_σ = sin(σ) for any angle σ, the propulsion moment is

$$M_p^b=\left[\begin{array}{c}{M}_x^b\\ M_y^b\\ M_z^b\end{array}\right]=\left[\begin{array}{c}\ell c_{\pi /4}(T_2+T_3-T_1-T_4)\\ \ell c_{\pi /4}(T_1+T_2-T_3-T_4)\\ Q_1+Q_3-Q_2-Q_4\end{array}\right]$$

where ℓ is 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.

The force $f_{RM}^i$ and the moment $M_{RM}^b$ 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, 1R and 2R; see Fig 2. Fig 2 also shows the robotic manipulator’s center of gravity.

Fig 2. UAM ideal workspace.

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 1R2R robotic manipulator as an equivalent robotic manipulator composed of one revolute joint 1′R and one prismatic joint 1P, as illustrated in Fig 3. This idea was partially developed in [15] by modeling the 1R2R robotic 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.

Fig 3. UAM ideal workspace with revolute and prismatic joints.

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.

Table 1. Link parameters.

i	α i−1	a i−1	d i	θ i
1	α0 = 0	a0 = 0	d1 = 0	θ 1
2	${\alpha }_1=\frac{\pi }{2}$	a1 = 0	d2 = 0	θ2 = 0
3	α2 = 0	a2 = 0	d3 = L	θ3 = 0

With α_i the angle from zⁱ to zⁱ⁺¹ measured around xⁱ, a_i is the distance from zⁱ to zⁱ⁺¹ measured along xⁱ, d_i is the distance from xⁱ⁻¹ to xⁱ measured along zⁱ, and θ_i is the angle from xⁱ⁻¹ to xⁱ measured around zⁱ.

Fig 4. UAM link variables.

From the link parameters, the following rotation matrices are obtained

$$\begin{array}{c}\hfill \begin{array}{c}\left({}_1{}^{0}R\right)=\left[\begin{array}{ccc}{c}_{{\theta }_1}& -s_{{\theta }_1}& 0\\ s_{{\theta }_1}& c_{{\theta }_1}& 0\\ 0& 0& 1\end{array}\right],\left({}_2{}^{1}R\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right]\\ \left({}_3{}^{2}R\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}\end{array}$$ (2)

the rotation matrices use the notation introduced in [14]. Hence, (ⁱ⁺¹iR) is the rotation matrix from reference frame i to i + 1. Moreover, (ⁱ⁺¹iR)^⊤ = (ⁱi + 1R) and ${(}^{i+1}i+2R){(}_i^{i+2}R)={(}_i^{i+1}R)$.

Also, the distance $P_{i+1}^i$ between frame i and frame i + 1 measured from frame i is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill P_1^0=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]P_2^1=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]P_3^2=\left[\begin{array}{c}0\\ 0\\ L\end{array}\right]\end{array}\end{array}$$ (3)

Now, without any loss of generality, the following simplifications are considered to tailor the robotic manipulator dynamic model.

The revolute joint 1′R is massless so that the mass of the revolute joints 1R and 2R is concentrated at the distal end of equivalent prismatic link P₁. Hence,

$$m_{1^{\prime }R}=0,m_P=m_{1R}+m_{2R}$$

with m_1′R the mass of the revolute joint 1′R, m_P the mass of the revolute joint 1P and m_1R, m_2R the mass of the revolute joints 1R and 2R, respectively. Moreover, the position $P_C^i$ of the center of mass of link i expressed in the i-th reference frame, is defined by the following vectors

$$\begin{array}{c}\hfill P_C^1=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],P_C^2=\left[\begin{array}{c}0\\ 0\\ L\end{array}\right]\end{array}$$ (4)

moreover, the inertia tensors for each point mass are $I_C^1=0$ and $I_C^2=0$.

Since the robotic manipulator is attached to a flying base, it is necessary to obtain the rotation matrix between the reference frame 0x⁰y⁰z⁰ and the body reference frame 0x^by^bz^b. Note that both frames are rigidly attached but have different configurations, see Fig 4. The corresponding rotation matrix is the following.

$$\left({}_b{}^{0}R\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right]$$

Thus, the boundary conditions for the flying base are [17–19]

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\mathrm{\Omega }}^0& =& \left({}_b{}^{0}R\right)\mathrm{\Omega }+{\dot{\theta }}_0{\widehat{z}}^0\hfill \\ \hfill {\dot{\mathrm{\Omega }}}^0& =& \left({}_b{}^{0}R\right)\dot{\mathrm{\Omega }}+\left({}_b{}^{0}R\right)\mathrm{\Omega }\times {\dot{\theta }}_0{\widehat{z}}^0+{\ddot{\theta }}_0{\widehat{z}}^0\hfill \\ \hfill {\dot{V}}^0& =& \left({}_b{}^{0}R\right)[{\dot{V}}^b+\mathrm{\Omega }\times V^b+\mathrm{\Omega }\times P_0^b+\mathrm{\Omega }\times (\mathrm{\Omega }\times P_0^b)-g{R}^{\top }{e}_3]\hfill \end{array}\end{array}$$

where Ω⁰ and V⁰ are the angular and translational velocities of the frame 0x⁰y⁰z⁰, and $V^b=R^{\top }\dot{X}$ is the translational velocity of the quadrotor expressed in body axes. Moreover, $P_0^b=0\in {\mathbb{R}}^3$ is the distance from the frame 0x⁰y⁰z⁰ to the body frame and θ₀ is the angle of rotation of frame 0x⁰y⁰z⁰ with respect to frame 0x^by^bz^b around the ${\widehat{z}}^0$ axis. The term R^⊤ge₃ is introduced to consider the gravitational acceleration. In the following, Ωⁱ, ${\dot{\mathrm{\Omega }}}^i$, Vⁱ and ${\dot{V}}^i$ are the rotational and translational velocities and accelerations of joint i expressed in the frame 0xⁱyⁱzⁱ, respectively.

The frame 0x⁰y⁰z⁰ is rigidly attached to the body frame so that ${\theta }_0={\dot{\theta }}_0={\ddot{\theta }}_0=0$, thus the boundary conditions reduce to

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\mathrm{\Omega }}^0& =& \left({}_b{}^{0}R\right)\mathrm{\Omega }\hfill \\ \hfill {\dot{\mathrm{\Omega }}}^0& =& \left({}_b{}^{0}R\right)\dot{\mathrm{\Omega }}\hfill \\ \hfill {\dot{V}}^0& =& \left({}_b{}^{0}R\right)({\dot{V}}^b+\mathrm{\Omega }\times V^b-g{R}^{\top }{e}_3)\hfill \end{array}\end{array}$$

The angular velocities are propagated to frames 0x¹y¹z¹ and 0x²y²z² following the outward iteration of the RNEA method, see Appendix A, as follows. Therefore, the angular velocity for frame 0x¹y¹z¹ attached to a revolute joint with i + 1 = 1 is given by (44)

$$\begin{array}{c}\hfill {\mathrm{\Omega }}^1=\left({}_0{}^{1}R\right){\mathrm{\Omega }}^0+{\dot{\theta }}_1{\widehat{z}}^1\end{array}$$

meanwhile, for frame 0x²y²z² attached to the prismatic joint with i + 1 = 2, it follows that (45)

$$\begin{array}{c}\hfill {\mathrm{\Omega }}^2=\left({}_2{}^{1}R\right){\mathrm{\Omega }}^1\end{array}$$

Once again, from the RNEA outward iteration, the angular and translational accelerations are propagated as follows. For the frames 0x¹y¹z¹ and 0x²y²z² with i + 1 = 1 and i + 1 = 2, respectively, one obtains (46) and (47),

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{\mathrm{\Omega }}}^1& =& \left({}_0{}^{1}R\right){\dot{\mathrm{\Omega }}}^0+\left({}_0{}^{1}R\right){\mathrm{\Omega }}^0\times {\dot{\theta }}_1{\widehat{z}}^1+{\ddot{\theta }}_1{\widehat{z}}^1\hfill \\ \hfill {\dot{V}}^1& =& \left({}_0{}^{1}R\right)[{\dot{\mathrm{\Omega }}}^0\times P_1^0+{\mathrm{\Omega }}^0\times ({\mathrm{\Omega }}^0\times P_1^0)+{\dot{V}}^0]\hfill \end{array}\end{array}$$

For the prismatic joint, one has (48) and (49),

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{\mathrm{\Omega }}}^2& =& \left({}_1{}^{2}R\right){\dot{\mathrm{\Omega }}}^1\hfill \\ \hfill {\dot{V}}^2& =& \left({}_1{}^{2}R\right)[{\dot{\mathrm{\Omega }}}^1\times P_2^1+{\mathrm{\Omega }}^1\times ({\mathrm{\Omega }}^1\times P_2^1)+{\dot{V}}^1]+2{\mathrm{\Omega }}^2\times \dot{L}{\widehat{z}}^2+\ddot{L}{\widehat{z}}^2.\hfill \end{array}\end{array}$$

Taking into account $P_1^0$ and $P_2^1$ defined in Eq (3) the accelerations ${\dot{V}}^1$ and ${\dot{V}}^2$ reduce to

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{V}}^1& =& \left({}_0{}^{1}R\right){\dot{V}}^0\hfill \\ \hfill {\dot{V}}^2& =& \left({}_1{}^{2}R\right){\dot{V}}^1+2{\mathrm{\Omega }}^2\times \dot{L}{\widehat{z}}^2+\ddot{L}{\widehat{z}}^2\hfill \end{array}\end{array}$$

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)

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{V}}_C^1& =& {\dot{\mathrm{\Omega }}}^1\times P_C^1+{\mathrm{\Omega }}^1\times ({\mathrm{\Omega }}^1\times P_C^1)+{\dot{V}}^1\hfill \\ \hfill {\dot{V}}_C^2& =& {\dot{\mathrm{\Omega }}}^2\times P_C^2+{\mathrm{\Omega }}^2\times ({\mathrm{\Omega }}^2\times P_C^2)+{\dot{V}}^2\hfill \end{array}\end{array}$$

Considering (4) the acceleration of the revolute link center of mass reduces to

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{V}}_C^1& =& {\dot{V}}^1\hfill \end{array}\end{array}$$

The forces acting at the center of mass of each link are (51)

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill F^1& =& m_{R_1^{\prime }}{\dot{V}}^1=0\hfill \\ \hfill F^2& =& m_{P_1}{\dot{V}}_C^2\hfill \end{array}\end{array}$$

Finally, under the aforementioned considerations, the moments on each link are N¹ = N² = 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 f² exerted on link 2, by link 1 and the force f¹ exerted on link 1 by the robotic manipulator’s base are (52)

$$\begin{array}{ccc}\hfill f^2& =& F^2\hfill \\ \hfill f^1& =& \left({}_2{}^{1}R\right)f^2\hfill \end{array}$$ (5)

where it is assumed that f³ = 0. Additionally, the torque n² exerted on link 2 by link 1 and the torque n¹ exerted on link 1 by the robotic manipulator base are given by

$$\begin{array}{ccc}\hfill n^2& =& P_C^2\times F^2\hfill \\ \hfill n^1& =& \left({}_2{}^{1}R\right)n^2+P_2^1\times \left({}_2{}^{1}R\right)f^2\hfill \end{array}$$ (6)

where it is assumed that n³ = 0. Considering (3) and (4), it follows that

$$\begin{array}{ccc}\hfill n^1& =& \left({}_2{}^{1}R\right)n^2\hfill \end{array}$$

Finally, the robotic manipulator dynamic model is described by the following equations

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\tau }_R& =& {n^1}^{\top }{\widehat{z}}^1\hfill \\ \hfill f_P& =& {f^2}^{\top }{\widehat{z}}^2\hfill \end{array}\end{array}$$ (7)

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 f¹ and the moment n¹ can be expressed in the reference frame 0x⁰y⁰0z⁰ as follows

$$\begin{array}{ccc}\hfill f^0& =& {(}_1^{0}R)f^1\hfill \\ \hfill n^0& =& {(}_1^{0}R)n^1\hfill \end{array}$$ (8)

thus,

$$\begin{array}{ccc}\hfill f_{RM}^i& =& R{\left({}_b{}^{0}R\right)}^{\top }{f}^0\hfill \\ \hfill n_{RM}^b& =& {\left({}_b{}^{0}R\right)}^{\top }{n}^0\hfill \end{array}$$ (9)

In the following the angle θ₁ is replaced by θ_P measured as shown in Fig 5. Thus,

$${\theta }_1={\theta }_P-{\theta }_Q$$

Fig 5. Prismatic link angle with respect to the inertial frame.

Complex but straightforward computations show that the quadrotor translational dynamic model constrained to the plane 0xⁱzⁱ becomes

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill \overline{M}\left[\begin{array}{c}\ddot{x}\\ \ddot{z}\end{array}\right]& =& \overline{M}g\left[\begin{array}{c}0\\ 1\end{array}\right]-T_T\left[\begin{array}{c}{s}_{{\theta }_Q}\\ c_{{\theta }_Q}\end{array}\right]-\left[\begin{array}{c}{\overline{f}}_{R{M}_x}^i\\ {\overline{f}}_{R{M}_z}^i\end{array}\right]\hfill \end{array}\end{array}$$ (10)

where the following identity has been considered $R^{\top }\dot{X}={\dot{V}}^b+\mathrm{\Omega }\times V^b$, with $\overline{M}=m_Q+m_{P_1}$ and

$$\begin{array}{ccc}\hfill {\overline{f}}_{RM}^i& =& \left[\begin{array}{c}{\overline{f}}_{R{M}_x}^i\\ {\overline{f}}_{R{M}_y}^i\\ {\overline{f}}_{R{M}_z}^i\end{array}\right]\hfill \\ & =& m_{P_1}\left[\begin{array}{c}{c}_{{\theta }_P}(L{\ddot{\theta }}_P+2\dot{L}{\dot{\theta }}_P)+s_{{\theta }_P}(\ddot{L}-L{\dot{\theta }}_P^2)\\ 0\\ s_{{\theta }_P}(L{\ddot{\theta }}_P+2\dot{L}{\dot{\theta }}_P)+c_{{\theta }_P}(L{\dot{\theta }}_P^2-\ddot{L})\end{array}\right]\hfill \end{array}$$

The rotational dynamics constrained to the 0xⁱzⁱ plane becomes

$$\begin{array}{c}\hfill J_{yy}{\ddot{\theta }}_Q=M_y^b-n_{R{M}_q}^b\end{array}$$ (11)

where

$$\begin{array}{ccc}\hfill n_{RM}^b& =& \left[\begin{array}{c}{n}_{R{M}_p}^b\\ n_{R{M}_q}^b\\ n_{R{M}_r}^b\end{array}\right]\hfill \end{array}$$

with $n_{R{M}_p}^b=n_{R{M}_r}^b=0$ and the following identities are instrumental $c_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}-s_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}=c_{{\theta }_P}$, $s_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}+c_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}=s_{{\theta }_P}$

$$\begin{array}{ccc}\hfill n_{R{M}_q}^b& =& m_P{L}^2{\ddot{\theta }}_P+2m_{P}L\dot{L}{\dot{\theta }}_P+m_{P}gL{s}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}-\frac{m_p}{M}L{c}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}{\overline{f}}_{R{M}_x}\hfill \\ & & +\frac{m_p}{M}L{c}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}{\overline{f}}_{R{M}_z}\hfill \end{array}$$

Finally, from (7) the robotic manipulator dynamic becomes

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill \left[\begin{array}{cc}{m}_P{L}^2& 0\\ 0& m_P\end{array}\right]\left[\begin{array}{c}{\ddot{\theta }}_P\\ \ddot{L}\end{array}\right]+& & \\ \hfill \left[\begin{array}{cc}{m}_{P}L\dot{L}& m_{P}L{\dot{\theta }}_P\\ -m_{P}L{\dot{\theta }}_P& 0\end{array}\right]\left[\begin{array}{c}{\dot{\theta }}_P\\ \dot{L}\end{array}\right]+& & \\ \hfill \left[\begin{array}{c}{m}_{P}Lg{s}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}\\ -m_{P}g{s}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}\end{array}\right]+& & \\ \hfill \left[\begin{array}{cc}-\frac{1}{M}{c}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}& \frac{1}{M}{c}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}\\ -\frac{m_P}{M}{s}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}& \frac{m_P}{M}{s}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}\end{array}\right]\times \left[\begin{array}{c}{\overline{f}}_{R{M}_x}^i\\ {\overline{f}}_{R{M}_z}^i\end{array}\right]& =& \left[\begin{array}{c}{\tau }_R\\ f_P\end{array}\right]\hfill \end{array}\end{array}$$ (12)

Summarizing, the UAM dynamic model that will be considered for control design is described by Eqs (10), (11) and (12).

3 Decentralized robust control strategy

The control design is divided into two control loops: the inner loop is a state feedback controller that customizes the robotic manipulator effects on the aerial vehicle as an exogenous disturbance (C4D), generating a decentralized UAM dynamic model. In contrast, the outer control loop uses the decentralized model and independently applies control strategies for the quadrotor and robotic arm. We first present the C4D state feedback and then the estimator of the quadrotor exogeneous moments and forces (EQEMF).

Since the quadrotor acts as a flying base for the manipulator, the quadrotor and the robotic manipulator dynamics are interconnected. Hence, assuming that the forces and moments from the inward iteration are exogenous signals is not trivial, being this development and analysis contributions of the present work. Therefore, the first step is to show that the manipulator dynamics can be considered exogenous disturbances for the quadrotor dynamics and vice versa. Since the robotic manipulator dynamic is fully actuated, its interaction with the quadrotor can be characterized as exogenous, thus allowing to design a control scheme to compensate for its effects.

The following controller is proposed to make the disturbances on the quadrotor from the manipulator exogenous and vice versa.

$$\begin{array}{ccc}\hfill M_y^b& =& m_{P}gL{c}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}+\frac{m_p}{M}L{s}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}{\overline{f}}_{R{M}_x}\hfill \\ & & +\frac{m_p}{M}L{s}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}{\overline{f}}_{R{M}_z}+{\overline{M}}_b^y\hfill \end{array}$$ (13)

$$\begin{array}{ccc}\hfill {\tau }_R& =& -\frac{1}{M}(c_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}{\overline{f}}_{R{M}_x}^i-c_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}{\overline{f}}_{R{M}_z}^i)\hfill \\ & & +m_{P}Lg{c}_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}+{\overline{\tau }}_R\hfill \end{array}$$ (14)

$$\begin{array}{ccc}\hfill f& =& -\frac{m_P}{M}(s_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}{\overline{f}}_{R{M}_x}^i-s_{({\theta }_P-{\theta }_Q)}{s}_{{\theta }_Q}{\overline{f}}_{R{M}_z}^i)\hfill \\ & & +m_{P}g{c}_{({\theta }_P-{\theta }_Q)}{c}_{{\theta }_Q}+{\overline{f}}_P\hfill \end{array}$$ (15)

The UAM dynamics (10)–(12) with the inner loop controller RC4D (13)–(15) results in

$$\begin{array}{cc}\hfill \overline{M}\ddot{\mathbf{X}}& =\overline{M}g\mathbf{e}-T_T{\mathbf{r}}_{{\theta }_Q}+{\delta }_T\hfill \\ \hfill {\mathbf{J}}_{\mathbf{M}}\left[\begin{array}{c}{\ddot{\theta }}_Q\\ {\ddot{\theta }}_P\\ \ddot{L}\end{array}\right]& =\left[\begin{array}{c}{\delta }_R\\ -2m_{P}L\dot{L}{\dot{\theta }}_P-m_{P}Lg{s}_{{\theta }_P}\\ -m_{P}L{\dot{\theta }}_P^2+m_{P}g{c}_{{\theta }_P}\end{array}\right]+{\mathbf{I}}_{\mathbf{3}\times \mathbf{3}}\left[\begin{array}{c}{\overline{M}}_b^y\\ {\overline{\tau }}_R\\ {\overline{f}}_P\end{array}\right]\hfill \end{array}$$ (16)

where

$$\mathbf{X}=\left[\begin{array}{c}x\\ z\end{array}\right],\mathbf{e}=\left[\begin{array}{c}0\\ 1\end{array}\right],{\mathbf{r}}_{{\theta }_Q}=\left[\begin{array}{c}{s}_{{\theta }_Q}\\ c_{{\theta }_Q}\end{array}\right],{\delta }_T=\left[\begin{array}{c}{\overline{f}}_{R{M}_x}^i\\ {\overline{f}}_{R{M}_z}^i\end{array}\right],$$

$${\mathbf{J}}_{\mathbf{M}}=\left[\begin{array}{ccc}{J}_{yy}& 0& 0\\ 0& m_p{L}^2& 0\\ 0& 0& m_p\end{array}\right],{\mathbf{I}}_{\mathbf{3}\times \mathbf{3}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

and

$${\delta }_R=m_P{L}^2{\ddot{\theta }}_P+2m_{P}L\dot{L}{\dot{\theta }}_P+m_{P}gL{s}_{{\theta }_P}.$$

Remark 1 It is important to underscore, that the combination of both the coordinate change and the controller is what ensures that δ_T and δ_R can be treated as external disturbances for the quadrotor dynamics comming from the robotic manipulator. Furthermore, the proposed change of coordinates paved the way to the such controller design. More importantly, the proposed design allows the online estimation of the signals δ_T, δ_R, as it is described in the following developments.

3.1 Quadrotor exogeneous disturbances estimator

The estimator of external forces and moments for a quadrotor was introduced in [13]. This estimator was also employed in the previous version of this study in [15]. The estimator is based on the Immersion and Invariance method [20]. As it will be evident in the following developments, if the disturbance is not exogenous, the estimator assumptions fail to be fulfilled.

First, the quadrotor dynamics is rewritten as

$$\begin{array}{ccc}\hfill {\dot{\zeta }}_1& =& {\overline{f}}_1({\zeta }_1,{\mathbf{r}}_{{\theta }_Q},T_T)+{\delta }_T\hfill \\ \hfill {\dot{\zeta }}_2& =& {\overline{f}}_2\left({\overline{M}}_b^y\right)+{\delta }_R\hfill \end{array}$$

with ${\zeta }_1=\overline{M}\dot{\mathbf{X}}$, ${\zeta }_2=J_{yy}{\dot{\theta }}_Q$,

$$\begin{array}{ccc}\hfill {\overline{f}}_1& =& {\displaystyle \overline{M}{g}^e{e}_3-T_T{\mathbf{r}}_{{\theta }_Q}{{}^{b}e}_3}\hfill \\ \hfill {\overline{f}}_2& =& {\displaystyle {\overline{M}}_b^y}\hfill \end{array}$$

The external forces and moments errors are defined as

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\tilde{\delta }}_1& =& {\delta }_T-{\eta }_1+{\beta }_1\left({\zeta }_1\right)\hfill \\ \hfill {\tilde{\delta }}_2& =& {\dot{\delta }}_T-{\eta }_2+{\beta }_2\left({\zeta }_1\right)\hfill \\ \hfill {\tilde{\delta }}_3& =& {\ddot{\delta }}_T-{\eta }_3+{\beta }_3\left({\zeta }_1\right)\hfill \\ \hfill {\tilde{\delta }}_4& =& {\delta }_R-{\eta }_4+{\beta }_4\left({\zeta }_2\right)\hfill \end{array}\end{array}$$ (17)

where η_i, i = 1, 2, 3, 4 are the estimator states β_i(ζ₁), i = 1, 2, 3, β₄(ζ₂) are functions defined on the design process.

Note that ${\text{lim}}_{t\to \infty }{\tilde{\delta }}_i=0,i=1,2,3$, implies that the following relationships hold

$$\begin{array}{ccc}\hfill {\displaystyle \underset{t\to \infty }{\text{lim}}{\eta }_1-{\beta }_1\left({\zeta }_1\right)}& =& {\displaystyle {\delta }_T,\underset{t\to \infty }{\text{lim}}{\eta }_2-{\beta }_2\left({\zeta }_1\right)={\dot{\delta }}_T}\hfill \\ \hfill {\displaystyle \underset{t\to \infty }{\text{lim}}{\eta }_3-{\beta }_3\left({\zeta }_1\right)}& =& {\displaystyle {\ddot{\delta }}_T,\underset{t\to \infty }{\text{lim}}{\eta }_4-{\beta }_4\left({\zeta }_2\right)={\delta }_R}\hfill \end{array}$$

Defining the dynamics of the estimator states as follow

$$\begin{array}{ccc}\hfill {\dot{\eta }}_1& =& {\displaystyle {\eta }_2-{\beta }_2\left({\zeta }_1\right)+\frac{\partial {\beta }_1}{\partial {\zeta }_1}[{\overline{f}}_1+{\eta }_1-{\beta }_1\left({\zeta }_1\right)]}\hfill \\ \hfill {\dot{\eta }}_2& =& {\displaystyle {\eta }_3-{\beta }_3\left({\zeta }_1\right)+\frac{\partial {\beta }_2}{\partial {\zeta }_1}[{\overline{f}}_1+{\eta }_1-{\beta }_1\left({\zeta }_1\right)]}\hfill \\ \hfill {\dot{\eta }}_3& =& {\displaystyle \frac{\partial {\beta }_3}{\partial {\zeta }_1}[{\overline{f}}_1+{\eta }_1-{\beta }_1\left({\zeta }_1\right)]}\hfill \end{array}$$

and choosing β_i(ζ), i = 1, 2, 3 such as

$$\begin{array}{c}\hfill \frac{\partial {\beta }_1}{\partial {\zeta }_1}=-{\mathrm{\Gamma }}_1,\frac{\partial {\beta }_2}{\partial {\zeta }_1}=-{\mathrm{\Gamma }}_2,\frac{\partial {\beta }_3}{\partial {\zeta }_1}=-{\mathrm{\Gamma }}_3,\end{array}$$ (18)

with Γ_i, i = 1, 2, 3 positive definite matrices, the external forces estimator dynamics become

$$\begin{array}{c}\hfill {\tilde{\delta }}_1^{\left(3\right)}+{\mathrm{\Gamma }}_1{\ddot{\tilde{\delta }}}_1+{\mathrm{\Gamma }}_2{\dot{\tilde{\delta }}}_1+{\mathrm{\Gamma }}_3{\tilde{\delta }}_1={\delta }_T^{\left(3\right)}\end{array}$$ (19)

Following the same procedure for the external moments estimator, its dynamics results in

$$\begin{array}{c}\hfill {\dot{\tilde{\delta }}}_4+{\mathrm{\Gamma }}_4{\tilde{\delta }}_4={\dot{\delta }}_R\end{array}$$ (20)

with

$${\displaystyle {\dot{\eta }}_4=\frac{\partial {\beta }_4}{\partial {\zeta }_2}[{\overline{f}}_2+{\eta }_4-{\beta }_4\left({\zeta }_2\right)]}$$

and

$$\begin{array}{c}\hfill \frac{\partial {\beta }_4}{\partial {\zeta }_2}=-{\mathrm{\Gamma }}_4\end{array}$$ (21)

being Γ₄ a positive defined matrix.

3.2 Quadrotor position and attitude control

The control design for the quadrotor position starts by defining the trajectory tracking error as

$$\tilde{\mathbf{X}}=\mathbf{X}-{\mathbf{X}}_d$$

where X_d is the reference position. Then we have

$$\begin{array}{c}\hfill \ddot{\tilde{\mathbf{X}}}=g\mathbf{e}-\frac{T_T}{\overline{M}}{\mathbf{r}}_{{\theta }_Q}+\frac{{\delta }_T\left(t\right)}{\overline{M}}-{\ddot{\mathbf{X}}}_d\end{array}$$ (22)

The vertical dynamics are directly controlled with T_T, meanwhile the horizontal dynamics on the axis 0xⁱ are underactuated and controlled by modifying θ_Q.

First, we rewrite the term $T_T{\mathbf{r}}_{{\theta }_Q}$ as follows

$$\begin{array}{c}\hfill \frac{T_T}{\overline{M}}{\mathbf{r}}_{{\theta }_Q}={\displaystyle \frac{T_T}{\overline{M}{\mathbf{r}}_{{\theta }_{Q_d}}^{\top }{\mathbf{r}}_{{\theta }_Q}}}\left[\right({\mathbf{r}}_{{\theta }_{Q_d}}^{\top }{\mathbf{r}}_{{\theta }_Q}\left){\mathbf{r}}_{{\theta }_Q}\right]\end{array}$$

where ${\mathbf{r}}_{{\theta }_{Q_d}}$ the desired value for ${\mathbf{r}}_{{\theta }_Q}$. Now, the following term is added and subtracted

$$\begin{array}{c}\hfill \frac{1}{\overline{M}}{\displaystyle \frac{T_T}{{\mathbf{r}}_{{\theta }_{Q_d}}^{\top }{\mathbf{r}}_{{\theta }_Q}}}{\mathbf{r}}_{{\theta }_{Q_d}}.\end{array}$$

and hence (22) becomes

$$\begin{array}{c}\hfill \ddot{\tilde{\mathbf{X}}}=g{\mathbf{e}}_2+{\displaystyle \frac{{\delta }_T}{\overline{M}}}-{\ddot{\mathbf{X}}}_d-{\displaystyle \frac{1}{\overline{M}}}{\displaystyle \frac{T_T}{{\mathbf{r}}_{{\theta }_{Q_d}}^{\top }{\mathbf{r}}_{{\theta }_Q}}}{\mathbf{r}}_{{\theta }_{Q_d}}-{\displaystyle \frac{1}{\overline{M}}}\mathrm{\Theta }\end{array}$$

The term Θ is defined as

$$\mathrm{\Theta }={\displaystyle \frac{T_T}{{\mathbf{r}}_{{\theta }_{Q_d}}^{\top }{\mathbf{r}}_{{\theta }_Q}}}\left[\right({\mathbf{r}}_{{\theta }_{Q_d}}^{\top }{\mathbf{r}}_{{\theta }_Q}){\mathbf{r}}_{{\theta }_Q}-{\mathbf{r}}_{{\theta }_{Q_d}}]$$

Let us define the control input T_T and ${\mathbf{r}}_{{\theta }_{Q_d}}$ in the following form

$$\begin{array}{c}\hfill T_T=u^{\top }{\mathbf{r}}_{{\theta }_Q},{\mathbf{r}}_{{\theta }_{Q_d}}=\frac{u}{\Vert u\Vert }\end{array}$$ (23)

where u = [u_x u_z]^⊤ is a new control input. The final controller is defined through u as follows

$$\begin{array}{c}\hfill u=\overline{M}(K_{PX}\tilde{\mathbf{X}}+K_{DX}\dot{\tilde{\mathbf{X}}}+g{e}_3+{\ddot{\mathbf{X}}}_d)+{\eta }_1-{\beta }_1\end{array}$$ (24)

with η₁ − β₁ the exogenous estimation (19) of the disturbance. The closed loop dynamics results in

$$\begin{array}{c}\hfill \ddot{\tilde{\mathbf{X}}}=-K_{PX}\tilde{\mathbf{X}}-K_{DX}\dot{\tilde{\mathbf{X}}}+{\displaystyle \frac{{\tilde{\delta }}_1}{\overline{M}}}-{\displaystyle \frac{1}{\overline{M}}}\mathrm{\Theta }\end{array}$$

where K_PX and K_DX are positive definite gain matrices.

Due to the underactuated nature of the, translational dynamics, the desired angle θ_Qd is defined geometrically in Fig 6, where u_x is the component on the direction of the 0xⁱ axis of the control vector u, thus, one has

$$\begin{array}{c}\hfill {\theta }_{Qd}=\text{arcsin}\left(\frac{u_x}{\Vert u\Vert }\right)\end{array}$$ (25)

Then, the quadrotor attitude control input can then be defined as

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\overline{M}}_b^y& =& J_{yy}(-K_{PQ}({\theta }_Q-{\theta }_{Q_d})-K_{DQ}({\dot{\theta }}_Q-{\dot{\theta }}_{Q_d})+{\ddot{\theta }}_{Qd})-({\eta }_4-{\beta }_4)\hfill \end{array}\end{array}$$ (26)

where K_PQ and K_DQ are positive gains and η₄ − β₄ is the exogenous estimation (20) of the disturbance on the attitude dynamics.

Fig 6. The angle ${\theta }_{Q_d}$.

3.3 Robot arm controller based on the equivalent model

The controller design is completed with the following control inputs for the revolute and the prismatic joints, for the equivalent manipulator.

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\overline{\tau }}_R& =& 2m_{P}L\dot{L}{\dot{\theta }}_P+m_{P}Lg{s}_{{\theta }_P}\hfill \\ & & -m_P{L}^2(K_{PA}({\theta }_P-{\theta }_{P_d})+K_{DA}({\dot{\theta }}_P-{\dot{\theta }}_{P_d})-{\ddot{\theta }}_{Pd})\hfill \end{array}\end{array}$$ (27)

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\overline{f}}_P& =& m_{P}L{\dot{\theta }}_P^2+m_{P}g{c}_{{\theta }_P}\hfill \\ & & -m_P(K_{PP}(L-L_d)+K_{DP}(\dot{L}-{\dot{L}}_d)-{\ddot{L}}_d)\hfill \end{array}\end{array}$$ (28)

where K_PA, K_DA, K_PP and K_DP are positive gains, and ${\theta }_{P_d}$ and L_d the desired trajectories for θ_P and L, respectively.

3.4 UAM closed loop dynamics

To sum up everything, the controller and estimator proposed provide the following UAM closed loop dynamics

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\tilde{X}}}_1=& {\tilde{X}}_2\hfill \\ \hfill {\dot{\tilde{X}}}_2=& -K_{PX}{\tilde{X}}_1-K_{DX}{\tilde{X}}_2\hfill \\ & -\frac{u^{\top }{\mathbf{r}}_{{\theta }_Q}}{m\text{cos}\left({\tilde{\theta }}_{Q1}\right)}\left\{\text{cos}\left({\tilde{\theta }}_{Q1}\right)\right[\begin{array}{c}\text{cos}({\tilde{\theta }}_{Q1}+{\theta }_{Q_d})\\ \text{sin}({\tilde{\theta }}_{Q1}+{\theta }_{Q_d})\end{array}]\hfill \\ & -\left[\begin{array}{c}\text{cos}({\theta }_{Q_d}\\ -\text{sin}\left({\theta }_{Q_d}\right)\end{array}\right]\}+\frac{1}{m}{\mu }_1\chi \hfill \\ \hfill {\dot{\tilde{\theta }}}_{Q1}=& {\tilde{\theta }}_{Q2}-\frac{u^{\top }S[(K_{DX}-{\mathrm{\Gamma }}_1){\mu }_1+(K_{DX}-I){\mu }_2]\chi }{u^{\top }u}\hfill \\ \hfill {\dot{\tilde{\theta }}}_{Q2}=& -K_{PQ}{\tilde{\theta }}_{Q1}-K_{DQ}{\tilde{\theta }}_{Q2}-F(u,\chi ,{\tilde{\delta }}_4)\hfill \\ \hfill {\dot{\tilde{\theta }}}_{P1}=& {\tilde{\theta }}_{P2}\hfill \\ \hfill {\dot{\tilde{\theta }}}_{P2}=& -K_{PA}{\tilde{\theta }}_{P1}-K_{DA}{\tilde{\theta }}_{P2}\hfill \\ \hfill {\dot{\tilde{L}}}_1=& {\tilde{L}}_2\hfill \\ \hfill {\dot{\tilde{L}}}_2=& -K_{PP}{\tilde{L}}_1-K_{DP}{\tilde{L}}_2\hfill \\ \hfill \dot{\chi }=& A_{\chi }\chi +{\mu }_3^{\top }{\delta }_T^{\left(3\right)}\hfill \\ \hfill {\dot{\tilde{\delta }}}_4=& -{\mathrm{\Gamma }}_4{\tilde{\delta }}_4+{\dot{\delta }}_R\hfill \end{array}\end{array}$$ (29)

where ${\tilde{X}}_1=\tilde{\mathbf{X}}$, ${\tilde{X}}_2=\dot{\tilde{\mathbf{X}}}$, ${\theta }_{Q1}={\theta }_Q-{\theta }_{Q_d}$, ${\theta }_{Q2}={\dot{\theta }}_Q-{\dot{\theta }}_{Q_d}$, ${\theta }_{P1}={\tilde{\theta }}_P$, ${\theta }_{P2}={\dot{\tilde{\theta }}}_P$, ${\tilde{L}}_1=\tilde{L}$, ${\tilde{L}}_2=\dot{\tilde{L}}$, ${\chi }_1={\tilde{\delta }}_1$, ${\chi }_2={\dot{\tilde{\delta }}}_1$, ${\chi }_3={\ddot{\tilde{\delta }}}_1$, and

$$A_{\chi }=\left[\begin{array}{cccc}-{\mathrm{\Gamma }}_1& I& 0& 0\\ -{\mathrm{\Gamma }}_2& -{\mathrm{\Gamma }}_1& I& 0\\ -{\mathrm{\Gamma }}_3& -{\mathrm{\Gamma }}_2& -{\mathrm{\Gamma }}_1& 0\\ 0& 0& 0& -{\mathrm{\Gamma }}_4\end{array}\right],$$

moreover,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F(u,\chi ,{\tilde{\delta }}_4)& =\frac{1}{{\left(u^{\top }u\right)}^2}\left({\Vert u\Vert }^2{u}^{\top }S\right[({\overline{K}}_D+\mathrm{\Gamma }2){\mu }_1+\hfill \\ & (I+{\mathrm{\Gamma }}_1){\mu }_2+2{\mu }_3]\chi -\hfill \\ & \left\{2u^{\top }\right[(K_D-{\mathrm{\Gamma }}_1){\mu }_1+(K_D-I){\mu }_2]\chi \hfill \\ & u^{\top }S[(K_D-{\mathrm{\Gamma }}_1){\mu }_1+(K_D-I){\mu }_2]\chi \left\}\right)-{\tilde{\delta }}_4\hfill \end{array}\end{array}$$

where

$$\begin{array}{ccc}\hfill {\mu }_1& =& \left[I_{3\times 3}{0}_{3\times 3}{0}_{3\times 3}\right]\hfill \\ \hfill {\mu }_2& =& \left[0_{3\times 3}{I}_{3\times 3}{0}_{3\times 3}\right]\hfill \\ \hfill {\mu }_3& =& \left[0_{3\times 3}{0}_{3\times 3}{I}_{3\times 3}\right]\hfill \end{array}$$

with I_3×3 and 0_3×3, the identity and zero matrix in ${\mathbb{R}}^{3\times 3}$, respectively.

Function $F(u,\chi ,{\tilde{\delta }}_4)$ accounts 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 0xⁱ axis, this is, through the ${\theta }_{Q_d}$ computed as

$$\begin{array}{c}\hfill {\dot{\theta }}_{Qd}=\frac{u^{\top }S\dot{u}}{{\Vert u\Vert }^2}\end{array}$$ (30)

$$\begin{array}{c}\hfill \begin{array}{cc}& {\ddot{\theta }}_{Qd}=\frac{\left(u^{\top }S\ddot{u}\right)u^{\top }u-2\left(u^{\top }\dot{u}\right)u^{\top }S\dot{u}}{{\left(u^{\top }u\right)}^2}\hfill \end{array}\end{array}$$ (31)

with

$$\begin{array}{c}\hfill \dot{u}=\overline{M}(K_P{\tilde{X}}_2+K_D{\tilde{X}}_3)+{\eta }_2-{\beta }_2+K_D{\tilde{\delta }}_1-{\dot{\tilde{\delta }}}_1\end{array}$$ (32)

$$\begin{array}{c}\hfill \ddot{u}=\overline{M}(-{\overline{K}}_P{\tilde{X}}_2-{\overline{K}}_D{\tilde{X}}_3)+{\eta }_3-{\beta }_3+{\overline{K}}_D{\tilde{\delta }}_1+{\dot{\tilde{\delta }}}_1+{\ddot{\tilde{\delta }}}_1+{\tilde{\delta }}_3\end{array}$$ (33)

where ${\tilde{X}}_3=-K_P{\tilde{X}}_1-K_D{\tilde{X}}_2$, ${\overline{K}}_P=K_D{K}_P$ and ${\overline{K}}_D=K_P+K_D^2$.

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 ${\delta }_T^{\left(i\right)}\left(t\right)$, i = 1, 2, 3 and δ_R(t) and ${\dot{\delta }}_R\left(t\right)$ are in ${\mathcal{L}}_{\infty }$, t ≥ 0.

Let us define $\tilde{x}≔\text{col}({\tilde{X}}_1,{\tilde{X}}_2)$, ${\tilde{\theta }}_Q≔\text{col}\left({\tilde{\theta }}_{Q1},{\tilde{\theta }}_{Q2}\right)$, ${\chi }_a≔\text{col}(\chi ,{\tilde{\delta }}_4)$ and Δ_δ ≔ col(${\mu }_3^{\top }{\delta }_T^{\left(3\right)},{\dot{\delta }}_M$). Thus, from (29), the $\tilde{x}$, ${\tilde{\theta }}_Q$ and χ_a dynamics in compact form become

$$\begin{array}{ccc}\hfill \dot{\tilde{x}}& =& A_x\tilde{x}+{\mathrm{\Psi }}_x\left({\tilde{\theta }}_Q\right)·\frac{u^{\top }{\mathbf{r}}_{{\theta }_Q}}{m}+D_x{\chi }_a\hfill \\ \hfill {\dot{\tilde{\theta }}}_Q& =& A_Q{\tilde{\theta }}_Q+{\mathrm{\Psi }}_Q\left(u\right)·{\chi }_a\hfill \\ \hfill {\dot{\chi }}_a& =& A_{{\chi }_a}{\chi }_a+{\mathrm{\Delta }}_{\delta }\hfill \end{array}$$ (34)

where Ψ_(⋅) 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

$$\begin{array}{c}\hfill A_x≔\left[\begin{array}{cc}0& I_2\\ -K_P& -K_D\end{array}\right];A_Q≔\left[\begin{array}{cc}0& I_2\\ -K_{PR}& -K_{DR}\end{array}\right];\end{array}$$

where K_P, K_D, K_PR and K_PD are positive definite matrices and let $A_{{\chi }_a}≔\text{Blockdiag}(A_{\chi },-{\Gamma }_4)$. Thus, the main stability result is stated in the following proposition.

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 ${\tilde{L}}_i$ and ${\tilde{\theta }}_{Pi}$ , 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 $\mathbf{e}≔\text{col}\left(\tilde{x},{\tilde{\theta }}_Q,{\chi }_a\right)$ such that (34) can be rewritten in matrix form as

$$\begin{array}{c}\hfill \dot{\mathbf{e}}=\mathcal{A}\mathbf{e}+\left[\begin{array}{c}0\\ 0\\ {\mathrm{\Delta }}_{\delta }\end{array}\right],\mathcal{A}≔\left[\begin{array}{ccc}{A}_x& {\mathrm{\Delta }}_x\frac{u^{\top }{\mathbf{r}}_{{\theta }_Q}}{m}& D_x\\ 0& A_Q& {\mathrm{\Psi }}_Q\\ 0& 0& A_{{\chi }_a}\end{array}\right],\end{array}$$ (35)

where we have omitted all the arguments for compactness. Notice that, by design Ψ_x(0) = 0 and hence the whole term can be factored by Δ_x as in (35). The results follow noting that the block-triangular matrix $\mathcal{A}$ is 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 Rodríguez 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

$$\begin{array}{ccc}\hfill \mathrm{\Sigma }& :& \dot{\chi }=f\left(\chi \right)+g\left(\chi \right)\nu \hfill \end{array}$$ (36)

$$\begin{array}{ccc}\hfill \Pi & :& \dot{\tilde{\chi }}=\tilde{f}\left(\tilde{\chi }\right)+\tilde{g}\left(\tilde{\chi }\right)\tilde{\nu },\hfill \end{array}$$ (37)

with $\chi ,\tilde{\chi }\in {\mathbb{R}}^{{\kappa }_1}$ , $\nu ,\tilde{\nu }\in {\mathbb{R}}^{{\kappa }_2}$ are equivalent if there exist:

• i. A diffeomorphism

  $$\begin{array}{c}\hfill \tilde{\chi }=\mathrm{\Phi }\left(\chi \right)\end{array}$$ (38)
• ii. A static state feedback

  $$\begin{array}{c}\hfill \nu ={\alpha }_u\left(\chi \right)+{\beta }_u\left(\chi \right)\tilde{\nu },\end{array}$$ (39)

  with β_u(χ) a nonsingular square matrix, such that the transformation of Σ under (Φ, α_u, β_u) is equal to Π.

Fig 7 shows the link variables γ₁ and γ₂ of the R1, R2 manipulator. Hence, for the dynamic systems addressed in this work, Definition 1 can be applied considering that the system Σ corresponds to the two revolute joints manipulator; thus, $\chi ={\left[{\gamma }_1{\gamma }_2{\dot{\gamma }}_1{\dot{\gamma }}_2\right]}^{\top }$, ν = [τ₁ τ₂]^⊤. At the same time, Π is the revolute-prismatic joints manipulator system; this is $\tilde{\chi }={\left[{\theta }_{P}L{\dot{\theta }}_P\dot{L}\right]}^{\top }$, $\overline{\nu }={\left[{\overline{\tau }}_R{\overline{f}}_P\right]}^{\top }$.

Fig 7. Angles γ₁ and γ₂ on green.

It is possible to verify that the diffeomorphism (38) and the static state feedback (39) can be defined as follows

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\theta }_P\\ L\\ {\dot{\theta }}_P\\ \dot{L}\end{array}\right]=\mathrm{\Phi }({\gamma }_1,{\gamma }_2,{\dot{\gamma }}_1,{\dot{\gamma }}_2)\end{array}$$ (40)

$$\left[\begin{array}{c}{\tau }_1\\ {\tau }_2\end{array}\right]={\beta }_u({\gamma }_1,{\gamma }_2)\left[\begin{array}{c}{\overline{\tau }}_R\\ {\overline{f}}_P\end{array}\right]$$

where

$$\begin{array}{c}\hfill \mathrm{\Phi }({\gamma }_1,{\gamma }_2,{\dot{\gamma }}_1,{\dot{\gamma }}_2)=\left[\begin{array}{c}\frac{1}{2}\sqrt{d_l}\\ \frac{\pi }{2}+{\gamma }_1+\text{arcsin}\left(\frac{l_2\text{sin}\left({\gamma }_2\right)}{\sqrt{d_l}}\right)\\ -\frac{l_1{l}_2{s}_{{\gamma }_2}\dot{{\gamma }_2}}{\sqrt{d_l}}\\ {\dot{\gamma }}_1-\frac{{\dot{\gamma }}_2{l}_2(2c_{{\gamma }_2}^2{l}_1{l}_2-d_l{c}_{{\gamma }_2}-2l_1{l}_2)}{d_l\sqrt{c_{{\gamma }_2}^2{l}_2^2-l_2^2+dl}}\end{array}\right]\end{array}$$ (41)

$${\beta }_u({\gamma }_1,{\gamma }_2)=\left[\begin{array}{cc}1& 0\\ \frac{l_2}{2L}\text{sin}\left(\frac{\pi }{2}+{\gamma }_2-\alpha \right)& \frac{l_2}{2}\text{sin}(\alpha -{\gamma }_2)\end{array}\right]$$ (42)

with

$$\alpha =\text{arcsin}\left(\frac{l_2\text{sin}\left({\gamma }_2\right)}{\sqrt{d_l}}\right)\text{and}{d}_l=4l_1^2+l_2^2+4l_1{l}_2{c}_{{\gamma }_2}.$$

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

$$\left[\begin{array}{c}{\theta }_{Pd}\\ L_d\\ {\dot{\theta }}_{Pd}\\ {\dot{L}}_d\end{array}\right]=\mathrm{\Phi }({\gamma }_{1d},{\gamma }_{2d},{\dot{\gamma }}_{1d},{\dot{\gamma }}_{2d}),$$

The physical paramethers are sumarized on Table 2, while the gains values are on Table 3.

Table 2. UAM physical parameters.

Parameter	Value
M	1 Kg
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 ⋅ m2
UAV inertia	0.048 Kg ⋅ m2

Table 3. UAM controller gains.

Gain	Value
Γ1	5.5
Γ2	5.5
Γ3	10.5
Γ4	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

The diagram in Fig 8 illustrates how the simulations were implemented for the manipulator.

Fig 8. Robot arm simulation implementation.

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.

Fig 9. Wind disturbance profile defined by wind magnitude and direction.

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

Figs 10 and 11 depict the time evolution of the UAV translational axis errors, $\tilde{x}$ and $\tilde{z}$, respectively, while Fig 12 depicts the UAV attitude error, ${\tilde{\theta }}_Q$.

Fig 10. UAV translational x axis error.

Fig 11. UAV translational z axis error.

Fig 12. UAV attitude error.

Figs 13 and 14 show the end effector position error ${\tilde{x}}_{EE}=x_{EE}-x_{EEd}$ and ${\tilde{z}}_{EE}=z_{EE}-z_{EEd}$, respectively. As can be observed, all error signals converge to a zero neighborhood, as the theoretical analysis predicted.

Fig 13. End effector x axis position error.

Fig 14. End effector z axis position error.

The equivalent robotic manipulator errors $\tilde{L}$ and ${\tilde{\theta }}_P$ are shown in Figs 15 and 16, while the original robotic manipulator errors ${\tilde{\gamma }}_1={\gamma }_1-{\gamma }_{1d}$ and ${\tilde{\gamma }}_2={\gamma }_2-{\gamma }_{2d}$ are 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.

Fig 15. Equivalent manipulator prismatic joint error, $\tilde{L}$.

Fig 16. Equivalent manipulator revolute joint error, ${\tilde{\theta }}_P$.

Fig 17. Two revolute joints manipulator joint error, ${\tilde{\gamma }}_1$.

Fig 18. Two revolute joints manipulator joint error, ${\tilde{\gamma }}_2$.

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 performance in all simulations better,

$$\begin{array}{ll}{F}_1\hfill & ={\displaystyle {\int }_0^t\parallel \tilde{X}}\left(\tau \right)\parallel d\tau \hfill \\ F_2\hfill & ={\displaystyle {\int }_0^t{\tilde{\theta }}_Q{\left(\tau \right)}^{2}d\tau }\hfill \\ F_3\hfill & ={\displaystyle {\int }_0^t\left(\widetilde{{\gamma }_1}{\left(\tau \right)}^2+\widetilde{{\gamma }_2}{\left(\tau \right)}^2\right)}d\tau \hfill \\ F_4\hfill & ={\displaystyle {\int }_0^t\left({\tilde{x}}_{EE}{\left(\tau \right)}^2+{\tilde{z}}_{EE}{\left(\tau \right)}^2\right)}d\tau \hfill \\ F_5\hfill & ={\displaystyle {\int }_0^t\left(\tilde{L}{\left(\tau \right)}^2+{\tilde{\theta }}_P{\left(\tau \right)}^2\right)}d\tau \hfill \end{array}$$ (43)

Fig 19. UAV total trust, T_T and moment, ${\overline{M}}_b^y$ input controls.

Fig 20. Two revolute joints manipulator control inputs, τ₁ and τ₂.

Table 5 presents the values for each measurement F_i, i = 1, ⋯, 5 correspondent to each simulation.

Table 5. Simulations with random wind disturbances, measuring function F_i, i = 1, 2, 3, 4, 5.

Simulation	F 1	F 2	F 3	F 4	F 5
1	4.42	6.19	6.29	6.03	2.40
2	3.78	7.96	6.11	5.76	2.48
3	3.76	7.73	5.90	5.40	2.27
4	3.67	7.85	5.87	5.37	2.29
5	4.35	5.33	6.48	5.77	2.21
Average	4.0011	7.0153	6.1364	5.6683	2.3364
Standard deviation	0.3597	1.1819	0.2591	0.2794	0.1091

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 21. Disturbances estimated by the proposed estimation strategy.

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 22. Simulation sequence.

5 Conclusions

This work proposed a control algorithm for an Unmanned Aerial Manipulator. Analyzing the effects on a flying platform generated by a two-revolute manipulator was simplified using an equivalent revolute-prismatic joints manipulator. This approach permitted compensating for the known dynamics and decoupling the UAV dynamics from the remaining robotic manipulator dynamics. Thus, the remaining manipulator dynamics were treated as external forces and moments acting on the quadrotor. The resulting dynamical UAM structure permits designing a disturbance estimator based on the Immersion and Invariance technique. Then, a PD-like controller with disturbance compensation is proposed to solve the trajectory tracking problem for the UAM. A formal stability analysis of the resulting closed-loop dynamics is presented.

Numerical simulations in a realistic simulator are presented to evaluate the proposed control strategy. The realistic simulator considers wind profiles acting on the UAM. For future work, this work has established a solid base for an extension of the results to find the equivalence between n-degrees of freedom revolute joints manipulator and an R-P type manipulator, this approach would simplify the disturbances analysis on more general UAM configurations.

6 Appendices

A Recursive Newton-Euler Algorithm

In this work the RNEA reported in [12] is followed. The angular velocity propagation is computed from the following equations. For the revolute joint

$$\begin{array}{c}\hfill {\mathrm{\Omega }}^{i+1}={(}_i^{i+1}R){\mathrm{\Omega }}^i+{\dot{\theta }}_{i+1}{\widehat{z}}^{i+1}\end{array}$$ (44)

and for the prismatic joint

$$\begin{array}{c}\hfill {\mathrm{\Omega }}^{i+1}={(}_i^{i+1}R){\mathrm{\Omega }}^i\end{array}$$ (45)

The angular and translational accelerations are propagated as follows. For a revolute joint

$$\begin{array}{c}\hfill {\dot{\mathrm{\Omega }}}^{i+1}={(}_i^{i+1}R){\mathrm{\Omega }}^i+{(}_i^{i+1}R){\mathrm{\Omega }}^i\times {\dot{\theta }}_{i+1}{\widehat{z}}^{i+1}+{\ddot{\theta }}_{i+1}{\widehat{z}}^{i+1}\end{array}$$ (46)

and

$$\begin{array}{c}\hfill {\dot{V}}^{i+1}={(}_i^{i+1}R)[{\dot{\mathrm{\Omega }}}^i\times P_{i+1}^i+{\mathrm{\Omega }}^i\times ({\mathrm{\Omega }}^i\times P_{i+1}^i)+{\dot{V}}^i]\end{array}$$ (47)

For a prismatic joint

$$\begin{array}{c}\hfill {\dot{\mathrm{\Omega }}}^{i+1}={(}_i^{i+1}R){\mathrm{\Omega }}^i\end{array}$$ (48)

and

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{V}}^{i+1}& =& {(}_i^{i+1}R)[{\dot{\mathrm{\Omega }}}^i\times P_{i+1}^i+{\mathrm{\Omega }}^i\times ({\mathrm{\Omega }}^i\times P_{i+1}^i)+{\dot{V}}^i]\hfill \\ & & +2{\mathrm{\Omega }}^{i+1}\times {\dot{d}}_{i+1}{\widehat{z}}^{i+1}+{\ddot{d}}_{i+1}{\widehat{z}}^{i+1}\hfill \end{array}\end{array}$$ (49)

Finally, the link center of mass acceleration is computed as

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\dot{V}}_C^{i+1}& =& {\dot{\mathrm{\Omega }}}^{i+1}\times P_C^{i+1}+{\mathrm{\Omega }}^{i+1}\times ({\mathrm{\Omega }}^{i+1}\times P_C^{i+1})+{\dot{V}}^{i+1}\hfill \end{array}\end{array}$$ (50)

and the forces and moments acting at the center of gravity are

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill F^{i+1}& =& m_{i+1}{\dot{V}}_C^{i+1}\hfill \\ \hfill N^{i+1}& =& J_{i+1}{\dot{\mathrm{\Omega }}}^{i+1}+{\mathrm{\Omega }}^{i+1}\times J_{i+1}{\mathrm{\Omega }}^{i+1}\hfill \end{array}\end{array}$$ (51)

This completes the outward RNEA iteration.

The inward RNEA starts computing the forces acting on each link as

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill f^i& =& F^i+{(}_{i+1}^{i}R)f^{i+1}\hfill \\ \hfill n^i& =& N^i+{(}_{i+1}^{i}R)n^{i+1}+P_C^i\times F^i+P_{i+1}^i\times {(}_{i+1}^{i}R)f^{i+1}\hfill \end{array}\end{array}$$ (52)

Data Availability

All relevant data are within the manuscript.

Funding Statement

Y.E. Tlatelpa-Osorio 702178 CONSEJO NACIONAL DE CIENCIA Y TECNOLOGÍA https://conahcyt.mx/.