# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 1 Introduction  
Aerial manipulations systems (AMSs) have drawn significant attention due to their numerous applications ranging from inspection and maintenance to structure assembly [1, 2]. In particular, aerial manipulation brings several advantages over ground-based manipulation such as mobility, access and reachability [3]. The platforms of AMSs are naturally rotor-type [2]. For instance, both quadrotors [4, 5, 6] and hexrotors [7] are popular choices for such unmanned aerial vehicles (UAVs) in the literature. The used arms in AMSs are typically rigid-link manipulators with limited degrees-of-freedom (DOFs) [8, 9, 10, 11, 12, 13, 14]. In the presence of established methods, the dynamics of AMSs with rigid manipulators (AMS-RMS) have already been analyzed [8, 9, 10] and a number of control methods ([7, 8, 15, 16, 17, 18, 19]) have been proposed. However, AMS-RMS have several issues, highlighted in Table 1. For instance, AMS-RMS are not compliant with their environment or shape of the objects, hence requiring exact information about the object shape and position, environmental constraints, and so on [20]. Therefore, AMS-RMS are often operated at a reduced speed to ensure successful grasps [21]. Accordingly, preference is given to manipulators with few DOFs, limiting their capability to compensate for disturbances caused by the vehicle positioning errors. Moreover, external contact forces are avoided as much as possible in order to maintain stability [21]. Such conventional AMSs can hardly be utilized for interactive missions in confined and unstructured environments [22]. As a remedy for the aforementioned issues, AMS with continuum robotic (CR) arms, namely aerial continuum manipulation system (ACMS), [20] (Figure 1) and soft robots [21] have been introduced in the literature. The advantages of CRs and ACMSs have been highlighted in Table 1 and therecent report [20]. For instance, mechanical compliance of ACMS can boost safety when manipulators physically interact with the environment [23; 24; 25]. Despite the preliminary analysis of ACMSs in [20], there are still many open problems related to their modeling. For example, a previously reported modeling approach [20] for the continuum arm and UAV platform was based on the decoupled dynamics assumption, which decreases the accuracy of the model and introduces certain challenges to the whole ACMS control design. In addition, some simplifications become unavoidable within a decoupled approach when considering the effects of the arm on UAV and vice versa. The focus of this work will be placed on coupled modeling of ACMS dynamics. While coupled modeling of AMS-RMs can be found in the literature ([16; 26]), coupled dynamic modeling of ACMS and its verification remain to be done and constitute the main contribution of this paper. Generally, there are four methods for dynamic modeling of continuum arms in the literature including those relying on models of Cosserat rod [27], constant deformation [28], center of gravity [29] and Lagrangian [30; 31; 32]. Our previous published paper on decoupled modeling and control of ACMS [20] was based on Cosserat rod theory. Although this method can be considered as an accurate modeling approach for continuum robots, it is computationally expensive because it needs an iterative solver to satisfy the robot's boundary conditions at each iteration of the program. Therefore, its hardware and software implementation for real time purposes is really problematic. Furthermore, previously proposed Cosserat rod-based controllers in the literature [33; 34; 35] can only control the robot in x and/or y directions using two tendons per each direction. Thus, the previous method could not handle 3D motion control of the arm. In addition, the previous Cosserat rod-based work can only model the arm. Therefore, Cosserat rod theory cannot be chosen for ACMS coupled dynamic modeling.