# Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems
## 1 Introduction
As highlighted in [29], control design for Cosserat rod theory is really challenging and only a few papers [33; 34; 35] have been published to address the control design of the continuum arm using this method. Writing continuum robots formulations to facilitate the control design process is one of the main motivations of introducing a constant deformation method [29]. Recently, some control approaches utilized this method to control continuum arms [36; 37; 38]. However, we used this method earlier to model the continuum arm and noticed that it is even computationally more expensive than Cosserat rod theory. Furthermore, similar to the Cosserat rod theory, it is presented for arm modeling. Therefore, it can be selected for decoupled ACMS modeling design not for coupled approach. Finally, the center of gravity approach [29] has the same assumption as our method. However, presented formulations for this method are too complicated, resulting in difficulty of adding the effects of the arm on UAV and vice versa. Therefore, it can be concluded that in the first three continuum arm modeling methods, execution time, hardware implementation, real time 3D control and coupling with UAV are problematic. The only remaining method that is also suitable for coupled modeling is the Lagrangian method. However, existing Lagrangian techniques [30; 31; 32] did not present a general formulation and mostly used Taylor expansion and many curve fittings to simplify equations. Therefore, an appropriate Lagrangian method is needed to address these CR modelling problems in the first step. Additionally, the coupling effect within an aerial manipulation platform should be considered to derive the whole dynamic model of ACMSs. Our previous work [20] focused on the decoupled modelling and control of ACMSs with quadcopters as the base platform. Within this research, we concentrated on the decoupled modelling approach, which did not consider all interactions between the arm and its base platform. To solve this limitation and to provide a more accurate and general dynamic model, here we consider coupled modelling of ACMS with a multi-rotor-based platform. Another limitation of the previous work relates to the type of UAV used in modeling. Commonly used UAVs, in general, and quadrotors, in particular, are non-holonomic and underactuated. These multi rotors with parallel thrusts cannot independently control UAV's position and orientation (pose) simultaneously [39]. Moreover, they cannot be used for dexterous application with arbitrary generalized forces [40]. Therefore, using an under-actuation platform may degrade the performance of the ACMS missions. To tackle this issue, one strategy is to exploit tilt rotors to achieve non-parallel thrust vectors besides having at least six actuators. A fully actuated UAV has capacity to instantaneously resist arbitrary forces and torques [39] besides the ability to hover at arbitrary orientations [41]. For aerial manipulation tasks, resisting any arbitrary wrench with fast response of exerting forces is a significant factor. Since variable tilting propellers may not be fast enough to deal with external wrenches [39], in this work, fixed-angle propellers are selected for aerial continuum manipulation. While most of previous works on dexterous aerial platforms adopted six propellers, we formulated a general model for a dexterous multirotor platform that is independent of the number and angles of propellers. In this work, the modeling approach is complemented with a control technique to demonstrate the effectiveness of the proposed modeling technique for not only dynamic simulations but also control purposes. It is a commonly held view that AMS models, even for rigid robotic arms, are highly nonlinear, coupled and complex. Additional complication comes from the integration of continuum arm models that are costly for real-time control. As highlighted in Table 1, control of CRs alone is more difficult than rigid arms and CR's modeling and control are currently defined as a pending challenge in the robotics community.