I Introduction
Unmanned aerial vehicles are able to greatly extend the workspace of a robotic arm [2021BodTogSie], which opens up new possibilities in aerial object manipulation and transportation [2021gOllTogSuaLeeFra]. However, the major shortcoming of rigid robot arms is their limited capability to interact with humans. The lack of compliance and smooth motions forces these robots to operate in wellstructured and fencedoff environments. In regards to the floating base, common quadcopters and hexacopters with fixed propellers are underactuated due to the orientation of their thrusters in the same direction, which makes the modeling and control more complex when using them as the flying base for such aerial manipulators. We propose the combination of a soft robotic arm mounted on a fully actuated Omnidirectional Micro Aerial Vehicle (OMAV) [allenspach_OMAV] (see Figure 1) to tackle these challenges and to counteract the deficiencies of current aerial manipulators.
Robotics researchers have tried in recent years to improve physical humanrobot interactions for rigid robots by introducing compliance to their structure, either directly in the joints [hwisu_compliance_joint], in the links [dissertation_compliant_links], or in the control software [schumacher2019_compliant_control]. Analogously, compliant joints were proposed for robots mimicking animals [hutter_starleth_compliance_actuator]. Controlling these advanced rigidbodied robots in an optimal and robust fashion opens up new challenges and patterns [hutter_starleth_compliance_control].
The aforementioned issues and considerations on the lack of inherent compliance gave rise to the emergence of soft robotics. Soft robotics concentrates on building robots that mimic nature as close as possible. As opposed to “classical” robotics, soft robotics shifts paradigms in the areas of material selection, actuator design, fabrication, and construction.
A direct counterpart to the conventional rigid robot arms is a controllable soft continuum arm [katzschmann2015softarm2D]. A follow up work [katzschmann2020softarm2Dcontrol] extended the notions to modelbased feedback control both in parameterspace and Cartesian operationalspace, enhancing the capabilities of such soft arms. The soft continuum arms were recently elevated from 2D to 3D using either a modelbased feedback control approach based on an Augmented Rigid Body Model (ARBM) [katzschmann2019softarm3Dcontrol], or a reducedorder finite element method model based on proper orthogonal decomposition and a state observer [katzschmann2019softarm3DFEM].
Similar to classical rigidbodied systems, the workspace of the soft robotic arm can be extended by mounting it to a flying robot. Instead of choosing a traditional drone, we liberate the soft arm from its fixed base by mounting it to an OMAV for increased dexterity and maneuverability as seen in Figure 1. An OMAV, as opposed to commercially available quadcopters and hexacopters with fixed propeller orientation toward the same direction, is a fully actuated flying system and is capable of exerting forces and torques in any arbitrary direction [2021fHamUsaSabStaTogFra]. This feature not only allows the soft continuum arm to move from a confined workspace to the free space – theoretically reaching any point with any arbitrary orientation in 3Dspace – but also makes the control more robust and agile. The OMAV contributes to the agility and flexibility of such a flying platform and the exchange of the rigid robot arm with a soft arm adds the desired compliance and inherent safety that cannot be achieved otherwise. This platform opens a still unexplored solution in the current scenario of aerial manipulation.
In this paper, we derive a mathematical model from the PCC and ARBM approach and unify it with a floating base robot. Moreover, we propose a hierarchical taskprioritizing controller architecture that enables the user to intuitively define highlevel tasks in the operational space. The proposed architecture is able to regulate the system to a fixed point, track a trajectory with a given orientation, and exploit the nullspace of the highpriority tasks to execute additional noninterfering background motions.
Consequently, this work contributes with:

A mathematical parametrisation and model of the unified flying platform consisting of the OMAV and the soft robotic arm;

A hierarchical taskprioritising controller capable of tracking trajectories while exploiting the nullspace of the higher priority tasks;

A validation of the hierarchical controller in various simulations, such as nullspace motion, disturbance recovery and trajectory tracking.
Ii Background
We give an overview of the relevant modeling approaches for soft continuum manipulators with focus on the PCC and ARBM hypotheses and the stateoftheart control methods. We then introduce the aerial robotic research field to motivate our choice in using an OMAV.
Iia Soft continuum modeling
Since soft actuators show high flexibility and compliance, they virtually possess infinite degrees of freedom (DOF), which is challenging from the modeling perspective. Researchers have proposed various techniques for dimensional reduction to render the challenge feasible, for example the
RitzGalerkin models for continuum manipulators [sadati2018control] and the Cosserat approach for soft robots [renda2017discrete].In this paper we use the PCC and ARBM hypotheses. The PCC model approximates the segments of a soft robotic arm with constant bending curvature and treats it as one piece. An example is presented in Figure 2 for a fourway segmented soft arm. While PCC is merely a kinematic approximation, previous works [katzschmann2015softarm2D, katzschmann2019softarm3Dcontrol, katzschmann2020softarm2Dcontrol] have confirmed the validity and efficacy of this approach in closedloop controlled realworld scenarios. The advantage of the PCC and ARBM approach is that modeling approaches and control architectures from the rigid body literature can be directly applied to continuum arms with little effort. For the sake of brevity, we do not include here the fundamentals of PCC and ARBM, but we refer the interested reader to [katzschmann2020softarm2Dcontrol, katzschmann2019softarm3Dcontrol] for a more detailed derivation of this modeling technique.
IiB Omnidirectional micro aerial vehicle (OMAV)
OMAV aims to eliminate the underactuated systemcharacteristics thus empowering these flying vehicles with full motion range, six DOF and decoupled controllability between the translational and rotational dynamics. To obtain such property, different designs have been presented in [dandrea_omav] and [ryll_omav]. In this paper we consider a general tiltrotor OMAV construction optimized for flight efficiency and large payload capacity, which was introduced and modeled in [siegwart_OMAV].
To be able to exploit the abilities of such an agile machine, previous works developed suitable controllers focusing on maneuverability while maintaining flight efficiency at the same time. Decoupling the translation and rotational dynamics while still ensuring robust control made it possible to deploy and use the OMAV for more practical tasks, like contactbased inspections [siegwart_OMAV_force_control, 2019eTogTelGasSabBicMalLanSanRevCorFra].
The variable propeller tilting together with the aforementioned controller resulted in an increase in the system’s capabilities and maneuverability compared to the fixedpropeller quadrotor or hexarotor. For example, hovering in an arbitrary pose is possible with an OMAV but cannot be achieved with propellers having a fixed tilt angle and facing the same direction (as seen in the classical commercialgrade quad and hexacopters). Naturally, the modeling and controlling of an OMAV is significantly more challenging than that of a quad or hexarotor due to the increased capabilities and complexity.
IiC Unified flying platform with a manipulator
To the best of our knowledge, there is currently not any work that discusses the control of a flying manipulator consisting of an flying platform and a soft robotic arm manipulator. Most research in this field headed towards the direction of controlling a njoint rigid body manipulator arm mounted on a quadcopter, although most recent works started to explore rigidbody arms mounted on omnidirectional vehicles as well.
Even though we use a softrobot arm on the flying platform, our hierarchical control architecture is inspired by rigidarms, such as [kannan_control_uav_operational_space], where a 2joint rigid link arm manipulator was mounted on a drone. The proposed hierarchical control structure consisted of the outermost closedloop inverse kinematics algorithm layer and a position and attitude control loop inner layer. The results showed that the controller was stable and efficient with satisfactory performance. [caccavale_control_uav_robot_arm] demonstrated a similar control structure for a 5 DOF arm, further analysing and proving the stability mathematically. A more closely related work is [fishman2021softdrone], a quadrotor equipped with a soft tendondriven grasping mechanism attached to it. The control is conducted with an optimisationbased approach consisting of two submodules, separately optimizing for the quadrotor trajectory and the soft gripper movement.
Our aim is to combine and extend the previous approaches: we build upon the novel tiltrotor OMAV architecture and combine it with the soft continuum arm, we find a PCC/ARBMsupported unified parametrization and we design an analytic modelbased hierarchical feedback controller.
Iii Model
The model shall leverage all six DOF of the OMAV. Therefore, we propose a unified extended floatingbase parametrisation for the coupled system. This choice is analogous to the frequently used rigidbody counterpart, as described for example in [nakanishi2007floatingbase].
The soft continuum arm is modeled using the PCC and ARBM hypotheses. These hypotheses are based on the assumptions of nonextensible curvature segments and constant curvature along each one of these segments. The PCC approximation offers a suitable onetoone mapping between the kinematic properties of nonextensible and constant curvature segments and a realworld soft continuum arm segment. The ARBM approach the realworld arm dynamics by describing an augmented rigidrobot space with rigid links connecting translational and rotational joints [katzschmann2018softarm2Dcontrol]. Each segment has its own set of parameters for the bending angle and for the offplane rotation in the PCC space (see Figure 3). To match the kinematic and dynamic profile of the soft continuum arm, a suitable rigidbody augmentation is required. A tradeoff between computational costs resulting from model complexity and modeling accuracy puts limitations on the potential configuration of the chosen rigidbody robot acting dynamically equivalent to the soft arm.
In this paper, we propose a 7joint rigid robot configuration, which builds upon and extends the 5joint parametrisation presented in [yasu2021sopra]. Without and , the mass would have to lie on the virtual line connecting the start and end point of a PCC segment. Adding these two extra joints helps to extend the system’s capability by more accurately representing the shift of the center of mass of the soft arm during bending (see Figure 3).
In previous work [katzschmann2020softarm2Dcontrol, katzschmann2019softarm3Dcontrol], one PCC element was mapped to one actuated segment of the soft continuum arm. However, we decided to take a more recent approach [yasu2021sopra] and increase the fidelity of simulation by modeling the actuated segments each with PCC elements, summing up to a total of joints. The augmentation of one actuated segment is shown in Figure 4. This step increases the accuracy of the model at the cost of higher computational demands. Note that the variables , , and determine the dimensions of the augmented space.
The augmented soft continuum arm is extended with a floating base, i.e., the OMAV. The OMAV is modeled as a mass point with given mass and inertia properties. To represent rotations without a singularity, we use the quaternion elements
. The state of the system results in the following parametrisation vector:
(1) 
where represents the Cartesian coordinates of the OMAV’s centre of mass, while and are the first and second derivative of this quantity with respect to time. The vector denotes the augmentation of the constant curvature segments, as depicted in Figure 3.
The calligraphic prescripts – refers to the bodyfixed coordinate system attached to the center of mass of the OMAV and axis aligned with main axis of inertia, and is the inertial coordinate frame – denotes the coordinate system, in which the quantity is described. The double subscript in denotes the angular velocity of the coordinate frame with respect to the inertial frame and represents the angular acceleration. Note in (1) that the parametrisation vector , and the first and second derivative terms are from the vector space . This is due to the choice of a singularityfree rotation representation using quaternions with 4 elements.
Transforming the hybrid platform to the augmented rigidrobot space enables a fast and straightforward derivation of the Jacobian matrix for an arbitrary fixedpoint in the body coordinate system:
(2) 
where denotes a vector from the origin of the inertial frame to the origin of the floatingbase bodyfixed frame , is the local Jacobian written in the bodyfixed coordinate frame, and is the time derivative of the robot parametrisation vector without the floatingbase ( denotes the number of joints).
For the sake of brevity, the dependency of the terms on was omitted. From the last equation, the Jacobian is , where is the rotation matrix of the bodyframe with respect to the inertial frame and
denotes the crossproduct skew symmetric matrix created from the vector
. With the Jacobian, the inertia matrix , Coriolis and centrifugal vector , and the gravitational field vector can be derived in the augmented formulation using the equations (3.43), (3.44) and (3.45) from [robot_dynamics_lecture_notes_eth].Remark 1.
Instead of parameterizing the constant curvature segments with the offplane rotation and bending angles as shown in Figure 3, an alternative parameterization composed of taken from [yasu2021sopra] is applied:
(3) 
which avoids a singularity when the soft continuum arm is in its straight configuration. This parametrization is used later for the state vector .
The bridging between the augmented rigidbody space and the PCC space is described with the set of equations (11) from [katzschmann2019softarm3Dcontrol]:
(4) 
where is the coupled system’s parametrisation in the PCC space consisting of the parameters for the PCC segments, the 4 quaternion elements, and the 3 Cartesian coordinates of the floating base. maps the PCC parametrisation to the augmented space and is the Jacobian of , i.e., . Its upperleft
submatrix affecting the OMAV parametrisation from one space to the other is the identity matrix, since the mapping is onetoone.
Using the mapping above, the PCC and ARBM assumptions [katzschmann2019softarm3Dcontrol] define the systemmatrices as follows:
(5) 
where and are augmented space quantities and have dimensions as described above. Finally, the dynamics of the coupled system are given as (for the full derivation please refer to [katzschmann2018softarm2Dcontrol]):
(6) 
which is analogous to the standard rigidbody formulation. With , is the inertia matrix, is the vector containing the Coriolis and centrifugal terms, is the gravitational force equivalent. The stiffness of the soft continuum arm is considered in the stiffness matrix and the damping effects are expressed by the damping matrix . For the exact derivation and calculation of the stiffness and damping, please refer to [yasu2021sopra] section III. C. Both matrices have an upperleft block of zeros to account for the lack of stiffness or damping in the equations of motion of the OMAV. Note that the stiffness matrix is multiplied with a modified state vector . Since the first 6 elements of the vector are affected by the zero block of the stiffness matrix, the values in the modified state vector are irrelevant and thus set to zero, i.e., . This multiplication is necessary because the dimensions of the matrix multiplication would be inconsistent otherwise. is responsible for mapping external forces from the operational space to the joint space.
The OMAV rotor force generation and the soft robotic arm pressure actuation is on the right hand side of (6). The expanded expression for the modified allocation matrix is , where is the number of rotors. The squared rotor speeds multiplied with the instantaneous allocation matrix results in the body forces and torques, as described by (7) in [allenspach_OMAV]. The wrench is mapped to the augmented rigidbody space with the OMAV center of mass Jacobian . The last step is the transition from the augmented space to the PCC space, which is carried out using the space mapping Jacobian . The actuation inclusion of the soft continuum arm in the mathematical description follows analogously. is the conversion matrix between the chamber pressures and generalized forces, which acts on the pressure input and is derived in [yasu2021sopra]. Since the pressurisation of the soft arm chamber disregards the OMAV’s equation of motions, a selection matrix is required to transform the input vector to the correct dimensions. To keep the mathematical consistency of the derivation, is composed of two blocks: the upper region affecting the OMAV’s DOFs are zero, while the lower is the identity matrix.
Iv Control
This section introduces our proposed hierarchical control architecture designed for the coupled system. The controller is a prioritisationbased hierarchical controller and operates on the endeffector orientation, position, and OMAV orientation tasks. The OMAV position is not taken into account, since the only important position quantity is that of the endeffector. Nevertheless, OMAV orientation can help with avoiding singularities and inefficient configurations. The more important targets from the user’s perspective are assigned the highest priority and the ordering in priority is as follows:

Endeffector orientation: determined by the offset term that depends on the reference orientation and the controlled orientation , both represented as angleaxis.

Endeffector position: is the difference of reference Cartesian position and the current Cartesian position of the endeffector.

OMAV orientation: separately controlled thanks to the high number of DOFs and the hierarchical prioritisation approach. depends on the reference OMAV rotation and current orientation . Both rotations are represented in angleaxis.
The angular offset is calculated using the definitions from [robot_dynamics_lecture_notes_eth]. The current orientation (endeffector and OMAV frame) is rotated back to an arbitrary inertial frame , then rotated to the goal frame using the rotation equation and the orthonormality of rotation matrices:
(7) 
Consequently, the rotation matrix is converted to an angleaxis representation to describe an offset vector in .
The backbone of each task is an offsetdriven feedback term presented on the left half of Figure 5:
(8) 
For the sake of brevity, the matrix dependencies on the full system parametrization vector and its derivative were omitted. are the stiffness and damping tuning matrices for the individual tasks. are the endeffector rotational, translational, and OMAV rotational Jacobians in the PCC space, respectively.
To simultaneously ensure a task prioritisation and dynamic consistency, successive nullspace projection matrices are applied to the tasks, as described in [dietrich2015nullspace_projection]. A nullspace projector for a task with the Jacobian of a previous task (e.g., the endeffector orientation task given by the controller) is obtained by:
(9) 
where is the generalized inverse satisfying the criterion . To fulfill the criteria that lower priority tasks not interfere with higher priority tasks during the transient phase or the steady state, a dynamically consistent inverse weighted by the inertia matrix is adapted and denoted as:
(10) 
based on [dietrich2015nullspace_projection].
The torque calculated by the controller is directly fed into the plant. To avoid a steady state tracking error, a gravitational decoupling was added to the control scheme. For a more detailed description of the output allocation pipeline for the OMAV and for the soft continuum arm, please refer to [allenspach_OMAV] and [yasu2021sopra], respectively.
V Simulations
The simulation scenarios and tasks were designed to show some key capabilities of the coupled system, like continuous nullspace exploitation, disturbance rejection, or dynamic trajectory tracking. We believe that these simulations show the potential of a coupled soft robot arm and aerial drone system. In this section, we first describe the hardware and software setup together with the parameter values used for the simulations. Afterwards, simulation results are shown and discussed.
Va Simulation setup
The simulations were run on a laptop with a 4core CPU (Intel i76820HQ (4 x 2.7GHz) 6. Generation) and 16GB RAM memory. The operating system used was an Ubuntu 18.04 Bionic.
The code was written in C++ 17, with cmake version 3.20.0. The visualisation pipeline was built on ROS 1  Melodic Morenia [ros] and Gazebo
, an open source 3D robotics simulator (here only used for visualisation). The
Drake [drake] library runs the physics engine in the background for the augmented rigidbody robot system descriptions and simulations. The virtual control frequency was set to 100Hz (this sets an achievable target for the real platform as well) and the internal state update frequency of the system was 100kHz. Note that due to the offline nature of the simulation, no realtime performance was targeted and the code is therefore not necessarily performanceoptimised.Control task  

static  dynamic  
We use for the simulation a twosegment SoPrA soft continuum arm [yasu2021sopra] coupled to an OMAV with six dual propellers. The mass (including the batteries) of the OMAV is
and the moments of inertia are
. The length of each SoPrA arm segment is . The length of each connector piece between the segments is and is considered as an unactuated, rigid extension. The mass of the first segment is , the second segment is , and the connectors are , each. The arm’s diameter at the base is , between the first and second segment is , and at the tip is . The remaining system properties were calculated as stated in [yasu2021sopra].The control gain and damping matrices were finetuned empirically and the final values can be seen in Table I. The tuple of three numbers in the table is interpreted as . matrices have the dimension and are of . and are slightly different for the static regulation and dynamic trajectory tracking cases: since the accuracy in the regulation tasks is crucial, higher gains are proposed to ensure a fast, aggressive, and accurate enough control, whereas for motion tracking the safety (more compliance) and stability are the primary concerns leading to lower gains.
VB Results
The first experiment on Figure 6 demonstrates the exploitation of the motion’s nullspace. The OMAV was commanded to change its orientation around a given rotation axis by a certain amount of degrees in a continuous manner in the repeating cycle . Meanwhile, the position and orientation of the endeffector remained stable and close to the reference. After the transient behaviour, the endeffector orientation and position remain stable with negligible error to the reference, even though the OMAV was changing its orientation during the whole () simulation.
In another experiment, the recovery capability after a disturbance was tested. The simulation shown in Figure 7 shows that the system is able to recover fast from disturbances acting either on the OMAV or on the soft arm. After the disturbance is applied, the system is able to react fast and returns to the reference position and orientation without any overshooting. These disturbances are not modeled, thus active disturbance rejection is not possible and not desired when collaborating with humans where a compliant behavior is preferred over a stiff one.
The third simulation examined the dynamic trajectory tracking capabilities of the controller shown in Figure 8. While the system is capable of following the circular trajectory with a given orientation over time, there was a small lag introduced in the position tracking. The mean L2 norm of the position error is around . This is due to the fact that the control is governed by the offset terms introduced in Figure 5, which are arbitrary small at vanishing differences between the reference and control variable. It can be understood as follows: the controller needs a certain “minimal” difference between the reference and the controlled value to be “active” and effective enough. Note that we did not employ any model predictive control techniques.
Vi Conclusion and future work
In this paper we propose a unified mathematical description of a coupled flying system consisting of an OMAV and a soft continuum arm. We show how extending the floating base approach taken from the classical rigidbody robotics literature leads to an analogous formulation in the PCC space with the ARBM. Furthermore, we derive a hierarchical taskprioritisation control architecture tailored to the coupled system. The approach has been tested and evaluated in various simulation scenarios. The system’s architecture exhibits certain desirable characteristics, such as the exploration of nullspacemotion, disturbance recovery, and trajectory tracking with a given orientation of the endeffector and the OMAV.
We hope that with this work we can lay the foundation for future research in the area of soft continuum manipulators mounted to flying vehicles. We believe that their potential for industrial applications is tremendous in the evergrowing demand on semiautomated warehouses, where humans and robots would actively and efficiently collaborate to retrieve and disperse the goods. Future work will focus on conducting experiments on the realworld system, potentially with an additional gripper attached at the endeffector, to test some basic transport capabilities. Another focus will be placed on micro oscillation effects observed during control simulations using high gains. These effects could be addressed using either curvature space or hybrid control, where the gains for the OMAV and SoPrA can be adjusted independently.
Acknowledgment
We are grateful for Ing. Róbert Szász Sr.’s support in editing the video material accompanying this paper.
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