Aerial manipulation has been a growing research topic which aims to utilize the maneuverability of an aerial vehicle and the versatility of a robotic manipulator. Different from physically non-interacting passive tasks such as surveillance and remote sensing, an aerial manipulator can execute active tasks involving physical interaction such as grasping [kim2013aerial, kim2019sampling], valve turning [korpela2014towards], drawer opening [kim2015operating], contact inspection [jimenez2015aerial], transportation [lee2016estimation, byun2020line], and door opening [lee2020aerial_1].
Despite various demonstrations of aerial manipulation tasks involving contact with environments, they usually involved a relatively low level of changes in the dynamic characteristics, and they rarely dealt with the transition during the physical interaction explicitly. In fact, there is a lack of research on the stability of aerial manipulation before and after the physical interaction. When a mode switch entails a significant change in the system response, neglecting it can lead to destabilization. Therefore, it is necessary to systematically analyze dynamical modes and design a robust controller to more realistically embrace the whole operation.
As an example of aerial manipulation involving a drastic change in dynamics, this paper deals with the problem of pulling a plug from a socket using a multirotor equipped with a two degree-of-freedom (DOF) robotic arm. In this task, the large force exerted on the end-effector suddenly disappears after the plug is separated from the socket. For formal analysis of the stability, we first formulate hybrid automata[goebel2009hybrid] which enclose all the dynamic models and the operative modes that have their own control laws different from another. Then, we design disturbance-observer (DOB)-based controllers [back2009inner] for the respective operative modes and prove the stability and robustness of the formulated hybrid automata [marconi2009control].
I-a Related Works
There have been several works which explain aerial vehicles using the concept of hybrid automata. In [scholten2013interaction, darivianakis2014hybrid, Praveen2020inspection], the contact task using an aerial vehicle is divided into two operative modes, i.e. docking and free-flight, and each mode are controlled by a different controller from one another. However, these works did not conduct an analysis on stability and robustness and the transition between those modes was not explicitly discussed.
Some studies addressed the stability and robustness of an aerial vehicle involved in the physical interaction using the hybrid automata theory. In [marconi2012control], the stability of path-following control considering mode changes was investigated for a robust contact of a ducted-fan aerial vehicle on a vertical surface. In [Cabecinhas2016robust], the process of a multirotor landing on the slope was divided into several modes and the stability and robustness was proved in a similar way to [marconi2012control]. However, in such settings, the effect of dynamic change can be reduced by slowly approaching the wall or landing site. Thus, there is no guarantee that such control methods can maintain the stability and robustness of the aerial manipulation involving an abrupt change such as plug-pulling.
To the best of the authors’ knowledge, this is the first attempt to conduct a plug-pulling task using an aerial manipulator, which involves a significant mode change, and present a thorough analysis on the stability and robustness of the aerial manipulator using the hybrid automata theory. We propose a trajectory generation strategy and DOB control structure for each operative mode. Especially, for the situation of pulling the wire, we derive a dynamical model of the aerial manipulator constrained to the wire and the socket. In addition, we construct a DOB structure corresponding to the model of the plug-pulling aerial manipulator and prove the stability and robustness of the proposed controller.
In Section II, we briefly explain the concept of hybrid automata, describe notions utilized throughout the paper and introduce the aerial plug-pulling scenario. Section III formulates hybrid automata for the aerial manipulator conducting the plug task, and the trajectory generation and controller design is described in Section IV. Section V shows the stability and robustness analysis, and Section VI presents the experimental setup and results.
Ii Problem Setup
Ii-a Preliminary: Hybrid Automata
The following elements define hybrid automata [goebel2009hybrid] with the state variable and the control input .
Set of operative modes, , contains names of the control modes. With respect to , we let and denote the time when the mode begins and the desired time to terminate the mode , respectively.
Domain mapping, , means the possible region where and can evolve while maintaining a specific mode . It is expressed as .
Flow map, : , describes the dynamics in each operative mode .
Set of edges, , means all possible pairs of operative mode changes .
Guard mapping, , describes the conditions where the transition from to occurs. It is represented as .
Reset Map, , means the jump of the state variable . It is expressed as .
-th element of a column vector, the operator representing the cross product , the dimension of , the -th element a matrix , the block matrix of containing from -th to -th elements and the set where is a constant positive number. The Kronecker product is expressed as .
As in Fig. 2, we denote the frame of the inertial coordinate, multirotor, 1, 2 servo motor and the end-effector by , , , and respectively.
To express the state of an aerial manipulator, we define the generalized coordinate as consisting of the position of the multirotor , Euler angles where , and represent roll, pitch and yaw angles, and the angles of servo motors . Also, we let , , and denote , , and . To represent inputs, we use and to denote and where , and
mean a total thrust, moments with respect to, and and torque inputs exerted on the servo motors. We let denote a matrix which satisfies and the scalar the gravitational acceleration. Moreover, we set , and
as a desired trajectory, an estimate ofand the nominal value of .
We use , and to denote mass of the multirotor, the 1 and the 2 servo motor while diagonal matrices , , in are the moments of inertia of the corresponding components. Additionally, means the moment of inertia of the end-effector.
As in Fig. 2, the aerial manipulator tries to unplug in direction from the socket installed on the wall aligned with - plane. After the plug is separated from the socket, the vehicle quickly stabilizes its attitude in a short time and maintains the hovering state.
Iii Hybrid Automata of Aerial Plug-Pulling
We construct the elements of the hybrid automata listed in Section II-A for the aerial manipulator pulling the plug.
Iii-a Set of Operative Modes, = Wp, St, Ff
In (wire-pulling) mode, the aerial manipulator tries to unplug by pulling the wire in direction. In (stabilizing) mode, the vehicle quickly stabilizes its attitude immediately after the separation of the plug. In (free-flight) mode, the aerial manipulator returns to the original location and keeps the hovering state.
Iii-B Domain Mappings, , and
where is the interaction force acting on the end-effector due to the friction between the plug and the socket and is the force limit up to which the plug can resist from separating.
where means an space representing the flight envelope, i.e., all possible regions of the state and inputs for the flight experiment.
Iii-C Flow Maps, and
The derivation of is based on [kim2013aerial], but since the position of the end-effector is fixed, we newly derive it in the form of the Euler-Lagrange equation with which fully describes the dynamics of the wire-pulling aerial manipulator.
First we express position and angular velocity as
with the kinematic constraint By substituting (1) into the derivation process presented in [kim2013aerial], an Euler-Lagrange equation of the wire-pulling aerial manipulator model can be derived. Then, the obtained equation of motion can be analyzed in the form of flow map as follows.
where , , ,
and means the external disturbance applied to the wire-pulling aerial manipulator system. Thus, the variable evolves in correspondence with (2).
Iii-C2 and modes
The Euler-Lagrange equation for the multirotor equipped with the 2 DOF robotic arm is derived in [kim2013aerial]. Then, the flow map can be derived as follows.
where , , , , and means the external disturbance applied to the system in free flight.
Iii-D Set of Edges,
There would be only two possible edges in our scenario because the transition from ST or FF mode to WP mode does not occur unless the plug is attached to the socket again. Also, a change from to mode is impossible because the mode is primarily designed as an intermediate stage between the and modes.
Iii-E Guard Mappings, and
A transition from WP mode to ST mode occurs when the element of exceeds . Therefore, the guard map is defined as . Also, since (, ) in the ST mode are user-defined values while they are computed from the user-defined values of (, ) in the FF mode, an undesirable abrupt change in (, ) would provoke a failure in attitude control. Therefore, the guard map is defined as where is defined as the threshold of for the mode change.
Iii-F Reset maps, and
If the operative mode changes from WP to ST, there would be jumps in due to the sudden disappearance of the force exerted on the end-effector. However, since we cannot know the exact magnitude of the jumps in , we denote it by which satisfies . Then, the reset map is expressed as . On the other hand, the change from ST to FF does not entail any jump in because they evolve under the same dynamics. Therefore, the reset map can be derived as .
Iv Trajectory Generation and Controller Design
Iv-a Trajectory Generation
It is assumed that and exactly follow and respectively and the desired values that are not defined at each mode are set to be the same as the current values.
Iv-A1 WP mode
In this mode, the aerial manipulator tries to tilt its body with respect to in order to exercise a pulling force to the socket. Therefore, is given as below,
where means the maximum absolute value of the pitch angle. It prevents a sudden transition to mode by gradually tilting the vehicle’s body.
Iv-A2 ST mode
This mode is proposed for compensating the overshoot invoked by the transition of the dynamical model and avoiding an abrupt change in and . In order to simultaneously minimize the overshoot and make and close to zero, the time interval needs to be reasonably small. Therefore, and are set as
where coefficients , and satisfy the conditions , and .
Iv-A3 FF mode
In FF mode, and are set to fly back to the original position as follows:
Iv-B Nominal Model for Each Mode
Servo motors are usually controlled by the given desired position, not torque. Therefore, the equations of motion that eliminate the term are derived for each model.
Iv-B1 WP mode
In (2), is computed as
Therefore, the model with respect to is obtained with known values , , and an observable value as follows.
with block matrices , , and . Then based on this, the nominal model for the WP mode can be obtained as follows.
with the nominal input . The total thrust is calculated in the DOB controller introduced in [lee2020aerial_2].
Iv-B2 ST and FF mode
The nominal model for ST and FF mode is derived in [lee2020aerial_2] as below.
where and mean the center of mass of and respectively. The other notations are defined in [lee2020aerial_2].
Iv-C Controller Design
Iv-C1 WP mode
If we design the nominal input to make the solution of (9) adequately follows , the compensation of model discrepancy, , is conducted by the DOB structure presented in [back2009inner]. The overall diagram is shown in Fig. 3 and the detailed DOB control law is formulated as below.
where , , and
with the positive constants , and the small positive constant . In this DOB structure, we use a saturation function defined with the conditions below.
: is a globally bounded function.
for where .
From the conditions above, the quasi-steady state range of satisfies the equation while avoiding the saturation [back2009inner].
Iv-C2 ST mode
For the ST mode, we only apply the DOB structure for the fully-actuated system introduced in [kim2017robust].
Iv-C3 FF mode
We will utilize the same controller presented in [lee2020aerial_2] for the FF mode.
V Stability and Robustness Analysis
In this section, an analysis of the stability and robustness will be presented. During the analysis, the term maneuver which means in a particular mode will be used. and mean the solution from the actual flow map and the nominal flow map. Additionally, tr is defined as a set including all values of in the given time interval.
V-a Preliminary Definitions for the Analysis on Hybrid Automata
Definition 1 (-robust -single maneuver in )).
For , and , a maneuver satisfies
Here, means the time interval where evolves in the mode within .
Definition 2 (-robust approach maneuver in )).
For , and , a maneuver satisfies
Let and be compact sets defined as and
Then, for any there exists such that
Definition 3 (-robust coverage set, ).
A set of elements which satisfies
Definition 4 (-robust transition maneuver).
For , , , and , which is a union of a -robust approach maneuver before the switching time and of a set of -robust single maneuvers after transition with the property .
V-B Robustness of the Nominal maneuver
V-B1 -robust transition maneuver
With the assumption that the aerial manipulator is in a quasi-equilibrium state while perching on the wall, the pulling force is equal to [kim2015operating]. Thus, increases with the gradually decreasing trajectory of as generated in (4). As a result, does not reach before closely approaches . Moreover, as depicted in Fig. 3(a), there is no possibility of a transition from to mode because is defined as . Therefore, the maneuver is proved to be a -robust -single maneuver in .
can also become a -robust approach maneuver when there exists a time instant that a change from to occurs in a finite time. Thanks to the relation that equals to , we can easily find a sufficient condition where means the maximum value of . The inequality above infers that can reach with the given trajectory of . Additionally, since the direct change from to mode never occurs as mentioned above, the claim that is a robust approach maneuver is proved.
After the transition from to mode, the reset map of is uncertain. However, we can avoid the situation where the value of becomes an element of the switch from to does not occur before the time reaches the value of by the condition in . Therefore, a maneuver turns out to be a -robust -single maneuver in after the mode transition. From the analyses above, the union of tr and tr is proved to be a -robust transition maneuver.
V-B2 -robust transition maneuver
As proved in the previous section, the maneuver is a -robust -single maneuver in . Also, there must be a change from to mode in a finite time since defined in mode reaches when reaches . Also, since the transition from ST to WP mode is impossible, also turns out to be a -robust approach maneuver in .
Accordingly, the property that and are not the elements of also guarantees that a maneuver is a -robust -single maneuver in . Thus, it is proved that the union of tr and tr becomes a -robust transition maneuver.
V-C Analysis on Stability and Robustness at Each Mode
V-C1 Wire-pulling mode
Prior to formulating a theorem about the stability and robustness of the mode, there need some remarks and assumptions as below.
, , are continuous and bounded in . In addition, let where is a known compact set.
Let be the nominal solution of (9). For given , the solution evolves in a bounded set if the initial condition is in a compact set , and initiated in is locally asymptotically stable.
According to [kim2017robust], the aerodynamic effects, such as drag or buoyancy forces, are negligible in a near hovering condition due to the small size of the multirotors. Also, frictional torque and force applied on the end-effector are at least and bounded in because they are functions of and .
The terms , , , , , , , , and are vectors and matrices which consist of , and . Thus, these terms are at least and bounded in . Moreover, by the assumption 2, is also a vector that is and bounded in . Since , , and have the terms described above, vectors and matrices shown in (9) are all at least and bounded in .
Then finally, from the assumptions and remarks stated above, we can formulate a theorem on the relationship similar to the theorem introduced in [lee2020aerial_2] and prove it.