In our work, we address the articulated model with more than two links. To achieve the aerial transformation by such a multilinked model, a comprehensive investigation on modeling is important. In this section, we first describe the approximation method to obtain the simplified multilinked model, and then present the actuator allocation for both under-actuated and fully actuated models.
3.1. Approximation Model
As shown in
Figure 2, the kinematic model of the proposed aerial robot is composed from a chained link structure. We assume the number of links is
; then the vectoring of joint angles
can be defined as
. The thrust vectoring apparatus, as shown in
Figure 2B, consists of two vectoring angles (
) and dual rotors that generate two thrust forces (
). Therefore, there are four control inputs,
, and
, in each vectoring apparatus, and we developed two different usage patterns for the under-actuated and fully actuated models, respectively. It is also notable that the vectoring apparatus is not necessarily deployed in each link module, as shown in
Figure 2A. Therefore, the number of rotors,
, can be different from
.
Then, the dynamic model of such multilinked model w.r.t. the entire CoG frame
can be written as follows:
where the first equation denotes the dynamic motion of the entire linear momentum, which is described in the inertial frame
, whereas the second equation denotes the dynamic motion of the entire rotational momentum, which is described in the CoG frame of the entire multibody model (i.e.,
). The third equation corresponds to the joint motion.
is a three-dimensional vector expressing gravity.
and
on the left sides of (
1) and (
2) are the total linear and angular momentum, respectively, which are both affected by the joint angles, vectoring angles, and their velocities, whereas
and
on the right sides are the total wrench obtained from all vectored thrust forces. The allocation from the vectored thrust forces from this wrench is the key to achieving the flight control, which is described in
Section 3.2 in detail.
In Equation (
3),
denotes the inertial matrix, whereas
is the term related to the centrifugal and Coriolis forces in joint motion. The symbol "
" stands for "segment" in multilinks.
and
are the Jacobian matrices for the frames of the
i-th rotor and the
i-th segment’s CoG, respectively.
is the vector of joint torque, and
denotes the three dimensional force generated by each vectored rotor.
The entire dynamics model summarized in (
1)∼(
3) shows the high complexity due to the joint motion, and thus the real-time feedback control based on such a nonlinear model is significantly difficult. Therefore, a crucial quasi-static assumption is introduced in our work to simplify the dynamics; i.e., all the joints are actuated well and slowly by servos (
). Then, the joint velocity and acceleration can be omitted regardless of the joint motion. Under this assumption, the original dynamic model can be approximated as follows:
where
,
, and
are the position, attitude, and angular velocity of the CoG frame calculated based on the forward-kinematics from the root link states (i.e.,
,
,
, and
) with joint angles
.
Equations (
4) and (
5) still show the properties of the time-variant model because
changes over time and affects both the cog position
and the overall rotational inertia
. (
6) shows the equilibrium between the joint torque, thrust force, and gravity, which can help us to obtain the desired joint torque from the thrust force. Given that we assume that joints are well controlled by the feedback position control of servos, it is indicated that there is no necessity to perform torque control based on (
6). By ignoring (
6), the whole model for control can be finally considered as a single rigid body by our approximation.
3.2. Actuator Allocation
Allocation from the three-dimensional thrust forces to the CoG wrench provides the connection from the whole body feedback control to the actuators that includes the thrust force and vectoring angles and .
The force gap (
) from the dual rotors, as shown in
Figure 2B, would induce a moment load on the servo that controls the vectoring angle
. Since this vectoring servo should be compact and thus weak, it is considered difficult to dynamically control both the vectoring angle
and the thrust forces
and
at the same time.
Therefore, we developed two different strategies as follows: (1) a dual-rotor mode that allows different forces for and , but the vectoring angle is constant in control framework; (2) a virtual-single-rotor mode that assigns the same force for and to avoid the moment load on vectoring angle , and thus, we can use as the control input.
3.2.1. Dual-Rotor Mode
The force and torque related to the
j-th rotor module can be written as:
where ∗ denotes the index of the rotor, which is either 1 or 2.
is the thrust unit normal,
is the ratio of rotor thrust to its drag, and
is the rotational direction of each rotor.
is the rotor position that is affected by the joint angles
and the vectoring angle
. Then, the relationship between the target wrench and rotor thrust can be expressed by
where
is the number of rotor apparatus, and
and
correspond to (7) and (8).
3.2.2. Virtual-Single-Rotor Mode
Given that the dual rotors generate the same thrust forces, there is no moment that occurs in the vectoring angle
. Then, we can count the pair of rotors as an integrated rotor that generates a combined uni-directional thrust
. In addition, the drag moment and gyroscopic moment can be ideally counteracted. Then, the force
and torque
related to the
j-th rotor module can be written as:
where
. Definitions of the frames
,
, and
can be found in
Figure 2B.
in (
12) is the position of the frame
, which depends on the joint angles
and the vectoring roll angle
because there is an offset from
to
, as shown in
Figure 2C.
Then, the total wrench in the CoG frame can be given by
where
and
correspond to (
11) and (
12).
If allocation matrix
Q in (
13) is full-rank, an arbitrary wrench
can be achieved by the control input of
,
, and
, which can be considered fully actuated. If a model has more than two rotor vectoring apparatuses, the full pose control can be achieved by using this allocation mode. However, only two apparatuses imply the bijection between six control input (i.e.,
,
,
,
,
, and
) and full pose motion
, which can easily result in a control input that exceeds the valid range, especially for the thrust force
. However, for a model with more than three vectoring apparatuses, there is the redundancy in control input. Therefore, we apply the dual-rotor mode for model with two vectoring apparatuses and the virtual-single-rotor mode for other cases.