Autonomous maneuvering flight control of rotor-flying robots (RFR) is a challenging problem due to the highly complicated structure of its model and significant uncertainties regarding many aspects of the field. As a consequence, it is difficult in many cases to decide whether or not a flight maneuver trajectory is feasible. It is necessary to conduct an analysis of the flight maneuvering ability of an RFR prior to test flight. Our aim in this paper is to use a numerical method called algorithm differentiation (AD) to solve this problem. The basic idea is to compute the internal state (i.e., attitude angles and angular rates) and input profiles based on predetermined maneuvering trajectory information denoted by the outputs (i.e., positions and yaw angle) and their higher-order derivatives. For this purpose, we first present a model of the RFR system and show that it is flat. We then cast the procedure for obtaining the required state/input based on the desired outputs as a static optimization problem, which is solved using AD and a derivative based optimization algorithm. Finally, we test our proposed method using a flight maneuver trajectory to verify its performance.

The autonomous rotor-flying robot (RFR) is one of the frontier research topics in the field of robotics. Extensive research has been conducted on issues related to RFR, including flight control [

Flight maneuvering is a very important ability for an RFR system because it can greatly extend the system’s field of applications. However, the flight maneuver control of an RFR is a challenging and absorbing problem in the field of autonomous RFR research [

Most work on flight maneuver control, such as [

In this paper, based on a full state dynamics model, we investigate the feasibility analysis problem of a flight maneuvering trajectory. This is actually an inverse computation problem, that is, to compute both the interval states and the inputs based on predetermined position, yaw angle, and their higher-order derivatives. In our work, we model it as a static optimization problem and then use a derivative-based optimization algorithm to obtain an accurate solution of it. We use algorithm differentiation (AD) to obtain precise higher-order derivatives of the nonlinear cost function. The advantages of our proposed algorithm are as follows:

The remainder of this paper is organized as follows. We first present a model of the RFR system and show that it is flat by taking the three positions and the yaw angle as a group of flat outputs. This allows us to model the trajectory evaluation problem as an optimization problem by taking into account the dynamical model, which is solved by using the AD algorithm along with a derivative-based optimization algorithm. Finally, in order to test our proposed method, we use it to analyze a flight maneuvering trajectory.

Usually, the complete dynamical model of an RFR system can be divided into three parts: the actuator dynamics, the aerodynamics, and the rigid body dynamics. In this paper, only the rigid body dynamics as shown in the following equation (

Sketch of quadrocopter.

Thus, define inputs as

It can be easily shown that system (

That means system (

AD is a numerical method that is effective to accurately compute the derivatives of some complicated nonlinear function. In this section, we will briefly introduce how to conduct the inverse computation of a nonlinear flat system using AD algorithm [

Consider the following nonlinear system:

It has been shown that the inverse computation problem of system (

Derivatives of

For this purpose, we firstly use the following Taylor series to denote the state vector

It has been shown that each Taylor coefficient

Thus, if

Furthermore, based on (

Through (

Define a new matrix as

From [

Thus,

Furthermore, if we denote

then the same process can be used to compute

Also, it is not difficult to show that the derivatives of output with respect to the states can be denoted as follows:

Up to now, we have shown that, with a predefined state

Compute the output vector

Based on (

Update the state

Go to Step

If the system is nonautonomous, that is, it can be rewritten as following form,

Thus, the inverse computation can be conducted using the preceding algorithm.

As for the proposed quadrocopter system introduced in Section

In this section, a so-called “Circle Maneuver,” that is, the RFR system flies along a circle as shown in the following equation (

Circle maneuver trajectory.

The parameters of the RFR system are as follows:

The trajectory feasibility is decided by some constrains on both inputs and states. Here the following constrains are considered.

In this paper, the software toolbox LIEDRIVERS [

Control input:

State:

State:

From these figures, we can obtain the following results.

For input constraints (Figures

For angular velocity constrains (Figure

Based on the attitude constrains (Figure

Finally, from the preceding results, it can be seen that the circle maneuvers are really a highly dynamical maneuver, and the quantitative analysis prior to the flight test is useful and necessary to evaluate whether or not a maneuver trajectory is feasible or even dangerous.

In this paper, the flight maneuvering trajectories evaluation problem is researched, and the algorithm differentiation (AD) is used to realize the inverse computation of the rotor-flying robot (RFR) system. This paper starts from constructing nonlinear dynamical model of an RFR system. Then, we show that the RFR system model is flat taking translational positions and yaw angle as flat outputs. After that, the scheme of AD based inverse computation is introduced, that is, computing the internal states and inputs based on the desired flat outputs. Finally, with the proposed scheme, a typical flight maneuver, that is, the circle maneuver, is analyzed.

The advantages of our proposed algorithm are as follows:

The method proposed in this paper can be used to not only evaluate the feasibility and validity of the maneuver flying trajectories, but also realize the tracking control algorithm, where the computed internal states and inputs are useful for attenuating the online computational burden of the tracking controller and thus improving the closed loop performance. This will be one of our future’s works.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to present their thanks to Professor Klaus Röbenack at Technique University of Dresden (Germany) for the related discussion and suggestions on AD algorithm. This work is supported by the National Natural Science Foundation of China (Grants no. 61035005 and no. 61203340).