General pseudospectral method is extended to the special Euclidean group SE(3) by virtue of equivariant map for rigid-body dynamics of the aircraft. On SE(3), a complete left invariant rigid-body dynamics model of the aircraft in body-fixed frame is established, including configuration model and velocity model. For the left invariance of the configuration model, equivalent Lie algebra equation corresponding to the configuration equation is derived based on the left-trivialized tangent of local coordinate map, and the top eight orders truncated Magnus series expansion with its coefficients of the solution of the equivalent Lie algebra equation are given. A numerical method called geometric pseudospectral method is developed, which, respectively, computes configurations and velocities at the collocation points and the endpoint based on two different collocation strategies. Through numerical tests on a free-floating rigid-body dynamics compared with several same order classical methods in Euclidean space and Lie group, it is found that the proposed method has higher accuracy, satisfying computational efficiency, stable Lie group structural conservativeness. Finally, how to apply the previous discretization scheme to rigid-body dynamics simulation and control of the aircraft is illustrated.
The aircraft is usually regarded as a single rigid body in three-dimensional space. The dynamics of a rigid body is an important problem modeling the time evolution of aircraft and other vehicles [
The difficulty of developing a numerical method on Lie group arises as the group is not the well-known Euclidean space
Since accuracy, time performance, conservativeness, and numerical stability of the aircraft rigid-body dynamics need to be considered comprehensively in aircraft simulation and control, we do not intend to adopt any of the previous methods but use the same idea of equivariant map to develop a new method on SE(3) for driving the evolution of aircraft dynamics. We resort to pseudospectral method, which is widely used in fluid mechanics, quantum mechanics, linear and nonlinear waves, aerospace, and other fields by virtue of its high accuracy, spectral (or exponential) convergence rates, requirement for less computer memory under the same precision condition, and so forth [
In this work, we establish a completely rigid-body dynamics model of aircraft in body-fixed frame on SE(3). With respect to kinematics, due to the fact that applying the general pseudospectral method directly to the configuration equation of the rigid-body dynamics could not preserve the Lie group structure of the solution of the equation, drawing on the equivariant map, we transform the configuration equation on SE(3) into an equivalent equations in a Lie algebra space and accordingly give the top eight orders reduced truncated Magnus series expansion with its coefficients
The rest of the paper is organized as follows. In Section
The special Euclidean group SE(3) is the semidirect product of SO(3) and
The navigation equations of the aircraft are given by [
Denote
Also, the kinematic equations are given by [
Denote
Therefore, from the preceding equation and (
The force equations of the aircraft are given by [
One has
Denote
The moment equations of the aircraft are given by [
where,
Denote
Therefore, from the preceding equation and (
General pseudospectral method will be used for computing the velocities in the Lie algebra. For this purpose, we briefly describe the basic principle of general pseudospectral method [
Firstly, we equally divide a time interval
Thus, accordingly, (
Let
Equation (
Through expressing the derivative of the Lagrange polynomials at the collocation points in differential matrix form
We can write (
Based on the previous equations, we establish defect equations
According to different selection methods of collocation points, pseudospectral methods can be divided into the standard method and the orthogonal method. Common collocation points in the orthogonal method are those obtained from the roots of either Chebyshev polynomials
First recall and rewrite the aircraft rigid-body dynamics model Equations (
For the kinematics Equation (
For the dynamics equation (
In order to transform (
Let
First, from the definition of an action of
It is known that there is a local coordinate map
In the case where
The theorem 3.6 of [
Finally, we need to determine what
Let
The following commutative diagram describing the equivariant maps of composition
Also, [
In order to obtain the explicit expression of
For deriving the previous formula, we differentiate (
It seems from the equivariant map and the previous formula derivation that solving equation (
In the context of rigid-body motion, the right trivialization corresponds to the differential equation with tangent vectors
The previous formula can be rewritten as an infinite sum
It is worthwhile to note that, unlike right trivialization in [
Based on the binary tree theory, we can find the following
Then, we use Gauss pseudospectral method to solve (
Next, through transforming (
The same as the univariate case, we use the following quadrature formulae (
One has
Through computing
There are several choices of local coordinate map
In the same way, after
As mentioned above, here, we can directly use general pseudospectral in Section
To begin with, we write (
By computing the velocity deviation between adjacent iterative steps, we determine whether or not to terminate the iterative process. It is noteworthy that we must ensure that computing configurations and velocities at the same Gauss points.
Finally, after the final satisfying iterative results of the velocities at the Gauss points are obtained, we use the following formula to compute velocity at the endpoint:
In accordance with the aforementioned basic principle of geometric pseudospectral method, we develop a 4th-order algorithm for the rigid-body dynamics evolution over time.
Due to the fact that (
By the way, the reason why we choose the previous metric is its intuitive physical meaning; taking SE(2) as an example, the 2-norm
Turning to order condition of the algorithm, there are something worthy of our attention. Firstly, if we consider the time symmetry of the Magnus series expansion, the number of terms belong to (
Initialize time interval
Step 2.2.1. Set the number of iterations, threshold of iterations deviation and let current step be 1; Step 2.2.2. Let initial value of both Step 2.2.3. Child loop:
Step 2.2.4. End child loop; Step 2.2.5. Use the final iterative results
Step 2.2.6. Compute configuration
Step 2.2.7. Compare
In this section, the performance of Algorithm
Parameters of numerical test.
Time |
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Mass |
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Moment of inertia |
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Initial orientation |
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Initial position |
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Initial linear velocity |
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Initial angular velocity |
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Magnus series terms corresponding to the top eight orders rooted trees.
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1 |
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1 |
3 |
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1/2 |
4 |
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1/4 |
4 |
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1/12 |
5 |
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1/24 |
5 |
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1/24 |
7 |
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1/24 |
5 |
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1/8 |
5 |
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0 |
6 |
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1/16 |
6 |
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1/48 |
8 |
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1/48 |
6 |
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1/48 |
6 |
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0 |
7 |
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1/48 |
7 |
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1/144 |
7 |
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1/48 |
7 |
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0 |
6 |
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1/48 |
6 |
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1/144 |
8 |
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0 |
6 |
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0 |
6 |
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−1/720 |
The top eight orders reduced truncated Magnus series expansion.
Order |
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Reduced Magnus series expansion |
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2 | 1 | 1 |
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4 | 2 | 3 |
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6 | 3 | 5 |
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8 | 4 | 7 |
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For making the numerical test more comparable, we select four 4th-order numerical methods used to compare with geometric pseudospectral method (GPM), including ode45 [
In order to intuitively present the difference between these methods, the 240 seconds evolving trajectories of the free-floating rigid body for a large step size are shown in Figure
Position (
In order to quantitatively compare the proposed method with the other three methods, deviation statistics and runtime statistics over 10 runs using a wide range of initial conditions on a 240 seconds evolution are provided in Figure
Accuracy and computational efficiency of GPM compared with PM, RKMK, and RKi.
Average position deviation
Average orientation deviation
Average runtime per update
Finally, we examine the conservativeness of Lie group structure. Considering that configuration is parameterized by unit quaternion in ode45 and implicit RK method, we only compare the proposed method with explicit RKMK and Gauss pseudospectral method. As mentioned, the group element of SO(3) satisfies
Lie group structural deviation.
Gauss pseudospectral method
Explicit RKMK
Geometric pseudospectral method
It is seen from the numerical test in the previous section that the proposed method has better accuracy, stable structural conservativeness, and satisfying computational efficiency. Thus, it is able to meet the fidelity and timeliness requirements of aircraft dynamics simulation. In view of the aircraft dynamics is complex and underactuated, we adopt optimal control to generate flyable trajectories of aircraft. As mentioned in Section
Depending on the specified scenario, cost functions include minimum control effort, minimum time, obstacle avoidance, and the quickest manoeuver. For further details regarding the optimization setup, one can consult to [
Simulation results for the aircraft avoiding obstacles.
A completely left invariant rigid-body dynamics model of aircraft on SE(3) is established. For the left invariance of rigid-body dynamics model in body-fixed frame, an equivalent differential equation on
For the future work, we will further analyze its performance in aircraft simulation and control. For presence of a large number of tedious commutator operator in multivariate quadrature, we will try to find an alternative method to simplify the calculation process so that extend our method to the higher order. Moreover, we will extend the proposed method to a broader kind of left invariant rigid-body dynamics systems in engineering.
The top eight orders truncated Magnus series expansion of
Great thanks are due to the reviewers for valuable comments on our revision of the paper. This work was supported by the National Natural Science Foundation of China (61005077), the National Basic Research Program of China (6138101001), and the National Defense Foundation of China (403060103).