Geometric Pseudospectral Method on SE(3) for Rigid-Body Dynamics with Application to Aircraft

Abstract

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.

1. Introduction

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 [1]. In particular, in aircraft simulation and control, its authenticity affects the fidelity of the simulation and also determines whether it can truly reflect the performance of the designed controller to a certain extent. The rigid-body dynamics has fundamental motion invariants; for example, the flow of a Hamiltonian system is symplectic [2, 3], the total energy of a rigid body is conserved in the absence of nonconservative forces [4], and the momentum map associated with a symmetry of a rigid body is preserved [5]. Furthermore, the exact flow of a rigid body always stays on a configuration manifold, since the combined configurations of the translation and rotation of the rigid body construct a special Lie group called the special Euclidean group SE(3) [6]. The invariants often manifest through geometric characteristics of exact flow of a rigid body, such as symplecticity, first integrals, symmetry, and Lie group structure. However, for most rigid bodies, obtaining the analytical solution is quite difficult. Thus, under the condition of non analytical solution, it is desired to develop a numerical method whose iterates preserve the previous fundamental invariants [7]. For this purpose, the research area of development of numerical methods for rigid-body differential equations that preserve geometric properties of the numerical solution has been of great interest in recent years [8]. Simos [8, 9], Monovasilis et al. [3, 10], and Kalogiratou [11] construct several different multistep symplectic integrators based on symplectic geometry in order to preserve the Hamiltonian energy of rigid bodies as the harmonic oscillator, the pendulum, two-body problem, and orbital problem. Lee et al. [5] and Hussein et al. [12] adopt variational integrators to evolve the rigid-body dynamics, which ensure that while the total energy of the rigid body is conserved under conservative forces, the momentum associated with a symmetry of the system is also conserved. Iserles et al. [13] and Lee et al. [5] extend numerical methods in Euclidean space into Lie group, so that numerical flow of the rigid-body dynamics has Lie group structural conservativeness, which is also the main focus of this paper. Retaining motion invariants under discretization has been proven not only a nice mathematical property, but also the key to improved numerics, as they capture the right dynamics (even in long-time integration) and exhibit increased accuracy [2, 7, 13, 14]. Therefore, for driving the evolution of rigid-body dynamics, it is very important to develop a numerical method with preservation of the previous geometric characteristics.

The difficulty of developing a numerical method on Lie group arises as the group is not the well-known Euclidean space ℝ^𝑛; consequently, the rigid-body dynamics evolving on SE(3) cannot be properly integrated by conventional numerical method, including the popular Runge-Kutta schemes, developed for ℝ^𝑛. For this reason, one resorts to other means to drive the rigid-body dynamics forward while preserving the Lie group structure of the configuration. It can be roughly divided into two categories: classical Lie group methods which discretize the continuous equations of motion and Lie group variational integrators which discretize the variational principles of mechanics. Classical Lie group methods include the Munthe-Kaas (MK) method [15], the Crouch-Grossmann (CG) method [16], the Newmark-type method [17], and the commutator-free (CF) method [18], and so forth. The MK method is based on a differential equation on the Lie algebra and uses a single evaluation of the exponential map. The CG method updates the group elements by multiple evaluations using the exponential map. For detailed information, one can consult to [13]. Lie group variational integrators are proposed in recent years whose idea is to discretize the variational principles of mechanics: Hamilton’s principle or Lagrange-d’Alembert principle [1, 5, 14]. Both methods have their own advantage and disadvantage. The former has a better time performance, but its accuracy is not an advantage under the same order condition. The latter preserves exactly energy, momentum, symplectic structure, and group structure [12, 19] and offers a particularly robust and efficient framework for simulation [14], attitude control [4], motion control [1], and trajectory generating [14] of the aircraft and other vehicles, however; it has no advantage of accuracy, and its performance is subjected to the Newton-type solver for solving the equivalent vector equation [5, 14, 20]. Regardless of the classical Lie group methods or Lie group variational integrators, they are all based on Kenth Engø’s idea about equivariant map which transforms the differential equations evolving on a Lie group into an equivalent differential equation evolving on a Lie algebra corresponding to the Lie group [21].

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 [22]. Furthermore, it is commonly used in optimal control of the aircraft and other vehicles [23] and thereby lay the foundation for our simulation and control application. Unlike variational integrator must analytically derive the first-order discrete necessary optimality, which, more often than not, is nontrival for most optimal control problems [23], typical pseudospectral method can be used to transcribe a continuous optimal control problem into a discrete nonlinear programming problem (NLP), and it has been shown that solving the NLP derived from the pseudospectral transcription of the Gauss form is exactly equivalent to solve the discrete first-order necessary conditions [23, 24]. However, when applied to rigid body dynamics directly, general pseudospectral method cannot preserve Lie group structure. For this purpose, a numerical method on SE(3) called geometric pseudospectral method is proposed in this work, which provides satisfactory accuracy, computational efficiency and preserves the essential Lie group structure. To our knowledge, Moulla et al. [25] was the first to pose the concept of “geometric pseudospectral method” not long ago. They suggested a polynomial pseudospectral method that preserves the geometric structure of port-Hamiltonian systems, the phenomenological laws, and the conservation laws without introducing any uncontrolled numerical dissipation. However, their method was designed only for port-Hamiltonian systems having a special structure, that is, the Dirac structure, thus it could not be directly extended to the systems on SE(3). Thereby, how to apply the pseudospectral method to a dynamics system on SE(3) and preserve essential Lie group structure of the system is still an open question, which is the main topic in this paper.

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 𝛼(𝜅) (refer to the appendix) of the solution of the equivalent equation under left trivialization. For the condition of each term in 𝑢^[2𝑞](𝑡) being multivariate quadrature, we apply the general pseudospectral method to compute quadrature weights, so that obtaining the values of 𝑢^[2𝑞](𝑡) at the collocation points and the endpoint (collectively referred to discrete points). Finally, we compute configurations at the discrete points via coordinate map. It is worth mentioning that the theories with respect to Lie group methods in the extensive literatures [13–15] are almost derived under right trivialization and thus are not suitable for our method, since our modeled aircraft configuration model is left invariant. To this end, we provide a relatively complete set of left trivialization framework for developing geometric pseudospectral method on SE(3). With respect to dynamics, considering the Lie algebra space is isomorphism to ℝ^𝑛, we use the general pseudospectral method directly to compute the velocities at the same discrete points as that of configuration. Under the same 4th-order and the same large step size, the proposed method is compared with implicit RK, explicit RKMK, and Gauss pseudospectral method. Through a comprehensive comparison of accuracy, computational efficiency, and Lie group structural conservativeness, the proposed method has higher accuracy, satisfying computational efficiency, stable Lie group structural conservativeness. The proposed method is coordinate-free, no need to switch chart and special handling of singularities. Finally, we give a way of applying the proposed numerical method to rigid-body dynamics simulation and control of the aircraft.

The rest of the paper is organized as follows. In Section 2, a completely left invariant rigid-body dynamics model of the aircraft on SE(3) is established. Geometric pseudospectral method on SE(3) is developed in Section 3. In Section 3.1, for completeness, the basic idea of the general pseudospectral method in Euclidean space is roughly introduced. In Section 3.2, geometric pseudospectral method on SE(3) is developed under left trivialization, and a 4th-order geometric pseudospectral algorithm is given. In Section 3.3, numerical tests are carried out on a free-floating rigid-body model in comparison with four other numerical methods to validate numerical accuracy, computational efficiency, and structural conservativeness of the proposed method. Then, how to apply the proposed method to rigid-body dynamics simulation and control of the aircraft is illustrated in Section 4. Finally, the conclusions and future work are outlined in Section 5.

2. Aircraft Rigid-Body Dynamics Model on SE(3)

The special Euclidean group SE(3) is the semidirect product of SO(3) and ℝ³, whose group element 𝑔_𝑎𝑏=(𝑅_𝑎𝑏,𝑝_𝑎𝑏) consists of rotational component 𝑅_𝑎𝑏∈SO(3) and translational component 𝑝_𝑎𝑏∈ℝ³ of the body-fixed frame {𝑏}, relative to the space frame {𝑎}, where SO(3)={𝑅_𝑎𝑏∈ℝ^3×3∣𝑅_𝑎𝑏𝑅^𝑇_𝑎𝑏=𝐼,det𝑅_𝑎𝑏=+1} is called the special orthogonal group [6]. In this section, we will regard the aircraft as a single rigid body in three-dimensional space and establish a rigid-body dynamics model of the aircraft on SE(3).

2.1. Kinematics Model

The navigation equations of the aircraft are given by [26] 𝑥̇𝑦̇𝑧̇=𝑅_𝑎𝑏(𝑡)𝑢𝑣𝑤, (1) where, 𝑥̇, 𝑦̇, and 𝑧̇ are the north, east, and vertical components of the aircraft velocity, respectively, in the locally level geographic frame on the surface of the Earth, 𝑢, 𝑣, and 𝑤 denote 𝑋-axis component, 𝑌-axis component, and 𝑍-axis component of the aircraft airspeed, respectively, in the body-fixed frame, and 𝑅_𝑎𝑏(𝑡) is a sequence of 𝑍𝑌𝑋 Euler rotations from the body-fixed frame to the locally level geographic frame, given by𝑅_𝑎𝑏(𝑡)=cos𝜃cos𝜓−cos𝜙sin𝜓+sin𝜙sin𝜃cos𝜓sin𝜙sin𝜓+cos𝜙sin𝜃cos𝜓cos𝜃sin𝜓cos𝜙cos𝜓+sin𝜙cos𝜃sin𝜓−sin𝜙cos𝜓+cos𝜙sin𝜃sin𝜓−sin𝜃sin𝜙cos𝜃cos𝜙cos𝜃. (2)

Denote [𝑥,𝑦,𝑧]^𝑇 and [𝑢,𝑣,𝑤]^𝑇 by 𝑝^𝑎_𝑎𝑏(𝑡) and 𝜐^𝑏_𝑎𝑏(𝑡), respectively, then we have 𝑝̇^𝑎_𝑎𝑏(𝑡)=𝑅_𝑎𝑏(𝑡)⋅𝜐^𝑏_𝑎𝑏(𝑡). (3)

Also, the kinematic equations are given by [26] 𝜙̇𝜃̇𝜓̇=𝑝+tan𝜃𝑞sin𝜙+𝑟cos𝜙𝑞cos𝜙−𝑟sin𝜙𝑞sin𝜙+𝑟cos𝜙cos𝜃, (4) where, 𝜙, 𝜃, and 𝜓 are roll angle, pitch angle, and yaw angle, respectively, which are commonly referred to as Euler angles; 𝑝, 𝑞, and 𝑟 denote the roll rate, pitch rate, and yaw rate, respectively, about 𝑋-body axis, 𝑌-body axis, and 𝑍-body axis. Then, we have 𝑝𝑞𝑟=𝜙̇𝜃̇𝜓̇=̇̇̇̇−̇̇.10−sin𝜃0cos𝜙sin𝜙cos𝜃0−sin𝜙cos𝜙cos𝜃𝜙−𝜓sin𝜃𝜃cos𝜙+𝜓sin𝜙cos𝜃𝜃sin𝜙+𝜓cos𝜙cos𝜃 (5)

Denote [𝑝,𝑞,𝑟]^𝑇 by 𝜔^𝑏_𝑎𝑏(𝑡), and bring in a operator ⋅̂∶ℝ³→𝔰𝔬(3), for all 𝑦∈ℝ³, satisfying 𝜔^𝑏_𝑎𝑏(𝑡)⋅𝑦=𝜔^𝑏_𝑎𝑏(𝑡)×𝑦, where × is vector cross-product; we have 𝜔^𝑏_𝑎𝑏(𝑡)=^{0−𝑟𝑞𝑟0−𝑝}_−𝑞𝑝0 and obtain 𝑅̇_𝑎𝑏(𝑡)=𝑅_𝑎𝑏(𝑡)⋅𝜔^𝑏_𝑎𝑏(𝑡). (6)

Therefore, from the preceding equation and (3), we can construct a kinematics model (configuration) 𝑔̇‌_𝑎𝑏(𝑡)=𝑔_𝑎𝑏(𝑡)⋅𝑉^𝑏_𝑎𝑏(𝑡), namely, 𝑅̇_𝑎𝑏(𝑡)𝑝̇^𝑎_𝑎𝑏(𝑡)=𝑅00^𝑎𝑏(𝑡)𝑝^𝑎_𝑎𝑏(𝑡)𝜔01^𝑏𝑎𝑏(𝑡)𝜐^𝑏_𝑎𝑏(𝑡)00, (7) where 𝑔_𝑎𝑏(𝑡) and 𝑉^𝑏_𝑎𝑏(𝑡) are, respectively, called the homogeneous representation of 𝑔_𝑎𝑏(𝑡) and that of 𝑉^𝑏_𝑎𝑏(𝑡)=[𝜔^𝑏_𝑎𝑏(𝑡),𝜐^𝑏_𝑎𝑏(𝑡)]^𝑇∈𝔰𝔢(3), 𝔰𝔢(3)=𝔰𝔬(3)×ℝ³ is the Lie algebra corresponding to SE(3) where 𝔰𝔬(3) is the Lie algebra corresponding to SO(3) [6]. The right hand of (7) is the infinitesimal generator of the action corresponding to 𝑉^𝑏_𝑎𝑏(𝑡), and it is left invariant under left multiplication by constant matrices [27]. These properties determine the trivialization way of local coordinate map in Section 3.2.1.

2.2. Dynamics Model

The force equations of the aircraft are given by [28] 𝑢̇𝑣̇𝑤̇=+𝑟⋅𝑣−𝑞⋅𝑤𝑝⋅𝑤−𝑟⋅𝑢𝑞⋅𝑢−𝑝⋅𝑤−sin𝜃sin𝜙⋅cos𝜃cos𝜙⋅cos𝜃𝑔+𝑞⋅𝑠𝑚𝐶_𝑋,𝑡𝑞⋅𝑠𝑚𝐶_𝑌,𝑡𝑞⋅𝑠𝑚𝐶_𝑍,𝑡+𝑇𝑚00, (8) where, 𝑢̇, 𝑣̇, and 𝑤̇ denote 𝑋-axis component, 𝑌-axis component, and 𝑍-axis component, respectively, of the aircraft acceleration in the body-fixed frame, 𝑚 is the aircraft mass, 𝑠 is the wing reference area, 𝑞 is the free-stream dynamic pressure, 𝑇 is the engine thrust, and 𝐶_𝑋,𝑡, 𝐶_𝑌,𝑡, and 𝐶_𝑍,𝑡 denote total 𝑋-axis force coefficient, total 𝑌-axis force coefficient, and total 𝑍-axis force coefficient, respectively.

One has 𝐶_𝑋,𝑡=𝐶_𝑋𝛼,𝛽+𝑐𝑞2𝑉_𝑇𝐶_𝑋𝑞(𝛼),𝐶_𝑌,𝑡=𝐶_𝑌+𝑏𝛼,𝛽2𝑉_𝑇𝐶_𝑌𝑟(𝛼)𝑟+𝐶_𝑌𝑝(𝛼)𝑝,𝐶_𝑍,𝑡=𝐶_𝑍𝛼,𝛽+𝑐𝑞2𝑉_𝑇𝐶_𝑍𝑞(𝛼). (9)

Denote [𝑞𝑠𝐶_𝑋,𝑡,𝑞𝑠𝐶_𝑌,𝑡,𝑞𝑠𝐶_𝑍,𝑡]^𝑇 and [𝑇,0,0]^𝑇 by 𝐹^𝑏_𝐴(𝑡) and 𝐹^𝑏_𝑇(𝑡), respectively; the former represents aerodynamics, and the latter represents thrust. Then, we have 𝑚𝜐̇^𝑏_𝑎𝑏(𝑡)=−𝑚𝜔^𝑏_𝑎𝑏(𝑡)×𝜐^𝑏_𝑎𝑏(𝑡)+𝑚𝑔𝑅^𝑇_𝑎𝑏(𝑡)𝑒₃+𝐹^𝑏_𝐴(𝑡)+𝐹^𝑏_𝑇(𝑡). (10)

The moment equations of the aircraft are given by [26]𝑝̇𝑞̇𝑟̇=𝐽_𝑥𝑧𝐽_𝑥−𝐽_𝑦+𝐽_𝑧𝑝𝑞−𝐽_𝑧𝐽_𝑧−𝐽_𝑧𝐽_𝑦+𝐽²_𝑥𝑧𝑟𝑞+𝐽_𝑧𝑞𝑠𝑏𝐶_𝑙,𝑡+𝐽_𝑥𝑧𝑞𝑠𝑏𝐶_𝑛,𝑡𝐽_𝑥𝐽_𝑧−𝐽²_𝑥𝑧𝐽_𝑧−𝐽_𝑥𝑝𝑟−𝐽_𝑥𝑧𝑝²−𝑟²+𝑞𝑠𝑐𝐶_𝑚,𝑡𝐽_𝑦𝐽_𝑥−𝐽_𝑦𝐽_𝑥+𝐽²_𝑥𝑧𝑝𝑞−𝐽_𝑥𝑧𝐽_𝑥−𝐽_𝑦+𝐽_𝑧𝑟𝑞+𝐽_𝑥𝑧𝑞𝑠𝑏𝐶_𝑙,𝑡+𝐽_𝑥𝑞𝑠𝑏𝐶_𝑛,𝑡𝐽_𝑥𝐽_𝑧−𝐽²_𝑥𝑧, (11)

where, 𝑏 is wing span, 𝐽_𝑥, 𝐽_𝑦, and 𝐽_𝑧 are moment of inertia about 𝑋-axis, 𝑌-axis, and 𝑍-axis, respectively, 𝐽_𝑥𝑧 is cross-product of inertia, and 𝐶_𝑙,𝑡, 𝐶_𝑚,𝑡, and 𝐶_𝑛,𝑡 denote total rolling-moment coefficient, total pitching-moment coefficient, and total yawing-moment coefficient, respectively. The remaining parameters can refer to [26] 𝐶_𝑙,𝑡=𝐶_𝑙𝛼,𝛽+Δ𝐶_{𝑙,𝛿∘𝑎=20}𝛿_𝑎20+Δ𝐶_{𝑙,𝛿∘𝑟=30}𝛿_𝑟+𝑏302𝑉_𝑇𝐶_𝑙𝑟(𝛼)𝑟+𝐶_𝑙𝑝(𝛼)𝑝,𝐶_𝑚,𝑡=𝐶_𝑚𝛼,𝛽𝜂_𝛿ℎ𝛿_ℎ+𝐶_𝑍,𝑡𝑥_𝑐𝑔,ref−𝑥_𝑐𝑔+𝑐𝑞2𝑉_𝑇𝐶_𝑚𝑞(𝛼),𝐶_𝑛,𝑡=𝐶_𝑛𝛼,𝛽−𝐶_𝑌,𝑡𝑥_𝑐𝑔,ref−𝑥_𝑐𝑔𝑐𝑏+Δ𝐶_{𝑛,𝛿∘𝑎=20}𝛿_𝑎20+Δ𝐶_{𝑛,𝛿∘𝑟=30}𝛿_𝑟+𝑏302𝑉_𝑇𝐶_𝑛𝑟(𝛼)𝑟+𝐶_𝑛𝑝(𝛼)𝑝. (12)

Denote [𝑞𝑠𝑏𝐶_𝑙,𝑡,𝑞𝑠𝑐𝐶_𝑚,𝑡,𝑞𝑠𝑏𝐶_𝑛,𝑡]^𝑇 by 𝑇_𝐵(𝑡), then we have 𝐽𝜔̇^𝑏_𝑎𝑏(𝑡)=−𝜔^𝑏_𝑎𝑏(𝑡)×𝐽𝜔^𝑏_𝑎𝑏(𝑡)+𝑇_𝐵(𝑡), (13) where 𝐽=^{𝐽𝑥0−𝐽𝑥𝑧0𝐽𝑦0}_{−𝐽𝑥𝑧0𝐽𝑧} is referred to as the inertia matrix of the rigid body.

Therefore, from the preceding equation and (10), we can construct a dynamics model 𝑉̇^𝑏_𝑎𝑏(𝑡)=Π⁻¹⋅(𝐵(𝑉^𝑏_𝑎𝑏(𝑡))⋅𝑉^𝑏_𝑎𝑏(𝑡)+𝐹(𝑡,𝑔_𝑎𝑏(𝑡))) in 𝔰𝔢(3), namely, 𝜔̇^𝑏_𝑎𝑏(𝑡)𝜐̇^𝑏_𝑎𝑏(𝑡)=𝐽00𝑚𝐼₃⁻¹⋅𝐽𝜔^𝑏_𝑎𝑏(𝑡)00−𝑚𝜔^𝑏_𝑎𝑏(𝑡)𝜔^𝑏_𝑎𝑏(𝑡)𝜐^𝑏_𝑎𝑏(𝑡)+𝑇_𝐵(𝑡)𝑚𝑔𝑅^𝑇_𝑎𝑏(𝑡)𝑒₃+𝐹^𝑏_𝐴(𝑡)+𝐹^𝑏_𝑇(𝑡). (14)

3. Geometric Pseudospectral Method on SE(3)

3.1. General Pseudospectral Method in Euclidean Space

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 [24]. Let us take solving the following differential equation in ℝ^𝑛 at 𝑡_𝑓, for example, 𝑥̇(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑥𝑡₀=𝑥₀∈ℝ^𝑛,𝑡₀≤𝑡≤𝑡_𝑓. (15)

Firstly, we equally divide a time interval [𝑡₀,𝑡_𝑓] into several time subintervals [𝑡_𝑘,𝑡_𝑘+1] having a length ℎ and transform the time interval 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] to the time interval 𝜏∈[−1,1] via the affine transformation 𝑡𝑡=^𝑘+1−𝑡_𝑘2𝑡𝜏+^𝑘+1+𝑡_𝑘2. (16)

Thus, accordingly, (15) over [𝑡_𝑘,𝑡_𝑘+1] becomes 𝑑𝑥(𝜏)=𝑡𝑑𝜏^𝑘+1−𝑡_𝑘2𝑓(𝜏,𝑥(𝜏)),−1≤𝜏≤1. (17)

Let 𝑐₁<⋯<𝑐_𝑁 be collocation points in [−1,1] and 𝑐₀=−1. We approximate the solution of (17) by the following formula: 𝑥(𝜏)≈𝑋(𝜏)=^𝑁_𝑖=0𝑥_𝑖⋅𝐿_𝑖(𝜏), (18) where 𝐿_𝑖(𝜏)=∏^𝑁_{𝑗=0,𝑗≠𝑖}(𝜏−𝑐_𝑗)/(𝑐_𝑖−𝑐_𝑗) is the Lagrange polynomials, satisfying the isolation property 𝐿_𝑖𝑐_𝑗=𝛿_𝑖𝑗=1,𝑖=𝑗,0,𝑖≠𝑗. (19)

Equation (18) together with the isolation property leads to the fact that 𝑥_𝑖=𝑥𝑐_𝑖 (20) thus 𝑋(𝑐_𝑖)=𝑥(𝑐_𝑖).

Through expressing the derivative of the Lagrange polynomials at the collocation points in differential matrix form 𝐷=[𝑑_𝑗𝑖]∈ℝ^{𝑁×(𝑁+1)}, 𝑑_𝑗𝑖=𝐿̇_𝑖𝑐_𝑗=^𝑁_𝑙=0∏^𝑁_{𝑘=0,𝑘≠𝑖,𝑙}𝑐_𝑗−𝑐_𝑘∏^𝑁_{𝑘=0,𝑘≠𝑖}𝑐_𝑖−𝑐_𝑘,𝑗=1,…,𝑁,𝑖=0,…,𝑁. (21)

We can write (17) at the collocation points as a set of differential algebra equations as follows: ^𝑁_𝑖=0𝑑_𝑗𝑖⋅𝑋𝑐_𝑖=𝑡_𝑘+1−𝑡_𝑘2𝑓𝑐_𝑗,𝑋𝑐_𝑗,𝑗=1,…,𝑁. (22)

Based on the previous equations, we establish defect equations 𝜍_𝑗=^𝑁_𝑖=0𝑑_𝑗𝑖⋅𝑋𝑐_𝑖−𝑡_𝑘+1−𝑡_𝑘2𝑓𝑐_𝑗,𝑋𝑐_𝑗,𝑗=1,…,𝑁 (23) and apply iterative algorithms to the previous equations so that determining the approximation to 𝑥(𝜏) at the collocation points. Finally, according to the following formula, we obtain the approximation to (15) at the endpoint 𝑡_𝑘+1 of time subinterval [𝑡_𝑘,𝑡_𝑘+1]: 𝑥𝑡_𝑘+1≈𝑋𝑡_𝑘+𝑡_𝑘+1−𝑡_𝑘2^𝑁_𝑗=1𝜔_𝑗⋅𝑓𝑐_𝑗,𝑋𝑐_𝑗, (24) where 𝜔_𝑗=∫¹₋₁𝐿_𝑗(𝑡)d𝑡 is quadrature weight corresponding to the collocation points. This approximate solution will be initial value of 𝑥(𝑡) over [𝑡_𝑘+1,𝑡_𝑘+2] and so on.

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 𝑇_𝑁(𝑡) or Legendre polynomials 𝑃_𝑁(𝑡) belonging to the orthogonal polynomial [29]. The benefit of using the orthogonal method over the standard method is that the quadrature approximation to a definite integral is extremely accurate [24]. Furthermore, according to whether the endpoint is or is not a collocation point, pseudospectral methods fall into three general categories [30]: Gauss methods, neither of the endpoints −1 or 1, is a collocation point; Radau methods, at most one of the endpoints −1 or 1, are a collocation point; Lobatto methods, both of the endpoints −1 and 1, are collocation points. In the rest of this paper, we will develop our geometric pseudospectral method based on Legendre-Gauss points, abbreviated as Gauss points in the following, for two main reasons: firstly, there is an intimate relationship between Gauss method and some implicit RK method whose coefficients in the Butcher Tableau can be used by our method for computing configuration [2]; secondly, when Gauss method is used for transcribing the continues optimal control problem into a discrete nonlinear programming problem, it does not suffer from a defect in the optimality conditions at the boundary points due to the endpoints being not collocation points [23].

3.2. Geometric Pseudospectral Method on SE(3)

First recall and rewrite the aircraft rigid-body dynamics model Equations (7) and (14) in Section 2 as follows: 𝑔̇‌_𝑎𝑏(𝑡)=𝑔_𝑎𝑏(𝑡)⋅𝑉^𝑏_𝑎𝑏(𝑡),𝑉̇^𝑏_𝑎𝑏(𝑡)=Π⁻¹⋅𝐵𝑉^𝑏_𝑎𝑏(𝑡)⋅𝑉^𝑏_𝑎𝑏(𝑡)+𝐹𝑡,𝑔_𝑎𝑏(𝑡)≜𝑓𝑉^𝑏_𝑎𝑏(𝑡),𝑔_𝑎𝑏(𝑡). (25)(26)

For the kinematics Equation (25), it is well known that the solution of (25) stays on SE(3) for all 𝑡≥𝑡₀. How to use pseudospectral method to solve (25) while preserving the structural feature of the differential equation under discretization of 𝑔_𝑎𝑏(𝑡) is the main problem to be solved in this section. In ℝ^𝑛, both the solution space and the tangent space are linear vector space. Classical numerical methods, including general pseudospectral method, just rely on domain space consisting of vectors. However, the special Euclidean group is a nonlinear manifold, so using classical numerical methods directly to solve differential equation on SE(3) will be not able to preserve its Lie group structure. Reference [21] indicated that any differential equation in the form of an infinitesimal generator on a homogeneous space (a manifold with a transitive Lie group action) is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogenous space. Also, a Lie algebra corresponding to the Lie group is a vector space with the additional structure of a commutator. For the previous reasons, the Lie algebra is the natural choice of space for our geometric pseudospectral method. First, we will apply the equivariant map to transform (25) evolving on SE(3) into an equivalent differential equation evolving in 𝔰𝔢(3). Next, we use the Gauss pseudospectral method to solve the equivalent differential equation in order to obtain 𝑝th-order truncated approximation of the equation at the Gauss points and the endpoints, followed by mapping the approximate solution back to configuration space via local coordinate map 𝑓.

For the dynamics equation (26), because the velocity 𝑉^𝑏_𝑎𝑏(𝑡) belongs to 𝔰𝔢(3), and actually 𝔰𝔢(3) is isomorphic to the ℝ⁶, we can use Gauss pseudospectral method directly on (26) to compute the velocities at the Gauss points and the endpoint.

3.2.1. Computing Configurations at the Gauss Points and the Endpoint

In order to transform (25) on SE(3) into an equivalent differential equation evolving on 𝔰𝔢(3), we briefly describe the basic idea of equivariant map.

Definition 1 (equivariant map). Let ℳ and 𝒩 be manifolds, and let 𝐺 be a Lie group which acts on ℳ by Φ_𝑔∶ℳ→ℳ and on 𝒩 by Ψ_𝑔∶𝒩→𝒩. A smooth map 𝑓∶ℳ→𝒩 is called equivariant with respect to these actions if, for all 𝑔∈𝐺, 𝑓∘Φ_𝑔=Ψ_𝑔∘𝑓 (27) that is, the following diagram commutes of equivariant map 𝑓: (28)
First, from the definition of an action of 𝐺 on ℳ, we can obtain an equivariant map Φ_𝑦∶𝐺→ℳ with respect to the left translation action 𝐿_𝑔 of 𝐺 on itself and an action 𝐺 of Φ_𝑔 on ℳ as follows: Φ_𝑦∘𝐿_𝑔=Φ_𝑔∘Φ_𝑦. (29)
It is known that there is a local coordinate map 𝑓∶𝔤→𝐺 on 𝐺 with the Lie algebra 𝔤; the most typical one is exponential map exp∶𝔤→𝐺. At this point, we need to find an action 𝐵_𝑔 of 𝐺 on 𝔤 such that 𝑓 will be an equivariant map with 𝐵_𝑔 and the left action of 𝐺 on itself, namely, 𝑓∘𝐵_𝑔=𝐿_𝑔∘𝑓. (30)
In the case where 𝑓 is the exponential map, 𝐵_𝑔 is nothing else than the well-known Baker-Campbell-Hausdorff (BCH) formula:𝐵_𝑔(𝑢)=log𝑔⋅exp(𝑢), (31) where log∶𝐺→𝔤 is called the logarithm map. Since composition of two equivariant maps is also an equivariant map, we can construct an equivariant map Φ_𝑦∘𝑓 from 𝔤 to ℳ with respect to the action 𝐵_𝑔 on 𝔤 and Φ on ℳ. Diagram commutes of composition Φ_𝑦∘𝑓 are as follows: (32)
The theorem 3.6 of [21] stated that if 𝜙 is an equivariant map, then the infinitesimal generators of the action with respect to the same element 𝜉∈𝔤 are 𝜙-related vector fields. Thus, the infinitesimal generators 𝜉_𝔤 and 𝜉_ℳ of the flows 𝐵_{exp(𝑡𝜉)} and Φ_{exp(𝑡𝜉)}, respectively, on 𝔤 and ℳ are Φ_𝑦∘𝑓-related; that is, 𝜉_ℳ∘Φ_𝑦∘𝑓=𝑇Φ_𝑦∘𝑇𝑓∘𝜉_𝔤. (33)
Finally, we need to determine what 𝜉_𝔤 is, and the following theorem gives the conditions that it needs to meet.

Theorem 2 (see [21]). Let 𝑓∶𝔤→𝐺 be a coordinate map on 𝐺 and Φ_𝑦∘𝑓 equivariant with respect to the flows 𝐵_{exp(𝑡𝜉)} and Φ_{exp(𝑡𝜉)}. The infinitesimal generator of 𝐵_𝑔 satisfying (33) is 𝜉_𝔤(𝑢)=d𝑓_𝑢⁻¹(𝜐). d𝑓∶𝔤→𝔤 is the trivialization of 𝑇𝑓, defined as d𝑓_𝑢=𝑇𝑅_𝑓(𝑢)⁻¹∘𝑇𝑓_𝑢.

The following commutative diagram describing the equivariant maps of composition Φ_𝑦∘𝑓 (left) and its infinitesimal generator (right) sums up the previous processes: (34)

Also, [21] stated that any differential equation on ℳ which can be written as an infinitesimal generator fits into the previous framework, including all Lie type equations, isospectral flows, and rigid frames. It is noted that, as mentioned in Section 2.1, the right hand of (7) or (25) is an infinitesimal generator on SE(3), and here we assume that the solution 𝑔_𝑎𝑏(𝑡) of (25) can be written as the following form [13]: 𝑔_𝑎𝑏(𝑡)=𝑔_𝑎𝑏𝑡₀⋅𝑓(𝑢(𝑡)). (35)

In order to obtain the explicit expression of 𝑢(𝑡), we need to solve the following differential equation: 𝑢̇(𝑡)=d𝑓⁻¹_−𝑢𝑉^𝑏_𝑎𝑏(𝑡). (36)

For deriving the previous formula, we differentiate (35) as follows: dd𝑡𝑔_𝑎𝑏(𝑡)=dd𝑡𝑔_𝑎𝑏𝑡₀(𝑢(𝑡=⋅𝑓))𝑔_𝑎𝑏𝑡₀⋅d𝐿_{𝑓(𝑢(𝑡))}∘d𝑓_{−𝑢(𝑡)}(𝑢̇(𝑡=))𝑔_𝑎𝑏𝑡₀⋅𝑓(𝑢(𝑡))⋅d𝑓_{−𝑢(𝑡)}(𝑢̇(𝑡)). (37) Comparing (37) with (25) and (35), we have 𝑉^𝑏_𝑎𝑏(𝑡)=d𝑓_{−𝑢(𝑡)}(𝑢̇(𝑡)). (38) Taking the inverse of previous formula, we obtain (36).

It seems from the equivariant map and the previous formula derivation that solving equation (25) can be transformed into a research on the equivalent equation (36), and the manifold ℳ is SE(3) at this point. d𝑓⁻¹_−𝑢(𝑉^𝑏_𝑎𝑏(𝑡)) has different forms, according to the different choices of local coordinate map 𝑓. In the case where 𝑓 is the exponential map, the vector field d𝑓⁻¹_−𝑢 on 𝔤 can be represented as [13] dexp⁻¹_−𝑢𝑉^𝑏_𝑎𝑏(𝑡)=𝑉^𝑏_𝑎𝑏(𝑡)+12ad_𝑢𝑉^𝑏_𝑎𝑏(𝑡+_∞𝑘=2)𝐵_𝑘𝑘!ad^𝑘_𝑢𝑉^𝑏_𝑎𝑏(𝑡), (39) where the adjoint operator ad_𝑢∶𝔤→𝔤 is Lie bracket or commutator, ad_𝑢(𝑉^𝑏_𝑎𝑏(𝑡))=[𝑢,𝑉^𝑏_𝑎𝑏(𝑡)], satisfying recursive expression ad^𝑘_𝑢(𝑉^𝑏_𝑎𝑏(𝑡))=[𝑢,ad_𝑢^𝑘−1(𝑉^𝑏_𝑎𝑏(𝑡))], and 𝐵_𝑘 are Bernoulli numbers [31] as follows: 𝐵_𝑘=161,−,130142,−,53066,−6912730,…,𝑘=2𝑁,𝑁∈ℤ⁺,0,𝑘=2𝑁+1,𝑁∈ℤ⁺. (40)

In the context of rigid-body motion, the right trivialization corresponds to the differential equation with tangent vectors 𝐴𝑌∈𝔛(ℳ), 𝐴∈𝔤, and 𝑌∈𝐺, and the left trivialization corresponds to the differential equation with tangent vectors 𝑌𝐴∈𝔛(ℳ),𝐴∈𝔤, and 𝑌∈𝐺. The former represents the change of configuration in a space frame, and the latter represents the change of configuration in the body-fixed frame. Here, let d𝑓 be the left trivialization of 𝑇𝑓, since the configuration model established in Section 2.1 is in the body-fixed frame. In the case of d𝑓 being the left trivialization and 𝑓 being the exponential map, we can find an explicit approximation to 𝑢(𝑡) called the Magnus series expansion via applying Picard iterations to (36) [31] as follows: 𝑢(𝑡)=^𝑡₀𝑉^𝑏_𝑎𝑏𝜉₁𝑑𝜉₁+12^𝑡₀^𝜉1₀𝑉^𝑏_𝑎𝑏𝜉₂𝑑𝜉₂,𝑉^𝑏_𝑎𝑏𝜉₁𝑑𝜉₁+112^𝑡₀^𝜉1₀𝑉^𝑏_𝑎𝑏𝜉₂𝑑𝜉₂,^𝜉1₀𝑉^𝑏_𝑎𝑏𝜉₂𝑑𝜉₂,𝑉^𝑏_𝑎𝑏𝜉₁×𝑑𝜉₁+⋯. (41)

The previous formula can be rewritten as an infinite sum 𝑢(𝑡)=∑^∞_𝑘=0𝐻_𝑘(𝑡), where each 𝐻_𝑘(𝑡) is a linear combination of terms that have equal numbers 𝑘 of integrals (exclude the outermost integral from 0 to 𝑡) and commutators. We denote each term of 𝐻_𝑘(𝑡) by 𝐶_𝜅(𝜉) and associate it with a binary tree 𝜅∈𝕋_𝑘, where 𝕋_𝑘 is a set which includes all trees with the same number 𝑘 of integrals and commutators. Then, we can rewrite (41) as 𝑢(𝑡)=^∞_{𝑘=0𝜅∈𝕋𝑘}𝛼(𝜅)^𝑡₀𝐶_𝜅(𝜉)𝑑𝜉,𝑡≥0, (42) where 𝛼(𝜅)=𝐵_𝑘𝑘!^𝑘_𝑖=1𝛼𝜅_𝑖,𝑘∈ℤ⁺. (43)

It is worthwhile to note that, unlike right trivialization in [13], here, we take left-trivialized tangent of local coordinate map, thus 𝐵_𝑘=12𝐵,for𝑘=1,_𝑘,for𝑘≠1. (44)

Based on the binary tree theory, we can find the following 𝑝th-order approximate solution of (36), 𝑢^[𝑝](𝑡)=^𝑝−1_{𝑚=0𝜅∈𝔽𝑚}𝛼(𝜅)^𝑡₀𝐶_𝜅(𝜉)𝑑𝜉, (45) where 𝔽_𝑚 is the set of all 𝑚-power rooted trees. Thereby, we give the Magnus series terms with its coefficients 𝛼(𝜅) under left trivialization in Table 2, which correspond to the top eight orders rooted trees.

Then, we use Gauss pseudospectral method to solve (45), so that we get the approximate solution of (36) at the Gauss points 𝑡_𝑘<𝜏₁,…,𝜏_𝑁<𝑡_𝑘+1 and the endpoint 𝑡_𝑘+1≜𝑡_𝑘+ℎ. To begin with, assume that all of velocities 𝑉^𝑏_𝑎𝑏(𝜏_𝑖) at the Gauss points are known previously; we use the Lagrange polynomials at these points to approximate the velocity at any time 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] as follows: 𝑉^𝑏_𝑎𝑏(𝑡)≈^𝑁_𝑖=1𝐿_𝑖𝑡−𝑡_𝑘ℎ⋅𝑉^𝑏_𝑎𝑏𝜏_𝑖. (46)

Next, through transforming (46) into [0,1] via an affine transformation 𝜉=(𝑡−𝑡_𝑘)/ℎ, and inserting it to (45), we have 𝑢^[𝑝](𝑡)=𝐼₁(𝑡)+12𝐼₂(𝑡)+14𝐼₃(𝑡)+1𝐼12⁴(𝑡)+1𝐼24⁵(𝑡)+1𝐼24⁶(𝑡)+18𝐼₈(𝑡)+⋯, (47) where each term 𝐼(𝑡) is the following form of multivariate quadrature: 𝐼(𝑡)=^𝑡₀^𝜉𝑚2₀^𝜉𝑚3₀⋯_{𝜉𝑚𝑠0}ℒℎ⋅𝑉^𝑏_𝑎𝑏𝜉₁,…,ℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑠𝑑𝜉_𝑠⋯𝑑𝜉₂𝑑𝜉₁=_𝑆ℒℎ⋅𝑉^𝑏_𝑎𝑏𝜉₁,…,ℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑠𝑑𝜉_𝑠⋯𝑑𝜉₂𝑑𝜉₁, (48) where 𝑆={𝜉₁,…,𝜉_𝑠∈ℝ|𝜉₁∈[0,𝑡],𝜉_ℓ∈[0,𝜉_𝑚ℓ],𝑚_ℓ∈{1,2,…,ℓ−1},ℓ=2,3,…,𝑠} and ℒ denotes multivariate function.

The same as the univariate case, we use the following quadrature formulae (49) and (50) to evaluate the approximate solution of (48) at 𝜏₁,…,𝜏_𝑁 and 𝑡_𝑘+1, respectively: 𝐼𝜏_𝑖=𝑆^𝑖ℒℎ⋅𝑉^𝑏_𝑎𝑏𝜉₁,…,ℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑠−1𝑑𝜉_𝑠−1⋯𝑑𝜉₂𝑑𝜉₁≈_{𝑘∈𝐶𝑁𝑠−1}𝑎^𝑖_𝑘ℒℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑘1,…,ℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑘𝑠−1,𝑎^𝑖_𝑘=𝑆^𝑖𝑠−1_ℓ=1𝐿_𝑘ℓ𝜉_ℓ𝑑𝜉_ℓ,𝑘₁,𝑘₂,…,𝑘_𝑠−1∈𝑘, (49) where 𝑆^𝑖={𝜉₁,…,𝜉_𝑠−1∈ℝ∣𝜉₁∈[0,𝑐_𝑖],𝜉_ℓ∈[0,𝜉_𝑚ℓ],𝑚_ℓ∈{1,2,…,ℓ−1},ℓ=2,3,…,𝑠−1}, and 𝐶^𝑁_𝑠−1 is the set of all combinations of (𝑠−1)-tuples 𝑘 from the set {1,2,…,𝑁}, excluding 𝑠−1 repeats of the same number in the set.

One has 𝐼𝑡_𝑘+1=𝑆ℒℎ⋅𝑉^𝑏_𝑎𝑏𝜉₁,…,ℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑠𝑑𝜉_𝑠⋯𝑑𝜉₂𝑑𝜉₁≈_{𝑘∈𝐶𝑁𝑠}𝑏_𝑘ℒℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑘1,…,ℎ⋅𝑉^𝑏_𝑎𝑏𝜉_𝑘𝑠𝑏_𝑘=𝑆^𝑠_ℓ=1𝐿_𝑘ℓ𝜉_ℓ𝑑𝜉_ℓ,𝑘₁,𝑘₂,…,𝑘_𝑠∈𝑘, (50) where 𝑆={𝜉₁,…,𝜉_𝑠∈ℝ|𝜉₁∈[0,1],𝜉_ℓ∈[0,𝜉_𝑚ℓ],𝑚_ℓ∈{1,2,…,ℓ−1},ℓ=2,3,…,𝑠} and 𝐶^𝑁_𝑠 is similar to 𝐶^𝑁_𝑠−1.

Through computing 𝐼(𝑡) of (47) term by term, we obtain 𝑝th-order approximate solution 𝑢^[𝑝](𝑡) of 𝑢(𝑡) at the discrete points. As seen from (48), 𝐼(𝑡) depends on the velocities at the Gauss points, and the velocities need to be solved in an iterative manner which will be explained in next section. Thereby, the solutions of configuration at the Gauss points need to be also determined in an iterative manner. Finally, we use 𝑢^[𝑝](𝜏_𝑖) at the Gauss points to compute configuration corresponding to the same points via the following formula: 𝑔_𝑎𝑏𝜏_𝑖=𝑔_𝑎𝑏𝑡_𝑘⋅𝑓𝑢^[𝑝]𝜏_𝑖. (51)

There are several choices of local coordinate map 𝑓∶𝔤→𝐺, such as exponential map and Cayley map [21]. We incline to the Cayley map, since it is very easy to compute, despite it is only an approximation to the integral curve defined by exponential map as follows: cay_SE(3)(𝑢)=cay_SO(3)(𝜔)𝐴_cay(𝜔)⋅𝜐01cay_SO(3)(𝜔)=𝐼₃+44+‖𝜔‖²(𝜔)𝜔+²2, (52) where ‖⋅‖ is a standard Euclidean norm, and 𝐴_cay(𝜔)=24+‖𝜔‖²2𝐼₃+𝜔. (53)

In the same way, after 𝑢^[𝑝](𝑡_𝑘+1) is obtained via evaluating 𝐼(𝑡_𝑘+1) termwise, we use (52) to update configuration 𝑔_𝑎𝑏(𝑡_𝑘+1) at the endpoint 𝑡_𝑘+1, 𝑔_𝑎𝑏𝑡_𝑘+1=𝑔_𝑎𝑏𝑡_𝑘⋅𝑓𝑢^[𝑝]𝑡_𝑘+1. (54)

3.2.2. Computing Velocities at the Gauss Points and the Endpoint

As mentioned above, here, we can directly use general pseudospectral in Section 3.1 to compute velocities at the Gauss points 𝜏₁,…,𝜏_𝑁 and the endpoint 𝑡_𝑘+1. However, due to (26) being implicit, we have to use iterative algorithm to solve it. When computing configurations in the preceding section, we assumed that 𝑉^𝑏_𝑎𝑏(𝜏_𝑖),𝑖=1,…,𝑁 in (46) are known in advance. Hence, we need to give their solutions, and we are going to do it in this section.

To begin with, we write (26) at all of the Gauss points in a compact matrix form, 𝑉^𝑏(_𝑎𝑏^new)𝜏₁𝑉^𝑏(_𝑎𝑏^new)𝜏₂⋮𝑉^𝑏(_𝑎𝑏^new)𝜏_𝑁=𝑡_𝑘+1−𝑡_𝑘2𝐷⁻¹⋅^{Π−1⋅𝐵𝑉𝑏(𝑎𝑏old)𝜏1⋅𝑉𝑏(𝑎𝑏old)𝜏1𝜏+𝐹1,𝑔(old)𝑎𝑏𝜏1Π−1⋅𝐵𝑉𝑏(𝑎𝑏old)𝜏2⋅𝑉𝑏(𝑎𝑏old)𝜏2𝜏+𝐹2,𝑔(old)𝑎𝑏𝜏2}_{⋮Π−1⋅𝐵𝑉𝑏(𝑎𝑏old)𝜏𝑁⋅𝑉𝑏(𝑎𝑏old)𝜏𝑁𝜏+𝐹2,𝑔(old)𝑎𝑏𝜏𝑁}−𝐷⁻¹⋅𝑑₁₀𝑑₂₀⋮𝑑_𝑁0⋅𝑉^𝑏_𝑎𝑏𝑡_𝑘, (55) where 𝐷∈ℝ^𝑁×𝑁 is the submatrix of 𝐷 in Section 3.1, and it is a reversible matrix, 𝑑𝐷=¹¹𝑑₁₂⋯𝑑_1𝑁𝑑₂₁𝑑₂₂⋯𝑑_2𝑁𝑑⋮⋮⋱⋮_𝑁1𝑑_𝑁2⋯𝑑_𝑁𝑁. (56)

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: 𝑉^𝑏_𝑎𝑏𝑡_𝑘+1=𝑉^𝑏_𝑎𝑏𝑡_𝑘+𝑡_𝑘+1−𝑡_𝑘2×^𝑁_𝑖=1𝜔_𝑖⋅𝑓𝑉^𝑏(_𝑎𝑏^new)𝜏_𝑖,𝑔^(new)_𝑎𝑏𝜏_𝑖. (57)

3.2.3. 4th-Order Algorithm of Geometric Pseudospectral Method

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.

Implementation and the Order Condition. Legendre-Gauss point 𝜏_𝑖 is the root of Legendre polynomial 𝑃_𝑁(𝜏)=(1/2^𝑁𝑁!)(𝑑^𝑁/𝑑𝜏^𝑁){[𝜏²−1]^𝑁} of degree 𝑁. In fact, a convenient way to compute the Legendre-Gauss points is via the eigenvalues of the following tridiagonal Jacobi matrix: 𝐽=0𝛽₁𝛽₁0𝛽₂⋱⋱⋱⋱0𝛽_𝑁−1𝛽_𝑁−10,𝛽_𝑛=𝑛√4𝑛²−1. (58)

Due to the fact that (25)-(26) are implicit, therefore, we have to use iterative algorithm to update the velocities and configurations at the Gauss points. In the interactive processes, we need to compute the velocity deviation and configuration deviation, respectively, between the adjacent interative steps. For velocity deviation, since the Lie algebra space 𝔰𝔢(3) in which the velocities belong is isomorphic to ℝ⁶, so we are able to use minus directly. However, for configuration deviation, because configuration space is a nonlinear manifold, we introduce the following natural error of configuration [27]: 𝑒=𝑔^(step+1)_𝑎𝑏⁻¹𝑔^(step)_𝑎𝑏=𝑅^(step+1)𝑇𝑅^(step)𝑅^(step+1)𝑇𝑝^(step)−𝑝^(step+1).01 (59)

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 ‖(𝑅^(step+1))^𝑇𝑅^(step)‖₂ denotes angle difference Δ𝜃=𝜃^(step+1)−𝜃^(step) between configurations, and the 2-norm ‖(𝑅^(step+1))^𝑇(𝑝^(step)−𝑝^(step+1))‖₂ denotes distance difference Δ𝑑=(𝑥^(step+1)−𝑥^(step))²+(𝑦^(step+1)−𝑦^(step))² between configurations. The last one required to pay attention to is to ensure computing configurations and velocities at the same Gauss points.

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 (47) can be further reduced [13]. According to the time symmetry, function 𝑢^[𝑝](𝑡) can be expanded as the odd power of time 𝑡, and then 𝑢^[2𝑞](𝑡)=𝑢_2𝑞−1(𝑡)+𝒪(𝑡^2𝑞+1). For this purpose, we provide the top eight orders reduced truncated Magnus series expansion in Table 3 which have been used in step 2.2.3.3 and step 2.2.6.1 of Algorithm 1. Secondly, let the Magnus series expansion be truncated up to the 𝑝th-order trees for 𝑔_𝑎𝑏(𝑡_𝑘+1) and 𝑝−1th-order trees for the intermediate stages 𝑔_𝑎𝑏(𝜏_𝑖), then the resulting schemes have 𝑝th-order [32]. Moreover, [2] stated that the interpolatory quadrature formula with 𝑁 Gauss points is 2𝑁th-order. In summary, our 4th-order geometric pseudospectral method only depends on two Gauss points.

Step 1. Initialization:  Initialize time interval 𝑡0,𝑡𝑓, step length ℎ, configuration 𝑔𝑎𝑏𝑡0, velocity 𝑉𝑏𝑎𝑏𝑡0, and let 𝑘 be 0; Step 2. Main loop:  Step 2.1. Compute the Gauss points 𝜏1,𝜏2 in 𝑡𝑘,𝑡𝑘+1, corresponding differentiation matrix 𝐷2×3  and quadrature weights 𝜔1,𝜔2;  Step 2.2. Compute velocities 𝑉𝑏𝑎𝑏𝜏1 and 𝑉𝑏𝑎𝑏𝜏2 at the Gauss points:     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 𝑔(step)𝑎𝑏𝜏1 and 𝑔(step)𝑎𝑏𝜏2 be 𝑔𝑎𝑏𝑡𝑘, that of both 𝑉𝑏(step)𝑎𝑏𝜏1 and 𝑉𝑏(step)𝑎𝑏𝜏2 be 𝑉𝑏𝑎𝑏𝑡𝑘;     Step 2.2.3. Child loop:    Step 2.2.3.1. According to the following equations, obtain 𝑉̇𝑏(step)𝑎𝑏𝜏1,𝑉̇𝑏(step)𝑎𝑏𝜏2𝑇,        𝑉̇𝑏(step)𝑎𝑏𝜏1𝑉̇𝑏(step)𝑎𝑏𝜏2=𝑡𝑘+1−𝑡𝑘2Π−1⋅𝐵𝑉𝑏(step)𝑎𝑏𝜏1⋅𝑉𝑏(step)𝑎𝑏𝜏1+𝐹𝜏1,𝑔(step)𝑎𝑏𝜏1Π−1⋅𝐵𝑉𝑏(step)𝑎𝑏𝜏2⋅𝑉𝑏(step)𝑎𝑏𝜏2+𝐹𝜏2,𝑔(step)𝑎𝑏𝜏2−𝑑01𝑑02𝑉𝑏𝑎𝑏𝑡𝑘    Step 2.2.3.2. Update velocities 𝑉𝑏(step+1)𝑎𝑏𝜏1,𝑉𝑏(step+1)𝑎𝑏𝜏2𝑇 at the Gauss point,        𝑉𝑏(step+1)𝑎𝑏𝜏1𝑉𝑏(step+1)𝑎𝑏𝜏2=𝐷−1𝑉̇𝑏(step)𝑎𝑏𝜏1𝑉̇𝑏(step)𝑎𝑏𝜏2    Step 2.2.3.3. Compute 4th-order reduced truncated Magnus series expansion 𝑢[4]𝜏1,𝑢[4]𝜏2𝑇 at the Gauss points, 𝑢[4]𝜏1𝑢[4]𝜏2=14ℎ⋅𝑉𝑏(step)𝑎𝑏𝜏1+14−√36ℎ⋅𝑉𝑏(step)𝑎𝑏𝜏2−125−√723ℎ242⋅𝑉𝑏(step)𝑎𝑏𝜏1,𝑉𝑏(step)𝑎𝑏𝜏214+√36ℎ⋅𝑉𝑏(step)𝑎𝑏𝜏1+14ℎ⋅𝑉𝑏(step)𝑎𝑏𝜏2−12−5−√723ℎ242⋅𝑉𝑏(step)𝑎𝑏𝜏1,𝑉𝑏(step)𝑎𝑏𝜏2    Step 2.2.3.4. Using the Cayley map, update configurations 𝑔(step+1)𝑎𝑏𝜏1,𝑔(step+1)𝑎𝑏𝜏2𝑇at the Gauss points, 𝑔(step+1)𝑎𝑏𝜏1𝑔(step+1)𝑎𝑏𝜏2=𝑔𝑎𝑏𝑡𝑘⋅cay𝑢[4]𝜏1𝑔𝑎𝑏𝑡𝑘⋅cay𝑢[4]𝜏2    Step 2.2.3.5. Compute the configuration deviation and velocity deviation respectively between the adjacent steps,    and let the larger one between them be iterative deviation at the current step, norm𝑔(err)𝑎𝑏=logSE(3)𝑔(step+1)𝑎𝑏𝜏1−1⋅𝑔(step)𝑎𝑏𝜏1logSE(3)𝑔(step+1)𝑎𝑏𝜏2−1⋅𝑔(step)𝑎𝑏𝜏22, norm𝑉𝑏(err)𝑎𝑏=𝑉𝑏(step+1)𝑎𝑏𝜏1−𝑉𝑏(step)𝑎𝑏𝜏12𝑉𝑏(step+1)𝑎𝑏𝜏2−𝑉𝑏(step)𝑎𝑏𝜏22∞    Step 2.2.3.6. Substitute 𝑔(step+1)𝑎𝑏 and 𝑉𝑏(step+1)𝑎𝑏 at 𝜏1 and 𝜏2 into 𝑔(step)𝑎𝑏 and 𝑉𝑏(step)𝑎𝑏 respectively, at the same time, let step plus 1;    Step 2.2.3.7. Compute the difference between iterative deviation and specified tolerance,    so that determine whether or not to terminate the iterative process.     Step 2.2.4. End child loop;     Step 2.2.5. Use the final iterative results 𝑔(step)𝑎𝑏 and 𝑉𝑏(step)𝑎𝑏 in child loop to compute the velocity 𝑉𝑏𝑎𝑏𝑡𝑘+1 at the endpoint 𝑡𝑘+1, 𝑉𝑏𝑎𝑏𝑡𝑘+1=𝑉𝑏𝑎𝑏𝑡𝑘+𝑡𝑘+1−𝑡𝑘22𝑖∑=1𝜔𝑖⋅Π−1⋅𝐵𝑉𝑏(step)𝑎𝑏𝜏𝑖𝑉𝑏(step)𝑎𝑏𝜏𝑖+𝐹𝜏𝑖,𝑔(step)𝑎𝑏𝜏𝑖     Step 2.2.6. Compute configuration 𝑔𝑎𝑏𝑡𝑘+1 at 𝑡𝑘+1:    Step 2.2.6.1. Compute 4th-order reduced truncated Magnus series expansion 𝑢[4]𝑡𝑘+1 at 𝑡𝑘+1, 𝑢[4]𝑡𝑘+1=12ℎ⋅𝑉𝑏(step)𝑎𝑏𝜏1+𝑉𝑏(step)𝑎𝑏𝜏2+√3ℎ122⋅𝑉𝑏(step)𝑎𝑏𝜏1,𝑉𝑏(step)𝑎𝑏𝜏2    Step 2.2.6.2. Use the Cayley map to update configuration 𝑔𝑎𝑏𝑡𝑘+1,        𝑔𝑎𝑏𝑡𝑘+1=𝑔𝑎𝑏𝑡𝑘⋅cay𝑢[4]𝑡𝑘+1     Step 2.2.7. Compare 𝑡𝑘+1 with 𝑡𝑓, so that determine whether or not to terminate the main loop; Step 3. End loop.

Algorithm 1 

4th-order geometric pseudospectral algorithm on SE(3).

3.3. Numerical Test on Free-Floating Rigid Body

In this section, the performance of Algorithm 1 is tested through evolution of the following free-floating rigid body dynamics equations (60)-(61) on SE(3). Note that, under the condition of no external force, (60)-(61) are equivalent to (25)-(26). Table 1 summarizes the parameters used in the numerical test as follows: 𝑅̇_𝑎𝑏(𝑡)𝑝̇^𝑎_𝑎𝑏(𝑡)=𝑅00^𝑎𝑏(𝑡)𝑝^𝑎_𝑎𝑏(𝑡)𝜔01^𝑏𝑎𝑏(𝑡)𝜐^𝑏_𝑎𝑏(𝑡)𝜔̇00_𝑏𝑎𝑏,(𝑡)𝜐̇^𝑏_𝑎𝑏(𝑡)=𝐽00𝑚𝐼₃⁻¹𝐽𝜔^𝑏_𝑎𝑏(𝑡)00−𝑚𝜔^𝑏_𝑎𝑏(𝑡)𝜔^𝑏_𝑎𝑏(𝑡)𝜐^𝑏_𝑎𝑏(𝑡). (60)(61)

Table 1 

Parameters of numerical test.

Table 2 

Magnus series terms corresponding to the top eight orders rooted trees.

Table 3 

The top eight orders reduced truncated Magnus series expansion.

For making the numerical test more comparable, we select four 4th-order numerical methods used to compare with geometric pseudospectral method (GPM), including ode45 [33], implicit 4th-order Runge-Kutta (RKi), explicit 4th-order RKMK (RKMK) [13], and 4th-order Gauss pseudospectral method (PM) [24], where iterative number and tolerance of the child loop of GPM and PM are set to 10 and 10⁻¹⁴, respectively. Among the previous methods, ode45 is the build-in integrator from MATLAB which is a variable-step method and whose results would be used as the ground truth with a specified tolerance 10⁻¹⁴. Both RKi and PM are the methods in Euclidean space. RKMK is an approximate Runge-Kutta method on Lie group [15]. It is worth mentioning that each test must be carried out under the same step size.

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 1. As seen from the evolution versus time, the positions of PM is closest to the ground truth, and that of GPM take second place, but that of RKi and RKMK deviate from the true values to varying degrees. The potential reason resulted in accuracy of GPM being lower than PM is that position of PM is computed by iterative operator with specified iterative numbers and tolerance directly, but position of GPM is truncated to a specific order before iterative operator. Despite inferior to PM, the proposed method is obviously superior to RKi and RKMK in position accuracy. As can be seen, error accumulates faster in translational components 𝑝^𝑎_𝑎𝑏 than in rotational components 𝑅_𝑎𝑏. This could be explained by the fact that rotation is decoupled from translation and hence has its own error dynamics, whereas translation suffers from small cumulative errors in rotation. In addition, we also give the total computational time in the first line and the average time per step in parenthesis. Obviously, RKMK needs the least time since it is an explicit method. Both GPM and PM need less time than RKi. From orientation, linear velocity, and angular velocity, there is no obvious difference, so we need to further compare these methods quantitatively.

Figure 1 

Position (𝑝𝑎𝑎𝑏), orientation (𝑅𝑎𝑏), linear velocity (𝜐𝑏𝑎𝑏), and angular velocity (𝜔𝑏𝑎𝑏) versus time and total computational time with average time per step for the free-floating rigid body when step size is equal to 0.25 s.

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 2. Figure 2(a) shows the position deviation against timesteps. It conforms that the accuracy of the GPM is superior to RKi and RKMK under the same large step size, although it is inferior to PM in average position deviation, and the relationship of position deviation between the four methods is consistent with that of Figure 1. In Figure 2(b), we give the orientation deviation against timesteps by calculating the metric ‖log_SO(3)(𝑅^𝑇_ode45𝑅_𝑎𝑏)‖₂ of SO(3) [27], where 𝑅_ode45 denotes the orientation of the free-floating rigid body which is computed by ode45. It should be noted that 𝑅_ode45 transforms from the unit quaternion [26], since in ode45 unit quaternion is selected for parameterizing orientation at this point. Figure 2(c) shows the averaged computation time of four methods. As can be seen, computational efficiency of GPM is inferior to that of RKMK, because RKMK is an explicit method. However, our method requires less computation time than RKi and is nearly twice as efficient as RKi on average.

(a) Average position deviation

(b) Average orientation deviation

(c) Average runtime per update

Figure 2 

Accuracy and computational efficiency of GPM compared with PM, RKMK, and RKi.

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 𝑅_𝑎𝑏𝑅^𝑇_𝑎𝑏=𝐼, and we compute deviation ‖𝐼_3×3−𝑅^𝑇𝑅‖_∞ for evaluating structural conservativeness of the algorithms [5]. Figure 3 illustrates that Gauss pseudospectral method in Euclidean space has no conservativeness of Lie group structure, since it is not a Lie group method; RKMK and geometric pseudospectral method on Lie group are able to preserve Lie group structure with accuracy approaching to machine accuracy.

(a) Gauss pseudospectral method

(b) Explicit RKMK

(c) Geometric pseudospectral method

Figure 3 

Lie group structural deviation.

4. Simulation and Control Application of Aircraft

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 1, pseudospectral method of the Gauss form can be easily applied to optimal control of the aircraft or other vehicles and satisfies optimization condition. Reference [23] shows that solving the NLP derived from the pseudospectral transcription of the Gauss form is exactly equivalent to solve discrete first-order necessary conditions. The KKT conditions of the NLP are exactly equivalent to the discrete first-order necessary conditions of the Bolza problem. And, KKT multipliers can be mapped to the costate of the continuous-time optimal control problem via costate mapping principle. Because our geometric pseudospectral discretization scheme is also based on Gauss points, we can use it to transcribe a continuous aircraft optimal control problem into a discrete nonlinear programming problem (NLP). Then, the resulting NLP can be solved by many well-known optimization techniques [34, 35]. For this purpose, we follow the Gauss pseudospectral transcription approach introduced in [36]. The optimal control problem can be formulated as in Algorithm 2.

Compute: 𝑔𝑎𝑏,𝑉𝑏𝑎𝑏,𝑇,𝛿𝑎,𝛿𝑟,𝛿𝑒 at 𝜏𝑗, 𝑗=1,…,𝑁+1 Minimizing: Cost Function Subject to: 𝑔0=𝑔𝑖,𝑉0=𝑉𝑖, 𝑔𝑁+1=𝑔𝑓,𝑉𝑁+1=𝑉𝑓, 𝑖,𝑓 denote initial time and terminal time. and (i) dynamics constraints: (25), (26) (ii) path constraints (iii) decision variables bounds:      𝑔𝜏𝑖∈𝑔𝑙,𝑔𝑢,𝑉𝜏𝑖∈𝑉𝑙,𝑉𝑢 𝑇𝜏𝑖∈𝑇𝑙,𝑇𝑢,𝛿𝑎∈𝛿𝑎𝑙,𝛿𝑎𝑢, 𝛿𝑟∈𝛿𝑟𝑙,𝛿𝑟𝑢,𝛿𝑒∈𝛿𝑒𝑙,𝛿𝑒𝑢     where 𝑙,𝑢 denote lower limit and upper limit.

Algorithm 2 

Problem formulation of the NLP.

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 [36]. In this work, referring to the geometric optimal control framework proposed in [37], rotation component 𝑅_𝑎𝑏 of 𝑔_𝑎𝑏 is not treated as our decision variable, for three reasons: firstly, during the optimization, 𝑅_𝑎𝑏 can be reconstructed from 𝜔^𝑏_𝑎𝑏 internally through evaluating (45), (51), and (54); secondly, the Lie group structural conservativeness could not be considered by NLP solvers, unless additional constraint with respect to structural conservativeness is introduced, which may affect optimality; moreover, computational efficiency is improved. We take obstacle avoidance as our example, with optimal control which is implemented using the combination of GPOPS [36] in MATLAB and SNOPT package in TOMLAB version 7.7 [38]. Figure 4 shows simulation results for an aircraft avoiding obstacles. Their detailed analysis is the subject of on-going work and will be presented in a future paper.

Figure 4 

Simulation results for the aircraft avoiding obstacles.

5. Conclusions

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 𝔰𝔢(3) is given under left trivialization. Accordingly, based on Magnus series expansion and exponential map, the solutions of configuration equation at the Gauss points and the endpoint are obtained. Velocities on 𝔰𝔢(3) at the discrete points are computed based on general pseudospectral method, since 𝔰𝔢(3) is isomorphic to ℝ⁶. Thereby, a 4th-order numerical method called geometric pseudospectral method is developed. Through a series of numerical tests and comparison with other numerical methods on Lie group and Euclidean space, the proposed method has a comprehensive advantage in accuracy, computational efficiency, and Lie group structural conservativeness. Finally, the way of applying our method to aircraft simulation and optimal control is illustrated.

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.

Appendix

The top eight orders truncated Magnus series expansion of 𝑢(𝑡) under left trivialization (see Tables 2 and 3).

Acknowledgments

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).