Research On Motion Planning And Guidance For UAV Based On Geometric Mechanical Models

The motion guidance is the basis for guiding the unmanned aerial vehicles(UAV) to perform tasks. The ways of motion guidance based on the traditional route planning and manual instruction have been unable to meet the requirement of the diverse mission and maneuver in complex environments. It is necessary to study more accurate methods for guiding the UAV. Based on the geometric mechanics and the geometric control theory,the dissertation studies the key technologies in practical offline or online motion guidance problems: modeling, motion planning and motion guidance. The main work and contributions are summarized as follows:Applying the differential geometry and the geometric mechanics synthetically, three different methods of modeling the UAV’s geometric dynamics are proposed. The models are highly accurate, coordinate-free and global. 1) Combined with rigid body dynamics and matrix group, the Newton forms of the UAV’s geometric mechanical models on SE(3) are established. 2) The Hamilton-Pontryagin(HP) variational principle is extended to the d’Alembert-Pontryagin(DP) variational principle which would be applied to the mechanical system with non-conservative force or control, and further used to derive the Hamilton forms of the UAV’s geometric mechanical models. 3) The UAV’s geometric mechanical models on the Frenet-Serret frame are established based on differential geometry curve theory. While the modeling methods are not the same, the three models all have the intrinsic geometrical invariance, such as group structure, energy/momentum conservation, symplectic structure, etc, which ensures that the configurations are high accurate.The UAV’s attitudes in the models are represented in terms of the special orthogonal group SO(3), and they are parameterized by a rotation matrix. The method for parameterizing the attitude not only avoids the singularities and confusions in Euler angles, but also avoids the ambiguity in the unit quaternion. Moreover, the method has no numerical dissipation which coexisted in Euler angles and the unit quaternion. It is thought that representing a3-dimensional attitude using 9 real elements with 6 constraints is inefficient. This redundancy is eliminated by using the exponential map. The three modeling methods provide highly accurate, coordinate-free and global models for the following study of the motion guidance.The Legendre-Gauss pseudospectral method in Euclidean space is extended to the geometric pseudospectral method on the special Euclidean group SE(3), and an offline motion planning method is proposed based on the fact that it discretizes NLP problems by geometric pseudospectral method. 1) Based on the equivariant map and configuration’s left invariance, the Legendre-Gauss pseudospectral method(LGP) in Euclidean space is extended to the geometric pseudospectral method on SE(3), which respectively computes configurations and velocities at the Gauss points and the endpoint by two different collocation strategies: first, considering that directly discretizing the UAV’s kinematical models by LGP results in loss of accuracy of the approximate solution, such as,numerical dissipation or energy dissipation, the equivalent equations in Lie algebra space se(3) are derived based on the left-trivialized tangent of local coordinate map, and their solutions are mapped back into the configuration space via the local logarithm map. Second,considering the Lie algebra space where the dynamics belong is isomorphism to R6, the velocities could be computed by LGP directly. Results from numerical simulation show that: the configuration accuracy of the proposed method outperforms that of the same order method RK4 i and a typical Lie group method known as RKMK. Although the position accuracy is lower than that of LGP, the proposed method does well in convergence and its attitude accuracy is higher than that of LGP. Moreover, the computation efficiency of the proposed method is about two times as that of RK4 i, and is slightly better than LGP.In addition, the approximation accuracy of the Lie group structure of the configurations can achieve 10-14. 2)The differential dynamical defect constraint is discretized by GPM,and the offline motion planning problem with differential dynamical constraints on SE(3)is transcribed as the classical nonlinear programming(NLP) problems. The simulation results show that: the generated plan has high accuracy and the proposed method need less collocated points, because the geometric models have better attitude accuracy than the classical models, and GPM has better convergence performance than other methods,along with the transcribed NLP problems being small-scale.The Lie group variational integrator is derived by discrete DP variational principle,and an online motion planning method with differential dynamics constraints is proposed based on Lie group variational optimal control. 1) The discrete HP principle is promoted as the discrete DP principle that would be applied to the UAV’s discrete geometric mechanical models(or, Lie group variational integrator, LGVI) with non-conservative force and control. The derivation is simpler than some existing methods. Results from numerical simulation show that: LGVI could approximate the exact flow well and conserve energy, momentum and the Lie group structure, but the classical RK4 i would have numerical dissipation and energy dissipation. 2) A discrete mechanical optimal control(DMOC) problem with LGVI as its differential dynamics constraints is described, and the first-order necessary optimality condition(or, two-point boundary value problems, TPBVPs) of the DMOC is derived by using discrete variation method on Lie group.3) Referring to Continuation/Forward Difference Approximation-Generalized Minimum Residual(C/FD-GMRES) method, an online receding horizontal motion planning method is developed for solving the TPBVPs. Compared with the classical nonlinear root findings, Newton-Armijo method or RK4 i under the same condition, the proposed method has better time performance and better convergence performance, because LGVI has outperforming numerical accuracy under low time resolution, at the same time, C/FD-GMRES has better convergence velocity and would not be sensitive with the initial guess.Under the leader-follower formation pattern, a method for UAV formation rendezvous is developed based on double-geodesic geometric control law. 1) The mathematical descriptions of three-dimensional formation rendezvous are provided, where the impact angular constraint in missile guidance was mapped to a flight path angle of the follower in formation rendezvous, and an additional azimuth angular constraint was introduced in.2) A motion guidance method for formation rendezvous was developed based on twogeodesic geometric control law, and the corresponding curvature command, torsion command and thrust command are derived, where the orientation deviation between the leader and the follower was measured using an element of the special orthogonal group SO(3),and the element was mapped to an element in the corresponding Lie algebra space so(3)by local coordinate mapping. The simulation experiments for multi-UAVs formation rendezvous was carried out under the leader flying straightly and making a turn respectively.The results from experiments indicate that the follower could track the orientation of the leader availably and converge to a specified formation configuration.