基于Radau伪谱法的重复使用运载器再入轨迹优化

为了提高数值解法的收敛速度,本文利用Radau伪谱法求解重复使用运载器的再入轨迹优化问题.该方法在一组Legendre-Gauss-Radau点上构造全局Lagrange插值多项式对状态变量和控制变量进行逼近,在动力学方程中状态变量对时间的导数可由插值多项式的导数来近似,故可将动力学方程约束转化为在Legendre-Gauss-Radau点上的代数微分方程约束.因此,可将连续时间的最优控制问题转化为有限维的非线性规划(NLP)问题,之后通过稀疏NLP求解器SNOPT即可对其进行求解.最后的仿真结果显示,通过该方法优化后的再入轨迹成功满足过程约束与边界约束.由于该方法的高效率和高精度特性,可将其应用于轨迹快速优化工程实际问题中.

Re-entry trajectory optimization using Radau pseudospectral method

To increase the convergence rate of the numerical method, we employ the Radau pseudospectral method (RPM) in solving the optimal re-entry trajectory for the reusable launch vehicle. In this method, a finite base of global Lagrange interpolating polynomials is used to approximate the states and control at a set of Legendre-Gauss-Radau points. The time derivative of the state in the dynamic equations is approximated by the derivative of the interpolating polynomial, therefore they can be converted to the differential-algebraic equations at the Legendre-Gauss-Radau points. Consequently, the continuous-time optimal control problem is transcribed to a finite-dimensional nonlinear programming (NLP) problem. Then, the resulting NLP problem is solved by a sparse nonlinear programming solver named SNOPT. Finally, simulation results show that the optimized re-entry trajectory satisfies the path constraints and the boundary constraints successfully. The results indicate that the RPM can be applied to fast trajectory-generation problems in practical engineering due to its high efficiency and high precision.