The Homotopy Methods In The Low-thrust Transfer Trajectory Optimization Problems

Carrying out various and complex space missions to improve scientific and economic returns is an important development tendency for spaceflight.However,the traditional chemical propulsion technology severely restricts the development of the aerospace industry due to the disadvantages of small specific impulse and low propulsion efficiency.Represented by the electric propulsion technology,the low thrust propulsion technology has the advantage of high specific impulse,which can significantly reduce the fuel consumption,increase the payload and extend the operating life,and it is the key technology to complete low-cost and high-return space missions.The excellent performance of the low thrust propulsion technology has been verified in many space missions,and it has an attractive prospect in the future.Due to the strong nonlinearity and high sensitivity caused by low thrust magnitude and long flight duration,the low-thrust transfer trajectory optimization problems are extremely difficult to solve.Based on the idea of numerical continuation,the principle of homotopy methods is that a given problem is embedded into a family of subproblems parameterized by a homotopic parameter,and the solution to the original problem is obtained by iteratively solving these subproblems.With the advantages of simple structure and easy convergence,homotopy methods are widely used to overcome these difficulties.However,the research of homotopy methods in the trajectory optimization is still in the development stage,and it mainly focuses on simple application,but lacks of research on homotopy principles and key technologies.This thesis systematically studies the homotopy methods on the three key steps: constructing the homotopy function,tracking the homotopy path,and finding the solution.The main research contents of this thesis are:(1)Based on the characteristic of the low-thrust transfer trajectory optimization problems,the guidance of constructing homotopy functions is provided,and multiple homotopy methods for various space missions are proposed.(2)The typical geometries of homotopy paths are studied,and three methods are proposed to overcome singularities of homotopy paths,including the probability-one homotopy,the parameter bounding homotopy and the double-phase homotopy method.(3)Considering that there are many local solutions to the optimization problem,a homotopy method for finding the best solution is proposed,and the minimum-time problem and minimum-fuel problem are respectively solved.The main innovations of this thesis are summarized as follows:(1)Combining the homotopy of performance index,dynamic model and constraint conditions,multiple homotopy methods are proposed to reduce the high initial guess sensitivity caused by the long flight duration and the discontinuous bang-bang control.Besides,the quadratic homotopy method is proposed,which combines the quadratic homotopy of performance index and the linear homotopy of dynamical model.By analytically solving the initial problem,the solution process is simplified,and the calculation efficiency is improved.(2)The probability-one homotopy method and parameter-bounding homotopy method are proposed to constructing a continuous zero curve.The sufficient conditions of zero curve are derived analytically,and the homotopy function that satisfies these conditions is constructed to guarantee the existence of the zero curve.In the parameter-bounding homotopy method,the bounding homotopy function is constructed by adding the auxiliary function and penalty function to the general homotopy function,and the continuous zero curve can be obtained by connecting several homotopy path branches.(3)The double-homotopy method for constructing a discontinuous zero curve is proposed to overcome the singularities that the homotopy paths return to the initial boundary or wander to infinity.In the first layer,a general homotopy function is constructed to form the connection between the initial problem and the original problem.In the second layer,the jumps between different homotopy path branches are implemented.Compared with the probability-one homotopy method and parameter-bounding homotopy method,the zero curve of the doublehomotopy is discontinuous and monotonous,and it has a much shorter curve length,which indicates a higher efficiency.(4)The two-phase homotopy method for global optimization is proposed to address the problem of multiple local solutions in the low-thrust transfer trajectory optimization.The parameterbounding fixed-point homotopy in the first phase is able to achieve many local solutions to the initial problem.By tracking many homotopy paths starting from these solution points in the second phase,many solutions to the original problem is obtained,and then the best solution is identified.