共形几何代数框架下的无人机目标定位理论研究

As the latest development of geometric algebra, conformal geometric algebra shows great advantages in the motion description of rigid bodies and geometric modeling and calculation. This project takes conformal geometric algebra as the mathematical tool and the target location of unmanned aerial vehicle ( UAV ) as the research content to conduct exploratory studies. Research content mainly includes establishing a mathematical model of target location described by conformal geometric algebra according to the basic principle of bearing-only target location , and deducing the objective function, using homonymous points and lines in sequential images and combining with geometric constraints of object, image and optical center to build an attitude estimation model of UAV which mixes the geometrical characteristics under the framework of conformal geometric algebra, improving the accuracy of target location through optimizing estimated attitude value of UAV, taking conformal geometric algebra as a theoretical basis to establish a location model of moving targets based on the geometric constraints of ground objects, applying nonlinear filtering algorithm to estimate the target position, inferring the theoretical lower bound of location error and developing software simulation and hardware experiment platform to verify the validity of the algorithm and evaluate the algorithm performance. This project is a research subject of application foundation, whose research result will provide a new idea for the theoretical research of target location of UAV and gradually push the target location of UAV based on conformal geometric algebra toward engineering application.

共形几何代数作为几何代数的最新发展，在刚体运动描述、几何建模与计算方面表现出很大的优势。本项目以共形几何代数为数学工具，以无人机目标定位为研究内容，进行探索性研究，研究内容主要包括：从纯方位目标定位原理出发，建立共形几何代数描述的无人机目标定位数学模型，并推导待求目标函数；利用序列图像中同名的点、直线等特征，结合成像时物、像、光心间的几何约束，在共形几何代数框架下建立融合多种几何特征的无人机姿态估计模型，通过优化无人机姿态估计值来提高目标定位精度；以共形几何代数为理论基础，建立基于地物几何约束的运动目标定位模型，利用非线性滤波算法估计目标位置，推导定位误差理论下限；研制一套软件仿真和硬件实验平台，验证算法的有效性和评估算法性能。作为应用基础研究课题，本项目的研究成果将为无人机目标定位的理论研究提供新的思路，逐步推动基于共形几何代数的无人机目标定位理论走向工程应用。

本项目以共形几何代数为数学工具，以无人机目标定位为研究内容，建立了基于共形几何代数的无人机目标定位数学模型，使用螺旋马达表示刚体运动，避免了复杂的三角函数运算，在保证目标定位精度的基础上降低了计算复杂度；在共形几何代数框架下建立了融合多种几何特征的无人机姿态估计模型，推导了姿态估计模型的线性与非线性求解方法，并评估了姿态估计精度、算法鲁棒性以及计算复杂度；建立了基于地物几何约束的运动目标定位模型，并采用非线性滤波算法优化运动目标位置，推导定位误差的克拉美-罗下限，分析目标在不同运动模态下的定位精度以及鲁棒性；研制了一套软件仿真和硬件实验平台，能够实现目标定位算法的功能验证与性能评估。本项目的研究成果有良好的应用前景，形成了一套完整的基于共形几何代数的无人机目标定位新方法，具有定位精度高、计算复杂度低的优势，为无人机目标定位的片上实现奠定了技术基础。