This technical report presents the construction and analysis of polynomial navigation functions for motion planning in 3-D workspaces populated by spherical and cylindrical obstacles. The workspace is modeled as a bounded spherical region, and obstacles are encoded using smooth polynomial implicit functions. We establish conditions under which the proposed navigation functions admit a unique non-degenerate minimum at the target while avoiding local minima, including in the presence of pairwise intersecting obstacles. Gradient and Hessian analyses are provided, and the theoretical results are validated through numerical simulations in obstacle rich 3-D environments.

翻译：本技术报告介绍了在布满球形与圆柱形障碍物的三维工作空间中，用于运动规划的多项式导航函数的构建与分析。工作空间被建模为有界球形区域，障碍物通过光滑多项式隐函数进行编码。我们建立了所提出的导航函数在目标点处存在唯一非退化极小值且避免局部极小值的条件，该条件在成对相交障碍物存在时依然成立。报告提供了梯度与海森矩阵分析，并通过障碍物密集的三维环境数值仿真验证了理论结果。