论文标题
通过搭配和Magnus扩展解决Cosserat杆模型
Solving Cosserat Rod Models via Collocation and the Magnus Expansion
论文作者
论文摘要
为连续机器人选择运动学模型通常涉及在准确性和计算复杂性之间做出权衡。一种常见的建模方法是使用cosserat杆方程，这些方程已被证明是许多类型的连续机器人的准确性。然而，这种方法仍然显示出明显的计算成本，尤其是当许多Cosserat杆通过运动学约束耦合时。在这项工作中，我们提出了一种数值方法，该方法结合了局部杆曲率上的正交搭配，并通过Magnus扩展将Cosserat杆运动学方程的正向整合结合在一起，从而使平衡形状可以写成矩阵指数的产物。我们提供了最大步长的结合，以确保Cosserat棒的Magnus扩展的收敛，在模拟中与其他方法进行比较，并证明了第四和第六阶Magnus扩展以及不同数量的套在一起点之间的速度和准确性之间的权衡。我们的结果表明，所提出的方法可以找到Cosserat杆方程的准确解决方案，并且在计算速度上可能具有竞争力。
Choosing a kinematic model for a continuum robot typically involves making a tradeoff between accuracy and computational complexity. One common modeling approach is to use the Cosserat rod equations, which have been shown to be accurate for many types of continuum robots. This approach, however, still presents significant computational cost, particularly when many Cosserat rods are coupled via kinematic constraints. In this work, we propose a numerical method that combines orthogonal collocation on the local rod curvature and forward integration of the Cosserat rod kinematic equations via the Magnus expansion, allowing the equilibrium shape to be written as a product of matrix exponentials. We provide a bound on the maximum step size to guarantee convergence of the Magnus expansion for the case of Cosserat rods, compare in simulation against other approaches, and demonstrate the tradeoffs between speed and accuracy for the fourth and sixth order Magnus expansions as well as for different numbers of collocation points. Our results show that the proposed method can find accurate solutions to the Cosserat rod equations and can potentially be competitive in computation speed.
