Continuum robots suffer large deflections due to internal and external forces. Accurate modeling of their passive compliance is necessary for accurate environmental interaction, especially in scenarios where direct force sensing is not practical. This paper focuses on deriving analytic formulations for the compliance of continuum robots that can be modeled as Kirchhoff rods. Compared to prior works, the approach presented herein is not subject to the constant-curvature assumptions to derive the configuration space compliance, and we do not rely on computationally-expensive finite difference approximations to obtain the task space compliance. Using modal approximations over curvature space and Lie group integration, we obtain closed-form expressions for the task and configuration space compliance matrices of continuum robots, thereby bridging the gap between constant-curvature analytic formulations of configuration space compliance and variable curvature task space compliance. We first present an analytic expression for the compliance of a single Kirchhoff rod. We then extend this formulation for computing both the task space and configuration space compliance of a tendon-actuated continuum robot. We then use our formulation to study the tradeoffs between computation cost and modeling accuracy as well as the loss in accuracy from neglecting the Jacobian derivative term in the compliance model. Finally, we experimentally validate the model on a tendon-actuated continuum segment, demonstrating the model's ability to predict passive deflections with error below 11.5\% percent of total arc length.

翻译：连续体机器人在内外力作用下会产生大变形。准确建模其被动柔顺性对于实现精确的环境交互至关重要，尤其在直接力传感不可行的场景中。本文聚焦于推导可建模为Kirchhoff杆的连续体机器人的柔顺性解析表达式。与先前研究相比，本文方法无需依赖常曲率假设来推导构型空间柔顺性，也不使用计算成本高昂的有限差分近似来获得任务空间柔顺性。通过曲率空间模态近似与李群积分，我们获得了连续体机器人任务空间与构型空间柔顺性矩阵的闭式表达式，从而弥合了构型空间柔顺性的常曲率解析公式与变曲率任务空间柔顺性之间的差距。我们首先给出单根Kirchhoff杆的柔顺性解析表达式，随后将这一公式扩展到计算腱驱动连续体机器人的任务空间与构型空间柔顺性。接着，利用所提公式研究计算成本与建模精度之间的权衡，以及忽略柔顺性模型中雅可比导数项导致的精度损失。最后，通过腱驱动连续体节段的实验验证该模型，结果表明模型预测被动变形的误差低于总弧长的11.5%。