AI 中文总结
针对肌腱驱动连续体机器人，提出降阶笛卡尔框架，通过泰勒-伽辽金降阶和残差校正实现高效精确的全形状传播，比逐点GVS求解快约11倍。
AI 中文摘要
许多肌腱驱动连续体机器人（TDCR）的规划和控制任务需要完整的笛卡尔骨干几何形状。我们提出了一种用于平面、轴向可压缩TDCR的降阶笛卡尔框架，该框架沿预设的肌腱力和肌腱位移轨迹传播平衡构型。骨干由两个全局位置场表示。在精确变分之后，泰勒-伽辽金降阶将预设的空间属性和分布载荷压缩为离线力矩向量，从而产生解析的降阶残差和雅可比矩阵，无需在线空间求积或数值微分。解析微分和残差校正产生一阶速率系统，在规则分支上的初始平衡对齐后，每次速率评估只需一次固定维度的线性求解。在涵盖可变肌腱路径、非均匀几何、轴向压缩及其组合效应的四个模拟案例中，传播的笛卡尔形状和分布应变与逐点几何可变应变（GVS）平衡解紧密匹配。残差校正抑制了所测试步长下的传播漂移，同时仅使未校正欧拉法的平均更新时间增加约0.98%。所提出的方法平均每次更新需0.508毫秒，比逐点GVS求解快约11倍。位移驱动实验产生最大归一化平均骨干位置误差为1.02%，最大末端执行器位置误差为0.28%。这些结果支持沿预设驱动路径的高效且准确的笛卡尔全形状预测。
英文摘要
Many planning and control tasks for tendon-driven continuum robots (TDCRs) require the complete Cartesian backbone geometry. We present a reduced Cartesian framework for planar, axially compressible TDCRs that propagates equilibrium configurations along prescribed tendon-force and tendon-displacement trajectories. The backbone is represented by two global position fields. Following exact variation, a Taylor-Galerkin reduction condenses prescribed spatial properties and distributed loads into offline moment vectors, yielding analytic reduced residuals and Jacobians without online spatial quadrature or numerical differentiation. Analytical differentiation and residual correction yield first-order rate systems requiring one fixed-dimensional linear solve per rate evaluation after initial equilibrium alignment on a regular branch. Across four simulated cases covering variable tendon routing, nonuniform geometry, axial compression, and their combined effects, the propagated Cartesian shapes and distributed strains closely match pointwise geometrically variable-strain (GVS) equilibrium solutions. Residual correction suppresses propagation drift across the tested step sizes while adding only about 0.98% to the mean update time of uncorrected Euler. The proposed method requires 0.508 ms per update on average, approximately 11 times faster than pointwise GVS solves. Displacement-driven experiments yield a maximum normalized mean backbone position error of 1.02% and a maximum end-effector position error of 0.28%. These results support efficient and accurate Cartesian full-shape prediction along prescribed actuation paths.
