﻿ 手术机器人的运动学标定方法与实验
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 哈尔滨工程大学学报  2020, Vol. 41 Issue (7): 1080-1086  DOI: 10.11990/jheu.201904065 0

引用本文

YU Lingtao, YU Xiaoyan, TANG Zexu, et al. Kinematic calibration method and experiments for surgical robots[J]. Journal of Harbin Engineering University, 2020, 41(7): 1080-1086. DOI: 10.11990/jheu.201904065.

文章历史

Kinematic calibration method and experiments for surgical robots
YU Lingtao , YU Xiaoyan , TANG Zexu , WANG Lan
College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: To improve the positioning accuracy of surgical robots, in this paper, we propose a kinematic calibration method that is based on a three-dimensional position detection device. We designed a new type of position detection device based on draw-wire sensors, and then used spatial analytical geometry to establish a mathematical model of the device. To obtain the structural parameter errors, we used an accuracy-measurement experimental platform. Using the kinematics model of the surgical robot, we then established absolute-error and distance-error models, and used the distance-error model to obtain the kinematics parameter errors. We then conducted a kinematics parameter calibration experiment for a 3-degree-of-freedom minimally invasive surgical robot. For 20 groups of different movement distances, we found the average relative error to decrease from 5.157 9% to 1.419 6% after calibration, and the positioning accuracy of the surgical robot to improve by 72%. The experimental results verify the effectiveness of the proposed method and detection device.
Keywords: surgical robot    draw-wire sensor    position detection    self-calibration    parameter calibration    absolute error    distance accuracy

1 位置检测装置数学模型 1.1 结构及硬件设计

1.2 数学模型

 $\left\{ \begin{array}{l} {L_1} = TE + A{O_a} \cdot \angle A{O_a}E\\ {\left( {{x_e} + {l_a}} \right)^2} + {\left( {{y_e} - r\cos \theta } \right)^2} + {\left( {{z_e} - r\sin \theta } \right)^2} = {r^2}\\ {z_t}{y_e} - {y_t}{z_e} = 0\\ \left( {{x_e} + {l_a}} \right)\left( {{x_t} - {x_e}} \right) + \left( {{y_e} - r\cos \theta } \right)\left( {{y_t} - {y_e}} \right) + \\ \left( {{z_e} - r\sin \theta } \right)\left( {{z_t} - {z_e}} \right) = 0 \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} {L_2} = TG + A{O_b} \cdot \angle A{O_b}G\\ {\left( {{x_f} - r\cos \beta } \right)^2} + {\left( {{y_f} - {l_b}} \right)^2} + {\left( {{z_f} - r\sin \beta } \right)^2} = {r^2}\\ {z_t}{x_f} - {x_t}{z_f} = 0\\ {\left( {{x_f} - r\cos \beta } \right)^2}\left( {{x_t} - {x_f}} \right) + \left( {{y_f} - {l_b}} \right)\left( {{y_t} - {y_f}} \right) + \\ {\left( {{z_f} - r\sin \beta } \right)^2}\left( {{z_t} - {z_f}} \right) = 0 \end{array} \right.$ (2)
 $\left\{ \begin{array}{l} {L_3} = TG + A{O_c} \cdot \angle A{O_c}G\\ {\left( {{x_g} - {l_c}} \right)^2} + {\left( {{y_g} - r\cos \delta } \right)^2} + {\left( {{z_g} - r\sin \delta } \right)^2} = {r^2}\\ {z_t}{x_g} - {x_t}{z_g} = 0\\ {\left( {{x_g} - {l_c}} \right)^2}\left( {{x_t} - {x_g}} \right) + {\left( {{y_g} - r\cos \delta } \right)^2}\left( {{y_t} - {y_g}} \right) + \\ {\left( {{z_g} - r\sin \beta } \right)^2}\left( {{z_t} - {z_g}} \right) = 0 \end{array} \right.$ (3)
 $\left\{ \begin{array}{l} {L_4} = TG + A{O_d} \cdot \angle A{O_d}G\\ {\left( {{x_h} - r\cos \sigma } \right)^2} + {\left( {{y_h} - {l_d}} \right)^2} + {\left( {{z_h} - r\sin \sigma } \right)^2} = {r^2}\\ {z_t}{x_h} - {x_t}{z_h} = 0\\ {\left( {{x_h} - r\cos \sigma } \right)^2}\left( {{x_t} - {x_h}} \right) + {\left( {{y_h} - {l_d}} \right)^2}\left( {{y_t} - {y_h}} \right) + \\ {\left( {{z_h} - r\sin \sigma } \right)^2}\left( {{z_t} - {z_h}} \right) = 0 \end{array} \right.$ (4)
2 位置检测装置误差分析 2.1 装置精度测量实验

 Download: 图 4 相同位置点实验数据 Fig. 4 Relative error of the same point precision test
 Download: 图 5 不同位置点实验数据 Fig. 5 Relative error of the different points precision test
2.2 实验结果分析

3 位置检测装置自标定 3.1 自标定误差模型

 $\left( {{z_i} + \Delta {z_t}} \right)\left( {{y_e} + \Delta {y_e}} \right) - \left( {{y_t} + \Delta {y_t}} \right)\left( {{z_e} + \Delta {z_e}} \right) = 0$
3.2 自标定误差参数辨识

3.3 自标定后精度测量实验

 Download: 图 6 自标定后精度测量相对误差 Fig. 6 Relative error of precision test after self-calibration
4 手术机器人运动学参数标定 4.1 运动学模型

 $^{i - 1}{\mathit{\boldsymbol{T}}_i} = \left[ {\begin{array}{*{20}{l}} {\cos {\theta _i}}&{ - \sin {\theta _i}\cos {\alpha _i}}&{\sin {\theta _i}\sin {\alpha _i}}&{{a_i}\cos {\theta _i}}\\ {\sin {\theta _i}}&{\cos {\theta _i}\cos {\alpha _i}}&{ - \cos {\theta _i}\sin {\alpha _i}}&{{a_i}\sin {\theta _i}}\\ 0&{\sin {\alpha _i}}&{\cos {\alpha _i}}&{{d_i}}\\ 0&0&0&1 \end{array}} \right]$

 $^4{\mathit{\boldsymbol{T}}_7}{ = ^4}{\mathit{\boldsymbol{T}}_5}{ \cdot ^5}{\mathit{\boldsymbol{T}}_5}{ \cdot ^6}{\mathit{\boldsymbol{T}}_7} = \left[ {\begin{array}{*{20}{l}} {{n_x}}&{{o_x}}&{{a_x}}&{{p_x}}\\ {{n_y}}&{{o_y}}&{{a_y}}&{{p_y}}\\ {{n_z}}&{{o_z}}&{{a_z}}&{{p_z}}\\ 0&0&0&1 \end{array}} \right]$ (5)

4.2 绝对误差模型

 $\Delta \mathit{\boldsymbol{p}} = \mathit{\boldsymbol{J}} \cdot \Delta \mathit{\boldsymbol{q}}$
 $\Delta \mathit{\boldsymbol{q}} = {\left[ {\Delta {\alpha _5}\;\Delta {\alpha _6}\;\Delta {\alpha _7}\;\Delta {a_5}\;\Delta {a_6}\;\Delta {a_7}\;\Delta {d_5}\;\Delta {d_6}\;\Delta {\theta _7}} \right]^T}$
 $\Delta \mathit{\boldsymbol{p}} = {\left[ {{x_m} - {x_t}\;{y_m} - {y_t}\;{z_m} - {z_t}} \right]^T}$
 $J = \left[ {\begin{array}{*{20}{l}} {\frac{{\partial {p_x}}}{{\partial {\alpha _5}}}}&{\frac{{\partial {p_x}}}{{\partial {\alpha _6}}}}&{\frac{{\partial {p_x}}}{{\partial {\alpha _7}}}}&{\frac{{\partial {p_x}}}{{\partial {a_5}}}}&{\frac{{\partial {p_x}}}{{\partial {a_6}}}}&{\frac{{\partial {p_x}}}{{\partial {a_7}}}}&{\frac{{\partial {p_x}}}{{\partial {d_5}}}}&{\frac{{\partial {p_x}}}{{\partial {d_6}}}}&{\frac{{\partial {p_x}}}{{\partial {d_7}}}}\\ {\frac{{\partial {p_y}}}{{\partial {\alpha _5}}}}&{\frac{{\partial {p_y}}}{{\partial {\alpha _6}}}}&{\frac{{\partial {p_y}}}{{\partial {\alpha _7}}}}&{\frac{{\partial {p_y}}}{{\partial {a_5}}}}&{\frac{{\partial {p_y}}}{{\partial {a_6}}}}&{\frac{{\partial {p_y}}}{{\partial {a_7}}}}&{\frac{{\partial {p_y}}}{{\partial {d_5}}}}&{\frac{{\partial {p_y}}}{{\partial {d_6}}}}&{\frac{{\partial {p_y}}}{{\partial {d_7}}}}\\ {\frac{{\partial {p_z}}}{{\partial {\alpha _5}}}}&{\frac{{\partial {p_z}}}{{\partial {\alpha _6}}}}&{\frac{{\partial {p_z}}}{{\partial {\alpha _7}}}}&{\frac{{\partial {p_z}}}{{\partial {a_5}}}}&{\frac{{\partial {p_z}}}{{\partial {a_6}}}}&{\frac{{\partial {p_z}}}{{\partial {a_7}}}}&{\frac{{\partial {p_z}}}{{\partial {d_5}}}}&{\frac{{\partial {p_z}}}{{\partial {d_6}}}}&{\frac{{\partial {p_z}}}{{\partial {d_7}}}} \end{array}} \right]$

4.3 距离误差模型

 $\Delta l\left( {i,i + 1} \right) = \left[ {{b_x},{b_y},{b_z}} \right] \cdot \left( {{\mathit{\boldsymbol{J}}_{i + 1}} - {\mathit{\boldsymbol{J}}_i}} \right) \cdot \Delta \mathit{\boldsymbol{q}}$ (6)

 $\left[ {{b_x},{b_y},{b_z}} \right] = \left[ {\frac{{{x_t}\left( {i + 1} \right) - {x_t}\left( i \right)}}{{{l_t}\left( {i,i + 1} \right)}},\frac{{{y_t}\left( {i + 1} \right) - {y_t}\left( i \right)}}{{{l_t}\left( {i,i + 1} \right)}},\frac{{{z_t}\left( {i + 1} \right) - {z_t}\left( i \right)}}{{{l_t}\left( {i,i + 1} \right)}}} \right]$

4.4 手术机器人标定实验

 Download: 图 9 机器人标定前后运动距离相对误差 Fig. 9 Relative error of distance before and after calibration

5 结论

1) 本文设计一种基于拉线传感器的三维位置检测装置，并提出一种手术机器人运动学标定方法。该方法的优势在于：仅需自制位置检测装置和机器人的角度/位移传感器即可实现运动学标定，无需其他高精度标定设备，成本低且易实现。

2) 建立的自标定误差模型对装置的数学模型进行修正，使得装置测量精度提升61.07%，满足手术机器人标定实验的使用要求。

3) 使用3自由度微创腹腔手术机器人进行标定实验，标定后的平均相对误差下降了72%，实验结果表明本文提出的标定方法与检测装置可以提高手术机器人的定位精度，具有明显的实用价值。

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