Development of a novel hand−eye calibration for intuitive control of minimally invasive surgical robot

Abstract

Robotic-assisted surgical system has introduced a powerful platform through dexterous instrument and hand−eye coordination intuitive control. The knowledge of laparoscopic vision is a crucial piece of information for robot-assisted minimally invasive surgery focusing on improved surgical outcomes. Obtaining the transformation with respect to the laparoscope and robot slave arm frames using hand−eye calibration is essential, which is a key component for developing intuitive control algorithm. We proposed a novel two-step modified dual quaternion for hand−eye calibration in this study. The dual quaternion was exploited to solve the hand−eye calibration simultaneously and powered by an iteratively separate solution. The obtained hand−eye calibration result was applied to the intuitive control by using the hand−eye coordination criterion. Promising simulations and experimental studies were conducted to evaluate the proposed method on our surgical robot system. We extensively compared the proposed method with state-of-the-art methods. Results demonstrate this method can improve the calibration accuracy. The effectiveness of the intuitive control algorithm was quantitatively evaluated, and an improved hand−eye calibration method was developed. The relationship between laparoscope and robot kinematics can be established for intuitive control.

Keywords

minimally invasive surgery / hand−eye calibration / intuitive control / surgical robot / dual quaternion

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Nomenclature

A	Matrix of laparoscope motion
B	Matrix of robot motion
K	Matrix for solving hand‒eye equation
L	Modified matrix for hand‒eye equation solution
M	Matrix vector
N	Number of experimental test data groups
$P_{\mathrm{B}\mathrm{L}}$	Robot laparoscope slave arm base frame
$P_{\mathrm{E}}$	Robot laparoscope slave arm end-effector frame
$P_{\mathrm{L}}$	Laparoscope frame
$P_{\mathrm{O}}$	Calibration object frame
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\mathit{P}$	Translation vector from surgical robot slave instrument arm base frame to the instrument frame
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{e}\mathrm{y}\mathrm{e}}{\mathit{P}}_i$	Position of surgical instrument in the surgeon eye frame in time step i
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{i}\mathrm{m}\mathrm{g}}{\mathit{P}}_i$	Instrument position with respective with the laparoscope camera image frame
${}_{\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}}^{\mathrm{e}\mathrm{y}\mathrm{e}}{\mathit{P}}_i$	Position of surgeon hand in the surgeon eye frame in time step i
${}_{\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}}^{\mathrm{m}\mathrm{a}\mathrm{s}}\mathit{P}$	Translation vector from master system base frame to the surgeon hand frame
${\mathit{q}}_A$, ${\mathit{q}}_X$, ${\mathit{q}}_B$	Dual quaternion representations of Eq. (2)
qAR, qXR, qBR	Rotation components
qAR0, qBR0	Components of the ${\mathit{q}}_{A\mathrm{R}}$ and ${\mathit{q}}_{B\mathrm{R}}$, respectively
qAt, qXt, qBt	Translation components
${}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}^{\mathrm{l}\mathrm{a}\mathrm{p}}\mathit{R}$	Rotation matrix component of ${}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}^{\mathrm{l}\mathrm{a}\mathrm{p}}\mathit{T}$
${}_{\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}}^{\mathrm{m}\mathrm{a}\mathrm{s}}\mathit{R}$	Rotation matrix component of homogeneous transformation matrix from master system base frame to the surgeon hand frame
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\mathit{R}$	Rotation matrix component of homogeneous transformation matrix from surgical robot slave instrument arm base frame to the instrument frame
${}_{\mathrm{l}\mathrm{a}\mathrm{p}}^{\mathrm{i}\mathrm{m}\mathrm{g}}\mathit{R}$	Rotation matrix component of homogeneous transformation matrix from laparoscope camera image frame to robot laparoscope slave arm end tool frame
${}_{\mathrm{m}\mathrm{a}\mathrm{s}}^{\mathrm{e}\mathrm{y}\mathrm{e}}\mathit{R}$	Rotation matrix component of ${}_{\mathrm{m}\mathrm{a}\mathrm{s}}^{\mathrm{e}\mathrm{y}\mathrm{e}}\mathit{T}$
${\mathit{t}}_A$, ${\mathit{t}}_B$, ${\mathit{t}}_X$	Translation component of the homogeneous transformation matrix in AX = XB
${\mathit{t}}_{\mathrm{G}\mathrm{T}}$	Ground-truth value
${}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}^{\mathrm{l}\mathrm{a}\mathrm{p}}\mathit{T}$	Transformation matrix from robot laparoscope end tool frame to robot instrument slave arm base frame
${}_{\mathrm{E}}^{\mathrm{L}}\mathit{T}$	Transformation between $P_{\mathrm{L}}$ and $P_{\mathrm{E}}$
${}_{\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}}^{\mathrm{e}\mathrm{y}\mathrm{e}}\mathit{T}$	Homogeneous transformation from surgeon eye frame to surgeon hand frame
${}_{\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}}^{\mathrm{m}\mathrm{a}\mathrm{s}}\mathit{T}$	Homogeneous transformation matrix from master system base frame to the surgeon hand frame
${}_{\mathrm{i}\mathrm{m}\mathrm{g}}^{\mathrm{e}\mathrm{y}\mathrm{e}}\mathit{T}$	Transformation matrix from surgeon eye frame to the laparoscope camera image frame
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\mathit{T}$	Homogeneous transformation matrix from surgical robot slave instrument arm base frame to the instrument frame
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{e}\mathrm{y}\mathrm{e}}\mathit{T}$	Homogeneous transformation from surgeon eye frame to instrument frame
${}_{\mathrm{i}\mathrm{n}\mathrm{s}}^{\mathrm{i}\mathrm{m}\mathrm{g}}\mathit{T}$	Homogeneous transformation from laparoscope camera image frame to instrument frame
${}_{\mathrm{l}\mathrm{a}\mathrm{p}}^{\mathrm{i}\mathrm{m}\mathrm{g}}\mathit{T}$	Transformation matrix from laparoscope camera image frame to robot laparoscope slave arm end tool frame
${}_{\mathrm{m}\mathrm{a}\mathrm{s}}^{\mathrm{e}\mathrm{y}\mathrm{e}}\mathit{T}$	Homogeneous transformation from surgeon eye frame to master system base frame
${}_{\mathrm{B}\mathrm{L}}^{\mathrm{E}}\mathit{T}(i)$	Motion i of transformation between robot laparoscope slave arm base frame $P_{\mathrm{B}\mathrm{L}}$ and the robot laparoscope slave arm end-effector frame $P_{\mathrm{E}}$
${}_{\mathrm{O}}^{\mathrm{L}}\mathit{T}(i)$	Motion i of transformation between calibration object frame $P_{\mathrm{O}}$ and the laparoscope frame $P_{\mathrm{L}}$
X	Target homogeneous transformation matrix
${\text{ω}}_A$, ${\text{ω}}_B$, ${\text{ω}}_X$	Rotation component of the homogeneous transformation matrix in AX = XB
${\text{ω}}_{\mathrm{G}\mathrm{T}}$	Ground-truth value
${\gamma }_1$, ${\gamma }_2$	Parameters for dual quaternion solution
$\mathit{\tau }$, $\mathit{\upsilon }$	Parameters for dual quaternion solution of Eq. (9)
$\kappa$	Defined ratio variable for solving equation
${\mathit{\alpha }}_A$, ${\mathit{\beta }}_A$, ${\mathit{\alpha }}_B$, ${\mathit{\beta }}_B$	Lie group members of the transformation
$\lambda$	Master–slave position mapping scale factor
${\mathrm{\Delta }}_{\mathrm{r}}$	Rotation matrix error
${\mathrm{\Delta }}_{\mathrm{t}}$	Translation vector error
${\theta }_{\mathrm{t}}$	Translation error between calculated value and ground-truth value
${\theta }_{\omega }$	Rotation error between calculated value and ground-truth value

Acknowledgements

The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. This study was supported by the State Key Laboratory of Robotics and Systems, China (Grant No. SKLRS202009B).

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