Dynamic virtual fixture on the Euclidean group for admittance-type manipulator in deforming environments
- Dongwen Zhang^{1, 2},
- Qingsong Zhu^{1},
- Jing Xiong^{1} and
- Lei Wang^{1}Email author
https://doi.org/10.1186/1475-925X-13-51
© Zhang et al.; licensee BioMed Central Ltd. 2014
Received: 3 December 2013
Accepted: 24 March 2014
Published: 27 April 2014
Abstract
Background
In a deforming anatomic environment, the motion of an instrument suffers from complex geometrical and dynamic constraints, robot assisted minimally invasive surgery therefore requires more sophisticated skills for surgeons. This paper proposes a novel dynamic virtual fixture (DVF) to enhance the surgical operation accuracy of admittance-type medical robotics in the deforming environment.
Methods
A framework for DVF on the Euclidean Group SE(3) is presented, which unites rotation and translation in a compact form. First, we constructed the holonomic/non-holonomic constraints, and then searched for the corresponded reference to make a distinction between preferred and non-preferred directions. Second, different control strategies are employed to deal with the tasks along the distinguished directions. The desired spatial compliance matrix is synthesized from an allowable motion screw set to filter out the task unrelated components from manual input, the operator has complete control over the preferred directions; while the relative motion between the surgical instrument and the anatomy structures is actively tracked and cancelled, the deviation relative to the reference is compensated jointly by the operator and DVF controllers. The operator, haptic device, admittance-type proxy and virtual deforming environment are involved in a hardware-in-the-loop experiment, human-robot cooperation with the assistance of DVF controller is carried out on a deforming sphere to simulate beating heart surgery, performance of the proposed DVF on admittance-type proxy is evaluated, and both human factors and control parameters are analyzed.
Results
The DVF can improve the dynamic properties of human-robot cooperation in a low-frequency (0 ~ 40 rad/sec) deforming environment, and maintain synergy of orientation and translation during the operation. Statistical analysis reveals that the operator has intuitive control over the preferred directions, human and the DVF controller jointly control the motion along the non-preferred directions, the target deformation is tracked actively.
Conclusions
The proposed DVF for an admittance-type manipulator is capable of assisting the operator to deal with skilled operations in a deforming environment.
Background
In a deforming anatomic environment, minimally invasive surgery (MIS) therefore requires more sophisticated skills for the surgeons, the motion of an instrument suffers from complex geometrical and dynamic constraints. One approach for enhancing the accuracy and safety of human operation is to use robot controller to regulate the instrument motion under either hands-on or teleoperation control. These are referred to as virtual fixtures (VF), which is kind of task-dependent and computer-generated mechanisms to limit movement into restricted regions [1–3] or influence movement along desired paths [4–8]. Extension of VF into deforming environment also refers to dynamic virtual fixture [9], where the geometric constraint moves continuously, as a result of changes in the physical environment or task being undertaken [10, 11].
An important case of VFs for surgical application is guidance VF, Bettini et al. [5, 6] used vision information to assist VF construction, and they discussed the application in vitreoretinal surgery. Marayong et al. [7] demonstrated a geometrical constraint with varying compliance, which is described for the general spatial case. Compliant control is significantly necessary to overcome the uncertainties associated with registration errors, variations in anatomy [12]. Li et al. [13] presented an anatomy generated VF for sinus surgery, which employed a constrained optimization framework to incorporate task goals, anatomy-based constraints, forbidden zones, etc. Castillo-Cruces et al. [14] presented an admittance controller with autonomous error compensation, a clear distribution of responsibilities between surgeon and robotic system is preserved. Becker et al. [15] derived a virtual fixture framework for active handheld micromanipulators, the forces were replaced with an actuated tool tip that is permitted by the high-bandwidth position measurements. These work all used admittance controller to implement VFs, the active robots comply with the manual force that applied on the end-effector or telemanipulated joystick. However, such first-order admittance controller with manual/autonomous error compensation is inadequate to adapt to dynamic environment. Bebek et al. proposed an intelligent control algorithm for robotic-assisted beating heart surgery, the robotic tools actively cancel the relative motion between the surgical instruments and the point of interest on the beating heart [16]. However, motions along the preferred directions are not regulated.
Traditional VFs deal with rotation and translation separately in ℝ^{3} space, the interconnection between them is not involved in VF design. Synergy of orientation and position is important for MIS, it requires that the orientation should be ready before the position is reached. Otherwise, a delay is needed to instrument motion when the orientation and position are out of synchronous, it could be awful for real-time operation, collisions could probably appears in the inner anatomic operation space. Bullo and Murray [17] proposed a generalized proportional derivative (PD) laws on Euclidean Group SE(3), which used logarithmic feedback control and dealt with rotation and translation simultaneously with a compact form. Besides, the geometric properties of Lie group and Lie algebra facilitate spatial compliance and stiffness matrix synthesis and decomposition. Shuguang Huang [18, 19] examined the structure of spatial stiffness by evaluating the eigenscrew featured rank-1 primitives stiffness matrices.
In our previous works, we studied the spatial compliance/stiffness matrices synthesis for admittance and impedance controlled devices respectively [20].
The proposed VFs could not effectively eliminate the tracking error drifts over the deforming frequency growth, thus we hope to improve the dynamic performance of the VF for admittance controlled device. In this paper, we study the application of VF in deforming environment, and propose a novel framework of DVF for admittance-type device on SE(3). We construct the holonomic constraint/non-holonomic constraints and then search for the corresponded reference tasks to make a distinction between preferred and non-preferred directions. Different control strategies are employed to deal with the tasks along the distinguished directions. The paper is organized as follows: the first section outlines the mathematical preliminaries of Lie group and Lie algebra for robot control, spatial compliance matrix synthesis and DVF on SE(3) for admittance-type device are proposed in Methods the last two sections describe a hardware-in-the-loop experiment and discussions.
Methods
Mathematical preliminaries
the operator $\widehat{*}:{\mathrm{\mathbb{R}}}^{3}\to \mathit{so}\left(3\right)$ is a cross-product matrix that transforms cross-product operation into matrix multiplication, so that $\widehat{\mathit{x}}\mathit{y}=\mathit{x}\times \mathit{y}$ for all x, y ∈ R^{3}. Elements of se(3) can also be represented as vector pair (ω, v), ω, v ∈ ℝ^{3}.
Let V _{2} = (ω _{2}, v _{2}) ∊ se(3) denote the spatial velocity of a rigid body in frame M _{2}, g_{12} ∊ SE(3) represents the right coordinate transformation from frame M _{1} to frame M _{2}, and then the spatial velocity relative to frame M _{1}, denoted by V _{1} = (ω _{1}, v _{1}), is given by ${\mathit{V}}_{1}=\mathit{A}{\mathit{d}}_{{\mathit{g}}_{12}}{\mathit{V}}_{2}$. Let F _{2} = (m _{2}, f _{2}) ∊ se(3) * denotes the moment-force pair acting on a rigid body with respect to frame M _{2}, g _{12} ∊ SE(3) represents the right coordinate transformation from frame M _{1} to frame M _{2}, and then the spatial force in frame M _{1} is given by ${\mathit{F}}_{2}=\mathit{A}{\mathit{d}}_{{\mathit{g}}_{12}}^{*}{\mathit{F}}_{1}$.
where $\widehat{\mathit{\psi}}={log}_{\mathit{SO}\left(3\right)}\left(\mathbf{R}\right)$.
Dynamic constrains and reference search
In deforming anatomic environment, the motion of instrument suffers from complex geometrical and dynamic constraints, their translational and rotational degree of freedom are restricted. MIS operation requires more sophisticated skills for the surgeons, making a distinction between and preferred and non-preferred directions and employed different control strategy helps reduce operation burden for surgeons. This can be achieved by construction of the holonomic/non-holonomic constraints and searching for the corresponded reference trajectories. The surgeons have complete control over the preferred directions, which are featured by a set of unit motion screws; while the deviation relative to the reference can be compensated autonomously. For a fully actuated control systems (the number of independent control inputs equals to the number of position variables), the reference trajectory falls into the holonomic and non-holonomic according to the constraints applied to the end-effector.
(i). Holonomic constraint and the corresponded reference
in which ${log}_{\mathrm{SE}\left(3\right)}\left(\mathit{e}\right)=\left(\widehat{\mathit{\psi}},\mathit{q}\right)\in \mathit{se}\left(3\right)$, the norm of logarithmic error is a weighted distance of pure rotation and translation. Searching for the nearest reference along the task sequence can be generalized as the traversal searching the minimum point along the whole task sequence for initialization, local searching along the direction of norm reduction and the minimum is always reachable due to continuous restriction on rotation and translation.
(ii). Non-holonomic constraint and the corresponded reference
When the axis vectors n _{ z } and m _{ z } coincides, the rotation matrix of reference g _{ d } equals to that of robot state g according to Eq.(15), that means the user has complete control over the robot along the tangent plane and directs the task flow intuitively, the deviations relative to reference can be compensated manually or autonomously during human-robot cooperation.
Spatial compliance for virtual fixture on SE(3)
Four primitive tasks and the corresponding preferred motion screw
Primitive tasks | Allowable motion screw set |
---|---|
Fixed configuration | g _{ d } ∊ SE(3), S = log (g _{ d } ^{−1} g) |
Trajectory tracking g _{ d } ∊ SE(3) | $\begin{array}{l}{\mathit{S}}_{1}={\dot{\mathit{g}}}_{\mathit{d}}{\mathit{g}}_{\mathit{d}}^{-1}\\ {\mathit{S}}_{2}=log\left({\mathit{g}}_{\mathit{d}}^{-1}\mathit{g}\right)\end{array}$ |
Rotate around a line | S = (s,r × s) s: rotation axis, r: arbitrary point on s |
Move on plane | S _{1} = (0,v _{1}),S _{2} = (0,v _{2}),S _{3} = (s,r × s) span{v _{1},v _{2}}: continent plane, s: norm direction |
Dynamic virtual fixture for admittance-type device
Considering that the end-effector is of admittance type, which can be modeled as non-backdrivable kinematic system, and all joints are equipped with velocity-source actuators. The VF on admittance type device acts as anisotropic compliant wall, the end-effector complies with the manual force along preferred the directions. Bettini and Marayong [5, 7] combined the deviation error into the allowable motion subspace and redefined a new VF that makes it possible to reduce the deviations manually. However, it has been shown that manual compensation does not necessarily compensate for all deviations, especially when the VF is defined for translation and the deviation error is of orientation. Castillo-Cruces et al. [14] added linear error feedback term into the admittance controller to compensate rotation and translation deviations separately and autonomously along the non-preferred directions.
where the {B_{n}} are Bernoulli numbers, the symbols B_{x} denote the Lie bracket series [17]. We have
B_{X} X = X and ${\mathrm{B}}_{\mathrm{X}}\left[\begin{array}{l}\mathit{0}\\ \mathit{q}\end{array}\right]=\left[\begin{array}{l}\mathit{0}\\ \mathit{A}{\left(\mathit{\psi}\right)}^{-\mathit{T}}\mathit{q}\end{array}\right]$
where α(y) ≜ (y/2) cot (y/2), 0 < ‖ψ‖ < 1, q = q _{||} + q _{⟂} is the orthogonal decomposition of q along span {ψ} and {ψ}_{⟂}. Thus global exponential stability is proven also for translational part.
Experiments, results and discussion
The proposed DVF is designed for fully actuated robotic system with force/torque sensor. However, both force/torque sensor and admittance controlled device are not available. We designed a virtual proxy based hardware-in-the-loop simulation test bed to evaluate the proposed DVF. The PHANToM Omni device, from SensAble Technologies, is equipped with 3 degree-of-freedom (DOF) actuations and 6 DOF position measurements. It serves as a force source rather than a haptic device to drive a virtual admittance-type proxy alternatively. The device is actuated passively by human force that applied on the end-effector, while there is no haptic feedback to the operator.
where M(Θ) is the inertia matrix, $\mathbf{N}\left(\mathit{\Theta},\dot{\mathit{\Theta}}\right)$ is the Coriolis matrix, the gravity term C(Θ) is eliminated from the actual manual force, kinematics and dynamics details of the Phantom Omni device can be found in [30]. F^{ s } represents the manual force with respect to the inertia frame, J(Θ) is manipulator Jacobian matrix, U _{ DVF } ^{ Ad } refers to the dynamic tracking controller in Eq. (23).
In the simulation experiment, we simplify the admittance controller robot as a virtual proxy, manual force is estimated indirectly through robot dynamics rather than a real force sensor. The proxy workspace is not restricted compared to a real robot, measurement errors of joint angles are not involved, manual force is also inaccurate due to inertia parameters estimation. We will evaluate the proposed algorithm on a real admittance controlled device when all instruments are ready in the future.
Numerous surgical tasks require the surgeon to follow a predetermined path on a deformable anatomy tissue surface while maintaining the shaft orientation within a safe range, such as minimally invasive beating heart surgery, the area of interest deforms with heartbeat and respiratory motion, instrument motion should be regulated to satisfy the varying geometric constraints. We simplified this task as a non-holonomic constraint problem by moving a tool on a deforming ball, while the tool shaft along the radial direction, the axial rotation is not constrained. The radius r of the ball varies periodically, r = r _{0} + Δr sin ωt, r _{0} = 15 mm, Δr = 5 mm. The bandwidth of the position measurements is 26 Hz, which refers to the upper limit of tissue motion resulted from physiologic movements, such as heartbeat and respiration [16].
Three experiments were conducted to validate the efficiency of proposed DVF for admittance-type proxy.
Proportional feedback control experiment
In this experiment, we compare the performance of three control laws to stabilizes the robot state g at fixed state g _{ d } from any initial state g(0) = (R(0), p(0)). The deviation of the end-effector state relative to the reference is defined as e = g _{ d } ^{−1} g, the logarithmic coordinates refers to $log\left(\mathit{e}\right)=\left(\mathit{\psi},\mathit{q}\right)\in \mathrm{se}\left(3\right)$.
Law-1 takes the error definition and feedback control from Castillo-Cruces’s work [14], which treats translation and rotation separately. Law-2 applies proportional feedback actions along geodesic directions for both rotation and translation in SO(3) and R^{3} respectively, law-3 is the logarithmic feedback control. Law-1 is dependent on the choice of inertia frame, both law-2 and law-3 are independent of inertia frame choice. To compare the proportional feedback laws presented above, the same planar motion SE(2), initial state g _{0}, destination state g _{ d } and proportional parameters k _{ w }, k _{ v } are set for the three control laws in our experiment.
Residual errors of three proportional feedback control laws
Law-1 | Law-2 | Law-3 | |
---|---|---|---|
TTE (mm) | 0.0046 | 0.0050 | 0.012 |
RTE (rad) | 2.7108e-020 | 1.7837e-017 | 1.2123e-007 |
Dynamic tracking experiment
Impact of R/T ratio on dynamic tracking error
R-T ratio | 4 | 3 | 2 | 1 | 0.5 |
---|---|---|---|---|---|
Tracking error(mm) | 0.5347 | 0.5996 | 0.6487 | 0.6610 | 0.6569 |
Settling time(sec) | 0.1 | 0.2 | 0.35 | 0.6 | 1.3 |
Human-robot collaboration experiment
As seen in Figure 6, changes in the magnitude and direction of manual force had influence on rotational and translational errors. The operator dragged the tool along 8-shaped curve with three intensities of manual force{low, medium, high}, k _{ p } =10, deforming frequency 5 rad/s, the following index RFI and TFI, dynamic tracking error TTE and RTE are recorded in Table 3.
Results of shape-8 operation with assistance of DVF on admittance proxy
Intensity | R-T ratio | RFI | TFI | RTE (rad) | TTE (mm) |
---|---|---|---|---|---|
Low | 4 | 0.1296 | 0.6148 | 0.0093 | 0.1046 |
Low | 2 | 0.8138 | 0.8333 | 0.0260 | 0.1693 |
Low | 1 | 0.6554 | 0.8647 | 0.0575 | 0.2633 |
Low | 0.5 | 0.8008 | 0.9087 | 0.0856 | 0.3404 |
Medium | 4 | 0.1815 | 0.7453 | 0.0180 | 0.1567 |
Medium | 2 | 0.7633 | 0.9374 | 0.0452 | 0.1876 |
Medium | 1 | 0.6231 | 0.8815 | 0.0712 | 0.2847 |
Medium | 0.5 | 0.7000 | 0.9407 | 0.1226 | 0.3455 |
High | 4 | 0.1692 | 0.8519 | 0.0291 | 0.1140 |
High | 2 | 0.8190 | 0.9440 | 0.0727 | 0.1832 |
High | 1 | 0.7173 | 0.7725 | 0.0812 | 0.2728 |
High | 0.5 | 0.6133 | 0.8714 | 0.1755 | 0.3303 |
The correlation between manual force and proxy velocity
Corr. | m _{x} | m _{y} | m _{z} | f _{x} | f _{y} | f _{z} |
---|---|---|---|---|---|---|
ω_{x} | 0. 761 | −0.192 | −0.051 | −0.003 | −0.940 | −0.692 |
ω_{y} | −0.046 | 0.880 | 0.092 | 0.922 | −0.058 | 0.114 |
ω_{z} | −0.545 | −0.043 | 1.000 | 0.173 | 0.048 | 0.021 |
v_{x} | −0.127 | 0.889 | 0.134 | 0.998 | −0.040 | 0.106 |
v_{y} | −0.754 | 0.127 | 0.091 | −0.044 | 0.998 | 0.676 |
v_{z} | 0.006 | 0.039 | 0.016 | 0.079 | −0.029 | 0.012 |
Conclusions
This paper described a novel framework of DVF for admittance-type manipulators on the Euclidean Group SE(3) to assist the surgeons to deal with the dynamic tasks, which unites rotation and translation in a compact form We constructed the holonomic/ non-holonomic constraints, and then searched for the corresponded references to make a distinction between preferred and non-preferred directions. Different control strategies are employed to deal with the task along these directions. The DVF can improve the dynamic properties of human-robot cooperation in low-frequency deforming environment, and maintain synergy of orientation and translation during the operation. The experiments show that the DVF implemented on the virtual admittance-type proxy can assist the user to deal with the skilled operations in deforming environment. We will evaluate the proposed algorithm on a real admittance controlled device when all instruments are ready in the future.
Notation and nomenclature
a; scalar
f; force vector
m; moment vector
ω; angular velocity vector
v; translational velocity vector
R; rotation matrix
0; null vector
0; null matrix
q; Vector
$\widehat{\mathit{q}}$; cross-product matrix of vector q
SE(3); Lie group
SE(2); Displacement subgroup of Lie group
SO(3); Rotation group
se (3); Lie algebra
so(3); Lie algebra of rotation group
se(3)*; dual space of Lie algebra
so(3)*; dual space of so(3)
g; elements in SE(3)
$\mathbf{V}=\left(\widehat{\mathit{\omega}},\mathit{v}\right)$; matrix form velocity in se(3)
V = (ω, v); column vector form velocity in se(3)
$\mathbf{X}=\left(\widehat{\mathit{\psi}},\mathit{q}\right)$; the matrix form logarithmic coordinates of g
X = (ψ, q); column vector form logarithmic coordinates of g
* ^{ b }; element relative to the body frame
* ^{s}; element relative to the spatial frame
${\mathrm{Ad}}_{\mathit{g}},\phantom{\rule{0.5em}{0ex}}{\mathrm{Ad}}_{\mathit{g}}^{\ast}$; adjoint transformation
${\mathrm{ad}}_{\mathit{X}},\phantom{\rule{0.5em}{0ex}}{\mathrm{ad}}_{\mathit{X}}^{\ast}$; Lie bracket
exp _{ SE(3)}( * ); exponential map on SE(3)
log_{ S E(3)}(∗); logarithmic map on SE(3)
‖ ∗ ‖_{G}; norm on the matrix Lie Group SE(3)
Generally, we use italic small characters for scalars, italic bold for vectors , bold capitals for matrices, italic bold capitals for vectors of Lie algebra.
Declarations
Acknowledgements
This study was financed partially by the Projects of National Natural Science Foundation of China (Grant Nos. 60932001 and 61072031), the National 863 Program of China (Grant No. 2012AA02A604), the National 973 Program of China (Grant No. 2010CB732606), the Next generation communication technology Major project of National S&T (Grant No.2013ZX03005013), the Key Research Program of the Chinese Academy of Sciences, and the Guangdong Innovation Research Team Funds for Low-cost Healthcare and Image-Guided Therapy.
Authors’ Affiliations
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