## I Introduction

The da Vinci Surgical System (Intuitive Surgical,
Inc., Sunnyvale, CA) has received considerable attention in the context
of minimally invasive surgery, which involves procedures performed
through small incisions. The robot is teleoperated: the surgeon generates
motion commands on the *master side, *using a *master* interface;
then, the commands are translated into motion by the *slave *robot,
which interacts with the patient on the *slave side*.

The success of the da Vinci in adult laparoscopy has led to attempts to use it in surgical scenarios with workspaces more constrained than those in the initial target applications, such as infant surgery [1] and paranasal sinuses and skull base surgery [2]. However, these attempts have had limited success owing to the to large diameter and length of the da Vinci’s tools and its large operating-room footprint. The fixed remote center-of-motion (RCM) is also a limitation. Alternative designs try to compensate for some of those drawbacks in adult laparoscopy [3, 4].

Other robotic systems have been developed to provide robot aid in
areas in which the da Vinci is hindered by its design. For instance,
robots have been developed for procedures in restricted workspaces
such as brain microsurgery [5], eye surgery
[6], endonasal surgery [7], and
pediatric surgery [8]. These robotic systems have several
designs, such as serial linkage, as in the da Vinci system and others
[6, 5], parallel linkage [8],
and flexible tubes [7, 9]. There are also
many control methodologies for the autonomous generation of constrained
motion using *active constraints/virtual fixtures* [10, 11, 12, 13, 14, 15, 16, 17].

An in-depth survey on active constraints is presented by Bowyer *et
al*. [18], who show that most of the research
in the field of virtual fixtures for teleoperated robots has focused
on impedance control on the master side, along with techniques such
as *proxy and linkage simulation* and *reference virtual
fixtures*. Impedance control on the master side has been successful
in pose^{1}^{1}1*Pose* stands for combined position and orientation.
control of non-redundant robotic systems, such as the da Vinci, because
generating virtual fixtures on the master means the slave can be effectively
kept away from undesired interactions with the patient’s anatomy [19].

However, such techniques, if applied only on the master side, are not suitable when the slave robot is redundant because, even if the master’s and the slave’s end-effector poses are the same (with respect to their own reference frames), the slave robot may have infinite configurations in joint-space [20]. Consequently, some slave robot’s links can have harmful interactions with the patient, despite any feedback on the master.

As an alternative to master-side techniques, slave-side techniques
have also been proposed, some of which use conventional control algorithms
based on the Jacobian pseudoinverse and nullspace projection [20]
to generate an RCM [13, 14, 15]
or even more complex constrained workspaces [9].
Nevertheless, these standard techniques struggle to deal with *hard*
constraints^{2}^{2}2Hard constraints cannot be violated [18], in contrast
with *soft* constraints [11], in which
small violations are allowed for short periods of time. that are important in the medical field, such as joint and actuation
limits.

## Ii Related works

Initial approaches to constrained joint optimization in the generation of virtual fixtures [10, 11, 12] have been successful in providing constrained motion in complex scenarios, but have had issues such as being “computationally demanding and inconsistent for some constraint and cost functions” and having a “non-force-reflecting” master under teleoperation [18].

The computational demand resulting from the use of quadratic positional
constraints and the difficulty of balancing virtual fixture and teleoperation
terms in the objective function is reported by Kapoor *et al*.
[11]. A follow-up work by Li *et al*.*
*[12] has been shown to be computationally more efficient,
as long as there is a single tool moving in the workspace , which
is not the case in most surgical scenarios. Kwok *et al.* have
proposed *ad hoc* techniques for snake robots [21].
Lastly, several validation studies have focused on a single robotic
system in laparoscopic scenarios [10, 11, 12]
and in sinus surgery [12], or on two robotic systems
that follow a predefined trajectory in the contexts of deep neurosurgery
[16] and transnasal surgery [17].

A general framework for constrained motion control that does not depend
on specific robot designs can have several advantages. First, once
constraints are defined to achieve a desired behavior (e.g., avoiding
joint limits, preventing self-collisions and collisions with the workspace),
those same constraints can be applied to other type of robots to achieve
similar behavior. Second, the theoretical properties of the motion
controller (e.g., time response, closed-loop stability, computational
complexity) depend mostly on the framework, and a particular robotic
platform has little or no influence on the closed-loop behavior. Third,
researchers can focus on defining *new* relevant constraints
for a particular robot design using a coherent theoretical framework,
instead of resorting to *ad hoc *techniques. Thus, constrained
optimization allows for the the most generalizable solution, once
the aforementioned issues are solved.

### Ii-a Statement of contributions

In this letter, we propose a novel unified framework for robot control under teleoperation, which is presented in Section IV

. First, we tackle the issue of teleoperation in constrained optimization approaches by including only terms specific to teleoperation in the objective function, without mixing them with virtual fixture terms, which facilitates parameter tuning. Second, we use the objective function with the vector field inequality method (VFI)

[16, 17], which provides customizable dynamic active constraints using linear constraints in the control inputs (i.e., the joints’ velocities). Third, we add*Cartesian impedance*to our framework, effectively resolving lack of force-reflection on teleoperated robots under constrained optimization.

These three contributions allow us to perform teleoperation in complex scenarios, regardless of the to robot geometry. The generality of the proposed unified framework is tested in two bi-manual experiments using different robotic systems, as shown in Section V.

## Iii Mathematical Background

The proposed unified framework for surgical robot teleoperation uses
quadratic programming for closed-loop inverse kinematics. To generate
dynamic virtual fixtures, geometrical primitives are modeled using
dual quaternion algebra, and linear constraints are added to the quadratic
program using the VFI method. The basics of quaternions,^{3}^{3}3Although we use the formalism introduced in [16, 17],
which employs dual quaternion algebra to represent points, lines,
and planes as primitives to generate dynamic virtual fixtures and
to define the linear constraints in the control inputs, only a very
brief presentation of *quaternions* is sufficient for the purposes
of this letter. quadratic programming for closed-loop inverse kinematics, and the
vector field inequalities method are briefly explained in this section.
For more information, the reader is referred to [16, 17].

### Iii-a Preliminaries

Quaternions can be regarded as an extension of complex numbers. The quaternion set is , in which the imaginary units , , and have the following properties: . The bijective mapping is used to explicitly map the coefficients of a quaternion into a four-dimensional vector; more specifically, if , then . Similarly, the set has a bijective relation with . The quaternion represents the point , the mapping of which is given by , analogously to . The set of quaternions with unit norm is . Any can always be written as , where is the rotation angle around the rotation axis [22]. The conjugate of is given by .

### Iii-B Differential kinematics

Differential kinematics is the relation between task-space velocities and joint-space velocities, in the general form

in which is the vector of manipulator joints’ configurations, is the vector of task-space variables, and is a Jacobian matrix.

The Jacobians relating the robot’s joint velocities to its end-effector’s unit dual quaternion pose (), rotation (), and translation () can be found using dual quaternion algebra [23].

### Iii-C Quadratic programming for differential inverse kinematics

In closed-loop differential inverse kinematics, we first define a
task-space target and a task error .
Considering , , and a gain
, the standard form of the
inverse differential kinematics with *linear* constraints is

(1) | ||||

subject to |

where and .

It is common practice to add a damping factor to Problem 1 to give it some robustness to singularities [24]. Moreover, more than one robot can be controlled simultaneously in a centralized architecture. Suppose that robots should follow their own independent task-space trajectories. For , let each robot have joints, joint velocity vector , task Jacobian , and task error . With these changes, Problem 1 becomes

(2) | ||||

subject to |

where

, , and is a matrix of zeros with appropriate dimensions.

### Iii-D Vector field inequalities in the generation of dynamic virtual fixtures

The VFI method for dynamic elements [17] first requires a function that encodes the (signed) distance between two geometric primitives. Second, it requires a distance Jacobian and a residual relating the time derivative of the distance function and the joints’ velocities in the general form

(3) |

where the residual contains the distance dynamics unrelated to the joints’ velocities. The required distance function, distance Jacobians, and residuals for all relevant primitives used in this letter are shown in [17]. Lastly, the VFI method requires the definition of a safe distance .

With these definitions, and given , the signed distance dynamics for each pair of primitives is constrained by

(4) |

Constraint 4 assigns to each primitive a velocity constraint that actively filters the robot motion in the direction of the restricted zone boundary so that the primitives do not collide. At most, each primitive will converge to the boundary, and velocities tangential to the boundary itself are unaffected.

## Iv Proposed unified framework for teleoperation

The proposed framework is divided into two parts, with a constrained optimization algorithm that runs on the slave side and a Cartesian impedance feedback that runs on the master side. Both are explained in this section.

The technique proposed in this letter can be used to control any robotic system, as long as the forward kinematics model and Jacobian are available. Therefore, this includes serial-link, parallel-link, and even flexible robots [9].

### Iv-a Slave side: Constrained optimization

Existing approaches to constrained optimization have terms in the objective function for both trajectory tracking and virtual fixture generation, which is a major source of parameter tuning difficulties [11] and inconsistencies in constraints and cost functions [18]. To prevent issues related to having these mixed terms, the proposed technique includes only those terms related to trajectory tracking in the objective function.

Without loss of generality, suppose two identical slave robots are controlled through teleoperation, each by an independent master interface that generates a desired pose signal . In this letter, we propose the following constrained optimization problem

(7) | ||||

subject to |

where

in which , , and are the unweighted cost functions related to the end-effector translation, end-effector rotation, and joint velocities of the -th robot, respectively; furthermore, each -th robot has a vector of joint velocities , a translation Jacobian , a translation error , a rotation Jacobian , and a switching rotational error

(8) |

based on the dual quaternion invariant error [25], where and are the desired and current end-effector orientations, respectively. In addition, and is a positive definite damping matrix, usually diagonal. Lastly, are weights used to define the priorities between robots and between the translation and the rotation.

We use the linear constraints to avoid joints limits [24] and to generate active constraints using the VFIs [17]. Each parameter is explained in more detail in the following subsections.

#### Iv-A1 The translation and rotation weight,

The weight is used to balance translational and rotational gains. In our application, the translation error is usually on a millimeter scale or lower. Therefore, the rotation error may overtake the translation error, depending on the units used to represent distance. Adding the weight allows us to intuitively set that balance without other modifications to the optimization problem.

#### Iv-A2 The robot prioritization weight,

The weight is used to set a *soft*
priority between robotic systems. To understand this parameter, first
note that if Problem 7 has
a solution, the objective function will be optimized, *given
that the linear constraints are satisfied*. This means that the linear
constraints prevent any collisions, even if this causes the trajectory
tracking error of a particular robot to increase. In such cases, the
parameter can be used to weight the priority between the
two robots. If , then minimizing the trajectory tracking
error for robot 1 is favored over robot 2, effectively prioritizing
robot 1. The reverse is true for . No explicit priority
is given if .

#### Iv-A3 The joint weight matrix,

Whenever the robot is redundant and has a heterogenous structure, for instance a robotic manipulator with DOF attached to a customized forceps with DOF, the damping matrix can be written in the form

in which and are matrices used to increase the relative weights of joints we wish to have move less than others. For instance, given and , with , we can favor forceps motion over manipulator motion by setting .

#### Iv-A4 The switching unit quaternion controller

Because the group of unit quaternions double covers , both and represent the same orientation, which causes the unwinding problem [26]. In practice, this problem results in undesired motions whenever a continuous control law is employed. In order to see that, suppose that the orientation error is given only by ; if , then the orientation error is equal to . However, if , then the orientation error is equal to , although the current orientation is already the desired one. In that case, the robot moves unnecessarily until again reaching the new equilibrium point. A way to circumvent this problem is to use discontinuous or hybrid control laws [26], which in our case is done by switching the error. This way, if is closer to , the error is given by ; conversely, if is closer to , the error is given by .

### Iv-B Master side: Cartesian impedance

The existing approaches to constrained optimization in the generation of virtual fixtures are known to lack force reflection under teleoperation [18]. To compensate for this drawback, we add a Cartesian force feedback on the master side that is proportional to the current error on the slave side, in the form

(9) |

for each master–slave pair, where is the reflected force on the master side, are, respectively, stiffness and viscosity parameters, is the translation error of the slave, but seen from the point of view of the master, and is the linear velocity of that master interface. This proportional force feedback with viscosity allows the operator to “feel” any task-space directions in which the robot has difficulty moving.

##
V Experiments^{4}^{4}4See accompanying video.

In order to evaluate the technique proposed in this letter, we first present experiments to evaluate the effects of and the dynamic active constraints using the da Vinci Research Kit (dVRK) [27], which is a research-friendly robotic system comprising the same master and slave robotic systems of the da Vinci Surgical System. Second, we present a peg transfer experiment to evaluate the proposed framework in complex tasks. For this second experiment, a seven-DOF robot was operated by a medical doctor.

The software implementation was the same for both systems, namely
Ubuntu 16.04 x64 running ROS Kinetic Kame.^{6}^{6}6http://wiki.ros.org/kinetic/Installation/Ubuntu
Robot kinematics was implemented using the DQ Robotics^{7}^{7}7http://dqrobotics.sourceforge.net
library, and constrained convex optimization was implemented using
IBM ILOG CPLEX Optimization Studio^{8}^{8}8https://www.ibm.com/bs-en/marketplace/ibm-ilog-cplex
with Concert Technology.

### V-a dVRK experiments

The first set of experiments used the experimental setup shown in Fig. 1 and was devised to evaluate the effects of a change in the prioritization weight , while dynamic active constraints to prevent collisions between shafts were enabled. Three types of constraints were added: a shaft-to-shaft distance constraint, to prevent collisions between tool shafts; a plane-to-point constraint, to prevent collisions between the right tool and the peg transfer board; and a joint limit constraint. All were implemented using VFIs [17].

The experiment involved manipulating a triangle on a peg transfer
board, which is the same peg transfer board used in the Fundamentals
of Laparoscopic Surgery (FLS) curriculum.^{9}^{9}9http://www.flsprogram.org
For repeatability, before the task begun, the right tool was positioned
on a central peg and the triangle was placed on the bottom-right peg
closest to the right tool. Only the left tool was allowed to move.
The right tool was commanded to stay in a constant pose throughout
the procedure.

The user had to pick and place the triangle in a clock-wise motion, which required the triangle be transferred between five pegs. Reaching the four initial pegs should not induce any collisions between tools and were useful to show whether the prioritization was cumbersome outside of collision situations. The last target peg was the same as the first to close the peg transfer circle. Reaching the last peg required the left tool to push on the right tool’s shaft. The behavior of the system was evaluated under three different levels of prioritization, as shown in Table I. The other parameters are shown in Table II.

Same priority | Left tool higher priority | Left tool lower priority | |

0.5 | 0.99 | 0.01 |

Suppose the tracking error of the tools should be 10 mm in order to prevent a shaft–shaft collision. means that, in order to prevent a collision, both arms’ trajectory tracking errors are increased by the same amount, therefore, 5 mm each. means that the left tool has a tracking error of mm and the right tool an error of 9.9 mm. The reverse holds for a .

*In the case of the infant experiment, there was no active constraint relating both robots; therefore, a change in had no effect.

MS | |||||||
---|---|---|---|---|---|---|---|

dVRK Experiments | 0.99 | (Tab. I) | 1 | 1 | 350 | 10 | |

Infant Experiment | 0.99 | * | 80 | 1 | 100 | 10 |

, : proportional gain of the kinematic controller and the VFI, respectively.

: translation error to orientation error weight (Section IV-A1).

: robot prioritization weight (Section IV-A2).

: Cartesian impedance proportional and viscosity gains, respectively (Section IV-B).

MS: Motion scaling. A motion scaling of X means that a relative translation of the master was multiplied by X before being sent to the slave.

#### V-A1 Results and discussion

Snapshots of the peg transfer task using the dVRK are shown in Fig. 2, for each of the three experimental cases. Complete footage of each experiment is shown in the accompanying video.

In the first case (), the left tool could reach all off the pegs, as required by the task. The right tool autonomously evaded the left tool whenever the left tool was commanded to a region that would cause a collision. Although the positioning of the triangle on the last peg was possible, it required considerable force from the operator to push the right tool (0.35 N on the left tool for each millimeter of error on the right tool).

In the second case (), the left tool could reach all off the pegs, as required by the task. The force feedback on the left tool was weak (0.0035 N on the left tool for each millimeter of error on the right tool); therefore, the left tool could even place the triangle on the peg in which the right tool was initially located.

Finally, in the last case, (), the left tool was not able to reach all off the pegs in the prescribed order. The user could feel a strong force feedback (34.65 N on the left tool for each millimeter of error on the right tool) whenever forcing the left tool against the right tool.

These results show that the parameter can be used to prioritize tools in an intuitive manner. How to effectively use this in a surgical task is left to future work.

### V-B Infant peg transfer experiments

In this task, our target was to known whether a medical doctor could perform a difficult task under teleoperation in a constrained workspace. Therefore, an expert in manual laparoscopic pediatric surgery was invited to participate in this preliminary experiment.

The constraints in infant surgery are considerably more complex than those in adult laparoscopy, and the da Vinci was shown to be inadequate for this type of surgery [1]. In this context, we employed a surgical system that is being developed in parallel to this work.

Three types of constraints are required in infant surgery. First, medical doctors use the compliance of the infant’s skin to increase the reachable workspace. This compliance can be considered in our framework by generating an entry-sphere (shaft-to-point distance with safe distance larger than zero), rather than using an entry point. Second, the tool might move outside of the camera’s field-of-view owing to the small size of the workspace. Even though this situation is common in manual surgery, because medical doctors rely on their spatial perception of their bodies to locate tools, such out-of-bounds motion is highly undesirable in robot-aided surgery owing to safety concerns. In this context, a safety cuboid constraint was added for each individual robotic arm, as shown in Fig. 5. Lastly, joint limits were also considered.

As in the FLS curriculum, the medical doctor was asked to transfer the triangles from one side of the peg transfer board to the other.

#### V-B1 Results and discussion

The medical doctor participated in three trials, one of which is illustrated in Fig. 6. With very little experience using the proposed system, the medical doctor was able to perform a full peg transfer experiment in about 7 min. Overall, the medical doctor gave a high evaluation of the robotic system usability.

Qualitatively, after inspecting a video recording of the robot motion during the peg transfer experiment, it was visible that the entry-sphere constraint was properly maintained. There were no rib dislocations and no model motion, which happened when using the da Vinci [1].

Quantitatively, the tool shaft distance to the entry-sphere center is shown in Fig. 7, as measured from the robot’s encoders. The maximum distances between each robot shaft and the center of its entry-sphere were 2.54 mm and 2.41 mm, respectively. This means there was a maximum constraint violation of 0.5 mm. Understanding the source of this constraint violation is a topic of ongoing research. The culprit is thought to be the discrete time implementation of Problem 7.

Another important set of constraints was the planar constraints making up the cuboid workspace. Among the 12 plane constraints, the maximum constraint violation corresponded to the plane that impeded the right robot’s tool tip from being retracted from the model, as shown in Fig. 8. The magnitude of the violation was 0.142 mm, which is of a similar magnitude to the constraint violation of the entry-sphere. Other planes showed negligible constraint violations of under 0.1 mm. Because the right robot tool tip was kept at the border of that plane during most of the experiment, this indicates why a higher violation of that plane was observed.

These results show that a complex task, with several active constraints, can be performed smoothly under teleoperation by a medical doctor using the proposed framework. How well the framework can operate in estill more complex scenarios, including flexible tools, is a topic of ongoing research.

## Vi Conclusions and future work

In this letter, a novel unified framework for robot control under teleoperation was proposed. The method can be used to provide smooth teleoperation, regardless of the robot geometry and under workspace constraints. On the slave side, a constrained optimization algorithm provides virtual fixtures for collision avoidance and the avoidance of joint limits. On the master side, a Cartesian impedance algorithm allows the user to “feel” directions in which the robot has difficulty moving. The proposed framework is evaluated in two scenarios, with different robot geometries. First, we demonstrate a shaft–shaft collision avoidance with tool prioritization under teleoperation using the dVRK. Second, we show a peg transfer experiment performed by a medical doctor using a redundant robot system in an infant surgery scenario.

In future works, we plan to test the performance of the framework in the teleoperation of flexible robots to further validate its generalizability.

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