kHz for high-level control. For the second task, the robot can track 2D patterns on a surface while regulating the position of the robot in the gravitational direction. The residual errors in regulation control did not result in applying extensive forces on the tracking pad, due to the inherent compliance of CTRs.
2.5
A Low-Cost and Compact CTR Design
The robot built in the previous section exhibited high performance in teleoperated surgical tasks. However, the cost of this robot was around $20,000 and the size of it was 25×
20 × 90 cm (width, height, length), weighing roughly 20 kg. This bulky design does not allow it to be installed on any of the existing surgical robotic platforms; therefore, positioning and orienting this robot during surgical operations will be difficult. In addition, electrical wiring will be challenging, as it has 96 electric conductors between the drive unit and the control box, which must be protected and sealed for medical applications. In this section, a low-cost and compact CTR drive unit is proposed and implemented to address the above mentioned shortcomings. This new design minimizes the complexity in mechanical transmissions and electrical circuits, allowing this prototype to be more affordable and compact than the previous design.
As shown in Fig. 2.6, the actuation module of each tube was made of two stepper mo- tors and an aluminum bracket. A stainless steel linear guide rail (SS MGN12, RobotDigg, China) with three ball-bearing carriages were used to provide low-friction linear guidance and high-precision alignment between each motorized module. Three hollow-centered stepper motors (HS-Nema17, RobotDigg, China) were used for rotating the tubes. The holding torque of each motor is up to 0.4 Nm, which satisfies the specifications for driving Nitinol tubes in most medical applications. Therefore, the tubes can be directly attached to these rotary motors without the need of gearboxes or timing belts, eliminating the chances
2.5. A LOW-COST AND COMPACT CTR DESIGN 42
Figure 2.6: CAD design of a low-cost and compact CTR drive unit. (a) View of the main components of the proposed drive unit. (b) View of the overall robot.
of backlash. Three non-captive stepper motors (NC17HS3001-400T84, RobotDigg, China) with a threaded robot passing through their center axis were used for linear actuation. This threaded rod was the only mechanical transmission of the entire robot. Stepper drivers AMIS-30543 (ON Semiconductor, United States) were used to command these motors, which features 1/128 micro stepping, resulting in 0.014◦ rotary resolution and 0.00031 mm linear resolution. Each of these drivers can output 3.6 A of current without the need of cool- ing, and 6 A with forced air cooling. A 32-bit ARM core microcontroller board (Arduino Due, Arduino, United States) and an Ethernet shield (V2, Arduino, United States) were
2.5. A LOW-COST AND COMPACT CTR DESIGN 43
Figure 2.7: Assembly of the third prototype of CTR drive unit (except for the outer case) used to transmit the position and velocity commands from the main computer to the mo- tor drivers. In this design, for homing the robot, three photo interrupters (OPB930W51Z, Optek Technology, United States) were mounted next to the rotary motors and another three photo interrupters (GP1A51HRJ00F, Sharp Microelectronics, United States) were mounted next to the linear guide rail. These sensors can also be used for detecting the step(s) miss- ing in these motors during robot motion control. Four bumper switches (SS-5GL111-3, Omron Electronic Components, United States) were installed on the aluminum brackets and at one end of the guide rail to avoid collisions between the actuation modules. Based on the experimental results of the previous robotic setup, the errors in the low-level motor control were negligible compared to the total position error, as they mostly came from the inaccuracy in the kinematics of CTRs. Therefore, installing a high resolution encoder for each stepper motor was not necessary. Having said this, the errors of the robot’s tip need to be compensated for either by a position sensor or a human operator in teleoperation. Finally, the total cost of the entire robot drive unit was under $1200.
In order to access the performance of this prototype in an operating room environment, a plastic case (textured ABS) was designed to package and seal all the mechanical and elec- tronic components into one enclosure, as shown in Fig. 2.6. The overall appearance of the robot was designed to be similar to normal-use medical devices to ease the adaption of this
2.5. A LOW-COST AND COMPACT CTR DESIGN 44 technology by clinicians. The fully assembled robotic setup except for the outer case is shown in Fig. 2.7. One remaining problem is the sterilization process for this robot. After each use in animal trials, in order to detach tubes from the drive unit, these tubes have to pass through the hollow center of the stepper motors. This process could contaminate the motors’ shaft with blood or other body fluids. At this stage of development, this contami- nation is unavoidable, requiring the motors to be cleaned each time. Future work will focus on developing a sterilizable version of this robot.
BIBLIOGRAPHY 45
Bibliography
[1] http://www.mmsonline.com/articles/understanding-swiss-type-machining.
[2] Y.-L. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-time estimation of 3-d needle shape and de- flection for mri-guided interventions,”IEEE/ASME Trans. Mechatronics, vol. 15, no. 6, pp. 906–915, 2010.
[3] X. He, J. Handa, P. Gehlbach, R. Taylor, and I. Iordachita, “A submillimetric 3-dof force sensing instrument with integrated fiber bragg grating for retinal microsurgery,”
IEEE Trans. Biomed. Eng., vol. 61, no. 2, pp. 522–534, 2014.
[4] D.-Y. Lee, J. Kim, J.-S. Kim, C. Baek, G. Noh, D.-N. Kim, K. Kim, S. Kang, and K.- J. Cho, “Anisotropic patterning to reduce instability of concentric-tube robots,” IEEE Transactions on Robotics, vol. 31, no. 6, pp. 1311–1323, 2015.
[5] H. Azimian, P. Francis, T. Looi, and J. Drake, “Structurally-redesigned concentric-tube manipulators with improved stability,” inProc. IEEE/RSJ Int. Conf. Intell. Robots Syst.
46
Chapter 3
A Fast Torsionally Compliant Kinematic
Model of Concentric-Tube Robots
Concentric-tube robots have the potential to become an important surgical tool for robot- assisted percutaneous interventions. They can provide dexterous operation in a small con- strained environment. The kinematic model of a concentric-tube robot has been well de- veloped in terms of accuracy, but the computational cost places limitations on real-time im- plementation. In this chapter, we propose a new technique that will substantially improve the computational efficiency of evaluating the kinematics of a concentric-tube robot in the context of developing a control strategy without sacrificing the accuracy of the results. The model is validated by comparing the results obtained by computing the kinematic model corresponding to an experimental setup of a concentric-tube robot to which a force/torque sensor has been mounted at its base with those obtained directly from the experimental setup. The results indicate that it is feasible to compute the kinematics of the concentric- tube robot fast enough to allow the position/force control loop to be implemented at a rate of 1 kHz.
3.1. INTRODUCTION 47 Table 3.1: Nomenclature
{e1, e2, e3} World frame
s Arc length
i Tube index
{d1(s), d2(s), d3(s)} Body frame of the cross-section located ats r(s) = [x(s), y(s), z(s)]T Position vector of the cross-section located ats
R(s) Rotation matrix between a body frame and the world frame
n(s) = [nx(s), ny(s), nz(s)]T Stress vector
ui(s) = [uix(s), uiy(s), uiz(s)]T Bending curvature and torsion of theithtube ˆ
ui(s) = [ˆuix(s),uˆiy(s),uˆiz(s)]T Pre-curvature of theithtube v(s) = [vx(s), vy(s), vz(s)]T Shear strains and elongation
f(s) Distributed force vector
l(s) Distributed moment vector
Ki Stiffness matrix of theithtube
kix, kiy, kiz Bending and torsional stiffness of theithtube αi(s) Twist angle difference between theithand1st tube R(αi(s)) Rotation matrix between the body frames of
theithand1st tube
L Length of the curved section
l Length of the straight section
θ1(s), θ2(s), θi3(s) Euler angles of the rotation matrixR(s)
of theithtube
3.1
Introduction
The concentric-tube robots are a new type of continuum robots. A concentric-tube robot consists of several pre-curved elastic tubes inserted one inside another. By translating and rotating two consecutive tubes relative to each other, this kind of robot can achieve to fairly complex 3D shapes. The concentric-tube robot is suitable for surgical environments because it can offer more than 5-DOF (degrees of freedom) with dimensions as small as those of a needle (typically less than 3 mm in diameter). In recent years, various kinematic models of concentric-tube robots have been proposed based on different considerations. The model proposed in [1] is simple but has limited application, since it requires that the
3.1. INTRODUCTION 48 stiffness of the outer tubes is nearly infinite compared to that of the inner tubes. Torsionally rigid models were developed in [2], with the assumption that the tubes only experience bending. The position and orientation of the robot can be obtained analytically regardless of the ratios of stiffness between tubes, and the inverse kinematics also exists in closed form [3]. The importance of introducing torsion effects into the model was shown experi- mentally in [4]. A kinematic model that contains the torsion of the straight section of robots was proposed in [4]. Although the solution of this model has to be evaluated numerically, the Jacobian kinematics can be obtained in closed form [5]. A torsionally compliant model that includes the torsion effects of both straight and curved sections presents significant improvement with regard to accuracy [6, 7]. However, this model is computationally very expensive, because it involves solving a set of nonlinear differential equations with two- point boundary conditions. Additional calculations are needed because the solutions do not give the robots position and orientation directly. Other comprehensive models have also been developed by considering the friction effects in the tubes [8] or external loads [9]. The complexity of these models increases as more mechanical effects are included. Efforts have been made to achieve a trade-off between computational efficiency and numerical ac- curacy. In [10] a function approximation method was developed to implement a torsionally compliant model in real-time for position control with minimum loss of accuracy. How- ever, the approach requires pre-computation of a large dataset of position and orientation information over the entire workspace. A Fast Jacobian-based inverse kinematic algorithm was presented in [11], and it was shown that the computational time can be reduced to 40 ms for a 3-tube robots. In this chapter, we propose a technique which can significantly decrease the computation time to evaluate a torsionally compliant model in the context of implementing a kinematic control strategy for a concentric-tube robot. This is achieved by improving the model in the following three steps:
• Reformulating the torsionally compliant kinematic model with global variables. • Piecewise-linearization of the reformulated model.