5.5
Conclusions
In this chapter, a framework was proposed for analyzing the kinematic stability of concentric- tube robots, in presence of (a) straight sections at the proximal ends and (b) multi-tube body architecture. It was mathematically shown that the proposed stability conditions are related to the tube parameters including pre-curvatures, stiffness, length of straight sections, and length of curved sections. Simulation results (for two-tube and three-tube robots) were given in support of the theory. In the simulations, it was shown that changing the length of the straight sections and/or increasing the number of tubes can directly affect the robot’s kinematic stability. A good agreement was observed between the proposed theory and the simulated forward kinematics.
BIBLIOGRAPHY 107
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109
Chapter 6
Real-time Trajectory Tracking for
Externally-Loaded Concentric-Tube
Robots
Concentric-Tube robots can offer a suitable compromise between force and curvature con- trol. In a previous study by the authors, a real-time trajectory tracking scheme for an unloaded concentric-tube robot was developed. One of the practical barriers to the use of a concentric-tube robot in medical applications is compensation for the impact of en- vironmental forces which can cause drastic deterioration in tracking performance. In this chapter, by modifying the robots forward kinematics and Jacobian, a new method is de- veloped to facilitate tip tracking in real-time while accounting for an external load at the robots tip. By considering the tip deflection resulting from the external load, a novel dual- layer control architecture is proposed to compensate for this deflection during trajectory tracking. In order to measure the force exerted on the tip position of the robot, a new tech- nique is proposed that can move the sensing system from the distal tip to the proximal base. Experimental results are given to illustrate the effectiveness of the proposed method.
6.1. INTRODUCTION 110 Table 6.1: Nomenclature I
{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 n(s) = [nx(s), ny(s), nz(s)]T Stress vector represented in the body frame m(s) = [mx(s), my(s), mz(s)]T Bending moment vector represented
in the body frame
ui(s) = [uix(s), uiy(s), uiz(s)]T Bending curvature and torsion of theithtube
which are represented in the body frame
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 represented
in the body frame
f(s) Distributed force vector represented in the body frame
l(s) Distributed moment vector represented in the body frame
Ki Stiffness matrix of theithtube
kix, kiy, kiz Bending and torsional stiffness of theithtube L Length of the curved section
Θ(s) = θ1(s), θ2(s), θi3(s) Euler angles of the rotation matrixR(s)
of theithtube
r(s) = [x(s), y(s), z(s)]T Position vector of the cross-section located ats
represented in the world frame
M(s) Bending moment vector represented in the world frame
αi(s) Twist angle difference between theithtube and
the1st tube
RZ(αi(s)) Rotation matrix between the body frames of
theithand the1st tube
Fext External point force at the tip of the robot
represented in the world frame
Pi Position of the proximal end of theithtube H Mapping betweenPi to each sub-link’s length
˜
J Jacobian matrix of a sub-link
6.1
Introduction
A concentric-tube robot as a subset of continuum robots is composed of a sequence of telescoping pre-curved elastic tubes inserted one inside the next. In this flexible robotic
6.1. INTRODUCTION 111 Table 6.2: Nomenclature II
X Position of robot’s tip in the Cartesian space
Xmeas Measured position of robot’s tip in the Cartesian space q Joint variables of the robot
qdes Desired trajectory in joint space w External point wrench at the tip of the robot
F(q, w) Forward kinematics of the robot
J(q, w), C(q, w) Jacobian and Compliance matrices of the robot
∆ Deflection of the robot under loading
ˆ
∆ Estimated deflection of the robot under loading
Uint Controller’s input signal S Laplace variable
structure, axial rotation and translation of individual tubes with respect to each other can generate 3D curvatures. Thus, the shape of the robot can be controlled in order to guide it inside lumens, natural orifices, and other anatomical organs in a variety of medical appli- cations specially involving unreachable or confined surgical sites.
Accurate position control of surgical instruments is a vital need in Robotics-Assisted Min- imally Invasive Surgery (RAMIS); however the small size of the incision reduces the robot rigidity and, as a result, challenges positioning accuracy. As is known, flexible surgical tools (such as surgical needles [1]) have been widely used in percutaneous minimally invasive interventions. However, complex and inaccurate kinematics modeling of flex- ible/continuum mechanisms has imposed limitations on accurate control of the tip mo- tion in RAMIS. This topic has been the subject of recent publications for concentric-tube robots [2, 3], and is explored further in this study.
Modeling the shape of concentric-tube robots has evolved over the last few years. Histor- ically, a variety of mechanical phenomena, e.g., bending [4], torsion [2] [5], and friction between the tubes [6] have been gradually taken into account to improve modeling accu- racy. It has been shown that the assumption of zero external load in kinematic modeling can cause considerable error in estimating the robot configuration [7]. To deal with this
6.1. INTRODUCTION 112 issue, in [7] and [8] the forward kinematics of the robot are modified to include the effects of external loading. However, most of the developed models are mathematically complex and computationally expensive. To achieve a good balance between computational effi- cacy and numerical accuracy, the authors proposed a fast torsionally-compliant model [9], which was later utilized to develop a feasible strategy for real-time tip tracking in free mo- tion [3]. Human-in-the-loop architecture was developed in [2], and [10] respectively using an inverse kinematics scheme and an inverse Jacobian technique (with singularity avoid- ance) as local controllers. Implementation of Magnetic Resonance Imaging based position control (utilizing inverse kinematics) is given in [11]. In addition, more advanced control architectures such as stiffness control [12] have been recently studied in the literature.
In this chapter, the problem of real-time position control in the presence of undesirable robot deflection is studied. The deflection is caused by: (a) variations in external loading, and (b) alteration of kinematic behavior of the robot in the presence of external forces. The ultimate goal is to navigate the robot accurately regardless of external disturbances. This feature is important in medical applications, where the targeting accuracy strongly correlates with the performance of the administered therapy. For this goal, deflection of a concentric-tube robot in the presence of external disturbances is estimated; then, the desired trajectory is reshaped using the estimated robot deflection in order to compensate for the effect of varying loading. For this purpose, the interaction force between the robot tip position and environment should be used in the proposed control architecture. In this chapter a new scheme is proposed that uses force sensors at the proximal end (close to the driving unit) of the robot. This removes the need for measuring the force at the distal end (close to the robot’s tip position) while compensating for the internal forces between the tubes. This feature is motivated by the fact that having a sensor close to the robot’s tip is not very feasible in practical application.