Many clinicians and roboticists theorize the major limiting factor in the performance of robotic-assisted minimally invasive surgery (MIS) is the lack of haptic feedback (3).
When a surgeon is interacting with a tissue or organ, their own proprioceptive and cutaneous senses allow them to determine the appropriate amount of force to apply in order to minimize damage. This ability is removed when a robotic manipulator is interacting with the tissue. The force feedback also allows surgeons to distinguish diseased tissue from healthy areas via mechanical properties, identifying regions of lumps or tumors, and hardened or calcified portions of arteries (153;154). The field of haptics seeks to provide the user of a tele-operated or virtual system with a realistic experience of an actual environment by providing them with the most intuitive and easy to understand information via electromechanical devices. There are two basic types of haptic feedback, tactile and force. Tactile feedback recreates sensations felt by the skin. This would include things like texture, temperature and pulsation. Force feedback replicates what a person feels from a solid boundary or object. This includes things like weight and inertia (155). It has been hypothesized that a greater degree of immersion into a virtual environment can be achieved with even rudimentary haptic devices than with more complex and higher defined visual displays alone.
Haptics research is a well-established field. The first time the concept was
mentioned was by computer scientist Ivan Sutherland when he wrote that the ultimate display would allow users to feel virtual objects that could only be displayed graphically with a mechanical force-feedback device (156). This specifically refers to machine haptics in which the feedback is provided to a human user by an electromechanical device. In surgery, haptics can enhance simulations, provide vital information on tissue deformation and provide a surgeon with the feedback needed to perform complex tasks. However, sensing forces at the tip of small instruments in a confined space is a difficult design, computational, and mechatronic challenge.
Force sensors for surgical instruments would ideally be placed at the tip and provide 6 DOF measurement. However, challenges of biocompatibility, sterilization,
and temperature sensitivity make this a difficult engineering task (3). Many
on the robotic manipulator that is outside the body. This also creates challenges because the forces encountered by the sensor can include friction in the manipulator actuation system and torques applied by the instrument at the point of insertion that can be large enough to mask the small tissue interaction forces.
Another option for force estimation involves teleoperation controllers, where the difference in the desired and actual position of the robot inside the body is an indication of the amount of force applied to the environment. One main example
of this type of control method is the four-channel framework (157) where the
position and force of both the master (surgeon controlled input device) and the slave (robotic manipulator inside the body) are sensed and influence the control of the manipulator. One assumption in this method is that the master and slave inputs and forces affect the output with the same dynamics. This would not be the case with compliant mechanisms. Other examples include position-position and position-force teleoperation.
While none of the commercially available systems provide a level of force feedback that would decrease damage to tissue, academic systems that include force feedback
are being developed such as the RAVEN II at the University of Washington (158),
an enhanced da Vinci robot (159), a laparoscopic robot (160), the NeuroArm and
the LapRobot (161). Most of these systems use a method of measuring the force at
the individual joints of the robot to estimate the force at the tip, a method called actuation-based force sensing.
Actuation-based force sensing is another topic that is well studied for classical rigid-link manipulators but has not been widely studied for continuum robots. In conventional rigid-link designs, end-effector wrench is mapped to the joint torques
by the transpose of the Jacobian matrix (162) due to conservation of energy. For
continuum robots, this simple relationship does not hold true due to the elastic energy stored within the robot links. Gravagne showed in (151) that the principal of virtual work can account for the elastic energy storage of a flexible backbone to obtain the true linearized relationship between actuator torques and end effector forces. When this
energy storage is accounted for, accurate actuation-based wrench sensing is possible for multi-backbone continuum robots, as has been investigated in (163;78;164; 134) (termed “intrinsic wrench sensing” in those works). These works derive and validate a wrench-sensing algorithm for constant curvature manipulators via the principle of virtual work. For the general PCRs used in our research, we use a geometrically exact, nonlinear rod-mechanics model to accurately predict the tip force instead of a virtual-work framework under the constant-curvature assumption.
There has also been work in deflection-based force sensing with continuum robots (63) and catheters (165;166). However, if the device is very stiff in one direction, this approach can become quite ill-conditioned, resulting in high estimation errors. We show that this is also true for PCRs. Our analysis also concludes that the sensitivity to sensing errors is greatly decreased for actuation-based force sensing when compared to deflection-based force sensing.
The model framework presented in Chapter 3 can be arranged in several forms,
three of which can be used to sense forces at the tip of the manipulator. We compare the potential error propagation of these force sensing methods, including deflection- based force sensing, to arrive at the proposed method of actuation-based force sensing
and provide results demonstrating a much lower error sensitivity. The sensing
method takes 12 inputs of actuator positions (6) and actuator forces (6) to solve the static equilibrium and boundary condition equations for the entire manipulator and calculate the 6-DOF applied end-effector wrench.
Stiffness control of continuum robots is a small but growing area of research. The goal of a stiffness controller is to provide a command to the robot that incorporates the sensed tip force and changes the robot’s active stiffness. The use of stiffness control with continuum robots takes advantage of their passive mechanical stiffness to increase their effectiveness in the medical application. Compliant motion control of a manipulator subject to forces along its axis during insertion through a small lumen has been investigated in (167). In (81), a concentric tube robot uses the controller
the tip of the manipulator using deflection based force sensing. Another example of of stiffness control with tendon driven robots can be seen in (168). In this research, a passive stiffness controller has been developed for the PCR using the modeling
framework presented in Chapter 3. The controller is considered passive due to its
limitation of only decreasing the stiffness of the PCR. The derivation and results of this work are presented in Chapter 5.2.
Chapter 3
General Kinematic Modeling for
Flexible Manipulators
The ultimate goal of modeling a robotic manipulator is to control the joints in a way that provides accurate and dexterous motion of the end effector. A robot’s model must be based on the physical parameters of its material and construction. This is especially true for continuum manipulators whose pose cannot be fully defined in a closed form solution by link lengths and joint angles (60). This chapter will review prior work in continuum robot modeling, detail the Cosserat rod equations and their use in modeling parallel continuum manipulators and concentric tube robots and outline the preliminary and proposed work on manipulator modeling in this dissertation.
For continuum manipulators, the equations defining the deformation of flexible structures are evaluated numerically and involve solving the statics or dynamics
as opposed to the forward kinematics (72; 97; 67; 70). The barriers to control
of continuum robots come from the limits of solving these models in real time. The Cosserat rod models used in parallel continuum manipulators and concentric
tube robots have an advantage of simplification. Classical Cosserat rod models
assume no cross-section deformation and gain a great amount of efficiency due to this simplification (e.g. two orders of magnitude in (169) for an extended Cosserat
s
m(s )
2n(s )
2f(s), l(s)
-m(s )
1-n(s )
1Figure 3.1: Arbitrary section of flexible link with the internal force and moment, n and m, acting on the portion of the link. The distributed force and moment, f (s) and l(s), act along the entire arc length. These forces and moments are written in the global reference frame g(s).
model with cross-section inflation. Weak-form Cosserat rod models can be formulated and solved using a finite-element, finite-difference, or discrete-differential-geometry approach. Other systems use information related to the actuator length limits to create real-time implementations (88). With these techniques, real-time control of parallel continuum and concentric tube manipulators is possible (170;46).
In order to establish a basis for the modeling of parallel continuum manipulators and concentric tube robots, we will review the Cosserat rod theory presented in (171) and then provide an overview of the modeling algorithms. We generalize the forward kinetostatics model that was previously presented as a minimal representation in (75). This model is more intuitive and unifies all problem formulations that we identify by only changing the known variables and utilizing the same system of equations. The new model framework also allows for the identification and efficient calculation of matrices that are applicable to any continuum manipulator or soft robot: the Jacobian, end-effector compliance, input stiffness, and wrench reflectivity matrices.
3.1
Cosserat Rod Equations
The robot is constructed of n flexible links in parallel. The shape of the ith link is defined by its position pi(si) ∈ R3 and material orientation in the form of an
orthonormal rotation matrix, Ri(si) ∈ SO(3), forming a material-attached reference frame, gi(si) = Ri(si) pi(si) 0T 1 ∈ SE(3) (3.1)
as a function of arc length si ∈ [0 li]. The position and orientation evolve along the length of the rod according to the linear, vi(s) ∈ R3, and angular, u
i(s) ∈ R3, rates of change expressed in local or “body frame” coordinates of the material frame as follows:
p0i =Rivi, R0i =Riui,b
(3.2)
where 0 denotes a derivative with respect to si, and b denotes mapping from R3 to
so(3) as follows: b a =h a03 −a03 −aa21 −a2 a1 0 i (3.3) As in (172), we also use ∨ to denote the inverse mapping of b , i.e. (bu)∨ = u. The same symbol is overloaded to map a R6 to se(3) as follows:
b a = 0 −a6 a5 a1 a6 0 −a4 a2 −a5 a4 0 a3 0 0 0 0 . (3.4)
Therefore, if one knows the body-frame vi and ui vectors and an initial frame gi(0), the remaining frames can be obtained by integrating the differential equations in Equation (3.2). Since vi and ui are not constant with respect to si, numerical integration is required to obtain gi.
As derived in (171; 63; 67), the rates of change of the internal force vector ni
and internal moment vector mi with respect to the arc length si are described by
the classical Cosserat rod differential equations of static equilibrium. This is done by taking the derivative with respect to arc length of the static equilibrium balance
shown in the free body diagram shown in Figure3.1.
n0i = − fi
m0i = − p0i× ni− li,
(3.5)
where all vectors are expressed in global coordinates, and fi and li are distributed force and moment vectors, respectively, applied per unit length to rod i. Distributed self-weight and any other external forces are straightforward to include within fi and li (see Figure3.1).
The kinematic variables vi and ui are related to the material strain (shear,
extension, bending, and torsion) and can be used to calculate the internal force and moment vectors via a material constitutive law. The linear constitutive relationship used follows the form
ni = RiKse,i(vi− v∗i) , Kse,i = hAiGi 0 0 0 AiGi 0 0 0 AiEi i mi = RiKbt,i(ui− u∗i) , Kbt,i= hEiIi 0 0 0 EiIi 0 0 0 JiGi i (3.6)
and expresses ni and mi in global coordinates where v∗i and u ∗
i are the kinematic
variables of the rod in an assigned stress free reference state. For an initially straight rod, the appropriate reference state variables are v∗i = [0 0 1]T and u∗
i = [0 0 0]T. The matrices Kse,i and Kbt,icontain the stiffness terms for a radially symmetric rod cross- section, which could vary with arc length, involving the area Ai, Young’s modulus, Ei, shear modulus Gi, second area moment Ii (about the local x and y axes), and the polar area moment Ji about the local z axis.
Thus, for each rod, equations (3.2), (3.5), and (3.6) form a system of differential equations that describes the evolution of the state variables pi, Ri, ni, and mi with
respect to si. p0i =Rivi, vi = K−1se,iR T i ni+ v∗i R0i =Riubi, ui = K −1 bt,iR T imi+ u∗i n0i = − fi m0i = − p0i× ni− li (3.7)
The above equations describe the shape and the internal forces and moments of the ithlink. In the next section, we detail how the design of the distal end effector and the proximal base platform create geometric constraints. These constraints are enforced along with end-effector static equilibrium via a coupled set of boundary conditions that are simultaneously solved to obtain the configuration of the manipulator.