Task and Configuration Space Compliance of Continuum Robots via Lie Group and Modal Shape Formulations

Continuum robots suffer large deflections due to internal and external forces. Accurate modeling of their passive compliance is necessary for accurate environmental interaction, especially in scenarios where direct force sensing is not practical. This paper focuses on deriving analytic formulations for the compliance of continuum robots that can be modeled as Kirchhoff rods. Compared to prior works, the approach presented herein is not subject to the constant-curvature assumptions to derive the configuration space compliance, and we do not rely on computationally-expensive finite difference approximations to obtain the task space compliance. Using modal approximations over curvature space and Lie group integration, we obtain closed-form expressions for the task and configuration space compliance matrices of continuum robots, thereby bridging the gap between constant-curvature analytic formulations of configuration space compliance and variable curvature task space compliance. We first present an analytic expression for the compliance of a single Kirchhoff rod. We then extend this formulation for computing both the task space and configuration space compliance of a tendon-actuated continuum robot. We then use our formulation to study the tradeoffs between computation cost and modeling accuracy as well as the loss in accuracy from neglecting the Jacobian derivative term in the compliance model. Finally, we experimentally validate the model on a tendon-actuated continuum segment, demonstrating the model's ability to predict passive deflections with error below 11.5\% percent of total arc length

Dynamics for variable length multisection continuum arms

Variable length multisection continuum arms are a class of continuum robotic manipulators that generate motion by structural mechanical deformation. Unlike most continuum robots, the sections of these arms do not have (central) supporting flexible backbone, and are actuated by multiple variable length actuators. Because of the constraining nature of actuators, the continuum sections can bend and/or elongate (compress) depending on the elongation/contraction characteristics of the actuators being used. Continuum arms have a number of distinctive differences with respect to traditional rigid arms namely: smooth bending, high inherent compliance, and adaptive whole arm grasping. However, due to numerical instability and the complexity of curve parametric models, there are no spatial dynamic models for multisection continuum arms. This paper introduces novel spatial dynamics and applies these to variable length multisection continuum arms with any number of sections. An efficient recursive computational scheme for deriving the equations of motion is presented. This is applied in a general form based on structurally accurate and numerically well-posed modal kinematics that assumes circular arc deformation of continuum sections without torsion. It is shown that the proposed modal dynamics are highly scalable, producing efficient and accurate numerical results. The spatial dynamic simulation results are experimentally validated using a pneumatic muscle actuated multisection prototype continuum arm. For the first time this enables investigation of spatial dynamic effects in this class of continuum arms

Control of Elastic Soft Robots based on Real-Time Finite Element Method

International audienceIn this paper, we present a new method for the control of soft robots with elastic behavior, piloted by several actuators. The central contribution of this work is the use of the Finite Element Method (FEM), computed in real-time, in the control algorithm. The FEM based simulation computes the nonlinear deformations of the robots at interactive rates. The model is completed by Lagrange multipliers at the actuation zones and at the end-effector position. A reduced compliance matrix is built in order to deal with the necessary inversion of the model. Then, an iterative algorithm uses this compliance matrix to find the contribution of the actuators (force and/or position) that will deform the structure so that the terminal end of the robot follows a given position. Additional constraints, like rigid or deformable obstacles, or the internal characteristics of the actuators are integrated in the control algorithm. We illustrate our method using simulated examples of both serial and parallel structures and we validate it on a real 3D soft robot made of silicon

Modeling, Sensorization and Control of Concentric-Tube Robots

Since the concept of the Concentric-Tube Robot (CTR) was proposed in 2006, CTRs have been a popular research topic in the field of surgical robotics. The unique mechanical design of this robot allows it to navigate through narrow channels in the human anatomy and operate in highly constrained environments. It is therefore likely to become the next generation of surgical robots to overcome the challenges that cannot be addressed by current technologies. In CSTAR, we have had ongoing work over the past several years aimed at developing novel techniques and technologies for CTRs. This thesis describes the contributions made in this context, focusing primarily on topics such as modeling, sensorization, and control of CTRs. Prior to this work, one of the main challenges in CTRs was to develop a kinematic model that achieves a balance between the numerical accuracy and computational efficiency for surgical applications. In this thesis, a fast kinematic model of CTRs is proposed, which can be solved at a comparatively fast rate (0.2 ms) with minimal loss of accuracy (0.1 mm) for a 3-tube CTR. A Jacobian matrix is derived based on this model, leading to the development of a real-time trajectory tracking controller for CTRs. For tissue-robot interactions, a force-rejection controller is proposed for position control of CTRs under time-varying force disturbances. In contrast to rigid-link robots, instability of position control could be caused by non-unique solutions to the forward kinematics of CTRs. This phenomenon is modeled and analyzed, resulting in design criteria that can ensure kinematic stability of a CTR in its entire workspace. Force sensing is another major difficulty for CTRs. To address this issue, commercial force/torque sensors (Nano43, ATI Industrial Automation, United States) are integrated into one of our CTR prototypes. These force/torque sensors are replaced by Fiber-Bragg Grating (FBG) sensors that are helically-wrapped and embedded in CTRs. A strain-force calculation algorithm is proposed, to convert the reflected wavelength of FBGs into force measurements with 0.1 N force resolution at 100 Hz sampling rate. In addition, this thesis reports on our innovations in prototyping drive units for CTRs. Three designs of CTR prototypes are proposed, the latest one being significantly more compact and cost efficient in comparison with most designs in the literature. All of these contributions have brought this technology a few steps closer to being used in operating rooms. Some of the techniques and technologies mentioned above are not merely limited to CTRs, but are also suitable for problems arising in other types of surgical robots, for example, for sensorizing da Vinci surgical instruments for force sensing (see Appendix A)

Modeling, Analysis, Force Sensing and Control of Continuum Robots for Minimally Invasive Surgery

This dissertation describes design, modeling and application of continuum robotics for surgical applications, specifically parallel continuum robots (PCRs) and concentric tube manipulators (CTMs). The introduction of robotics into surgical applications has allowed for a greater degree of precision, less invasive access to more remote surgical sites, and user-intuitive interfaces with enhanced vision systems. The most recent developments have been in the space of continuum robots, whose exible structure create an inherent safety factor when in contact with fragile tissues. The design challenges that exist involve balancing size and strength of the manipulators, controlling the manipulators over long transmission pathways, and incorporating force sensing and feedback from the manipulators to the user. Contributions presented in this work include: (1) prototyping, design, force sensing, and force control investigations of PCRs, and (2) prototyping of a concentric tube manipulator for use in a standard colonoscope. A general kinetostatic model is presented for PCRs along with identification of multiple physical constraints encountered in design and construction. Design considerations and manipulator capabilities are examined in the form of matrix metrics and ellipsoid representations. Finally, force sensing and control are explored and experimental results are provided showing the accuracy of force estimates based on actuation force measurements and control capabilities. An overview of the design requirements, manipulator construction, analysis and experimental results are provided for a CTM used as a tool manipulator in a traditional colonoscope. Currently, tools used in colonoscopic procedures are straight and exit the front of the scope with 1 DOF of operation (jaws of a grasper, tightening of a loop, etc.). This research shows that with a CTM deployed, the dexterity of these tools can be increased dramatically, increasing accuracy of tool operation, ease of use and safety of the overall procedure. The prototype investigated in this work allows for multiple tools to be used during a single procedure. Experimental results show the feasibility and advantages of the newly-designed manipulators

Design and Modeling of Multi-Arm Continuum Robots

Continuum robots are snake-like systems able to deliver optimal therapies to pathologies deep inside the human cavity by following 3D complex paths. They show promise when anatomical pathways need to be traversed thanks to their enhanced flexibility and dexterity and show advantages when deployed in the field of single-port surgery. This PhD thesis concerns the development and modelling of multi-arm and hybrid continuum robots for medical interventions. The flexibility and steerability of the robot’s end-effector are achieved through concentric tube technology and push/pull technology. Medical robotic prototypes have been designed as proof of concepts and testbeds of the proposed theoretical works.System design considers the limitations and constraints that occur in the surgical procedures for which the systems were proposed for. Specifically, two surgical applications are considered. Our first prototype was designed to deliver multiple tools to the eye cavity for deep orbital interventions focusing on a currently invasive intervention named Optic Nerve Sheath Fenestration (ONSF). This thesis presents the end-to-end design, engineering and modelling of the prototype. The developed prototype is the first suggested system to tackle the challenges (limited workspace, need for enhanced flexibility and dexterity, danger for harming tissue with rigid instruments, extensive manipulation of the eye) arising in ONSF. It was designed taking into account the clinical requirements and constraints while theoretical works employing the Cosserat rod theory predict the shape of the continuum end-effector. Experimental runs including ex vivo experimental evaluations, mock-up surgical scenarios and tests with and without loading conditions prove the concept of accessing the eye cavity. Moreover, a continuum robot for thoracic interventions employing push/pull technology was designed and manufactured. The developed system can reach deep seated pathologies in the lungs and access regions in the bronchial tree that are inaccessible with rigid and straight instruments either robotically or manually actuated. A geometrically exact model of the robot that considers both the geometry of the robot and mechanical properties of the backbones is presented. It can predict the shape of the bronchoscope without the constant curvature assumption. The proposed model can also predict the robot shape and micro-scale movements accurately in contrast to the classic geometric model which provides an accurate description of the robot’s differential kinematics for large scale movements

On the Statics, Dynamics, and Stability of Continuum Robots: Model Formulations and Efficient Computational Schemes

This dissertation presents advances in continuum-robotic mathematical-modeling techniques. Specifically, problems of statics, dynamics, and stability are studied for robots with slender elastic links. The general procedure within each topic is to develop a continuous theory describing robot behavior, develop a discretization strategy to enable simulation and control, and to validate simulation predictions against experimental results.Chapter 1 introduces the basic concept of continuum robotics and reviews progress in the field. It also introduces the mathematical modeling used to describe continuum robots and explains some notation used throughout the dissertation.The derivation of Cosserat rod statics, the coupling of rods to form a parallel continuum robot (PCR), and solution of the kinematics problem are reviewed in Chapter 2. With this foundation, soft real-time teleoperation of a PCR is demonstrated and a miniature prototype robot with a grasper is controlled.Chapter 3 reviews the derivation of Cosserat rod dynamics and presents a discretization strategy having several desirable features, such as generality, accuracy, and potential for good computational efficiency. The discretized rod model is validated experimentally using high speed camera footage of a cantilevered rod. The discretization strategy is then applied to simulate continuum robot dynamics for several classes of robot, including PCRs, tendon-driven robots, fluidic actuators, and concentric tube robots.In Chapter 4, the stability of a PCR is analyzed using optimal control theory. Conditions of stability are gradually developed starting from a single planar rod and finally arriving at a stability test for parallel continuum robots. The approach is experimentally validated using a camera tracking system.Chapter 5 provides closing discussion and proposes potential future work