Linear Parameter Varying Control of a Robot Manipulator for Aortic Valve Implantation

In this paper, we propose a linear parameter varying (LPV) control design approach for trajectory tracking in a robotic system, intended to be involved in an image-guided teleoperated cardiac surgery. The robot is eventually aimed to guide a 3 degree-of-freedom medical tool (a catheter) inside the left ventricle (LV) and achieve the implantation of a prosthetic aortic valve. The successful delivery of the valve from the apical entrance to the aortic annulus strongly depends on the precise navigation of the catheter such that its probable collision with the LV’s changing environment is avoided. The LPV control strategy is utilized here due to its ability to capture the nonlinearities of the designed robot manipulator and adapt in real-time based on the varying end effector’s angle. The simulation studies demonstrate promising results achieved for a guaranteed safe navigation through LV. INTRODUCTION In the last decade, employment of biomedical robots in surgical applications has attracted much attention of many researchers in this field. A broad range of medical procedures from highly precise brain surgeries to minimally invasive cardiac interventions and MRI guided teleoperated operations fall into aforementioned applications [1]. In minimally invasive robotassisted medical interventions, where robot’s miniature end effectors make it possible for surgeons to reach the target through minute incision points, significant achievements including shortened recovery time justify deployment of robots [2]. On the other hand, in teleoperated surgeries when due to the existence ∗Address all correspondence to this author. of biomedical apparatuses like imaging systems, the operator can not touch the body, or when surgeons have to perform on a patient from a distance, surgical robots play an important role in medical procedures [3]. Compensation of motion artifact caused by the respiration or heart beating is another reason behind letting teleoperated surgical systems be involved in beating heart operations [4]. Being equipped with such system, operators are enabled to work in a virtual environment adroitly to navigate the catheter to reach the target in a safe and accurate way. Master and slave robots form the very basic elements of a typical teleoperated surgery [5, 6]. The slave robot is directly in contact with patient and manipulates the catheter. Meanwhile, the surgeon actuates the master robot using visual feedbacks from the imaging system and provides commands to the slave unit. The master robot can further contribute in the process by providing the surgeon with tactile sensing while feeding the forces exerted on the end effector, back to the operator’s hand [7]. This latter capability changes the master robot to a haptic device which prevents the operator from applying excessive forces that might be brutal for the tissues. There is a rich literature on various designs for haptic devices suitable for different applications. For a robot to satisfy the performance requirements, the design of an advanced controller is as important as the robot mechanical design or the selection of actuators and sensors. The controller should be able to provide robot with proper control effort such that the expected stability criterion and desirable performance requirements are met. Based on the dynamics of the robot and to accomplish the task it is designed for, many different control objectives along with a variety of control methods could be considered. In this paper, we propose a control strategy for trajectory tracking of a slave robot intended to navigate a medical catheter inside the heart’s left ventricle. The designed robot’s highly nonlinear dynamics along with the challenging mission of guiding catheter in heart’s dynamic environment to ensure that it never collides with the boundary, necessitates the implementation of a sophisticated control scheme. The slave robot should be capable of maintaining translational, rotational and bending displacements close to the surgeon’s needs. Due to the environment uncertainties and external disturbances, a robust control strategy is essential for robot to quickly and effectively follow the reference trajectory provided by the master robot. In this paper, we study the use of a linear parameter varying (LPV) control approach [8] that fits within our application of interest due to the nonlinear dynamics of the designed robot manipulator acting as the slave unit. Simulation results shows acceptable specifications regarding both stability criteria and performance requirements. MECHANISM In this paper, we are interested in the design and control of a robot manipulator that acts as the slave unit in a teleoperated medical intervention and in particular guiding a medical catheter inside the left ventricle. The design configuration is shown in Fig. 1. This manipulator is a coherent component of a robotic assisted system for implanting a prosthetic aortic valve in beatingheart cardiac surgery. The system consists of several sections. A brief description of the system’s various components is presented next. The interested reader is referred to [9] for more details. The surgical robot is divided into two proximal and distal manipulators. While former part is compensating for the motion of body due to respiration and heart beating, the slave robot (the distal part) placed at top of the compensatory manipulator navigates the catheter inside the left ventricle. Reference trajectories are provided by the master unit that also feeds the haptic sensing back to the surgeon’s hands. To make the loop closed, the imaging system provides operator with the visual feedback. The slave robotic unit is aimed to insert the catheter inside the left ventricle through the apical entrance and guide it along a non-straight trajectory all the way towards aortic valve. This non-straight path is depicted in Fig. 2. The considered three degrees of freedom including translation, rotation and bending in slave robot ensure the safe navigation of catheter inside the left ventricle. Fig. 3 illustrates the main parts of the slave robot in detail. The catheter in Fig. 2(b) is placed in the tool holder shown in Fig. 3. The translational DOF is actuated by two DC motors 1 and 2 which slide the medical tool back and forth. DC motor 3 is actuated to rotate the catheter inside an identified safe region that the catheter is allowed to be. Using DC motor 4, valve component in Fig. 2(b) is bent and oriented toward aorta. A cable from the point XT,t in Fig. 2(b) passes through the tool holder and wraps around the shaft of the DC motor 4. The bending part of the system, i.e., valve component in Fig. 2, bends when DC Figure 1. THE DESIGN OF SLAVE ROBOT Figure 2. THE BIOMEDICAL ROBOT PATH WITHIN THE LV motor is active. In the present paper, we will not consider the actuation of the the bending part. DYNAMIC MODEL OF THE ROBOT Due to the structure of the designed robot, the dynamic equation of each part is decoupled from others. Obtaining the dynamic equations corresponding to the rotational and bending degrees of freedom is straightforward. In the remainder of this section, we will only discuss the development of dynamic equation for the translational part and corresponding control algorithm. Using the Euler-Lagrange equation [10], one can derive the dynamic model of the translational motion of the mechanism. Since the translational part consists of two symmetric components, for the sake of simplicity, we only show the derivation of the dynamic equation corresponding to one side in Fig. 4. Half of the handle weight is modeled as the concentrating massm2. In the following equations, m1 and L are mass and length of arms, Figure 3. TOP VIEW OF THE DESIGNED BIOMEDICAL ROBOT respectively, and Ii is the moment of inertia with respect to the mass center. Figure 4. THE ROBOT CONFIGURATION The total kinetic energy of the structure is: K = 3 ∑ i=1 ( 1 2 miv 2 ci + 1 2 Iiω 2 i ) = 1 4 m1L θ̇ +(m1 +2m2)L 2 sin θ+ Iθ̇ (1) The total potential energy on the working plane is assumed to be zero. Hence, the Euler-Lagrange equation reduces to: