Comparison of Transient Performance in the Control of Soft Tissue Grasping Xiaolong Yu, Howard Jay Chizeck and Blake Hannaford Abstract— In robot-assisted surgery, surgical tools interact with tissues that have nonlinear mechanical properties. For situations where a pre-specified trajectory of tool positions (or applied forces) is desired, there are many controller de- signs that might be used. Four candidates are comparatively evaluated here, via computer simulation involving a nonlinear model of soft tissue behavior during grasping actions. The parameters for this model were obtained experimentally (in earlier work). The four candidate controllers are: (1) a well- tuned PID controller; (2) feedback linearization in combination with deadbeat control; (3) an optimal open-loop control law obtained via minimization of a quadratic cost function; and (4) a model predictive controller. Simulation trials are used to compare the transient performance of these candidate controllers under different assumptions regarding input and output noises. The conditions where each of the candidates is best are characterized. Index Terms— Robot-Assisted Surgery, Transient Control, Trajectory Following, Soft Tissue Grasping, PID control, Feed- back Linearization, Deadbeat Control, Model Predictive Con- trol. I. I NTRODUCTION Commercially available robot-assisted surgery systems are essentially open loop devices. The surgeon provides the control function [1]. It might be useful to have the system incorporate feedback control to improve the precision and accuracy of end effector positions and applied forces, in the face of disturbances (including respiratory and other patient motions). A more ambitious goal would be the automatic execution of tasks such as grasping, cutting and suturing. There are several factors that complicate the application of automatic control to robotic surgery. These include: a) The nonlinear (and time-varying) properties of soft tissues; b) Sensor, actuator and other system noises and distur- bances; and c) The combination of primitive operations to accomplish more complex surgical actions is a hybrid system (discrete events plus continuous dynamics) that in- corporates controlled, but partially random transitions between different modes. In this study, we focus on one specific primitive operation – the grasping of soft tissue, where a trajectory of desired tool positions is specified. Four different control architectures are evaluated via computer simulation using a nonlinear model of the tissue. The parameters for this model are based upon values obtained in earlier work [2]. The four candidate controllers are: (1) a well-tuned PID controller; (2) feedback This work is supported by the US Army Telemedicine and Advanced Technologies Research Center (TATRC). The authors are with the De- partment of Electrical Engineering, University of Washington, Seattle, WA 98195-2500. linearization in combination with deadbeat control; (3) an optimal open-loop control law obtained via minimization of a quadratic cost function; and (4) a model predictive controller. We examine which of four candidate controllers results in the the best transient performance when a somewhat realistic model of the load and a reference trajectory that is appropriate for actual surgery is used, and where constraints on control effort (reflecting motor limitations) are imposed. In section II the soft tissue model and the desired grasping trajectory are described. In section III a metric is proposed to evaluate the transient trajectory tracking performance. In section IV the four candidate control methods are described. In section V simulation results are presented and compared. II. BACKGROUND A. Mathematical Model of Soft Tissue During Grasping A wide variety of living soft tissues have an exponential- like biomechanical response to applied force [3]. Fig. 1 depicts a model for soft tissue under the operation of grasping (i. e. , squeezing) by a mechanical device. There is a static exponential relation between the force and position. This nonlinear mass-spring-damper model is described by the following differential equation [4]: u = m d 2 p dt 2 + d dp dt + α (e βp - 1) (1) where u is the force applied to tissue; p is the position of robot end effector, incontact with the tissue; m is the lumped mass of the robot end effector and tissue; d is the viscosity of tissue; α,β are parameters related to the stiffness of the tissue. In the above model, it is assumed the that robot is a mass attached to the tissue. All of the parameters are assumed to be known. There are many possible refinements to this type of soft tissue model, reviewed in [3]; but this is a single model which captures several essential properties. Eq. (1) can be formulated as the following single input, single output system: ˙ x = f (x)+ g(x)u y = h(x) (2)