REGISTROS ELECTROFISIOLÓGICOS: Técnica del “patch-

In this section, we introduce additional notation related to deformable body simulation and our planning algorithm and give a brief overview of the approach.

4.2.1

Notation and Definitions

In relation to simulation, there are a wide variety of methods available for representing and simulation a deformable agent. Among the simplest methods is a spring-mass sys- tem, where the body is described by a collection of point masses, or particles, connected by springs and organized into a lattice structure. In practice, mass-spring systems are easy to construct and can be simulated at interactive rates on current commodity hard- ware. More accurate physical models treat deformable objects as a continuum. One of the most commonly used continuous models is the finite element methods (FEM). The object is decomposed into elements joined at discrete node points and a function that solves the equilibrium equation is computed for each element. The computational requirements of FEM can be high (as a function of model complexity) and it is difficult to use them for complex models in real-time applications. In fact, a spring-mass system can be seen as a discrete version of FEM.

The deformable robot,r, is represented as a time dependent set ofnparticles,P(t) = {p1(t),p2(t), . . . ,pn(t)}, connected by a set of springs E. Each spring si,j ∈ E attaches

pi topj and additionally has a rest lengthl, a spring constantks a damping constantkd, a stressstress, and a threshold valueδs. Briefly, rest length and the two constants define

Figure 4.2: Path Planning of Catheters in Liver Chemoembolization: The deformable catheter (robot), represented by 10K triangles, is 1.35mm in diam- eter and approximately 1,000mm in length. The obstacles including the arteries and liver consist of more than 83K triangles. The diameter of the arteries varies in the range 2.5-6mm. Our goal is to compute a collision free path from the start to end configuration for the deformable catheter. The free space of the robot is constrained. The path computed by our motion planner is shown in Fig. 4.9.

the behavior of the spring and elasticity of the body, while the stress and threshold are used to define material constraints. Finally, the set of positions of the particles X(t) = {x1(t),x2(t), . . . ,xn(t)} represent the agent’s configuration, q, at time t and the set of states of particles (positions and velocities) T ={(x1,v1)T,(x2,v2)T, . . . ,(xn,vn)T} is the agent’s state, s.

Since the shape of the body can change over time, we define other terms over the body which are used to generate physically-plausible motion. We define a deformation energy functionE(r) of the robot. E(r) simulates the potential energy of elastic solids, and is a measure of the amount of deformation from its rest state. The robot can be interactively deformed as contacts occur with the obstacles in the environment. This deformation may change the volumeV(r), as well as the energyE(r). To simulate physically plausible deformation, we need to find a new configuration of r, that preserves the total volume of the deformed robot, while minimizing the the total energy of the system.

Our approach uses a roadmapextracted from the medial axis to answer simple plan- ning queries and provide a initial trajectory for the robot to follow. Again, the following is done by selecting or specifying a control point or a series of control points on or in the robot. The roadmap, G, consists of a set of milestones, M, and a set of links, L. Our initial trajectory using this roadmap is the sequence of connected milestones and their links.

Problem Formulation: Given these definitions, we restate our problem as: Find a sequential set of robot configurationsq(t0), ..,q(t1) such that noq(ti) intersects any ob- stacle in Oand q(ti) satisfies the non-penetration and volume preservation constraints, whereq(t0) is the initial configuration of the robot, and q(t1) is the final configuration.

4.2.2

Extension to Deformable Agents

We need to determine a set of constraints which will be sufficient to solve planning prob- lems for a deformable body. We introduce constraints designed to guide the deformable

robot through the environment to the desired goal configuration. Using global analysis from any roadmap algorithm (e.g. PRM) to compute a possible path for a point robot, we define both geometric and physical constraints that move the deformable robot to avoid both static and moving obstacles, and also follow an estimated path to the goal. Finally, we identify some computational bottlenecks of the process and propose methods to optimize the performance of these portions. The main components of our physics- based motion planning approach include:

1. Guiding Path: Our framework allows the use of any estimated path computed by either quick global analysis of the environment, such random sampling [KSLO96,

FGLM01,LK00] or user guided input [BSA00]. Since we can use a path for a point- sized robot, we can extract the medial axis of the workspace as our roadmap. 2. Constraints: To achieve the desired results without robustness issues, we de-

termine a set of hard and soft constraints to guide our sampling process. Along with the standard hard constraints (obstacle non-penetration and staying within the environment boundaries), we additionally define soft-body and volume preser- vation constraints. Global volume preservation is relaxed for performance, and instead we define an internal pressure on the body and use this to help locally preserve the body’s volume. Necessary soft constraints include a path following constraint and collision response to guide the robot along its initial trajectory and to keep the robot from intersecting bodies. Thus, our approach will compute the new path by taking into consideration the body’s deformable properties and the interaction of the flexible robot with the obstacles in the workspace.

3. Discretization of Continuum: As for modeling the deformation of the robot or the environment, there exists many possible approaches. For example, these may include physically-based free-form deformation (FFD), mass-spring systems, boundary element methods (BEM), and finite element methods (FEM). The choice

of discretization impacts performance and also how the soft-body constraints are enforced. For coding simplicity and runtime performance, we chose an implementa- tion based on the spring-mass system to validate our basic approach. However, we can easily incorporate any discretization technique within this algorithmic frame- work.

4. Proximity Queries: We can take advantage of methods proposed in [HCK+99,

GRLM03] that uses graphics hardware to quickly perform proximity queries on the workspace or to provide dynamically updated discretized distance fields for obstacle avoidance involving deformable models.