Mobile Robots Perception & Navigation Part 16 pptx

VFH+: Reliable Obstacle Avoidance for Fast Mobile Robots, IEEE International Conference on Robotics and Automation ICRA’98, Belgium, pp.. World Modeling and Position Estimation for a Mob

Robot Mapping and Navigation by Fusing Sensory Information 591

active stereo-vision system or other type of sensors Due to the complementary error characteristics with respect to range and angular resolution, fusion of stereo-vision data with ultrasonic range information improves mapping precision significantly In addition, the vision can support the classification of dynamic and modeling of obstacles in 3D,

3 Consider building 3D based grid map,

4 It is essential to develop a powerful incremental integration between the geometric and topological mapping approaches supported by belief values This should have simultaneous support for localization, path planning and navigation,

5 Multi-robot sharing map building Merging accurately topological maps, or metric maps or hybrid maps created by different mobile robots In addition, the key challenge here is, how representation (i.e., any form of world model and mapping) can be effectively distributed over the behavior structure?

6 Due to the limitations associated with grid and topological based mapping, it is necessary to find new techniques to efficiently integrate both paradigms

6 References

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2 Cart and Pendulum Problem

Design and implementation of a system is followed by vigorous testing to examine the quality of the design This is true in the case of designing control systems One the classical systems to test quality and robustness of control scheme is inverted pendulum In recent years, the mechanism of an inverted pendulum on a moving cart has been used extensively and in many different types The cart and pendulum mechanism has become even more popular since the advent of intelligent control techniques This mechanism is simple, understandable in operation, and stimulating It has a non-linear model that can be transformed into linear by including certain condition and assumption in its operation For the above reasons, inverted pendulum’s performance has become a bench mark for testing novel control schemes In this chapter the focus is on the driving power in balancing the inverted pendulum which is an electrical motor Traditionally, DC motors are used for this type of tasks However, in this chapter the focus is on AC electrical motors for producing the torque required for the horizontal movements of the inverted pendulum A simplified control model for the AC motor is used which includes the motor's equivalent time constant

as the crucial parameter in producing rapid responses to the disturbances In the modeling

of fuzzy controllers for the inverted pendulum, the input to the pendulum block is considered to be a torque This torque is produced by an electrical motor which is not included in the model That is, the torque is output of the motor A disadvantage in this modeling is that the electrical motor dynamics is not built-in in the control system independently On the other hand, not including the electrical motor in the control scheme

of the pendulum mechanism provides the freedom to alter the electrical motor and examine the performance of the pendulum with different types of the drive Here, a simplified model

of an AC electrical motor is incorporated into the system The electrical motor receives its

inputs as current or voltage and produces a torque as output to control the balance of the mechanism

The new approach in modeling a fuzzy control system assists in achieving a number of goals such as: examining use of AC motors in producing rapid response, selecting sensitive parameters for an optimum high performance electrical motor capable to stabilize the inverted pendulum system, designing a Takagi-Sugeno type fuzzy controller, and comparing the effectiveness of inclusion of fuzzy controller along with the conventional control scheme

3 Conventional Controllers & Fuzzy Controllers

The conventional approach in controlling the inverted pendulum system is to use a PID (Proportional, Integral, and Derivative) controller In order to model the system the developer would have to know every technical detail about the system and be able to model

it mathematically Fuzzy Logic control (FLC) challenges this traditional approach by using educated guesses about the system to control it (Layne & Passino 2001) Passino states that differential equations are the language of conventional control (PID), while “rules” about how the system works is the language of fuzzy control (Passino and Yurkovich, 1998) Fuzzy logic has found its way into the everyday life of people, since Lotfi Zedah first introduced fuzzy logic in 1962 In Japan, the use of fuzzy logic in household appliances is common Fuzzy logic can be found in such common household products as video cameras, rice cookers and washing machines (Jenson 2005) From the weight of the clothes, fuzzy logic would be able to determine how much water as well as the time needed to effectively wash the clothes Japan developed one of the largest fuzzy logic projects, when they opened the Sendai Subway in 1987 (Kahaner 1993) In this subway, trains are controlled by fuzzy logic Fuzzy Logic is a subset of traditional Boolean logic Boolean logic states that something is either true or false, on or off, 0 or 1 Fuzzy logic extends this into saying that something is somewhat true, or not completely false In fuzzy logic there is no clear definition as to what is exactly true or false Fuzzy logic uses a degree of membership (DOM) to generalize the inputs and outputs of the system (Lin and Lee 1996) The DOM ranges from [0 1], where the degree of membership can lie anywhere in between

The majority of Inverted pendulum systems developed using fuzzy logic, are developed using a two dimensional approach, where only the angle and angular velocity of the pendulum’s arm are measured The following research will show why this method is insufficient for the development of an inverted pendulum on a limited size track To have an efficient fuzzy controller for an inverted pendulum, the system must also include inputs for the position of the cart that the pendulum is balanced upon and the velocity of the cart Two-dimensional fuzzy controllers are very simple examples of fuzzy control research Many of them will balance the inverted pendulum, but are not in control of the cart’s position on the track Adeel Nafis proposed a two-dimensional fuzzy controller to balance the Inverted pendulum on a track (Nafis 2005) Tests showed that the controller would balance the pendulum but neglected to control the position of the cart and eventually the cart’s position would exceed the length of the track Another FLC was proposed by Passino; again this cart had the same result as the previous FLC (Passino and Yurkovich, 1998) Control of the system requires that the cart holding the pendulum be moved by some mechanism For simulation purposes, in this experiment a field oriented AC motor was used (Bose 1997)

Intelligent Control of AC Induction Motors 597

4 Effect of Number of Inputs on Designing Fuzzy Logic Controllers

In a simple control mechanism there is one input and one output Fuzzy Logic Controllers can have more than one input Two-input FLC’s are easy to implement and receive great performance responses from simulations Layne (Layne & Passino 2001) modeled a fuzzy controller that had great performance balancing the pendulum but the cart’s positioning was unstable, making it an impractical rule set for real life implementation Two-input FLC’s are the most commonly researched inverted pendulum systems One of the most commonly researched types fuzzy controllers is two-input inverted pendulum systems The 2-input system receives angle θ and angular velocity ω as its inputs The system uses 5 membership functions for each input, and another 5 for the outputs which is the Force The system consists of 25 (that is 5 to power 2; 52) rules Table 1 shows the rule base for the inverted pendulum system According to Table 1 a value of NL represents a negative large angle or angular velocity, and PL represents a positive large angle/angular velocity As Table 1 indicates, if there is a situation where the angle is Zero (ZE) and the angular velocity

is PS then the rule NS will be fired Where, NL, NS, ZE, PS, PL are linguistic values of negative large, negative small, zero, Positive small, and positive large

Table 1 Rule-base Matrix for the Inverted Pendulum

A simulation that runs for 2 seconds is shown in Figure 1 The pendulum has an initial angle

of 0.2 radians (dashed line) When the simulation is run, the angle of pendulum balances quickly, in about 1 second, but the position of the cart is not controlled (continuous line) so the cart’s position will eventually drift off into the end of the track, even though the pendulum’s arm is balanced

Figure 1 Variation of angle θ (rad) and position X (m) of pendulum vs time t (s)

The benefit of adding two more inputs to the system to control the X-position of the cart and the velocity of the cart will greatly benefit the stability of the system There is a cost for better stability; this is a greater computation time, and greater complexity in the model The cost of adding more inputs increases exponentially with the number of inputs added The above two-input system used five membership function for each input used; this resulted in

a 25 (i.e 52) rule base By adding two more inputs to the system, the systems rule base would grow to 625 (i.e 54) rules Development time for a rule base this size can be very time consuming, both in development and in computational time Bush proposed using an equation to calculate the rules, rather than taking the time to develop the rules individually (Bush 2001) The system was a 54 system with 17 output membership functions (OMF) The equation used was:

Output Membership Function = I + (J – 1) + (-K + 5)+ (L+5) (1) This equation results in values ranging between 1 and 17 This corresponds to the OMF that

is to be used in the calculation of the output The performance of the system using this approach is not consistent with that of the original simulation, given by the author of the above Equation 1 (Bush 2001) The force given to the cart holding the pendulum was found not to be enough to balance the pendulum and the system failed within a small amount of time It can be concluded that this system would be a good starting point for one to base a large rule set on, but the system would need some tweaking of the rules and membership functions to get to balance the system effectively The final FLC controller that was modeled for simulation was a Takagi-Sugeno type fuzzy controller All the previous FLC’s modeled were of Mamdani type A Takagi-Sugeno type fuzzy controller (Mathswork, 2002), (Liang & Langari, 1995), (Johansen et al 2000), (Tanaka et al 2003) varies from the traditional Mamdani type controller by using linear or constant OMF’s instead of triangular, trapezoidal, Gaussian or any other method the developer decided to use The system uses 4-inputs with only 2 input membership functions for each This resulted in a 24, 16 rule system The linear output membership functions are calculate using the equation

) x

* c ( ) x

* c ( ) x

* c ( ) x

* c ( c Function Membership

Intelligent Control of AC Induction Motors 599

The control of all 4 parameters with only 2 membership functions causes the system to run very quickly The down side to this quick response is that it takes more time for the system

to stabilize when there are so few membership functions The system will overshoot the targeted position and eventually come to rest The settling time of this system takes more time than any other system

Figure 2 is the result of the simulation The pendulum is started with an initial disturbance

of 0.2 radians As shown, the fuzzy controller overcompensates for this initial disturbance and sends the pendulum’s angle (dashed line) in an opposite direction in an attempt to balance it, this is the overshoot It takes approximately 5 seconds for the pendulums arm to balance

5 Mathematical Modeling of Field Oriented AC Induction Motors

The motor chosen for the simulation is an AC motor The motor is modeled, Figure 3, using field oriented control scheme (Bose 1997)

ds e qs qs

s qs

dt

d i R

qs e ds ds

s ds

dt

d i R

dr r e qr qr

s

dt

d i

qr r e dr dr

r

dt

d i

) (

5

qr m qs s qs

i L i L

i L i L

qs m qr r qr

i L i L

i L i L

ϕ

ϕ

(9)

r dr qr

qr

dt

d

ϕ ϕ ϕ

qs r r

r m r

e

L

R L

ω

) ( 5

0

=

− + r r m ds

dt

d

ϕ τ

Where:τr = Lr Rr is the rotor time constant

Figure 3 Magnetic Flux Control Scheme in Induction Motors

The field-oriented scheme makes control of AC machine analogous to that of DC machine This is achieved by considering the d-q model of the AC machine in the reference frame rotating at synchronous speed ωe In this model ids and iqs are current components of the stator current on d-q axis, where ids component is aligned with the rotor field The rotor flux and torque can be controlled independently by ids and iqs, shown in Figure 4 The electric torque Te is proportional to the quadrature-axis current iqs, component of the stator current Is, and the rotor flux ψr can be controlled by the direct-axis current ids, of Is, where:

Intelligent Control of AC Induction Motors 601

The transfer function of this AC motor yields angular velocity (ǚ) as the motor shaft output

In the simulation, ǚ was easily converted into the force on the cart The motor responded well, reaching its maximum force exerted on the cart in less than 2.5 seconds

6 Discussion and Results

The simulation consists of four main components, the fuzzy controller, AC motor, the cart and the inverted pendulum, Figure 5 The cart passes the fuzzy controller four parameters θ,

ω, X, V Based on these four parameters the fuzzy controller outputs a voltage to the motor The motor in turn calculates the force that will be exerted on the cart The system then calculates the new values for parameters θ, ω, X, V and the cycle will be repeated

Figure 5 Schematic diagram of fuzzy controller for the inverted pendulum

The fuzzy controller used in the simulation, with the AC motor included, is a 24 FLC as described above The system runs identical to the 24 system only the settling time for the simulation, with the motor included, is larger Figure 6 shows the results of the simulation using the same fuzzy controller as (Sugeno 2002) with the AC motor included in the simulation

The AC motor has a delay, where it takes the motor a given time to reach a maximum force This in turn causes the simulation take longer to reach steady state Parameters used in the simulation of the motor are listed in Table 2

Table 2 Parameters of the Model Motor

Torque

Multi-input - +

Figure 6 shows that it takes approximately 12 seconds for the pendulum’s angle to become steady, and even longer for the cart’s position to stabilize The difference in the response time of this system can be found in the motor The motor has a time constant which delays the motor’s response time to an inputted voltage A typical AC motor has a time constant larger than that of a DC motor The shorter the time constant of the motor, the quicker the system will respond Therefore, it can be expected that it takes longer for AC motor to balance the pendulum

Figure 6 Variation of angle θ (rad) and position X(m) of the pendulum with time t(s) The simulation shows that the system responds well even with a motor attached to the system The cost of implementing a motor into the simulation is response time for the pendulum to stabilize Simulations done without the addition of the AC motor can not be considered for real life implementation because the motor is needed to investigate the response time that the system will observe in real life

In a series of tests carried on without the use of fuzzy controller, it was revealed that the pendulum can hardly overcome any disturbances If the disturbance is very small, it takes twice longer for the pendulum to balance again in an upright position

Performances of vector control AC induction motors are comparable to that of DC motors; however, AC motors are rugged and low cost Therefore, whenever possible, usage of AC motors will greatly reduce the capital cost of equipment and devices

7 Conclusion

In this chapter design of a Fuzzy Logic Controller for a multi-input output system is described It demonstrates a trade-off between precision which requires complex design and simplification which achieves less precise system There is no absolute solution in developing fuzzy logic controllers Designer of a FLC system must consider whether precision will be sacrificed for performance and simplicity The 52 system developed in this work was very simple and computed quickly The drawback of this initial design was that

Intelligent Control of AC Induction Motors 603

precision was compromised The 24 system was also very simple and ran quickly but the performance of the system was not satisfactory The settling time for the cart pendulum was required to be quicker The 54 system was very complex and performance was slow, but if tuned correctly, a system of this size would be very precise

Implementation of the system requires a high performance AC motor Simulation results showed that the system would work for this type of motor Having a smaller time constant

in the AC motor would result in a shorter response time of the system The FLC would need

to be fine tuned for other types of motors

With the AC motor implemented in the simulation model, the system did not react as well

to high disturbances as it did when the motor was neglected in the simulation, or a DC motor was used This indicates that the system will react well to small disturbances and be able to recover from them quickly As the results indicates, in order for this system to handle large disturbances a motor with high performance dynamics need to be used that has a very small time constant Use FLC made significant improvement to the controllability of the inverted pendulum by improving the response time

8 References

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Optimal Path Planning of Multiple

Mobile Robots for Sample

J.C Cardema and P.K.C Wang

University of California Los Angeles, California

U.S.A

1 Introduction

In the exploration of a planetary surface such as that of Mars using mobile robots, rock and soil-sample collection and analysis are essential in determining the terrain composition and in searching for traces of ancient life (Malin & Edgett, 2000) Several missions to Mars have already been sent In the 1997 Mars Pathfinder mission (Mars Pathfinder Homepage), the Sojourner rover used an alpha-proton-X-ray spectrometer (APXS) to analyze rock and soil sample compositions It also had a simple onboard control system for hazard avoidance, although the rover was operated remotely from Earth most of the time The method for rock and soil-sample collection is as follows After landing, the rover used its black-and-white and color imaging systems to survey the surrounding terrain The images were sent back to Earth, and analyzed by a team of geologists to determine where interesting samples might be found Based on that information, the next destination for the rover was selected and the commands to get there were sent to the rover via radio with transmission delays ranging from 10 to 15 minutes (depending on the relative orbital positions of Earth and Mars) The set of commands were sent out over a day with the rover moving only a small distance each time This was done to allow the mission control to constantly verify the position, with time to react to unforeseen problems When the rover finally reached its destination and analyzed the sample, it spent another day transmitting the information back to Earth The cycle was repeated as soon as the geologists had decided on the next destination for the rover Clearly, an automated system for rock and soil sample collection would expedite the process In the 2004 Mars Exploration Rover (MER) mission, the Spirit and Opportunity rovers (Mars Spirit & Opportunity Rovers Homepage) featured an upgraded navigation system Imagery from a stereo camera pair was used to create a 3-D model of the surrounding terrain, from which a traversability map could be generated This feature gave the mission controllers the option of either directly commanding the rovers or

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This chapter is an enhanced version of the paper by J.C Cardema, P.K.C Wang and G Rodriguez,

“Optimal Path Planning of Mobile Robots for Sample Collection”, J Robotic Systems, Vol.21, No.10, 2004,

pp.559-580.

allowing them to autonomously navigate over short distances Consequently, the rovers were often able to traverse over 100 meters a day (Biesiadecki & Maimone, 2006) The rovers were also programmed to autonomously select interesting soil samples, but this feature was seldom used Nevertheless, this was a significant first step toward fully automating the soil-sample collection process

In this study, an attempt is made to formulate the path planning problem for single and multiple mobile robots (referred to hereafter as “rovers” for brevity) for sample collection as

a mathematical optimization problem The objective is to maximize the value of the mission, which is expressed in the form of a mission return function This function contains the performance metric for evaluating the effectiveness of different mission setups To the best

of our knowledge, the problem of sample collection has not yet been studied in conjunction with optimal path planning There are many considerations in the mathematical formulation of this problem These include planetary terrain surface modeling, rover properties, number of rovers, initial starting positions, and the selection of a meaningful performance metric for rovers so that the performance of single versus multiple rovers in representative scenarios can be compared The basic problem is to find a sample-collection path based on this performance metric The main objective is to develop useful algorithms for path planning of single or multiple planetary rovers for sample collection Another objective is to determine quantitatively whether multiple rovers cooperating in sample collection can produce better performance than rovers operating independently In particular, the dependence of the overall performance on the number of rovers is studied

To clarify the basic ideas, we make use of the Mars rover rock and soil-sample collection scenario in the problem formulation and in the numerical study

2.1 Planetary Surface Modeling

Assuming that a region on the planetary surface has been selected for detailed scientific study, the main task is to develop a suitable terrain surface model for rover path-planning Initially, a crude surface model for the selected spatial region may be constructed from the aerial planetary survey data obtained by fly-by spacecraft or observation satellites such as the Mars Orbiter Once the rovers are on the planetary surface, more refined models (usually localized model) may be constructed from the image-data generated by on-board cameras Although the refined models may be useful for scientific studies, they may not be useful for practical optimal path planning Therefore we resort to approximate models that simplify the mathematical formulation and numerical solution of the optimal path-planning problems In our model, we assume that the area of the spatial domain for exploration is sufficiently small so that the curvature of the planetary surface can be neglected Moreover, the surface is sufficiently smooth for rover maneuvers

Optimal Path Planning of Multiple Mobile Robots for Sample Collection on a Planetary Surface 607

2.1.1 Approximate Surface Model

Let Ω be a bounded spatial domain of the two-dimensional real Euclidean space 2

R and the representation of a point in 2

R with respect a given orthonormal basis be denoted byx.

Let f =f x( ) be a real-valued continuous function defined on Ω Let

denote the graph of f , which represents the planetary surface

under consideration In this work, we use a polygonal approximation for the planetary surface G via triangulation that partitions f G into adjacent, non-overlapping triangular fpatches, where each edge of a triangular patch is shared by exactly two triangular patches except on the boundaries of G It has been proved that every Cf 1-surface defined on Ωwith a sufficiently smooth boundary has a triangulation, although an infinite number of triangular patches may be required (Weisstein) Here we make use of the Delaunay triangulation, which produces a set of lines connecting each point in a given finite point set

to its neighbors Furthermore, it has the property that the triangles created by these lines have empty circumcircles (i.e the circumcircles corresponding to each triangle contains no other data points) The Delaunay triangulation of G is a polygonal approximation of the foriginal planetary surface It can also be thought of as the projection of the planetary surface onto a mesh space The domain of the triangulation is a mesh space denoted by Ωˆ ⊂ Ω ,where Ωˆ is the discrete version of Ω The resulting polygonal approximation of the planetary surface G will be denoted by ˆf G This approximate surface model will be used f

in formulating the optimal path-planning problem Although one might use other forms of approximation for G that lead to smoother approximate surfaces, our choice provides fsignificant simplification of the optimal path-planning problem, since the paths are restricted to lie on the edges of the triangular patches

2.1.2 Rock and Soil-sample Properties

Rock and soil samples have different values to geologists based on the questions they are trying to answer For example, in Mars exploration, sedimentary rocks are important since two of the primary questions about early Martian geological history are whether liquid water could exist on its surface and, if so, whether liquid water ever took the form of lakes

or seas (Malin & Edgett, 2000) According to Malin and Edgett, outcrop materials are interpreted as Martian sedimentary rock, and they are of particular interest to geologists for answering these questions The outcrop materials occur in three types: layered, massive, and thin mesas, which differ in visual tone, thickness, texture, and configuration The locations of these outcrops are limited to specific regions mostly between ± 30 degrees latitude One of the regions with large outcrop occurrence is in Valles Marineris The terrain

in a portion of this region is used for our case study The three types of outcrops are speculated to come from different Martian ages The rover should have the capability of identifying and distinguishing these different types To model a portion of Valles Marineris,

we assume that the rock and soil samples are randomly distributed over various regions An appropriate model should allow for the specification of any rock and soil-sample distribution on the given Mars terrain A simple way to do this is to divide the terrain into sub-regions and assign weights to determine how many samples to uniformly

sub-distribute within each sub-region Moreover, we assume there are a finite number of samples, each with an assigned value in a prescribed range A high sample value implies high scientific value In practical situations, this task may be accomplished by a careful study of the aerial survey data

2.2 Single Rover Case

2.2.1.2 Mission time limit

The mission length is an important consideration in path planning For the 1997 Pathfinder mission, the planned mission duration was 30 days The algorithm for path planning should verify that the time duration for sample collection is within the prescribed mission time limit (denoted by τm ax), which in turn determines the maximum terrain coverage

There should be a clarification about the distinction between the overall mission time and the time it takes to execute a planned path In practical situations, it would be difficult to plan a path for the entire mission duration It would be more reasonable to plan paths of shorter duration that can be executed at specific intervals during the mission However, to simplify our formulation, we do not make this distinction and assume that we can plan a path for the entire mission duration

2.2.1.3 Sample analysis time

As mentioned earlier, the Sojourner rover in the Pathfinder mission was equipped with a spectrometer (APXS) for rock and soil-sample analysis The sensor head of the APXS was placed on the sample for 10 hours during the analysis To account for this, the sample analysis time τwaitis introduced into our model It represents the amount of time required to analyze the sample To simplify the model, we assume that τmax is the same for every sample, regardless of its type With the inclusion of the sample analysis time, the rover is

Optimal Path Planning of Multiple Mobile Robots for Sample Collection on a Planetary Surface 609

forced to consider more carefully which rock and soil samples to collect while still operating within the time limit

2.2.1.3 Mission return function

To characterize the performance of a rover in rock and soil sample collection, we need to choose an appropriate mission return function to quantify the rover’s performance throughout the mission The mission return function used in this study is simply the sum of the collected sample values

2.2.1.4 Terrain risk

Letθmaxdenote the angle of the maximum traversable slope corresponding to the maximum

allowable tilt angle of the rover before it topples over To determine if a point on the surface

is too risky to traverse, the terrain slopes at that point in all directions are computed If the magnitude of the slope angle in any direction exceeds θmax, that point is deemed un-

traversable Usually, the rover is more susceptible to tipping over sideways than forwards

or backwards, although the dimensions of the rover are not considered in this study

2.2.1.5 Terrain properties

Ideally, the terrain surface in the spatial region chosen for exploration should be sufficiently smooth to facilitate rover maneuverability but also has features to indicate the possible presence of interesting samples For rover traversability, the terrain texture and hardness are also important (Seraji, 2000) Terrains that are rough and rocky are avoided in favor of smoother ones Terrain risk depends on both the terrain texture and hardness

(x i+, (f x i+)) (i.e The slopes of ˆG fromf x and a x to all their neighboring points satisfy the b

maximum traversable slope angle constraint θ )

2.2.2.2 Definition 2

(Admissible path): A path DZ composed of connected segments in G is said to be admissible if ˆf

each segment is admissible

Each admissible path can be represented by an ordered string of points that will be denoted

by S ⊂ Ωˆ The string of points S may include repeated points since partial backtracking along the path is allowed To account for different sample values in the model, the samples are individually indexed Each sample σk has a corresponding value λk , where k is the

index number We assume that the sample values as well as the sample distribution D ss( )x

on the terrain defined below are known a priori.

2.2.2.3 Definition 3

(Sample distribution): The sample distribution D ss=D ss( )x is a set-valued function defined as follows: If at a point x∈Ω there are ˆ , msamples with indices J x={k x,1,k x,2, ,k x m, }, then ( )

ss

D x = J x If there are no samples at x∈ Ω , then ˆ D ss( )x is an empty set

The entire set of all samples indices is denoted by E Along each admissible path ƥ with ss

the corresponding string S , there is a set of collectable samples C that includes all the sssamples contained in S as defined below:

D x

Γ Γ

the maximal attainable set from x 0

Evidently, we can find the admissible paths and their collectible sample sets associated with

a given A( ;t x0,0) Once we determine the samples to collect along the path, we can

introduce the notion of an admissible tour.