# Planning with Discrete Harmonic Potential Fields

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- Journal of Dynamic Systems Measurement and Control 01/1981; 103(2). · 0.76 Impact Factor
- 01/1950; Houghton Mifflin Company.
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**ABSTRACT:**In this paper we introduce a motion planning method which uses an artificial potential field obtained by solving Laplace's differential equation. A potential field based on Laplace's equation has no minimal point; therefore, path planning is performed without falling into local minima. Furthermore, we propose an application of the motion planning method for recursive motion planning in an uncertain environment. We illustrate the robot motion generated by the proposed method with simulation examples.Advanced Robotics 01/1992; 7:449-461. · 0.51 Impact Factor

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17

Planning with Discrete Harmonic Potential

Fields

Ahmad A. Masoud

KFUPM

Saudi Arabia

1. The Harmonic Potential Field Planning Approach: A Background

Utility and meaning in the behavior of an agent are highly contingent on the agent’s ability

to semantically embed its actions in the context of its environment. Such an ability is cloned

into an agent using a class of intelligent motion controllers that are called motion planners.

Designing a motion planner is not an easy task. A planner is the hub that integrates the

internal environment of an agent (e.g. its dynamics and internal representation etc.), its

external environment (i.e. the work space in which it is operating, its attributes and the

entities populating it), and the operator’s imposed task and constraints on behavior as one

goal-oriented unit. Success in achieving this goal in a realistic setting seems closely tied to

the four conditions stated by Brooks which are: intelligence, emergence, situatedness, and

embodiment (Brooks, 1991). While designing planners for agents whose inside occupies a

simply-connected region of space can be challenging, the level of difficulty considerably

rises when the planner is to be designed for agents whose inside no longer occupies a

simply-connected space (i.e. the agent is a distributed entity in space). This situation gives

rise to sensitive issues in communication, decision making and environment representation

and whether a centralize top-down mode for behavior generation should be adopted or a

decentralized, bottom-up approach may better suit the situation at hand.

To the best of this authors’ knowledge, the potential field (PF) approach was the first to be

used to generate a paradigm for motion guidance (Hull,1932); (Hull, 1938). The

paradigm began from the simple idea of an attractor field situated on the target and a

repeller field fencing the obstacles. Several decades later, the paradigm surfaced again

through the little-known work of Loef and Soni which was carried out in the early 1970s

(Loef , 1973); (Loef & Soni, 1975). Not until the mid-1980s did this approach achieve

recognition in the path planning literature through the works of Khatib (Khatib , 1985),

Krogh (Krogh, 1984), Takegaki and Arimoto (Takegaki & Arimoto, 1981), Pavlov and

Voronin (Pavlov & Voronin, 1984) ,Malyshev (Malyshev, 1980), Aksenov et al. (Aksenov et

al., 1978), as well as Petrov and Sirota (Petrov, Sirota ,1981); (Petrov & Sirota, 1983).

Andrews and Hogan also worked on the idea in the context of force control (Andrews &

Hogan, 1983).

Despite its promising start, the attractor-repeller paradigm for configuring a potential field

for use in navigation faced several problems. The most serious one is its inability to

guarantee convergence to a target point (the local minima problem). However, the problem

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was quickly solved. One solution uses a configuration that forces the divergence of the PF

gradient to be zero almost everywhere in the workspace, hence eliminating the local minima

problem. The approach was named the harmonic potential field (HPF) planning approach.

A basic setting of this approach is shown in (1) below:

∇2V(X)=0 X (1)

subject to: V(XS) = 1, V(XT) = 0 , and ∂

∂

V= 0

n

at X = Γ,

where Ω is the workspace, Γ is its boundary, n is a unit vector normal to Γ, Xs is the start

point, and XT is the target point.

Although a paradigm to describe motion using HPFs has been available for more than three

decades, it was not until 1987 that Sato (Sato, 1987) formally used it as a tool for motion

planning (an English version of the work may be found in (Sato, 1993)). The approach was

formally introduced to the robotics and intelligent control literature through the

independent work of Connolly et al. (Connolly et al., 1990), Prassler (Prassler, 1989) and

Tarassenko et al. (Tarassenko & Blake, 1991) who demonstrated the approach using an

electric network analogy, Lei (Lei, 1990) and Plumer (Plumer, 1991) who used a neural

network setting, and Keymeulen et al. (Decuyper & Keymeulen, 1990); (Keymeulen &

Decuyper, 1990) and Akishita et al. (Akishita et al., 1990) who utilized a fluid dynamic

metaphor in their development of the approach. Cheng et al. (Cheng & Tanaka,

1991);(Cheng, 1991);(Kanaya et al., 1994) utilized harmonic potential fields for the

construction of silicon retina, VLSI wire routing, and robot motion planning. A unity

resistive grid was used for computing the potential. In (Dunskaya & Pyatnitskiy 1990) a

potential field was suggested whose differential properties are governed by the

inhomogeneous Poisson equation for constructing a nonlinear controller for a robotic

manipulator taking into consideration obstacles and joint limits.

Harmonic potential fields (HPFs) have proven themselves to be effective tools for inducing

in an agent an intelligent, emergent, embodied, context-sensitive and goal-oriented behavior

(i.e. a planning action). A planning action generated by an HPF-based planner can operate

in an informationally-open and organizationally-closed mode; therefore, enabling an agent

to make decisions on-the-fly using on-line sensory data without relying on the help of an

external agent. HPF-based planners can also operate in an informationally-closed,

organizationally-open mode (Masoud, 2003) ;(Masoud & Masoud, 1998) which makes it

possible to utilize existing data about the environment in generating the planning action as

well as illicit the help of external agents . A hybrid of the two modes may also be

constructed. Such features make it possible to adapt HPFs for planning in a variety of

situations. For example in (Masoud & Masoud, 2000) vector-harmonic potential fields

were used for planning with robots having second order dynamics. In (Masoud, 2002) the

approach was configured to work with a pursuit-evasion planning problem, and in

(Masoud & Masoud, 2002) the HPF approach was modified to incorporate joint constraints

on regional avoidance and direction. The decentralized, multi-agent, planning case was

tackled using the HPF approach in (Masoud, 2007). The HPF approach was also found to

facilitate the integration of planners as subsystems in networked controllers containing

sensory, communication and control modules with a good chance of yielding a successful

behavior in a realistic, physical setting (Gupta et al., 2006).

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Although a variety of provably-correct, HPF-based planning techniques exist for the

continuous case, to the best of this author’s knowledge, there are no attempts to formally

use HPFs to synthesize planners that work with discrete spaces described by weighted

graphs. Planning in discrete spaces is of considerable significance in managing the

complexity in high dimensions (Aarno et al., 2004); (Kazemi et al. , 2005). It is also important

for dealing with inherently discrete systems such as robustly planning the motion of data

packets in a network of routers (Royer & Toh, 1999).

In this work a discrete counterpart to the continuous harmonic potential field approach is

suggested. The extension to the discrete case makes use of the strong relation HPF-based

planning has to connectionist artificial intelligence (AI). Connectionist AI systems are

networks of simple, interconnected processors running in parallel within the confines of the

environment in which the planning action is to be synthesized. It is not hard to see that such

a paradigm naturally lends itself to planning on weighted graphs where the processors may

be seen as the vertices of the graph and the relations among them as its edges. Electrical

networks are an effective realization of connectionist AI. Many computational techniques

utilizing electrical networks do exist (Blasum et al., 1996);(Duffin, 1971);(Bertsekas,

1996);(Wolaver, 1971) and the strong relation graph theory has to this area (Bollabas,

1979);(Seshu, Reed, 1961) is well-known. This relation is directly utilized for constructing a

discrete counterpart to the BVP in (1) used to generate the continuous HPF. The discrete

counterpart is established by replacing the Laplace operator with the flow balance operator

represented by Krichhoff current law (KCL) (Bobrow, 1981). As for the boundary conditions,

they are applied in the same manner as in (1) to the boundary vertices. The discrete

counterpart is supported with definitions and propositions that helps in utilizing it for

developing motion planners. The utility of the discrete HPF (DHPF) approach is

demonstrated in three ways. First, the capability of the DHPF approach to generate new,

abstract, planning techniques is demonstrated by constructing a novel, efficient, optimal,

discrete planning method called the M* algorithm. Also, its ability to augment the

capabilities of existing planners is demonstrated by suggesting a generic solution to the

lower bound problem faced by the A* algorithm. The DHPF approach is shown to be useful

in solving specific planning problems in communication. It is demonstrated that the discrete

HPF paradigm can support routing on-the-fly while the network is still in a transient state. It

is shown by simulation that if a path to the target always exist and the switching delays in

the routers are negligible, a packet will reach its destination despite the changes in the

network which may simultaneously take place while the packet is being routed. An

important property of the DHPF paradigm is its ability to utilize a continuous HPF-based

planner for solving a discrete problem. For example, the HPF-based planner in (Masoud,

2002) may be adapted for pursuit-evasion on a grid. Here a note is provided on how the

continuous, multi-agent, HPF-based planner in (Masoud, 2007) may be adapted for solving

a form of the sliding block puzzle (SBP).

2. HPF and Connectionism:

Learning is the main tool used by most researchers to adapt the behavior of an agent to

structural changes in its environment (Tham & Prager, 1993),(Humphrys, 1995),(Ram et al.,

1994). The overwhelming majority of learning techniques are unified in their reliance on

experience as the driver of action selection. There are, however, environments which an

agent is required to operate in that rule-out experience as the only mechanism for action

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selection. Learning, or the acquisition of knowledge needed to deal with a situation, may be

carried-out via hieratical, symbolic reasoning. Despite the popularity of the symbolic

reasoning AI approach, its fitness to synthesize autonomous and intelligent behavior in such

types of environments is being seriously questioned (Brooks, 1990); (Brooks, 1991).

Representing an environment as a group of discrete heterogeneous entities that are glued

together via a hierarchical set of relations is a long standing tradition in philosophy and

science. There is, however, an opposing, but less popular, camp to the above point of view

stressing that representations should be indivisible, and homogeneous. Distributed

representations have already found supporters among modern mathematicians, system

theorists, and philosophers. Norbert Wiener said "The identity of a body is more like the

identity of a flame than that of a stone; it is the identity of a structure, not of a piece of

matter" (Wiener, 1950);(Wiener, 1961). In (Lefebvre,1977) Lefebvre viewes an entity or a

process as a wave that glides on a substrate of parts where the relation between the two is

that of a system drawn on a system. In (Campbell , 1994) Campbell argues against the

hypothesis that geometrical symbols are used by creatures, to model the environment that

they want to navigate. He postulate the existence of a more subtle and distributed

representation of the environment inside an agent. With this in mind, the following

guidelines are used for constructing a representation:

1. A representation is a pattern that is imprinted on a substrate of some kind.

2. The substrate is chosen as a set of homogeneous, simple, automata that densely covers

the agent's domain of awareness. This domain describes the state of the environment

and is referred to as state space.

3. The representation is self-referential. A self-referential representation may be

constructed using a dense substrate of automata that depicts the manner in which an

agent acts at every point in state space. Self-referential representations are completely at

odd with objective representations. They are a product of the stream of philosophy and

epistemology (theory of knowledge) (Glasserfeld, 1986), (Lewis, 1929), (Nagel & Brandt,

1965), (Masani, 1994) which stresses that ontological (absolute or objective) reality does

not exist, and any knowledge that is acquired by the agent is subjective (self-referential.)

4. In conformity with the view that objective reality is unattainable, a representation is

looked upon as merely a belief. Its value to an agent is in how useful it is, not how well

it represents its outside reality. Therefore, a pattern that evolves as a result of a self-

regulating construction is at all phases of its evolution a legitimate representation.

A machine is a two-port device that consists of an operator port, an environment port, and a

construction that would allow a goal set by the operator, defined relative to the environment

to be reached. CYBERNETICS (Wiener, 1950); (Wiener, 1961), or as Wiener defined it:

"communication and control in the animal and the machine," is based on the conjectures

that a machine can learn, can produce other machines in its own image, and can evolve to a

degree where it exceeds the capabilities of its own creator. It is no longer necessary for the

operator to generate a detailed and precise plan to convert the goal into a successful motor

action. The operator has to only provide a general outline of a plan and the machine will fill

in the "gaps"; hence confining the operator's intervention to the high-level functions of the

undergoing process. Such functions dictate goals and constrain behavior. The machine is

supposed to transform the high-level commands into successful actions. CYBERNETICS

unifies the nature of communication and control. It gives actions the soft nature of

information. To a cybernist a machine that is interacting with its environment is an agent

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that is engaging in information exchange with other agents in its environment. In turn, a

machine consists of interacting subagents, and is an interactive subagent in a larger

machine. A controller which forces an agent to comply with the will of the operator is seen

as an encoder that translates the requests of the operator to a language the agent can

understand. Therefore, an action is a message, and a message is an information-bearing

signal or simply information. Accepting the above paves the way for a qualitative

understanding of the ability of a machine to complement the plan of the operator. Let an

information theoretic approach (Shannon, 1949); (Gallager, 1968) be used to examine two

agents that are interacting or, equivalently, exchanging messages. Assume that the activities

of the first agent has Ix equivalence of information, and that the second has Iy. Although

what is being contributed by the interacting agents is equal to Ix+Iy (self-information), the

actual information content of the process is Ix+Ixy+Iy, where Ixy is called mutual

information. While the measure of self information is always positive definite (Ix=-log(Px)

,Iy=-log(Py)), the measure of mutual information (Ixy= log(Px,y/(Px.Py)) is indefinite (Px

and Py are the probability of x and y respectively, and Px,y is their joint probability). In an

environment where carefully designed modes of interaction are instituted among the

constituting agents, the net outcome from the interaction will far exceed the sum of the

individual contributions. On the other hand, in non-cooperative environments the total

information maybe much less than the self-information (an interaction that paralyzes the

members makes Px,y/ 0, and Ixy 6-4). It has been shown experimentally and by simulation

that sophisticated goal-oriented behavior can emerge from the local interaction of a large

number of participants which exhibit a much more simplistic behavior. This has motivated a

new look at the synthesis of behavior that is fundamentally different from the top-bottom

approach which is a characteristic of classical AI. Artificial Life (AL) (Langton, 1988)

approaches behavior as a bottom-up process that is generated from elementary, distributed,

local actions of individual organisms interacting in an environment. The manner in which

an individual interact with others in its loca1 environment is called the Geno-type. On the

other hand, the overall behavior of the group (Phenotype, or P-type) evolves in space and

time as a result of the interpretation of the Geno-type in the context of the environment. The

process by which the P-type develop under the direction of the G-type is called

Morphogenesis (Thorn, 1975).

To alter its state in some environment an agent (from now on is referred to as the operator)

needs to construct a machine that would interface its goal to its actions. The machine (or

interface) function to convert the goal into a sequence of actions that are imbedded into the

environment. These actions are designed to yield a corresponding sequence of states so that

the final state is the goal state of the operator. The action sequence is called a plan and it is a

member of a field of plans (Action field) that densely covers state space so that regardless of

the starting point, a plan always exist to propel the agent to its goal. To construct a machine

of the above kind the operator must begin by reproducing itself by densely spreading

operator-like micro-agents at every point in state space (Figure-1). The only difference

between the operator-agent and an operator-like micro-agent is that the state of the

Operator evolves in time and space while the state of the micro-agent is stagnant and

immobilized to one a priori known point in state space. The second part of machine

construction is to induce the proper action structure over the micro-agent group. It is

obvious that a hierarchical, holistic, centralized approach for inducing structure over the

group entails the existence of a central planning agent/s that is/are not operator-like.

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Including such an agent in the machine violates organizational closure, i.e. the restrictions on

intelligent machines receiving no influx of external intelligence to help them realize their goals.

In other words, the agent must be able to lift itself from its own bootstraps. By restricting the

forms of the agents constituting the machine to that of the operator, an AL approach does not

require the intervention of any external intelligence to help in the construction of the machine.

The AL approach, which is decentralized by definition (i.e. no supervisor is needed), requires a

micro-agent to locally constrain its behavior (Genotype, or G-type behavior) using the

information derived from the states of the neighboring micro-agents (Figure-1). Unlike

centralized approaches where each micro-agent has to exert the "correct" action in order to

generate a group structure that unifies the micro-agents in one goal-oriented unit, the AL

approach only requires the micro-agents only not to exert the "wrong" action that would

prevent the operator from proceeding to its goal. Obviously, not selecting the wrong action is

not enough, on its own, for each micro-agent to restrict itself to one and only one admissible

action that would constitute a proper building block of the global structure that is required to

turn the group into a functional unit. In an AL approach, the additional effort (besides that of

the G-type behavior) needed to induce the global structure on the micro-agents is a result of

evolution in space and time under the guidance of the environment. This interpretation or

guidance is what eventually limits each micro-agent to one and only one action that is also the

proper component in a functioning group structure.

Figure 1. An interacting collective of micro-agents & Layers of functions in a micro-agent

To construct a machine that operates in an AL mode, the operator must have the means to:

1. Reproduce itself at every point in state space.

2. Clone the geno-type behavior in each member of the micro-aqent group.

3. Factor the environment in the behavior generation process.

The HPF approach lends itself to the above guidelines for the construction of an AL-driven,

intelligent machine. The potential field is used to induce a dense collective of virtual agents

covering the workspace. The interaction among the agents is generated by enforcing the

Laplace equation. The environment is factored into the behavior generation process by

enforcing the boundary conditions. More details can be found in (Masoud, 2003);(Masoud &

Masoud, 1998).

3. The DHPF approach:

This section provides basic definitions and propositions that serve as a good starting point

for the understanding and utilization of the DHPF approach.

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Definition -1: Let G be a non-directed graph containing N vertices. Let the cost of moving

from vertex i to vertex j be Cij (Cij=Cji). Let a potential Vi be defined at each vertex of the

graph (i=1,..,N), and Iij be the flow from vertex i to vertex j defined as:

V -V

i

j

I =

ij

Cij

, (2)

where Vi > Vj .Note that equation-2 is analogous to ohm’s law in electric circuits (Bobrow,

1981). Let T and S be the target and start boundary vertices respectively.

A discrete counterpart for the BVP in (1) is obtained if at each vertex of G (excluding the

boundary vertices) the balance condition represented by KCL is enforced:

∑

Iij

j

and

VS = 1, VT = 0.

=0

i=1,..,N, i ≠T, i≠S

(3)

Definition-2: Let the equivalent cost between any two arbitrarily chosen vertices, i and j, of

G (Ceqij) be defined as the potential difference applied to the i-j port of G (ΔV) divided by

the flow, I, entering vertex i and leaving vertex j (figure-2)

ΔV

I

Ceq =

ij

(4)

Proposition-1: The equivalent cost of a graph, G, that satisfies KCL at all of its nodes (i.e. an

electric network) as seen from the i-j port (vertices) is less than or equal to the sum of all the

costs along any forward path connecting vertex i to vertex j. Note that if the proposition

holds for forward paths, it will also hold for paths with cycles.

Figure 2. Equivalent cost of a graph as seen from the i-j vertices

Proof: Since the graph is required to satisfy KCL, then any internal flow in the edges of the

graph is less than or equal to the external flow I (figure-3).

Figure 3. KCL splits input current into smaller or equal components

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Consider a forward sequence of vertices connecting vertex i to vertex j: i→i+1→...→i+L-1→j.

The potential difference at the k’th edge (ΔVk) in the selected path is: ΔVk = Vi+k-1 - Vi+k. The

potential difference between i and j may be written as:

ΔV = ΔV1 + ΔV2 + .... + ΔVk+ .....+ ΔVL (5)

= C1I1 + C2I2 + ..... + CKIK+ ....+ CLIL ,

where for simplicity Ck is used to denote Ci+k-1,i+k , and Ik is used instead of Ii+k-1,i+k . Now

divide both sides by the flow I:

1k

I

L

I

i,j1kL

V

I

I

I

II

CeqC ...... C....... C

Δ

==++++

(6)

Since Ik/I ≤ 1, we have:

Ceqi,j ≤ C1+....+Ck+....CL . (7)

Proposition-2: If G satisfies the conditions in equation-3, then the potential defined on the

graph (V(G)) will have a unique minimum at T (VT) and a unique maximum at S (VS).

Proof: The proof of the above proposition follows directly from KCL and the definition of a

flow (equation-2). If at vertex i Vi is a maximum, local or global, with respect to the potential

at its neighboring vertices, all the flows along the edges connected to i will be outward (i.e.

positive). In this case KCL will fail. Vice versa if Vi is a minimum.

Proposition-3: Traversing a positive, outgoing flow from any vertex in G will generate a

sequence of vertices (i.e. a path) that terminates at T. Vice versa, traversing a negative,

ingoing flow from any vertex in G will generate a sequence of vertices (i.e. a path) that

terminates at S.

Proof: Assume that a hop is going to be made from vertex i to vertex i+1 based on a selected

outgoing, positive flow Ii,i+1 (Figure-4).

Figure 4. positive flow-guided vertex transition

From the definition of a flow we have:

Vi - Vi+1 = C i,i+1AIi,i+1 . (8)

Since we are dealing with positive costs, and the flow that was selected is positive, we have:

Vi - Vi+1 > 0, or Vi > Vi+1. (9)

By guiding vertex transition using positive flows, a continuous decrease in potential is

established. This means that the unique minimum, VT, will finally be achieved. In other

words, the path will converge to T. The proof of the second part of the preposition can be

easily carried out in a manner similar to the first part.

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Proposition-4: A path linking S to T generated by moving from a vertex to another using a

positive flow cannot have repeated vertices (i.e. it contains no loops).

Proof: If the flow used to direct vertex transition loops back to a previously encountered

vertex, it will keep looping back to that vertex trapping the path into a cycle and preventing

it from reaching the target vertex. Since in proposition-3 convergence to T of a positive flow

driven path is guaranteed, no cycles can occur and the path cannot have repeated vertices.

Definition-3: Since a positive flow path (PFP) beginning at S is guaranteed to terminate at T

with no repeated vertices in-between, the combination of all PFPs define a tree with S as the

top parent vertex and bottom, offspring vertices equal to T. This tree is called the harmonic

flow tree (HFT).

Proposition-5: The HFT of a graph contains all the vertices in that graph.

Proof: Since all PFPs of a graph start at S and terminate at T, the boundary vertices have to

be a part of the HFT. Since at the remaining intermediate vertices the flow balance relation

in equation-2 (KCL) is enforced, positive flow must enter each of these vertices; otherwise,

the balance relation cannot be enforced. Therefore, every intermediate vertex has to belong

to a PFP which makes it also a vertex in the HFT tree.

Proposition-6: The HFT of a graph contains the optimal path linking S to T.

Proof: If the optimum path of a graph linking S to T is not a branch of its HFT, then some of

the hops used to construct that path were driven by negative flows. It is obvious that the

optimum path cannot be constructed, even partially, using negative flows. If negative flows

are used, convergence cannot be guaranteed. Even if countermeasures are taken to prevent

a cycle from persisting, vertices will get repeated and the path will contain loops that can be

easily removed to construct a lower cost path. However, a proof that negative flows cannot

be used in constructing a least cost path may be obtained using the weighted Jensen

inequality:

⋅⋅≥

11LL

wf(x )+....+wf(x )

⋅⋅

11LL

f(wx +....+wx ) (10)

where f( ) is a convex function and w1+...+wL=1, wi≥0. Equality will hold if and only if

x1=x2=..=xL.

Assume that the optimal path connecting S to T contains L hops (L+1 vertices). The cost of

that path may be written as:

⋅⋅⋅

12L12L

12L

1

I

1

I

1

I

C=C +C +....+C =ΔV+ΔV....+ΔV, (11)

where ΔVl’s and Il’s are defined in the proof of proposition-1. Since KCL is enforced we

have:

1L

ΔV +....+ΔV =V -V =1-0=1.

ST

(12)

Also, notice that

inequality may be applied to the cost function above,

1/

l1

f(I )=I is a convex function of Il. Therefore, the weighed Jensen

⋅⋅≥⋅⋅

11LL11LL

ΔV f(I )+....+ΔVf(I )f(ΔV I +....+ΔV I ). (13)

or

≥

⋅⋅

∑

i=1

∑

i=1

∑

i=1

L

i

LL

2

iiii

11

C=

ΔV I C I

. (14)

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For the case of an electric network Ik and the corresponding ΔVk are related using ohm’s

law. For a positive cost having a negative ΔVk implies a negative Ik. Therefore, the inequality

will still hold for negative values of ΔVk. This variational inequality may be used to establish

a lower bound on the cost of a path connecting the start and target vertices. It is interesting

to notice that the lower bound on the cost is equal to the inverse of the power consumed in

traversing a path. According to the inequality, a zero difference between the two can only

occur if all the flows along the path are the same. In general, the more variation in the value

of the flows along the path the more is the deviation from the optimal cost will be. It is

obvious that the extreme case of mixing positive and negative flows in constructing the path

cannot lead to the optimum solution.

Proposition-7: The optimum path (or any PFP for that matter) must contain at most N

vertices.

Proof: It is obvious that if the path contains more than N vertices, an intermediate vertex in

the path is repeated. Based on the previous propositions, this cannot happen. Therefore

transition from S to T must be obtained in N hops or less.

4. The M* Algorithm:

It is not hard to see that the DHPF belongs to connectionist AI. Even in its raw form, as

shown by preposition-3, the approach is capable of tackling planning problems in a

provably-correct manner. While connectionist AI and symbolic reasoning AI are considered

to be two separate camps in artificial intelligence, there is a trend to hybridize the two in

order to generate techniques that combine the attractive properties of each approach. In this

section it is demonstrated that the DHPF approach can work in a hybrid mode. The reason

for that is: the flow induced by the connectionist system has a structure which can be

reasoned about to enhance or add to the basic capabilities of the DHPF approach. This is

demonstrated in this section by suggesting the M* algorithm for finding the minimum cost

path between S and T on a graph. By interfacing the connectionst layer to a symbolic layer,

the quality of the path which the connectionist stage is guaranteed to find is enhanced.

4.1 The M*: direct realization:

The followings are the steps for directly realizing the M* procedure:

01. Write the KCL equations for each vertex of the graph

∑ ij

j

I =0 i = 1,...,N (15)

02. From the KCL equations derive the vertex potential update equations:

N

ii,kk

k=1

k i

V=b V

≠∑

(16)

03. Initialize the variables: VS=1; VT=0; Vi=1/2 i=1,..,N i≠S, i≠T (17)

04. Loop till convergence is achieved performing the operations:

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N

ii,kk

k=1

k i

V=b V

≠∑

i=1,..,N, i≠S, i≠T (18)

05. Compute the flows

06. Using the flows construct the HFT of the graph

07. Starting from the last parent nodes, for each node retain the branch with lowest cost and

delete the others

08. Move to the parent nodes one level up and repeat step 7.

09. Repeat step 8 till the top parent node S is reached

10. The remaining branch connected to S is the optimal path linking S to T.

4.2 An Example

Consider the weighted graph shown in figure-5. It is required that a minimum cost path be

found from the start vertex S=1 to the target vertex T=5. The transition costs are: C16=1,

C14=3, C23=4, C34=7, C26=1, C37=5, C35=2, C47=6, C67=9, C57=5. The KCL equations are:

Vertex-1:

1416

14 16

V -V

C

V -V

C

+=0 Vertex-2:

1416

14 16

V -V

C

V -V

C

+=0

Vertex-3:

32343537

2334 3537

V -V

C

V -V

C

V -V

C

V -V

C

+++=0 Vertex-4:

414347

14 3447

V -V

C

V -V

C

V -V

C

++=0

Vertex-5:

5357

3557

V -V

C

V -V

C

+=0 Vertex-6:

616267

162667

V -V

C

V -V

C

V -V

C

++=0

(19)

Vertex-7:

.

73747576

3747 57 67

V -V

C

V -V

C

V -V

C

V -V

C

+++ =0

The update equations may be derived as:

46

11,441,66

1 1416

1V

C

V

C

V =[+]=b V +b V

K

,

36

2 2,33 2,66

223 26

1V

C

V

C

V =[+]=b V +b V

K

2457

33,223,443,553,77

323343537

1V

C

V

C

V

C

V

C

V =[+++]=b V +b V +b V +b V

K

137

4 4,11 4,334,77

4 1434 47

1V

C

V

C

V

C

V =[++ ]=b V +b V +b V

K

,

37

5 5,33 5,77

5 3557

1V

C

V

C

V =[+ ]=b V +b V

K

(20)

127

6 6,11 6,22 6,77

6 1,62,66,7

1V

C

V

C

V

C

V =[++ ]=b V +b V +b V

K

3456

77,337,447,557,66

73747 5767

1V

C

V

C

V

C

V

C

V =[+++]=b V +b V +b V +b V

K

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where

11416

111

=[+]

KCC

2 23 26

111

=[+]

KCC

32334 3537

11111

=[+++]

KCCCC

414 34 47

1111

=[++]

KCCC

5 3557

111

=[+]

KCC

127

6 16 2667

1V

C

V

C

V

C

=[++]

K

7 3747 5767

11111

=[+++]

KCCCC

Setting V1=1, V5=0 and applying the procedure described above we obtain the vertices

potential: V1=1, V2=0.74673, V3=0.33753, V4=0.70006, V5=0, V6=0.84902, V7=0.41093 . The

flows may be computed as: I14=0.09998, I16=0.15098, I23=0.1023, I47=0.048189, I43=0.05179,

I35=0.16877, I62=0.1023, I67=0.048677, I73=0.014679, I75=0.082186. The graph, flows and

corresponding HFT are in figure-5.

Figure 5. The graph and corresponding flows and HFT

Now start reducing the HFT,

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Level-1 parent node branch removal

Level-2 parent node branch removal

Level-3 parent node branch removal

As can be seen from figure-6 the optimum path: 5→ 3→2→6→1 with a cost 8 was obtained

after only two levels of branch removal. The HFT need not be computed explicitly in order

to carry out the branch removal procedure. The M* algorithm may be implemented

indirectly using a procedure that successively relaxes the graph till only the optimal path is

left.

4.2 The M* indirect realization:

In this subsection a successive relaxation procedure for implementing M* is suggested. The

rapid growth of an HFT with the size of a graph makes it impractical to apply the algorithm

on the tree directly. Here a procedure that allows us to operate on the HFT indirectly by

successively relaxing the graph is suggested. In order to apply the procedure the following

terms need to be defined (figure-7): positive flow index (PFI) of a vertex: number of edges

connected to the vertex with outward positive flows. Negative flow index (NFI) of a vertex:

is the number of edges connected to the vertex with inward negative flows.

Figure 7. PFI and NFI of a vertex

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The procedure is:

0-compute the PFI and NFI for each vertex of the graph,

1-starting from the target vertex and using the negative flows along with the NFIs and PFIs

of the vertices, detect the junction vertices and label them based on their levels,

2- at the encountered junction vertex clear NB buffers where NB=PFI of the junction vertex,

3- starting from the junction vertex, trace forward all paths to the target vertex traversing

vertices with positive flows and PFIs=1,

4- excluding the lowest cost path, delete all the edges in the graph connecting the junction

vertex to the first, subsequent vertices in the remaining paths,

5- decrement the PFI of the junction vertex by the number of edges removed from the graph,

6- decrement the NFIs of the first subsequent vertices from step 4 by 1,

7- if the NFI of any vertex in the graph from step 4 is equal zero and the vertex is not a start

vertex, delete the edge in the graph connecting that vertex to the subsequent vertex and

reduce the NFI of the subsequent vertex by 1,

8- repeat 7 till all the reaming vertices in the paths with PFIs=1 have NFIs >0,

9- go to 1 and repeat till there is only one branch left in the graph with two terminal vertices

having NFI=0, PFI=1 and NFI=1, PFI=0. This is the optimum path connecting the start

vertex to the target vertex.

In figure-8 the procedure is applied, in a step by step manner, to gradually reduce the

graph in figure-5 to the optimum path connecting vertex 1 to 5.

initially traced path: 5→7

constructed paths: 7→5 cost=5

7→3→5 cost=5+2=7 (eliminate edge 7→3 in graph)

initially traced path: 5→3→2→6

constructed paths: 6→7→5 cost=9+5=14 (eliminate edge 6→7 in graph)

6→2→3→5 cost=1+4+2=7

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initially traced path: 5→7→4

constructed paths: 4→7→5 cost=6+5=11 (eliminate edge 4→7 in graph)

4→3→5 cost=7+2=9

Vertex 7 NFI=0, not a start vertex (eliminate edge 7→5 in graph)

initially traced path: 5→3→2→6→1

constructed paths: 1→4→3→5 cost= 3+7+2=12 (eliminate edge 164 in graph)

1→6→2→3→5 cost=1+1+4+2=8

Vertex 4 has an NFI=0 and is not a start vertex (eliminate edge 4 → 3)