In this section, the problem ofsurvivability is formulated in terms of the fulfilment of theneedsof the system. Before proceeding, the following assumptions are made:
ASSUMPTION4.1 Needs are denumerable and computable, i.e. it is possible to enumerate the needs of a system and represent their intensities as numerical values.
ASSUMPTION4.2 Needs have different levels of priority, i.e. as in Maslow’s Hierarchy of needs, and discussed in section 3.8.4.
ASSUMPTION4.3 Changes in the intensities of needs are a reflection of changes in the internal state and external environment.
Table 4.2: Operational requirements and theneedsof an autonomous systems.
Operational Requirements
Autonomous Systems Needs
Safe Navigation Sustenance, Safety, Accomplishment: The need to ensure the system has sufficient power, is safe from cul-de-sacs, remains on driveable (traversable) terrain, avoid obstacles, arrive at the destination (and hence achieve its mission).
Situation Awareness Safety, Awareness, Cognition: The need to be aware of and understand the situation (i.e. events occurring within the environment, presence of danger, presence of other entities in the environment).
Interaction (human, machine, environment)
Awareness, Accomplishment, Cognition: For effective interac- tion, the system has to be aware of the situation, its tasks, and the rules of interaction with human operators (or users), other autonomous systems, and its workspace.
Survivability Sustenance, Safety, Awareness, Accomplishment, Cognition:
Survival entails the maximization of all the needs of an autonomous system, in order for it to be capable of meeting all other operational requirements and persist in its operation.
Let the system’s internal state be represented by xr and its observations of the environment bezr. The ego-state of the system (both internal and external states) is given byx= [xTr,zTr]T, subject to a dynamic model of the following form:
˙
x=f(x,u,d) (4.3)
whereuis the control input, anddis the disturbance.
Let aneed be represented by the tuple < li, ni > where li is a linguistic variable bearing the description of theneed(e.g.“need for power"),ni∈[0,1]∩Ris a measure of the intensity of theith need(i.e. value approaches zero if theneedis fulfilled, or approaches unity if theneedis intense). LetN be the set of all possibleneeds, andFP :N →[0,1]∩R be a function that evaluates the priority of a specificneed. Assumptions (4.1) and (4.2) permit the definition of a state vector representing a hierarchy ofneeds, given by the following:
DEFINITION4.1 (Needs vector) Aneeds vectoris an ordered list ofmneedsgiven by:
n= [n0, n1, . . . , nm−1, nm]T; ni∈[0,1]∩R. (4.4) ranked by descending order of the priority of eachneedsuch that∀i, j∈Z+∪ {0}, i < j→ FP(ni)≥FP(nj).
DEFINITION4.2 (Needs fulfilment) Theneeds fulfilment vectoris the complement ofn, denoted byn¯. For example, we can letn¯ = [1]1×m−n. Theneeds fulfilment vectorcan also be represented by a vector formed by the probabilities of fulfilling eachneed, i.e. each element¯ni ,P(neednican be fulfilled).
DEFINITION4.3 (Needs egostate) Theneeds egostate vectoris the composition ofnand
¯
n. The egostate describes fully, the needs of a system and their corresponding fulfilment levels, and is denoted as:
˜
n= [nn¯]T (4.5)
= [n0, n1, . . . , nM−1, nM, n¯0, n¯1, . . . , ¯nM−1, n¯M]T (4.6)
= [˜n0, n˜1, . . . , n˜2M]T. (4.7) wheren˜iis a direct relabelling of the elements inn˜, to simplify the notation. n˜has|n|+|n¯| elements, i.e.|n˜|= 2M.
DEFINITION4.4 (Desired needs fulfilment) Adesired needs vector, expressed as:
n∗= [n∗1, n∗2, ..., n∗M]T, (4.8) represents the maximum level of needs that a system should reach. Its complement is the desired needs fulfilment vector:
¯
n∗= [¯n∗1, n¯∗2, ..., n¯∗M]T, (4.9) representing the optimal level of needs achievement for a system. To attain the desired level of needs, the following conditions, which are contrapositives of each other, must be met: (i)∀ ni ∈ n, ni ≤ n∗i, (ii) ∀ ¯ni ∈ n¯, n¯i ≥ n¯∗i. This means that for each need, ni
should be kept below the maximum allowed value ofn∗i, which means that its fulfilment
¯
nishould be maintained aboven∗i.
From assumption 4.1, theneeds vectornrcan be computed from the internal state of the robotxrand observation of the environmentzr:
nr=H(xr,zr) =H(x), (4.10) whereH :R|x|→[0,1]1×mis a transformation from the states of the robot to a vector of real values. The mapping functionH can be implemented as distinct functions (one for each element innr) or as a lookup table or matrix. For instance, thesustenance needcan be represented asnsus = 1−current power/maximum power.
LetArbe the set of all possibleactionsa robot can perform, andarbe a vector of|Ar| elements, where each elementai ∈[0,1]indicates the activation level of theithactionαi.
Each actionαi ∈ Ar is a function that generates a control signaluthat is sent to the actuators of the robot. From assumption 4.3, a state-space formulation of the dynamic behaviour ofneedscan be expressed as:
˙
nr=F(nr,ar), (4.11)
which is analogous to equation (4.3), since theneeds vector nr is obtained fromx and the actionsar results in the control signalu. Such a formulation has some important theoretical implications:
• Equation (4.11) describes how the needs of a system would change, given its previous level of needs, and the past decisions made i.e. the history of actions that was executed).
• If equation (4.11) is controllable, then by definition, for any n0,n1, at least one finite sequence ofactions{a0,a1, . . . ,aN}exists such that a system with staten0at timek= 0would be forced to staten1 at timek=N, i.e. any trajectory innr can be realized in finite time, from arbitrary starting positions, by a finite sequence of actionsthat lead to arbitrary end points.
• Thesurvivabilityof a system relates to its ability to maximize the fulfilment of its needs. This is the ability to withstand, and recover from any deprivation of the needs.
• Therefore, by computing the controllability of the system, one can assess the survivability of the system, namely the ability of the system to withstand and recover from anyneedsdeficiency in finite time, with a finite sequence ofactions.
The above implications leads to the following formal definition ofsurvivability:
DEFINITION4.5 (Survivability) A system is survivable if and only if the state equation [equation (4.11)]formed by theneedsandactionsof the system (i.e. theneeds-action state equation) is controllable.
In the case when the system represented in equation (4.11) is linear, it can be expressed as the following:
˙
nr=Anr+Bar, (4.12)
whereAis the(m×m)system needs matrix, and Bis the(m×l)needs-action matrix (wherelis the number of actions). The controllability of the state equation (4.12) can be determined with any of the following conditions (Ogata, 1997):
1. The(m×ml)controllability matrixU= [B AB A2B ... Am−1B]has full rank.
2. The controllability grammian:
W(0, t) = Z t
0
eAτBBTeATτdτ= Z t
0
eA(t−τ)BBeAT(t−τ)dτ is non-singular for allt >0.
3. The matrix[A−λI B]has full row-rank at every eigenvalueλofA.
Definition 4.5 implies that, survivability can be achieved, if designers can determine a comprehensive set of needs and actions such that the resulting equation (4.11) is controllable.