A Class of Co-Design Problems with Cyclic Constraints and Their Solution

Andrea Censi

Abstract— Co-design problems in the field of robotics involvethe trade-off of resources usage (cost, execution time, energy,etc.) with mission performance, under recursive constraints thatinvolve energetics, mechanics, computation, and communication.This paper shows that a large class of co-design problems havea common structure, as they are described by two posets, a“function space” poset and a “resources trade-off space” poset,and the design constraints can be expressed as two maps inopposite directions between the two posets. Finding the mostresource-economical solution is then equivalent to finding theleast fixed point of the composition of those two maps. If the twomaps are monotone, results from order theory allow concludinguniqueness and systematically deriving an optimal design or acertificate for infeasibility.

I. INTRODUCTION

Everything comes together in the field of robotics.The design of an autonomous robots involves: the choice

of the mechanical platform, the choice of actuators, the choiceof sensors, the choice of the energy source, the choicesof algorithms (perception, planning, and control). Each ofthese subproblems corresponds to a discipline in itself, withits design trade-offs of achievable performance vs limitedresources. Furthermore, the interaction of those trade-offswith application-specific objectives and customer-specificpreferences creates additional irreducible complexity.

Some of the complexity arises from the fact that limitedresources (cost, power, computation, communication) alsoconstrain the function that can be implemented.

Example 1 (Cost). According to Rod Brooks, the main designconstraint for the Roomba was obtained from a survey thatasked: “What is the maximum price of a widget you can buywithout asking your spouse for permission?”. The response,$200, defined the feasible function that could be implemented.

The resource constraints vary with the application domain.Domestic robotics is cost-sensitive; defense applications arenot. In astronautics applications, there are typically strictconstraints on mass and volume. Kiva Systems’ warehouserobots can be bulky, but should be cheap.

Network constraints are a good example of application-dependent variability. In Kiva’s automated warehouse, thereis a high-speed dedicated wireless network at ~300 Mbps.NASA’s Deep Space Network allows communicating with theMars Rovers, 220 million km away, at a maximum rate of12 Kbps. Marine roboticists have it worse. Using acousticmodems, it is hard to obtain reliable transmission rates over100 bytes/s at a distance of 1 mile.

Andrea Censi <censi@mit.edu> is with the Laboratory for Information andDecision Systems (LIDS) at the Massachusetts Institute of Technology.

5

P F I R

parameters function implementation resources

exec eval

F I R

functionality implementations resources

h

trade-offs

P(R)

execute evaluate

realize

constrain

realize constrain

“Carry payload W! at distance D.”

“Need power P! and time T(P).”

power

choice of battery

choice of sensor

timeW T(P)

Fig. 1. This paper considers co-design problems in which there are non-trivialrecursive constraints between the “function” and the “resources” used. Forexample, when designing a UAV, the payload determines how much energyis needed, or more generically, the trade-off of power and time; conversely,the energy needed determines what battery to use, and therefore the payload.

Actuation, energetics, sensing, control, computation, andcommunication are all tied together by co-design constraints.Take sensing as an example. Automotive applications are notenergy-sensitive, so it is possible to use active sensors suchas LIDAR. Applications like robotic insects have extremelylimited power budgets (50 mW for the Harvard Robobee), sopassive sensors are needed for unthetered operation.

Many works in robotics can be interpreted as either“maximizing performance” or “minimizing resources”, whichare conflicting objectives. Methods that are evaluated on abenchmark typically belong to the “maximize performance”category. Clearly, we want our robots to be faster, more agile,etc. However, it is also important to minimize resource usage.

There are many examples of works which had an explicitminimality objective; the following belong to the sensingdomain. Minimality of vision sensors has been investigated byMilford [1] for the task of place localization. Fuller et al. [2]show that 4 defocussed photoreceptors in general position aresufficient to perform attitude stabilization for a robotic insect,thus realizing a “minimal” solution. The most comprehensiveattempt to systematize the field to obtain minimal robotconfigurations sufficient to perform a task is due to O’Kane andLavalle [3]. Soatto [4] and Lavalle [5] investigated the sameproblem of obtaining minimal “representations” sufficientfor a task, from the two complementary views of stochasticand nondeterministic uncertainty. Unfortunately, while theseworks provided neatly designed solutions for special cases,they cannot be easily generalized.

The discussion for perception could be repeated for actu-ation, energetics, and computation. For example, we mightdiscuss what is the minimal number of actuators to perform atask [6] (minimize resources) vs what is the fastest achievable

speed by a legged robot [7] (maximize performance).What is the general pattern? “Minimize resources” or

“maximize performance” are two halves of the same coin.This paper is an effort towards realizing a symmetric theory,with the two extrema as special cases.

This paper describes a class of co-design problems thatcan express explicit co-design constraints between functionsand resources, treated in a completely symmetric way. Theformalizations involves the definition of a “function space”and a “resource space” and the definition of two maps fromfunction to resources and vice versa (Fig. 1). These two spacescome naturally with a partial order structure. In this setup, theco-design problem of finding the function to be implementedwith the least resources usage can be cast as the search forthe least fixed point of the composition of the two maps fromfunctions to resource space. A large subclass of co-designproblems have an additional monotonicity property. Thisallows to use results of order theory to conclude uniquenessof the solution, and a systematic design procedure for findingthe optimal design or a certificate of infeasibility.

II. BACKGROUND

This section includes standard background material onpartial orders that leads to Kleene’s fixed-point theorem (The-orem 2), which will do the heavy lifting for us. Davey andPriestley [8] is a possible reference text for this material.

Definition 1. A partially-ordered set (poset) is a pair (P,≤)where P is a set and ≤ is a reflexive, antisymmetric, andtransitive relation on P × P .

Definition 2 (Minimal elements). For a poset (P,≤) and asubset S ⊆ P , let min≤ S be the minimal elements of S.

Definition 3 (Monotonicity). A function f : A→ B betweentwo posets (A,≤A) and (B,≤B) is monotone iff a1 ≤A a2

implies f(a1) ≤B f(a2).

We will make use of Kleene’s theorem (Theorem 2), a resultof order theory that is well known in the fields of ComputerScience and Embedded Systems, as it is used for definingdenotational semantics (see, e.g., [9]) and for defining thesemantics of models of computations (see, e.g., [10]). Forreaders from other engineering backgrounds, the best way toappreciate the results is to see them as the parallel to Banach’stheorem on metric spaces, transcribed below as Theorem 1.

Definition 4. A metric space (X, d) is complete iff everyCauchy sequence in X has a limit in X .

Definition 5. A map f : X → X on a metric space (X, d)is a contraction if there exists a constant q ∈ [0, 1) such thatd(f(x1), f(x2)) ≤ q d(x1, x2) for all x1, x2 ∈ X .

Theorem 1 (Banach’s Fixed Point Theorem). Let f be acontraction on a complete metric space (X, d). Then f hasa unique fixed point in X , and this point can be found bycomputing limn→∞ fn(x0) from any x0 ∈ X .

The equivalent of complete metric space (Def. 4) in ordertheory is that of complete partial order (Def. 7).

Definition 6 (Directed set). A poset A is directed if eachpair of elements in A has an upper bound. In other words, forall a, b ∈ A, there exists c ∈ A such that a ≤ c and b ≤ c.Definition 7. A poset is a directed complete partial order(DCPO) if each of its directed subsets has a supremum (leastof upper bounds). It is a complete partial order (CPO) if italso has a bottom element ⊥.

Example 2 (Completion of R+ to R+). This is a simple

construction that illustrates the principle and is needed for thesuccessive examples. The set of real numbers R is not a CPO,because it lacks a bottom element. The set of nonnegativereals R+ = {x ∈ R | x ≥ 0} has a bottom element ⊥ = 0,however, it is not a DCPO because some of its directed subsetsdo not have a supremum. For example, take R+, which is asubset of R+. Then R+ is directed, because for each a, b ∈ R+,there exists c = max{a, b} ∈ R+ for which a ≤ c and b ≤ c.One way to make (R+,≤) a CPO is by adding to it an artificialtop element >. Define R+

, R+∪{>}, and extend the partialorder ≤ so that > ≤ > and a ≤ > for all a ∈ R+. Now R+

is a CPO, because each directed subset has a supremum.

The equivalent property to contraction (Def. 5) is Scott-continuity. (In metric spaces, contraction implies continuity.)

Definition 8 (Scott continuity). A function f between twoDCPOs P and Q is called Scott-continuous if for eachdirected D ⊆ P , the image f(D) ⊆ Q is directed,and f(supD) = sup f(D).

Kleene’s fixed-point theorem (Theorem 2) is the analogousof Banach’s (Theorem 1). It ensure the existence of a leastfixed point and gives a constructive procedure to find it.

Definition 9. A least fixed point of f : A → A is theminimum (if it exists) of the set of fixed points:

lfp(f).= min≤A

{a ∈ A | f(a) = a}.

Theorem 2 (Kleene’s fixed-point theorem ). Let (L,≤) be acomplete partial order, and let f : L→ L be Scott-continuous.

Then f has a least fixed point, which is the supremum ofthe sequence ⊥ ≤ f(⊥) ≤ f(f(⊥)) ≤ · · · ≤ fn(⊥) ≤ . . .

Many variations of this results are available, as it was orig-inally stated in different fields with different hypotheses [11].Kleene’s theorem might appear to be weaker, but is possible toprove Banach’s theorem directly from Kleene’s theorem [12].

III. RESOURCE-AWARE CO-DESIGN FOR ILLUSIONISTS

The theory proposed by this paper is applicable to manydomains, so it is better to start with an example that introducesthe concepts needed with a neutral vocabulary.

Fig. 2 shows an illusion designed and performed by Britishmagician Derren Brown. He describes it as follows [13, p.20]:

The effect is that a thought-of card is divined, disappears from the deck,and arrives burnt and smoking in the performer’s mouth in place of thecigarette he had been smoking throughout.• The card is merely looked at in a ribbon spread. The performer is facingaway. The deck is reassembled. Yet he states his aim to divine it, without

touching the deck. This will get everyone’s attention. It does seem impossible.Climax One – the card is named. The spectators sit back.• The magician says it was never there in the deck. He lets the participantargue that he saw it. The performer coolly blows a smoke-ring, smiling tohimself. All eyes are now on the squared deck. The magician spreads it outagain. True, the card isn’t there. Suddenly there is confusion. The spectatoris sure he saw it. Climax Two.• The performer splutters and the cigarette seems to be causing him trouble.It can be seen to have changed. He removes it and unrolls it. A resolution tothe card’s disappearance is given, but the weirdness has escalated irretrievably.There are no answers.

Illusionists distinguish between the “effect” and the “methods.”The text cited above is a description of the effect; it is,more precisely, a description of what is perceived by theaudience. An effect corresponds roughly to what we mightcall a “specification” in engineering.

6

1. Spectator thinks of a card 2. The card is divined

3. The card has disappeared 4. The card reappears burnt

Fig. 2. A performance of the “Smoke” illusion by Derren Brown, whichcan be watched at [14]. Effect and method are described in [13].

1

Effects Methods

a regular deck

Effects Methods Gimmicks

a shuffable deckan examinable deck

Effects Methods Gimmicks“what do I need?”

“what can I do?”

≤

Fig. 3.

Any effect can be realized in mul-tiple ways, called “methods” inthe jargon of magicians (Fig. 3).There are indeed manifold ways toproduce a card from your mouth,and from the point of the audience, they are almost indistin-guishable.

From the point of view of the performer: once the effect isfixed, there are many alternative methods that can be evaluatedalong many axes. For example, a method might require certainskills or dexterity. One way to evaluate different methods is bythe “gimmicks” that are needed to perform them. For example,one might ask: is the deck used by Brown a regular deck?Could he perform the same effect using a deck provided bythe audience?

Define the set Gimmicks and suppose there is a map fromMethods to Gimmicks (Fig. 4).

1

Effects Methods

a regular deck

Effects Methods Gimmicks

a shuffable deckan examinable deck

Effects Methods Gimmicks“what do I need?”

“what can I do?”

≤

Fig. 4. The set of Gimmicks has naturally the structure of a partial order,with the element ⊥ denoting the absence of any gimmick.

We can give the set of Gimmicks the structure of a poset.The bottom element ⊥ denotes the absence of any gimmick;every object is as it appears and can be examined and handled

by the audience. The precedence relation ≤ describes whethera gimmick can simulate another.

This is an example of an ordered chain in that order:⊥ = (An ordinary deck of cards)≤ (A deck that can be shuffled by the audience)≤ (A deck that can be cut, but not shuffled)≤ (A deck that appears ordinary when spread out

by the magician in a controlled manner).

The “design” problem for a magician is, fixed the effect,choose the method that uses the least gimmicks to achieve thedesired effect. We can define a map from Effects to subsets ofGimmicks, associating to each effect the minimal gimmicksthat are needed to perform (Fig. 5).

1

Effects Methods

a regular deck

Effects Methods Gimmicks

a shuffable deckan examinable deck

Effects Methods Gimmicks“what do I need?”

“what can I do?”

≤

Fig. 5.

Suppose now that you invite Derren Brown to dinner in yourhouse, and you ask him to perform a trick. Now in his mindBrown must solve a different problem: given the gimmicks thatare available to him, which might be none, he must choose thebest trick to perform. This decision is a map from Gimmicks toEffects (Fig. 5). The two arrows that ask “what do I need?” and“what can I do?” are the essential ingredients that characterizeco-design problems. We then see that illusionism displaysqualitatively the same problems as engineering. In the theoryto be developed, the names Effects, Methods, Gimmicks, arereplaced by Functions, Implementations, and Resources:

illusionism: Effects Methods Gimmicksgeneric domain: Function Implementations Resources

Remark 1. Albeit one tries to be as culturally sensitive aspossible, it is impossible to choose names familiar to all fields.For example, the distinction of “function” vs “implementation”is familiar to the Embedded Systems field; but, in ControlTheory, the distinction is “behavioral” vs “functional”, givingthe term “function” exactly the opposite meaning (!).

IV. LEAST FIXED-POINT FORMULATION OF CO-DESIGN

2

functions implementations

P1: Find an implementation

functions implementations resources

P2: Find the most efficient implementation

Fig. 6.

Let F be a set called func-tion space and let be I be a setcalled implementation space. As-sume there is a map

exec : I→ F

that determines the function from the implementation (Fig. 6).The mnemonics for exec is “execution”.

The basic design problem one can consider is:

Problem 1. Given a function f ∈ F, find an implementa-tion i ∈ I such that exec(i) = f, or a proof that none exist.

Example 3. For a compiler, the “function” is (the semanticsof) the human-readable code, and the “implementation” is themachine-readable code (plus the machine to execute it).

Example 4 (Functions in robotics). The function is the robot’stask and associated performance constraints. In classicalrobotics, the performance constraints are easy to describe(e.g., “repeatability” of a motion). In modern applications, thetask description could be very rich (e.g., an autonomous carneeds to be “safe”—how to measure “safe”?).

A. Resources usage

A common class of design problems is one where imple-mentations are evaluated according to the “resources” thatthey use (cost, power consumption, etc.). These resources aretypically multidimensional and only partially ordered.

Let the set of resources be a poset (R,≤R), with theconvention that “less” is better, in the sense that if r1 ≤R r2

then r1 is preferable to r2. The map eval : I→ R determinesfor each implementation in I the resources used (Fig. 7). Themnemonics for eval is “evaluation.”

2

functions implementations

P1: Find an implementation

functions implementations resources

P2: Find the most efficient implementation

Fig. 7. There is a map eval : I→ R that associates to each implementationthe resources usage, which live in a poset (R,≤R). The map h : F → P(R)associates to a function the minimal resources needed to be implemented.

Example 5 (Example 3, continued). In the compilation pro-cess, the resources are execution time or memory usage.

Example 6 (Resources in robotics). Resource constraintsinclude: energy, power, cost, weight, volume, shape, band-width, memory, computation, mission time, charging time,the availability of a maintenance crew. . . As discussed in theintroduction, which one of these resources are abundant orscarce is extremely dependent on the application.

With the introduction of resources, Problem 1 becomes:across all implementations of the same function, find the onesthat use minimal resources.

Problem 2. Given a function f ∈ F, find the implementa-tion i ∈ I that minimizes resources usage:

min ≤Rr, (1)

s.t. exec(i) = f,

eval(i) = r. (2)

In (1), min ≤Rindicates the set of non-dominated minima,

which in general might have more than one element. Let P(R)be the power set of R, which is the set of all subsets. Definethe map h : F → P(R) as the one that associates to eachfunction f ∈ F the minimal elements of the set of resourcesthat correspond to feasible implementations:

h : F → P(R), (3)f 7→ min≤R

{eval(i) | i ∈ I, exec(i) = f}.B. Resource-aware functions

So far, the function f to be implemented has been consideredfixed. In robotics, the complexity of many co-design problems

arises from the recursive constraint that the function to beimplemented depends on the resources usage.

Call ϕ : R→ F the map that describes the function to beimplemented given the resources usage (Fig. 8).

3

function implementation resourcesexec eval

F

R

P3: Adapt the function (but the resources don’t change)

function implementation resources

F

R

P4: Fully joint problem

Fig. 8. We consider “resource-aware” design problems in which the functionto be implemented depends on the resources usage.

Example 7 (Which battery for my drone?). A larger batteryprovides more energy, but it adds to the weight to betransported, and thus the energy needed. This is a typicalexample of a recursive co-design constraint. In this case thefunction space F consists of specifications of the kind

F = {"Carry a payload W " |W ≥ 0},while the resource space is

R = {"The energy needed isE" | E ≥ 0}.The implementations space I is the set of all possible platformsdesign. The map h : F → R associates to a certain payload(function) the energy needed to transport it; the map ϕ : R→F associates to a certain energy the payload needed to storeit. This example will be expanded in Sec. VI. (For a realisticanalysis, refer to [15].)

The solution of a co-design problem is a triple of function-implementation-resources (f, i, r) that satisfy three inter-relatedconstraints, given by exec, eval, and ϕ.

Problem 3. Find the tuples (f, i, r) that are solutions of

min ≤Rr, (4)

s.t. exec(i) = f,

eval(i) = r, (5)ϕ(r) = f. (6)

Make the following temporary assumption:

Assumption 1. For each f ∈ F, h(f) is a singleton.

Then Problem 3 can be rewritten as a fixed point on f only.We are looking for a pair (f, r) such that ϕ(r) = f and h(f) =r. That is, we are looking for f that is a fixed point of ϕ ◦ h :F → F and that minimizes the cost h(f).

Problem 4. Find the function f:

min ≤Rh(f), (7)

s.t. f = (ϕ ◦ h)(f). (8)

C. Fixed-point formulation in the resources space R

The downside of the function-space formalization in Prob-lem 4 is that it does not highlight the role of the resources;for example, we would like to use the partial order on theresources to characterize which parameter are the best tochoose. We thus prefer an alternative formulation, in whichthe fixed point is described in the resource space R.

We are looking for the resources r such that the corre-sponding function f = ϕ(r) can be realized with resourcesusage h(ϕ(r))) = r. Thus the fixed point in the resourcesspace is described by

r = (h ◦ ϕ)(r). (9)

Note that if we knew of an r0 that satisfies (9), we couldsimply obtain the corresponding function as f0 = ϕ(r0). Inthis sense (9) is equivalent to (8).

Among all r that satisfies (9), we would like to choose theone that is minimum, meaning that we are looking for theleast fixed point of the function F = h ◦ ϕ : R→ R:

min ≤R{r ∈ R | F (r) = r}. (10)

The condition can be further relaxed by noting that, if thefunction corresponding to resources r is r′ ≤R r, we would behappy to accept r′ as well. Thus the equality F (r) = r in (10)can be relaxed to F (r) ≤R r, which leads to the followingformulation.

Problem 5. Find the least fixed points of F = h ◦ ϕ :

min ≤R{r ∈ R | F (r) ≤R r}. (11)

The formulation now highlights the space of resources andits natural structure given by the order ≤R, and we haveestablished the correspondence with the problem of finding aleast fixed point.

D. Fixed-point formulation in the trade-off space P(R)

7

abstract!functions

concrete!parameters

Fig. 9.

We now relax Assumption 1. Theequivalent of (11) can be writtenfor P(R), the power set of R (Fig. 9).A good name for P(R) is the “re-source trade-off space”.

The partial order ≤R on R induces one on P(R).

Lemma 1. Let (R,≤R) be a poset. Then (P(R),≤P(R)) is aposet, defining ≤P(R) so that

S1 ≤P(R) S2 ≡ ↑S1 ⊇ ↑S2.

where ↑S is the upper closure of S.

7

abstract!functions

concrete!parameters

Fig. 10. S1 ≤P(R) S2

The intuition is that if S1 ≤P(R)

S2, the choices in S1 dominates anychoice of S2 (Fig. 10).

We can now rewrite Problem 5 asthe search for a fixed point in theresources trade-off space P(R).

Problem 6. Find the least fixed points on theposet (P(R),≤P(R)) for the function F = (h ◦ ϕ):

min ≤P(R){S ∈ P(R) | F (S) ≤P(R) S}. (12)

E. The poset structure of the function space F

To obtain algorithmic results, one needs additional assump-tions about the structure of the function space F and the map h.Assume that the functions of interest (range of h) have thestructure of a poset with a partial order ≤F. By convention, “alarger function is better”, in the sense that if f1 ≤F f2, then f2is preferable to f1.

7

abstract!functions

concrete!parameters

Fig. 11.

It is perhaps simpler to imagine F

as a large space of “abstract func-tions” of which a subset is indexed by“concrete parameters” in a poset. The“instantiation” map from parameters tofunctions then induces a partial orderfor the function space (Fig. 11).

Example 8 (Mission requirements). A requirement for a UAVcould be “Travel at distance D, leave a payload W , and comeback.” Then the parameters are (D,W ) and the function spaceis F = R+ × R+.

Example 9 (Algorithm parameters). Perception algorithmshave many parameters to tune (e.g., number of iterations,number of features) which slightly change the function of thealgorithm, and those parameters have a natural partial order.

V. SYSTEMATIC SOLUTION FOR A CLASSOF MONOTONE CO-DESIGN PROBLEMS

The developments in Sec. IV can be summarized as follows:

Definition 10. A resource-aware co-design problem is adesign problem described by:1) A function space F.2) An implementation space I.3) A resource space that is a poset (R,≤R).4) A map exec : I→ F.5) A map eval : I→ R.6) A map ϕ : R → F that gives an additional constraintbetween the resources used and the minimum functionalityto be implemented. This is the “co-design” constraint.The optimization problem (already written as Problem 3)

consists in finding the tuples (f, i, r) that satisfy the constraintsgiven by exec, eval, ϕ and are minimal with respect to ≤R.

We call “monotone” the problems in which one can makemonotonicity assumption about ϕ and h.

Definition 11. A co-design problem is monotone if• The spaces F and R are two CPOs (Def. 7).• The map ϕ : R→ F is monotone and Scott-continuous.• The map h : F → P(R) (defined as in (3)) is monotoneand Scott-continuous with respect to the partial order on P(R)defined in Lemma 1.

There is a systematic design procedure for monotone problems.

Proposition 1 (Solution of monotone co-design problems).1) Solving a monotone co-design problem is equivalent tofinding an element of P(R) that is the least fixed point ofan endomap F = h ◦ ϕ on P(R):

min ≤P(R){S ∈ P(R) | F (S) ≤P(R) S}. (13)

2) A (unique by definition) least fixed point always exists.3) If the least fixed point is S = {>R}, the design problemis “infeasible”, in the sense that it needs “infinite resources”.(The problem is not “infeasible” in the mathematical sense,because the optimization problem always has a solution.)

4) The least fixed point can be found by computing theascending Kleene chain for the map (h ◦ ϕ) : P(R)→ P(R)starting from S0 = {⊥R} and iterating Sk+1 = (h ◦ ϕ)(Sk).

Proof. If ϕ : R→ F is Scott-continuous and h : F → P(R)is Scott-continuous then the composition (h ◦ ϕ) is Scott-continuous. All conclusions then follow easily from Kleene’stheorem (Theorem 2) applied to the map (h ◦ ϕ).

Remark 2 (Non-monotone co-design problems). It is easyto come up with examples of monotone co-design problems.Are there co-design problems that are not monotone? Sometechnologies have non-monotone efficiency curves. An internalcombustion engine at constant load has a peak efficiencyas a function of velocity. There is a non-monotone relationbetween vehicle speed (m/s or mph) and fuel economy (km/lor mpg) [16]. Consequently, if one takes the function spaceto be parametrized as “Travel at speed V from point A toB”, and the resource space as “Fuel consumed F ”, then itmight not be true that there is a monotone relation between Vand F .

VI. EXAMPLES

The rest of the paper consists of examples:1) In an aerial robotics scenario, Example 10–12 describethe co-design constraints among actuation, energetics, andsensing. Numerical examples are provided.

2) Example 13 describes co-design constraints deriving fromdesigning a controller under finite computation resources.

3) In a marine robotics scenario, Example 14 describes co-design constraints between function and cost (as a resource),from the point of view of customer acceptance.

Example 10. 1The choices of sensors, actuators, and batteriesfor an UAV must respect two main co-design constraints: 1)a larger battery weights more; 2) the available power mustbe allocated to sensing and actuations. These are a system ofnonlinear constraints (Fig. 12).

10

Fig. 12. Nonlinear constraints for actuation, sensing, energetics.

The parameters of interest and their units are in Table I.TABLE I

symbol unitB g Battery weightρ J/g Energy densityT s Mission timeW0 g Payload without batteryW g Total payload, equal to W =W0 +BPa W Power for actuation

By the end of Example 12, we will have considered the fullnonlinear model. We consider first additional assumptions toobtain a linear problem:

1For this example, the software is available as IPython notebooks athttp://purl.org/censi/2015/mocdp1/notebooks.

1) The mission time T does not depend on the payload.2) The power for actuation is linear in the payload: Pa = αW .What is the minimum size of the battery for this task?

Manual solution: The constraint is that the stored energy onboard (ρB) exceeds the energy needed (PaT ) for the mission:

ρB ≥ PaT (energy constraint) (14)

Because Pa = αW = α(W0 +B), the constraint becomes

B ≥ T αρ

(W0 +B), (15)

and because we want the minimum B, this becomes B =T αρ (W0 + B). This is a linear equation that can be easily

solved in closed form. The solution is in R+ iff

ρ > Tα. (16)

This makes sense: there is a solution only if the energy densityis large enough—the battery should “carry its own weight”.

Solution as monotone co-design problem: A systematicsolution to the problem can be obtained in 6 steps:1) Choice of function space: Choose F as:

F = {Carry a payloadW |W ∈ R+}. (17)

Note that it is necessary to use the completion R+for the

reasons explained in Example 2.2) Choice of implementation space: In this simple example,it is not necessary to choose an implementation space I.3) Choice of resource space: Choose R as:

R = {"Total energy needed is E" | E ∈ R+}. (18)

4) Definition of the map ϕ: After the previous two choices,the rest follows easily. One needs to write down the function h :F → R that maps the payload W to the energy E. Accordingto the previous assumptions, to carry a payload W theactuation power needed is Pa = αW . The energy is E =TPa = TαW . The map h is then

ϕ : W 7→ TαW.

5) Definition of the map h: Conversely, the function ϕ :R→ F maps an energy E to a payload W . The intermediatevariable is the size of the battery. To have available theenergy E one needs a battery of weight B = E/ρ. Thepayload as a function of the energy requirement is W =W0 +B = W0 + E/ρ. The map ϕ is then

ϕ : E 7→W0 + E/ρ.

6) Iteration: Now we can solve by iterating the map F = h◦ϕ,written explicitly as

F : E 7→ Tα(W0 + E/ρ),

starting from E0 = 0. Will the sequence converge?Here is a classical analysis, first. If the condition (16)holds then F is a contraction, and by Banach’s fixed pointtheorem (Theorem 1) the sequence converges to a unique fixedpoint. If (16) does not hold, then it is easy to show that thesequence is unbounded and it does not converge in R+ as it

“goes to infinity”.Using Prop. 1, based on Kleene’s theorem (Theorem 2), the

conclusions are more elegant. Because F is a Scott-continuousfunction on a dcpo R = R+

, the least fixed point exists andthe iteration always converges to it. Wait!—this is apparentlya contradiction, as it does not mention the condition (16) at all.In fact, it might be that that the least fixed point is >, whichwe added to R+ to play the role of “infinity” (Example 2). Inthat case, the sequence of iterates that converges to > is acertificate for the infeasibility of the design problem.

8

Fig. 13.

Example 11 (Taking into account sensingpower). In this second example, we con-sider a slightly more complicated case, inwhich we have to decide how much powerto allocate to sensing, assuming that thecompletion time is a function of sensing power.

The additional symbols are listed in Table II. Call Ps thesensing power, so the total power consumption is Ptot = Pact +Ps. Assume that the function T (Ps) is monotone decreasingin Ps (Fig. 13). No other assumptions on Ps are needed.

TABLE II

symbol unitPs W Sensing powerPtot W Total power consumption, Ptot = Pact + PsT (Ps) s Mission time

The energy constraint (14) now becomes

B ≥ 1

ρT (Ps)(α(W0 +B) + Ps).

Compared to the simplified case in Example 10, now theoptimization problem involves two variables (Ps and B) andthe constraints are nonlinear, so a direct analytical solution isnot possible.

Following the same procedure as before we obtain:1) Choice of function space: Choose F as in (17).2) Choice of implementation space: The implementationspace is indexed by the sensing power Ps:

I = {"Use a sensor with power Ps" | Ps ∈ R+}. (19)

3) Choice of resource space: Choose R as in (18).4) Definition of the map h: This map answers the question:“How much energy do I need for carrying a payload W ?”.Write the energy E, now depending on Ps:

E(Ps) = T (Ps)(αW + Ps).

The minimum energy is obtained by minimizing over Ps:

ϕ : W 7→ minPs

T (Ps)(αW + Ps).

The map ϕ is monotone in W , because a larger W increasesthe objective of the cost function.5) Definition of the map ϕ: This map answers the question“How much payload do I need for having available energy E?”.Like before, the answer is ϕ : E 7→W0 + E/ρ.

6) Iteration: The solution is obtained by simply iterating themaps ϕ and h starting from E0 = 0. If the iteration convergesto a finite fixed point (E,W ), then it is guaranteed to be the

most energy-efficient solution. If the sequence diverges toinfinity, then the co-design problem is infeasible.

Fig. 15a shows the results of running the optimizationalgorithm as a function of the energy density ρ. Refer tothe caption for the value of the parameters.

Example 12 (Explicit acknowledgement of the time-energytrade-off). This example shows how one can use the con-struction of the trade-off space P(R) to explicitly handle thetrade-off between resources, which in this case are time andenergy. Suppose that we want to find the minimum energysolution given a constraint on the execution time:

T ≤ Tmax. (20)

1) Choice of function space: Choose F as in (17).2) Choice of implementation space: Choose I as in (19).3) Choice of resource space: We now explicitly acknowledgethe trade-off of power and time, rather than considering onlythe total energy. Define

R = {"Need energy E and time T " | (E, T ) ∈ R+ × R+}.

4) Definition of map h: Now the map h takes a payload Winto the subset of P(R) that describes the set of possible(energy, time) tuples that make it possible to transport W :

h : R+ → P(R+ × R+)

W 7→ min≤R

{〈T (Ps)(αW + Ps), T (Ps)〉 | Ps ∈ R} .

The “min” operator discards the dominated solutions (Fig. 14a).It can be verified that h is Scott-Continuous; this means that,while for a fixed W there is a trade-off of time vs energy,the trade-off is monotone, in the sense of Lemma 1, as thepayload W increases (Fig. 14b).

energy E (J)

time

T(s

)

dominated solutions

minimal solutions

(a)energy E (J)

time

T(s

)

increasing payload W

(b)Fig. 14.

5) Definition of map ϕ: The map ϕ : P(R)→ R+is defined

as follows:

ϕ : S 7→W0 +1

ρmin{E | (E, T ) ∈ S, T ≤ Tmax}.

The set S ∈ P(R) is a collection of achievable (energy, time)operating points. The map ϕ selects the solution correspondingto the minimum energy under the time constraint (20). It canbe checked that this map is Scott-Continuous with respect tothe two partial orders ≤P(R) on P(R) and ≤ on R+

.6) Iteration: The optimization algorithm consists in iteratingϕ and h on the set P(R) starting from the initial point S0 ={(0, 0)}. Concretely, this means that the state of the iterationis a subset of R. In this case, it is a vector of (energy, time)tuples. Some numerical examples of the optimization areshown in Fig. 15b.

0.00.51.01.52.02.5

sens

ing

pow

er(W

)

01020304050607080

payl

oad

(g)

0 5 10 15 20 25 30

energy density (J/g)

10.610.811.011.211.411.611.812.012.2

time

(s)

(unfeasible)

(unfeasible)

(unfeasible)

less sensing poweras energy density increases

less battery on board

more time needed

(a) Solution for the variation described in Example 11.

0.00.51.01.52.02.5

sens

ing

pow

er(W

)

01020304050607080

payl

oad

(g)

0 5 10 15 20 25 30

energy density (J/g)

10.610.811.011.211.411.611.812.012.2

time

(s)

(unfeasible)

(unfeasible)

(unfeasible) kink due to explicitconstraint on time

(b) Solution for the variation described in Example 12.

Fig. 15. Numerical results for the examples. The numerical values of theparameters used are: W0 = 1, T (Ps) = 10 + 1/

√Ps, Tmax = 11.5.

7

abstract!functions

concrete!parameters

Fig. 16.

Example 13 (Control and computationco-design). This example illustrateshow the theory applies to co-designproblems that relate control and com-putation.

Suppose we need to control a physi-cal plant P , a directed system from thecommands ut to the observations yt, for example describedby a state-space model like xt = f(xt,ut), yt = h(xt).A generic controller C is similarly defined, as a directedsystem with input yt and output ut that is in feedback withthe plant (Fig. 16). Suppose we have a map γ that associatesto each plant P a controller CP = γ(P ) that is best in agiven class. The conclusions hold for any control objective;for example, we can require stability, or stricter performanceobjectives.

Let comptime(C) be the computation time associatedto evaluating the output of a controller C for every newobservation yt. The computation time is not an intrinsicproperty of C, but it rather depends on the computationsubstrate that one has available. If comptime(CP ) is notnegligible with respect to the time constant of the process, then

the controller might fail, because instead of P it is effectivelyinteracting with another plant obtained from P by addingdiscrete sampling and delay.

Let delayδ be a line delay of δ. Let sample∆ be the mapthat takes a process P into its sampled version sample∆(P )at sampling interval ∆. If the computation time is T , then theprocess to be controlled is P = sampleT (delayT ◦ P ).

The problem is to find a fixed point (P , T ) that satisfies{T = comptime(γ(P )),

P = sampleT (delayT ◦ P ).

We can easily analyze the problem using the framework ofresource-aware co-design. Define the function space as

F = {"Find a control input for the systemP = sample∆(delayδ ◦ P )" | (∆, δ) ∈ (R+)2}.

Take for the partial order ≤F the usual partial order on (R+)2.The resources set R model the time required for computation:

R = {"Need computation time T " | T ≥ 0}.The implementation space I is the class of controllersconsidered indexed by appropriate parameters that relate tocomputation (e.g. the number of samples used by an RRTplanner, or the resolution of a grid-based method). Considerthe map h : F → R that maps to each choice of (∆, δ) thecomputation time T :

h : (∆, δ) 7→ comptime(γ(sample∆(delayδ ◦ P ))).

Then this map is monotone, because as delay/sampling intervalgrow, the control problem is more difficult. Conversely,consider the map ϕ : R → F that associates to eachcomputation time T the requirement that one must control aplant with delay and sampling equal to T :

ϕ : T 7→ "Find a control input for sampleT (delayT ◦ P )".

Then also the map ϕ is monotone. It follows from Prop. 1that the co-design problem can be solved iteratively:• Construct the sequence {Tk} starting from T0 ← 0.– Let Pk ← sampleTk

(delayTk◦ P ).

– Find the controller Ck ← γ(Pk).– Benchmark the controller and set Tk ← comptime(Ck).

If the sequence Tk converges, then the limit T∞ is theminimum delay/sampling time that makes the problem feasibleand achieves the best performance. If the sequences diverges,the co-design problem is infeasible.

Example 14 (Seabed surveying). This example takes intoaccount a partial order of resources (time, energy, cost).Consider the scenario of seabed surveying using a AUV [17],formalized as an area coverage problem. Many other scenariosare equivalent, such as the problem of autonomous lawnmowing, and aerial inspection for search and rescue. Assumethat the vehicle moves at velocity v [m/s] with respect tothe seabed, at a fixed depth, such that the field of viewis r [m] (Fig. 17a). The function of the system is to sweepan area A [m2] (Fig. 17b). We will characterize the resources

(time, energy, cost) and then ask under what condition thedesign is feasible based on the customer’s preferences. 9

(a) Mapping the seabed (b) Sweeping strategy ! (perfect localization)

(a) Seabed surveying using an AUV

9

(a) Mapping the seabed (b) Sweeping strategy ! (perfect localization)

(b) Sweeping strategy

Fig. 17. The AUV seabed surveying scenario. Figures adapted from [17],with permission of the authors.

All the quantities in this problem are related by cyclic co-design constraints (Fig. 18). Read the diagram from the left.The area A is a parametrization of the function. Neglecting theturning time, and assuming there are no currents, the constraintbetween velocity v, time T , and field of view r is vTr ≥ cA,for a constant c. The actuation power Pa [W] is a function ψof v. We can assert that this function ψ is monotonicallynondecreasing, even without any knowledge of hydrodynamics.This gives the constraint Pa ≥ ψ(v). If there is a maximumvelocity vmax, let ψ(v) = > for v ≥ vmax. Similarly, thesensing power Ps is monotonically nondecreasing in r. Theenergetics constraint is E ≥ (Pa+Ps)T . We can also considerthe cost of the sensor, as a monotonically nondecreasingfunction of r. For clarity, the costs for the other componentsare neglected. Together, these relations define a constraint ofthe kind

〈E, T, $〉 ≥ h(A).

9

(a) Mapping the seabed (b) Sweeping strategy ! (perfect localization)

Fig. 18. Formalization of part of the co-design constraints involving thefunction (sweeping an area) and the resources (time, energy, money).

The other half of the story are the customer’s preferences.Suppose we create a system that sweeps an area A for a certaintime, energy, and cost. Fixed 〈T,E, $〉, what is the minimumarea for which the customer accepts the system? In general,there is a nonlinear constraint of the kind A ≥ ϕ(T,E, $) fora monotonic function ϕ, which describes the customer. Puttingthe two constraints together gives a fixed-point constraint ofthe type A ≥ (ϕ ◦ h)(A), which one can solve to find theminimum function A (if it exists) at which the system isaccepted by the customer, and the corresponding minimumresources needed h(A).

VII. CONCLUSIONS

The discipline of robotics is uniquely defined by theirreducible complexity of its co-design constraints.

This paper discussed a class of co-design problems, inwhich there is a monotone constraint between resources usageand functionality to implement. Based on elementary resultsof order theory, mainly Kleene’s theorem, it is possible toconclude the existence and uniqueness of a solution, and obtaina systematic design procedure. The optimization algorithmobtained is practical, easy to understand for practitioners, trivialto implement, and is guaranteed to give the optimal solution,or a certificate of infeasibility if a solution does not exist.

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