Robot Base Disturbance Optimization

with Compact Differential Evolution Light�

Giovanni Iacca, Fabio Caraffini, Ferrante Neri, and Ernesto Mininno

Department of Mathematical Information Technology,P.O. Box 35 (Agora), 40014 University of Jyvaskyla, Finland

{giovanni.iacca,fabio.caraffini,ferrante.neri,ernesto.mininno}@jyu.fi

Abstract. Despite the constant growth of the computational power inconsumer electronics, very simple hardware is still used in space applica-tions. In order to obtain the highest possible reliability, in space systemslimited-power but fully tested and certified hardware is used, thus reduc-ing fault risks. Some space applications require the solution of an opti-mization problem, often plagued by real-time and memory constraints.In this paper, the disturbance to the base of a robotic arm mountedon a spacecraft is modeled, and it is used as a cost function for an on-line trajectory optimization process. In order to tackle this problem in acomputationally efficient manner, addressing not only the memory sav-ing necessities but also real-time requirements, we propose a novel com-pact algorithm, namely compact Differential Evolution light (cDElight).cDElight belongs to the class of Estimation of Distribution Algorithms(EDAs), which mimic the behavior of population-based algorithms bymeans of a probabilistic model of the population of candidate solutions.This model has a more limited memory footprint than the actual popula-tion. Compared to a selected set of memory-saving algorithms, cDElightis able to obtain the best results, despite a lower computational overhead.

1 Introduction

Some real-world problems, due to real-time, space, and cost requirements, imposethe solution of an optimization problem, sometimes even complex, on a devicewith limited memory and computational resources. This situation is typical, forinstance, in mobile robots and real-time control systems, where all the compu-tation is performed on board of an embedded system. Some examples of thisclass of problems can be found in home automation, mobile TLC devices, smartsensors and biomedical devices. Among these applications, space control systemsrepresent an interesting exception. Despite the constant growth of the computa-tional power in consumer electronics, very simple and dated hardware is indeedstill used on spacecrafts. It must be remarked that the computational deviceson board of a spaceship should reliably work without any kind of rebooting for

� This research is supported by the Academy of Finland, Akatemiatutkija 130600,Algorithmic Design Issues in Memetic Computing and Tutkijatohtori 140487, Algo-rithmic Design and Software Implementation: a Novel Optimization Platform.

C. Di Chio et al. (Eds.): EvoApplications 2012, LNCS 7248, pp. 285–294, 2012.c© Springer-Verlag Berlin Heidelberg 2012

286 G. Iacca et al.

months or, in some cases, even for years. In this sense, the use of tremendouslysimple but fully tested and certified hardware allows a high reliability of thecomputational cores, thus reducing fault risks. For example, since over twentyyears, National Aeronautics and Space Administration (NASA) employs, withinthe space shuttles, IBM AP-101S computers [12]. These computers constitute anembedded system for performing the control operations. The memory of compu-tational devices is only 1 Mb, i.e. much less capacious than any modern device.Thus, the necessity of having an efficient control notwithstanding the hardwarelimitations arises.

In these cases, an optimization algorithm should perform the task withoutlimited requirements of memory and computational resources. Unfortunately,high performance algorithms are usually fairly complex structures which makeuse of a population of candidate solutions (requiring a high memory footprint)and other components such as learning systems or classifiers, see e.g. [13,18].

In order to address problems characterized by a limited hardware, compactEvolutionary Algorithms (cEAs) have been designed. A cEA is an EA belong-ing to the class of Estimation of Distribution Algorithms (EDAs) [7]. Compactalgorithms mimic the behavior of population-based algorithms but, instead ofstoring and processing an entire population, make use of a probabilistic rep-resentation of the population. In this way, the algorithm saves at least a partof the advantages of population-based algorithms but requires a much smallermemory with respect to their corresponding population-based versions. Thus, arun of these algorithms requires a small amount of memory compared to theircorrespondent standard EAs. Recently, a compact algorithm based on Differ-ential Evolution, called cDE, has been proposed in [9]. This algorithm encodesthe population within a probabilistic distribution and employs the standard DElogic for generating new trial solutions and selecting the most promising searchdirections. The cDE algorithm showed a performance superior to other compactalgorithms for a large set of test problems. Some work has been performed toenhance the performance of the cDE algorithm, for example embedding it in amemetic framework [10,11], or combining multiple cDE core [5,6].

In this work we propose a modified cDE scheme for minimizing the disturbanceto the base of a robotic arm working in a non-gravitational environment onboard of a spacecraft. The proposed algorithm, namely compact DifferentialEvolution light (cDElight), includes a light mutation, which allows a uniquesolution sampling instead of the multiple sampling typical of cDE frameworks,and a light crossover, which allows to save on the random number generation.Simulation results show that the proposed algorithm has the same performanceas the original cDE but with a lower computational overhead, thus leading toimportant advantages for real-time online optimization.

The remainder of this paper is organized as follows. Section 2 describes themechanical model of the robotic arm and the fitness function. Section 3 describesthe proposed cDElight. Section 4 shows the simulation results. Section 5 givesthe conclusion of this work.

Robot Trajectory Optimization with cDElight 287

2 Base Disturbance Optimization in Space Robotic Arms

Space robots are crucially important in current space operations as they canprevent humans from having to perform extremely risky operations, e.g. extra-vehicular activities such as reparations outside the spacecraft. Due to the enor-mous distances, the robot cannot be fully remotely controlled manually from theEarth because the communication delay between the command and the execu-tion of the robot operation can likely be unacceptable in several cases. For thisreason an efficient real-time control system is essential. The absence of gravityplays an important role in the dynamics of the robot and must be taken intoaccount when the control system is designed. In this case of study, a robotic armconnected to a base, e.g. a spacecraft or a satellite, is considered. In a nutshell,the control system aims to perform the robot movements in order to minimizethe disturbances, i.e. inertial movements, on the base. More specifically, each newtrajectory step is optimized online using the look-ahead optimized algorithm fortrajectory planning proposed in [14].

Space robots are highly nonlinear, coupled multi-body systems with nonlinearconstraints. Moreover, the dynamic coupling between the manipulator (roboticarm) and the base usually affects the performance of the manipulator. The dy-namic coupling is important to understand the relationship between the robotjoint motion and the resultant base motion, and it is useful in minimizing fuelconsumption for base attitude control. The measure of dynamic coupling hasbeen formulated in [17].

Let us consider a manipulator composed of nb links (bodies) interconnected byjoints and connected by means of an external joint to the base. With reference toFig. 1, let Vi and Ωi be linear and angular velocities of the ith body of the manip-ulator arm with respect to the inertial reference system ΣI , and let vi and ωi belinear and angular velocities of the ith body of the manipulator arm with respectto the base ΣB. Thus, we can obtain that the velocities of the ith body are:

Vi = vi + V0 +Ω0 × riΩi = ωi +Ω0

(1)

Fig. 1. Space robotic arm scheme

288 G. Iacca et al.

where the operator × represents the outer product of R3 vectors. V0 and Ω0

are, respectively, linear and angular velocities of the centroid of the base withrespect to ΣI . The variable ri represents the position vector related to the ith

body pointing towards the center of the base with reference to ΣB, see [17] fordetails. The velocities in the base of the coordinates of the reference system ΣB

can be calculated as: [viωi

]= Ji(q)q (2)

where q is the vector of the angular positions of each joint of the manipulatorarm (see q1, q2, and q3 in Fig. 1) and Ji(q) is thus the Jacobian of the ith body ofmanipulator arm. The Jacobian can be then decomposed into two sub-matricesrelated to its translational and rotational movements [17]:

Ji(q) =

[JTi(q)JRi(q)

](3)

The total linear (P ) and angular (L) momenta of the entire robotic arm can beexpressed as:

P = m0V0 +

nb∑i=1

(mivi) (4)

L = I0Ω0 +m0RB × V0 +

nb∑i=1

(Iiωi +miri × vi) (5)

where Ii and mi are the inertia momentum and mass of each body composing therobot manipulator, and RB is a positioning vector pointing towards the centroidof the base with reference to ΣI . Equations (4) and (5) can then be combined:

[PL

]= HB

[V0

Ω0

]+Hmq. (6)

The details about the structures of the matrices HB and Hm are given in [4]. Ina free-floating situation (due to the fact that both robot and base are in outerspace), there are no external forces or momenta. If we consider the gravitationalforce to be negligible, linear and angular momenta are conserved. We assume thatthe initial state of the system is stationary, so that the total linear and angularmomenta are zero. Hence, from Eq. (6), the mapping relationship between themanipulator joint motion and the base motion is thus given by:

[V0

Ω0

]= −HB

−1Hmq. (7)

For a given trajectory that must be followed by the robot, the optimizationproblem under investigation consists of detecting the angular positions qi, an-gular velocities qi, and angular accelerations qi of each joint i in each knot kidentifying the trajectory such that the disturbance on the base is minimized.The fitness to be minimized is, in our study, the integral over time of the norm

Robot Trajectory Optimization with cDElight 289

of the acceleration vector on the base. The acceleration values can be derivedby Eq. (7). Since the trajectory must be continuous, the function describing theposition of the joints over time must also be continuous. In order to satisfy thiscondition, we modeled each function qi (t) (where t is the time) as a set of 5th

polynomial splines and imposed the continuity of qi (t), qi (t), and qi (t). Eachspline is a polynomial of the 5th order because six conditions are imposed bythe physics of the phenomenon (continuity of the function, its first and secondderivative in the knots). With reference to Fig. 1, considering that the robot ma-nipulator contains three joints, the trajectory is marked by two knots, and foreach joint it is necessary to control angular position, velocity and acceleration,our fitness function depends on 3 · 2 · 3 = 18 variables.

3 Compact Differential Evolution Light

In order to solve the online trajectory optimization problem described in the pre-vious section, we considered a compact algorithm which proved to be successfulin a wide range of landscapes, namely the compact Differential Evolution (cDE)[9]. More specifically, we introduced two minor modifications into the originalcDE framework, in order to make the algorithm computationally less expensivewithout compromising its efficiency. Most importantly, these modifications allowto use cDE in real-time environments.

The resulting algorithm, named compact Differential Evolution light (cDE-light), see Algorithm 1, consists of the following steps. Without loss of generality,let us assume that the decision variables are normalized so that each search in-terval is [−1, 1]. At the beginning of the optimization, a 2 × n matrix, where nis the problem dimension, is generated. This matrix PV = [μ, σ] is called proba-bility vector. The initial values of μ and σ are set equal to 0 and 10, respectively.The value of σ is empirically set large enough to simulate a uniform distribution.

A solution xe, called elite, is then sampled from PV . More specifically, thesampling mechanism of a design variable xr [i] associated to a generic candidatesolution x from PV consists of the following steps. For each i-th design variableindexed, i = 1, · · · , n, a truncated Gaussian Probability Distribution Function(PDF) characterized by a mean value μ [i] and a standard deviation σ [i] isassociated. The formula of the PDF is:

PDF (truncNorm (x)) =e− (x−μ[i])2

2σ[i]2

√2π

σ [i](erf

(μ[i]+1√2σ[i]

)− erf

(μ[i]−1√2σ[i]

)) (8)

where erf is the error function [3]. From the PDF, the corresponding CumulativeDistribution Function (CDF) is constructed by means of Chebyshev polynomi-als according to the procedure described in [2]. It must be observed that thecodomain of CDF is [0, 1]. In order to sample the normalized design variable x[i]from PV , a random number r = rand(0, 1) is sampled from a uniform distribu-tion. Finally, the inverse function of CDF, representing x[i] in correspondence ofr, is then calculated and scaled in the original decision space.

290 G. Iacca et al.

counter t = 0{** PV initialization **}initialize μ = 0 and σ = 1 · 10sample elite from PVwhile budget condition do

{** Mutation Light**}generate x′

off from a modified PV : PV ′ = [μ,√

(1 + 2F 2)σ2]

{** Crossover Light**}xoff = elitegenerate istart = round (n · rand (0, 1))xoff [istart] = x′

off [istart] {** Deterministic Copy**}xoverL = round

(log(rand(0,1))

log(Cr)

)

i = istart + 1j = 1while i �= istart AND j ≤ xoverL + 1 do

xoff [i] = x′off [i]

i = i + 1j = j + 1if i == n then

i = 1end if

end while{** Elite Selection **}[winner, loser] = compete (xoff , elite)if xoff == winner then

elite = xoff

end if{** PV Update **}update μ and σ according to Eq. 12t = t + 1

end while

Algorithm 1. cDElight pseudo-code

At each step t, cDElight generates an offspring by means of two operatorscalled mutation light and crossover light, which represent two modifications ofthe original cDE framework. Mutation light consists of the following steps. Ac-cording to a DE/rand/1 mutation scheme, three individuals xr, xs, and xt areneeded to generate a provisional offspring x′

off computing x′off = xt+F (xr − xs).

Instead of sampling three separate individuals, like the original cDE algorithmdoes, the light mutation generates x′

off by performing only one sampling. In orderto do that, we intentionally confuse Gaussian and truncated Gaussian PDF, andwe apply the properties of the algebraic sum of normally distributed variables.In this way, given that {xr , xs, xt} ∼ N (

μ, σ2), and considering the DE/rand/1

scheme, it is possible to sample directly the x′off from the following modified

Gaussian distribution:

x′off ∼ N (

μ, σ2)+ F

(N (μ, σ2

)−N (μ, σ2

))=

= N (μ+ F (μ− μ) , σ2 + F 2

(σ2 + σ2

))= N (

μ, (1 + 2F 2)σ2) (9)

It must be remarked that this operation is not fully equivalent to sampling threesolutions from a truncated Gaussian distribution but it is an approximationwhich neglects the tails of the distribution outside the interval [−1, 1].

Similarly to mutation light, crossover light is a computationally light versionof the exponential crossover used in the cDE schemes, described in [10]. Instead

Robot Trajectory Optimization with cDElight 291

of generating a set of random numbers until the condition rand (0, 1) ≤ Cr is nolonger satisfied, only the length of the section of solutions to be inherited, i.e. howmany genes should be swapped, is randomly generated. Obviously, since only onerandom number is generated, this light exponential crossover has a lighter com-putational overhead with respect to a traditional exponential crossover. Whencrossover is applied, the offspring xoff is initialized to the current elite, and onerandomly selected decision variable is deterministically copied from x′

off to thecorresponding position in xoff . In order to simplify the notation let us considerthe deterministic copy of the first gene and the probabilistic copy of the othergenes as independent events. The probability that m genes, on the top of thefirst one, are copied from x′

off to xoff is Crm. More formally the discrete proba-bility Pr that the crossover length xoverL is equal to m is known as geometricdistribution and is given by:

Pr (xoverL = m) = Crm (10)

where m = 1, 2, . . . , n − 1. In order to extract the number of genes m to becopied, it is enough to apply the inverse formulas and obtain:

xoverL ∼ round (logCr (rand (0, 1))) = round

(log (rand (0, 1))

log (Cr)

)(11)

where the last equality is due to the change of base of a logarithm. In other words,crossover light consists of performing the deterministic copy and subsequentlythe copy of xoverL genes, where xoverL is determined by formula (11). Tofurther control the crossover effect, cDElight makes use of parameter, namelythe proportion of genes undergoing exponential crossover αm, so that, for achosen αm, the crossover rate can be set as Cr = 1

nαm√2, as proposed in [10]. In

this way, it is possible to estimate the proportion of genes to be inherited fromthe provisional offspring independently on problem dimension.

Finally, when xoff is generated, its fitness is computed and compared withthe fitness of the elite (see the function compete() in Alg. 1). On the basis ofthis comparison, a winner solution (solution displaying the best fitness) and aloser solution (solution displaying the worst fitness) are detected. The winnersolution biases the virtual population by affecting the PV values, according tothe following update rules:

μt+1 = μt + 1Np

(winner − loser)

σt+1 =√(σt)

2+ (μt)

2 − (μt+1)2+ 1

Np(winner2 − loser2)

(12)

where Np is a parameter, namely virtual population size. Details for constructingEg. 12 are given in [8]. In addition to the PV values, also the elite is updated,according to a persistent elitism scheme [1].

4 Simulation Results

In order to minimize the fitness function described in Section 2, the followingmemory saving algorithms have been implemented and compared:

292 G. Iacca et al.

− Simplified Intelligence Single Particle Optimization: ISPO [19], with accel-eration A = 1, acceleration power factor P = 10, learning coefficient B = 2,learning factor reduction ratio S = 4, minimum threshold on learning factorE = 1e− 5, and particle learning steps PartLoop = 30;

− Non uniform Simulated Annealing (nuSA) [16] with mutation exponent B =5, temperature reduction ratio α = 0.9, temperature reduction period Lk =3, and initialSolutions= 10;

− compact Differential Evolution (cDE) [9] with rand/1 mutation, exponen-tial crossover and persistent elitism. The cDE algorithm has been run withvirtual population size Np = 300, scale factor F = 0.5, and αm = 0.25;

− compact Differential Evolution light (cDElight), as described above, withNp = 300, scale factor F = 0.5, and αm = 0.25.

We decided to compare the proposed cDElight to ISPO and nuSA because thesetwo methods employ a completely different logic. On the other hand, cDE hasbeen included in the experiments because it represents the original version ofcDElight. For each competing algorithm, the parameter setting suggested inthe original paper was used. It must be remarked that, in order to perform afair comparison, cDE and cDElight employ the same parameter setting. All thealgorithms can be considered memory saving heuristics, as they require a fixedamount of memory slots which does not depend on the problem dimension. Inother words, if one of these algorithms is used to tackle a large scale problem,although the slot length is proportional to the dimension of the problem, theseslots do not increase in number. More specifically, ISPO and nuSA are typicalsingle solution algorithms, requiring only two memory slots, one for the currentbest solution and the other for a trial candidate solution. On the other hand, thecDE schemes are memory-wise slightly more expensive as they require, on thetop of the two slots for single solution algorithms, two extra slots for the virtualpopulation PV . This compromise is made in order to have the advantages of apopulation-based search and a still low memory usage.

0 2000 4000 6000 8000 100000

500

1000

1500

2000

2500

3000

Fitness function call

Fitn

ess

valu

e

6000 7000 8000 9000 10000140

150

160

170

180

190

200

210

cDEISPOcDElightnuSA

Fig. 2. Average performance trend of the algorithms for the space robot application.The inset figure shows a close-up of the fitness trend after 6000 fitness evaluations.

Robot Trajectory Optimization with cDElight 293

For each algorithm, 30 independent runs have been performed. The bud-get of each single run has been fixed equal to 10000 fitness evaluations. InFig. 2, the average performance trend of the algorithms considered in this studyis represented. Table 1 shows the obtained numerical results. Average final fitnessvalues are computed for each algorithm over the 30 runs available. The best re-sult is highlighted in bold face. In order to strengthen the statistical significanceof the results, the Wilcoxon Rank-Sum test has also been applied according tothe description given in [15], where the confidence level has been fixed to 0.95.With respect to Table 1, the results of the Wilcoxon test for cDElight against theother algorithms are displayed. The symbol ”+” indicates that cDElight statis-tically outperforms both ISPO and nuSA, while the symbol ”=” indicates that,on the basis of the Wilcoxon Rank-Sum test, the null hypothesis is accepted, i.e.cDE and cDElight have similar performance. However, as we have seen in theprevious section, the computational overhead of cDElight is definitely lower.

Table 1. Compared results on the space robot application

cDE W ISPO W nuSA W cDElight

1.480e+02 ± 5.55e+00 = 1.959e+02 ± 1.08e+02 + 1.709e+02 ± 7.19e+00 + 1.440e+02 ± 1.08e+00

5 Conclusion

In this paper we modeled the disturbance to the base of a robotic arm mountedon a spacecraft, and we used a measure of the acceleration at the base as costfunction to be minimized. In order to avoid waste of fuel and disturbances to thespacecraft orbit, an online optimization of the robot trajectory is proposed. Aspecifically designed compact algorithm, namely compact Differential Evolutionlight (cDElight), was used to tackle this problem in a computationally efficientmanner. Using cDElight, it was possible to address not only the memory savingnecessities of the hardware typically used in space applications, but also the real-time requirements of the specific application. Compared to the other memorysaving meta-heuristics considered in this study, cDElight proved to obtain thebest results, still having a limited computational overhead and a reduced memoryfootprint. In our view, the proposed cDElight algorithm represents an interestingoptimization solution when a limited hardware (memory and CPU) is availableand a fast response of the system is required.

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