ADMM-based Adaptive Sampling Strategy for Nonholonomic Mobile Robotic Sensor Networks
IIEEE SENSORS JOURNAL, VOL. XX, NO. XX, XXXX 2017 1
ADMM-based Adaptive Sampling Strategy forNonholonomic Mobile Robotic Sensor Networks
Viet-Anh Le,
Student, IEEE , Linh Nguyen,
Member, IEEE , and Truong X. Nghiem,
Member, IEEE
Optimizingsampling metricsAdaptive sampling problem Nonholonomicdynamics andother constraints Nonholonomicmobile sensorTaking mea-surementsGaussian Processtraining andprediction AgentsCentral stationPredicted field and corresponding sampling paths of mobile sensors + Abstract — This paper discusses the adaptive sampling problemin a nonholonomic mobile robotic sensor network for efficientlymonitoring a spatial field. It is proposed to employ Gaussian pro-cess to model a spatial phenomenon and predict it at unmeasuredpositions, which enables the sampling optimization problem to beformulated by the use of the log determinant of a predicted covari-ance matrix at next sampling locations. The control, movementand nonholonomic dynamics constraints of the mobile sensorsare also considered in the adaptive sampling optimization prob-lem. In order to tackle the nonlinearity and nonconvexity of theobjective function in the optimization problem we first exploit thelinearized alternating direction method of multipliers (L-ADMM)method that can effectively simplify the objective function, thoughit is computationally expensive since a nonconvex problem needsto be solved exactly in each iteration. We then propose a novelapproach called the successive convexified ADMM (SC-ADMM)that sequentially convexify the nonlinear dynamic constraints sothat the original optimization problem can be split into convexsubproblems. It is noted that both the L-ADMM algorithm and our SC-ADMM approach can solve the sampling optimizationproblem in either a centralized or a distributed manner. We validated the proposed approaches in 1000 experiments ina synthetic environment with a real-world dataset, where the obtained results suggest that both the L-ADMM and SC-ADMM techniques can provide good accuracy for the monitoring purpose. However, our proposed SC-ADMM approachcomputationally outperforms the L-ADMM counterpart, demonstrating its better practicality.
Index Terms — Adaptive sampling, Gaussian Process, mobile sensor networks, nonholonomic, ADMM.
I. I
NRODUCTION
In applications of monitoring environmental spatial fields,such as exploring ecosystems on land and in ocean, observingchemical concentration and monitoring air pollutants andindoor climates [1], [2], a mobile robotic sensor network(MRSN) incorporated by a machine learning based modelrepresenting the spatial phenomenon have been widely useddue to its universal capability of monitoring the spatial field,exploring the environment and predicting the field at unob-served locations. However, due to the resource constraintsin the network including limited numbers of the sensorsand robots, communication, memory, computation, power,time and motion dynamics, the fundamental yet challengingproblem of how to optimally drive the mobile sensors forefficient monitoring is still not fully solved. In this problem,the sensors are expected to take measurements on the most
V-A. Le and T. Nghiem are with the School of Informatics, Computing,and Cyber Systems, Northern Arizona University, Flagstaff, AZ 86011,USA (e-mail: { vl385,truong.nghiem } @nau.edu).L. Nguyen is with the School of Engineering, Information Technologyand Physical Sciences, Federation University Australia, Churchill, VIC3842, Australia (e-mail: [email protected]).This work has been submitted to the IEEE Sensors Journal forpossible publication. Copyright may be transferred without notice, afterwhich this version may no longer be accessible. informative sampling paths so that the prediction uncertaintyat unobserved positions of interest is minimal. The problem isalso known as adaptive sampling (see [3], [4] for reviews).The adaptive sampling problem has been variously formu-lated and considered in the literature. For instance, Xu et al. in [5] proposed to minimize the Fisher information matrixbased objective function to find the optimal sampling locationsfor a MRSN. The authors in [6] considered a minimizationproblem where the cost function is derived from the averageof the prediction variances over the prespecified target pointsin order to obtain the next sampling locations for the mobilesensors. In [7], the maximum a posterior estimation wasexploited to design an adaptive sampling strategy for a MRSNby minimizing the prediction error variances. Likewise, [8]proposed that each individual sensing agent in a networkdetermines the sampling path by maximizing the predictedvariance at its next location. While Tiwari et al. in [9] consid-ered a decentralized multi-robot system in which each robot isallocated and responsible for monitoring a local sensing zone,the work [10] discusses a method to partition a MRSN intoseveral small groups such that each group is independent offinding its locations. Nevertheless, none of the aforementionedworks have considered the constraints, e.g. dynamics, of aMRSN in the adaptive sampling problem. a r X i v : . [ c s . R O ] J a n ome other works have imposed the movement constrainton a MRSN when designing its sampling scheme. For instance,Marchant et al. , in [11] employed a Bayesian optimizationapproach to derive a sampling strategy, where the traveleddistances of the mobile robots were taken into account tobalance a trade-off between exploration and exploitation. In[12], a problem formulation consisting of an objective functionthat maximizes both the mean value for exploitation and vari-ance estimate for exploration and subjecting to the constraintsfor collision avoidance was given. Some other works usedconcepts from the information theory to formulate the adaptivesampling optimization problem [13], [14]. In our previousstudies [15], [16], the conditional entropy and the posteriorvariances were utilized to formulate an adaptive samplingproblem for a resource-constrained MRSN, while the single-integrator dynamics and the collision avoidance scheme weretaken into consideration. The optimization problems in thoseworks were efficiently resolved by the grid based greedyalgorithm. However, to the best of our knowledge that mobileplatforms with nonholonomic dynamics are popularly used ina MRSN and that the adaptive sampling problem under thenonholonomic dynamics constraint and the movement con-straint (i.e., to avoid physical collision among robots) is not yetconsidered. Therefore, in this work we propose to incorporatethese two constraints into the adaptive sampling optimizationproblem in a MRSN so that it can be practically implementedin environmental monitoring applications. Furthermore, thesingle-integrator model proposed in our previous works [15],[16] is not controllably feasible to drive the nonholonomicmobile sensors through the environment. It is noted thatthe grid based greedy algorithm impractically handles theadaptive sampling problem for a nonholonomic MRSN giventhe movement constraint due to difficulty in defining a searchregion for a mobile sensor at each moving step.In this paper we present two approaches to tackle theadaptive sampling problem in a MRSN subject to the bothnonholonomic dynamics and movement constraints. First, weexploit a state-of-the-art optimization method called the lin-earized alternating direction method of multipliers (L-ADMM)for nonconvex nonsmooth optimization presented in [17] toaddress the problem. Nonetheless, it is discussed that the L-ADMM algorithm is limited by its indicator function that isnondifferentiable, which causes the method to be computation-ally expensive. Thus, we propose a novel approach called thesuccessive convexified ADMM (SC-ADMM) that dexterouslyexploits both the distributed proximal characteristic of theADMM paradigm [18] and the successive convexificationprogramming (SCP) [19]. It is noted that in this work weemploy the non-parametric Gaussian process (GP) model topresent a spatial field as the trained GP model can be usedto effectively predict the field at unobserved positions ofinterest. Thus, the objective function of the sampling problem,which is formed through the conditional entropy [15], can berepresented by the log determinant of a predicted covariancematrix obtained by the trained GP model. Nevertheless, itis demonstrated that the objective function is nonconvex andhighly complicated, which leads to computational intractabilityin the optimization. In both L-ADMM and SC-ADMM, the first-order approximation is employed to simplify the log deter-minant objective function. Additionally, by the use of the SCP[19], the proposed SC-ADMM algorithm can also sequentiallyconvexify the nonlinear dynamic constraints in a small trustregion around a nominal solution. That is, the nonconvexand highly complicated adaptive sampling problem with thenonholonomic and movement constraints can be transformedinto the convex subproblems that can be efficiently addressedby any convex optimization toolboxes.Furthermore, both the L-ADMM algorithm and the pro-posed SC-ADMM approach can be implemented in a eithercentralized or distributed manner. In the centralized sce-nario all the computation is conducted at the central stationwhile in the distributed scenario each individual mobile robottakes charge of its own nonholonomic dynamics, control andmovement constraints. It is noted that exploiting the parallelcomputing can significantly reduce the computation time andthe SC-ADMM algorithm always computationally outperformsthe L-ADMM technique in either the computation paradigm.All the proposed approaches in this work were evaluated inthe synthetic experiments using the realistic dataset, wherethe obtained results highly demonstrate their effectivenessin monitoring environmental phenomena. In particular, theSC-ADMM algorithm presents to be attracted by practicalimplementation in real-time systems.In summary, the main contributions of this paper are three-fold:1) An adaptive sampling optimization problem for aresource-constrained MRSN is derived in which the non-holonomic dynamics of sensing robots are taken intoaccount.2) Two ADMM-based algorithms, L-ADMM presented in[17], and our proposed SC-ADMM, are employed toeffectively address the complex and non-convex adaptivesampling optimization problem in continuous domain.3) Both algorithms allows the computation for solving theoptimization problem to be distributed to all sensingagents.The rest of this paper is organized as follows. Section IIintroduces a nonholonomic MRSN for efficiently monitoringa spatial field, where the adaptive sampling optimizationproblem subject to the nonholonomic and movement con-straints is formulated. Section III then presents how to addressthe adaptive sampling optimization problem by either theL-ADMM algorithm or the proposed SC-ADMM approach.The evaluation of the proposed approaches in the syntheticenvironment is discussed in Section IV before the conclusionsare drawn in Section V. II. N
ONHOLONOMIC M OBILE S ENSOR N ETWORKS FOR E NVIRONMENTAL M ONITORING
In the environmental monitoring applications using aMRSN, it is expected that the robotic sensors adaptivelyconduct sampling at the most informative positions so thattheir collective measurements can be utilized in a data-drivenmodel, such as a GP, for efficiently predicting the environ-mental field at unobserved locations. However, when movingn a sampling path, a mobile sensor is constrained not onlyby its minimum distance to other robots to avoid physicalcollision but also the nonholonomic dynamic configuration,which limits its motions. In order to design an efficientsampling strategy for the mobile sensors, in this section theadaptive sampling optimization problem in the MRSN ismathematically formulated given those constraints.
A. Nonholonomic Mobile Robotic Sensor Networks
Let us consider M networked mobile spatial-field sensors.For the simplicity purpose, we assume that all the sensors areidentical and take measurements of an environmental field atdiscrete time steps while their mobile robots navigate throughthe environment space. Given the nonholonomic constraints,the dynamics of a mobile robot i ∈ V , V = { , . . . , M } , aredescribed by the following kinematic unicycle model, ˙ s x,i = cos( θ i ) v i ˙ s y,i = sin( θ i ) v i ˙ θ i = ω i , (1)where ( s x,i , s y,i ) is the position vector of the robot on a plane, θ i is the heading angle, v i and ω i are the linear and angularvelocities, respectively. And the nonholonomic dynamics (1)are held by the following constraint, ˙ s x,i sin( θ i ) − ˙ s y,i cos( θ i ) = 0 . Let ∆ T denote a sampling time, the discrete model of (1) canbe specified as follows, s x,i,t +1 = s x,i,t + ∆ T cos( θ i,t ) v i,t s y,i,t +1 = s y,i,t + ∆ T sin( θ i,t ) v i,t θ i,t +1 = θ i,t + ∆ T ω i,t . (2)where x i,t = [ s x,i,t , s y,i,t , θ i,t ] T , s i,t = [ s x,i,t , s y,i,t ] T and u i,t = [ v i,t , ω i,t ] T are defined as the state vector, positionvector and control input vector of a robot i at time t . Themobile platform’s dynamics (2) can be aggregated by x i,t +1 = f d ( x i,t , u i,t ) . (3)We denote x t = [ x i,t ] i ∈V ∈ R × M , u t = [ u i,t ] i ∈V ∈ R × M , s t = [ s i,t ] i ∈V ∈ R × M and s t = [ s k ] k =0 ,...,t ∈ R × M ( t +1) ,while assume that a MRSN is operated in a compact 2DEuclidean space of interest Q ⊂ R , i.e., s i,t ∈ Q forall i ∈ V . Moreover, due to the limitation on the robotactuation, the control input is bounded by u i,t ∈ U i where U i := { u ∈ R | u min ≤ u ≤ u max } .At time step t , a noisy measurement of the environmentalfield of interest taken by the mobile sensor i at the location s i,t is modeled as y i,t = h ( s i,t ) + w i , (4)where h : R → R is a latent function of the spatial field at s i,t while w i is a Gaussian zero-mean independent and identicallydistributed noise. All the sensor measurements at time step t are denoted by y t = [ y i,t ] i ∈V ∈ R M and the collectivemeasurements from time step 0 to time step t are denoted by y t = [ y k ] k =0 ,...,t ∈ R M ( t +1) .The measurements of the spatial phenomenon collectedafter each time step are limited as compared with the infinitenumber of locations in the space of interest. Thus, in themonitoring applications, it is expected to employ the collective (cid:15) Ω i,t Fig. 1 : A constrained movement region Ω i,t (shaded area) ofa mobile sensor i at time t (red cross). t + Ht t + 2 t + 1 t + H − ... Currentmeasurementtime step Nextmeasurementtime stepTime steps for control only,no measurement is taken
Fig. 2 : The timeline for our adaptive sampling strategy:assume current measurements have been taken at time step t , a sequence of control signals at time step t , t + 1 , . . . , t + H − are computed to drive the robots to new positionsat time step t + H where new measurements are collected.measurements to predict the spatial field at unmeasured posi-tions. Statistically, this paradigm can be obtained by the use ofthe data-driven GP model G h,t specified by a mean function m ( s i,t ; θ t ) and a covariance function cov ( s i,t , s j,t ; θ t ) . Thehyperparameters θ t can be trained based on the data set D t = ( s t , y t ) through optimization such as maximizingthe likelihood θ (cid:63)t = argmax θ t Pr ( y t | s t , θ t ) [20].Another constraint considered in the MRSN model in thiswork is the physical collision avoidance among the mobilesensors when they navigate through the environment. To math-ematically formulate this constraint, we employ the Voronoitheory [21]. The Voronoi partition of the mobile agent i attime t is defined by V i,t := { q ∈ Q | (cid:107) q − s i,t (cid:107) ≤ (cid:107) q − s j,t (cid:107) , ∀ j (cid:54) = i } . Moreover, due to geometrical shapes of the robots and themodelling errors caused by the dynamics approximations,we consider the allowable movement region Ω i,t , which isconstituted by shrinking the Voronoi cell V i,t by a small safetythreshold (cid:15) > , as demonstrated in Fig. 1. B. Adaptive Sampling Problem for Nonholonomic MRSN
It is defined that at the time step t , the mobile sensorstake measurements of the environmental phenomenon at theircurrent locations. The goal is to find optimal sampling lo-cations of the network at the time t + H and a sequenceof control inputs from t to t + H − using discretizedpredictive models with control horizon of H > , as can beillustrated by the timeline in Fig. 2. Therefore, measurementsare taken periodically every H time steps. Due to the resourceconstraints in the network including limited numbers of thesensors and robots, communication, memory, computation,power, time and motion dynamics, it is required that theampling locations lie on the most informative sampling paths.To address this problem, we exploit the prediction capabilityof the GP model G h,t given the collective observations y t topredict the environmental field at unmeasured locations. Thepredictions can be utilized to optimize the sampling positions.Statistically, the mean vector and covariance matrix of theposterior distribution at the possible next sampling locations s t + H are given by µ ˆy t + H | y t = m ( s t + H ) + Σ t + H, t Σ − t (cid:0) y t − m ( s t ) (cid:1) Σ ˆy t + H | y t = Σ t + H,t + H − Σ t + H, t Σ − t Σ Tt + H, t , where m ( s t ) and m ( s t + H ) are the mean vectors of the latentvariables at s t and s t + H , respectively. The M × M matrix Σ t + H,t + H is the covariance matrix at s t + H , Σ t + H, t is thecross-covariance matrix between ˆy t + H and y t , and Σ t isthe covariance matrix of y t . For the simplicity purpose, let Σ ( s t + H ) denote Σ ˆy t + H | y t henceforth.The main objective in the MRSN for monitoring an envi-ronmental phenomenon is minimize the prediction uncertaintyat unmeasured positions given the collective measurements. In[15], [22] it was proved that minimizing the prediction uncer-tainty at unobserved locations is equivalent to maximizing theconditional entropy of the spatial field at the next samplinglocations. In other words, in the context of the GP model,the optimization criterion for finding the most informativesampling paths can be calculated through the log determinantof the covariance matrix. The adaptive sampling optimizationproblem for a general MRSN in a spatial monitoring applica-tion can be formulated as follows, s ∗ t + H = argmax s t + H log det Σ ( s t + H ) . (5)Nonetheless, under the nonholonomic dynamics, movementand control constraints as presented in Section II-A, theoptimization problem (5) can be rewritten in a constrainedoptimization problem format by minimize { s t + j +1 , u t + j } j ∈I t f ( s t + H )+ M (cid:88) i =1 f i ( { u i,t + j , s i,t + j +1 } j ∈I t ) subject to x i,t + j +1 = f d ( x i,t + j , u i,t + j ) , u i,t + j ∈ U i , s i,t + j +1 ∈ Ω i,t , ∀ j ∈ I t , ∀ i ∈ V , (6)where f ( s t + H ) = − log det (cid:0) Σ ( s t + H ) (cid:1) is the sampling met-ric, f i ( { u i,t + j , s i,t + j +1 } j ∈I t ) is a convex function of the con-trol cost for each robot with I t = { , . . . , H − } . For example,in this work f i = (cid:80) H − j =0 (cid:107) u i,t + j (cid:107) Q i + (cid:107) u i,t + j − u i,t + j − (cid:107) R i that controls both velocity and acceleration of the robot i forsmoother movement. It is noted that the notation (cid:107) ν (cid:107) (cid:3) = ν T (cid:3) ν denotes the (cid:3) -norm of vector ν with respect to thepositive semidefinite matrix (cid:3) .It can be seen that (6) is a highly nonconvex and complexoptimization problem where the nonconvexity is presented inboth the objective and constraint functions. That is, solvingthe problem by the grid-based methods, e.g., [15], may beimpractical. Moreover, complexity of the objective functioncauses (6) to be computationally intractable. For instance, thepopular nonlinear programming solver Ipopt [23] failed to solve it. Therefore, in this work we employ another algorithmin the optimization domain and propose a new approachto effectively address this highly nonconvex and complexproblem. III. A
DAPTIVE S AMPLING STRATEGY USING D ISTRIBUTED O PTIMIZATION ALGORITHMS
To solve the adaptive sampling problem (6) by distributedoptimization algorithms, we first re-write the problem in asplitting form. Define z = [ s Ti,t + H ] Ti ∈V ∈ R M and w i =[ x Ti , u Ti ] T ∈ R H where x i = [ x Ti,t + j +1 ] Tj ∈I t ∈ R H and u i = [ u Ti,t + j ] Tj ∈I t ∈ R H are the vectors collecting states andcontrol variables of the robot i over a horizon. The problem(6) can be represented in the following splitting form, minimize { w i } i ∈V , z f ( z ) + M (cid:88) i =1 f i ( w i ) (7a)subject to w i ∈ C i,t , ∀ i ∈ V (7b) E i z = F i w i , ∀ i ∈ V . (7c)where E i and F i are transformation matrices that extract s i,t + H from z and w i , respectively. The nonconvex set C i,t represents all the constraints in (6) and can be defined by aset of equality constraints g i,j,t ( w i ) = 0 , ∀ j ∈ J i,eq,t andinequality constraints h i,j,t ( w i ) ≤ , ∀ j ∈ J i,ieq,t where J i,eq,t and J i,ieq,t are the sets of the equality and inequalityconstraint indices, respectively. Based on the definitions ofall the constraints in (6), it is noted that for all i ∈ V , g i,j,t ( w i ) , ∀ j ∈ J i,eq,t and h i,j,t ( w i ) , ∀ j ∈ J i,ieq,t arecontinuously differentiable.This section presents two approaches based on distributedADMM framework to tackle the sharing problem (7), andtherefore, the adaptive sampling optimization problem (6).The first method is derived from the work [17] where thelinearized ADMM (L-ADMM) algorithm is proposed fornonconvex nonsmooth optimization problems. Though the L-ADMM technique can find a solution for the problem (6), it isstill computationally expensive since a nonconvex optimizationproblem is involved in each algorithmic iteration. Therefore,in the second method, we propose a novel algorithm calledsuccessive convexified ADMM (SC-ADMM) that dexterouslyexploits both the distributed proximal characteristic of theADMM paradigm [18] and the successive convexificationprogramming [19] to avoid solving non-convex optimizationproblems. The proposed SC-ADMM can address the problem(6) more efficiently than the L-ADMM algorithm in terms ofcomputation time, as demonstrated in Section IV. A. L-ADMM for Adaptive Sampling Problem
In L-ADMM framework, we minimize the augmented La-grangian function of the problem (7) that is defined by L ( z , { w i } i ∈V , µ ) = f ( z ) + M (cid:88) i =1 (cid:16) f i ( w i ) + I C i,t ( w i )+ ( E i µ ) T ( E i z − F i w i ) + ρ (cid:107) E i z − F i w i (cid:107) (cid:17) , (8)here µ ∈ R M is a vector of the associated dual variables and ρ is a regularization parameter, while I C i,t ( w i ) is an indicatorfunction of the set C i,t , for all i ∈ V .Given the nonconvex augmented Lagrangian function (8),the classical ADMM algorithm [24] can solve the problem (7)by performing the following steps. w ( k +1) i = argmin w i ∈ C i,t f i ( w i )+ ρ (cid:13)(cid:13)(cid:13)(cid:13) F i w i − E i ( z ( k ) + µ ( k ) ρ ) (cid:13)(cid:13)(cid:13)(cid:13) , ∀ i ∈ V (9a) z ( k +1) = argmin z f ( z ) + ρ (cid:13)(cid:13)(cid:13)(cid:13) z − (cid:18) v ( k +1) − µ ( k ) ρ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) (9b) µ ( k +1) = µ ( k ) + ρ (cid:16) z ( k +1) − v ( k +1) (cid:17) , (9c)where v = [( F i w i ) T ] Ti ∈V ∈ R M .It can be seen that solving the optimization problem (9b),which involves the log determinant of a covariance matrix inthe function f ( · ) , is computationally intractable. To addressthis issue, the authors of the work [17] proposed in their L-ADMM method to employ the first-order approximation of f to find z . The steps to solving the problem (7) by the L-ADMM technique are presented as follows, w ( k +1) i = argmin w i ∈ C i,t f i ( w i )+ ρ (cid:13)(cid:13)(cid:13)(cid:13) F i w i − E i ( z ( k ) + µ ( k ) ρ ) (cid:13)(cid:13)(cid:13)(cid:13) , ∀ i ∈ V (10a) z ( k +1) = v ( k +1) − ρ + L ) (cid:16) ∇ f ( v ( k +1) ) + µ ( k ) (cid:17) (10b) µ ( k +1) = µ ( k ) + ρ (cid:16) z ( k +1) − v ( k +1) (cid:17) , (10c)where L is a positive constant. To run the L-ADMM algorithmin a distributed manner, at each iteration, each mobile robotis required to solve the optimization problem (10a) and thensend its correspondingly obtained sampling location s i,t + H tothe central station where the consensus z ( k +1) is computed.The dual variables µ ( k +1) are then also updated. The algorithmiterations are repeated until the convergence or the maximumnumber of the iterations is reached. It is noticed that [17]provides some criteria to choose the parameters ρ and L sothat the L-ADMM algorithm can converge to the Karush-Kuhn-Tucker points of the original nonconvex and nonsmoothproblem (7). The distributed L-ADMM method is summarizedin Algorithm 1. B. Successive Convexified ADMM Algorithm
The main disadvantage of the L-ADMM algorithm [17] isthat the indicator function I C i,t ( w i ) in (8) is nondifferentiableand hence can not be linearized, leading to the nonconvexoptimization problem (10a) at each iteration. Therefore, wepropose a novel SC-ADMM method based on the sequentialconvexification programming [19] to convexify the non-convexproblem (10a) by linearizing the nonholonomic dynamics ina small trust region around a nominal solution. In otherwords, by the use of the first-order approximations, both thenon-linear dynamic constraints and the log determinant ofthe predicted covariance matrix are linearized. The proposedapproach can significantly reduces the computation time in Algorithm 1
Distributed L-ADMM algorithm
Require: z (0) , µ (0) , (cid:15) res , k max , ρ , L for k = 0 , . . . , k max do Central station sends the query point E i ( z ( k ) + µ ( k ) /ρ ) to agent i , ∀ i ∈ V Agent i computes w ( k +1) i by (10a), ∀ i ∈ V , in parallel Agent i sends v ( k ) i = F i w ( k ) i to the central station Central station collects v ( k ) i and forms v ( k ) = [ v Ti ] Ti ∈V Central station computes z ( k +1) by (10b) Central station updates µ ( k +1) by (10c) if (cid:13)(cid:13) z ( k +1) − s ( k +1) (cid:13)(cid:13) < (cid:15) res then Stop and return w ( k +1) return w ( k max ) Algorithm 2
Distributed SC-ADMM algorithm
Require: z (0) , µ (0) , (cid:15) res , k max , ρ , L , β fail , β succ , (cid:15) , (cid:15) , r min , r max . for k = 0 , . . . , k max do Central station sends the query point E i ( z ( k ) + µ ( k ) /ρ ) to agent i , ∀ i ∈ V Agent i computes w ( k +1) i by (13a) and (14), ∀ i ∈ V ,in parallel Agent i sends v ( k ) i = F i w ( k ) i to the central station Agent i computes δ ( k +1) i then adjusts the trust region r i based on Remark 1 Central station collects v ( k ) i and forms v ( k ) = [ v i ] Ti ∈V Central station computes z ( k +1) by (13b) Central station updates µ ( k +1) by (13c) if (cid:13)(cid:13) z ( k +1) − v ( k +1) (cid:13)(cid:13) < (cid:15) res then Stop and return w ( k +1) return w ( k max ) solving the nonconvex optimization problem (10a) as com-pared with the L-ADMM algorithm.Specifically, instead of considering the indicator functionof the constraint sets, we encode the inequality and equalityconstraints by the exact penalty functions [19] leading to thenonconvex penalty problem corresponding to (7) as follows, minimize { w i } i ∈V , z f ( z ) + M (cid:88) i =1 (cid:16) f i ( w i )+ (cid:88) j ∈J i,eq λ i,j | g i,j ( w i ) | + (cid:88) j ∈J i,ieq τ i,j max (0 , h i,j ( w i )) (cid:17) subject to E i z − F i w i = , ∀ i ∈ V , (11)where for each mobile sensor i , λ i,j , ∀ j ∈ J i,eq and τ i,j , ∀ j ∈J i,ieq are the large penalty weights.The augmented Lagrangian function of the problem (11)can be defined by: L ( z , { w i } i ∈V , µ ) = f ( z ) + M (cid:88) i =1 (cid:16) f i ( w i )+ (cid:88) j ∈J i,eq λ i,j | g i,j ( w i ) | + (cid:88) j ∈J i,ieq τ i,j max (0 , h i,j ( w i ))+ ( E i µ ) T ( E i z − F i w i ) + ρ (cid:107) E i z − F i w i (cid:107) (cid:17) . (12)In the proposed algorithm, we employ the first-order approx-mation of the non-linear terms in the w-minimization steps toform the convex subproblems. It is noted that the linearizationis computed by the | · | and max(0 , · ) functions, leading to thefollowing iterative computation steps, w ( k +1) i = w ( k ) i + ∆ ( k +1) i , ∀ i ∈ V (13a) z ( k +1) = v ( k +1) − ρ + L ) (cid:16) ∇ f ( v ( k +1) ) + µ ( k ) (cid:17) (13b) µ ( k +1) = µ ( k ) + ρ (cid:16) z ( k +1) − v ( k +1) (cid:17) (13c) ∆ ( k +1) i in (13a) can be determined by solving the followingconvex optimization subproblem. ∆ ( k +1) i = argmin ∆ i f i ( w ( k ) i + ∆ i )+ (cid:88) j ∈J i,eq λ i,j (cid:12)(cid:12)(cid:12) g i,j ( w ( k ) i ) + ∇ g i,j ( w ( k ) i ) T ∆ i (cid:12)(cid:12)(cid:12) + (cid:88) j ∈J i,ieq τ i,j max (cid:16) , h i,j ( w ( k ) i ) + ∇ h i,j ( w ( k ) i ) T ∆ i (cid:17) + (cid:13)(cid:13)(cid:13) F i ( w ( k ) i + ∆ i ) − E i (cid:16) z ( k ) + µ ( k ) /ρ (cid:17)(cid:13)(cid:13)(cid:13) subject to (cid:107) ∆ i (cid:107) ≤ r i , (14)where r i is a small trust-region radius that specifies the localneighborhood around the nominal solution in which the convexoptimization subproblem (14) is valid. This radius can beadapted by comparing the actual cost value and the predictedcost value, which are respectively given by J i ( w ( k +1) i ) = f i ( w ( k +1) i ) + (cid:88) j ∈J i,eq λ i,j (cid:12)(cid:12)(cid:12) g i,j ( w ( k +1) i ) (cid:12)(cid:12)(cid:12) + (cid:88) j ∈J i,ieq τ i,j max (cid:16) , h i,j ( w ( k +1) i ) (cid:17) and ˜ J i ( ∆ ( k +1) i ) = f i ( w ( k ) i + ∆ ( k +1) i )+ (cid:88) j ∈J i,eq λ i,j (cid:12)(cid:12)(cid:12) g i,j ( w ( k ) i ) + ∇ g i,j ( w ( k ) i ) T ∆ ( k +1) i (cid:12)(cid:12)(cid:12) + (cid:88) j ∈J i,ieq τ i,j max (cid:16) , h i,j ( w ( k ) i ) + ∇ h i,j ( w ( k ) i ) T ∆ ( k +1) i (cid:17) . Remark 1: (The adjustment rule for the trust-region ra-dius ) We compare the difference between the actual cost andthe predicted cost, i.e., δ ( k +1) i = J i ( w ( k +1) i ) − ˜ J i ( ∆ ( k +1) i ) ,with some predefined thresholds < (cid:15) < (cid:15) < (cid:15) < + ∞ toadjust the trust region r i according to the following rule. • If δ ( k +1) i > (cid:15) , the approximation is considered highly in-accurate, then the solution is rejected and r i is contractedby a predefined factor β fail < . • If (cid:15) > δ ( k +1) i > (cid:15) , the approximation is consideredinaccurate but acceptable, then the solution is accepted.However, r i is still contracted by β fail . • If (cid:15) > δ ( k +1) i > (cid:15) , the approximation is sufficientlyaccurate, then the solution is acceptted and r i is main-tained. • If δ ( k +1) i < (cid:15) , the approximation is accurate, then thesolution is accepted and r i is enlarged by a predefinedfactor β succ > . This rule is an adapted version of the original adjustment rule in [19].
X (m) Y ( m ) Fig. 3 : A generated ground truth of the indoor temperaturefield. • r i is constrained by a upper bound r max and a lower bound r min .The SC-ADMM approach can be computed in a distributedfashion where each individual robot calculates its own non-holonomic dynamics, control and movement constraints beforesending the results to the central station. The approximatedlinearization of both the objective and constraint functionsand the parallel computing allows the SC-ADMM method tosignificantly reduce its computation time. In other words, theproposed algorithm is highly practically scalable, particularlyin a large-scale network.The distributed SC-ADMM approach for solving the non-holonomic adaptive sampling optimization problem (6) issummarized in Algorithm 2, where the adjustment rule isapplied for computing the trust-region radius. Although theconvergence analysis of SC-ADMM algorithm have not beenderived in literature, the algorithm can be converged in prac-tice. Remark 2:
In the algorithm design, we assume that the con-vex constraints are also linearized to facilitate the formulationand design. However, in practice, they are handled explicitly(in their original form) by the convex optimization solver.
IV. S
IMULATION R ESULTS AND D ISCUSSION
To validate effectiveness of the proposed approach, weconducted several experiments in a synthetic environmentusing the Intel Berkeley Research Lab’s temperature data[25]. The synthetic experiments were executed on a DELLcomputer with a . Intel Core i5 CPU and
RAM,where the Python programming language was employed asa platform to implement the algorithms. For the GP model,it was proposed to use the constant mean and the squaredexponential covariance functions. More importantly, to presentan environmental field in the experiments, a ground truthGP model of an indoor spatial temperature phenomenon wastrained based on the 54 temperature measurements gatheredby the 54 sensors in the Intel Berkeley Research Lab. Itis noted that the ground truth model, as demonstrated inFig. 3, was utilized for the two purposes: (1) generatingsensor measurements of the temperature field and (2) verifyingpredictions.In all the synthetic experiments, 5 networked mobile sensorswere expected to efficiently monitor the indoor temperature
10 20 30 400102030 00 . . . . (a) Time step 5 . . . . (b) Time step 10 . . . . (c) Time step 15 . . . . (d) Time step 5 . . . . (e) Time step 10 . . . . (f) Time step 15 Fig. 4 : An example of the predicted variances in the entire environment and the sampling trajectories of the mobile sensorsobtained by the SC-ADMM algorithm ((a), (b) and (c)) and L-ADMM algorithm ((d), (e) and (f)) at some specific time instants.The starting locations of the mobile robots are shown in the white circles.field in an environment of size
40 m -by-
30 m . The linearvelocity for each mobile sensor was bounded between v min = − ( m / s ) and v max = 2 ( m / s ) while its angular velocity wasbounded between ω min = − π ( rad / s ) and ω max = π ( rad / s ).The parameters in the control cost functions were chosenby Q i = diag([0 . , . and R i = diag([1 . , . .Moreover the sampling time and the length of the controlhorizon were set to ∆ T = 0 . and H = 10 , respectively.It is noticed that the initial locations of the mobile sensorswere arbitrarily chosen at each experiment. More importantly,in order to address the adaptive sampling problem (6), boththe L-ADMM and SC-ADMM algorithms were implemented.While some common parameters between the two algorithmswere set to ρ = 0 . , L = 0 . , (cid:15) res = 10 − and k max = 100 ,the specific parameters for the SC-ADMM approach were setto λ i,j = τ i,j = 10 , β fail = 0 . , β succ = 2 . , (cid:15) = 1 , (cid:15) = 10 , (cid:15) = 10 , r min = 10 − and r max = 1 . , respectively.It is understood that at the beginning of the monitoringprocess, all the mobile sensors have no information aboutthe spatial temperature phenomenon. That is, they can startat any random locations. To demonstrate that our proposedalgorithm is always valid, we conducted the 1000 experimentsgiven the arbitrary starting positions of the robotic sensors.Overall, the obtained results show that the network of the5 mobile sensors efficiently monitored the temperature field.In other words, the proposed approach drove the roboticsensors on the most informative sampling paths while theprediction uncertainty of the spatial phenomenon in the entireenvironment was significantly reduced after every samplingstep where the sensors took measurements. For instance, thesampling trajectories of the mobile sensors at one randomexperiment example are illustrated in Fig. 4. These samplingpaths were obtained by both the L-ADMM algorithm and our proposed SC-ADMM technique. It is noted that the trajectoriesare plotted on the background with the predicted variancesof the spatial temperature field in the entire environment inorder to highlight how considerably the robot navigationsreduced the prediction uncertainty of the temperature. Giventhe color bars, it can be seen that at each sampling step eachrobotic sensor tended to move to the location with the highestprediction uncertainty in its allowable movement region so thatthe predicted variance was minimized.Furthermore, the predicted temperature fields in the wholeenvironment at the 3 particular time steps of 5, 10 and 15,respectively, in the demonstrated example are depicted in Fig.5. Given the sampling trajectories shown in Fig. 4, over timeit can be seen that the corresponding predicted fields in Fig.5 were gradually approaching to the ground truth presented inFig. 3. Though at the time step of 5 the predicted temperatureobtained by the L-ADMM method is slightly better than thatobtained by our proposed SC-ADMM algorithm as comparedwith the ground truth, at the time steps of 10 and 15, differ-entiation of the predicted temperature results obtained by boththe techniques is hardly seen in Fig. 5. Therefore, we furthersummarized the results at every sampling step for the 3 otherdifferent validation metrics including the average logarithm ofpredicted variances (ALPVs), the root mean squared errors(RMSEs) and the maximum absolute errors (MAEs). Theseresults were computed against the ground truth in Fig. 3 andare demonstrated in Fig. 6. It can be clearly seen that fromthe time step of 8 onwards the proposed SC-ADMM algorithmoutperforms the L-ADMM method though at the time step of15 this outperformance is trivial. It can be explained by thefact that at the time step of 15 the two sampling networksreached to a large number of 80 measurements, leading to thepredicted temperature fields obtained by both the algorithms as
10 20 30 400102030 16171819 (a) Time step 5 (b) Time step 10 (c) Time step 15 (d) Time step 5 (e) Time step 10 (f) Time step 15
Fig. 5 : An example of the predicted means in the entire environment obtained by the SC-ADMM algorithm ((a), (b) and (c))and L-ADMM algorithm ((d), (e) and (f)) at some specific time instants.demonstrated in Figures 5c and 5f, respectively, to be highlycomparable with the ground truth.We now summarize all the results for the 3 different vali-dation metrics obtained by both the algorithms in all the 1000experiments and present them in the box plot format in Fig. 7.Similar to the aforementioned discussion for a single experi-ment example, from Fig. 7 overall difference in the predictionresults obtained by the L-ADMM algorithm and our proposedSC-ADMM approach is trivial although it is noticed that thepredicted spatial field is highly comparable to the ground truth.Thus, in practice, for the purpose of prediction accuracy, it issuggested to solve the adaptive sampling optimization problemin a MRSN for environmental monitoring applications bythe either L-ADMM or SC-ADMM techniques. Nonetheless,for practicality, we investigated computational complexity ofboth the algorithms by summarizing their computing time inall the 1000 experiments. The summarized computation timeis shown by the boxplots in Fig. 8, where we consideredboth the scenarios of using the distributed and centralizedoptimization. It is noted that in the centralized scenario all thecomputation was conducted at the central station while in thedistributed scenario each individual robot calculated its ownnonholonomic dynamics, control and movement constraintsbefore sending the results to the central station, as presentedin Section III. Fig. 8 obviously shows that the distributedparadigm is computationally better than the centralized modeland that the proposed SC-ADMM algorithm runs much fasterthan the L-ADMM counterpart given the same system set-up.That is, the highly computational efficiency of the SC-ADMMapproach leads to preferences in using the proposed algorithmto practical implementation in real-time systems.
V. C
ONCLUSIONS
The paper has presented a discussion about the adaptivesampling in a nonholonomic MRSN for effectively monitoringan environmental spatial field. To the best of our knowledge, itis the first time the paper has taken all the control, movementand nonholonomic dynamic constraints of the mobile sensorsinto consideration, which makes the sampling optimizationproblem highly nonlinear, nonconvex and complex. We havefirst discussed how to solve the optimization problem bythe L-ADMM algorithm and discovered the computationalcomplexity in the algorithm due to its indicator functiondefinition. We have then proposed a novel but efficient SC-ADMM approach that exploits the first-order approximationto tractably handle non-convexity and high complexity ofthe objective function and the successive convexification pro-gramming to sequentially convexify the nonlinear dynamicconstraints. Thus, the proposed SC-ADMM approach cancomputationally effectively and accurately address the adap-tive sampling optimization problem as compared with the L-ADMM counterpart. We have implemented both the L-ADMMand SC-ADMM algorithms in the 1000 experiments in asynthetic environment using the realistic indoor temperaturedataset. Though the obtained results demonstrate that boththe algorithms can provide accurate prediction of the spatialtemperature field at the unobserved locations, the SC-ADMMmethod run much faster, which promises its potential practi-cality. R EFERENCES [1] T. Arampatzis, J. Lygeros, and S. Manesis, “A survey of applicationsof wireless sensors and wireless sensor networks,” in
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