Reconfigurable-Intelligent-Surface-Assisted Indoor Millimeter-Wave Communications for Mobile Robots

Reconfigurable intelligent surfaces (RISs) and millimeter-wave (mmWave) communications have been considered for providing wireless connectivity to mobile robots used in industrial plants and other indoor environments. However, the existing works have not sufficiently studied how the number and deployment locations of RISs should be optimized for serving a mobile robot. In this article, we study RIS-assisted mmWave communications for a robot moving around fixed obstacles in an indoor industrial environment. For a fixed total number of reflecting elements, we formulate an optimization problem to minimize the transmission energy consumption of the access point (AP) while ensuring the robot’s received signal-to-noise ratio (SNR) above a threshold throughout its journey by jointly optimizing the number, positions, and phase shifts of RISs and the beamforming vectors of the AP. To solve the formulated nonconvex optimization problem, we devise an iterative algorithm that decomposes it into two subproblems (i.e., optimizing the phase shifts of RISs and the beamforming vector of the AP, and optimizing the number and locations of RISs) and solves them alternately. Simulation results show that the proposed algorithm converges fast and can obtain the best number and locations of RISs that lead to a transmission energy consumption of the AP much lower than the benchmark schemes.

Reconfigurable-Intelligent-Surface-Assisted Indoor Millimeter-Wave Communications for Mobile Robots Zhiyu Liu , Yang Liu, Member, IEEE, and Xiaoli Chu , Senior Member, IEEE Abstract-Reconfigurable intelligent surfaces (RISs) and millimeter-wave (mmWave) communications have been considered for providing wireless connectivity to mobile robots used in industrial plants and other indoor environments.However, the existing works have not sufficiently studied how the number and deployment locations of RISs should be optimized for serving a mobile robot.In this article, we study RIS-assisted mmWave communications for a robot moving around fixed obstacles in an indoor industrial environment.For a fixed total number of reflecting elements, we formulate an optimization problem to minimize the transmission energy consumption of the access point (AP) while ensuring the robot's received signal-to-noise ratio (SNR) above a threshold throughout its journey by jointly optimizing the number, positions, and phase shifts of RISs and the beamforming vectors of the AP.To solve the formulated nonconvex optimization problem, we devise an iterative algorithm that decomposes it into two subproblems (i.e., optimizing the phase shifts of RISs and the beamforming vector of the AP, and optimizing the number and locations of RISs) and solves them alternately.Simulation results show that the proposed algorithm converges fast and can obtain the best number and locations of RISs that lead to a transmission energy consumption of the AP much lower than the benchmark schemes.

I. INTRODUCTION
A S THE Industrial Internet of Things (IIoT) continues to expand, wirelessly connected mobile robots are expected to be in increasingly widespread use, carrying out various tasks [1].Due to the presence of machines, furniture, and other things in an industrial environment, the Line-of-Sight (LoS) link between an access point (AP) and a mobile robot is likely to be blocked while the robot is moving around them.This makes it difficult to employ millimeter-wave (mmWave) communications for mobile robots because non-LoS (NLoS) mmWave links suffer from severe penetration losses [2], [3].The authors are with the Department of Electronic and Electrical Engineering, The University of Sheffield, S1 4ET Sheffield, U.K. (e-mail: zliu160@sheffield.ac.uk; liuyang91e@gmail.com;x.chu@sheffield.ac.uk).
Digital Object Identifier 10.1109/JIOT.2023.3289772 To increase the possibility of a receiver seeing a LoS link and to enable continuous connectivity, reconfigurable intelligent surfaces (RISs) have been employed to reflect signals from a transmitter to a receiver [4].An RIS is a planar surface composed of reconfigurable passive reflecting elements, each being capable of controlling the amplitude and phase shift of their reflected signal [5], and has attracted a lot of interest from researchers.Wu and Zhang [6] minimized the transmit power of the AP by jointly optimizing the phase shifts of elements on a single RIS and the beamforming vectors at the AP while meeting the downlink signal-to-interference-plusnoise ratio (SINR) requirement.Their results demonstrated that the transmit power at the AP can be scaled down by 1/N 2 , where N denotes the number of elements on the RIS.Du et al. [7] maximized the downlink received signal-to-noise ratio (SNR) in a single RIS-aided multiple-input-single-output (MISO) mmWave communication system by optimizing the phase shifts of the RIS and found that the RIS achieved a lower outage probability than amplify-and-forward (AF) relaying.Lin et al. [8] maximized the downlink received signal power in an RIS-aided MISO system by jointly optimizing the active beamforming at the AP and passive beamforming at the RIS.We note that only a single RIS was considered in [6], [7], and [8] and their results cannot be readily extended to cases of multiple RISs.
Depoying multiple RISs opens the door to more design variables, such as the number and deployment positions of RISs.In [9], the simulation results showed that a distributed deployment of multiple RISs across the service area of a base station (BS) can improve the mean rate of all the users as compared with the full-duplex (FD) relaying; and when centralizing the fixed total number of elements on fewer RISs, the mean user rate will increase but the fairness among spatially distributed users in terms of their achievable rates will degrade.Zhang and Zhang [10] found that for a fixed total number of elements, deploying one RIS achieves a larger capacity region for two single-antenna users of different data rate requirements than deploying two RISs.In [11], the results showed that deploying two RISs can increase the downlink received signal power by a factor of O(N 4 ), while that of deploying a single RIS is O(N 2 ), where N is the total number of reflecting elements.Zhang et al. [12] showed that with a fixed total number of elements, the distributed deployment of multiple RISs (at the same height on a vertical 2-D plane) can achieve a higher downlink ergodic capacity than deploying a single RIS under outdated channel state information in a single-inputsingle-output (SISO) system.Pala et al. [13] maximized the downlink sum rate of all users in a multi-RIS-aided MISO system by jointly optimizing the beamforming vector at the BS, the phase shifts at the RISs and the block-length for ultrareliable low-latency communication (URLLC) subject to a maximum allowed packet error probability.Zhao et al. [14] jointly optimized the locations of UAVs each carrying an RIS, the phase shifts of the RISs and the BS transmit power to minimize the number of RISs required for meeting a downlink received SNR requirement.However, the above studies on the deployment of multiple RISs all assumed that the user devices were stationary.Their results cannot be readily extended to the cases of moving users, because the LoS/NLoS status of the links from the RISs and BSs to the users may change as the users move around obstacles.Moreover, in [10], [11], and [12], only two RISs were considered, while the locations of them were not optimized.
Recently, RIS-assisted communications for moving objects have been studied.Tatino et al. [15] maximized the uplink average data rate of a robot moving from an initial position to a target position within a limited time by jointly optimizing the phase shifts of an RIS deployed nearby, the trajectory and the beamforming vector of the robot.In [16], the trajectory and speed of a UAV (carrying an RIS) and the phase shifts of the RIS were jointly optimized to maximize the average downlink rate of a vehicle that moves along a road in an urban area.Agrawal et al. [17] analyzed the coverage probability of a single-antenna vehicle served by a UAV carried BS and a single RIS deployed on the exterior wall of a building while the vehicle drives along a straight motorway for different positions of the UAV.In [18], for a vehicular network assisted by a UAV carried RIS, Yu et al. maximized the minimum downlink throughput among all the vehicles served by a BS by jointly optimizing the phase shifts of the RIS, transmission scheduling, transmission power of the BS, and the UAV trajectory.In [15], [16], [17], and [18], only a single RIS was considered.In [19], the vehicles moving along a straight motorway were served by a BS carried by a UAV and multiple RISs deployed on a vertical 2-D plane that is parallel to motorway, where each RIS served a different vehicle.The authors minimized the sum outage probability of all the vehicles by optimizing the transmit power of the BS, but the locations or number of RISs were not optimized.We note that in the above works, the LoS/NLoS status of each link was statistically characterized by an independent LoS probability, but the impact of blockage(s) on the wireless links seen by a moving user was not explicitly modeled.
In this article, we study RIS-assisted mmWave communications for a mobile robot in an indoor industrial environment with large obstacles.Aiming to minimize the transmission energy consumption of the AP, we investigate whether a fixed number of passive elements should be distributed to a large number of small RISs or centralized to a small number of large RISs, while maintaining reliable downlink communications for a robot moving around the obstacles along a known trajectory within a given time limit.The contributions of this article are summarized as follows.
1) We present a system model for the RIS-assisted mmWave downlink transmission from an AP on the ceiling to a robot moving around obstacles at fixed locations inside an industrial plant, where one or multiple RISs are deployed on a selected wall.At each position along its trajectory, whether the robot sees an LoS or NLoS link from an RIS or the AP is explicitly modeled for the considered indoor industrial environment.This is different from the existing works [15], [16], [17], [18], [19] where the effects of blockages on the links from the RISs or APs to the moving users (e.g., vehicles or robots) were not explicitly modeled.2) We formulate an optimization problem to minimize the transmission energy consumption of the AP while guaranteeing that the received SNR at the robot is kept above a threshold throughout its journey by jointly optimizing the number, positions, and phase shifts of RISs and the beamforming vectors of the AP.The difficulties of this optimization problem mainly lie in the unfixed number of variables that are coupled in both the objective function and the constraint on the received SNR in complicated ways.This is different from the existing works where only a single RIS was considered [6], [7], [8], [15], [16], [17], [18], the locations or numbers of RISs were not optimized [10], [11], [12], [19], or users were assumed to be stationary [6], [7], [8], [9], [10], [11], [12], [13], [14].3) Since the formulated joint optimization problem is nonconvex and difficult to solve directly, we decompose it into two subproblems and solve them alternately.More specifically, for given number and locations of RISs, we devise a total energy consumption optimization (TECO) Algorithm to minimize the AP's transmission energy consumption by iteratively optimizing the phase shifts of each RIS and the beamforming vector of the AP.For given phase shifts of the RISs and beamforming vectors of the AP, we devise a genetic algorithm to find the optimal number and deployment locations of RISs that minimize the AP's transmission energy consumption.Then, we propose an RIS locations and number (RISLN) Algorithm that iteratively calls the TECO Algorithm and the genetic algorithm until the decrease of the AP transmission energy consumption is below a threshold.4) Our simulation results demonstrate that the proposed algorithms converge fast and can find the suitable number and best deployment locations of RISs that minimize the AP's transmission energy consumption while maintaining reliable communications between the AP and the robot.The simulation results also show that the transmission energy consumption of the AP decreases when the fixed total number of reflecting elements are shared by more distributed RISs, but the reduction in AP transmission energy consumption stops when the number of RISs becomes too large.The remainder of this article is organized as follow.In Section II, we introduce the system model, LoS/NLoS status indicator vector and channel model.In Section III, we formulate the AP's transmission energy consumption minimization problem and propose algorithms to solve it.In Section IV, we provide simulation results.Finally, we conclude this article in Section V.

A. System Model
As shown in Fig. 1, we consider an industrial indoor environment containing several parallel rows of interior obstacles (e.g., large machines or shelves deployed in parallel across the industrial plant) in a Cartesian coordinate system, where one of the ground corners of the plant is set as the origin of the horizontal plane.The plant has a rectangular floor area of l f × l p , where l f is the length of the wall perpendicular to the rows of obstacles, and l p is the length of the wall parallel to the rows of obstacles.We model the parallel obstacles as evenly spaced identical convex cubes each of length L, width W, and height H, where the distance between any two adjacent obstacle is l a , and the number of obstacles is given by I ≤ l f /(l a +W).A robot moves on the horizontal ground plane around the parallel rows of obstacles from a starting point q s to a destination point q d along a fixed trajectory (as indicated by the dashed line on the ground plane in Fig. 1).The trajectory keeps a distance of l (< l a /2) meters from the closet obstacle and forms a U shaped route between any two adjacent obstacles.The obstacles locate along the robot's trajectory from q s to q d are denoted by O 1 , O 2 , . . ., O I , respectively.The corners of the obstacles closest to the start point are denoted by q O 1 , q O 2 , . . ., q O I and are used to represent the locations of the corresponding obstacles.
We consider the downlink from an AP mounted on the ceiling to the single-antenna robot.The AP is equipped with a uniform linear array (ULA) of M (>1) antennas.The location of the AP is denoted by q A .There are K RISs deployed on a wall perpendicular to the obstacles to reflect signals from the AP to the robot.Each RIS is equipped with (N/K) passive reflecting elements, where N is the total number of passive reflecting elements of all the RISs.We assume that the reflecting elements on the RISs are all passive [5], [6], [7].The deployment of RISs ensures the LoS links between the AP and themselves, and their locations are higher than the receive antenna on the robot.The locations of RISs are given by q The passive reflecting elements of each RIS form a uniform rectangular array (URA).For simplicity, the location of the central element of the AP's or an RIS's array is considered as the location of the AP or the RIS, which is used to calculate the distance between the AP or the RIS and another communication node [6].Each RIS is connected to a smart controller that adjusts its phase shifts and reflection amplitudes via a separate wired link.
The time required for the robot to move from the start point to the destination along the fixed trajectory is divided into T + 1 timeslots, t = 0, . . ., T. Timeslot t has a duration t , which is sufficiently short so that the channel conditions can be assumed to be fixed within the timeslot.In timeslot t, the robot moves over a distance of D t = v t t , where v t is the speed of the robot in timeslot t.We use the location of the robot antenna to represent the location of the robot at any time.The trajectory of the robot can be represented by q = [q 0 , q 1 , . . ., q T ], where q t = [x t , y t , z r ] denotes the position of the robot in the tth timeslot, z r is the height of the robot antenna above ground, q 0 = q s and q T = q d .

B. LoS Indicator Matrix
The AP-robot and RIS k -robot links (k = 1, . . ., K) may become NLoS when they are blocked by obstacles as the robot moves along the trajectory.We define an I × (K + 1) LoS indicator matrix at timeslot t where μ (t) i0 = 1 if the LoS link between the AP and the robot is not blocked by the ith obstacle at timeslot t, otherwise, μ RIS k and the robot is not blocked by the ith obstacle at timeslot t, otherwise, μ (t)  ik = 0. We use a ray-slope method, as shown in Algorithm 1, to identify intersections between the K + 1 links and the I obstacles at timeslot t [20], and obtain the LoS indicator matrix.
We note that the system model in Section II-A can be generalized by allowing each obstacle to have its distinct length, width, and height, and by allowing different distances between different pairs of adjacent obstacles.In this way, the modeled obstacles can be of various sizes and do not necessarily locate along a straight line.The proposed Algorithm 1 can still obtain the LoS indicator matrix under such a more general setting of obstacles as long as the trajectory of the robot and the locations and sizes of the obstacles are predetermined.
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Algorithm 1 LoS Indicator Vector
Obtain the slope of the link between the AP and the robot/ the link between the k-th RIS and the robot with the input q a , q R k , and q t .5: Obtain the slope of the link between AP/RIS and the corner of the i-th obstacle with the input q A , q R k , and Obtain μ (t)  ik by comparing the obtained slopes. 7: end for 8: end for

C. Channel Model
The channels from the AP to RIS k , from the AP to the robot, and from RIS k to the robot at timeslot t are denoted by where ρ (<1) denotes the LoS path loss at the reference distance of 1 m, d Ak denotes the distance between the AP and RIS k , τ Ak is the Time of Arrival (ToA) from the AP to RIS k , c r (V AoA Ak ) and c t (V AoD Ak ) are, respectively, given by where V AoA Ak and V AoD Ak are the Angle of Arrival (AoA) at RIS k from the AP and the Angle of Departure (AoD) at the AP toward RIS k , respectively, d A is the spacing between any two adjacent antennas at the AP, and λ AI is the wavelength of the signal transmitted from the AP to RIS k .
Similarly, h AR is given by where AR denotes the distance between the AP and the robot at timeslot t, β AR is the path loss exponent of the NLoS channel between the AP and the robot, τ (t)  AR is the ToA from the AP to the robot at timeslot t, and h ∼ CN (0, 1) is a 1 × M vector whose elements follow independent complex Gaussian distribution with zero mean and unit variance, and c t (V AoD(t) AR ) is given by where V

AoD(t) AR
is the AoD at the AP toward RIS k at timeslot t, and λ (t) AR can be expressed as (7) where f c is the carrier frequency of the AP transmission, v t is the velocity of the robot at timeslot t, φ (t) AR is the angle between the AP signal's AoA at the robot and the direction of the robot's motion, and c is the speed of light.
Similarly, h (t)  kR is given by where d (t)  kR denotes the distance between RIS k and the robot at timeslot t, β kR is the path loss exponent of the NLoS channel between RIS k and the robot, τ (t) kR is the ToA from RIS k to the robot, and c t (V AoD(t) kR ) is given by where V

AoD(t) kR
is the AoD at the AP at timeslot t, d I is the spacing between any two adjacent antennas at an RIS.λ (t) kR is the wavelength of the signal reflected by RIS k to the robot at timeslot t and can be expressed as (10) where φ (t)  kR is the angle between the AoA of the signal reflected by RIS k to the robot and the direction of the robot's motion.
The channel from the AP to the robot at timeslot t is given by where the diagonal matrix k,n ∈ [0, 1] denote the phase shift and the reflection coefficient amplitude of the nth element of the kth RIS, respectively.For simplicity, we assume η (t) k,n = 1 ∀n, k, t, since the reflection coefficient amplitudes are typically set at their maximum value [6], and accordingly (t)  k = diag(e jθ (t) k,1 , . . ., e jθ (t) k,(N/K) ).We consider linear transmit precoding at the AP.At timeslot t (t = 0, . . ., T), the signal transmitted by the AP can be expressed as u (t) = w (t) s (t) , where s (t) is the information symbol with a zero mean and unit variance (i.e., normalized power), w (t) = √ P (t) ŵ(t) is the AP's beamforming vector at timeslot t, ŵ(t) ∈ C M×1 denotes the transmit beamforming direction vector, and P (t) is the transmit power of the AP at timeslot t.For given phase shifts (t) k of the kth RIS at timeslot t, k = 1, . . ., K, ŵ(t) is given by [22] The signal received by the robot at timeslot t is expressed as where n (t) ∼ CN (0, σ 2 ) denotes the additive white Gaussian noise (AWGN) at the robot at timeslot t.Accordingly, the SNR received at the robot at timeslot t is given by

III. PROBLEM FORMULATION AND SOLUTION ALGORITHMS
We formulate an optimization problem to minimize the total transmission energy consumption at the AP by jointly optimizing the number and locations of RISs, reflection-coefficients at RISs, and the beamforming vector at the AP, subject to a minimum received SNR constraint at the robot throughout its trajectory, i.e., (P1): where γ > 0 is the minimum SNR requirement of the robot, K is the number of RISs, is the position of the kth RIS, k = 1, . . ., K, P max is the maximum transmit power at the AP, and h ce is the height of the ceiling.The objective function T t=0 w (t) 2 is the total transmission energy consumption of the AP during the robot's journey.Constraint C1 imposes the value range of the phase shift for each element on the RISs, C2 ensures that the received SNR of the robot at each time slot is above the minimum SNR required, C3 ensures that at least one RIS will be deployed, C4 specifies the upper and lower bounds of the transmit power at the AP, and C5 is the constraint on the location of each RIS.

A. TECO Algorithm
We note that it is hard to solve (P1) directly due to the varying number of variables involved.We first study the case with the number K and the locations of RISs fixed, for which (P1) reduces to (P2) : min Since w (t) and k are independent for different t, the objective function of (P2) is minimized when each w (t) 2 in the summation from t = 0 to t = T is minimized.According to the above and substituting w (t) = √ P (t) ŵ(t) into ( 16), (P2) can be equivalently rewritten as K ,P (t) , ŵ(t) Although (P3) is still a nonconvex problem and is difficult to solve directly, since ŵ(t) 2 = 1, we can obtain the optimal transmit power of the AP by substituting ( 14) into (15b) and solving the resulting inequality for P (t) .Thus, the optimal transmit power of the AP is given by By substituting ( 18) into (P3) and assuming that the AP's maximum transmit power P max is sufficiently large (and hence C4 can be ignored for analytical tractability), we have Please note that C4 will be considered in our proposed Algorithm 2 (which will be presented later in this section) to ensure that the optimized transmit power of the AP does not go beyond its maximum allowed transmit power.
For a given ŵ(t) , (P4) is equivalent to According to the triangle inequality, we have The equality in (21) holds if and only if arg h where arg(x) gives the phases of the elements of x.
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For given 2 , . . ., (t) K (and thus given ŵ(t) , and substituting (21) under the condition of ( 22) into (P5), we have According to (22), we can show that the optimal solution to (P6) is given by (24) which suggests that the phase shifts of RIS 1 should be tuned according to the beamforming of the AP and the phase shifts of RIS k , k = 2, . . ., K.
Following the similar procedures as above, the optimal phase shifts of RIS k (k = 2, . . ., K) can be obtained as: where For given Based on ( 14) and (15b), we solve (P7) and obtain the optimal transmit beamforming direction vector ŵ(t) opt of the AP as [22] We can obtain the optimal transmit power P (t) opt of the AP by substituting (24), (25), and (28) into (18).
Based on the above solutions to the two subproblems (P5) and (P7), we propose an iterative algorithm that solves (P5) and (P7) alternately in each iteration to minimize the total energy consumption at the AP, i.e., the TECO algorithm as presented in Algorithm 2. In each iteration, according to (21) and ( 22), calculating (24) or (25) guarantees that the objective function of (P5) is nondecreasing, and calculating (28)

Algorithm 2 TECO Algorithm
Input: h (t)  kR , G (t)  k , h ( k(s) , k = 1, . . ., K, using (24) or (25).Obtain ŵ(t) opt(s) according to (28).Obtain opt(s) according to (18).9: opt(s) , P max 10: until guarantees that the objective function of (P7) is nondecreasing [22].As a result, after each iteration, the objective function value of (P4) is nonincreasing.Since there is a lower bound of the objective function of (P4) given by C2, the objective function of (P4) will decrease after each iteration until its convergence.In Algorithm 2, solving (P5) has a complexity of where N is the total number of passive reflecting elements of all the RISs, while solving (P7) has a complexity of O(M), where M is the number of antennas at the AP.Thus, Algorithm 2 has a complexity of O(N I (M +N)), where N I is the number of iterations required for convergence.

B. RISLN Algorithm
In order to optimize the number and locations of RISs, we divide the wall area where the RISs are deployed into a grid of ) equal rectangular units, where l f is the length of the wall, l h is the height of the wall, z r is the height of the robot antenna, and d 2 u is approximately the size of each RIS.We assume that each RIS is deployed in one of the units and no more than one RIS can be deployed in any unit.Hence, the location of the kth RIS, q Rk ∀k ∈ {1, . . ., K}, can be uniquely represented by a sequence of b ) binary bits.Given I obstacles in the considered indoor environment, we can show that there is no need to deploy more than I + 1 RISs to ensure the robot seeing an LoS link from at least one RIS or the AP at all times along its trajectroy.For each K ∈ {1, . . ., I + 1}, we employ Algorithm 2 and the Genetic Algorithm [23] to optimize the locations of the K RISs as follows.First, set the iteration index s = 1, and randomly initialize a population of N ind individuals, where N ind is an even number and each individual contains a binary bit sequence q R = [q R 1 , q R 2 , . . ., q R K ] that represents the locations of K RISs.Then, Algorithm 2 is used to obtain ŵ(t) opt , (t) opt , t ∈ {0, . . ., T}, k ∈ {1, . . ., K} for each individual and T t=0 P (t)  opt is taken as the fitness value of the corresponding individual.
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Update s = s + 1, and a new population of N ind individuals are generated from the current population by the Roulette wheel selection scheme [23], where each individual has a probability of selection proportional to its fitness value.Next, the population generated by Selection is randomly divided into (N ind /2) pairs of individuals.For each pair, generate a random number r following a uniform distribution in (0, 1) and swap the (Kb/2) th to the Kbth bits between the two individuals if r ≤ P c , where 0 < P c < 1 is the Crossover probability.Afterwards, for each individual in the new population generated by Crossover, generate a random number r following a uniform distribution in (0, 1) and flip every binary bit of the corresponding individual if r ≤ P m , where 0 < P m < 1 is the Mutation probability.
After the optimized locations of K RISs have been obtained for all K ∈ {1, . . ., I + 1}, the value of K that is associated with the lowest value of T t=0 P

(t)
optK is identified as the optimal value of K and its corresponding q RoptK , P

IV. SIMULATION RESULTS
In this section, we present numerical results to verify the proposed algorithms.The parameter values used in the simulation are listed in Table I unless otherwise specified.
Fig. 2 shows the transmission energy consumption of the AP versus the iteration number in Algorithm 3. The fluctuation of the curve is due to the use of the genetic algorithm that randomly generates a new population of candidate solutions in each iteration.By generating new populations, the genetic algorithm introduces genetic diversity so as to escape local optima by exploring different regions of the solution space, but the AP's transmission energy consumption of a randomly generated new population is not necessarily lower than that of the population obtained in the previous iteration.We can see that Algorithm 3 converges after the 19th iteration.Set the iteration index s = 1, and randomly initialize a population of N ind individuals, where each individual is a binary bit sequence q R = q R 1 , q R 2 , . . ., q R K .Run Algorithm 2 for t = 0, . . ., T for each individual and take T t=0 P (t) opt as its fitness value.Selection: Obtain a new population of N ind individuals from the current population by the Roulette Wheel Selection scheme [23].

7:
Crossover: The new population is randomly divided into N ind 2 pairs.For each pair, randomly generate r ∈ (0, 1) and swap the (Kb/2) -th to the Kb-th bits between the two individuals if r P c .

8:
Mutation: For each individual in the population generated by Crossover, randomly generate r ∈ (0, 1) and flip every bit of it if r P m .9: Run Algorithm 2 for each individual in the population generated by Mutation.The individual that has the smallest fitness value returns ŵ(t) optK(s ) , (t) koptK(s ) and P (t)  optK(s ) , t ∈ {0, . . ., T}, k ∈ {1, . . ., K}. 10: until T t=0 P (t) optK(s −1) − T t=0 P (t) optK(s ) < ε.In the simulation, we compare the performance of Algorithm 3 with the following four benchmark schemes: 1) the evenly distributed RIS deployment scheme proposed in [24], where the RISs are evenly spaced on the wall at the same height, and the number and the height of RISs are the optimal results obtained by Algorithm 3; 2) the centralized deployment (i.e., a single RIS) with the optimal location of the RIS obtained by Algorithm 3 for K = 1; 3) the centralized deployment of one RIS at the centre of the wall; and 4) the case without deploying any RIS.For all the RIS deployment schemes under comparison, the RISs are deployed on the same wall as shown in Fig. 1.The specific locations of RISs for the four RIS deployment schemes are indicated in Fig. 3. Fig. 4 plots the AP's transmission energy consumption versus the minimum required SNR at the robot.The optimal number of RISs obtained by the proposed algorithm is K opt = 3.As shown in Fig. 4, the transmission energy consumption at the AP increases when the minimum required  SNR at the robot increases, since a higher target SNR at the robot requires higher transmit power at the AP at each timeslot.The optimal RIS deployment obtained by Algorithm 3 achieves the lowest energy consumption at the AP among  all the considered schemes.Compared with the evenly distributed RIS deployment scheme, deploying a single RIS at the optimal location obtained by Algorithm 3 for K = 1 leads to a lower energy consumption at the AP.This is because Algorithm 3 optimizes the RIS location while considering the exact locations of obstacles and the robot.Between the two schemes both without optimizing the RIS location, the evenly distributed deployment outperforms deploying a single RIS at the center of the wall because the multiple distributed RISs make it more likely for the robot to see an LoS link from one of the RISs along the whole trajectory than a single RIS.The case with no RIS always has the highest transmission energy consumption at the AP among all the considered schemes.This indicates that the deployment of RIS can decrease the transmission energy consumption at the AP.
In Fig. 5, we plot the transmission energy consumption at the AP versus the total number of elements on RISs.The optimal number of RISs obtained by the proposed algorithm is K opt = 3.In addition to the four benchmark shcemes considered in Fig. 4, we also plot a curve for the even deployment scheme [24] with K = 2K opt .The AP's transmission energy consumption decreases with the total number of elements on RISs under all the considered schemes, because a larger number of elements leads to a larger beamforming gain of the RISs.The proposed algorithm performs the best because the optimized deployment locations of RISs ensure that the robot sees an LoS link from an RIS or the AP throughout its trajectory and the AP can transmit at the lowest possible power at each timeslot.The even deployment scheme for K = K opt achieves a lower transmission energy consumption of the AP than the even deployment scheme for K = 2K opt .This indicates that further distributing the fixed total number of elements to more RISs cannot further lower the AP's transmission energy consumption if the whole trajectory has already been covered with LoS links, while less elements per RIS will reduce the beamforming gain of each RIS.
In Fig. 6, we plot the transmission energy consumption at the AP versus the carrier frequency of the mmWave signal.The optimal number of RISs obtained by the proposed algorithm is K opt = 3.For all the considered schemes, the transmission energy consumption at the AP increases with the carrier frequency of the signal because the signal strength decays faster over distance at a higher carrier frequency.The performance gain of the proposed algorithm over the other schemes becomes larger with the increase of the carrier frequency.This shows the importance of optimizing RIS deployment for indoor downlink transmissions in high mmWave bands.
In Fig. 7, we plot the transmission energy consumption at the AP versus the number of obstacles.The transmission energy consumption at the AP increases with the number of obstacles for all the considered schemes because of the robot's longer trajectory.The proposed algorithm for K = 1 and the even deployment scheme have the same AP's transmission energy consumption when the number of obstacle is one, because deploying one RIS is enough for the robot to maintain an LoS link from the RIS or the AP along the whole trajectory.The even deployment scheme outperforms the proposed algorithm for K = 1 when the number of obstacles is four or five, because the centralized deployment scheme keeps the robot seeing an LoS link from the RIS or the AP for fewer timeslots.

V. CONCLUSION AND FUTURE WORK
In this article, we have studied multiple-RIS-assisted mmWave communications for a robot moving around obstacles along a predefined trajectory inside an industrial building.To minimize the transmission energy consumption at the AP while ensuring the received SNR at the robot above a threshold throughout its journey, we have proposed the TECO Algorithm and RISLN Algorithm that jointly optimize the number, locations and phase shifts of RISs and the beamforming vector of the AP.Simulation results demonstrate that the proposed algorithms converge fast and can significantly reduce the transmission energy consumption at the AP as compared to the benchmark schemes that either do not deploy any RIS or do not optimize the number or locations of RISs.Distributing the fixed total number of reflecting elements to more RISs leads to a lower transmission energy consumption of the AP, since more distributed RISs provide more LoS links to the robot while it moves along its trajectory.The reduction in AP transmission energy consumption stops when the number of RISs becomes too large.This is because when all sections of the robot trajectory have been covered by LoS links from either the AP or an RIS, further dividing the fixed total number of reflecting elements into more RISs will reduce the beamforming gain of each RIS.For the considered scenario of I obstacles, properly deploying I RISs is sufficient to ensure that the robot sees an LoS link from at least one RIS or the AP at all times along its trajectory.The AP's transmission energy consumption increases with the minimum SNR required at the robot and the carrier frequency of the mmWave signal, but decreases with a larger total number of elements on the RISs.
In our future work, we will study the AP transmission energy consumption minimization problem where the robot's trajectory can be jointly optimized with the deployment of the AP and RISs.We will also extend our work to more complex scenarios, e.g., multiple robots moving along different trajectories.This will require allocating the available RISs and AP beamforming vectors to different robots while considering potential co-channel interference among the links serving neighboring robots.It will also be interesting to consider the energy consumption of RISs when the reflecting elements are not fully passive.

Manuscript received 22
February 2023; revised 4 June 2023 and 18 June 2023; accepted 22 June 2023.Date of publication 26 June 2023; date of current version 25 December 2023.This work was supported in part by the European Union's Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie Actions Grant 778305.(Corresponding author: Xiaoli Chu.)

Fig. 4 .
Fig. 4. AP's transmission energy consumption versus the minimum required SNR at the robot.

Fig. 5 .
Fig. 5. AP's transmission energy consumption versus the total number of elements on RISs.

Fig. 6 .
Fig. 6.AP's transmission energy consumption versus the carrier frequency of the signal.

Fig. 7 .
Fig. 7.AP's transmission energy consumption versus the number of obstacles.

TABLE I PARAMETER
VALUES USED IN THE SIMULATION