There has been considerable research on a team of unmanned vehicles carrying out a wide range of tasks such as search and rescue, surveillance, agriculture, and environment monitoring (Yan and Mostofi (2014); Tokekar et al. (2016)). Communication between such a team of robots and a remote operator or within the robotic network itself, is often crucial for the successful completion of these tasks. For instance, consider a scenario where a robot has collected information about its environment and needs to transmit this information to a remote operator or another robot. In order to do so, it first needs to establish a connection with the remote operator or the other robot. The robot may not be able to do so at its current location and may need to move to establish a connection, exploiting the spatial variations of the channel quality. This paper then answers the following question: what are the statistics of the distance traveled along a given path until connectivity?
There has been considerable recent interest in the area of connectivity in robotic systems. For instance, in Zavlanos et al. (2011), the connectivity of a network is maximized using a graph-theoretic analysis while in Yan and Mostofi (2012), connectivity is optimized using a more realistic channel model. There has also been work on path planning to enable connectivity (Muralidharan and Mostofi (2017c); Caccamo et al. (2017); Zeng and Zhang (2017); Chatzipanagiotis and Zavlanos (2016); Muralidharan and Mostofi (2017a); Yan and Mostofi (2012)) as well as on communication-aware sensing (Yan and Mostofi (2014)).
However, a mathematical characterization of the statistics of the distance traveled until connectivity is lacking in the literature, which is the main motivation for this paper. We refer to this problem as the first passage distance (FPD) problem, analogous to the concept of first passage time (Siegert (1951)). We next summarize the contributions of the paper.
Statement of contributions:
We mathematically characterize the probability density function (PDF) of the FPD as a function of the underlying channel parameters of the environment, such as shadowing, path loss, and multipath fading parameters. We do so for two cases: 1) when ignoring the multipath component (which could be of interest when the robot looks for an area of good connectivity as opposed to a single spot, or when multipath is negligible), and 2) when considering the multipath component. In both cases, we first develop an exact characterization of the statistics of the FPD for the setting with straight paths. We utilize tools from the stochastic equation literature to characterize the FPD while ignoring the multipath component, and develop a recursive characterization for the case when we include multipath. We then mathematically characterize a more general space of paths for which the analysis holds, based on properties of the path such as its curvature.
Note that the PDF of the FPD can be directly computed via a high dimensional integration, as we will discuss in Section 4.1. However, this direct computation is infeasible for moderate distances. Our proposed theoretical framework is not only computationally efficient but also brings a foundational analytical understanding to the FPD and can significantly affect networked robotic operation design.
The paper is organized as follows. In Section 2, we formally introduce the problem and briefly summarize the channel’s underlying dynamics. In Section 3, we characterize the statistics of the distance traveled until connectivity while ignoring the multipath component. In Section 4, we characterize the statistics of the FPD while including the effect of multipath in the analysis. Finally, in Section 5, we validate our mathematical characterizations through extensive simulation with real channel parameters from downtown San Francisco.
2 Problem Setup
Consider a robot traveling along a given trajectory that needs to get connected to either a remote operator or another robot, as shown in Fig. 1a. In order for the robot to successfully connect with the remote operator, the receptions need to satisfy a Quality of Service (QoS) requirement such as a target Bit Error Rate, which in turn results in a minimum required received Signal to Noise Ratio, or equivalently a minimum required channel power, given a fixed transmission power. We denote this minimum required received channel power as in this paper. This paper then asks the following question: What is the distance traveled by the robot along the path before it gets connected to the remote operator? More specifically, we are interested in mathematically characterizing the probability density function (PDF) of this distance, for a given path, as a function of the underlying channel parameters, such as path loss, shadowing and multipath fading parameters, as well as the parameters of the path, such as its curvature.
2.1 Channel Model
In the communication literature, channel power is well modeled as a multi-scale random process with three major dynamics: path loss, shadowing and multipath fading (Rappaport (1996)). Let represent the received channel power (in the dB domain) at location with the remote operator located at the origin. can then be expressed as where is the distance-dependent path loss with representing the path loss exponent, and and
are random variables denoting the impact of shadowing and multipath respectively (in dB). The multipath component or small-scale fading represents fluctuations in the channel power in the order of a wavelength, while the shadowing component or large-scale fading represents fluctuations of the channel power after the signal is locally averaged over multipath, thus reflecting the impact of larger objects such as blocking buildings.is best modeled as a Gaussian random process with an exponential spatial correlation, i.e., where is the shadowing power and is the decorrelation distance (Rappaport (1996)). As for multipath, a number of distributions such as Nakagami, Rician and lognormal have been found to be a good fit (in the linear domain) (Rappaport (1996); Hashemi (1994)).
Consider the case where the robot is traveling along a path. Let be the distance traveled by the robot along this path. With a slight abuse of notation, in the rest of the paper we let represent the channel power when the robot has traveled distance along the path, as marked in Fig. 1a. We thus have .
3 Characterizing the FPD Without Considering Multipath
We start our analysis by ignoring the multipath and only considering the shadowing and path loss components of the channel, i.e., we want to be above . This assumption allows us to better analyze and understand the FPD, and paves the way towards our most general characterization of the next section, which includes multipath as well. Moreover, the analysis also has practical values of its own, and would be relevant to the case where the robot is interested in finding a general area of good connectivity as opposed to a single good spot. In this section, we will characterize the statistics of the distance traveled until connectivity for this scenario. We begin by analyzing straight paths in Section 3.1, where we utilize the stochastic differential equation literature (Gardiner (2009)) in our characterization. We then extend our analysis to a more general space of paths in Section 3.2.
3.1 Straight Paths: Stochastic Differential Equation Analysis
In this section, we characterize the PDF of the distance traveled until connectivity for straight-line paths. Consider a robot situated at a distance from a remote operator or from another robot to which it needs to be connected, and moving in the direction specified by the angle , as shown in Fig. 1b. The angle is measured clockwise with respect to the line segment connecting the remote operator and the robot, as can be seen in Fig. 1b, and denotes the direction of travel chosen by the robot.
represents the channel power when the robot is at distance along direction , as marked in Fig. 1b. We thus have , where
and is a zero mean Gaussian process with the spatial correlation of , with . Note that is also a function of and . We drop ’s dependency on them in the notation as the analysis of the paper is carried out for a fixed and .
As we shall see, becomes an Ornstein-Uhlenbeck process, one of the most studied types of Gauss-Markov processes (Ricciardi and Sato (1988); Ricciardi and Sacerdote (1979), Leblanc and Scaillet (1998); Gardiner (2009)
). Ornstein-Uhlenbeck process appears in many practical scenarios, such as Brownian motion, financial stock markets, or neuronal firing (Ricciardi and Sacerdote (1979); Leblanc and Scaillet (1998)), and thus has been heavily studied in the literature. In this paper, we shall utilize this rich literature (Gardiner (2009); Di Nardo et al. (2001)) to mathematically characterize the FPD to connectivity for a mobile robot.
We begin by summarizing the definitions of a Gaussian process and a Markov process.
Definition 1 (Gaussian Process)
A Gaussian process is completely specified by its mean function and its covariance function . We use the notation to denote the underlying process.
Definition 2 (Markov Process)
(Papoulis and Pillai (2002)) A process is Markov if
for all and for all , where denotes the probability of the argument.
Definition 3 (Gauss-Markov Process)
(Mehr and McFadden (1965)) A stochastic process is Gauss-Markov if it satisfies the requirements of both a Gaussian process and a Markov process.
We next state a lemma that shows when a Gaussian process is also Markov, which we shall utilize to prove that the channel shadowing power is Gauss-Markov.
A Gaussian process is Markov if and only if , for all .
See Doob (1949) for the proof.
The channel shadowing power and the channel power are Gauss-Markov processes.
Remark 1 (see Gardiner (2009))
The Ornstein-Uhlenbeck process is a Gauss-Markov process with the covariance function , where and are constants. Thus, we can see that is an Ornstein-Uhlenbeck process.
In order to gain more insight into the stochastic process , we next discuss the transition PDF , where , as well as the stochastic differential equation governing , both of which we shall subsequently use in our characterization of the PDF of the FPD.
3.1.1 The Underlying Stochastic Differential Equation
The transition PDF characterizes the distribution of given
. This is a normal density characterized by a mean and variance of (see 10.5 ofKay (1993))
The transition PDF explicitly shows the spatial dependence of the channel power . As stated in Di Nardo et al. (2001),
satisfies the partial differential equation known as the forward Fokker-Planck equation:111The Fokker-Planck equation of Di Nardo et al. (2001) is stated for a general Gauss-Markov process. Here we adapted it for our specific Gauss-Markov process .
with the associated initial condition of , where , and is as stated in (1), with its derivative:
The Fokker-Planck equation shows the evolution of the probability density with the traveled distance given .
Moreover, as shown in Gardiner (2009), the channel power can be represented as a stochastic differential equation:222Gardiner (2009) provides the stochastic differential equation for the Ornstein-Uhlenbeck process, from which we can easily obtain (4).
where is the Wiener process and and are as defined before.
In (3.1.1) and (4), and are known as the drift and the diffusion components respectively. The drift is a pull towards the mean, and the diffusion component is a function of the shadowing variance and the decorrelation distance. Then, in an increment , we can think of the channel power spatially evolving with a deterministic rate , in addition to a random Gaussian term with the variance .
Next, we utilize our established lemmas to derive the PDF of the FPD.
3.1.2 First Passage Distance
Consider the random variable . This denotes the FPD of the process to the connectivity threshold , with the initial value . Further, let represent the PDF of the FPD. In the following theorem, we characterize this PDF.
The PDF of FPD satisfies the following non-singular second-kind Volterra integral equation:
represents the FPD for a given initial value of . In many scenarios, we are instead interested in characterizing the FPD for the initial state being a random variable bounded from above by , i.e., we are interested in characterizing the FPD when the starting position is not connected. This is known as the upcrossing FPD in the general first passage literature (Di Nardo et al. (2001)). We next extend our analysis to derive the PDF of the upcrossing FPD. Let the random variable denote the -upcrossing FPD of to the boundary given that the initial state satisfies , where is a fixed real number. The -upcrossing FPD, , can be characterized as follows:
where is the FPD given the initial value , as defined earlier, and
is the PDF of with denoting the PDF of . Moreover, the -upcrossing FPD density is similarly related to the FPD density as follows: .
Note that we have required . This is due to the fact that the mathematical tools we shall utilize are not well-defined for . However, can be chosen arbitrarily small.
In the following theorem, we derive an expression for , the PDF of the -upcrossing FPD.
The PDF of the -upcrossing FPD, , satisfies the following non-singular second-kind Volterra integral equation:
where is as defined in (3.1),
with representing the error function, and
The proof is obtained by adapting Theorem of Di Nardo et al. (2001) to our particular Gauss-Markov process form.
In terms of implementation, the functions and in Theorem 3.1 and Theorem 3.2 can be easily computed. The PDF of the FPD () and the PDF of the -upcrossing FPD () can then be computed from the integral equations (5) and (7) respectively. In particular, Simpson rule provides the basis for an efficient iterative algorithm for evaluating these integrals (See Section 4 of Di Nardo et al. (2001)).
Remark 4 (Computational complexity)
The direct computation of involves a high dimension integration, as we will discuss in Section 4.1. For a discretized path of steps, this direct computation would have a computational cost exponential in , i.e. for some constant . In contrast, the computation cost of using Theorem 3.2 is . Moreover, Theorem 3.2 is also an elegant characterization of the -upcrossing FPD that can be utilized for analysis and design of robotic operations.
3.2 Approximately-Markovian Paths
In this section, we characterize the space of paths (beyond straight paths) that results in approximately-Markovian processes. As we saw in Section 3.1, the channel shadowing component along a straight line is a Gauss-Markov process. This allowed us to characterize the statistics of the distance to connectivity for a mobile robot traveling along a straight path. A general non-straight path is not Markovian since the covariance function does not satisfy Lemma 1. In this section, we characterize the space of paths for which the channel shadowing power along the path is approximately a Gauss-Markov process. This allows us to immediately apply the stochastic differential equation analysis of Section 3.1 to characterize the statistics of the distance until connectivity for these paths.
Consider the scenario in Fig. 2 (top), where we have discretized the path, with denoting the shadowing power at the current location and indicating the channel shadowing power at previously-visited points.333Note that the discretization step size of the path must be small for the derivations of Theorem 3.2 to be valid. In Section 3.2, we saw that a Gauss-Markov process satisfies the Fokker-Planck equation of (3), which provides us with the result of Theorem 3.2. The Fokker-Planck equation in turn requires the property that for its derivation (through the Chapman-Kolmogorov equation (Gardiner (2009))). Thus, we say a path is approximately-Markovian, if at every point on the path, we have that is close to . We will characterize this closeness precisely in Section 3.2.2 using the Kullback-Leibler (KL) divergence metric.
Our key insight is that the approximate Markovian nature is related to the curvature of a path, which is a measure of how much the path curves, i.e., how much it deviates from a straight line. For instance, a straight line has a curvature of . Thus, we would expect that paths with small enough curvature would result in approximately-Markovian processes. We will precisely characterize what we mean by this in Section 3.2.4.
We first describe an outline of our approach for characterizing the space of approximately-Markovian paths. At every point on the path, instead of checking for the conditional distribution given all the past points on the path, which is cumbersome, we consider all past points on the path within a certain distance of the current point, i.e., within a ball centered at the current point. In other words, to check the approximately-Markovian property, we evaluate instead of . Fig. 2
(top) shows an illustration of this. This makes sense since the shadowing component has an exponential correlation function. Thus, if the radius of the ball is large enough, the points outside of the ball will have a negligible impact on the estimate at the center of the ball. We will characterize this radius in Section3.2.3. Thus, our strategy is to roll a ball along the path, as shown in Fig. 2 (bottom), and to check if the approximate Markovian property holds at each point along the path. We then characterize two conditions that can ensure that a path will be approximately-Markovian. The first is that, at any point on the path, if we travel backward along the path it should not loop either within the ball or such that it re-enters the ball. We refer to such looping as -looping ( being the radius of the ball), and examples of this are shown in figures 3a and 3b. Equivalently, a path is called -loop-free if there is no -looping. The second condition is that the maximum curvature of the path should be smaller than a certain bound, which will be characterized later in Section 3.2.4. If the -loop-free condition is satisfied, then the only part of the path that lies within the ball would lie in the shaded region of Fig. 3c, and if the maximum curvature of the path is small enough, then the path will be approximately-Markovian. We will formulate this precisely in Section 3.2.4.
We start by mathematically characterizing the -looping condition in detail.
3.2.1 -Loop-Free Constraint
We define -loop-free paths as paths where neither of the two following scenarios occurs at any point on the path. The first is when traveling backward along a path, the path loops within the ball itself. More precisely, when traveling backward along the path, let the initial direction of travel be along the negative x-axis. We say that the path loops within the ball if at any point (still inside the ball), the direction of travel has a component along the positive x-axis (e.g., Fig. 3a). The second scenario is when the path re-enters the ball once it leaves it. These two scenarios, which we collectively refer to as -looping, are illustrated in figures 3a and 3b. Such -looping behavior can possibly invalidate the approximate Markovian nature of the path.
We next relate the -loop-free condition to the curvature of the path. We first review the precise definition of curvature.
Definition 4 (Curvature)
(Kline (1998)) The curvature of a planar path parameterized by arc-length is defined as
where is the unit tangent vector at .
When traveling backward along a path, consider the segment of the path inside the ball, before the path exits the ball. Let refer to this segment, as shown in Fig. 3c. Moreover, let refer to its length. The following lemma characterizes some important properties of .
For a path with maximum curvature and a ball with radius , the path segment satisfies the following properties:
See Appendix A.1 for the proof.
Then, a sufficient condition for a -loop-free path is given as follows.
Lemma 3 (-loop-free path)
Consider a planar path parameterized by arc length, i.e., denotes the arc length. Let be the maximum curvature of the path. The path is -loop-free if it satisfies and
for and for all .
From Lemma 2, we know that if , the path cannot loop within the ball, preventing the condition of Fig. 3a. Moreover, from Lemma 2, it can easily be confirmed that for is the euclidean distance from the center to a point on the part of the path that has left the ball. Thus, if , for the path cannot re-enter the ball (i.e., scenario of Fig. 3b is not possible).
Any path can be reparameterized by arc length. Details on this can be found in Eberly (2008).
We next characterize the similarity or dissimilarity between the true distribution and its Markov approximation using the KL divergence metric. We then utilize this to obtain sufficient conditions on the ball radius and the curvature of a path for the approximate Markovian nature to hold.
3.2.2 Approximately-Markovian: KL Divergence metric
Consider a path as shown in Fig. 2 (top). Let be the channel shadowing power on the current location and be the channel shadowing power on the past locations along the path. From Section 2.1, we know that are jointly Gaussian random variables. The distribution of is then given as , where
with , and (see 10.5 of Kay (1993)). Moreover, , where is the location corresponding to . Let denote the coefficients of the mean. We then have .
We want to approximate this distribution with the Markovian distribution where and , with , and being the step size of the path. We first characterize the difference between the means, given as , where . is thus a zero-mean Gaussian random variable , where
We will compare how close the true distribution and its approximation are using the KL divergence metric. We first review the definition of KL divergence.
Definition 5 (KL Divergence)
(Cover and Thomas (2012)) The KL divergence between two distributions and is defined as
KL divergence is a measure of the distance between two distributions (Cover and Thomas (2012)). We will utilize this as a measure of the goodness of the approximation: the smaller the KL divergence, the better the approximation. The following lemma gives us the expression for this KL divergence.
The KL divergence between and its approximation is given as
See Robert (1996) for the proof.
Since and are functions of , they are random variables. Thus,
becomes a Chi-squared random variable with one degree of freedom since(Lancaster and Seneta (2005)), and the KL divergence of (11) becomes a random variable. More specifically, from (11), we know that the KL divergence is a scaled Chi-squared random variable with an offset term. We use the mean
and the standard deviationof the KL divergence to capture the deviation of the Markov approximation from the true distribution. The smaller these values are, the better the approximation is. In our approach, we set maximum tolerable values for the mean and the standard deviation as and respectively. Then, we say that the distribution is approximately-Markovian for the parameters and if we satisfy and .
We next consider the setting with points in space, as shown in Fig. 4a, where we have the current point (), the previous point () and a general point in space (). We are interested in mathematically characterizing the impact of on the estimate at the current point, i.e., how good an approximation is for the true distribution . As we shall see, we will utilize this analysis in such a way that it serves as a good proxy for the general point analysis. Specifically, we will utilize it to obtain bounds on the ball radius as well as on the maximum allowed curvature of a path in Section 3.2.3 and Section 3.2.4 respectively. Let , , and , as shown in Fig. 4a, where is the location of the general point. Moreover, .
The following lemma characterizes the mean and standard deviation of the KL divergence between the true distribution and its approximation for the point analysis.
The mean and standard deviation of the KL divergence between the true distribution and its approximation for the point analysis of Fig. 4a is given as
See Appendix A.2 for the proof.
Any point on the path can belong to three possible regions: ) the shaded region within the ball of Fig. 3c, ) within the ball but outside the shaded region, and ) outside the ball. If the path is -loop-free, then no point of the path lies within region (i.e., within the ball but outside the shaded region). We next characterize the minimum ball radius and the maximum allowed curvature of a path such that the impact of any point () in region and on the estimate at the center of the ball is negligible.
3.2.3 Ball Radius
We next utilize our analysis to determine the ball radius . We wish to select the minimum such that the impact of any point outside the ball on the approximation is within the tolerable KL divergence parameters and , i.e., the KL divergence between the true and the approximating distribution (in the point analysis) satisfies and .
The following lemma characterizes what the minimum ball radius should be.
The minimum ball radius such that any point outside the ball satisfies the maximum tolerable KL divergence parameters and for the point analysis, is given by
where and .
See Appendix A.3 for the proof.
3.2.4 Curvature Constraint
We next utilize the point analysis to determine the maximum curvature of a path such that it is approximately-Markovian, i.e., it satisfies the KL divergence constraint and .
Consider the scenario in Fig. 4b. For a given maximum curvature , any valid point of the path must lie within the shaded region of the figure, where the boundary corresponds to circular paths with curvature . We wish to find the maximum allowed curvature such that the impact of any point within the shaded region on the approximation is within the tolerable KL divergence parameters and , i.e., the KL divergence between the true and the approximating distribution (in the point analysis) satisfies and . The following lemma characterizes this maximum allowed curvature as the solution of an optimization problem.
The maximum allowed curvature such that any past point on the path within the ball of radius satisfies the maximum tolerable KL divergence parameters and for the point analysis, is the solution to the following optimization problem:
and , , , .
See Appendix A.4 for the proof.
Ideally, we would have preferred to use the KL divergence between the approximation and the true distribution where we condition on all the past points on the path within the ball radius, as opposed to using just the point with the maximal impact. However, such an analysis does not lend itself to a neat characterization of the maximum allowed curvature. Through simulations, we have seen that the point analysis, as described in Lemma 7, serves as a good proxy for the past points case on a circular path (which has a maximum curvature everywhere for a given ). For instance, for parameters , and m, the KL divergence mean and standard deviation when considering all the past points of the path within the ball are and respectively. This is comparable to the values and obtained for the point analysis from Lemma 7.
Finally, we put together all our results to provide sufficient conditions for an approximately-Markovian path.
Lemma 8 (Approximately-Markovian Path)
Let be a path parameterized by its arc length. The path is approximately-Markovian for given maximum tolerable KL divergence parameters and for the point analysis, if it satisfies the following conditions:
is -loop-free for ball radius (as characterized by Lemma 3),
curvature for all ,
Consider a given path. For a given and , we can check if the path satisfies the conditions of Lemma 8. If it does, we can then directly use the results of Section 3.1 to obtain the PDF of the FPD for the path. Note that even if the path does not satisfy the conditions, the path may still be approximately-Markovian as the conditions of Lemma 8 are sufficient conditions.
4 Characterizing FPD Considering Multipath
The previous section analyzed the FPD to the connectivity threshold when the multipath component was ignored. In this section, we show how to derive the FPD density in the presence of the multipath fading component, and for the most general channel model of . We begin by analyzing straight paths in Section 4.1, where we derive the PDF of the FPD using a recursive formulation. We then extend our analysis to a larger space of paths in Section 4.2.
4.1 Straight Paths: A Recursive Characterization
We first characterize the PDF of the distance traveled until connectivity for straight paths. We consider the scenario described in Section 3.1, where a robot situated at a distance from a remote operator to which it needs to be connected, moves in a straight path in the direction specified by the angle , as shown in Fig. 1b. represents the channel power when the robot is at distance along direction , as marked in Fig. 1b.
Recall that we define connectivity as the event where . The connectivity requirement is then given as , considering all the channel components. In this case, the approach of Section 3.1 is not applicable anymore as we no longer deal with a Markov process. Even if the multipath component was taken to be a Gauss-Markov process (which could be a valid model for some environments (Hashemi (1994))), the resultant channel power would not be Markovian, as can be verified from Lemma 1. In this section, we assume that the robot measures the channel along the chosen straight path in discrete steps of size . We assume that is such that the multipath random variable is uncorrelated at the distance apart (this is a realistic assumption as multipath decorrelates fast (Malmirchegini and Mostofi (2012))). We then index the channel power and shadowing components accordingly, i.e., let and . The probability of failure of connectivity at the end of steps (given the initial failure of connectivity) can then be written as