Int J Soc Robot (2012) 4:15–27 DOI 10.1007/s12369-011-0102-2
O R I G I N A L PA P E R
Regression Analysis of Multi-Rendezvous Recharging Route in Multi-Robot Environment Soheil Keshmiri · Shahram Payandeh
Accepted: 24 July 2011 / Published online: 8 September 2011 © Springer Science & Business Media BV 2011
Abstract One of the crucial issue in the field of autonomous mobile robotics is the vitality of energy efficiency of robots and the entire system they form. By efficiency here we refer to ability of robots (or the system in which they are deployed) to maintain their survival throughout the course of the operation so as to provide themselves with the opportunity of attaining energy once needed. In this paper, issue of recharging of a group of autonomous worker robots in their working environment has been addressed. To deliver the objective, a tanker robot’s planner, capable of determining an energy supply route based on regression analysis techniques, has been implemented. Specifically we have examined the practicality of ordinary and weighted least squares (OLS and WLS respectively) as well as orthogonal least absolute values (ORLAV) regressions for recharging route computation (hence the terms Least Square Recharging Route (LSRR) and Orthogonal Recharging Route (ORR)). Studies were conducted (while examining OLS and WLS techniques) to analyze the effect of various uncertainties which may exist in location information of the robots with regards to the recharging route. It has been proven that ORLAV based planner may result to a recharging route that minimizes the cumulative sum of worker robots distance traversal during the recharging process, irrespective of tanker location. Simulations in both, environment with and without obstacles, have been conducted to examine the practicality of the techniques in contrast with fixed charging station sce-
S. Keshmiri (B) · S. Payandeh Experimental Robotics Laboratory, School of Engineering Science, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6 e-mail:
[email protected] S. Payandeh e-mail:
[email protected] nario. Appropriate graphs, diagrams and tables, representing the results obtained in simulations are provided for illustrative comparisons among different techniques. Keywords Multi-robot systems · Motion planning · Least square regression · Orthogonal regression · Recharging route
1 Introduction Every mechanism, whether biological or otherwise, which is capable of interacting with its environment, requires to maintain its energy for survival. Issue of equipping robots with the capability of maintaining their energy (autonomously or with the aid of other party such as recharging station, etc.) has been topic of several ongoing research projects. Depending on nature of one such recharging station, different strategies have been adapted. In its earlier attempt, the problem was addressed via introduction of fixed charging station in robots’ environment, thereby instructing robots to move between their current working locations and the charging station [1–3]. Such approach, however, will leave the system with the short-come of expending power for being recharged instead of performing their designated task(s). It may also suffer the limitation of determining the energy threshold for individuals as robots start spreading over the working environment since the increase of distance between the robot and fixed charging station increases the amount of energy to be expended by robots to reach the recharging location. Providing robots with opportunity of being recharged in their deployment units (i.e. displaceable charging station) would not only increase the energy preservation of the entire system, but would also save the computational resources
16
which would have been required for navigating robots between power station and their work zone. In this context, recharging paradigm might be sub-categorized as a special case of motion planning in a dynamic environment [4–6] (where robots are instructed to meet a target location(s) following the designated paths with a minimum requirement of avoiding collision with one another) and rendezvous problem [7–10] (in which group of robots are instructed to gather in some rendezvous location). In [11] a distributed algorithm to determine one such meeting location for robots with limited visibility has been demonstrated. Some approaches like one presented in [12] address the issue of finding single rendezvous location such that the individual travel cost is minimized. Making charging station displaceable can be delivered via the introduction of a special-purpose tanker robot, primarily designed to perform the task of refueling worker robots. Zuluaga and Vaughan [13] describes one such special-purpose tanker robot, capable of finding and delivering energy to robot(s) in need of energy. Searching/communication overhead, however, might be an issue in such scenario if workers are spread over a large environment. Zebrowski et al. [14] introduces a distributed heuristic for an initially unknown single workers/tanker rendezvous location in an obstacle-free environment that minimizes the total traveling cost of the system. In such approach, complication of finding one such rendezvous location would be increased as the workers population grows. Litus et al. [15] utilizes concept of Fermat-Torricelli point [16] to find an optimal set of meeting places for the tanker to rendezvous with multiple worker robots given an ordering of meetings. The robots field of operation is considered to be free of obstacles. The present paper examines the statistical technique known as regression analysis for computing a recharging route for a group of robots in their working environment. A special-purposed robot (referred to as tanker robot hereafter) is devised with a recharging route planner that computes a set of rendezvous locations, using individual robotic agents (referred to as worker robots hereafter) locations’ information. The proposed planner might be categorized as a centralized planner since tasks of route generation and instructing worker robots regarding their corresponding recharging locations along with the approximate rendezvous times are performed by tanker. Coordinates information of worker robots may provide the planner with the opportunity of modeling the recharging task in term of agents’ distribution over the field of operation. The planner will take into account the distribution of the worker robots in the environment and construct the recharging route via manipulation of worker robots’ current locations. Such approach may reduce the task of tanker during the recharging process from searching for robots in the field to a simple traversal of charging route, along which
Int J Soc Robot (2012) 4:15–27
worker robots will be met for recharging. The recharging route is constructed online, through modification of current worker robots’ location information and hence multirendezvous in nature. The multi-location property of regression route eliminates the distance traversal compromise that might be imposed on some of the worker robots in single rendezvous recharging scenario. We primarily focus on minimizing the total distance traversal of worker robots during the recharging process. We study the ordinary and weighted least squares (OLS and WLS respectively) as well as orthogonal least absolute values (ORLAV) regression techniques for such recharging route construction. Furthermore, both environmental setups i.e. environments with and without obstacles, are taken into consideration while analyzing the performance. The remaining of the paper is organized as follows: Problem description is presented in Sect. 2. Assumptions adapted while implementing approaches (both LSRR as well as ORR) are provided in Sect. 3. Section 4 discusses the LSRR planner in details. Section 5 explains the short-comes that are encountered while using least square method and investigates some advantages of choosing orthogonal least absolute values over least square techniques. ORR planner (ORLAV based planner) is elaborated in Sect. 6. In Sect. 7, properties of ORR planner are presented. Section 8 demonstrates the simulation results. Further discussions and conclusion are given in Sects. 9 and 10 respectively. A brief introduction to linear regression is provided in the Appendix.
2 Problem Description Problem of multi-robot, multi-rendezvous recharging route can be expressed as: Given a set of locations information for a group of worker robots, how to utilize these information to construct a linear recharging route that passes through tanker-workers rendezvous points for recharging and that minimizes the total sum of distances traveled by worker robots to accomplish the task.1 One way to formulate the problem is to model the worker robots’ relocation cost functions as the weighted distances of the robots to their corresponding rendezvous locations along the recharging route. Following definition provides the formalized statement of recharging route planner problem. Definition 1 (Recharging route planner problem) Given worker robots locations ri ∈ Rd (d ≥ 2), i = 1, . . . , n and 1 The solution to the 2-dimensional analog of the problem in location theory (e.g. access to a linear resource such as a highway or utility) is referred to as a “1-line median”.
Int J Soc Robot (2012) 4:15–27
17
the relocation cost function Ci : Rd → R, i = 1, . . . , n, find the recharging route that satisfies min
n X
(1)
Ci (ri , pi )
Upon receiving their respective estimated recharging time, only those workers that may not have enough energy to travel back to their designated recharging rendezvous locations will come to complete halt.
i=1
where
4 LSRR Planner i th
ri : Current Location of robot. pi : Recharging location for i th robot along the recharging route. Here, Ci (x, y) gives the cost of i th worker relocation from its current position x to its designated rendezvous location y for recharging i.e. Ci (ri , pi ) = wi kpi − ri k,
i = 1, . . . , n
(2)
3 Basic Assumptions Following assumptions have been adapted while implementing both LSRR as well as ORR recharging route planners: 1. Tanker robot is considered to be the main source of energy. 2. Tanker robot does not require refueling. 3. Recharging process of the worker robots will be started once the mean energy value of the entire flock reaches ¯ a pre-determined energy threshold level i.e. when PnE ≤ ε 1 where ε is the energy threshold level and E¯ = n i=1 ei , with ei and n being the current energy level of i th robot and total number of robotic agents, respectively. 4. Tanker will initiate the recharging process by traversing the route, starting from the route’s end-point that’s closer to its current resting location. 5. Once the route is constructed, every worker will be informed by tanker regarding its approximate recharging time i.e. time in which the worker is expected to meet the tanker at specified recharging location. Tanker calculates every worker’s estimated recharging time, using formula below: Td (τ, pi ) i = 1 P Recharging = Td (τ, p1 ) + ni=1 Td (pi−1 , pi ) (3) Ti + TE (pi−1 ) Otherwise
where
Recharging
: i th worker robot’s rendezvous time with tanker for recharging. τ : Tanker location. pi : Recharging location for i th robot. Td (p, q): Time required to travel between p and q locations. TE (p): Time required to perform recharging at location p. Ti
Tanker will be communicated by the worker robots about their respective locations in which they have reached the predefined energy lower-bound level. The information is used by tanker to construct the regression route as follow: Considering the robots as data points, each such data would consist of (x, y) pairs of measurements that describe the robots’ respective locations in the environment. Here we also assume x and y coordinates’ information as independent and dependent variables respectively. Another words, the planner gives higher priority to the x-coordinate’s information which in turn result to a one-dimensional (with regards to Cartesian coordinate system) constructed regression route. Having the recharging route to be presented by equation of the line: (4)
Y = aX + b
where, (X, Y ) = {(xi , yi )}, i = 1, . . . , n be the set of worker robots’ coordinates and n equals number of worker robots. We attempt to find the values of two parameters namely a (slope of the line) and b (line interception with y-coordinate) so as to minimize the discrepancy between measured and calculated values yi and y(xi ) where yi : the y-coordinate of i th robot y(xi ) = axi + b: Approximate functional relationship between xi and y(xi ), i = 1, . . . , n i.e. robots’ new ycoordinate values along the recharging route, with n being the total number of robots. With Gaussian assumption, for any estimated values of the parameters a and b, the probability of obtaining the observed set of measurements can be calculated as [19]: P (x, y) =
n µ Y i=1
¶ P y −y(xi ) 2 1 ] ) (− 1 n [ i e 2 i=1 σi √ σi 2π
In which quantities
1 σi2
(5)
serve as weighing factors. This is to
say that each data point is weighted inversely by its variance σi2 . Reason for choosing such criterion as weighing factor is due to the fact that distribution information may not have been measured with same precision. As an example, locations information of some robots may not be as precise as others due to noise or communication failure. Such
18
Int J Soc Robot (2012) 4:15–27
discrepancy in information precision can be represented by assuming a population distribution with same mean value µ but different standard deviation σi . The goal of finding the optimum fit to data can be then expressed as finding a and b values in (4) so as to minimize the exponential term (i.e. goodness-of-fit parameter or χ 2 ) in (5). In other words, obtaining the smallest sum of the squares or least-squares fit. To construct the linear regression route, slope a and line interception with y-coordinate i.e. b are required to be determined. Setting the partial derivatives of the exponential term in (5) with regards to parameters a and b, to zero, will yield to [19]: n n n X X X 1 xi yi = a + b 2 2 2 σ σ σ i=1 i i=1 i i=1 i
(6)
n n X X xi2 xi + σ 2 i=1 σi2 i=1 i
(7)
n X xi yi i=1
σi2
=a
Solving (6) and (7) for parameters a and b, we will have: ! Ã n n n n 1 X xi2 X yi2 X xi X xi yi − (8) a= 2 2 2 2 ∆ σ σ σ σ i i i i i=1 i=1 i=1 i=1 Ã n ! n n n 1 X 1 X xi yi X xi X yi b= − ∆ σ2 σi2 σ2 σ2 i=1 i i=1 i=1 i i=1 i #2 " n n n X xi X 1 X xi2 − ∆= σ2 σ2 σ2 i=1 i i=1 i i=1 i
(9)
(10)
Once a and b parameters are known, these values can be exploited to determine the workers-tanker recharging rendezvous locations: yi = axi + b,
i = 1, . . . , n
sets of observations based on which the behavior of response variables (also known as dependent variable) can be predicted by the predictor(s) (i.e. regressor or independent variable). This basic assumption requires us to pre-define a certain dependency among the position coordinates of the robots by considering one as function of the other. It further assumes that predictor or independent variable is free of error and confines all the errors to dependent or response variables [21]. Another demerit in using least square linear regression is when outliers are among the observations (in our case, robots that are located in further distances with regards to the rest of the flock). Since the objective function in least square evaluates squares of vertical distances to the hyperplane, such outlying observations are unduly heavily weighted [22]. To overcome such limitations that are due to the nature of information involved, use of orthogonal least absolute values (ORLAV) for recharging route construction has been examined. The technique is also referred to as Euclidean minimum sum of absolute errors (EMSAE), Euclidean regression (OR) and total least squares (TLS), interchangeably in the literature. Some of the advantages of using ORLAV are: 1. No choice of dependent and independent variables is required: Since the estimators of the orthogonal regression are obtained by minimizing the orthogonal distances of data points to the fitted hyperplane, it is especially applicable for situations in which the dependent and independent variables cannot be pre-determined. 2. It is resistant to outliers [23] since it minimizes the sum of orthogonal distances of each data point to the resulting hyperplane as opposed to minimizing the sum of squares of vertical distances in least square linear regression. 3. It is ideally suited for situations in which data under analysis is corrupted by noise [21].
(11)
where n is number of worker robots in the field. As it can be seen, the regression route and hence the rendezvous locations are determined through the readjustment of the y-coordinate of worker robots’ current locations. This would result in a horizontally constructed regression route that has its interception with y-coordinate.
5 Preliminary: from LSRR to ORR Planner One of the assumptions which is required to be met by the observations in order to fit the least square regression analysis is the dependency of observational data. This is to say that, in order for data to be studied under linear regression, there should exist a response-predictor relation among
6 ORR Planner Analogous to LSRR planner scenario, tanker will be communicated by worker robots about their respective locations in which they have reached the predefined energy lowerbound level. Such information consists of x, y pairs of measurements and describes the robots respective locations in the environment. For a line to be a 1-line median (i.e. a line that minimizes sum of orthogonal least absolute values of corresponding data points to it), it requires to satisfy a necessary condition of spreading the set of points into two sets P of approximately same weights, i.e. if W = ni=1 wi , T + = {i : axi + byi > −c} and T − = {i : axi + byi < −c}, denote total weight of all data points, and sets comprising data points above and below the line respectively, then following
Int J Soc Robot (2012) 4:15–27
19
inequality is required to be preserved [24]: X X 1 , ≤ W 2 + −
i∈T
i = 1, . . . , n
i∈T
(12)
Having the recharging route been represented by equation of the line ax + by + c = 0
(13)
ORR planner (Algorithm 1) calculates a, b and c parameters so as to minimize the weighted sum of distances of worker robots from (13). Such line will always pass through at least two of the data points. Recalling (12) that indicates division of worker robots into two sets (i.e. robots that are above the recharging route and those below the route), total weighted sum of orthogonal distances becomes ¶ ·µ X X 2 2 −1 2 (a + b ) wi x i a wi x i − i∈T −
i∈T +
+
µX
i∈T +
wi y i −
X
i∈T −
¶ µX X ¶ ¸ wi y i b + wi − wi c i∈T +
i∈T −
(14)
Algorithm 1: ORR Planner Data: (ri , rj ) worker robots’ data points pairs for iterative line generation. begin i ← 1; k ← 1; while i ≥ n − 1 do for j = i + 1 : n do find ak , bk and ck for the line ak x + bk y + ck = 0 that passes through (ri , rj ) for m = 1 : n do if ak xm + bk ym < −ck then T + ← {m} else T − ← {m} S ← {(ak , bk , ck )} k←k+1 Sort S in an ascending order of ak values. return s ∈ S that satisfies P −1 P min[(ai2 + bi2 ) 2 [( i∈T + wi xi − i∈T − wi xi )ai + P P (Pi∈T + wi yi P − i∈T − wi yi )b Pi + ( i∈T + wi +( i ∈ T + wi − i ∈ T − wi )]] end
Fig. 1 Collinear worker robots and the corresponding recharging route
7 Properties of ORR Planner As it has been demonstrated, the ORR planner minimizes sum of orthogonal distances of worker robots from the recharging route. It differs from the LSRR planner which is primarily based on minimization of sum of squared of vertical distances of worker robots from the route. There are several advantages in such shift from vertical to orthogonal distance minimization as presented below. Lemma 1 (Collinear case) The weighted sum of Euclidean distances of collinearly arranged worker robots ri , i = 1, . . . , n from the recharging route is always zero. Proof In such scenario (as depicted in Fig. 1) current locations of robots and their corresponding rendezvous locations for recharging coincide. Hence n X i=1
Ci (ri , pi ) = 0
(15) ¤
Lemma 2 (Charging location along the recharging route) If charging station/tanker location is along the linearly constructed recharging route, then sum of Euclidean distances traveled by worker robots to their corresponding recharging locations along the route will be smaller than sum of distances of worker robots to charging station/tanker location. Proof Referring to Fig. 2, let L represent the linearly constructed recharging route that minimizes weighted sum of Euclidean distances of worker robots ri , i = 1, . . . , n. Let f be the position of charging station. In 4ri pi f : wi kpi − ri k ≤ wi kf − ri k
(16)
where wi kf − ri k: weighted distance of i th robot to charging station. wi kpi − ri k: weighted Euclidean distance of i th robot from its current to recharging locations. which results in: n X i=1
wi kpi − ri k ≤
n X i=1
wi kf − ri k
(17) ¤
20
Int J Soc Robot (2012) 4:15–27
i = 1, . . . , n, therefore n X i=1
wi kpi − ri k ≤
n X i=1
wi kpi0 − ri k
(19)
Using (18) and (19), we have n X i=1
Fig. 2 Worker robots and the corresponding recharging route. Recharging station/tanker location is labeled f
wi kpi − ri k ≤
n X i=1
wi kf − ri k
(20) ¤
It can be shown that result obtained in Theorem 1 also holds for the case in which charging station/tanker location is the Fermat-Torricelli point of worker robots’ locations (see Corollary 1, below). Before succeeding to the corollary, however, we first restate the following lemma along with a definition and two claims (without proof) since they will be required for proving the corollary. Lemma 3 Relative to the Euclidean metric there exists a 1line median which contains at least two points from the set {(x1 , y1 ), . . . , (xn , yn )} [24]. Definition 2 Given n points p1 , . . . , pn ∈ Rd (d ≥ 1), find x0 ∈ Rd which minimizes the function2 [25]
Fig. 3 Worker robots and the corresponding recharging route. Recharging station/tanker location is labeled f
Theorem 1 Sum of Euclidean distances traveled by worker robots ri , i = 1, . . . , n to their corresponding recharging locations is smaller than sum of distances traveled by workers to attend single charging station, irrespective of charging station/tanker location in the environment. Proof Referring to Fig. 3, let L be the recharging route that minimizes the sum of orthogonal distances of worker robots ri , i = 1, . . . , n and f be the location of charging station/tanker. We draw a line L0 that passes through f and is parallel to L. Using Lemma 7 and considering 4ri pi0 f : n X i=1
wi kpi0 − ri k ≤
n X i=1
wi kf − ri k
(18)
where wi kpi0 − ri k: weighted Euclidean distance of i th robot from its current to recharging locations along the L0 . wi kf − ri k: weighted distance of i th robot from charging/tanker location. L, however, is (by its definition) the line that minimizes sum of orthogonal distances of worker robots ri ,
f (x) =
n X i=1
wi kpi − xk x ∈ Rd (d ≥ 1)
(21)
Claim 1 The function f is convex [25]. Claim 2 The function f is a strictly convex function if and only if the points p1 , . . . , pm are not collinear [25]. Corollary 1 Given a set of worker robots locations ri , i, . . . , n and charging station f , sum of orthogonal distances traveled by worker robots to recharge along the ORR based recharging route will be smaller than sum of distances to the recharging point f whose location is the Fermat-Torricelli point of the worker robots’ locations.
Proof – Floating Case: (i.e. when the Fermat-Torricelli point is not coincidental with either of worker robots locations). Referring to Fig. 3 it is easy to see that the analogue of Theorem 1 holds and therefore (20) is also true for the case of recharging point determined by Fermat-Torricelli. – Absorb Case: (i.e. when the location of the FermatTorricelli recharging point coincides with one the worker robots location). 2 f is the function that returns the Fermat-Torricelli point of n given points.
Int J Soc Robot (2012) 4:15–27
21
Fig. 4 Worker robots and the corresponding recharging route. Fermat– Torricelli based recharging location is coincidental with r3
Figure 4 depicts one such scenario in which the recharging location is coincidental with robot r3 . Claims 1 and 2 ensure that the absorbed worker robot location would definitely lie within the convex hull of the rest of worker robots’ locations. Using aforementioned claims and Theorem 1, we have: n−{3} X i=1
wi kpi − ri k ≤
n−{3} X i=1
wi kr3 − ri k
(22)
where n is number of worker robots. To satisfy the condition in Theorem 1, it is required to show that cost of relocating one such absorbed worker robot to its designated recharging location will be less than cost of relocating those worker robots that are coincidental with the recharging route (Lemma 3), hence the total sum of orthogonal distances, less than the sum of distance traversal to charging location determined by Fermat-Torricelli. Re0 ferring to Fig. 4, let L and L be the recharging route and the line that passes through the fixed charging location f and is parallel to L, respectively. Recalling Lemma 3, let R1 and R2 be the two worker robots whose locations are coincidental with the recharging route. As a result, sum of orthogonal distances of R1 and R2 to recharging route will be zero, i.e. 2 X i=1
Fig. 5 (Color online) ORR Based Recharging Route. Locations of workers and tanker robots are shown by circles and square, respectively
Equation (26) states that despite having one of the robot (in this case r3 ) in absorbed situation and resultantly not required to be relocated for recharging, the total distance traversal by robots Ri that are along the recharging route, is greater than that of the distance cost incurred by relocation of robot r3 , i.e. n−{3} X i=1
≤ n X i=1
wi kpi − ri k + w3 kr3 − p3 k + n X i=1
wi kf − ri k +
wi kpi − ri k ≤
n X i=1
2 X i=1
2 X i=1
wi kPi − Ri k
wi kf − Ri k
wi kf − ri k +
2 X i=1
=⇒ wi kf − Ri k (27) ¤
8 Case Studies wi kPi − Ri k = 0
(23)
With Pi be the allocated recharging location for worker Ri . In 4Ri p3 f (r3 and f are coincidental) we have w3 kp3 − r3 k ≤ w1 kf − R1 k
(24)
w3 kp3 − r3 k ≤ w2 kf − R2 k
(25)
Using (24) and (25) w3 kp3 − r3 k ≤
2 X i=1
wi kf − Ri k
(26)
To verify the validity of the proposed approaches, they were examined in simulation environment. While performing the simulation, robots’ field of operation was defined to be a 16 × 16 rectilinear polygon. We examined our approach against fixed charging station as well as recharging route constructed using least square linear regression [20]. Recharging routes generated using ORR as well as LSRR planners are presented in Fig. 5 and Fig. 6, respectively. Red-colored circles are the current locations of worker robots. The fixed charging station (which is the location of tanker robot as well) is shown as blue-colored square in bottom left corner of the figures. In Fig. 5 the ORR based planner recharging route is presented. As it can be seen in
22
Int J Soc Robot (2012) 4:15–27
– Weights are considered to be equal to inverse of squared root of each data point independent variable i.e. √1x . i
Fig. 6 (Color online) OLS/WLS Based Recharging Routes. Worker Robots are presented by circles. Tanker location is shown by the square in bottom left corner
the figure, route generated by minimizing the orthogonal distances of worker robots to the recharging route passes through at least two of the current worker robots positions. The linear regression routes in Fig. 6 are constructed based on method of least-square using different weighing schemes. The individual recharging location for each robot is in fact the projection of the current location of the robot that has been vertically (i.e. along the y-coordinate) re-adjusted to fit the line. Two environmental setups, namely obstacle-free environment and environment with obstacles (i.e. partitioned environment) have been considered while collecting the results. For LSRR and ORR based recharging routes, two types of results have been collected: 1. Total worker robots distance traversal during the recharging process, without considering distance traveled by the tanker robot. 2. Total distance traversal at system level, taking into account distance traveled by the tanker. For route generated using LSRR planner, two different scenarios have been considered: 1. Weighted Least Squares (WLS): where the measures of uncertainties on worker robots locations’ information have been taken into account. We have applied the following weighing principles [26–28]: – Data points are inversely weighted by their respective variances 12 , where σi2 is computed using σi
σi2
=
Pn
− y¯i )2 n−1
i=1 (yi
– y1i is considered to be the weight. – Data points are weighted by the magnitude of xyii . Using different weighing factors would, in essence, provide the route planner with more flexibility while calculating the rendezvous locations. Referring to Fig. 6, for instance, it can be seen how the behavior of the recharging route changes as the weighing factor is modified. Such flexibility on constructing the route can be used to address some of the uncertainties that may exist in workers’ positioning information and/or their respective energy levels as well as terrain condition, etc. As shown in the figure, route constructed using wi = 1 (i.e. least uncertainty in workers’ positioning information) fits the recharging route so as to provide best traversal for higher density distribution of the workers, giving less value to the outlier-ed worker robot(s). 2. Ordinary Least Squares (OLS): in which all the uncertainties are considered to be equal and set to 1. In such a case (8), (9) and (10) might be rewritten as: ! Ã n n n n X X 1 X 2X (29) xi yi xi yi − xi a= ∆ i=1
i=1
i=1
i=1
à n ! n n X X X 1 xi yi − xi yi b= n ∆ i=1
∆=n
n X
xi2
i=1
i=1
à n !2 X xi −
(31)
i=1
OLS can be considered as situation in which complete distribution information of robots’ locations are available. The x and the y coordinates of worker robots have been treated as independent and dependent variables respectively while constructing the recharging route based on LSRR planner (both OLS as well as WLS). We have examined the total distance traversal for recharging in fixed charging station as well as recharging based on LSRR and ORR planners. For this purpose, individual as well as cumulative sum of distance traversal of robots are required. The total distance traversal of all worker robots is calculated as: dtotal =
n X
di
i=1
where (28)
(30)
i=1
n: Total number of worker robots.
(32)
Int J Soc Robot (2012) 4:15–27
23
di : Euclidean distance between current and recharging locations of the i th robot. For 2nd case, distance traversal by the tanker has been calculated based on starting point of the recharging process on the regression route. Given the end points of the recharging route, recharging process is initiated from the end point closer to tanker i.e. min(dstart , dend )
(33)
Table 1 Distance traversal mean value, standard deviation as well as median for different weighing schemes of least square linear regression, fixed charging station and ORR planner Workers Robot Mean STD
Tanker Robot
Median Mean STD Median
ORLAV
4.09
1.26
5.00
wi = 1
9.88
2.59
9.00
20.35 3.88 20.00
2.98 10.90
20.98 3.48 20.44
9.77
2.62
9.00
20.07 3.72 20.00
10.01
2.71
9.05
20.00 3.67 20.00
15.51
4.11 13.00
wi =
wi =
1 σi2 √1 xi 1 yi yi xi
10.95
20.21 3.50 20.00
dstart and dend represent Euclidean distances of tanker to the regression route’s end points. Function (33) returns the smallest value of its two arguments. Total distance traveled by the tanker is then:
wi =
dtanker = droute + min(dstart , dend )
Figure 7 shows total distance traversal of worker robots for recharging. Total distance traversal at system level and after introduction of distance traveled by tanker during the recharging process is depicted in Fig. 8. Table 1 summarizes distance traversal mean values for different strategies. As shown in the table, ORR based recharging route planner outperforms both, fixed charging station as well as least square based recharging strategies. This is true in particular for case of worker robots distance traversal for recharging.
(34)
where droute : Length of the regression route. The total system distance traversal during the recharging process is then: dsystem = dtanker + dtotal
(35)
with dtanker and dtotal representing distance traveled by tanker and cumulative sum of distances traveled by worker robots, respectively. 8.1 Obstacle-Free Environment 18 test runs, with 5 workers and 1 tanker, have been examined based on following configurations: 1. Worker robots are distributed randomly. 2. Worker robots are placed to the farthest distances possible to fixed charging station/tanker’s resting location. 3. Workers are located as close as possible to fixed charging station/tanker’s resting location. 4. Worker robots are placed as close as possible to each other but farthest to fixed charging station/tanker location. While performing the simulation, fixed charging station/ tanker robot was relocated to the top and the bottom left most and right most corners as well as center of the environment. Above configurations were then examined against each of these relocations. For instance, fixed charging station/tanker was placed in top left most corner of the field and then all worker robots configurations described above (1 through 4) were evaluated. We have used the same workers and tanker configurations for both, fixed charging station as well as recharging strategy based on regression route techniques (both LSRR as well as ORR based recharging routes). Size of field of operation has been kept the same for all tests.
wi =
Fixed Charging Station 65.54 16.80 76.00
20.01 3.68 20.00 –
–
–
8.2 Environment with Obstacle While performing the recharging task in presence of obstacles, tanker robot was devised with a kernel-based navigation controller [29]. The recharging route planners (both LSRR as well as ORR) were also modified so as to calculate the piecewise recharging route. As shown in Fig. 9, worker robots are grouped as per their positioning information within their deployment units, thereby calculating the routes as per workers’ distributions. Positioning information of the obstacles within the field was assumed to be known and was used by tanker’s path planner to compute a collision-free traversal between different recharging routes. Number of worker robots have been increased to 11 as opposed to 5 in case of obstacle-free environment. Results obtained have been provided in Table 2.
9 Discussions Practicability of regression analysis in constructing a route for recharging a group of worker robots in their deployment units has been studied. In this context, the recharging route construction may also be considered as a sub-domain of broader topic of motion planning in dynamic environment since all the robots, whether tanker or worker, are required to reach a target location (it is multi-target-location for the tanker in our case) from their current positions and
24
Int J Soc Robot (2012) 4:15–27
Fig. 7 Workers Distance Traversal in fixed charging station, least square regression and ORR planner. Different weighing schemes have been used while using least square regression for recharging route con-
struction. (1) Workers farthest to tanker closest to each other, (2) Workers closest to tanker, (3) Workers distributed Randomly, (4) Workers farthest from tanker
Fig. 8 Workers and Tanker Distance Traversal while relocating the Tanker in fixed charging station, least square regression and ORR planner. Different weighing schemes have been used while using least
square regression for recharging route construction. (1) Workers farthest to tanker closest to each other, (2) Workers closest to tanker, (3) Workers distributed Randomly, (4) Workers farthest from tanker
Int J Soc Robot (2012) 4:15–27
25
Fig. 9 (Color online) OLS Based Recharging Routes with Wi = 1. Worker Robots are presented by circles. Tanker location is the square in bottom left corner. Obstacles are shown in black Table 2 Piecewise recharging in environment with obstacles. Distance traversal mean value, standard deviation as well as median for different weighing schemes of least square linear regression, fixed charging station and ORR planner Workers Robot Mean
Tanker Robot
STD Median Mean STD Median
ORLAV
7.13
4.59
6.98 26.21 6.42 28.00
wi = 1
14.54
3.23
12.08 26.35 5.98 28.21
21.53
5.96
13.62 33.58 7.14 26.50
15.77
4.55
16.00 33.74 5.51 26.09
19.06
4.71
11.15 28.16 8.94 23.76
28.65
9.11
21.43 36.18 8.18 26.00
wi = wi =
wi = wi =
1 σi2 √1 xi 1 yi yi xi
Fixed Charging Station 142.16 45.98 156.15 –
–
–
perform a designated tasks (recharging for the workers and supplying the energy for the tanker robots) while avoiding collision with each other. It has been shown that such route can be constructed based on locations’ information of worker robots for whom the supply of energy is needed. It has further been proven that total distance traversal of such group of worker robots for recharging will be minimized if the recharging route is the one that minimizes the orthogonal distance of every individual worker robot from the recharging route. It has also been proved that irrespective of charging station/tanker location, the above conclusion will be held. As illustrated through simulation, the ORR planner, in particular, is capable of constructing such recharging route that minimizes the sum of distances traversed by worker robots for recharging. It has also been observed that sum of worker robots’ distance traversal and resultantly, en-
ergy consumption, required for recharging are minimized. Both methods introduced also eliminate the need for worker robots to travel between their work zone and some stationary, fixed power station. Table 1 shows that overall performance of the system equipped with LSRR planner (irrespective of weighing schemes adapted) is also better than fixed charging station strategy. This is true in particular, for the case of worker robots distance traversal. Considering the different weighing schemes that have been adapted while using LSRR planner, however, it is clear that the differences between the LSRR results are statistically insignificant. An analysis of Table 1 entries shows that all the LSRR worker robot results are within one standard deviation of each other. Thus, their differences might be considered statistically insignificant. As a result, drawing conclusions on weighing schemes’ performances is not warranted. The ORLAV planner, however, shows a significant shift in performance improvement when compared to fixed charging station as well as LSRR and irrespective of weighing schemes adapted in the latter case. In case of environment comprising obstacles (Table 2), same conclusion as of above might be drawn while analyzing the table’s entries. As shown in Table 2, the LSRR results are all within one standard deviation of each other, and hence their difference might be interpreted as statistically insignificant. Furthermore, an analysis of Table 2 while considering the worker robots’ distance traversal, shows that the ORLAV results are within one standard deviation of the LSRR results using wi = 1 and wi = √1x which is clearly i not the case for Table 1, i.e. obstacle-free environment. Such peculiar behavior of the planner in environment with obstacles might be due to the sub-grouping of the worker robots for constructing recharging route. In other words, whereas worker robots positioning information in obstacle-free environment are considered all together and as one single group, it is necessary for the planner (in presence of obstacles) to partition workers into separate sub-groups before applying the piecewise regression. As a result, worker robots that are members of one such sub-group, are spatially closer to each other (due to the constraint imposed by the presence of the obstacles) and hence distribution density of the sub-group is reduced. However, further studies and analysis of the system performance in environment with obstacles are required in order to technically as well as theoretically address and clarify such behavior. Although methods presented in this paper, have shown capability in demonstrating some convincing results, they are, by no means, considered as a definitive solution for this issue, but an starting point. For instance, our simulations do not consider the constraints imposed by the environmental condition such as shape of terrain, occurrence of moving obstacles and alike. Such techniques, however, show solutions to certain type of condition and might be effective and
26
Int J Soc Robot (2012) 4:15–27
Fig. 10 Data Points and their corresponding projection along the regression line
practical for real-life, complex environments, if employed in conjunctions with other approaches that have been proposed in the literature.
satisfying the coordination problem among the tankers during the process of task allocation.
Appendix: Linear Regression: A Brief Introduction 10 Conclusion and Future Direction Recharging of multiple robots, using a special-purpose tanker, devised with a regression route planner has been examined. It has been proven that total distance traversal of such group of worker robots for recharging will be minimized if the recharging route is the one that minimizes the orthogonal distance of every individual worker robot from the recharging route as oppose to sum of squared of vertical distances as in case of least square regression. It has also been proven that irrespective of charging station location, the above conclusion will be held. Despite the capability of current implementation for calculating best fitted energy supply line for the system, there are several consideration that are required to be addressed in future. Determining the starvation state of the entire system is a crucial point by which the effectiveness of the approach might be influenced significantly. We believe computing the five number summary that takes into account range, median and quartile values of total system energy level (in addition to mean value) would provide a more reliable estimate upon which the recharging process might be started. Another measure that might introduce a significant improvement over the entire system recharging process is the manipulation of regression route construction by computing the equation of the recharging route so as to have one of its starting points, laying on the tanker location. It may also be interesting to introduce multiple tankers and address the recharging issue of a bigger crowd of worker robots while
Regression Analysis is a method of statistical analysis that deals with investigation of relationship between two or more variables related in a non-deterministic fashion [17]. The primarily usage of regression analysis and its corresponding techniques is to model and analyze the numerical data comprising values of a dependent variable (also called response variable or measurement) and one or more independent variables (also referred to as predictor or explanatory variable). Simple linear regression is a regression analysis in which the expected value of dependent variable is assumed to be a linear function of explanatory variable. It uses the method of least squares to interpret the relationship between the explanatory and response variables by fitting a straight line that minimizes the sum of the squares of vertical distances of data points from the line. This is to say that least squares regression, in fact, minimizes the sum of the squares of deviations of data points from the straight line that best fits the data. Deviations are generally measured along the ycoordinate [18]. The best fit in the least squares sense is then the instance of the model for which the sum of squared residuals has its least value, a residual being the difference between an observed value and the value given by the model. To represent data graphically (e.g. scatter plot) the independent variable is shown along the x-coordinate whereas the dependent variable forms the y-coordinate values. Figure 10 illustrates concept of linear regression, using least squares. Blue-colored circles are the worker robots’ that are scattered over the Cartesian space. Green-colored squares that are connected via a red dotted line, are in fact the projection
Int J Soc Robot (2012) 4:15–27
of same data points into a straight line such that their vertical distances to the line is minimized.
References 1. Walter WG (1963) The living brain. Norton, New York 2. Silverman M, Nies DM, Jung B, Sukhatme GS (2002) Staying alive: a docking station for autonomous robot recharging. In: Proceedings IEEE international conference on robotics and automation, Washington, DC, pp 1050–1055 3. Oh S, Zelinsky A (2000) Autonomous battery recharging for indoor mobile robots. In: Proceedings of the Australian conference on robotics and automation. [online]. Available: citeseer.ist.psu.edu/oh00autonomous.html 4. Keshmiri S, Payandeh S (2009) An overview of mobile robotic agents motion planning in dynamic environments. In: 14th IASTED international conference on robotics and applications, Cambridge, MA, pp 152–159 5. Foka AF, Trahanias PE (2010) Probabilistic autonomous robot navigation in dynamic environments with human motion prediction. Int J Soc Robot 2(1):79–94 6. Sisbot EA, Marin-Urias LF, BroquÃre X, Sidobre D, Alami R (2010) Synthesizing robot motions adapted to human presence. Int J Soc Robot 2(3):329–343 7. Cortes J, Martinez S, Bullo F (2006) Robust Rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions. IEEE Trans Autom Control 51(8):1289–1298 8. Lin AMJ, Anderson BDO (2003) The multi-agent rendezvous problem. In: Proceedings of the 42th IEEE conference on decision and control, Maui, Hawaii, pp 1508–1513 9. Schlude K (2003) From robotics to facility location: contraction functions, Weber point, convex core. Technical report 403, online: http://www.inf.ethz.ch/personal/schlude/contractionfunction.pdf 10. Dudek G, Roy N (1997) Multi-robot rendezvous in unknown environments or, what to do hen you’re lost at the zoo. In: Proceedings of the AAAI national conference workshop on online search, Providence, RI. 11. Ando H, Oasa Y, Suzuki I, Yamashita M (1999) A distributed memoryless point convergence algorithm for mobile robots with limited visibility. IEEE Trans Robot Autom 15(5):818–828 12. Lanthier MA, Nussbaum D, Wang TJ (2005) Calculating the meeting point of scattered robots on weighted terrain surfaces. In: Proceedings of the Australasian symposium on theory of computing. Australian Computer Society, Darlinghurst, pp 107–118 13. Zuluaga M, Vaughan R (2005) Recharging robot teams: a tanker approach. In: Proceedings of the 2005 international conference on advanced robotics, Seattle, WA 14. Zebrowski P, Litus Y, Vaughan R (2007) Energy efficient robot rendezvous. In: Proceedings of the 4th Canadian conference on computer and robot vision, Montreal, QC, pp 139–148 15. Litus Y, Vaughan RT, Zebrowski P (2007) The frugal feeding problem: energy-efficient, multi-robot, multi-place rendezvous. In: Proceedings of the IEEE international conference on robotics and automation, Roma, Italy, pp 27–32
27 16. Kuptiz Y, Martini H (1997) Geometric aspects of the generalized Fermat-Torricelli problem. J Bolyai Soc Math Stud 6:55–129 17. Devore JL (2004) Probability and statistics for engineering and the sciences, Thomson (Brooks/Cole), 6th edn. Pacific Grove, California 18. Jeffrey A (2005) Mathematics for engineers and scientists, 6th edn. Chapman & Hall/CRC Press, Boca Raton 19. Bevington PR, Robinson DK (2003) Data reduction and error analysis, 5th edn. McGraw-Hill, Boston 20. Keshmiri S, Payandeh S (2009) Multi-robots, multi-locations recharging paradigm: a regression route technique. In: Proceedings of the 14th IASTED international conference, robotics and applications, Cambridge, MA, USA, pp 160–165 21. Van Huffel S, Lemmerling P (2002) Total least squares and error-in-variables modeling: analysis, algorithms and applications. Kluwer Academic, Dordrecht 22. Bargiela A, Hartley JK (1993) Orthogonal linear regression algorithm based on augmented matrix formulation. J Comput Optim Res 20(9):829–836 23. Cavalier TM, Melloy BJ (1991) An iterative linear programming solution to the Euclidean regression model. J Comput Optim Res 18(8):655–661 24. Megiddo N, Tamir A (1983) Finding least-distances line. SIAM J Algebr Discrete Methods 4:207–211 25. Boltyanski V, Martini H, Soltan V (1999) Geometric methods and optimization problems. Kluwer Academic, Boston 26. Neter J, Wasserman W, Kutner MH (1990) Applied statistical models: regression, analysis of variance, and experimental designs, 3rd edn. Irwin, Homewood 27. Chatterjee S, Price B (1991) Regression analysis by example, 2nd edn. Wiley series in probability and mathematical statistics. Wiley, New York 28. Kutner MH, Nachtsheim CJ, Neter J, Li W (2005) Applied linear statistical models, 5th edn. McGraw-Hill, Irwin 29. Keshmiri S, Payandeh S (2010) Robot navigation controller: a non-parametric regression approach. In: IFAC workshop on intelligent control system (WICS2010), Sinaia, Romania
Soheil Keshmiri is a Ph.D. student in Experimental Robotics Laboratory at Simon Fraser University, Canada. He received a B.S. degree in computer science from, Pune University, Pune, India, in 2004 and a M.S. degree in computer science from, Hamdard University, New Delhi, India, in 2007. He was a visiting research associate in Budapest University of Technology and Economics in Fall 2010. He is currently with IBM, Markham, ON, through an internship program. His current research interests include autonomous navigation, strategic planning and coordination of multi-robot systems. Shahram Payandeh is a Professor in School of Engineering Science at Simon Fraser University, Canada, since 1991. He received his Ph.D. degree in 1990 from the University of Toronto, Canada. His main area of research includes interaction and coordination of networked, cooperative dynamical systems. Prof. Payandeh co-authored the first book in the area of medical robotics and holds the first patent in this area.