A Greedy Strategy for Tracking a Locally Predictable Target among Obstacles

Transcription

1 Proceedins of the 2006 IEEE International Conference on Robotics and Automation Orlando, Florida - May 2006 A Greedy Stratey for Trackin a Locally Predictable Taret amon Obstacles Tirthankar Bandyopadhyay, Yuanpin Li, Marcelo H. An Jr., and David Hsu Department of Mechanical Enineerin National University of Sinapore Sinapore, , Sinapore Department of Computer Science National University of Sinapore Sinapore, , Sinapore Abstract Taret trackin amon obstacles is an interestin class of motion plannin problems that combine the usual motion constraints with robot sensors visibility constraints. In this paper, we introduce the notion of vantae time and use it to formulate a risk function that evaluates the robot s advantae in maintainin the visibility constraint aainst the taret. Local minimization of the risk function leads to a reedy trackin stratey. We also use simple velocity prediction on the taret to further improve trackin performance. We compared our new stratey with earlier work in extensive simulation experiments and obtained much improved results. I. INTRODUCTION The taret trackin problem considers motion strateies for an autonomous mobile robot to track a movin taret amon obstacles, i.e., to keep the taret within the robot sensor s visibility reion. Taret trackin has many applications. In home care settins, a trackin robot can follow elderly people around and alert careivers of emerencies. In security and surveillance systems, trackin strateies enable mobile sensors to monitor movin tarets in cluttered environments. Taret trackin is an especially interestin class of motion plannin problems. Just as in classic motion plannin [7], we must consider motion constraints resultin from both obstacles in the environment and the robot s mechanical limitations. In particular, the robot must not collide with obstacles. Taret trackin has the additional visibility constraints due to sensor limitations, e.., obstacles blockin the view of the robot s camera. The robot must move in such a way that the taret remains visible at all times. Both motion constraints and visibility constraints play important roles in taret trackin. Inspired by earlier work [5], we propose a reedy stratey for taret trackin. It uses the robot s sensor to acquire local information on the taret and the environment, and use this information to compute the robot s motion at each step. Thus, it does not need aprioriknowlede of the environment or localization with respect to (w.r.t.) a lobal map. The key element of the reedy stratey is a local function ϕ that ives a combined estimate of the immediate risk of losin the taret and the future risk. Our definition of ϕ is based on two important considerations: Vantae time, which is a combined estimate of the robot s ability to maneuver aainst the taret in both the current and future time. The taret s instantaneous velocity, which can be estimated locally, indicates the taret s future movement. We show that by leverain these two considerations, our new stratey achieves sinificant improvement in performance over the earlier stratey presented in [5]. In the followin, after a brief review of related work (Section II), we state the taret trackin problem formally (Section III) and present our solution (Section IV). We have implemented the new trackin stratey and compared it with an existin one in simulated environments. The simulation results are shown in Section V. Finally, we conclude with some remarks on future research directions (Section VI). II. RELATED WORK Trackin strateies differ reatly, dependin on whether the environment is known in advance. If both the environment and the taret trajectory are completely known, optimal trackin strateies can be computed by dynamic prorammin [8] or by piecin toether certain canonical curves [3], thouh usually at a hih computational cost. If only the environment is known, one can preprocess the environment by decomposin it into cells separated by critical curves. The decomposition helps to identify the best robot action as well as to decide the feasibility of trackin [11]. Often, neither the environment nor the taret trajectory is known in advance. One approach in this case is to move the robot so as to minimize an objective function that depends on the shortest distance for the taret to escape from the visibility reion of the robot s sensor, abbreviated as SDE [5], [9], [12]. Our work belons to this cateory. An important issue here is to balance the immediate risk aainst the future risk of losin the taret. The earlier work heavily replies on SDE in formulatin the risk function. In contrast, the concept of vantae time introduced here provides a more systematic way to interate various factors contributin to the escape risk and derive a risk function offerin better trackin performance. The use of local velocity estimation to predict taret motion further improves the performance of our trackin stratey. Taret trackin has also been studied jointly with other objectives, such as stealth [1], which requires the trackin robot to remain invisible to the taret, and robot localization [4]. A problem that is related, but complementary to that of taret trackin is covert path plannin [2], [10], which tries to minimize the robot s exposure to observers with known or partially known locations /06/$ IEEE 2342

2 F ap ede occulusion ede e r e taret I II (a) x obstacle ede Fi. 1. A robot trackin a taret in a planar environment. (a) A robot mounted with a laser rane finder tracks another robot. Courtesy of H.H. González-Baños. (b) The visibility set (shaded) for the robot located at x (b) r O O v r ˆr ˆt robot robot Fi. 2. The robot s vantae zone (shaded) w.r.t. a ap ede. In the robot s visibility set, reion I lies above the dashed line, and reion II lies below. III. PROBLEM FORMULATION The oal of taret trackin is to keep the taret within the visibility reion of the robot s sensor at all times. The robot and the taret are assumed to operate in a planar environment populated with obstacles (Fi. 1) For simplicity, both are modeled as points in the plane. The extension to the usual cylindrical robots is straihtforward. We assume that the robot has no prior knowlede of the environment, but is equipped visual sensors (e.., cameras and laser rane-finders) to acquire information on the local environment and identify the taret. We use the standard straiht-line visibility model for the robot s sensor. Let F denote the subset of the plane not occupied by obstacles. The taret is visible to the robot if the line of siht between them is free of obstacles, and the distance between them is smaller than D max, the maximum sensor rane. The visibility set V(x) of the robot at point x consists of all the points at which the taret is visible (Fi. 1b): V(x) ={x F xx F and d(x, x ) D max }, where d(x, x ) denotes the distance between x and x.ifthe robot must monitor the taret from a minimum distance away, we can impose the additional constraint d(x, x ) D min.in the followin, the visibility set is always taken w.r.t. the current robot position, and so we omit the arument x. The robot s motion is modeled with a simple discrete-time transition equation. Let x(t) denote the position of the robot at time t. If it chooses a velocity v(t) at time t, its new position x(t +1)after a fixed time interval Δt is iven by x(t +1)=x(t)+v(t)Δt. Here, we implicitly assume that sensin occurs every Δt time. This discrete model is effective as lon as Δt remains small. As we will see, our trackin stratey is very efficient. Based on the experience of previous work [5], we expect it to run at the rate of 10 Hz, sufficient for keepin Δt small in many common tasks. The robot has velocity bound V, but has no other kinematic or dynamic constraints. So, in one time step, it can reach anywhere inside a circle with center x(t) and radius V Δt, unless it is obstructed by obstacles. The taret s motion is modeled similarly, but has velocity bound V. Typically, V V. Otherwise, the taret can easily escape by runnin straiht ahead with maximum velocity, and the trackin problem is uninterestin. We further assume that at any time, the robot has an estimate of the taret velocity. We now state the problem formally: Problem 1: A trackin stratey computes a sequence of actions, namely, the velocities v(t),t=0, 1,..., for the robot so that at any time t, the taret position lies inside V. IV. THE GREEDY TRACKING STRATEGY Our main idea is to define a function ϕ that estimates the risk for the taret to escape from V. The visibility set V can be approximated by a eneralized polyon with linear and circular boundary edes. The boundary of the polyon consists of ap edes and obstacle edes (Fi. 1b). Obstacle edes come from parts of the obstacle boundaries that block the robot s line of siht. Gap edes lie inside F. Theyare further divided into two types: occlusion edes that result from occlusion by the obstacles and rane edes that result from maximum or minimum sensor rane limits. Both obstacle and occlusion edes are linear. Rane edes are circular. To escape from V, the taret must exit one of the ap edes. Below, we first define the escape risk w.r.t. a sinle ap ede and then combine them to form the overall risk function ϕ. The robot then chooses its action, i.e., the velocity v, to minimize ϕ. A. Escape Risk with respect to a Sinle Gap Ede Let us first consider the more interestin type of ap edes, occlusion edes. Suppose that is an occlusion ede. Let l denote the visibility line containin, andleto denote the obstacle vertex abuttin l (Fi. 2). We call O the occlusion point. Theriskϕ of escapin throuh depends on several quantities. Clearly, ϕ depends on e, the shortest distance from the taret to. The smaller the e is, the easier it is to escape throuh. In a subtle way, ϕ also depends on the robot s and the taret s relative positions w.r.t. O: specifically, r, whichis the distance from the robot to O, andr, the projected distance from the taret to O alon l. Ifr is much smaller than r,the robot can move the ap ede away from the taret much faster by rotatin l around O. To combine e, r, andr into a sinle risk estimate, we introduce the notion of vantae zone. The robot s vantae zone D() w.r.t. a ap ede is the set of points that are closer to 2343

3 than the robot is (Fi. 2). Aain, we omit the arument from D() when it is clear from the context which ap ede is involved. Geometrically, D is a band adjacent to with width r. It is so named, because if the taret is outside D, then the robot can always reach before the taret and prevent it from escapin throuh by runnin towards O, the closest point in from the robot 1. Thus, the robot should keep the taret out of D, and a ood estimate of the taret s escape risk is the amount of time t D that the tarets needs to reach the boundary of D. We call t D the vantae time. If the taret is inside D, t D is positive by convention and indicates the amount of time that the robot needs to push the taret out of D. Ifthetaret is outside D, t D is neative and indicates the amount of time that the robot can keep the taret away from D. The robot s velocity v can be decomposed into a radial component v r towards O and a tanential component v t perpendicular to v r (Fi. 2). The tanential component v t causes the robot to swin out and immediately increases e, thus reducin the current escape risk. The radial component v r causes the robot to approach O. It does not affect e immediately. Instead, it decreases r so that future tanential motion will increase e much faster. In this sense, v r reduces future escape risk. Since the robot s velocity is upper bounded, we must choose v r and v t carefully to balance the current and the future risk. For this, let us examine the effects of v r and v t on D. Fi.2showsthatv r shrinks the width r of D at the rate dr/dt = v r, (1) and that v t rotates D about O with anular velocity ω = v t /r. (2) Both components can be used to keep the taret out of D by reducin t D, but their effectiveness depends on the taret position w.r.t. to. There are two cases. a) Case I: The closest point in to the taret is interior to. This case corresponds to reion I marked in Fi. 2. The vantae time t D depends on r e, the distance between the taret and the boundary of D. The rate of chane of r e is iven by d(r e)/dt = dr/dt de/dt. (3) Substitutin (1) and (2) into (3), we have d(r e)/dt = v r (v e r ω) = v r +(r /r)v t v e, (4) where v e is the effective escape velocity, i.e., the taret velocity v projected in the direction perpendicular to. To compute t D exactly, we must interate (4) and solve the interal equation r 0 e 0 = td 0 (v r +(r /r)v t v e ) dt, where r 0 and e 0 denote the values of r and e at current time step. This is not possible, as we do not know the future taret actions. Instead, we estimate the taret escape risk ϕ 1 Recall the assumption that the robot s velocity bound is reater than that of the taret. by approximatin t D usin only information available at the current time step: r 0 e 0 ϕ = v r + v t (r 0 /r 0) v e, (5) where r 0 is the value of r at the current time step. Now, for a sinle ap ede, taret trackin reduces to minimizin the risk function ϕ (v r,v t ). We do this by differentiatin ϕ and computin its neated radient: ϕ = ϕ v eff ( r 0 r 0ˆt + ˆr ). (6) In (6), ˆt and ˆr are unit vectors in the tanential and radial directions, respectively, and v eff = v r + v t (r 0/r 0 ) v e is the effective velocity in the direction alon the shortest path from the taret to. In this case, v eff is perpendicular to. The robot s action v w.r.t. is simply ϕ. Eq. (6) shows that the direction of v is (1/ r0 2 + r 0 2 )(r 0ˆt+r 0ˆr). It depends only on r 0 and r 0, which, intuitively, measure the robot s and the taret s abilities to swin the visibility line l aainst each other. When r 0 is smaller than r 0, swinin is effective. Thus, the tanential component ets hiher weiht. When r 0 is larer than r 0, the opposite holds. The manitude of v acts as a weiht when there are multiple ap edes. It depends on all three quantities, r, r,ande. In particular, when e is small w.r.t. to a ap ede, ϕ becomes lare. Thus, ϕ becomes lare accordin to (6), and the correspondin action ets hiher weiht (see next subsection). b) Case II: The closest point in to the taret is an endpoint of. This case corresponds to reion II marked in Fi. 2, and the closest point in to the taret is the occlusion point O. Here, the tanential component v t has no effect on t D. Thus, ϕ = r 0 e 0 v r v e, (7) which can be obtained from (5) by settin r = 0. The correspondin radient of ϕ is then ϕ = ϕ v eff ˆr, (8) where v eff is still the effective velocity in the direction alon the shortest path from the taret to, but this time, it is directed towards O and is equal to v r v e. Toether, Eqs. (6) and (8) reveal that the robot s action is continuous over the entire domain, provided that so is the taret s action. This is an important advantae in practice. Let us now turn to rane edes. For lack of space, the detailed derivation is omitted. We redefine O as the closest point in a rane ede to the taret position. Since r and r are defined w.r.t. to O, their definitions chane accordinly, but the definition of e remains the same as before. With these, the risk function and the resultin robot action are the same as those in (7) and (8). B. Escape Risk with respect to All Gap Edes A visibility set may contain many ap edes. Based on the taret s motion patterns, we can identify the important ones and improve trackin performance. Let the headin probability 2344

4 old stratey new stratey old stratey new stratey Fi. 3. A scenario in which swinin provides little advantae. In this and all followin examples, the shaded reion indicates the visibility set w.r.t. the current robot position. A small blue circle marks the robot position. A filled black trianle mark the taret position. The associated arrows indicate the robot s and taret s velocity directions. When shown, the robot trajectory is marked by blue circles. A circle filled with cyan indicates that the taret it not visible at that time step. The taret trajectory is marked by black crosses. p be the probability of the taret headed to a ap ede. The overall risk ϕ is then computed as the expected risk over all the ap edes: ϕ = p ϕ. (9) Recall that ϕ is an estimate of the vantae time t D, and thus ϕ is the expected vantae time over all the ap edes. To find the robot action, we solve a simple optimization problem: min ϕ(v r,v t ) subject to v 2 r + v 2 t = V 2. (10) v r,v t We solve (10) by computin the neated radient of ϕ: ϕ = ( p ϕ )= p ϕ, (11) which is then scaled to manitude V to ive the action v for the robot. Eq. 11 shows that the overall action for the robot is a linear combination of the actions w.r.t. the individual ap edes, weihted by the headin probabilities. The benefits of the new risk function is best illustrated in comparison with a related risk function introduced in earlier work [5]. For occlusion edes, the old risk function is a monotonic function of the ratio r/e and completely inores r. To simplify the presentation, we assume that the robot and the taret have the same velocity bounds in all the followin examples. Consider the scenario in Fi. 3. The taret is closer to the ap ede than the robot, and is movin towards the ap ede. The robot will lose the taret unless it runs straiht towards the occlusion point. Since r is small, swinin in the tanential direction provides little advantae to the robot. Since the old risk function inores r, it fails to reconize this and enerates rouhly equal amount of motion in the tanential and the radial directions. The new risk function puts almost all the motion in the radial direction towards the occlusion point. Now consider another scenario (Fi. 4). The taret is very close the ap ede, and thus e is small. As a result, the old risk function becomes very lare and enerates almost a full swin for the robot in order to reduce the current escape risk. However, the situation is in fact not that critical at all. The taret is still a small distance away from the ap ede, leavin Fi. 4. A scenario in which too much swinin increases future risk. some time for maneuverin. More importantly, the robot is slihtly closer to the occlusion point than the taret. A small swin is sufficient to keep the taret visible. Too much swin in the tanential direction reduces the motion in the radial direction and increases the future escape risk. This eventually causes the old risk function to lose the taret. The new risk handles this situation much better. It always keeps the taret visible until it eliminates the ap ede in the end (Fi. 4). These two scenarios show conclusively that alon with r and e, the ratio (r /r), which measures the robot s and the taret s relative positions w.r.t. to the occlusion, plays an important role in risk estimation. This is one major reason why the new risk function performs better. C. Headin Probability Estimation To estimate headin probabilities, we need the current taret velocity v. At any time, we maintain an estimate of v by storin a short history of the taret trajectory and extrapolatin. Many other methods for velocity estimation are possible. For simplicity, the uncertainty in estimatin the direction θ of v is assumed to follow a Gaussian distribution f(θ). The variance of the Gaussian indicates our confidence in estimatin the taret behavior. Other distributions, even non-parametric ones, can be used instead of the Gaussian, dependin on the method of velocity estimation. Our method for computin headin probabilities is eneral and works with any distribution. It is natural to assume that the taret will exit a ap, if it is headed to. In other words, suppose that λ θ is the ray oriinatin from the current taret position and havin direction θ. The taret will exit if λ θ intersects. Thus, p can be estimated from the taret s estimated velocity distribution and the anle subtended by : P () = f(θ) dθ, Θ where θ lies in the anular rane Θ if and only if λ θ intersects (Fi. 5). This seems reasonable, unless we consider a ap 2345

5 ap zone old stratey new stratey 0 overlap reion Θ taret Θ G Θ G Fi. 5. Estimatin headin probabilities. ede subtendin zero anle, e.., the one marked as 0 in Fi. 5a. It is a distinct possibility that the taret may exit 0. We must relax our initial assumption and incorporate this situation. To do this, we expand every ap ede by a predefined distance δ and call the resultin reion the ap zone: G() ={x V d(x, ) δ}, where d(x, ) denotes the shortest distance from x to. Now the headin probability of depends on the anle subtended by its ap zone instead of itself. In eneral, adjacent ap zones may overlap, and the probability in overlappin reion must be split evenly amon all ap zones involved. Takin all these into account, we have the followin formula for computin the headin probability of : p = P (G) = f(θ)/h(θ) dθ, (12) Θ G where Θ G is the anular rane subtended by the ap zone G of and h(θ) is the number of ap zones that λ θ intersects. Note that h(θ) 1. The threshold δ for determinin the ap zone basically says that the taret may exit whenever it comes within a distance δ of. It can be chosen accordin to our understandin of taret behaviors. In our experiments, we chose δ to be the distance that the taret can reach with maximum velocity. This is an aressive choice, indicatin hih confidence in the taret motion model. Good velocity prediction helps the robot to focus on the important ap edes and improve trackin performance. Consider the example in Fi. 6. It compares our new trackin stratey with the one in [5], [9], which does not use velocity prediction. Each imae in Fi. 6 shows several small line sements rooted at the current robot position. Each sement corresponds to the headin probability of a ap ede. The lenth of the sement is proportional to the headin probability, and its orientation points to the ap ede associated with the headin probability. For the old stratey, all the ap edes are weihted with equal probabilities. For the new stratey, the ap ede to which the taret is headed has a distinctively lare headin probability, indicated by a lon sement. When the taret makes abrupt turns, the velocity prediction is usually inaccurate. Our trackin stratey may cause the robot to make the wron move. Consider the example in Fi. 7. The taret makes several abrupt turns. However, the velocity Fi. 6. Usin the estimated taret velocity information, the robot can focus on the important ap edes. Fi. 7. old stratey new stratey An example in which the taret makes abrupt turns. prediction is reasonable for most of the time. Despite the wron moves, our trackin stratey follows the taret to the end and performs better than the old stratey, which loses the taret midway. The advantae of velocity prediction is further confirmed by the experiments presented in Section V. D. Emerency Actions Our trackin stratey invokes two emerency actions, when the taret is danerously close to escape. We describe below the specific robot actions when the taret is in reion I of a ap ede. The actions for the other cases are simpler. We omit the details for lack of space. First, we estimate the escape time t esc based on e and the current velocities of the robot and the taret. If t esc is below a threshold, then the robot must reduce the immediate risk maximally by increasin its tanential motion w.r.t.. It does this by settin v = V ˆt. Next, if the taret indeed escapes from an occlusion ede, the best that the robot can hope for is to eliminate. Anaive way is to run directly towards the correspondin occlusion point by settin v = V ˆr. The fastest way of eliminatin requires knowlede of an obstacle ede lyin outside the robot s visibility set. In this case, the optimal robot actions is to swin out alon a suitably constructed circular path [6]. V. EXPERIMENTS IN SIMULATION We implemented our new trackin stratey in C++ and compared it with earlier work [5] in simulation. To have a fair comparison, we provided the old stratey the same emerency actions that our new stratey uses, thouh they are not in the 2346

6 TABLE I PERFORMANCE COMPARISON OF THE OLD AND THE NEW TRACKING STRATEGIES. Env. Total No. Old Stratey New Stratey Taret Steps No. Steps Visible (%) No. Times Lost (Steps Lost) No. Steps Visible (%) No. Times Lost (Steps Lost) Maze (43%) 2 (11,12) 74 (90%) 1 (8) City Blocks (49%) 6 (14, 15, 16, 15, 8, 10) 131 (84%) 2 (13, 12) oriinal work. Some of the comparison results have already been presented in Fis. 3, 4, 6, and 7. Here, we show two additional examples with more complex eometry (Fi. 8): a) Maze. This environment brins toether various eometric features, such as lon corridors, open spaces, and sharp turns. The taret takes a lon and windin path. Even with emerency actions, the old stratey loses the taret midway. The new stratey follows the taret to the end. It loses the taret once, but recovers it quickly throuh emerency actions. b) City blocks. This example mimics city blocks in an urban environment. The old stratey has lots of difficulty in this environment. It loses the taret many times for extended periods (see Table I) and fails to follow the taret to the end. The new stratey has much improved performance. Detailed performance statistics on these two environments are shown in Table I. Column 2 of the table lists the lenth of the taret trajectory in time steps. For the old stratey, column 3 lists the number of steps that the robot has the taret visible as well as the number as a percentae of the total number of taret steps. Column 4 lists the number of times that the taret is lost and recovered with emerency actions, as well as the durations for which the taret is lost. Columns 5 6 ive the same information for our new stratey. The comparison in these two environments shows that the new stratey (i) less likely loses the taret, (ii) has the taret visible for much loner total duration, and (iii) always follows the taret the end. All these indicate better performance. VI. CONCLUSION This paper presents a practical alorithm for taret trackin, an interestin class of motion plannin problems that combine the usual motion constraints with robot sensors visibility constraints. We introduced the notion of vantae time, which provides a systematic way to interate various factors contributin to the escape risk. It is used to formulate a risk function and construct a reedy trackin stratey. We compared our new stratey with earlier work in extensive simulation experiments and obtained much improved results. We believe that the improvements result from a better-formulated risk function that takes into account both the relative positions of the robot and the taret and the velocity of the taret. We plan to implement our alorithm and test it on real robots. For this, we will improve the robot motion model, incorporatin nonholonomic constraints for wheeled robots if necessary. We will also improve the robot sensor model by imposin limited viewin anles. Another interestin extension is to use this approach for multi-robot trackin [12]. REFERENCES [1] T. Bandyopadhyay, Y. Li, M. An Jr., and D. Hsu, Stealth trackin of an unpredictable taret amon obstacles, in Alorithmic Foundations old stratey Fi. 8. new stratey Two environments with complex eometry. of Robotics VI, M. Erdmannet al., Eds. Spriner-Verla, 2004, pp [2] E. Birersson, A. Howard, and G. Sukhatme, Towards stealthy behaviors, in Proc. IEEE/RSJ Int. Conf. on Intellient Robots & Systems, 2003, pp [3] A. Efrat, H. González-Baños, S. Kobourov, and L. Palaniappan, Optimal strateies to track and capture a predictable taret, in Proc. IEEE. Int. Conf. on Robotics & Automation, 2003, pp [4] P. Fabiani, H. González-Baños, J. Latombe, and D. Lin, Trackin a partially predictable taret with uncertainties and visibility constraints, J. Robotics & Autonomous Systems, vol. 38, no. 1, pp , [5] H. González-Baños, C.-Y. Lee, and J.-C. 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