Conditions for Target Tracking with Range-only Information

Transcription

1 Conditions for Target Tracking with ange-only Information Gaurav Chaudhary a, Arpita Sinha b,, Twinkle Tripathy b, Aseem Borkar b a KPIT Technologies, India b Systems and Control Engineering, IIT Bombay, India Abstract This paper addresses the problem of guiding a mobile robot towards a target using only range sensors. The bearing information is not available. The target can be stationary or moving. It can be the source of some gas leakage or nuclear radiation or it can be some landmark or beacon or any manoeuvring vehicle. The mobile robot can be a ground vehicle or an aerial vehicle flying at a fixed altitude. In literature, many different strategies are proposed which use the range only measurement but they involve estimation of different parameters or have switching control strategy which make them difficult to implement. We propose two sets of conditions, one for stationary target and another for both stationary and moving target. Any control strategy, that will satisfy these conditions, can bring the robot arbitrarily closed to the target. There are no restrictions on the initial conditions. Estimation of any parameter is not required. Some candidate controllers are presented that included continuous controllers and switching controllers. Simulations are carried out with these controllers to validate our result with and without measurement noise. Experimental results with ground mobile robot are presented. Keywords: Target tracking, range only measurement, mobile sensors, pursuit, localization Corresponding Author addresses: arpita.sinha@iitb.ac.in (Arpita Sinha), twinkle.tripathy@iitb.ac.in (Twinkle Tripathy), aseem@sc.iitb.ac.in (Aseem Borkar) Preprint submitted to Elsevier August 11, 2017

2 1. Introduction Target tracking refers to a generic problem of collecting information about a target. This is an active research topic in sensor networks [1], [2], mobile robotics [3], computer vision [4] and so on. This paper deals with the problem of guiding a mobile robot towards a target, which can be stationary or moving. In the last decade, we have seen the use of autonomous vehicles in various applications like search, surveillance, reconnaissance and target tracking. Single or multiple vehicles are employed for the purpose. In target tracking, the objectives can be to capture a target [5], circumnavigate it [6], or pursue it from a distance [7]. esearch also diversified in the availability of target information, for example, the pursuing vehicle may only know the distance to the target [5, 8], or the direction in which the target lies [9, 10, 11] or both the information [12, 13, 14, 15] (see [16] and the references within). The situation where range information is only available can arise in a GPS denied environment or in a situation where we measure the intensity of some signal to estimate the distance from the target. In such cases, [17] - [21] have employed estimation techniques to find the target position in order to capture it. The problem of capturing a target with range only information is solved in discrete time in [5] where the target position is estimated geometrically. The robot measures the distance to the target at two different positions and finds the position of the target from these measurements in an iterative way. The target can manoeuvre and it is considered to be captured once the pursuer is within a pre specified distance from it. In [6], a similar problem is solved in continuous time. A unicycle kinematics is assumed for the pursuer. However, the pursuer is supposed to circumnavigate a target which is stationary. The information available to the pursuer are the range and range-rate of the target. In this paper, we address the target capturing problem similar to [5] but in continuous time. The pursuer is assumed to have unicycle kinematics. Unlike [6], we assume manoeuvring target and the pursuer needs to capture it instead of circumnavigating it. The proposed strategy does not involve any estimation 2

3 technique to know the target position. The input to the robot is the lateral acceleration, while the linear velocity is kept constant. A set of conditions are found that the controller needs to satisfy in order to capture the target. The user can define the control function which can be continuous or switching. This distinguishes our work from [22], [23] where some specific switching controllers are used. For obvious reasons, the strategy works well when the target is stationary and this work can be applied in source localisation, for example, to find the point of leakage of some gas by sensing the intensity of the gas or to locate the source of nuclear radiation in a similar manner. The assumption that will be required is that the intensity (of the gas or nuclear radiation) is proportional to the distance between the source and the robot. To include the essence of practical limitations, we assumed that the lateral acceleration for the robot is bounded. The target is assumed to have an unicycle kinematics when it is not stationary. Since the range and range rate information are often corrupted with noise, we discuss the performance of the strategies in presence of noise. The preliminary results are presented in [24] and [25]. This paper is organised as follows. In Section 2, we define the problem that is addressed in the paper. Section 3 and 4 deal with the development and validation of the control strategy for stationery and moving target respectively. Section 6 concludes the paper. 2. Problem Statement 55 Consider a mobile robot that can measure the distance and the rate of change of distance from a given point. The problem is to guide the robot to that point. The point may be stationary or moving, as for example, a landmark or a target vehicle. We consider an unicycle model for the robot, the kinematics of which is given by ẋ r = v 1 cos α 1, ẏ r = v 1 sin α 1, α 1 = u (1) where (x r, y r ) is the instantaneous position of the robot, v 1 is the linear velocity of the robot, α 1 is the heading angle with respect to a fixed reference and u is the 3

4 Figure 1: Engagement geometry between the robot and the target control input to the robot. Thus, the angular velocity of the robot is controlled while the linear velocity is constant. We assume that the lateral acceleration of the robot is bounded as α 1 [ α 1 min, α 1 max ] We are considering two types of targets 1) stationary targets and 2) moving targets. In the case of moving target, we assume the target to have unicycle model with constant linear speed v 2. There is an upper bound on the maximum speed and the maximum turn rate the target can have. Let at some time t, the target be at (x t, y t ) with respect to some reference frame and the robot be at (x r, y r ). This is shown in Figure 1. Let the distance between the robot and the target, that is, the line-of-sight (LOS) distance be and the LOS angle be θ. The robot can measure and Ṙ, but not θ. Thus, u is some function of and Ṙ. We determine the conditions which this function 70 needs to satisfy so that it can steer the robot towards the target. When the distance between the robot and the target is less than c, we assume that the target is captured. We call c the capture radius and c > Stationary target 75 When the target is stationary, the position (x t, y t ) is fixed with respect to time. Considering Figure 1, the line-of-sight (LOS) between the robot at (x r, y r ) 4

5 Figure 2: Engagement trajectory in (V θ, V ) space and the target at (x t, y t ) is characterized by V = Ṙ = v 1 cos(α 1 θ) V θ = θ = v 1 sin(α 1 θ) (2) where V is the relative velocity component with respect to the robot along the LOS and V θ is the relative velocity component with respect to the robot perpendicular to the LOS. Squaring and adding the two equations in (2), we get V 2 + V 2 θ = v2 1. This implies a circle in (V θ, V ) plane with center at the origin and radius v 1 as shown in Figure 2. The motion of the robot will be such that the instantaneous (V θ, V ) point will always lie on this circle. We assume that the target is captured when c. The following theorem states the conditions under which capture is possible. Theorem 1. A robot, with kinematics (1), will be able to capture a stationary target from any initial position if it uses a control law given by u = f(v ) where f(v ) is some function of V which satisfies = 0, V = v 1 ; f(v ) > 0, V ( v 1, 0); > v1 c, V [0, v 1 ]. (3) 5

6 and df(v ) dv > 0, V ( v 1, v 1 ); (4) 90 Proof. In order to capture the target, should decrease, which implies V < 0. From Figure 2, a point (V θ, V ) will always moves on the circle. We study the movement of the point (V θ, V ) in the different sectors of the circle. Differentiating (2) V = V θ (f(v ) V θ ) (, Vθ = V f(v ) V ) θ (5) Case 1: Between points D and A, V ( v 1, 0) and from (3), f(v ) > 0. Then 95 from (5) V > 0 and V θ < 0 and the point (V θ, V ) will move from D to A as shown in Figure 2. Case 2: At point A, we have V = 0 and f(v ) > v1 c. From (5), V > 0, V θ = 0 and (V θ, V ) point will cross A in the direction as shown in Figure 2 Case 3: Between points A and B, V (0, v 1 ) and f(v ) > 0. From (5), V > and V θ > 0 and the movement of the point (V, V θ ) is as shown in Figure 2. Case 4: At point B, V = v 1, V θ = 0 and f(v ) > 0. From (5), V = 0 and V θ > 0 and the movement of (V, V θ ) point is as shown in Figure 2. Case 5: Between point B and C, V (0, v 1 ) and V θ (0, v 1 ). Since, f(v ) > v 1 c V θ, we have V < 0 and V θ > 0 and the movement of (V, V θ ) point is as shown in Figure 2. Case 6: At point C, V = 0, V θ = v 1 and f(v ) > v1 c > v1. From (5), and V θ = 0 and the movement of (V, V θ ) point is as shown in Figure 2. V < 0 Case 7: Between points C and D, V ( v 1, 0), V( θ (0, v 1 ) and f(v ) > 0. Let V = v 1 β for some β (0, 1) and g(β) = f( v 1 β) v1 1 β 2 c ). If g(β) > 0 then V < 0 and V θ < 0 for all c. Since f(v ) is an increasing function, there exists a β = β (say) such that g(β) > 0, β (0, β ]. Let E corresponds to the point V = v 1 β. In the region between C to E, g(β) > 0 and thus V < 0 and V θ < 0 for all c. Hence, the movement of (V θ, V ) between C and E is as shown in Figure 2. 6

7 115 For a (V θ, V ) point within E to D, nothing can be said about direction of movement of the point. However, it cannot leave this region because at E, if > c, V < 0 and this will force the point to remain within the region c and at D, V = 0 and V θ = 0, which is a stationary point and is discussed in next case. 120 Case 8: At point D, V = v 1, V θ = 0 and f(v ) = 0. From (5) V = 0 and V θ = 0. Thus, (V, V θ ) is a stationary point and will not move. Since, V θ = 0, the LOS angle do not change with time and V < 0, the LOS distance keeps on decreasing. This corresponds to the case when the robot is heading directly to the target location. 125 From the different cases discussed above, it can be seen that, irrespective of the initial position of the robot with respect to the target, the (V θ, V ) point will always end up in the region between E to D. Exceptions will occur when becomes less than c before the point E is reached. This can happen in the regions D to A and C to E, when V < 0 and this implies that capture 130 has occurred. When the (V θ, V ) point is within E to D, V < 0 and thus the distance between the robot and the target will keep on decreasing until capture occurs. The capture will occur in finite time since either or both V and V θ are bounded away from zero in all the cases. Thus, we have the freedom to design the control law f(v ). Care should 135 be taken such that f(v ) is monotonically increasing and it must lie in shaded region shown in Figure 3. emark 1. Since V and v 1 is known, from (2), we can obtain the values of (α 1 θ). However, we do not use the angles information to generate the control law since this will result in a non smooth controller. The proposed control law 140 does not use the measured value of instantaneous. It need not necessarily use the instantaneous value of V, since the requirement is to have f(v ) within the shaded region of Fig. 3. We can have f(v ) = αv1 c of V, except at V = v 1. with α > 1 independent emark 2. The capture occurs when c. Thus, by properly selecting c, 7

8 Figure 3: f(v ) v/s V in (3) 145 the robot can be made to come arbitrarily close to the target. However, more lateral acceleration will be required for smaller c as can be seen from the third inequality in (3) Examples 150 We present some candidate functions that can be used as the control input to the robot. S1. Quadratic Function: f(v ) = V 2 + k 1 V + k 2 (6) such that k 2 = max{a, b} and k 1 = v 1 + k2 v 1 where a v 2 1 and b > v1 c. This{ function is plotted in } Figure 4. The capture radius is given by c > 2V min 1 α 1max 2V, 2. If we assume v = 10 and c = 0.2, then ( α1max ) α = f(v ) = V V and we will have β = and corresponding value of V = S2. Linear Function such that k 2 > v1 c capture radius is given by c > f(v ) = k 1 V + k 2 (7) and k 1 = k2 v 1. This function is plotted in Figure 5. The 2V1 α. Assuming again v 1max 1 = 10 and c = 0.2, we have f(v ) = 6V + 60, β = and the corresponding value of V =

9 Figure 4: A candidate quadratic control input function Figure 5: A candidate linear control input function 9

10 Cases (D)-(A) at (A) (A)-(B) at (B) (B)-(C) at (C) (C)-(E) (E)-(D) at (D) x r y r α o 36 o 45 o 135 o 130 o 37 o 50 o 150 o 135 o V V θ t qc t lc Table 1: Initial conditions and time to capture for quadratic and linear control input 3.2. Simulations We carried out simulations when the robot is following the above control laws. The target is assumed at the origin and the different initial positions of the robot are considered. The initial conditions are selected such that each case corresponds to the different sectors in Figure (2). Also the initial distance between the target and the robot is same for all the cases, which is equal to o = 25. The initial position of the robot and the time taken to reach the target is tabulated in Table (1), where t qc corresponds to quadratic control law and t lc corresponds to linear control law. Next, we considered noisy measurements. Two types of measurement noise are considered. The first case assumes both and Ṙ is measured and the measurement is corrupted with Gaussian noise. We considered N (0, 0.5) in the range measurement and N (0, 5) in range rate measurement. Figure 6 shows that the robot with initial conditions as in Case 1 of Table 1 is able to capture the target using S1 control law. Similar behaviour are observed for S2 control law (plot is omitted due to page limitation). Since range rate is used in computing the control law, it is observed in simulation that if the range rate measurement has a Gaussian noise with standard deviation more than 6 units, then the robot is not able to capture the target successfully all the time. 10

11 Figure 6: ange with respect to time for quadratic control with measurement error in and Ṙ 30 Without noise Gaussian noise Uniform noise Time Figure 7: ange with respect to time for linear control with measurement error in In the second case, it is assumed that is measured with noise and the range rate Ṙ is derived from the measurement. We considered two types of noise - Gaussian noise N (0, 0.1) and uniformly distributed random noise in the range [ 0.5, 0.5]. Using S2 control law, Figure 7 shows the range with respect to time with and without noise. S1 control law generates similar results and are not presented here. We observed in simulation that the performance of the robot strongly depends on how Ṙ is obtained. This is because Ṙ is used in calculating the control law. We considered a moving average filter for the result given in Figure 7. It can be observed from both the Figures 6 and 7 that the time required to capture is more when the measurement is corrupted with noise which is also intuitive. 11

12 Figure 8: Different cases for Theorem 2 4. Manoeuvring target For manoeuvring targets, we assume that the target has unicycle kinematics with bounds on maximum linear and angular speed. Consider Figure 1. The 195 line-of-sight (LOS) between the robot at (x r, y r ) and the target at (x t, y t ) is characterised similar to the stationary target case as V = v 2 cos(α 2 θ) v 1 cos(α 1 θ) and V θ = v 2 sin(α 2 θ) v 1 sin(α 1 θ). We assume ω 1 = α 1 θ and ω 2 = α 2 θ. Let us define the target to robot velocity as η = v2 v 1 and let η [0, γ 1 ], where γ 1 is max velocity ratio up to which robot can capture the 200 target. Then, V = Ṙ = v 1(η cos ω 2 cos ω 1 ) (8) V θ = θ = v 1 (η sin ω 2 sin ω 1 ) (9) Let, γ > γ 1 and r < 0, r < γ 1. From (8), if cos ω 1 > γ, then V < 0. Let ρ (γ, 1] and consider cos ω 1 ρ. Then, from (8), V [ (γ + 1)v 1, (γ ρ)v 1 ]. When cos ω 1 γ, then V [ 2γv 1, (γ + 1)v 1 ]. We choose the value of ρ such that these bounds on V do not overlap, that is, (γ ρ)v 1 < 2γv 1. Simplifying 205 we get ρ > 3γ which implies γ < 1 3. Since V [(γ + 1)v 1, (γ + 1)v 1 ], let us 12

13 define S 1 = {V (1 + γ)v 1 V (γ + ρ)v 1 } (10) S 2 = {V (γ + ρ)v 1 < V < 2γv 1 } (11) S 3 = {V 2γv 1 V (1 + γ)v 1 } (12) Theorem 2. A robot with kinematic (1) will be able to capture the moving target from any initial condition if the control law is given by u = f(v, ) where f(v, ) is define as (γ+ 1 ρ 2 )v 1, V S 1 ; f(v, ) [ α 1 max, α 1 min ] V S 2 ; > v1(1+γ), V S 3 ; and γ (0, 1 3 ). (13) 210 Proof. We use an idea similar to Theorem 1. To capture the target we need V < 0 which can be assured if cos ω 1 > γ. So we will analyses how ω 1 is varying with time. Let y = sin ω 1 and z = cos ω 1. Then, from (1), (9) and (13), { ẏ = cos ω 1 f(v, ) V } θ (14) ( ż = sin ω 1 f(v, ) V ) θ (15) 215 With reference to Figure 8, we consider the following cases. Case 1: Between A and A1, r < cos ω 1 γ and 1 γ 2 sin ω 1 1. So, from (8), V [ (η + γ)v 1, (η r)v 1 ] S 3 and V θ [ (1 + η)v 1, (η 1 γ 2 )v 1 ]. This implies from (13) that f(v, ) V θ v1 (γ γ 1) and from (15), ż v1 (γ γ 1) 1 γ 2 < 0. It implies that any point between A and A1 on cos ω 1 curve will move towards A1 in finite time. Case 2: Between A1 and B1, 1 r 2 sin ω 1 1 r 2 and 1 cos ω 1 r. So V [ (η r)v 1, (η + 1)v 1 ] S 3 and V θ [ (η + 1 r 2 )v 1, (η + 1 r2 )v 1 ]. Therefore, from (13), f(v, ) V θ v1 (1 + γ η 1 r 2 ) > v (γ γ 1 ) implying ẏ r(γ γ 1 ) v1 < 0. Hence, any point between A1 and B1 on cos ω 1 curve will move to B1 in finite time. 13

14 Case 3: Between B1 and C, r < cos ω 1 γ and 1 sin ω 1 < 1 γ 2. So V [ (η + γ)v 1, (η r)v 1 ] S 3 and V θ [( η + 1 γ 2 )v 1, (1 + η)v 1 ]. This implies f(v, ) V θ v1 (γ γ 1). Then, from (15), ż v1 (γ γ 1) 1 γ 2 > 0. This implies that any point that lies between B1 and C on cos ω 1 curve will move towards C in finite time. Case 4: Between D and E, ρ cos ω 1 < 1 and 1 ρ 2 sin ω 1 < 0. So V S 1, V θ ( ηv 1, (η + 1 ρ 2 )v 1 ] and f(v, ) (γ+ 1 ρ 2 )v 1. Since (γ+ 1 ρ 2 )v 1 < ηv1, from (15), ż < 0 for any value of η [0, γ]. This implies that any point in the region will move towards point D Case 6: At E, cos ω 1 = 1 and sin ω 1 = 0. So ż = 0 and we have to use (14). In this region, V S 1, V θ [ v 1 η, v 1 η] and f(v, ) (γ+ 1 ρ 2 ) Thus, ẏ < 0 which implies that at E the point will move towards D. < v1η. Case 5: Between E and F, ρ cos ω 1 < 1 and 0 < sin ω 1 1 ρ 2. So V S 1, V θ [ (η + 1 ρ 2 )v 1, ηv 1 ) and f(v, ) (γ+ 1 ρ 2 )v 1. Since (γ+ 1 ρ 2 )v 1 (η+ 1 ρ 2 )v 1 in the region will move towards E., from (15), ż 0. This implies that at any point Points between C and D will remain within C and D irrespective of the behaviour of the control law in this region. However, the points between F and G can do the following - i) remain within F and G, ii) move towards F and cross it or iii) move towards G and cross it. When a point crosses beyond G, it will fall under Case 1. Hence we can conclude that any point will always end up in the region CD in Figure 8 where cos ω 1 > γ in finite time. This implies V (γ 1 γ)v 1 < 0 and hence finite time capture is inevitable. emark 3. From the proof of the Theorem 2 we observe that V S 3 for all points between A to C and V S 1 for all points between D to F in Figure 8. However, between C to D and F to G, V S 1 S 2. 14

15 Figure 9: f(v, ) v/s V plot Examples We present some candidate control laws. For a particular value of =, f(v, ) lies in shaded region shown in Figure 9. M1. Switching Control: f(v, ) = (γ+ 1 ρ 2 )v 1, V S 1 ; a, V S 2 ; kv 1(1+γ), V S 3 ; (16) 255 where a {[ α 1 max, α 1 min ], k > 1 and S i, i = 1, 2, 3 as defined in (12). Here, (γ+ (1 ρ c max 2 ))v 1 α 1min, kv1(1+γ) α 1max }. This control function is shown in Figure 10 for a particular. As mentioned in emark 3, the control does not switch along A to C and along F to D. It switches when it crosses the points C or D. It is shown in the proof of Theorem 2 that all points inside CD cannot leave that region. Within CD, the controller can switch between a and (γ+ 1 ρ 2 )v 1. However, V = Ṙ is bounded away from zero by a negative constant for all 260 points within CD (more precisely between CDEFG). So, will monotonically decrease until it reaches c. Thus, in finite time and finite number of switching of controller, the point will be trapped with CD and capture is ensured. Assuming v 1 = 10, γ = 0.3, ρ = 0.95, α max = 5, α min = 5, η = 0.25, 15

16 Figure 10: A candidate Switching control input function a = α 1 min and k = 1.001, the capture radius is c and the control law is , V [ 13, 6.5]; f(v, ) = 5, V ( 6.5, 6); , V [ 6, 13]; M2. Piecewise Continuous Control: k 1(γ+ 1 ρ 2 )v 1, V S 1 ; f(v, ) = mv + c V S 2 ; k 2v 1(1+γ), V S 3 ; where m = k2(1+γ)+k1(γ+ 1 ρ 2 ) ( 3γ+ρ), c = k2(1+γ)v1 +m(2γv 1 ), 1 k 1 (17) (18) α min c (γ+ 1 ρ 2 )v 1, 1 < k 2 αmaxc (1+γ)v 1 and S i, i = 1, 2, 3 as defined in (12). We chose the value of c { } (γ+ (1 ρ such that it satisfies the two inequalities. Here, c > max 2 ))v 1 α min, (1+γ)v1 α max By properly defining the parameters in (18), we can have piecewise continuous control law as shown in Figure 11 for a particular. Assuming v 1 = 10, γ = 0.3, ρ = 0.95, α max = 5, α min = 5, η = 0.25, we obtain m = , c = , k 1 = 1, k 2 = and c = and the control law is 6.5, V [ 13, 6.5]; f(v, ) = mv + c, V ( 6.5, 6); , V [ 6, 13]; (19) 16

17 Figure 11: A candidate Piecewise continuous control input function 4.2. Simulations We considered a target moving in a straight line.the target is assumed at the different initial positions and α 2 = 0. The initial position of the robot is at the origin. The initial distance between the target and the robot is same for all the cases, which is equal to 0 = 50. The initial conditions are selected such that each case corresponds to the different sections in Figure 8. Table 2 tabulates the time taken to reach the target where t sc and t pc correspond to the time taken to capture the target for switching control and piecewise control strategy respectively. It is observed that the time taken by both the strategies are comparable. The last two columns of the table refer to the same section in Figure 8, which highlight that the time taken can be significantly different when the initial conditions are different. When the target is moving in a circular path, we consider a particular initial condition given by x r0 = 0, y r0 = 0, α 10 = 210, α 1 = u, x t0 = 40, y t0 = 30, α 20 = 15, α 2 = The initial cos ω 1 point lies in the region between points F and G as shown in Figure 8. For switching control, the trajectory of the robot and the control is plotted in Figure 12, where we observe rapid switching of control law towards the end of the trajectory as expected from a non smooth controller. However, in case of piecewise continuous control, the trajectory of the robot and the control input are shown in Figure 13. It can been seen that 17

18 Cases (A)-(B) at (B) (B)-(C) (C)-(D) (D)-(E) at (E) (E)-(F) (F)-(G) (F)-(G) x t y t α 1 50 o o 25 o 50 o o 165 o 210 o α o 45 o 125 o 45 o 145 o 15 o 30 o 15 o 15 o V V θ cos ω sin ω t sc t pc Table 2: Initial conditions and time to capture for Switching and piecewise continuous control input the trajectory for both the control laws are similar while there is no switching in the second case. To observe the effect of measurement noise, we considered two types of measurement noise as discussed for the stationary target case. In the first one, it is assumed that the range measurement is corrupted with Gaussian noise N (0, 0.4) and range rate measurement is corrupted with Gaussian noise N (0, 4). We considered the initial conditions as mention in the previous paragraph but we assume that the target is moving in a straight line, that is, α 2 = 0. From Figure 14, it can be seen that the robot is able to capture the target using control law M1. The performance is similar for M2 and is omitted here. It is observed in simulation that, for the standard deviation beyond 5 units in range rate measurement, capture cannot be guaranteed by either of the control laws M1 and M2. In the second case, we assume that the range measurement is corrupted with noise and the range rate is derived from the noisy range measurement. We consider two types of noise - Gaussian noise N (0, 0.1) and uniformly random noise 18

19 Figure 12: Trajectory of robot and the lateral acceleration for switching control Figure 13: Trajectory of robot and the lateral acceleration for piecewise continuous control 19

20 Figure 14: with respect to time for switching control with measurement error in and Ṙ Without noise Gaussian noise Uniform noise time Figure 15: with respect to time for piecewise continuous control with measurements error in in the range [ 0.5, 0.5]. With M2 control law, Figure 15 shows that the target is capture but the time required is more when there is noise. Again, similar to the stationary target case, we assumed moving average filter to compute the range rate. It is observed in simulation that the performance of the control law strongly depends on the filter used. It is observed in both types of measurement noise that the time taken to capture is larger when there is measurement noise as compared to the measurements without noise. However, the increase in time for M2 control law is less than that for S1, the control law for stationary target. This is intuitive since the control law S1 and S2 directly depends on Ṙ whereas M1 and M2 are computed from only. Comparing the strategies 20

21 Figure 16: Comparison of strategies for a stationary target The strategy developed for moving target will also work when the target is stationary. We compare the performance of the strategy for moving target with that of the stationery target for a typical case. Consider x r0 = 15, y r0 = 20, x t0 = 0, y t0 = 0, v 1 = 10 and the maximum latex of 120 unit. We assume c = 0.2. The time taken by strategy S2 is t lc = 2.74 units, by M1 is t = units and by M2 is t = units (assuming ρ = 0.99, γ = 0.32). The trajectories of the robot using S2, M1 and M2 control laws are shown in Figure 16. The time taken by S1 is smaller than M1 and M2, whereas the time taken by M1 and M2 are comparable. 5. Hardward Implementation To validate the analytical results, we implemented the control laws on Firebird V and Spark V robotic research platforms (shown in Fig.5). These are manufactured by Nex obotics and both are differential wheel robots. Spark V has a wheel diameter of 7 cm and can achieve angular speeds in the range of rad/s to rad/s. It has been used as the target for all the experiments. Firebird V robot is capable of achieving angular speeds in the range of 2.75 rad/s to 6.67 rad/s with a wheel diameter of 51 mm and hence being faster has been used as a the pursuing robot. In the experiments, we used the visual feedback system to obtain the range and range rate information. The Visual Feedback System consists of four cameras mounted on the ceiling. A XBee 21

22 Spark V (Target) Firebird V (Pursuing obot) Colour Pattern Colour Pattern XBee Figure 17: obots used for the experiments Pro Series 1 trans-receiver module has also been interfaced with the robot to receive range and V information relative to the target at every 0.3 seconds. The robots are distinguished by the use of a unique colour pattern mounted on top as shown in Fig 5. The linear speed of the robot is 10 cm/s. The maximum turn rate is α max = 2 and minimum turn rate is α min = 2. We conducted four experiments - two with stationary target and two with moving target. We have implemented the switching control law for both the cases of stationary and moving target. It can be seen that in all the cases the robot reaches the capture region in finite time and the commanded control is within the bounds. There is no chattering in commanded control even for the switching control law. Case 1: (Linear control law for the stationary target) We considered f(r) = 0.05V with C = 30cm. The initial position of the robot is (166.62, 0.17) with heading angle The target is at ( , 59.49). The trajectories of the robot and the target have been shown in Figure 18(a). Figure 18(b) shows that the robot reaches the specified C. The plots of α commanded is shown in Figure 18(c). The trajectory is recorded in LinearStationary.avi 22

23 Distance(cm) > c =30 obot Trajectory Agent Trajectory obot initial position Target initial position obot final position Target final position Distance(cm) > (a) Trajectory of the robot 350 Euclidean distance between the robot and the target Distance(cm) > Time(sec) > (b) ange with respect to time 0.6 Commanded angular speed of the robot 0.5 Angular Speed(rad/s) > Time(sec) > (c) Commanded α with respect to time Figure 18: Stationary target capture for linear control law 23

24 Case 2: (Piecewise continuous law for moving target) We consider / V 6.5 f(r) = V / / V ( 6.5, 6) 91/ V with c = 60cm. The linear speed of the target is 3.35 cm/s. The initial position of the robot is ( , ) with heading angle The target is at ( 54.58, 52.85). The target is capture as shown in Figure 19(a). The range is plotted in Figure 19(b). The plots of α commanded is shown in Figure 19(c).The trajectory is recorded in PiecewiseMoving.avi Case 3: (Switching control law for moving target) We considered / V 6.5 f(r) = 0.05 V ( 6.5, 6) 100/ V 6 with c = 60cm. The linear speed of the target is 3.35 cm/s. The initial position of the robot is ( 74.49, 1.35) with heading angle The target is at (61.82, 0.23). The robot captures the target which is verified in the plot for range with respect to time as given in Figure 20(b). The trajectories of the robots are shown in Figure 20(a). The plots of α commanded is shown in Figures 20(c). The trajectory is recorded in SwitchingMoving.avi Case 4: (Switching control law applied to stationary target) We considered 8/ V 6 f(r) = 0.05 V ( 6, 0) 150/ V 0 with c = 50cm. In this case, γ = 0, ρ = 0.6 and a = The initial position of the robot is ( , 3.43) with heading angle The target is at (55.26, 47.65). The capture of the target by the robot is shown in Figure 21(a). Figure 21(b) shows that eventually reduces to C. The 24

25 Distance(cm) > C =60 obot Trajectory Agent Trajectory obot initial position Target initial position obot final position Target final position Distance(cm) > (a) Trajectory of the robot and the target 180 Euclidean distance between the robot and the target Distance(cm) > C = Time(sec) > (b) ange with respect to time 2 Commanded angular speed of the robot 1.5 Angular Speed (rad/s) > Time(sec) > (c) Commanded α with respect to time Figure 19: Moving target capture for piece wise continuous control law 25

26 Distance(cm) > obot Trajectory Agent Trajectory obot initial position Target initial position obot final position Target final position C = Distance(cm) > (a) Trajectory of the robot and the target 140 Euclidean distance between the robot and the target Distance(cm) > C = Time(sec) > (b) ange with respect to time 2 Commanded angular speed of the robot 1.5 Angular Speed(rad/s) > Time(sec) > (c) Commanded α with respect to time Figure 20: Moving target capture for switching control law 26

27 plots of α commanded is shown in Figures 21(c). The trajectory is recorded in SwitchingStationary.avi Conclusion In the paper, we presented the strategies that can bring a robot arbitrarily close to a target point when the robot can only measure the range and the range rate but has no information about the bearing angle to the target. Estimation of the target position is not necessary here. We derived the conditions on the controller when the target is stationary and when the target is moving. Sample control laws are presented. When the robot is using these functions as the control law, it is shown in simulation that the robot can capture the target, thus validating our analysis. It is up to the users to define the control law that satisfies the conditions stated in the theorems. A simple control law, as given in examples, is easy to implement and works even in presence of noise as seen in simulation. An extension of this work will involve mathematical analysis of the effect of noise in measurement. eferences [1] J. Chen, M. B.Salim and M. Matsumoto: A Single Mobile Target Tracking in Voronoi-based Clustered Wireless Sensor Network, Journal of Information Processing Systems, Vol.7, No.1, pp , March 2011 [2] Mazomenos, E.B. and eeve, J.S. and White, N.M., Tracking with rangeonly measurements using a network of wireless sensors, Sixth International Conference on Broadband Communications, Networks, and Systems, pp. 1-8, Madrid, Spain, Sept [3] S. Daingade and A. Sinha: Nonlinear Cyclic Pursuit Based Cooperative Target Monitoring, Distributed Autonomous obotic Systems, Springer Tracts in Advanced obotics, Vol 104, pp ,

28 100 Distance(cm) > C =50 obot Trajectory Agent Trajectory obot initial position Target initial position obot final position Target final position Distance(cm) > (a) Trajectory of the robot Euclidean distance between the robot and the target 150 Distance(cm) > Time(sec) > (b) ange with respect to time 0.04 Commanded angular speed of the robot Angular Speed(rad/s) > Time(sec) > (c) Commanded α with respect to time Figure 21: Stationary target capture for switching control law 28

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