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Advanced driver assistance system for AHS over communication links with random packet dropouts
Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Advanced driver assistance system for AHS over communication links with random packet dropouts Seshadhri Srinivasan, Ramakalyan Ayyagari n Department of Instrumentation and Control Engineering, National Institute of Technology—Tiruchirappalli, Tiruchirappalli, India
a r t i c l e i n f o
abstract
Article history: Received 2 March 2010 Received in revised form 2 August 2012 Accepted 15 August 2012
In this paper, we propose an advanced driver assist system (ADAS) for platoon based automated highway system (AHS) with packet loss in inter-vehicle communication. Using the concept of rigidity, we first show that vehicles in a platoon tend to fall apart in the event of a packet loss among vehicles. To overcome this, we propose an estimation based dynamic platooning algorithm which employs the state estimate to maintain the platoon. Communication among the vehicle is reduced by using minimum spanning tree (MST) in state estimation algorithm. Effectiveness of the proposed ADAS scheme is illustrated by simulation wherein, dynamic platoons of holonomic vehicles with integrator dynamics are considered. Simulation studies indicate that the proposed algorithm maintains the platoon up to a packet loss rate of 48%. State transmission scheme proposed in our algorithm has three significant advantages, they are: (1) it handles packet loss in inter-vehicle communication, (2) reduces the effect of error in measured output, and (3) reduces the inter-vehicle communication. These advantages significantly increase the reliability and safety of the AHS. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Automated highway system (AHS) Advanced driver assistance system (ADAS) Packet dropouts Modified Kalman filter (MKF) Minimum spanning tree (MST)
1. Introduction AHS have been studied recently as a solution to highway congestion problem. Vehicle of the future need to be compatible with AHS for congestion mitigation and safety. Among many solutions available for implementing AHS, platooning is the most promising option studied in the literature. A detailed review of AHS implementation using platooning is available in [1]. In [2], it has been argued that the platooning structure achieves a balance between safety and throughput. Platooning requires coordination among the vehicles in a platoon, as well as among platoons. Now, consider a platoon as in Fig. 1, where d is called the intra-platooning distance and b bearing or the angle of the followers with respect to their peers. The given platoon may be abstracted as a graph with the vehicles as the nodes and the edges representing the communication channel among the vehicles. As seen in Fig. 1, in order to maintain a platoon, vehicles not only need to coordinate but also monitor the other vehicles in the platoon or in other words know the relative position of other vehicles in the platoon. Coordinating and monitoring other vehicles in the platoon are beyond human reflexes and require automation. One of the major challenges impeding automation of AHS is the presence of packet losses in the communication channel which are used for transmitting coordination information among vehicles in a platoon. In the presence of packet losses coordination information like relative velocity, displacement, bearing, etc., is either unavailable or imprecise.
n
Corresponding author. E-mail addresses: [email protected] (S. Srinivasan), [email protected] (R. Ayyagari).
0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.08.020
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Fig. 1. Platoon of vehicles.
This increases the risk of collision and is against the very spirit of employing AHS. Thus, there is a need not only to automate AHS but also to account for the packet losses in the channel. In this paper, we propose an ADAS system for AHS over packet dropping links by using platooning concept. Implementation aspects of AHS using vehicle platooning have been studied in [1,2]. Decentralized hybrid controllers for AHS have been proposed in [2] and a game theoretic approach is used to design the controllers that are guaranteed to meet safety requirements. In our analysis, we employ a formation control framework for maintaining platoons over packetdropping links. One popular method widely used for formation control is the separation bearing control (SBC) and has been studied in the context of maintaining formation among robots in [5,7–11]. In [12], consensus problems for network of agents with fixed and switching topologies have been investigated. Switching topologies consider platoons which are dynamic alongside packet losses. In [14], cooperative control of vehicle formation problem in the presence of channel constraints has been investigated. Formation control using the concept of rigidity has been studied in [15,16]. Recently, in [19] flocking of multi-agents with a virtual leader has been investigated by relaxing the usual assumptions that: (1) all agents need to be informed and (2) virtual leader travels at constant velocity. An overview of the obtained results in formation control are presented in [6]. In [22], the estimation of over wireless networks is presented by considering packet losses and delays. It has been shown that for a densely distributed sensor network with packet loss the state estimate can be improved by recalculating the estimate with a delayed measurement or lost packet. Recently, a platooning algorithm for AHS over packet dropping links has been proposed in [23,24]. It has been shown that the estimate error covariance reduces in the event of packet loss by transmitting a linear combination of current data and past estimate. Research on whether transmitting estimate of the state or measured output to improve the reliability of the platoon has not been investigated in the past. Motivated by this, we investigate the problem of ADAS design using state estimate transmission for AHS subjected to packet loss in inter-vehicle communication. In this paper, we discuss the implementation aspects of ADAS for AHS using the platooning concept in the presence of packet losses. We first show that the presence of packet dropouts affects the safety of the vehicle using the concept of rigidity. Next, we propose an estimation based platooning algorithm which employs minimal communication. The minimal communication is established by constructing the minimum spanning tree (MST) using the concept of rigidity. Although the proposed algorithm is closely linked to [22], our investigation has two extensions: (1) transmitting state estimate in the place of measured output and (2) minimizing the communication using MST and still retaining the improvement in estimation error in [22]. Our results show that transmitting state estimate makes AHS more robust to packet loss than the output measurement transmission scheme. This is particularly true in the presence of errors in output measurement. Furthermore, the two assumptions of flocking algorithms presented above are also relaxed with our approach. The proposed ADAS is illustrated using simulation studies conducted using holonomic point vehicles with integrator dynamics. The paper is organized as six sections, including the Introduction. In Section 2, the problem is formulated. In Section 3, the methodology employed to find the MST and the MKF proposed in [4] are discussed. In Section 4, the ADAS algorithm is proposed and in Section 5 the proposed algorithm is implemented considering a platoon with packet losses. Conclusions are drawn from the obtained results and the future course of this investigation is presented in Section 6.
2. Problem formulation Consider the platoon say P of vehicles that are connected using communication links as in Fig. 2(a). P may be conveniently represented using an undirected graph GP 9fN, E, Wg, where N is the node/vertex set of the graph that represents the vehicles in the platoon and E is the edge set that denotes the communication link among the vehicles in a platoon and W is the set of weights and is the intra platoon distance between the vehicles in the platoon. The graph of the platoon is shown in Fig. 2(b), where the nodes represent the vehicles and the edges represent the communication links. Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
S. Seshadhri, R. Ayyagari / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Fig. 2. (a) Platoon of vehicles and (b) graphical representation.
Definition 1. A vehicle is said to undergo rigid motion along a trajectory, only when the Euclidean distance between the vehicles in the platoon remains constant all along the trajectory of the vehicle. Definition 2. The graph GP shown in Fig. 2(b) is said to be rigid, if for all position assignments of the nodes, each and every move of the vehicle preserves the distance between the position of any pair of vertices in a graph. This condition may be expressed as in (1)
E ¼ Jxi xj J ¼ Cij
8fi,jg 2 E
where Cij is the predetermined distance between i and j in the platoon or it is the intra-vehicular spacing in the platoon. It may be noted that the intra spacing between the vehicles is the same as that of separation in SBC, discussed in Section 1. As Stated otherwise rigid motion is the only kind of motion a vehicle in a platoon can undergo along any trajectory on which the length of the links remain constant. Thus it is possible to ‘‘maintain formation’’ by making sure that the intra platooning separation between the vehicles is maintained. The requirement on intra platooning separation may be given as
E ¼ Jxi ðtÞxj ðtÞJ ¼ Jxi ðtÞxj ðtÞJ ¼ C 8½t, tÞ 2 R þ , 8fi,jg 2 E
ð1Þ
Intuitively rigidity gives information regarding the minimum number of edges that are needed for maintaining the platoon. The question of whether or not a given graph is rigid has been studied for a long time. One approach to ascertain the rigidity of the 2D planar graph is proposed in [17]. The algorithm starts by verifying if the intra-vehicle distances are preserved among the vehicles along their trajectory. In other words, the algorithm tries to make the pairwise distances constant, in order to maintain the formation. Thus along such a trajectory the condition may be given as in (2) 1 Jx ðtÞxj ðtÞJ2 ¼ C ij 2 i
8fi,jg 2 E t Z0
Assuming smooth trajectory, we can differentiate (2) to get (3) d 1 Jxi ðtÞxj ðtÞJ2 ¼ ðxi ðtÞxj ðtÞÞT ðx_ i ðtÞx_ j ðtÞÞ 8fi,jg 2 E t Z 0 dt 2
ð2Þ
ð3Þ
A little manipulation leads to (4) RðqÞq_ ¼ 0
ð4Þ dn
d
where q ¼ ½x1 ,x2 , . . . ,xn 2 R 8xi 2 R , n is the cardinality of the vertex set and d is the dimension of the vector. It can be seen that the rigidity matrix given by R(q) is of the order Rmnd with m being the number of edges in the given graph. Now, let q0 be a feasible formation, then the graph GP is generically rigid if and only if [15–17] rankðRðq0 ÞÞ ¼ 3n6 if d ¼ 3 ¼ 2n3 if d ¼ 2
ð5Þ
Theorem 1. Given a platoon of vehicles P represented using a graph GP , the platooning can be achieved if and only if the graph GP is rigid. Proof. Consider the platoon in Fig. 2, one may visualize that the cardinality of N is 5 and the cardinality of the edge set is 6. Let us assume the link between vehicles 1 and 2 to fail or in other words the link between vehicles 1 and 2 to drop packets. It is easy to visualize that the edges 2 and 4 are free to fall apart without messing the intra-platooning distance. This can also be ascertained using Eq. (5). From the above discussion, we may conclude that packet loss affects the rigidity of the platoon. In the worst case, the failure of platoons may lead to catastrophic conditions as they tend to affect the safety of the vehicles leading to congestion. In addition to packet losses, the formation or the structure of graph (platoon) is also dynamic owing to vehicles entering, leaving or changing lanes. Human reflexes are not fast enough to meet the requirements of AHS, particularly with Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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dynamic platoons. Thus, there is a need to automate the vehicle to be compatible with AHS. In our analysis, we take up the problem of designing an ADAS for AHS with dynamic platoons by considering packet losses in the channel. The proposed algorithm computes the MST by considering the rigidity of the underlying graph, thus leading to minimal channel utilization. Furthermore, as the state estimates transmission is used as against measured output communication, computations required at individual nodes are significantly reduced. & 3. Computation of MST and MKF Consider the set of points d0 given in (6) as d0 ¼ ½d01 ,d02 , . . . ,d0n
d0i 2 Rdn
ð6Þ
Considering d0 to meet the rigidity constraints in (5), let us now define the relative error ri(t) r i ðtÞ ¼ xi ðtÞd0i One possible strategy to maintain the platoons is to use consensus equation on ri(t), instead of xi(t). Assuming the links to be healthy, we have X ðr i r j Þ ð7Þ r_i ðtÞ ¼ j2Ni ðtÞ
where Ni(t) is the set of one-hop neighbors. It can be seen that r_i ¼
d ðx d Þ ¼ x_i dt i 0i
with r i r j ¼ ðxi d0i Þðxj d0j Þ The platoon control equation can be given as X ðxi xj Þðd0i d0j Þ x_i ¼
ð8Þ
ð9Þ
j2N i ðtÞ
The main drawback of (9) is that the scheme requires more communication as all the vehicles must know the absolute orientation of all the other vehicles in the platoon or of its one-hope neighbors. Optimal utilization of the available bandwidth requires that the amount of transmitted information be minimum. The requirement for maintaining platoon using (9) is that the graph GP should be connected and rigidity should be maintained by computing the state estimate at the individual node. This condition translates to the use of an MST (minimum spanning tree) in the place of the entire network. The first step of the algorithm is to compute the MST of the platoon. It may be seen that the MST is obtained during each time epoch in order to account for packet dropouts and platoon dynamics. 3.1. Minimum spanning tree (MST) The first step of the algorithm is to construct an MST from the leader. One guideline for constructing the MST is by using the distance between the vehicles as the distance between the vehicles increases, delays and packet losses also tend to increase. The MST that gives minimum cost in terms of length also reduces the bandwidth utilization of the channel. Furthermore, it is to be seen that the rigidity of the graph is maintained either by computing the node estimate in the place of the actual measured output or by transmitting it. The MST constructed using a greedy path planning algorithm is shown in Fig. 3(b) and that for the platoon in Fig. 3(a). In place of the output communication, state estimates are computed at individual nodes (1, 2 and 5). Thus, the rigidity is maintained by computing the state estimates at the individual nodes. If the state estimates of 1, 2 and 5 are not computed, the rigidity of the graph is lost or in other words the vehicles tend to fall apart.
Fig. 3. (a) Weighted graph of the platoon, (b) MST with no packet dropout, (c) platoon with packet dropout in link between 5 and 3 and (d) MST with packet dropout.
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Algorithm. Considering a given graph G, weight matrix W and assuming that a token is passed from the vehicles in the platoon in order to verify the connectedness of the edge, the algorithm for constructing the Minimum Spanning Tree [MST] T, may be given as Communication graph with packet losses function ðGlin:fail Þ Input: n:total number of vehicles in the platoons token: token from vehicles m: no. of links for i ¼1:m if token(i)¼ ¼ 1 Q: add the edge to Q else if token(i)¼ ¼ 0 Q: remove edge from Q N(q):number of healthy links % used to find the healthy links end Return Q,N(q)
Step: 2 Tree algorithm. Input: a(v):min leader node edge in Q W ¼ [] Q,N(q) Initialize the tree T WhileðT o NðqÞ1Þ ðu,vÞ’Q :removeminðÞ a(v) be the set containing u b(v) be the set containing v if aðuÞabðvÞ, then add edge(v,u) to T Merge a(u) and b(v) into one cluster Return T
In the presence of packet dropout it is not always possible to transmit the data with minimum cost or in the optimal path as certain links may not be available. Then MST can be constructed from a reduced graph left after deleting the links that are known to drop packets or delay data. In order to identify loss links, it is assumed that all links transmit a token to the leader through all its one-hop neighbors to indicate the availability of the channel. During each time epoch the leader node starts the algorithm by updating the tokens. The MST algorithm considering packet dropout between 5 and 3 is shown in Table 1. 3.2. Estimation of node variables Estimation in the presence of packet dropouts has been investigated by researches in the recent past. In [20], static estimation over graphs has been presented and a BLUE (Best Linear Unbiased Estimator) is proposed. Two algorithms have also been proposed for estimation in sensor networks namely (1) Jacobi algorithm and (2) over-lap sub graph algorithm and it has been concluded that the OS algorithm has better convergence properties than the Jacobi algorithm. Kalman filtering in the presence of packet dropouts has been investigated in [4,3] and the data that are to be transmitted in the presence of packet dropout have been investigated in [18,21]. Estimation over graphs has been investigated in [22] in the context of wireless sensor networks. The algorithm presented in this paper is closely linked with the estimation Table 1 Algorithm for MST considering packet dropout. Iteration Edge /5,1S /3,4S /4,2S /1,3S
1 2 3 4 a
Node a (u) Node b (v) 5 5,1 5,1,3 5,1,3,4,2
1 3,4 4,2
Oa
O represents a null set.
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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methodology in [22]. In our analysis, we propose an optimal estimation algorithm with state estimate alongside the output measurement. After having obtained the MST the next step in the algorithm is to generate an estimate of all the node variables connected to the leader. In order to generate the node estimates of the graph, the vehicle dynamics can be defined as in (10) xðk þ 1Þ ¼ AxðkÞ þwk
ð10Þ
The measurement available from each node may be given as yi ðkÞ ¼ Hi xk þ vik ði ¼ 0,1, . . . ,N1Þ n
i
ð11Þ mi
n
vik
m
2 R are zero-mean Gaussian where xðkÞ 2 R is the state vector, y ðkÞ 2 R is the observation vector, W k 2 R and 0 0 0 random vectors with E½w w ¼ d Q , Q Z 0, E½v v ¼ d P , E½v v ¼ 08t,k and iaj and d ¼ 1 otherwise. We assume that, i k kj k kj k kj j j j pffiffiffiffi ðA, Q Þ is controllable. In order to construct an estimate of the node variable, a Kalman filter is used to provide a dynamic estimate of the node variables or the consensus variables being generated from various vehicles. It is to be noted that the Kalman filter should also account for the packet dropouts in the channel. Hence, in the place of a standard KF, we use a modified Kalman filter (MKF) as in [3]. By considering the arrival of the packet to be modeled to be random using a binary random variable gk as in [3,4], gk ¼ 1 if the packet is transmitted successfully and gk ¼ 0 otherwise. Considering the packet dropout the sensor noise is given by pðvk 9gk Þ ¼ Nð0,RÞ pðvk 9gk Þ ¼ Nð0, s2 IÞ
if gk ¼ 1 if gk ¼ 0
where s is the covariance of the sensor measurement. When packets are dropped, it can be seen that it is equivalent to having a covariance s-1 Let us now define x^ k9k ¼ E½xk 9zk , gk P k9k ¼ E½ðxk x^k Þðxk x^k ÞT 9zk , gk x^ k ffi x^ k þ 19k ¼ E½xk þ 1 9zk , gk P k ¼ P k þ 19k ¼ E½ðxk þ 1 xk ^þ 1 Þðxk þ 1 xk ^þ 1 ÞT 9zk , gk z^ k þ 19k ¼ E½zk þ 1 9zk , gk Time update equations of the Kalman Filter are given using (12) and (13) x^ k þ 19k ¼ Ak x^ k9k
ð12Þ
P k þ 19k ¼ Ak Pk9k LTk þ Q k
ð13Þ
The measurement update equations are given using (14)–(16) x^ k þ 19k þ 1 ¼ x^ k þ 19k þ gk þ 1 K k þ 1 ðzk Hk x^ k þ 19k Þ
ð14Þ
K k þ 1 ¼ Pk þ 19k HTk ðHk Pk þ 19k ÞHTk þ gk þ 1 Rj þ ð1gk þ 1 Rj Þs2j IÞ1
ð15Þ
P k þ 19k þ 1 ¼ P k þ 19k ðIgk K k þ 1 Hk Þ
ð16Þ
where sj and Rj are the covariances of the agents in the adjacency matrix. if measurement from the agent is available, then the variance is R else it is given by s. The missed measurement can be treated as a case with s-1, i.e. the covariance of the measurement tending to infinity. Letting s-1 in Eqs. (14)–(16), we have the modified Kalman filter equations (17)–(19) x^ k þ 19k þ 1 ¼ x^ k þ 19k þ gk þ 1 K k þ 1 ðzk þ 1 Hx^ k þ 19k Þ
ð17Þ
K k ¼ P k HT ðHT P k HT þ gk Rj þ ð1gk Þs2j IÞ1
ð18Þ
P k þ 19k þ 1 ¼ P k þ 19k ðIgk þ 1 K k þ 1 Þ
ð19Þ
Considering that the random variable gk is a binary random variable, if gk ¼ 1 it can be seen that the node estimates are that of the generic Kalman filter and when the random variable gk ¼ 0 it employs the MKF gain that is random and is Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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dependent on the random variable gk . It can be seen that xk þ 1 and Pk þ 1 are now random variables as they are dependent on the arrival sequence gk and hence can be computed only by simulation and not off-line [3]. 4. Implementation of ADAS Consider a platoon P with a leader node having a depth of say D. Consider that at time k, the node N i generates a i measurement packet yk and sends it with all previous measurements from all its sensors that are after kD to all its neighbors, D Z0 is a constant and it is assumed that all the sensor data are time stamped and this frame work is similar to that in [22]. The time-stamping is necessary to impart the leader node with information regarding the node and time of the data. Let Bik9kl ðl ¼ 0,1,2, . . . ,D1Þ be the measurement packets available at the leader node at time k. Considering Fig. 2(b), it may be seen that the depth of the leader node is 2 and it sends fy5k ,y1k1 ,y3k1 g to its one-hop neighbors 1, 3 and receives fy1k ,y2k1 ,y1k1 ,y5k1 g from node 1 and fy3k ,y4k1 ,y3k1 ,y5k1 g from node 3. Communication sequence in node 3 is given by (20) Bik9kl ¼ fy1k1 ,y2k1 ,y3k1 ,y4k1 ,y5k1 g Bik9k ¼ fy1k ,y3k ,y5k g
ð20Þ
It may be seen that there may be redundancy in data and it can be avoided by using the MST in place of the entire graph as in Fig. 3(a). It may be seen that this reduces the communication required, as the redundancy in communication is avoided. Now if packet dropouts are considered as in Fig. 3(a) and by using the MST in 3(d), the communication sequence available at the leader node is given as Bik9kl ¼ fy1k1 ,y2k1 ,y3k1 ,y4k1 ,y5k1 g Bik9k ¼ fy1k ,y5k g
ð21Þ
The basic idea behind the computation of the estimate is as follows. Since yki may arrive at time k either due to delay in the channel or due to packet dropout. We may improve the quality of estimation by recalculating xki , utilizing the newly available measurement yki . Once x^ ki is updated. The optimal estimate of the node variable xik can be computed using the new available measurement yki as in [22]. The following estimation summarizes the estimation pattern: ðx^ ki ,Pki Þ ¼ MKFðx^ ki1 ,P ki1 ,1,yki Þ ðx^ ki þ 1 ,P ki þ 1 Þ ¼ MKFðx^ ki ,P ki , gki þ 1 ,yki þ 1 Þ ^ ðx^ k ,P k Þ ¼ MKFðx^ k1 ,P k1 , gk ,yk Þ
ð22Þ
The estimation pattern as shown in Fig. 4 is the D-step MKF algorithm presented in this section. Once the estimate of the node variable is available, (9) is used to maintain the position moves of the vehicle. The algorithm for ADAS may then be given as: input n,m,w,token,B MST(n,e,e,token) Return T For i ¼1:m MKF(T,B) end Return X¼ [x1,x2,...,xn] Use (9)
Fig. 4. MKF iterations at time k (cf. [22]).
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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5. Results and discussions Assuming that there is a packet dropout in the link between vehicles 5 and 3 in the platoon in Fig. 2(a), the MST with packet loss is shown in Fig. 3(d). Vehicles are assumed to be placed along the edges of the planar graph with the x–y positions of the vehicles in ascending order given by ð1,1Þ,ð0,1Þ,ð1,0Þ,ð0,0Þ,ð2,0:5Þ respectively. One position move translates to position of vehicles taken in ascending order as ð2,1Þ,ð1,1Þ,ð2,0Þ,ð1,0Þ, and ð3,0:5Þ. Fig. 5 shows the position of vehicle 5 considering packet loss in the communication channel which is found to be ð2:3,0:5Þ along x-axis. This shows that the rigidity or the safety of platoon is significantly affected by packet loss as stated in Theorem 1. Simulation studies show that the vehicles tend to fall apart when the packet loss rate exceeds 7% as shown in Fig. 5. The highest resilience to packet loss was recorded in the communication channel between 5 and 3 and is around 7%. The lowest resilience was found in channel between vehicles 1 and 2 and is about 4.5%. Rest of the section presents the positions of vehicle considering packet loss in the least resilient link between vehicles 1 and 2 and assuming two position moves. The position of vehicle 2 using the proposed state estimation scheme with a packet loss rate of 20% against the actual position of the vehicle with the output transmission scheme without packet loss is shown in Fig. 6. Although initially vehicle tends to be moving away, using the estimation scheme it is able to converge to the actual position after 53 iterations or time-steps. Fig. 7 shows the covariance of the estimation error of the proposed algorithm against the Kalman filter that employs full measurement information. This shows that the performance at 20% packet loss rate is comparable with that of a Kalman filter with full information. The performance of the proposed ADAS algorithm with error introduced in the output measurement is shown in Fig. 8 and it can be verified that estimation error is significantly reduced. Here the estimation error is calculated by translating the state estimate into output estimates. The performance of the proposed algorithm for 30, 40 and 48% dropout rates are shown in Figs. 9, 10 and 11, respectively. Although an increase in packet loss results in more iterations for estimating the actual position, the estimation error is
Fig. 5. Desired position and actual position of the vehicle in the presence of 8% packet dropout using measurement communication scheme.
Fig. 6. Actual position vs estimated position of vehicle 2 with 20% packet dropouts.
Fig. 7. Covariance of the proposed scheme with 20% dropout vs Kalman filter.
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Fig. 8. Measurement error vs estimation error.
Fig. 9. Actual position vs estimated position of vehicle 2 with 30% packet dropouts.
Fig. 10. Actual position vs estimated position of vehicle 2 with 40% packet dropouts.
Fig. 11. Actual position vs estimated position of vehicle 2 with 50% packet dropouts.
relatively small. Simulation studies on this test problem indicate a resilience up to 48% using the proposed scheme above which the performance starts to degrade. One important attribute for any algorithm designed for distributed systems like IVHS is scaling that denotes the performance of the algorithm with an increase in number of vehicles alongside packet loss rate. Simulations show that the time-step required to form the estimates increases with the number of vehicles and packet loss rate as shown in Fig. 12. This indicates that the proposed scheme does not scale up well with the number of vehicles and for an increase in packet loss rate. On the other hand at low packet loss rates the performance is not significantly affected by the number of vehicles. Thus the algorithm scales up well for low packet loss rates. But for high packet loss rates the proposed algorithm does not scale up and this is one short-coming of this algorithm. Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Fig. 12. Iterations vs packet loss rate for D ¼3,5,9,12,15.
Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ymssp. 2012.08.020.
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Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Advanced driver assistance system for AHS over communication links with random packet dropouts Seshadhri Srinivasan, Ramakalyan Ayyagari n Department of Instrumentation and Control Engineering, National Institute of Technology—Tiruchirappalli, Tiruchirappalli, India
a r t i c l e i n f o
abstract
Article history: Received 2 March 2010 Received in revised form 2 August 2012 Accepted 15 August 2012
In this paper, we propose an advanced driver assist system (ADAS) for platoon based automated highway system (AHS) with packet loss in inter-vehicle communication. Using the concept of rigidity, we first show that vehicles in a platoon tend to fall apart in the event of a packet loss among vehicles. To overcome this, we propose an estimation based dynamic platooning algorithm which employs the state estimate to maintain the platoon. Communication among the vehicle is reduced by using minimum spanning tree (MST) in state estimation algorithm. Effectiveness of the proposed ADAS scheme is illustrated by simulation wherein, dynamic platoons of holonomic vehicles with integrator dynamics are considered. Simulation studies indicate that the proposed algorithm maintains the platoon up to a packet loss rate of 48%. State transmission scheme proposed in our algorithm has three significant advantages, they are: (1) it handles packet loss in inter-vehicle communication, (2) reduces the effect of error in measured output, and (3) reduces the inter-vehicle communication. These advantages significantly increase the reliability and safety of the AHS. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Automated highway system (AHS) Advanced driver assistance system (ADAS) Packet dropouts Modified Kalman filter (MKF) Minimum spanning tree (MST)
1. Introduction AHS have been studied recently as a solution to highway congestion problem. Vehicle of the future need to be compatible with AHS for congestion mitigation and safety. Among many solutions available for implementing AHS, platooning is the most promising option studied in the literature. A detailed review of AHS implementation using platooning is available in [1]. In [2], it has been argued that the platooning structure achieves a balance between safety and throughput. Platooning requires coordination among the vehicles in a platoon, as well as among platoons. Now, consider a platoon as in Fig. 1, where d is called the intra-platooning distance and b bearing or the angle of the followers with respect to their peers. The given platoon may be abstracted as a graph with the vehicles as the nodes and the edges representing the communication channel among the vehicles. As seen in Fig. 1, in order to maintain a platoon, vehicles not only need to coordinate but also monitor the other vehicles in the platoon or in other words know the relative position of other vehicles in the platoon. Coordinating and monitoring other vehicles in the platoon are beyond human reflexes and require automation. One of the major challenges impeding automation of AHS is the presence of packet losses in the communication channel which are used for transmitting coordination information among vehicles in a platoon. In the presence of packet losses coordination information like relative velocity, displacement, bearing, etc., is either unavailable or imprecise.
n
Corresponding author. E-mail addresses: [email protected] (S. Srinivasan), [email protected] (R. Ayyagari).
0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.08.020
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Fig. 1. Platoon of vehicles.
This increases the risk of collision and is against the very spirit of employing AHS. Thus, there is a need not only to automate AHS but also to account for the packet losses in the channel. In this paper, we propose an ADAS system for AHS over packet dropping links by using platooning concept. Implementation aspects of AHS using vehicle platooning have been studied in [1,2]. Decentralized hybrid controllers for AHS have been proposed in [2] and a game theoretic approach is used to design the controllers that are guaranteed to meet safety requirements. In our analysis, we employ a formation control framework for maintaining platoons over packetdropping links. One popular method widely used for formation control is the separation bearing control (SBC) and has been studied in the context of maintaining formation among robots in [5,7–11]. In [12], consensus problems for network of agents with fixed and switching topologies have been investigated. Switching topologies consider platoons which are dynamic alongside packet losses. In [14], cooperative control of vehicle formation problem in the presence of channel constraints has been investigated. Formation control using the concept of rigidity has been studied in [15,16]. Recently, in [19] flocking of multi-agents with a virtual leader has been investigated by relaxing the usual assumptions that: (1) all agents need to be informed and (2) virtual leader travels at constant velocity. An overview of the obtained results in formation control are presented in [6]. In [22], the estimation of over wireless networks is presented by considering packet losses and delays. It has been shown that for a densely distributed sensor network with packet loss the state estimate can be improved by recalculating the estimate with a delayed measurement or lost packet. Recently, a platooning algorithm for AHS over packet dropping links has been proposed in [23,24]. It has been shown that the estimate error covariance reduces in the event of packet loss by transmitting a linear combination of current data and past estimate. Research on whether transmitting estimate of the state or measured output to improve the reliability of the platoon has not been investigated in the past. Motivated by this, we investigate the problem of ADAS design using state estimate transmission for AHS subjected to packet loss in inter-vehicle communication. In this paper, we discuss the implementation aspects of ADAS for AHS using the platooning concept in the presence of packet losses. We first show that the presence of packet dropouts affects the safety of the vehicle using the concept of rigidity. Next, we propose an estimation based platooning algorithm which employs minimal communication. The minimal communication is established by constructing the minimum spanning tree (MST) using the concept of rigidity. Although the proposed algorithm is closely linked to [22], our investigation has two extensions: (1) transmitting state estimate in the place of measured output and (2) minimizing the communication using MST and still retaining the improvement in estimation error in [22]. Our results show that transmitting state estimate makes AHS more robust to packet loss than the output measurement transmission scheme. This is particularly true in the presence of errors in output measurement. Furthermore, the two assumptions of flocking algorithms presented above are also relaxed with our approach. The proposed ADAS is illustrated using simulation studies conducted using holonomic point vehicles with integrator dynamics. The paper is organized as six sections, including the Introduction. In Section 2, the problem is formulated. In Section 3, the methodology employed to find the MST and the MKF proposed in [4] are discussed. In Section 4, the ADAS algorithm is proposed and in Section 5 the proposed algorithm is implemented considering a platoon with packet losses. Conclusions are drawn from the obtained results and the future course of this investigation is presented in Section 6.
2. Problem formulation Consider the platoon say P of vehicles that are connected using communication links as in Fig. 2(a). P may be conveniently represented using an undirected graph GP 9fN, E, Wg, where N is the node/vertex set of the graph that represents the vehicles in the platoon and E is the edge set that denotes the communication link among the vehicles in a platoon and W is the set of weights and is the intra platoon distance between the vehicles in the platoon. The graph of the platoon is shown in Fig. 2(b), where the nodes represent the vehicles and the edges represent the communication links. Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
S. Seshadhri, R. Ayyagari / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Fig. 2. (a) Platoon of vehicles and (b) graphical representation.
Definition 1. A vehicle is said to undergo rigid motion along a trajectory, only when the Euclidean distance between the vehicles in the platoon remains constant all along the trajectory of the vehicle. Definition 2. The graph GP shown in Fig. 2(b) is said to be rigid, if for all position assignments of the nodes, each and every move of the vehicle preserves the distance between the position of any pair of vertices in a graph. This condition may be expressed as in (1)
E ¼ Jxi xj J ¼ Cij
8fi,jg 2 E
where Cij is the predetermined distance between i and j in the platoon or it is the intra-vehicular spacing in the platoon. It may be noted that the intra spacing between the vehicles is the same as that of separation in SBC, discussed in Section 1. As Stated otherwise rigid motion is the only kind of motion a vehicle in a platoon can undergo along any trajectory on which the length of the links remain constant. Thus it is possible to ‘‘maintain formation’’ by making sure that the intra platooning separation between the vehicles is maintained. The requirement on intra platooning separation may be given as
E ¼ Jxi ðtÞxj ðtÞJ ¼ Jxi ðtÞxj ðtÞJ ¼ C 8½t, tÞ 2 R þ , 8fi,jg 2 E
ð1Þ
Intuitively rigidity gives information regarding the minimum number of edges that are needed for maintaining the platoon. The question of whether or not a given graph is rigid has been studied for a long time. One approach to ascertain the rigidity of the 2D planar graph is proposed in [17]. The algorithm starts by verifying if the intra-vehicle distances are preserved among the vehicles along their trajectory. In other words, the algorithm tries to make the pairwise distances constant, in order to maintain the formation. Thus along such a trajectory the condition may be given as in (2) 1 Jx ðtÞxj ðtÞJ2 ¼ C ij 2 i
8fi,jg 2 E t Z0
Assuming smooth trajectory, we can differentiate (2) to get (3) d 1 Jxi ðtÞxj ðtÞJ2 ¼ ðxi ðtÞxj ðtÞÞT ðx_ i ðtÞx_ j ðtÞÞ 8fi,jg 2 E t Z 0 dt 2
ð2Þ
ð3Þ
A little manipulation leads to (4) RðqÞq_ ¼ 0
ð4Þ dn
d
where q ¼ ½x1 ,x2 , . . . ,xn 2 R 8xi 2 R , n is the cardinality of the vertex set and d is the dimension of the vector. It can be seen that the rigidity matrix given by R(q) is of the order Rmnd with m being the number of edges in the given graph. Now, let q0 be a feasible formation, then the graph GP is generically rigid if and only if [15–17] rankðRðq0 ÞÞ ¼ 3n6 if d ¼ 3 ¼ 2n3 if d ¼ 2
ð5Þ
Theorem 1. Given a platoon of vehicles P represented using a graph GP , the platooning can be achieved if and only if the graph GP is rigid. Proof. Consider the platoon in Fig. 2, one may visualize that the cardinality of N is 5 and the cardinality of the edge set is 6. Let us assume the link between vehicles 1 and 2 to fail or in other words the link between vehicles 1 and 2 to drop packets. It is easy to visualize that the edges 2 and 4 are free to fall apart without messing the intra-platooning distance. This can also be ascertained using Eq. (5). From the above discussion, we may conclude that packet loss affects the rigidity of the platoon. In the worst case, the failure of platoons may lead to catastrophic conditions as they tend to affect the safety of the vehicles leading to congestion. In addition to packet losses, the formation or the structure of graph (platoon) is also dynamic owing to vehicles entering, leaving or changing lanes. Human reflexes are not fast enough to meet the requirements of AHS, particularly with Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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dynamic platoons. Thus, there is a need to automate the vehicle to be compatible with AHS. In our analysis, we take up the problem of designing an ADAS for AHS with dynamic platoons by considering packet losses in the channel. The proposed algorithm computes the MST by considering the rigidity of the underlying graph, thus leading to minimal channel utilization. Furthermore, as the state estimates transmission is used as against measured output communication, computations required at individual nodes are significantly reduced. & 3. Computation of MST and MKF Consider the set of points d0 given in (6) as d0 ¼ ½d01 ,d02 , . . . ,d0n
d0i 2 Rdn
ð6Þ
Considering d0 to meet the rigidity constraints in (5), let us now define the relative error ri(t) r i ðtÞ ¼ xi ðtÞd0i One possible strategy to maintain the platoons is to use consensus equation on ri(t), instead of xi(t). Assuming the links to be healthy, we have X ðr i r j Þ ð7Þ r_i ðtÞ ¼ j2Ni ðtÞ
where Ni(t) is the set of one-hop neighbors. It can be seen that r_i ¼
d ðx d Þ ¼ x_i dt i 0i
with r i r j ¼ ðxi d0i Þðxj d0j Þ The platoon control equation can be given as X ðxi xj Þðd0i d0j Þ x_i ¼
ð8Þ
ð9Þ
j2N i ðtÞ
The main drawback of (9) is that the scheme requires more communication as all the vehicles must know the absolute orientation of all the other vehicles in the platoon or of its one-hope neighbors. Optimal utilization of the available bandwidth requires that the amount of transmitted information be minimum. The requirement for maintaining platoon using (9) is that the graph GP should be connected and rigidity should be maintained by computing the state estimate at the individual node. This condition translates to the use of an MST (minimum spanning tree) in the place of the entire network. The first step of the algorithm is to compute the MST of the platoon. It may be seen that the MST is obtained during each time epoch in order to account for packet dropouts and platoon dynamics. 3.1. Minimum spanning tree (MST) The first step of the algorithm is to construct an MST from the leader. One guideline for constructing the MST is by using the distance between the vehicles as the distance between the vehicles increases, delays and packet losses also tend to increase. The MST that gives minimum cost in terms of length also reduces the bandwidth utilization of the channel. Furthermore, it is to be seen that the rigidity of the graph is maintained either by computing the node estimate in the place of the actual measured output or by transmitting it. The MST constructed using a greedy path planning algorithm is shown in Fig. 3(b) and that for the platoon in Fig. 3(a). In place of the output communication, state estimates are computed at individual nodes (1, 2 and 5). Thus, the rigidity is maintained by computing the state estimates at the individual nodes. If the state estimates of 1, 2 and 5 are not computed, the rigidity of the graph is lost or in other words the vehicles tend to fall apart.
Fig. 3. (a) Weighted graph of the platoon, (b) MST with no packet dropout, (c) platoon with packet dropout in link between 5 and 3 and (d) MST with packet dropout.
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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Algorithm. Considering a given graph G, weight matrix W and assuming that a token is passed from the vehicles in the platoon in order to verify the connectedness of the edge, the algorithm for constructing the Minimum Spanning Tree [MST] T, may be given as Communication graph with packet losses function ðGlin:fail Þ Input: n:total number of vehicles in the platoons token: token from vehicles m: no. of links for i ¼1:m if token(i)¼ ¼ 1 Q: add the edge to Q else if token(i)¼ ¼ 0 Q: remove edge from Q N(q):number of healthy links % used to find the healthy links end Return Q,N(q)
Step: 2 Tree algorithm. Input: a(v):min leader node edge in Q W ¼ [] Q,N(q) Initialize the tree T WhileðT o NðqÞ1Þ ðu,vÞ’Q :removeminðÞ a(v) be the set containing u b(v) be the set containing v if aðuÞabðvÞ, then add edge(v,u) to T Merge a(u) and b(v) into one cluster Return T
In the presence of packet dropout it is not always possible to transmit the data with minimum cost or in the optimal path as certain links may not be available. Then MST can be constructed from a reduced graph left after deleting the links that are known to drop packets or delay data. In order to identify loss links, it is assumed that all links transmit a token to the leader through all its one-hop neighbors to indicate the availability of the channel. During each time epoch the leader node starts the algorithm by updating the tokens. The MST algorithm considering packet dropout between 5 and 3 is shown in Table 1. 3.2. Estimation of node variables Estimation in the presence of packet dropouts has been investigated by researches in the recent past. In [20], static estimation over graphs has been presented and a BLUE (Best Linear Unbiased Estimator) is proposed. Two algorithms have also been proposed for estimation in sensor networks namely (1) Jacobi algorithm and (2) over-lap sub graph algorithm and it has been concluded that the OS algorithm has better convergence properties than the Jacobi algorithm. Kalman filtering in the presence of packet dropouts has been investigated in [4,3] and the data that are to be transmitted in the presence of packet dropout have been investigated in [18,21]. Estimation over graphs has been investigated in [22] in the context of wireless sensor networks. The algorithm presented in this paper is closely linked with the estimation Table 1 Algorithm for MST considering packet dropout. Iteration Edge /5,1S /3,4S /4,2S /1,3S
1 2 3 4 a
Node a (u) Node b (v) 5 5,1 5,1,3 5,1,3,4,2
1 3,4 4,2
Oa
O represents a null set.
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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methodology in [22]. In our analysis, we propose an optimal estimation algorithm with state estimate alongside the output measurement. After having obtained the MST the next step in the algorithm is to generate an estimate of all the node variables connected to the leader. In order to generate the node estimates of the graph, the vehicle dynamics can be defined as in (10) xðk þ 1Þ ¼ AxðkÞ þwk
ð10Þ
The measurement available from each node may be given as yi ðkÞ ¼ Hi xk þ vik ði ¼ 0,1, . . . ,N1Þ n
i
ð11Þ mi
n
vik
m
2 R are zero-mean Gaussian where xðkÞ 2 R is the state vector, y ðkÞ 2 R is the observation vector, W k 2 R and 0 0 0 random vectors with E½w w ¼ d Q , Q Z 0, E½v v ¼ d P , E½v v ¼ 08t,k and iaj and d ¼ 1 otherwise. We assume that, i k kj k kj k kj j j j pffiffiffiffi ðA, Q Þ is controllable. In order to construct an estimate of the node variable, a Kalman filter is used to provide a dynamic estimate of the node variables or the consensus variables being generated from various vehicles. It is to be noted that the Kalman filter should also account for the packet dropouts in the channel. Hence, in the place of a standard KF, we use a modified Kalman filter (MKF) as in [3]. By considering the arrival of the packet to be modeled to be random using a binary random variable gk as in [3,4], gk ¼ 1 if the packet is transmitted successfully and gk ¼ 0 otherwise. Considering the packet dropout the sensor noise is given by pðvk 9gk Þ ¼ Nð0,RÞ pðvk 9gk Þ ¼ Nð0, s2 IÞ
if gk ¼ 1 if gk ¼ 0
where s is the covariance of the sensor measurement. When packets are dropped, it can be seen that it is equivalent to having a covariance s-1 Let us now define x^ k9k ¼ E½xk 9zk , gk P k9k ¼ E½ðxk x^k Þðxk x^k ÞT 9zk , gk x^ k ffi x^ k þ 19k ¼ E½xk þ 1 9zk , gk P k ¼ P k þ 19k ¼ E½ðxk þ 1 xk ^þ 1 Þðxk þ 1 xk ^þ 1 ÞT 9zk , gk z^ k þ 19k ¼ E½zk þ 1 9zk , gk Time update equations of the Kalman Filter are given using (12) and (13) x^ k þ 19k ¼ Ak x^ k9k
ð12Þ
P k þ 19k ¼ Ak Pk9k LTk þ Q k
ð13Þ
The measurement update equations are given using (14)–(16) x^ k þ 19k þ 1 ¼ x^ k þ 19k þ gk þ 1 K k þ 1 ðzk Hk x^ k þ 19k Þ
ð14Þ
K k þ 1 ¼ Pk þ 19k HTk ðHk Pk þ 19k ÞHTk þ gk þ 1 Rj þ ð1gk þ 1 Rj Þs2j IÞ1
ð15Þ
P k þ 19k þ 1 ¼ P k þ 19k ðIgk K k þ 1 Hk Þ
ð16Þ
where sj and Rj are the covariances of the agents in the adjacency matrix. if measurement from the agent is available, then the variance is R else it is given by s. The missed measurement can be treated as a case with s-1, i.e. the covariance of the measurement tending to infinity. Letting s-1 in Eqs. (14)–(16), we have the modified Kalman filter equations (17)–(19) x^ k þ 19k þ 1 ¼ x^ k þ 19k þ gk þ 1 K k þ 1 ðzk þ 1 Hx^ k þ 19k Þ
ð17Þ
K k ¼ P k HT ðHT P k HT þ gk Rj þ ð1gk Þs2j IÞ1
ð18Þ
P k þ 19k þ 1 ¼ P k þ 19k ðIgk þ 1 K k þ 1 Þ
ð19Þ
Considering that the random variable gk is a binary random variable, if gk ¼ 1 it can be seen that the node estimates are that of the generic Kalman filter and when the random variable gk ¼ 0 it employs the MKF gain that is random and is Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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dependent on the random variable gk . It can be seen that xk þ 1 and Pk þ 1 are now random variables as they are dependent on the arrival sequence gk and hence can be computed only by simulation and not off-line [3]. 4. Implementation of ADAS Consider a platoon P with a leader node having a depth of say D. Consider that at time k, the node N i generates a i measurement packet yk and sends it with all previous measurements from all its sensors that are after kD to all its neighbors, D Z0 is a constant and it is assumed that all the sensor data are time stamped and this frame work is similar to that in [22]. The time-stamping is necessary to impart the leader node with information regarding the node and time of the data. Let Bik9kl ðl ¼ 0,1,2, . . . ,D1Þ be the measurement packets available at the leader node at time k. Considering Fig. 2(b), it may be seen that the depth of the leader node is 2 and it sends fy5k ,y1k1 ,y3k1 g to its one-hop neighbors 1, 3 and receives fy1k ,y2k1 ,y1k1 ,y5k1 g from node 1 and fy3k ,y4k1 ,y3k1 ,y5k1 g from node 3. Communication sequence in node 3 is given by (20) Bik9kl ¼ fy1k1 ,y2k1 ,y3k1 ,y4k1 ,y5k1 g Bik9k ¼ fy1k ,y3k ,y5k g
ð20Þ
It may be seen that there may be redundancy in data and it can be avoided by using the MST in place of the entire graph as in Fig. 3(a). It may be seen that this reduces the communication required, as the redundancy in communication is avoided. Now if packet dropouts are considered as in Fig. 3(a) and by using the MST in 3(d), the communication sequence available at the leader node is given as Bik9kl ¼ fy1k1 ,y2k1 ,y3k1 ,y4k1 ,y5k1 g Bik9k ¼ fy1k ,y5k g
ð21Þ
The basic idea behind the computation of the estimate is as follows. Since yki may arrive at time k either due to delay in the channel or due to packet dropout. We may improve the quality of estimation by recalculating xki , utilizing the newly available measurement yki . Once x^ ki is updated. The optimal estimate of the node variable xik can be computed using the new available measurement yki as in [22]. The following estimation summarizes the estimation pattern: ðx^ ki ,Pki Þ ¼ MKFðx^ ki1 ,P ki1 ,1,yki Þ ðx^ ki þ 1 ,P ki þ 1 Þ ¼ MKFðx^ ki ,P ki , gki þ 1 ,yki þ 1 Þ ^ ðx^ k ,P k Þ ¼ MKFðx^ k1 ,P k1 , gk ,yk Þ
ð22Þ
The estimation pattern as shown in Fig. 4 is the D-step MKF algorithm presented in this section. Once the estimate of the node variable is available, (9) is used to maintain the position moves of the vehicle. The algorithm for ADAS may then be given as: input n,m,w,token,B MST(n,e,e,token) Return T For i ¼1:m MKF(T,B) end Return X¼ [x1,x2,...,xn] Use (9)
Fig. 4. MKF iterations at time k (cf. [22]).
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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S. Seshadhri, R. Ayyagari / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
5. Results and discussions Assuming that there is a packet dropout in the link between vehicles 5 and 3 in the platoon in Fig. 2(a), the MST with packet loss is shown in Fig. 3(d). Vehicles are assumed to be placed along the edges of the planar graph with the x–y positions of the vehicles in ascending order given by ð1,1Þ,ð0,1Þ,ð1,0Þ,ð0,0Þ,ð2,0:5Þ respectively. One position move translates to position of vehicles taken in ascending order as ð2,1Þ,ð1,1Þ,ð2,0Þ,ð1,0Þ, and ð3,0:5Þ. Fig. 5 shows the position of vehicle 5 considering packet loss in the communication channel which is found to be ð2:3,0:5Þ along x-axis. This shows that the rigidity or the safety of platoon is significantly affected by packet loss as stated in Theorem 1. Simulation studies show that the vehicles tend to fall apart when the packet loss rate exceeds 7% as shown in Fig. 5. The highest resilience to packet loss was recorded in the communication channel between 5 and 3 and is around 7%. The lowest resilience was found in channel between vehicles 1 and 2 and is about 4.5%. Rest of the section presents the positions of vehicle considering packet loss in the least resilient link between vehicles 1 and 2 and assuming two position moves. The position of vehicle 2 using the proposed state estimation scheme with a packet loss rate of 20% against the actual position of the vehicle with the output transmission scheme without packet loss is shown in Fig. 6. Although initially vehicle tends to be moving away, using the estimation scheme it is able to converge to the actual position after 53 iterations or time-steps. Fig. 7 shows the covariance of the estimation error of the proposed algorithm against the Kalman filter that employs full measurement information. This shows that the performance at 20% packet loss rate is comparable with that of a Kalman filter with full information. The performance of the proposed ADAS algorithm with error introduced in the output measurement is shown in Fig. 8 and it can be verified that estimation error is significantly reduced. Here the estimation error is calculated by translating the state estimate into output estimates. The performance of the proposed algorithm for 30, 40 and 48% dropout rates are shown in Figs. 9, 10 and 11, respectively. Although an increase in packet loss results in more iterations for estimating the actual position, the estimation error is
Fig. 5. Desired position and actual position of the vehicle in the presence of 8% packet dropout using measurement communication scheme.
Fig. 6. Actual position vs estimated position of vehicle 2 with 20% packet dropouts.
Fig. 7. Covariance of the proposed scheme with 20% dropout vs Kalman filter.
Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
S. Seshadhri, R. Ayyagari / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Fig. 8. Measurement error vs estimation error.
Fig. 9. Actual position vs estimated position of vehicle 2 with 30% packet dropouts.
Fig. 10. Actual position vs estimated position of vehicle 2 with 40% packet dropouts.
Fig. 11. Actual position vs estimated position of vehicle 2 with 50% packet dropouts.
relatively small. Simulation studies on this test problem indicate a resilience up to 48% using the proposed scheme above which the performance starts to degrade. One important attribute for any algorithm designed for distributed systems like IVHS is scaling that denotes the performance of the algorithm with an increase in number of vehicles alongside packet loss rate. Simulations show that the time-step required to form the estimates increases with the number of vehicles and packet loss rate as shown in Fig. 12. This indicates that the proposed scheme does not scale up well with the number of vehicles and for an increase in packet loss rate. On the other hand at low packet loss rates the performance is not significantly affected by the number of vehicles. Thus the algorithm scales up well for low packet loss rates. But for high packet loss rates the proposed algorithm does not scale up and this is one short-coming of this algorithm. Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020
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S. Seshadhri, R. Ayyagari / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
Fig. 12. Iterations vs packet loss rate for D ¼3,5,9,12,15.
Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ymssp. 2012.08.020.
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Please cite this article as: S. Srinivasan, R. Ayyagari, Advanced driver assistance system for AHS over communication links with random packet dropouts, Mech. Syst. Signal Process. (2012), http://dx.doi.org/10.1016/j.ymssp.2012.08.020