IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 24, NO. I, JULY 1994

R. W. Payne and D. A. Preece, “Identification keys and diagnostic tables: A review,” J. Royal Statist. SOC., vol. A143, no. 3, pp. 253-292, 1980. C. L. Chang and J. R. Slagle, “An admissable and optimal algorithm for search AND/OR graphs,” Art$ Intell., vol. 2, no. 2, pp. 117-128, 1971. A. Martelli and U. Montanari, “Optimizing decision trees through heuristically guided search,’’ Commun. ACM, vol. 21, no. 12, pp. 1025-1039, 1978. N. J. Nilsson, Principles of Artificial Intelligence. Palo Alto, CA: Tioga, 1980. A. Bagcbi and A. Mahanti, “Admissable heuristic search in AND/OR graphs,” Theoret. Comput. Sei., vol. 24, no. 2, pp. 207-219, 1983.

Performance Analysis of Token Bus LAN in Coordinating Multiple Robots

Qichao Yin and Yuan F. Zheng

Abstract-Using a Local Area Network (LAN) to coordinate multiple industrial robots is considered in this paper. The LAN uses the standard Manufacturing Automation Protocol (MAP) to control the access of the communication bus. Since the MAP based LAN was not designed for real-time applications, performance evaluation of the LAN becomes an important issue. A new evaluation method is proposed in thii paper. The method evaluates the efficiency of the LAN by a parameter called effective token passing ratio. This parameter is affected by the speed and the access protocol of the LAN as well as the task decomposition-and-allocation scheme for computing the coordination algorithms. Previous methods, however, only evaluate the speed and the access protocol of the LAN. As a result, the new method provides more options for improving the LAN based system. To guarantee real-time coordination of multiple robots, the lower bound of the effective token passing ratio is determined and the probability of successful coordination when the transmission errors are possible is also considered.

I. INTRODUCTION Industrial robots play an important role in manufacturing automa-

tion. Since many manufacturing tasks need two or more robots such as carrying large objects, assembling complex parts, macro-micro manipulation, etc., many studies have been devoted to coordination of multiple robots. Most of these works, however, have been focused on the theoretical aspects including kinematics, dynamics and control of multirobot systems [ 11-[6]. Practical implementation of a multirobot system has received relatively less attention.

A number of multirobot systems have been implemented by a few research groups. Zheng, Luh, and Jia proposed a mechanism to integrate two industrial robots into a coordinated system [7], in which a VAX- 11/750 was used as the central computer to supervise two robot controllers. In addition, a distributed operating system was designed to make the integrated system function as a unified whole. In an earlier work by Maimon and Nof, the concept of Activity Controller was proposed for coordinating multiple robots [8]. The function of the Active Controller was similar to the central computer of [7] which synchronized the sequence of robot operations and avoided conflicts between two coordinated robots.

Manuscript received September 8, 1991; revised January 8, 1993 and August 16,1993. This work was supported by the National Science Foundation under Grant DDM-8996238 and by the ONR under Grant ”14-90-J-1516.

The authors are with the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210 USA.

IEEE Log Number 9402083.

When two robots execute a coordination task, the motion of one robot depends on the motion of the other [7]. As a result, real- time communication becomes necessary. Traditional approaches for real-time communication are to establish dedicated communication channels between every pair of computers. For example, in [7], parallel interfaces were established between the two robot controllers and between the robot controllers and the central computer. More recently, Tarn et al. developed a multirobot system in which dedicated interfaces were used for real-time synchronization, and a local area network was used for off-line information exchange [9]. The drawback of this dedicated approach is complexity. As the number of computers increase in a multirobot system, the number of the interfaces will become too massive to handle.

In this paper, we propose to use a local area network (LAN) which is based on the Manufacturing Automation Protocol (MAP) [IO] to coordinate a multirobot system. Unfortunately, MAP based LAN’s were originally designed for off-line communication. Previous works even judged that the MAP protocol was not suitable for real-time applications. Shin et al. [ l l ] , [I21 argued that the MAP protocol was too time-consuming to be a real-time protocol. Shin further proposed a new bus access mechanism called polled bus to reduce communication delays. The polled protocol computed a poll number for any message generated by a manufacturing task according to the message’s deadline and the task’s priority. The message with the highest poll number was guaranteed to get control of the bus. It was argued that real-time communication has a higher success rate using the polled bus than using the MAP (121. In addition to the above works, many other methods have also been proposed. Gauthier et al. summarized these methods in [13]. In summary, the efforts of the previous methods were focused on either new structures or new protocols of the communication channels.

However, the MAP based LAN is too attractive to be rejected from real-time applications. The reason is twofold. First, the MAP is a standard in the manufacturing industry. It will be cost effective to use a single standard for both off-line and on-line communication. Secondly, as the advent of new technologies in the LAN, the delay caused by the protocol becomes less a problem. For example, a recent technology called Fiber Distributed Data Interface offering a rate of 100 Mbps which is ten times as fast as the old technol- ogy.

Using the LAN for coordinating multiple robots, one has to consider if the LAN can satisfy the time constraint imposed on the communication delays, i.e., the performance of the LAN. Many performance evaluation methods have been proposed, such as those in [12] and [14]-[18]. These methods evaluate the delays of individual messages and compute how soon a new message can reach its destination. Other evaluation methods ([ 191-[2 I]) use maximum mean throughput rates and average message transmission delays as the performance measures. All these methods are not suitable in our applications. Since the protocol of the LAN has been determined and is an industrial standard, it is not necessary to evaluate the protocol again. The importance is how the LAN will perform for a given coordination task.

For coordinating a multirobot system, the coordination algorithm is to be computed by a number of computers involved in the system. To do so, the algorithm must be decomposed into subtasks which need to be allocated to the multiple computers. The efficiency of this task decomposition-and-allocation scheme affects the performance of the system as much as the speed and protocol of the LAN. Likewise,

0018-9472/94$04.00 0 1994 IEEE

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Object frame

The leader robot The follower robot

II. ON THE COORDINATION OF MULTIPLE ROBOTS

In this section, we discuss the structure of an LAN based multirobot system. First, the computation for coordinating multiple robots is discussed. Then the standard of the MAP is briefly introduced. Finally, the performance evaluation problem of the LAN based system is raised.

A. Coordination of Multiple Robots

Consider an object that is handled by two robots as shown in Fig. 1. Assume that the desired trajectory of the object is represented by To(t) that is a 4 by 4 matrix. The relations between the coordinate frame of the object and the coordinate frames of the two end-effectors can be represented by two constrained matrices, 'To(t), i = 1, 2, which are determined by the dimension and shape of the object. The positions and orientations of the two end-effectors can then be expressed as the following transformation matrices:

Tz(t) = To(t)['To(t)l-l

where i = 1, 2. From (l), one can further obtain the joint trajectories y l ( t ) and y 2 ( t ) of the robots by inverse kinematics, i.e.,

Fig. 1. ' h o robots handling a common object.

to improve the performance of the LAN, one may either improve the speed of the LAN (no protocol modification is allowed since the MAP is selected) or the decomposition-and-allocation scheme. To improve the latter, a performance index other than those which only evaluate transmission delays must be provided.

We propose to use a parameter called effective token passing ratio (ETPR) to evaluate both the LAN and the decomposition-and- allocation scheme. This ratio is the proportion between the number of the message transmissions and the number of the tokens transmissions occurred in the time period that computes the coordination algorithm. The number of the token transmissions includes both the message tokens and the empty tokens. The message token is a token which is followed by a message transmission after the token is transmitted, and an empty token is a token that is not followed by a message transmission. As the ratio increases (decreases), the LAN becomes more efficient (inefficient) since less (more) time is wasted for passing the empty tokens. This ratio is related to both the speed of the LAN and the efficiency of the decomposition-and-allocation scheme. As a result, the parameter of ETPR is a performance index of the entire system not just the LAN.

One important advantage of this proposed method is that it is not necessary to determine communication delays of individual messages. Instead, one can just calculate the token passing ratio, and to compare it with its lower bound which is to be defined based on the time constraint imposed on the coordination task. The difference between the actual ETPR and its lower bound will inform us how much the speed of the LAN or the decomposition-and-allocation scheme needs to be improved, or what is the probability that a given coordination task can be successfully accomplished if the ETPR is higher than the lower bound.

The structure of this paper is as follows. In the next section, we briefly describe the computation involved in the coordination tasks. The MAP based LAN is also introduced in the same section. The major contribution of this paper is presented in Section IV in which the performance evaluation method is discussed in detail. The method will include both the response mode and no response mode of the MAP. Stochastic behaviors of the LAN will also be studied. In Section IV, case studies are presented to illustrate the use of the proposed method. Section V concludes the paper.

where i = 1, 2. The above two equations calculate the joint trajectories of the

two robots given the motion trajectory of the object. To execute the coordinated motion, the trajectories of the two robots are decomposed into a number of set-points. The robots travel through all the set- points to reach its destination. The distance between two neighboring set-points is very small. It takes only a small fraction of one second to complete the motion. This time period is fixed and is called set-point period. To carry out the coordinated motion, a supervisory computer can be used to compute the set-points of each robot. These set-points are then transmitted to the robot controllers to control individual robots. The required computation and communication for each set- point must be accomplished witin the set-point period. This is the only time constraint imposed on the computation and communication, and no other constraint is specified for computation of individual subtasks and communication of individual messages.

Lifting a large object is only a simple coordination task. For more complicated tasks such as assembling two small parts into a single unit, distributing a load among multiple robots and controlling the interaction force between the two robots, etc., [2], the coordination algorithm will become more complicated. The time constraint of the set-point period, however, remains unchanged.

B. The MAP Based LAN A MAP based LAN is equivalent to the IEEE 802.4 token

passing standard [22]. According to the standard, multiple stations are connected to a single bus (Fig. 2), and a token (a software entity) is passed around the stations. When a station receives the token, the station is allowed to send a message. After the station finishes the message transmission, or the token holding time is expired, whichever comes first, the token is passed to the next station. If a station has no message to transmit, it immediately passes the token to the next station. The token is passed from station to station in a pre-defined logical ring as shown in Fig. 2. In addition, the logical connection of the stations can be reconfigured. This means that a station can get in or out of the logical ring at any time. This makes the LAN very flexible in use.

1084 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 24, NO. 7, JULY 1994

Supewisory Computer

/ The Sh&W

/

I f f 4 /

Bus 1 / I \\

Robot Other Automatic Sensors Robot 1 Devices

Fig. 2. A multirobot system connected by a token bus LAN.

C. Critical Problems Using LAN f o r Coordination

Using the LAN for coordination of multirobot is a new topic of research and many problems need to be considered. The most important ones are the selection of the LAN and the design of the task decomposition-and-allocation scheme. Since the protocol of the LAN is already selected which is a standard, one can only improve the speed of the LAN or the decomposition-and-allocation scheme to improve the efficiency of the system. A number of methods for efficient task decomposition-and-allocation have been proposed in our earlier research [23] and will not be further addressed in this paper. The emphasis of this paper i s to propose an effective method to evaluate the LAN based system.

The proposed evaluation method will determine if the combination of the LAN and the decomposition-and-allocation scheme is satisfac- tory for a given coordination task (for convenience, in the remainder of this paper, the term LAN implies the LAN based multirobot system unless otherwise cited). Since the coordination tasks are periodic, the transmission delay of a single message or the computation time of a single subtask is not critical, but the total communication and computation time is. Based on t h i s consideration, we propose to use the ETPR as a performance index. The details will be discussed in the next section.

m. METHOD OF PERFORMANCE EVALUATION In a token passing LAN, the token is passed around the network

while the coordination task is being computed. Some of the tokens are message tokens and the others are empty tokens. Clearly, if most tokens are message tokens, the decomposition-and-allocation scheme is very efficient since not much time is wasted passing empty tokens. On the other hand, if the speed of the LAN is very high, the LAN can afford to pass more empty tokens since a message or token transmission now consumes less time. Based on this consideration, we use the the ratio between the number of the token transmissions, i.e., the ETPR, as the performance index of the LAN. This ratio is affected by both the speed of the LAN and the pattern of the message transmission. The latter is determined by the decomposition- and-allocation scheme.

For a given coordination task, one will have to derive a lower bound of the ETPR. If the ETPR of a coordination task is higher than the lower bound, the task can be completed by the LAN within a pre-defined set-point period; otherwise, an improvement must be made. The deduction of this lower bound will be discussed in this section.

A performance evaluation is not completed unless the stochastic behavior of the system is studied. This stochastic behavior is caused by transmission errors which occur randomly in a LAN. The trans-

mission error may fail the LAN even though the ETPR is higher than the lower bound. As a result, there is a probability or rate of success associated with the LAN. This probability of success issue will also be discussed in this section.

In the token passing standard, there are two modes of transmission: no response and response mode . In the former mode, no acknowl- edgment is sent by the receiving station after it receives a message. In the latter mode, an acknowledgment is sent by the receiving station. Our study on the performance evaluation will be conducted for both of the modes.

A. No Response Mode

In this subsection, we consider the no response mode. The response mode will be discussed in the next subsection.

In a token bus network, a station can do one of the following after it receives the token: a) passes the token to the next station if no message nees to be transmitted, and b) transmits the message and then passes the token. Assume that it takes time TO for an empty token to travel from station to station. TO can be represented as

L TO = + t, + d (3)

where LO is the length of the token frame, s is the speed (data rate) of the network, tp is the token processing time by the station, and d is the propagation delay. In a multirobot system, all the machines are situated in a small area. The propagation delay can be ignored. As a result, (3) can be written as

L To = 2 +t,. (4)

The time required for transmitting a message can be expressed as

where L1 is the length of the address and other control fields in the message frame [22]. Let T, represent the time constraint, i.e., the set-point period of a multirobot system. The following inequality must hold

n

i=l where n is the number of the message transmissions, and N is the number of the token transmissions in the period of computation. Replacing T, in (6) by (3, one obtains

i = l

or

NTo + n' L + - 1 " L,, < T,.

i = l

From (8), one can obtain

/

where R is the ETPR and R1 is the lower bound of the ETPR. The ETPR must satisfy (9) to coordinate the robots successfully. It can be seen that the lower bound is determined 6y both the speed of the LAN and the number of the message and token transmissions.

The above analysis does not consider the stochastic behavior of the system. Assume that the token can be retransmitted once if a token failure occurs (it is not reasonable to allow more token

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 24, NO. 7, JULY 1994 1085

retransmissions since token-recovery is time-consuming). However, no message-recovery is allowed in the no response mode [22]. That is, the system is failed if a message transmission error is detected.

Let Ptf represent the probability that a token transmission is failed, and Pmf represent the probability that a message transmission is failed. The probability for a transmission failure is assunmed to be independent. Thus, the probability that a token pass is successful with a possible retransmission can be represented as

(10)

The probability that a message transmission is successful can be represented as

P,t = (1 - Ptp) + Ptf(1- Ptp) = 1 - P?f.

Suppose that there are n message transmissions and N token transmissions for a given decomposition-and-allocation scheme. The probability that the coordination task can be successfully accom- plished by the LAN can be represented as

P, = (1 - P$)N(l- P,fy. (12)

Equation (12) includes all the possible cases of successful trans- missions. From (10) and ( l l ) , one may further get the probability that there are j token retransmissions given that the LAN is successful. This conditional probability can be expressed as

where 0 5 j 5 N . Because of the transmission errors, one has to reconsider the lower

bound of the ETPR since a token retransmission consumes extra time. To guarantee a high probability of success, the constraint time has to be reduced. Let a new time period denoted as T, and be devoted to recover the transmission errors. As a result, the new time constraint becomes: T, - T,. Thus, (8) should be changed to

L 1 "

i = i NTo + n$ + - 1 L ~ , 5 T, = T,.

From (14), one can obtain

Based on the error-recovery time T,, one may further calculate the probability of success for a given decomposition-and-allocation scheme. Given that (15) is satisfied, the probability of h or less token retransmissions in the period of computation satisfies the following inequality:

pcm 2 = 5 ( N ) 3 j=O j = O

. (1 - Ptf)N(Ptf)i(l - Pmf)n (16)

where

h = [$I. (17)

Note that T, in (17) represents the time delay caused by a token retransmission, and h is the largest integer which is less than or equal to T,/T,.

Since (16) is a conditional probability, the probability of success can be expressed by the following inequality:

P, = Pc,Ps 2 (1 - Ptf)"(l+ Ptf)"

The selection of T, depends on the minimum rate of success that one choses for the LAN. A large T, will increase the rate of success. However, it will make the lower bound of the ETPR higher which needs a higher speed of the LAN or a more efficient decomposition-and-allocation scheme.

If the time period required to complete all the transmissions is less than the set-point period, the difference between the two periods can be used to recover the transmission errors. Let this difference be- expressed as T,. One may have T, = T, - (NTo + n ( L l / s ) + ( l /s)Ey=lLmi) and g = rT,/T,]. In this case, the probability of success can be expressed as

P = (1 - Ptf)"(l + Ptf)N

From the above discussion, we can see that if an ETPR is very close to the lower bound, no extra time can be spent to recover the transmission errors. The rate of success will be limited.

B. Response Mode In the response mode, the receiving station must send an acknowl-

edgement to the sending station after it receives the message. For a transmitting station, the time required to transmit a message Lmt and to obtain the acknowledgment can be represented as

+ 2tp (20) T, = Lmz + L1 + LZ S

where Lz indicates the length of the acknowledgment frame and is a constant. For all the transmissions occurred in the period of computation, the following inequality must hold:

e T z + NTo 5 T,. (21) t = l

Replacing T, in (21) by (20), one obtains

or

Ll+LZ 1 rl NTo + n- + ; C L m . + 2ntp 5 T,. (23)

a=1

From (23), one can obtain

nTo = Ri (24)

T, - - C L m . + n ( ~ 1 + ~ 2 ) - 2nt, ) R = E >

- (*Ii where Rlis the lower bound of the ETPR.

For the same reasons as for the no respond mode, we need to consider the transmission errors. In the response mode, the token is allowed to be retransmitted once and so is a message.

It is clear that the probability that the token is transmitted suc- cessfully can still be expressed as (10). Similarly, the probability that a message is transmitted successfully with a possible retransmission can be represented as

P,, = (1 - P,f) + Pmf(l - P,f) = 1 - PLp. (25)

1086 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 24, NO. 7, JULY 1994

Then the probability of success can be expressed as time difference is totally devoted to the recovery of the transmission

Pa = (1 - Ptyy1- P&)". errors, the probability that the LAN is successful in executing a

(26) coordination task is The conditional probability that there are j token retransmissions

and k message retransmissions given that all the message and token transmissions are successful can be expressed as

where 0 5 j 5 N and 0 5 k 5 n. To cope with the transmission errors, a short time period, T, should

be saved for token and message retransmissions. This is equivalent to a reduction of the constraint time T,. Thus, (23) should be changed to

L1 +Lz 1 " NTo + n- + ; CLm; + 2nt, 5 T, - T,. (28)

t=1

From (28), one can obtain

n nTo R = - - > N -

i=l

= R i . (29)

muation (29) defines the new lower bound of the ETPR. If the ETPR is greater than the lower bound, the probability of success satisfies the following inequality:

h h ( j )

p c m 2 x z p j k s j=O k=O

where

and

T. in (3 1) represents the time delay caused by a token retransmission, and h is the largest integer which is less than or equal to Te/Ts . Td in (32) represents the time delay caused by a message retransmission, and h ( j ) is the largest integer which is less than or equal to (T. - jT,)/Td for the corresponding j.

Thus, the probability that the LAN can accomplish a given coor- dination task satisfies the following inequality:

For the same reason as for the no response mode, one may have a time difference T, which is expressed as T, = T, - (NTo + n((L1 + Lz)/s) + ( l / s ) C L I L m i + 2nt,),g = rTz/Tsl, and g ( j ) = [(T, - jT,)/Tdl where 0 5 j 5 g . Assuming that this

P = (1 - P,f)"(l+ P t f y ( 1 - Pm#"(l+ Pmf)"

(34)

In this section, we have presented a performance evaulation method for the LAN based system to coordinate multiple robots. A key point of this method is to find the number of the token transmissions and the number of the message transmissions. This can be done by running a transmission detecting software running on a computer that is attached to the network since in a bus structure, message or token transmissions are accessible to all the computers. Once these two numbers are obtained, one can use (15) or (29) to see if the time constraint imposed on the network can be satisfied. If the time constraint cannot be satisfied, either the communication speed needs to be improved, or the decomposition-and-allocation scheme needs to be refined. If the time constraint is satisfied, the probability of success can further be determined.

IV. CASE STUDIES

In this section, we will present case studies to illustrate how the proposed method can be used to evaluate a LAN based multirobot system.

We first consider a general case. Assume that there is a multi-robot system similar to the one as shown in Fig. 2. It has two robot con- trollers and one supervisory computer, but all the sensors and other automated devices are disabled. Assume that for a coordination task, a decomposition-and-allocation scheme has been designed which has the following parameters: a) all the messages have an equal length of Lm, = 200 bits, b) the length of the token and the address and control fields LO = L1 = 96 bits, c) the token passing time t , = 4 ps, d) the speed of the LAN s = 1 Mb/s, and e) the set-point period T, is 28 ms.

Based on the above data, we calculate the lower bound of the ETPR for different numbers of the message transmissions. Fig. 3 shows the results for both the response mode and the no response mode. From the figure, we can see that as the number of the message transmissions increases, the lower bound of the ETPR increases as well. This means that fewer empty tokens are allowed in the LAN. More significantly, the table informs how many empty tokens one may still have in order to effectively use the LAN. Based on the above information, one may chose to redesign the decomposition-and-allocation scheme to reduce the number of empty tokens, or to increase the speed of the LAN if the ETPR is lower than the lower bound. Fig. 4 shows the result when the speed of the LAN is increased to 10 Mbls. One can see that the lower bound of the ETPR is significantly reduced for the same number of the message transmissions.

The above information cannot be provided by the evaluation methods which are based on the delay of individual transmissions such as [12] and [14]-[18]. To use these methods, one will have to determine a time constraint on each individual transmission. This will be very difficult if not impossible. Furthermore, these previous methods only indicate how the speed of the protocol of an LAN needs to be improved but provides no information on how efficient the decomposition-and-allocation scheme is. Our method is practically more useful to the designers of a multirobot system than the previous methods.

The second study is on a special case. In this case a multirobot system coordinates two robots which are connected in serial. In a serial connection of two robots, the base of the first robot (the top robot) is held by the end-effector of the second robot (the base robot)

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 24. NO. 7, JULY 1994

0.1

o.w-

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0.07

0.06

0.05

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0.03

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Lower bound of the effective token passing ratio in the case of

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Fig. 4. s = 10 Mb/s.

Lower bound of the effective token passing ratio in the case of

(Fig. 5). The purpose of the serial connection is to increase the Cartesian velocity of the top robot which is responsible for interacting with the environments [24].

To generate large Cartesian velocities, a velocity decomposition algorithm is employed in [24]. %e algorithm decomposes .a desired velocity in the Cartesian space X ( t ) into two velocities X l ( t ) and X 2 ( t ) for the top and base robots, respectively. The two velocities can be expressed as

(35) X i ( t ) = Bi(B1 + &)- ‘ (X( t ) - iBzVd)

and

X z ( t ) = BZ(B1 + l ? ~ ) - ~ ( X ( t ) - iB1Vd) (36)

where

(37)

(38)

1 B~ = - J ~ , W ; ~ J T , c1

c z 1 1 T BZ = - Jze W2- Jze

and

(39)

Fig. 5. Two robots in serial connection.

In (35)-(39), f ,C l and Cz are constants; W1 and WZ are weight matrices; J1, = R(qz)Jl(ql) and Jze = L(ql)JZ(qz) in which R(qz) and L(q l ) are transformation matrices, and Jl(q1) and Jz(q2) are Jacobian matrices of the top and base robots, respectively; q 1 and qz represent the joint positions of the two robots both of which have six .degrees of. freedom.

X1 (t) and X:, (t) can be further transformed into X i (t) and X: (t) which are expressed in the base coordinate systems of the top and base robots, respectively:

and

Using the above two equations and calculating the inverse kine- matics, the joint velocities of the two robots can be updated every set-point period.

To implement the above coordination algorithm, the multirobot system consists of a supervisory computer and two robot controllers [25] which is similar to the one as shown in Fig. 2. Let SU denote the supervisory computer, TR denote the top robot, and BR denote the base robot. A task decomposition-and-allocation scheme for the above coordination task is given as follows:

1) TR sends its current joint positions q1 (t) to SC. The length of this message is L,, = 200 bits.

2) BR sends its current joint positions qZ(t) to SC. The length of this message is L,, = 200 bits.

3) SC computes for X : ( t ) and X ; ( t ) using (35)-(41). Then SC sends X i (t) to TR. The length of this message is L,, = 200 bits.

4) SC sends X l ( t ) to BR. The length of this message is L,, = 200 bits.

5) By using inverse kinematics, TR and BR compute for the joint velocities 41 (t) and Q2(t) , and start to move the robots.

6) Set t = t + T , and go to step 1. For the given configuration-and-scheme, the number of the message transmissions is n = 4 and the length of all the messages are Lmt = 200 bits, i = 1,2,3, and 4. Assume that the network parameters are s = l M b / s , t, = 4 p , LO = L1 = LZ = 96 bits, Ts = Td = 600 ps, Ptf = P,f = loe6, and T, = 28 ms.

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 24, NO. 7, JULY 1994 1088

If the network is implemented in the no response mode, the lower bound of the ETPR can be obtained by using (9) which is R1 = 0.0149. If the network is implemented in the response mode, the lower bound of the ETPR becomes RI = 0.0152 [using (24)].

Now consider the rate of success in the no response mode. Suppose that the total number of the token transmissions is N = 240. We further calculate the difference between the time required to complete all the transmissions and the set-point period, T,, as 2816ps. By using (19), we get the probability that the coordination is successful as P = 0.999992. Since the time is now devoted to the recovery of the transmission errors, this is equivalent that the time constraint is reduced to T, - T,. As a result, the lower bound is increased to RI = 0.0167. In the case of the response mode, T, can be computed as 2400 ps. By using (34), we get the probability that the coordination is successful as P = 0.999996. The corresponding lower bound can be obtained as R1 = 0.0167..

If the set-point period T, of a robot controller is reduced to 21 ms, the lower bound R1 becomes 0.0202 in the no response mode, and 0.0206 in the response mode. These two lower bounds are greater than the previous two, and the task decomposition- and-allocation scheme must be more efficient if the same rates of success need to be maintained. Or the speed of the LAN must be increased.

In this section, we have conducted case studies to illustrate how the proposed method can be applied to a multirobot system. From the study, we can see that our method provides more information on the performance of a LAN based system than the previous evaluation methods. The evaulation indicates whether a selected LAN and a decomposition-and-allocation scheme can successfully coordinate a multirobot system. If an improvement needs to be made, it can be on the speed of the LAN or on the decomposition-and-allocation scheme, but not on the protocol of the bus access mechanism.

V. SUMMARY Recent developments in manufacturing automation have brought

a great demand for integrated systems that can coordinate multiple robots and other CNC machines in real-time. Traditional approaches use dedicated channels for real-time communication in a multirobot system. In this paper, we have proposed the use of a LAN for coordination of multiple robots since the LAN is both economical and flexible. To further reduce the cost of implementation, we propose to use the standard MAP as the access mechanism of the token bus. To evaluate the performance of the LAN based system, we have developed a new method which is based on an index called the ETPR. To guarantee the success of coordination, the lower bound of the ETPR is derived.

This ETPR method does not evaluate the delay of individual transmissions, but the overall efficiency of the LAN based system. This overall efficiency is affected by the performance of the task decomposition-and-allocation scheme and the speed and protocol of the LAN. Consequently, our method provides more information to the designer of the multirobot system, based on which the designer may either improve the speed of the network or the decomposition- and-allocation scheme. Previous methods, on the other hand, only recommend modifications to the speed or the protocol of the net- work.

The stochastic behavior of the LAN is also considered. For a pre- defined task decomposition-and-allocation scheme and the set-point period, the probability of successful coordination is derived. It has been shown that if the ETPR is close to the lower bound, the rate of success will be low since transmission errors may fail the system. Finally, by case sudies, we have shown how the proposed method

can be applied to a multirobot system, and how advantageous the proposed method is in comparison with the previous methods.

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