International Journal of Digital Information and Wireless Communications (IJDIWC) 4(4): 493-498 The Society of Digital Information and Wireless Communications, 2014 (ISSN: 2225-658X)
Navigation Error Reduction in Swarm of UAVs Shibarchi Majumder1, Rahul Shankar2, Mani Shankar Prasad3 1,3 Amity Institute of Space Science & Technology, INDIA 2 Amity Institute of Nuclear Science & Technology, INDIA 1
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1. ABSTRACT For large area surveillance, a number of UAVs are preferred. In such situation, navigation becomes a prime factor, which reduces the chance of collision. UAVs primarily depend on GPS, which has an order of inaccuracy. Conventional correction methods like differential GPS needs extra hardware and cannot improve stochastic errors like multipath error and receiver error. In this paper we demonstrate a novel technique to reduce the GPS error, in a swarm of UAVs.
2. KEYWORDS GPS Error, Navigation Error, GPS Error Reduction, UAV Swarm, Position Correction, Error offset, Intra Swarm Communication;
3. INTRODUCTION A group of cooperative UAVs, also called a swarm of UAVs sometimes very effective for certain tasks like mapping, target searching etc. Portability, lesser take-off and landing requirements, easy handling along with low manufacturing and maintenance costs make it more desirable than expensive and large UAVs. However, the low cost and small size principle puts a limit to the onboard sensors, resulting in a significant inaccuracy in navigation with others. IMU (Inertial Measurement Unit) is a popular solution for such problem, but precise IMU’s are expensive and heavy. On the other hand, GPS, which is a less expensive solution, comes with a lower accuracy. Commercial GPS provides an accuracy of the order 15 to 100 meters. #
Correspondence:
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A widely used method to reduce GPS error is to put reference stations to correct the GPS data (DGPS), which limits the operational area as well as increases the cost. Unlike inertial navigation system, error distribution in GPS remains almost linear over time. However, it could be difficult to estimate some error parameters accurately due to its modeling constraints. Errors like SA (Selective Availability), Ionosphere delay, propagation delay remains same over a large area, while receiver error and multipath error varies with different hardware and location respectively. Receiver error could be caused by receiver thermal noise or receiver clock error and it’s hard to model receiver error due to its stochastic nature. In most cases, the clock is a quartz-crystal oscillator whose accuracy is of the order of 1-10 ppm. Multipath Propagation Error (MPE) is caused when objects in the vicinity of the receiver antenna reflect GPS signals, resulting in one or more secondary propagation paths. These secondary-path signals superimpose on the desired direct-path signal and distort its amplitude and phase. Errors in multipath cannot be reduced by DGPS, since they depend on local reflection geometry near the receiver antenna. In a receiver without multipath protection, C/A code ranging errors of 10m or more can be experienced. Multipath errors not only cause large code ranging errors but also severely degrade the ambiguity resolution process required for carrier phase ranging such as that used in precision surveying applications. This paper demonstrates a new technique to reduce error in GPS for networked swarm of UAVs, where the accuracy in position 493
International Journal of Digital Information and Wireless Communications (IJDIWC) 4(4): 493-498 The Society of Digital Information and Wireless Communications, 2014 (ISSN: 2225-658X) measurement is improved using the error in relative distance between every member UAV in the swarm.
sender UAV calculates the distance of other UAVs.
4. INTRA-SWARM COMMUNICATION To reduce the error in position measurement by correlating the GPS data with other members in the swarm, it is necessary to determine the relative distance between the members as well as to get the GPS provided data for every UAV. Figure 2: A timeline showing the various delays in packet transmission
4.1. Total Propagation Time ( )
Figure 1: An example of inter UAV communication. In a system of three UAV's, each UAV has two different communication links with other UAVs.
Where, is response pulse receiving time is request pulse transmitting time and is predefined delay time. Therefore, distance between two UAVs is, ( ) ( ) …where 4.2. Navigation Error Deduction
Each UAV of the swarm is equipped with a transponder which is capable of transmitting/receiving the GPS data with other UAVs. For easy identification, a simple data format of unique ID and time delay ( ) of transmission is used. To calculate the distance between each member in the swarm, each UAV is featured with a DME type transponder with a unique bandwidth for identification. As the operational area is too small for the propagation time of the EM pulse across the distances to be significant, a predefined delay time is assigned to every transponder. In a sequential manner, each UAV sends a request pulse and after receiving the pulse other UAVs wait for the pre-determined delay assigned to them and send back a response pulse. From the known bandwidth and delay assigned to each UAV, the
The DME measurement gives the real time relative distance measurement of the UAV’s in the vicinity. Although GPS data received through trans-receiver can also give a calculated relative distance, there may be in errors or inaccuracies in them (inherent due to C/A code errors).
Figure 3: Diagram showing measurement of error between measured and calculated distances
4.3. Calculated Distance Equation ( )
494
International Journal of Digital Information and Wireless Communications (IJDIWC) 4(4): 493-498 The Society of Digital Information and Wireless Communications, 2014 (ISSN: 2225-658X) …where and are distance measured and distance calculated respectively. Errors between all UAVs are calculated by the above equation. For a network of ‘n’ number of UAVs, number of error equations can be formed and error for each UAV can be calculated. 5. CORRECTION
and receiver clock error, the error in distances is Gaussian in nature. Given these parameters, a Gaussian model probability density function is associated with each UAV location data whose mean value is located at the GPS indicated center. Thus the probability of finding the UAV at a point whose distance from the indicated center is would be given by… (
)
The correction process is divided into three stages, namely the primary, secondary and the tertiary correction stages. However, not all UAVs in the swarm will follow through with all three stages since this would be inefficient in terms of processing load as well as accuracy of results as erratic values may creep into the system due to large number of corrections made.
( ) √ The value of is set for each UAV such that the chances of occurrence of the true location of the UAV outside the circle of radius tends to zero i.e.
5.1. Correction Criteria
5.3. Correction Stage I
The question of which swarm member should follow which correction stages is decided using the sum-distance rule. Corrections based on relative distances are most effective for the UAV which is surrounded on all directions by other UAVs and has the minimum summed up distance from others.
Stage I correction involves estimating the most probable location of the UAV within the margin of GPS error by isolating the region within which the UAV must be in order to agree with the distance measured by every other member. The location of each UAV can be corrected using the measured distance between and ( ) . For UAV another UAV , the
∑
( ) th
…where is the sum of the distances of the j UAV from all the UAVs and is its distance th from the i UAV. From the above calculation the middle UAV (whose value of is minimum) is determined. Once this is done, the criterion is applied as follows: ( ) ( ) 5.2. Preliminary Calculation Before the correction process, certain preliminary calculations must be done in order to facilitate them. As we are considering only multipath error
( ⃗)
∫
( )
( )
distribution function of another UAV ( ) is reconstructed at a point represented by the position vector ⃗⃗⃗⃗⃗⃗ from the GPS center of given by ⃗⃗⃗⃗⃗⃗ ( ) ̂ (7) Where ̂ is the unit vector in the direction of and ( ) is the error from the calculated distance equation between and . Since ( ) is always less than GPS error radius ( ) for any pair of members and , there will exist a region of overlap of the two probability distribution functions, which collectively give rise to a local sum distribution function as shown in Figure 4.
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International Journal of Digital Information and Wireless Communications (IJDIWC) 4(4): 493-498 The Society of Digital Information and Wireless Communications, 2014 (ISSN: 2225-658X) 5.4. Correction Stage II Stage II correction is performed on the middle UAV ( ) after all the members have corrected centers ( ) after execution of stage I. New error values are calculated between the calculated distances between members’ corrected centers( ), and their measured distances ( )… ( ) ( ) ( ) ( ) Let there be a point represented by the position vector with respect to ( ) … ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ( ) ̂ ( ) Figure 4: The figure shows the reconstruction of the probability distribution function of the location of near and the region of overlap
Similarly, the probability distribution functions of the locations of all UAVs in the swarm are reconstructed around the GPS center of and are summed up. Since we know that the true location of must lie within the region of overlap, this sum function is normalized within this region. Let be the set of all points within the overlap region. A new probability density function is assigned to the location. This is given by…
( ⃗)
∑ { ( ∫⃗ (∑
⃗
( ⃗) ) ( ⃗ ))
}
⃗
( )
Where ⃗⃗ is the position vector of a point from the center of and (⃗⃗) is the probability density function at that point. The corrected GPS centre ( ) of the UAV is given by the centre of gravity of the distribution… ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ( )
∫⃗ (∑ ∫⃗ (∑
( ( ⃗)
⃗ ))
( ⃗ ))
( )
Therefore, the corrected middle member’s location will now be given by: ∑ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ ( ) After the stage II correction, the error in location has been reduced to its minimum as reducible by our algorithm. Hence, this location is considered to be the true location. 5.5. Correction Stage III Now that
has a correct location pinpointed… ̂ ⃗⃗⃗⃗⃗⃗⃗⃗ ( ) ( )
Where… ̂
(̂
̂
)
( ) √ Where is the direction angle of as provided by the antenna, and ̂ and ̂ are the unit vectors along the positive and axes. If ⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the position vector of the location of as calculated from the current centre, then the position vector of its accurate location will be given by: ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ (⃗⃗⃗⃗⃗) ( ) Therefore, the GPS center of each non-middle member of the swarm is shifted by (⃗⃗⃗⃗⃗) to complete the error reduction process for the whole swarm. 496
International Journal of Digital Information and Wireless Communications (IJDIWC) 4(4): 493-498 The Society of Digital Information and Wireless Communications, 2014 (ISSN: 2225-658X) 6. TEST RESULTS In order to test the algorithm’s accuracy, five GPS receivers were placed at different locations (whose location coordinates are known) across the Amity University campus and the distances between these locations were calculated. The data was provided to the algorithm for error reduction. The results obtained are as shown in figures 5, 6 and 7.
Figure 7: The figure shows the results of GPS error reduction. Here, receiver 4 (refer figure 5) was taken as the middle member and the black triangles represent the corrected locations provided by the algorithm.
Figure 5: The figure shows the various locations across the Amity University campus where GPS receivers were placed. The coordinates shown were of locations manually pinpointed on google maps.
The results shown are of a single run of the algorithm. The error at each location is similar since stage three corrections is simply a reverse engineering of the location of the members of the swarm based on measured distance and directional data. The algorithm was run 20 more times and each time, absolute errors were calculated. The results are graphically represented in figure 8.
Figure 8: Absolute errors obtained for corrected positions of middle members (represented by grey squares) and average absolute errors in indicated positions (represented by black rhombi) Figure 6: In this figure, the red rhombi represent the GPS indicated positions of the receivers placed at locations represented by the yellow stars. In other words, the yellow stars are the true positions and the red rhombi are the indicated positions.
497
International Journal of Digital Information and Wireless Communications (IJDIWC) 4(4): 493-498 The Society of Digital Information and Wireless Communications, 2014 (ISSN: 2225-658X) Location
Indicated position
True position
Corrected position
1
77
77
77
2
77
77
77
3
77
4
77
5
77
7 7
77
7
Abs. error
7
3.
77
77
77
7 7
77
77
7
Table 1: The table shows the true coordinates (obtained from google maps), indicated coordinates (from the receiver), and the corrected coordinates of the positions of the receivers in serial order, along with the absolute error on the corrected position.
4.
5.
7. CONCLUSION
6.
In this paper a way to reduce the GPS position error has been explained. The above mentioned way is not only able to reduce the receiver error and multipath error but also able to operate in nonGPS-connectivity. During the scenario when the swarm must explore a region where there is no clear sky visibility, it was seen in our trials that if even a single UAV in the swarm had a low intrinsic GPS error, the accuracy of the estimated locations of all other members of the swarm are greatly heightened, thus proving very useful for isolated explorations. In the near future we shall attempt to perform corrections on GPS velocity for a networked swarm of UAVs.
7. 8.
9.
10.
11.
12.
8. ACKNOWLEDGEMENT This research was done during post graduation phase, as a part of UAV Systems in Institute of Space Science and Technology, Amity University Uttar-Pradesh, INDIA.
13.
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