5. Simulation results

Due to limitation of space, two simulation results are presented for consensus with collision avoidance on the unit sphere, more simulation results are in [18, 20]. The first experiment is to test formation acquisition with avoidance on the sphere. The second experiment is to test arbitrary reconfigurations on the sphere with collision avoidance. Three different communication topologies used are shown in Figure 4. In Figure 4, Topology 1 (left) is a fully connected communication graph with no leader, Topology 2 (center) is a cyclic communication graph with one leader, node 1, and Topology 3 (right) is a cyclic communication graph with no leader.

Optimization software Sedumi [22] and Yalmip [23] running inside Matlab R2009a, were used for solving all the problems. The simulations were done on an Intel R Core(TM)2 Duo P8600 @ 2.40GHz with 2 GB RAM, running Windows 7.

#### 5.1. Formation acquisition on the unit sphere with avoidance

In this experiment, ten vehicles converge to a formation on the sphere, which is realized by maintaining a relative spacing with each other while also avoiding a static obstacle, with <sup>α</sup> <sup>¼</sup> 30o. Angle <sup>β</sup>ij <sup>¼</sup> 20o is set to maintain the relative spacing between the vehicles ∀ i, j ¼ 1⋯10, i 6¼ j. The initial positions are:

> When relative spacing are specified between the vehicles, the motion obtained from this Laplacian is like the result obtained in [11]. However, when there is no relative spacing specified, the circular motion converges to a point. Using a circulant matrix such as that of Topology 3, one can vary the radius of the circular formation achieved <sup>r</sup> <sup>¼</sup> cos <sup>θ</sup>ij ; by setting θij equal for all i, j and varying its size with time. If the magnitude of angle θ is reduced, the radius of the circular formation structure obtained also reduces, and vice versa. Figure 7 (left) shows the result for setting <sup>θ</sup>ij <sup>¼</sup> 30o<sup>∀</sup> i, j for four vehicles. The center and right figures show the results for ten vehicles as <sup>θ</sup>ij moves gradually from 20<sup>o</sup> toward 0o. When <sup>θ</sup>ij <sup>¼</sup> <sup>0</sup><sup>∀</sup> i, j, the

> Figure 7. Four-vehicle formation acquisition using topology 3 with <sup>β</sup>ij <sup>¼</sup> 30o (left), and ten-vehicle formation acquisition,

Multi-Path Planning on a Sphere with LMI-Based Collision Avoidance

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39

Figure 6. Ten-vehicle formation acquisition using topology 1 (left), and using topology 3 (right).

5.2. Collision free reconfiguration on the unit sphere with avoidance of no-fly zones

This is a more traditional reconfiguration problem which we try to solve by using the consensus based protocols presented in this chapter. Three flying vehicles (e.g. UAVs), are required to fly from their initial positions to given final positions. There are cross-paths (inter-vehicle

vehicles rendezvous to a point.

using topology 3, with <sup>β</sup>ij <sup>¼</sup> 20o (center) and <sup>β</sup>ij <sup>¼</sup> <sup>0</sup><sup>o</sup> (right).


The result for Topology 1 is shown in Figure 6 (left), while the right figure shows the result obtained using Topology 3 – a cyclic graph which produces a circulant Laplacian L, whose dynamics leads to swirling motion. The proof is in [18].

Figure 6. Ten-vehicle formation acquisition using topology 1 (left), and using topology 3 (right).

The corresponding collective dynamics of x<sup>i</sup>

38 Advanced Path Planning for Mobile Entities

2.40GHz with 2 GB RAM, running Windows 7.

∀ i, j ¼ 1⋯10, i 6¼ j. The initial positions are:

dynamics leads to swirling motion. The proof is in [18].

5.1. Formation acquisition on the unit sphere with avoidance

5. Simulation results

x\_ i ðÞ¼� t ð Þt is

ð Þt

This configuration was applied in the reconfiguration experiment in Section 5.2. Practical application of this strategy to the problem of separation in air traffic control is presented in [18, 20].

Due to limitation of space, two simulation results are presented for consensus with collision avoidance on the unit sphere, more simulation results are in [18, 20]. The first experiment is to test formation acquisition with avoidance on the sphere. The second experiment is to test arbitrary reconfigurations on the sphere with collision avoidance. Three different communication topologies used are shown in Figure 4. In Figure 4, Topology 1 (left) is a fully connected communication graph with no leader, Topology 2 (center) is a cyclic communication graph with one leader, node 1, and Topology 3 (right) is a cyclic communication graph with no leader.

Optimization software Sedumi [22] and Yalmip [23] running inside Matlab R2009a, were used for solving all the problems. The simulations were done on an Intel R Core(TM)2 Duo P8600 @

In this experiment, ten vehicles converge to a formation on the sphere, which is realized by maintaining a relative spacing with each other while also avoiding a static obstacle, with <sup>α</sup> <sup>¼</sup> 30o. Angle <sup>β</sup>ij <sup>¼</sup> 20o is set to maintain the relative spacing between the vehicles

> <sup>x</sup><sup>1</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:3417 0:5555 0:<sup>7581</sup> <sup>T</sup> <sup>x</sup><sup>2</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:4960 0:1270 0:<sup>8589</sup> <sup>T</sup> <sup>x</sup><sup>3</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:3045 0:9497 0:<sup>0730</sup> <sup>T</sup> <sup>x</sup><sup>4</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:5735 0:7952 0:<sup>1967</sup> <sup>T</sup> <sup>x</sup><sup>5</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:8005 0:3867 0:<sup>4580</sup> <sup>T</sup> <sup>x</sup><sup>6</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:3727 0:7372 0:<sup>5637</sup> <sup>T</sup> <sup>x</sup><sup>7</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:0355 0:5117 0:<sup>8585</sup> <sup>T</sup> <sup>x</sup><sup>8</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:6553 0:7428 0:<sup>1371</sup> <sup>T</sup> <sup>x</sup><sup>9</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:9188 0:2446 0:<sup>3094</sup> <sup>T</sup> <sup>x</sup><sup>10</sup>ðÞ ¼ <sup>0</sup> ½ � <sup>0</sup>:0261 0:8773 0:<sup>4792</sup> <sup>T</sup>

The result for Topology 1 is shown in Figure 6 (left), while the right figure shows the result obtained using Topology 3 – a cyclic graph which produces a circulant Laplacian L, whose

Lt ð Þt ⊗ I<sup>3</sup> � �x<sup>i</sup>

ð Þt : (32)

ð Þt 0 0 Λ<sup>i</sup>

" #

Λi

Figure 7. Four-vehicle formation acquisition using topology 3 with <sup>β</sup>ij <sup>¼</sup> 30o (left), and ten-vehicle formation acquisition, using topology 3, with <sup>β</sup>ij <sup>¼</sup> 20o (center) and <sup>β</sup>ij <sup>¼</sup> <sup>0</sup><sup>o</sup> (right).

When relative spacing are specified between the vehicles, the motion obtained from this Laplacian is like the result obtained in [11]. However, when there is no relative spacing specified, the circular motion converges to a point. Using a circulant matrix such as that of Topology 3, one can vary the radius of the circular formation achieved <sup>r</sup> <sup>¼</sup> cos <sup>θ</sup>ij ; by setting θij equal for all i, j and varying its size with time. If the magnitude of angle θ is reduced, the radius of the circular formation structure obtained also reduces, and vice versa. Figure 7 (left) shows the result for setting <sup>θ</sup>ij <sup>¼</sup> 30o<sup>∀</sup> i, j for four vehicles. The center and right figures show the results for ten vehicles as <sup>θ</sup>ij moves gradually from 20<sup>o</sup> toward 0o. When <sup>θ</sup>ij <sup>¼</sup> <sup>0</sup><sup>∀</sup> i, j, the vehicles rendezvous to a point.

#### 5.2. Collision free reconfiguration on the unit sphere with avoidance of no-fly zones

This is a more traditional reconfiguration problem which we try to solve by using the consensus based protocols presented in this chapter. Three flying vehicles (e.g. UAVs), are required to fly from their initial positions to given final positions. There are cross-paths (inter-vehicle

x1

x2

x3

x4

x5

tion. The result is shown in Figure 8.

application at hand.

Author details

Innocent Okoloko

References

Address all correspondence to: okoloko@ieee.org

DOI: 10.1109/JPROC.2006.876930

DOI: 10.1109/JPROC.2006.887295

DOI: 10.2514/2.4721

obs <sup>¼</sup> ½ � <sup>0</sup>:<sup>5237</sup> � <sup>0</sup>:7208 0:<sup>454</sup> <sup>T</sup>

obs <sup>¼</sup> ½ � <sup>0</sup> � <sup>0</sup>:9877 0:<sup>1564</sup> <sup>T</sup>

obs <sup>¼</sup> ½ � <sup>0</sup>:<sup>5878</sup> � <sup>0</sup>:809 0 <sup>T</sup>

obs <sup>¼</sup> ½ � <sup>0</sup> � <sup>0</sup>:9511 0:<sup>309</sup> <sup>T</sup>

obs <sup>¼</sup> ½ � <sup>0</sup>:<sup>2939</sup> � <sup>0</sup>:<sup>9045</sup> � <sup>0</sup>:<sup>309</sup> <sup>T</sup>

Multi-Path Planning on a Sphere with LMI-Based Collision Avoidance

http://dx.doi.org/10.5772/intechopen.71216

41

The radii of the no-fly zones are equal to <sup>r</sup>, therefore <sup>β</sup>ij <sup>¼</sup> <sup>α</sup>ij <sup>¼</sup> 10o<sup>∀</sup> i, j ið Þ 6¼ <sup>j</sup> for this simula-

On a final note, we have attempted to solve the problem of consensus in a spherical coordinate system by solving in on the unit sphere. The same unit sphere was used in [11]. This is convenient because the results are easier to visualize and compute on the unit sphere. The results presented here can be applied directly to real-life planetary navigation problems such as horizontal separation of aircraft [18, 20], simply by transforming actual position vectors into unit vectors in the unit sphere, solving to obtain the solution trajectories, and transforming the solutions back to actual desired trajectories in the real-world coordinates. The unit of measurement for implementation will therefore depend on the

Department of Electrical Engineering, Universidad de Ingenieria y Tecnologia, Lima, Peru

[1] Beard RW, McLain TW, Nelson DB, Kingston D. Decentralized cooperative aerial surveillance using fixed-wing miniature UAVs. Proceedings of the IEEE. 2006;94(7):1306-1324.

[2] Leonard NE, Paley DA, Lekien F, Sepulchre R, Fratantoni DM, Davis RE. Collective motion, sensor networks and ocean sampling. Proceedings of the IEEE. 2007;95(1):48-74.

[3] Mesbahi M, Hadaegh H. Formation flying control of multiple spacecraft via graphs, matrix inequalities, and switching. Journal of Guidance, Control, and Dynamics. 2001;24(2):369-377.

Figure 8. Three-vehicle reconfiguration with collision avoidance and avoidance of no-fly zones.

collision constraints) in addition to no-fly zones (static obstacle constraints), between the initial and final positions. The initial positions are:

$$\begin{aligned} \mathbf{x}\_0^1(0) &= \begin{bmatrix} 0.8659 & 0 & -0.4999 \end{bmatrix}^T\\ \mathbf{x}\_0^2(0) &= \begin{bmatrix} 0.4165 & -0.5721 & 0.7071 \end{bmatrix}^T\\ \mathbf{x}\_0^3(0) &= \begin{bmatrix} -0.5878 & -0.809 & 0 \end{bmatrix}^T \end{aligned}$$

The desired final positions are:

$$\begin{array}{rcl} \mathbf{x}\_f^1(0) &=& [-0.4330 & -0.7499 & 0.4999]^T \\ \mathbf{x}\_f^2(0) &=& [-0.2939 & -0.9045 & -0.309]^T \\ \mathbf{x}\_f^3(0) &=& [0.9393 & -0.3052 & 0.1564]^T \end{array}$$

For inter-vehicle collision avoidance, they are required to maintain a minimum safety distance of<sup>r</sup> <sup>¼</sup> cos 10o units. Five no-fly zones are imposed on the vehicles at the following positions:

$$\begin{array}{rcl} \mathbf{x}\_{\text{obs}}^{1} &=& \begin{bmatrix} 0.5237 & -0.7208 & 0.454 \end{bmatrix}^{T} \\ \mathbf{x}\_{\text{obs}}^{2} &=& \begin{bmatrix} 0.2939 & -0.9045 & -0.309 \end{bmatrix}^{T} \\ \mathbf{x}\_{\text{obs}}^{3} &=& \begin{bmatrix} 0 & -0.9877 & 0.1564 \end{bmatrix}^{T} \\ \mathbf{x}\_{\text{obs}}^{4} &=& \begin{bmatrix} 0.5878 & -0.809 & 0 \end{bmatrix}^{T} \\ \mathbf{x}\_{\text{obs}}^{5} &=& \begin{bmatrix} 0 & -0.9511 & 0.309 \end{bmatrix}^{T} \end{array}$$

The radii of the no-fly zones are equal to <sup>r</sup>, therefore <sup>β</sup>ij <sup>¼</sup> <sup>α</sup>ij <sup>¼</sup> 10o<sup>∀</sup> i, j ið Þ 6¼ <sup>j</sup> for this simulation. The result is shown in Figure 8.

On a final note, we have attempted to solve the problem of consensus in a spherical coordinate system by solving in on the unit sphere. The same unit sphere was used in [11]. This is convenient because the results are easier to visualize and compute on the unit sphere. The results presented here can be applied directly to real-life planetary navigation problems such as horizontal separation of aircraft [18, 20], simply by transforming actual position vectors into unit vectors in the unit sphere, solving to obtain the solution trajectories, and transforming the solutions back to actual desired trajectories in the real-world coordinates. The unit of measurement for implementation will therefore depend on the application at hand.
