5. Simulation results

Due to limitation of space, we present three results for attitude multi-path planning in different coordinate frames. More results can be found in [7, 8, 16].

#### 5.1. Dynamic avoidance in different coordinate frames without consensus

In this experiment, SC<sup>1</sup> and SC<sup>2</sup> are attempting a reconfiguration to Earth (either changing orientation to Earth or pointing an instrument to Earth). The initial quaternions of SC<sup>1</sup> and SC<sup>2</sup> are q<sup>1</sup> <sup>0</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup> <sup>0</sup> <sup>¼</sup> ½ � <sup>0001</sup> <sup>T</sup>. The desired final quaternions are

$$\begin{aligned} q\_f^1 &= \begin{bmatrix} 0.2269 & 0.0421 & 0.9567 & 0.1776 \end{bmatrix}^T\\ q\_f^2 &= \begin{bmatrix} 0 & 0 & 0.9903 & 0.1387 \end{bmatrix}^T \end{aligned} \tag{34}$$

Three thrusters of SC<sup>1</sup> in F<sup>B</sup> SC<sup>1</sup> are Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints http://dx.doi.org/10.5772/intechopen.71580 57

$$\begin{aligned} \boldsymbol{w}\_{\text{obs}\_{1},1}^{\boldsymbol{B}} &= \begin{bmatrix} -0.2132 - 0.0181 & 0.9768 \end{bmatrix}^{\boldsymbol{T}} \\ \boldsymbol{w}\_{\text{obs}\_{1},2}^{\boldsymbol{B}} &= \begin{bmatrix} 0.314 & 0.283 - 0.906 \end{bmatrix}^{\boldsymbol{T}} \\ \boldsymbol{w}\_{\text{obs}\_{1},3}^{\boldsymbol{B}} &= \begin{bmatrix} -0.112 - 0.133 - 0.985 \end{bmatrix}^{\boldsymbol{T}} \end{aligned} \tag{35}$$

A single thruster of SC<sup>2</sup> in F<sup>B</sup> SC<sup>2</sup> is at

where q<sup>T</sup>

qi (t) T qi

qi

<sup>1</sup> ð Þ<sup>t</sup> qT

(t) = 1 or q(t)

following constraints:

qi <sup>k</sup>þ<sup>1</sup> <sup>¼</sup> qi

quaternion dynamics Eq. (12).

5. Simulation results

are q<sup>1</sup>

<sup>0</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup>

Three thrusters of SC<sup>1</sup> in F<sup>B</sup>

<sup>2</sup> ð Þ<sup>t</sup> <sup>⋯</sup>qT

56 Advanced Path Planning for Mobile Entities

Eq. (30) is the discrete time version of qi

4.4. Integration for consensus based Q-CAC

<sup>k</sup> � <sup>Δ</sup>tP<sup>i</sup>

qi kT q<sup>i</sup>

μ q<sup>i</sup>

angular velocity ω<sup>i</sup> to rotate the SCi optimally to q<sup>i</sup>

coordinate frames. More results can be found in [7, 8, 16].

qi

2 4

Once the next safe quaternion trajectory q<sup>i</sup>

ð Þ<sup>t</sup> qi

<sup>k</sup>þ<sup>1</sup> � qi k

j ð Þt � ��<sup>1</sup>

5.1. Dynamic avoidance in different coordinate frames without consensus

<sup>0</sup> <sup>¼</sup> ½ � <sup>0001</sup> <sup>T</sup>. The desired final quaternions are

q1

q2

SC<sup>1</sup> are

ð Þt T

ð Þ<sup>t</sup> <sup>μ</sup>I<sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>~</sup> <sup>i</sup>

T

protocol Eq. (29), norm constraints must be enforced as follows:

<sup>q</sup>(t) = <sup>n</sup> for SC, iff <sup>k</sup>q<sup>i</sup>

qi kT q<sup>i</sup>

<sup>y</sup> ð Þt are the quaternions of the y other neighboring SC, which SCi can

ðÞ¼ <sup>t</sup> 0 or <sup>q</sup>ð Þ<sup>t</sup> <sup>T</sup>

� � <sup>¼</sup> <sup>0</sup> (30)

<sup>k</sup>, quaternion consensus dynamics constraint (31)

5 ≥ 0 exclusion constraints (33)

safe has been determined, the control torque <sup>τ</sup><sup>i</sup> and

safe can be determined by using the normal

� � <sup>¼</sup> <sup>0</sup>, norm constraint (32)

<sup>f</sup> <sup>¼</sup> ½ � <sup>000</sup>:9903 0:<sup>1387</sup> <sup>T</sup> (34)

q\_ðÞ¼ t 0. This guarantees that

communicate with at time t. Moreover, since we are going to apply consensus quaternion

<sup>k</sup>þ<sup>1</sup> � qi k

Using semidefinite programming, the solutions presented previously be cast as an optimization problem, augmented with a set of LMI constraints, and solved for optimal consensus quaternion trajectories. We consider the algorithm in discrete time. Given the initial attitude

(0) of SCi, (i = 1, ⋯, n), find a sequence of consensus quaternion trajectories that satisfies the

3

Due to limitation of space, we present three results for attitude multi-path planning in different

In this experiment, SC<sup>1</sup> and SC<sup>2</sup> are attempting a reconfiguration to Earth (either changing orientation to Earth or pointing an instrument to Earth). The initial quaternions of SC<sup>1</sup> and SC<sup>2</sup>

<sup>f</sup> <sup>¼</sup> ½ � <sup>0</sup>:2269 0:0421 0:9567 0:<sup>1776</sup> <sup>T</sup>

ð Þt T q\_i

(0)k = 1 ∀ i.

$$\boldsymbol{v}\_{obs\_2}^{B} = \begin{bmatrix} 0.02981 & 0.0819 & 0.9962 \end{bmatrix}^{T} \tag{36}$$

We want vI obs<sup>2</sup> to avoid vI obs1:<sup>1</sup> by 50� , and avoid v<sup>I</sup> obs1:<sup>2</sup> and vI obs1:<sup>3</sup> by 30� while both are maneuvering to their desired final attitudes. Figure 3(a) shows the avoidance between thrusters of SC<sup>1</sup> and SC<sup>2</sup> during reorientation to Earth: SC<sup>2</sup> cannot reconfigure to the desired q<sup>2</sup> <sup>f</sup> due to the avoidance constraints. Note that vI obs2:<sup>1</sup>, <sup>v</sup><sup>I</sup> obs2:<sup>2</sup>, vI obs2:<sup>3</sup> are the points of intersections of vI obs1:<sup>1</sup>, vI obs1:<sup>2</sup>, vI obs1:<sup>3</sup> with SC2. Figure 3(b) satisfaction of avoidance constraints: the sudden jumps to and from �1 indicate times when any of vI obs1:<sup>1</sup>, vI obs1:<sup>2</sup>, <sup>v</sup><sup>I</sup> obs1:<sup>3</sup> lost intersection with the sphere of SC<sup>2</sup> and therefore was replaced with �v<sup>I</sup> obs1:i , i ¼ 1, ⋯, 3.

This experiment demonstrates that when both constraints are in conflict the avoidance constraint is superior to the desired final quaternion constraint. As seen from (a), SC<sup>2</sup> cannot reconfigure exactly to the desired q<sup>2</sup> <sup>f</sup> due to the satisfaction of the avoidance constraints. To resolve this, it is necessary to change either the position of SC<sup>2</sup> or SC1.

#### 5.2. Consensus-based dynamic avoidance in different coordinate frames

In this experiment, SC1, SC2, and SC<sup>3</sup> will maneuver to a consensus attitude. The initial positions are

$$\begin{aligned} \mathcal{F}\_{\text{SC}\_1}^{l} &= \begin{bmatrix} -\mathbf{2} & \mathbf{0} & \mathbf{2} \end{bmatrix}^T\\ \mathcal{F}\_{\text{SC}\_2}^{l} &= \begin{bmatrix} \mathbf{0}.5 & \mathbf{0} & \mathbf{2} \end{bmatrix}^T\\ \mathcal{F}\_{\text{SC}\_3}^{l} &= \begin{bmatrix} \mathbf{3} & \mathbf{0} & \mathbf{2} \end{bmatrix}^T \end{aligned} \tag{37}$$

Figure 3. Reconfiguration of two spacecraft with avoidance in different coordinate frames: (a) the trajectories, (b) the avoidance graph.

A set of initial quaternions were randomly generated, with the following data:

$$\begin{aligned} q\_0^I &= \begin{bmatrix} -0.5101 & 0.6112 \ -0.3187 & -0.5145 \end{bmatrix}^T \\ q\_0^2 &= \begin{bmatrix} -0.9369 & 0.2704 \ -0.1836 & -0.124 \end{bmatrix}^T \\ q\_0^3 &= \begin{bmatrix} 0.1448 - 0.1151 & 0.1203 & 0.9753 \end{bmatrix}^T \end{aligned} \tag{38}$$

q1

q2

q3

The differences of these quaternions are 30�

6. Conclusion

algorithms using rotorcraft.

Address all correspondence to: okoloko@ieee.org

Author details

Innocent Okoloko

References

<sup>f</sup> ¼ �½ � <sup>0</sup>:6926 0:<sup>6468</sup> � <sup>0</sup>:2798 0:<sup>1541</sup> <sup>T</sup>

<sup>f</sup> ¼ �½ � <sup>0</sup>:8364 0:<sup>4455</sup> � <sup>0</sup>:2303 0:<sup>2212</sup> <sup>T</sup>

(40)

59

http://dx.doi.org/10.5772/intechopen.71580

<sup>f</sup> ¼ �½ � <sup>0</sup>:9232 0:<sup>2138</sup> � <sup>0</sup>:1652 0:<sup>2733</sup> <sup>T</sup>

We presented a solution, which we previously developed, to the problem of attitude path planning for multiple spacecraft with avoidance of exclusion zones, by combining consensus theory and Q-CAC optimization theory. Using the solutions, a team of spacecraft can point to the same direction, or to various formation patterns, while they avoid an arbitrary number of attitude obstacles or exclusion zones in any coordinate frames. We also provided the proof of stability of the Laplacian-like matrix used for the attitude synchronization. Simulation results demonstrated the effectiveness of the algorithm. Current work is underway to implement the

Figure 5. Consensus-based attitude formation acquisition with avoidance. (a) Reorientation to consensus formation attitude with intervehicle thruster plume avoidance, (b) avoidance constraints graph, (c) attitude consensus graph.

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

[1] Blackwood G, Lay O, Deininger B, Gudim M, Ahmed A, Duren R, Noeckerb C, Barden B. The StarLight mission: A formation-flying stellar interferometer. In: SPIE 4852,

apart about the same axis.

Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints

In the direction of the initial attitude qi <sup>0</sup> of each SCi, a sensitive instrument vI cami is attached. Also, each SCi has a thruster pointing to the opposite of q<sup>i</sup> <sup>0</sup>. It is desired that v<sup>I</sup> cami avoids the thruster plumes emanating from each of the two other SC by 30� during the entire period of the maneuvers. From the generated initial quaternions, there is possibility of intersection of the thrusters of SC<sup>1</sup> and SC3, with SC2, and the thruster of SC<sup>2</sup> may damage SC<sup>1</sup> or SC<sup>3</sup> at any time k. Figure 4(a) shows the solution trajectories; (b) shows the avoidance graph, which shows that constraints are not violated; (c) shows the consensus graph. The final consensus quaternion is qf = [�0.8167 0.4807 � 0.2396 0.2112]<sup>T</sup> . This is the normalized average of the initial attitude quaternions, which proves that consensus is achieved.

#### 5.3. Consensus-based attitude formation acquisition with avoidance

To test the capability of the consensus algorithm in formation acquisition, SC1, SC2, and SC<sup>3</sup> will maneuver to a consensus formation attitude. Relative offset quaternions were defined to enable the sensitive instruments to point at 30� offsets from each other about the z-axis. The previous set of initial data for qi <sup>0</sup> and F<sup>I</sup> SCi were used. The relative offsets are

$$\begin{aligned} q\_1^{\text{off}} &= \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}^T\\ q\_2^{\text{off}} &= \begin{bmatrix} 0 & 0 & 0.2588 & 0.9659 \end{bmatrix}^T\\ q\_3^{\text{off}} &= \begin{bmatrix} 0 & 0 & 0.5 & 0.866 \end{bmatrix}^T \end{aligned} \tag{39}$$

Like the previous experiment, we want the sensitive instruments to avoid the thruster plumes emanating from each of the two other SC by an angle of 30� . The trajectories are shown in Figure 5(a) and (b) shows the avoidance graph; no constraints are violated, and (c) shows the consensus graph. The final consensus quaternions are

Figure 4. Consensus-based dynamic avoidance in different coordinate frames. (a) Reorientation to consensus attitude with intervehicle thruster plume avoidance, (b) avoidance constraints graph, (c) attitude consensus graph.

Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints http://dx.doi.org/10.5772/intechopen.71580 59

$$\begin{aligned} q\_f^1 &= \begin{bmatrix} -0.6926 & 0.6468 \ -0.2798 & 0.1541 \end{bmatrix}^T\\ q\_f^2 &= \begin{bmatrix} -0.8364 & 0.4455 \ -0.2303 & 0.2212 \end{bmatrix}^T\\ q\_f^3 &= \begin{bmatrix} -0.9232 & 0.2138 \ -0.1652 & 0.2733 \end{bmatrix}^T \end{aligned} \tag{40}$$

The differences of these quaternions are 30� apart about the same axis.

Figure 5. Consensus-based attitude formation acquisition with avoidance. (a) Reorientation to consensus formation attitude with intervehicle thruster plume avoidance, (b) avoidance constraints graph, (c) attitude consensus graph.

#### 6. Conclusion

A set of initial quaternions were randomly generated, with the following data:

<sup>0</sup> ¼ �½ � <sup>0</sup>:5101 0:<sup>6112</sup> � <sup>0</sup>:<sup>3187</sup> � <sup>0</sup>:<sup>5145</sup> <sup>T</sup>

<sup>0</sup> ¼ �½ � <sup>0</sup>:9369 0:<sup>2704</sup> � <sup>0</sup>:<sup>1836</sup> � <sup>0</sup>:<sup>124</sup> <sup>T</sup>

maneuvers. From the generated initial quaternions, there is possibility of intersection of the thrusters of SC<sup>1</sup> and SC3, with SC2, and the thruster of SC<sup>2</sup> may damage SC<sup>1</sup> or SC<sup>3</sup> at any time k. Figure 4(a) shows the solution trajectories; (b) shows the avoidance graph, which shows that constraints are not violated; (c) shows the consensus graph. The final consensus quaternion is

To test the capability of the consensus algorithm in formation acquisition, SC1, SC2, and SC<sup>3</sup> will maneuver to a consensus formation attitude. Relative offset quaternions were defined to

<sup>2</sup> <sup>¼</sup> ½ � <sup>000</sup>:2588 0:<sup>9659</sup> <sup>T</sup>

Like the previous experiment, we want the sensitive instruments to avoid the thruster plumes

Figure 5(a) and (b) shows the avoidance graph; no constraints are violated, and (c) shows the

Figure 4. Consensus-based dynamic avoidance in different coordinate frames. (a) Reorientation to consensus attitude

with intervehicle thruster plume avoidance, (b) avoidance constraints graph, (c) attitude consensus graph.

<sup>3</sup> <sup>¼</sup> ½ � <sup>000</sup>:5 0:<sup>866</sup> <sup>T</sup>

<sup>1</sup> <sup>¼</sup> ½ � <sup>0001</sup> <sup>T</sup>

<sup>0</sup> of each SCi, a sensitive instrument vI

SCi were used. The relative offsets are

<sup>0</sup>. It is desired that v<sup>I</sup>

. This is the normalized average of the initial attitude

offsets from each other about the z-axis. The

. The trajectories are shown in

(38)

(39)

cami is attached.

cami avoids the

during the entire period of the

<sup>0</sup> <sup>¼</sup> ½ � <sup>0</sup>:<sup>1448</sup> � <sup>0</sup>:1151 0:1203 0:<sup>9753</sup> <sup>T</sup>

qI

q2

q3

quaternions, which proves that consensus is achieved.

enable the sensitive instruments to point at 30�

Also, each SCi has a thruster pointing to the opposite of q<sup>i</sup>

thruster plumes emanating from each of the two other SC by 30�

5.3. Consensus-based attitude formation acquisition with avoidance

<sup>0</sup> and F<sup>I</sup>

q off

q off

q off

emanating from each of the two other SC by an angle of 30�

consensus graph. The final consensus quaternions are

In the direction of the initial attitude qi

58 Advanced Path Planning for Mobile Entities

qf = [�0.8167 0.4807 � 0.2396 0.2112]<sup>T</sup>

previous set of initial data for qi

We presented a solution, which we previously developed, to the problem of attitude path planning for multiple spacecraft with avoidance of exclusion zones, by combining consensus theory and Q-CAC optimization theory. Using the solutions, a team of spacecraft can point to the same direction, or to various formation patterns, while they avoid an arbitrary number of attitude obstacles or exclusion zones in any coordinate frames. We also provided the proof of stability of the Laplacian-like matrix used for the attitude synchronization. Simulation results demonstrated the effectiveness of the algorithm. Current work is underway to implement the algorithms using rotorcraft.

#### Author details

Innocent Okoloko

Address all correspondence to: okoloko@ieee.org

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

#### References

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Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints

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[17] Kuipers JB. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality. 1st ed. Princeton, NJ: Princeton University Press; 2002. 371

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**Chapter 4**

**Provisional chapter**

**Search-Based Planning and Replanning in Robotics and**

In this chapter, we present one of the most crucial branches in motion planning: searchbased planning and replanning algorithms. This research branch involves two key points: first, representing traverse environment information as discrete graph form, in particular, occupancy grid cost map at arbitrary resolution, and, second, path planning algorithms calculate paths on these graphs from start to goal by propagating cost associated with each vertex in graph. The chapter will guide researcher through the foundation of motion planning concept, the history of search-based path planning and then focus on the evolution of state-of-the-art incremental, heuristic, anytime algorithm families that are currently applied on practical robot rover. The comparison experiment between algorithm families is demonstrated in terms of performance and optimality. The future of

search-based path planning and motion planning in general is also discussed.

**Keywords:** A\*, RRT, holonomic path planning, trajectory planning, occupancy map, D\* Lite, incremental planning, heuristics planning, ARA\*, anytime dynamic A\*

Nowadays, as the rapid advances of computational power together with development of state-of-the-art motion planning (MP) algorithms, autonomous robots can now robustly plan optimal path in narrow configuration space or wide dynamic complex environment with high accuracy and low latency. These recent MP developments have a large impact in medical surgery, animation, expedition and many other disciplines. For instance, RRT [1] algorithm was applied for multi-arm surgical robot in [2]. Expedition robot GDRS XUV was implemented field D\* any-angle path planner [3] that enables the robot to optimally move in harsh environment. D\* [4] is implemented for Mars Rover prototypes and tactical mobile robots in [5]. Bug

**Search-Based Planning and Replanning in Robotics and** 

DOI: 10.5772/intechopen.71663

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

algorithms were implemented in multi-robot cooperation scenarios [6].

**Autonomous Systems**

**Autonomous Systems**

http://dx.doi.org/10.5772/intechopen.71663

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

An T. Le and Than D. Le

**Abstract**

**1. Introduction**

An T. Le and Than D. Le

**Provisional chapter**
