Geometric Insight into the Control Allocation Problem for Open-Frame ROVs and Visualisation of Solution
2. Control Allocation Problem
2.1. Problem Definition
- Guidance: subsystem that continuously computes the reference (desired) position, velocity and acceleration of an ROV to be used by the motion control system.
- Navigation: subsystem to determine position/attitude, course, travelled distance and (optionally) velocity and acceleration of an ROV.
- Control: subsystem to determine necessary control forces and moments to be provided by the ROV to satisfy certain control objective (in conjunction with the guidance system).
- STEP 1 (Regulation Task): Design a control law, which specifies desired virtual control input (normalised vector of forces and moments ) in the virtual control space;
- STEP 2 (Actuator Selection Task): Design control allocator, which finds the “best” feasible true control input (normalised command vector to be applied to individual actuators) in the true control space.
- —longitudinal axes (directed to front side),
- —transversal axes (directed to starboard),
- —normal axes (directed from top to bottom).
- is empty (i.e., no solution exists),
- has exactly one element (i.e., there is one unique solution),
- has more than one element (i.e., there are many solutions).
- If the intersection is a segment, there is an infinite number of solutions (each point that belongs to the segment is a solution),
- If the intersection is a point, there is only one solution,
- If the intersection is an empty set, no solution exists.
2.3. Geometric Insight into Problem
2.4. Choice of Norm
- In the case when is the unity matrix, the norm distributes the virtual control demand among the control inputs in a uniform way, while the solution utilises as few control inputs as possible to satisfy the virtual control demand.
- The solution varies continuously with the parameters (elements) of , while the solution does not. Change in a parameter (element) of will produce the change in the slope of . The solution will vary continuously with , while it can be shown that the solution will have discontinuity for some value of and the solution in the breakpoint is not unique.
- If is a non-singular, the problem has a unique solution for . For , this is not always the case, as discussed above. The reason lies in the fact that the sphere is a strictly convex set, while this is not the case for .
2.5. Choice of the Weighting Matrix
3. Control Allocation Solution: Hybrid Method
3.2. Weighted Pseudoinverse
- Find such that ,
- Find .
3.2.2. Approximation of Unfeasible Solution
- T-approximation of the unfeasible pseudoinverse solution, introduces direction error , i.e., vectors and have not the same direction. At the same time, the direction error for S-approximation , i.e., vectors and always have the same direction, but the magnitude error is greater than .
- The fixed-point method (Section 3.3) is able to improve the T- or S-approximation of the unfeasible weighted pseudoinverse solution . Approximations or can be used as the initial iteration and the algorithm will find the solution such that is a better approximation of than or . This feature is the main idea of the hybrid approach for control allocation.
3.3. Fixed-Point Method
3.4. Algorithm (Hybrid Method)
3.5. Extension of Concepts from “virtual” ROV to Open-Frame ROV
- If lies inside , then has an infinite number of points, and the control allocation problem has an infinite number of solutions.
- If lies on the boundary of , then is a single point, the unique solution for the control allocation problem.
- If lies outside , then is an empty set, i.e., no exact solution exists, only approximation.
4. Testing and Validation
4.1. Evaluation of the FTC in Virtual Environment
4.1.1. Partial Fault in HT2
4.1.2. Total Fault in HT2
4.2. Evaluation of the FTC in Real-World Environment
4.2.1. Path Following: Simulated Faults
4.2.2. Complex Tasks with Faulty Thruster
Conflicts of Interest
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|Virtual Control Input|
|Virtual ROV||Open-Frame ROV|
|Virtual Control Input|
|True Control Input|
|Control Effectiveness Matrix|
|Actuator Position Constraints|
|Control Allocation Problem (Inequalities (2) and (4) apply component-wise.)||For a given , find such that||For a given , find such that|
|Virtual Control Space||True Control Space|
|Initial Point||# Of Iterations||Last Iteration||Limit||Obtained Virtual Control Input||Desired Virtual Control Input||Direction Error||Magnitude Error|
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Omerdic, E.; Trslic, P.; Kaknjo, A.; Weir, A.; Rao, M.; Dooly, G.; Toal, D. Geometric Insight into the Control Allocation Problem for Open-Frame ROVs and Visualisation of Solution. Robotics 2020, 9, 7. https://doi.org/10.3390/robotics9010007
Omerdic E, Trslic P, Kaknjo A, Weir A, Rao M, Dooly G, Toal D. Geometric Insight into the Control Allocation Problem for Open-Frame ROVs and Visualisation of Solution. Robotics. 2020; 9(1):7. https://doi.org/10.3390/robotics9010007Chicago/Turabian Style
Omerdic, Edin, Petar Trslic, Admir Kaknjo, Anthony Weir, Muzaffar Rao, Gerard Dooly, and Daniel Toal. 2020. "Geometric Insight into the Control Allocation Problem for Open-Frame ROVs and Visualisation of Solution" Robotics 9, no. 1: 7. https://doi.org/10.3390/robotics9010007