1. Introduction
During the mission of a fleet, the number of available support stations is an important factor that limits the combat effectiveness of the fleet, so an effective security plan is the necessary basis of enhancing the combat effectiveness of the fleet [1-3]. In this case, the security plan of the support station is pre-established. If there is a new faulty aircraft, the security plan will be disturbed. Therefore, it is important that we try to reasonably dispatch the maintenance order and minimize the impact of disturbances to enhance the departure ability of the fleet [4-5].
In response to the disturbance of new faulty aircraft for the security plan of support stations, the solution is to use a reasonable model or algorithm to re-plan the order, resources, and time of the protection [6-7]. To solve the problem of disturbances in the scheduling process, the traditional solution is the FCFS-based algorithm [8-9] or weight-priority scheduling algorithm [10-11]. In addition to the traditional algorithms for solving the problem, there are many optimization methods or models for the perturbation problem in the scheduling process. For example, Teodorovic and Guberinic used the network flow model to study flight rescheduled issues in dealing with disturbances of the delayed flights, but this method only reduces the number of late-perturbed flights on the basis of local optimization and reduced airport dispatch efficiency [12]. Then, Teodorovic and Stojkovic [13] developed a heuristic that optimizes the front model to minimize the number of cancellations, but it also results in a corresponding increase in computational costs. Soumis considered the time variable and established the DAYOPS model to seek the optimized dispatch plan. However, this model requires a large amount of data to achieve a large amount of calculations of the objective function, which has the disadvantages that the perturbation cannot be solved in time and the availability is not high [14].
There are many optimization methods for solving the problem of disturbance in the process of scheduling, and these methods have achieved the purpose of resolving the disturbance. However, there are still some shortcomings, which are briefly listed in the following. First, to solve the disturbance problem, the optimization methods represented by FCFS emphasis on the partial adjustment do not propose a better strategy in the direction of system science. Second, the algorithms such as determining the weight of the priority scheduling can solve these disturbances, like the high weight coefficient and outstanding problem, but they do not adequately consider scheduling resources and scheduling time arrangements. Third, some methods ignore the calculation costs; these methods usually require large amounts of computational data or tedious calculations. Lastly, some models or algorithms are only theoretically feasible, and their availability is low.
In this paper, we will analyze the disturbance of the security plan of support station caused by new faulty aircraft, assuming that the status of the support station is always good, with no faults occurring, and the number of new faulty aircrafts is 1. The case is based on Allen’s mathematical relationship [15-16] and determines the Allen relationship between the maintenance time intervals for faulty aircrafts, combined with interference management thinking [17-18], and also determines the set of dominating maintenance solutions based on the Jachson dominance principle [19]. Finally, this paper determines the best maintenance program based on the branch-based algorithm and achieves the purpose of minimizing the disturbance [20-21].
2. Problem Description
2.1. The Fault Disturbance and General Handling Process
When a fleet dispatches, due to the number of the support stations, the support plans and maintenance plans must be developed in advance to ensure the mobilization of fleet efficiency. Every faulty aircraft has different fault reasons and different combat missions, so there are specific requirements of every faulty aircraft maintenance time for the support station. Therefore, formulating the maintenance support plan is key to improving the mobilization ability of the fleet.
If the maintenance support plan of the support station has been developed based on the existing faulty aircrafts, the new failed aircraft will disturb the maintenance support plan of the support station. The general processes of solving the disturbance problem are the following:
$\cdot$ Give priority to free support stations for the security and maintenance of new faulty aircrafts;
$\cdot$ Select the support station with a relatively small security task. If there is no free support station;
$\cdot$ Develop a new security and maintenance plan based on the principles of maintenance support.
The main content of this article is to propose a new strategy to re-establish the maintenance support plan after selecting the support station. The general processing flow to solve the disturbance problem is shown in Figure 1.
Figure 1
Figure 1.
The general processing flow of fault disturbance
2.2. The Definitions of Parameters and Variable
In this paper, in order to solve the problem of disturbance of faulty aircraft, the security plan of resource allocation and security is planned. The core of this paper focuses on arranging the time of each fault. Therefore, sometime parameters need to be defined as follows:
Table 1. Time parameters
| ${{r}_{j}}$ | The time of aircraft j coming to the support station |
|---|---|
| ${{p}_{j}}$ | The time needed to repair of aircraft j |
| ${{d}_{j}}$ | The latest maintenance time of aircraft j |
| ${{T}_{j}}=\max \left\{ 0,{{r}_{j}}-{{d}_{j}} \right\}$ | Maintenance delay time of aircraft j |
2.3. Basic Assumptions
In order to reflect the fault aircraft disturbance analysis and highlight the case of specificity, this paper makes the following assumptions:
1)The state of the support station is intact, and this paper does not consider the support station occur error.
2)In this paper, we only consider the disturbance of faulty aircraft.
3)We assume the disturbance of a single faulty aircraft for a single failure.
The kinds of resource of the support station are enough, and we use the maximum maintenance delay time as an optimization goal.
3. Determining the Dominant Maintenance Program Set based on the Allen Relationship
3.1. Determining the Allen Relationship of Aircrafts
It is assumed that the set of aircrafts to be repaired is $A=\left\{ 1,2,\cdot \cdot \cdot ,n \right\}$. A does not contain the aircraft that is being repaired when the new faulty aircraft arrives, and the new faulty aircraft number is n. For the maintenance task of the support station, if $\max \left[ T_{j}^{\max }\left( {{\sigma }_{1}} \right) \right]\le \max \left[ T_{j}^{\max }\left( {{\sigma }_{2}} \right) \right]$, we assume that the maintenance program
The time when the aircraft numbered j is delivered to the maintenance site is rj, and the latest maintenance time is dj. These two hours constitute a range[rj,dj].The relationship between the time intervals of the aircraft numbered i and the aircraft numbered j (Allen relation) is shown in Figure 2. The time interval for all aircraft in A constitutes a set $V=\left\{ j|{{r}_{j}},{{d}_{j}},{{p}_{j}} \right\}$.
Figure 2
Figure 2.
The Allen relationship
Definition 1Theset of aircrafts that needs to be repaired is $A=\left\{ 1,2,\cdot \cdot \cdot ,n \right\}$. If $\exists i\in A$ satisfies the condition that $\forall j\in A,\text{ }\left( i\ne j \right)$, all Allen relationships of during(i,j) cannot be established, and then the aircraft numbered i is called the top aircraft, denoted as ti.
Definition 2 Pi is a set of aircrafts. If any of the elements j satisfy the condition $j\in P\subset A$, the Allen relationship of during(i,j) can be established, and then Pi is called the top pyramid of the top aircraft i.
3.2. Determine the Maintenance Order Rules
Determine the top aircrafts according to the time they enter the support station to sort. If they access the support stations at the same time, use the latest repair time to complete the sort. If the latest repair time is equal, then use any order for numbering.
Let the top pyramid i be the pyramid corresponding to the top aircraft tα, u(j) be the first pyramid number that belongs to the non-top aircraft j, and v(j) be the last pyramid number that belongs to the non-top aircraft j.
This article proposes the following dominating maintenance rules to determine the dominant maintenance programs.
Rule 1 All top aircraft maintenance order is based on the number from small to large for maintenance.
Rule 2 Only the aircrafts belonging to the first top pyramid can be repaired before the first top aircraft, and their order of repair is in ascending order of time sent to the support station. If the time is equal, their order of repair is arbitrary.
Rule 3 Only the aircrafts belonging to the last top pyramid can be repaired after the last top aircraft, and the order of their maintenance is in ascending order of the latest maintenance time. If the time is equal, their order of repair is arbitrary.
Rule 4 Only the aircrafts belonging to the top pyramid Pk or subordinate to the top pyramid P(k+1) can be repaired between two successive top aircrafts tk and t(k+1).
Rule 4-1 The aircrafts that belong only to the top pyramid Pk and do not belong to the top pyramid P(k+1) should be repaired after the top aircraft ${{t}_{k}}.$ Their order of repair is in ascending order of time sent to the support station. If the time is equal, their order of repair is arbitrary.
Rule 4-2 The maintenance order of aircrafts belonging to both the top pyramid Pk and the top pyramid P(k+1) is in any order.
Rule 4-3 The final maintenance only belongs to the top pyramid P(k+1) and does not belong to the top pyramid Pk. Their order of repair is in ascending order of time sent to the support station. If the time is equal, their order of repair is arbitrary.
These rules determine the order of maintenance between some aircrafts. First, determine the order of maintenance between top aircrafts. If ${{t}_{k}}\prec {{t}_{(k+1)}}$, the top aircraft tk is repaired before the top aircraft ${{t}_{(k+1)}}$. Second, determine the order of maintenance between top aircrafts and non-top aircrafts, which is ${{t}_{\left( u\left( j \right)-1 \right)}}\prec j\prec {{t}_{\left( v\left( j \right)-1 \right)}}$.
According to the above dominance rules, the space for optional maintenance options is reduced. Obtain the dominant maintenance program set $Mdom$. Because the dominating maintenance program set $Mdom$ is based on dominance rules, $Mdom$ contains at least one optimal maintenance solution.
4. Determine the Maintenance Delay Interval for a Single Aircraft
Suppose that in the dominant maintenance program set $Mdom$ obtained by the above dominance rules, the minimum maintenance delay time and the maximum maintenance delay time of aircraft j are $T_{j}^{\min }$ and $T_{j}^{\max }$, respectively. They respectively correspond to the best dominance maintenance plan and the worst dominance maintenance plan for the aircraft j in the dominant maintenance program set $Mdom$. The two programs are expressed as $Reps_{j}^{\min }$ and $Reps_{j}^{\max }$.
Two sets of aircraftsare defined at the same time. The first set is $Pre_{j}^{\min }=\left\{ k|v\left( k \right)<u\left( j \right) \right\}$, which means that aircrafts belonging to $Pre_{j}^{\min }$ are repaired prior to aircraft j in all dominant maintenance programs. The second set is $Pre_{j}^{\max }=\left\{ k|u\left( k \right)<v\left( j \right) \right\}$, which represents that aircraft k satisfying $u\left( k \right)>v\left( j \right)$ is behind aircraft j in all dominant maintenance programs.
The best dominance maintenance program $Reps_{j}^{\min }\in Mdom$ for aircraft j is ${{\omega }_{1}}\prec {{t}_{1}}\prec \cdot \cdot \cdot \prec {{\omega }_{(u(j)-1)}}\prec {{t}_{(u(j)-1)}}\prec j$, where the aircrafts belonging to the set ${{\omega }_{k}}=\left\{ i\in Pre_{j}^{\min }|u\left( i \right)=k \right\}$ are repaired in ascending order of their time to the support station. The worst dominance maintenance program for aircraft j $Reps_{j}^{\max }\in Mdom$ is
${{t}_{1}}\prec {{\omega }_{1}}\prec \cdot \cdot \cdot \prec {{t}_{(v(j)-1)}}\prec {{\omega }_{(v(j)-1)}}\prec A\prec {{t}_{v(j)}}\prec B\prec j$
Where the aircrafts belonging to the set ${{\omega }_{k}}=\left\{ i\in Pre_{j}^{\max }|v\left( i \right)<v\left( j \right),v\left( i \right)=k \right\}$ are repaired in ascending order of their latest repaired time. A and B are sets of aircraft, and aircrafts belonging to the set $A=i\in Pre_{j}^{\max }|v\left( i \right)\ge v\left( j \right)$ and $di>dj$ are repaired in ascending order of their arrival time at the support station. Aircrafts belonging to the set $B=i\in Pre_{j}^{\max }|v\left( i \right)\ge v\left( j \right)$ and di≤dj are repaired in ascending order of their latest maintenance time.
Through the above method to determine the best dominance of the aircraft j repair program $Reps_{j}^{\min }$ and the worst dominance of the maintenance program $Reps_{j}^{\max }$, we can calculate the minimum maintenance delay time $T_{j}^{\min }$ and the maximum maintenance delay time $T_{j}^{\max }$ of aircraft j. The maintenance delay interval is $\left[ T_{j}^{\min },T_{j}^{\max } \right]$. Obviously, the maximum maintenance delay time $T_{j}^{*}$ of the optimal maintenance program satisfies $T_{j}^{\min }\le T_{j}^{\text{*}}\le T_{j}^{\max }$, $\forall j\in A$.
5. Determine the Optimal Solution based on the Branch Definition Method
By narrowing the scope of the dominant set $Mdom$, we can clear the bad dominance of maintenance programs, continue to reduce the single plane maximum maintenance delay time $Z=T_{j}^{\max }$, and ultimately get the best maintenance program or close to the best maintenance program. Select the single aircraft j corresponding to the maximum maintenance delay time of the maintenance program as the starting point $N0$ of the branch. The dominant maintenance program set of this aircraft is $M_{dom}^{0}={{M}_{dom}}$, and the set only includes the maintenance program of part of the aircraft before the aircraft j. Define a set ${V}'=\left\{ j|{{{{r}'}}_{j}}\ge {{r}_{j}},{{{{d}'}}_{j}}\ge {{d}_{j}},{{{{r}'}}_{j}}\ge {{r}_{j}} \right\}$ that is compatible with set $V=\left\{ j|{{r}_{j}},{{d}_{j}},{{p}_{j}} \right\}$, and then apply the branch definition method to traverse all sets and get the best maintenance program.
The steps of the branch definition method are shown as follows, and the initial values are $\underline{Z}=\max \left( T_{j}^{\min } \right),\forall j\in A$, $\overline{Z}=\max \left( T_{j}^{\max } \right),\forall j\in A$.
Step 1 For the branch node Ni ,this node corresponds to the dominant maintenance program set $M_{dom}^{i}$, and the maximum delay time of this set is ${{\overline{Z}}_{i}}.$ Choose the top aircraft as the pivot aircraft $\alpha $ and choose the non-top aircraft as the free aircraft ∆. Aircraft j meets the condition $T_{j}^{\max }={{\max }_{i\in A}}T_{i}^{\max }$.
Case 1: If aircraft j is a non-top aircraft, select the top aircraft corresponding to the last top pyramid v(j) of aircraft j as the pivot aircraft α. Aircrafts followed by pivot aircraft α are used as free aircrafts.
Case 2: If aircraft j is the top aircraft and its top pyramid contains a non-top aircraft, select the aircraft j as the pivot aircraft α, and the nearest aircraft to be repaired prior to the pivot aircraft α is selected as the free aircraft ∆.
Case 3: If aircraft j is the top aircraft and its top pyramid does not contain a non-top aircraft, select the top aircraft immediately before the aircraft j. Its top pyramid contains the non-top aircraft as the pivot aircraft α, and aircrafts followed by pivot aircraft α are used as free aircrafts.
Step 2 According to the preconceived dominance maintenance rules, the branch node Ni is divided into two branches: $N_{i+1}^{\alpha \prec \delta }\left( {{r}_{i}}\leftarrow {{r}_{\alpha }} \right)$ and $N_{i+1}^{\alpha \prec \delta }\left( {{d}_{i}}\leftarrow {{d}_{\alpha }} \right)$. Their dominant maintenance programs are $M_{dom}^{i+1}\left( N_{i+1}^{\alpha \prec \delta } \right)$ and $M_{dom}^{i+1}\left( N_{i+1}^{\delta \prec \alpha } \right)$. These two dominant maintenance programs meet the conditions $M_{dom}^{i+1}\left( N_{i+1}^{\alpha \prec \delta } \right)\cup M_{dom}^{i+1}\left( N_{i+1}^{\delta \prec \alpha } \right)\text{=}M_{dom}^{i}$.
Step 3 Calculate $\left[ \underline{{{Z}_{N_{i+1}^{\alpha \prec \delta }}}},\overline{{{Z}_{N_{i+1}^{\alpha \prec \delta }}}} \right]$ and $\left[ \underline{{{Z}_{N_{i+1}^{\delta \prec \alpha }}}},\overline{{{Z}_{N_{i+1}^{\delta \prec \alpha }}}} \right]$ based on the method of determining the minimum and maximum maintenance delay time for a single aircraft. If there is a branch where the minimum value of the interval is greater than $\overline{{{Z}_{i}}}$, discard this branch.
Step 4 If both branches are saved, $\overline{{{Z}_{i\text{+1}}}}\text{=}\min \left( \overline{{{Z}_{N_{i+1}^{\alpha \prec \delta }}}},\overline{{{Z}_{N_{i+1}^{\delta \prec \alpha }}}} \right)$. If there is only one branch, the value of $\overline{{{Z}_{i\text{+1}}}}$ equals the maximum value of this branch interval.
Step 5 When $\overline{{{Z}_{i}}}$ is no longer reduced, stop the branch; otherwise, repeat Steps 1-4.
Step 6 When $\overline{{{Z}_{i}}}$ is no longer reduced, the corresponding dominant maintenance program set is $M_{dom}^{*}$. According to the idea of interference management, select the maintenance program with the smallest deviation from the maintenance program before the new faulty aircraft arrives as the optimal maintenance program.
6. Case Study
Suppose that when the new faulty aircraft arrives at the support station, the support station is undergoing maintenance tasks and there are four other faulty aircrafts that need to be repaired, with these aircrafts being numbered in order of arrival time at the support station. Take the time r1 at which the aircraft 1 reaches the support station as time 0. The aircraft completes maintenance at time 12. When the maintenance is completed, the next aircraft will be repaired immediately.
Table 2. Time parameters of each aircraft (unit: minutes)
| Number | rj | dj | pj |
|---|---|---|---|
| 1 | 0 | 43 | 10 |
| 2 | 7 | 34 | 15 |
| 3 | 15 | 67 | 22 |
| 4 | 22 | 75 | 11 |
| 5 | 36 | 58 | 8 |
Table 3. Allen relationship for each aircraft time interval
| 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|
| 1 | equals(1,1) | during(2,1) | overlaps(1,3) | overlaps(1,4) | overlaps(1,5) |
| 2 | during(2,1) | equals(2,2) | overlaps(2,3) | overlaps(2,4) | precedes(2,5) |
| 3 | overlaps(1,3) | overlaps(2,3) | equals(3,3) | overlaps (3,4) | during(5,3) |
| 4 | overlaps(1,4) | overlaps(2,4) | overlaps (3,4) | equals(4,4) | during(5,4) |
| 5 | overlaps(1,5) | precedes(2,5) | during(5,3) | during(5,4) | equals(5,5) |
According to the definition of the top aircraft, we know that the two top aircrafts are t1=2 and t2=5 respectively, and the top pyramids they correspond to are P1={1} and P2={3,4}. According to the dominant maintenance rules, we can get ten dominant maintenance programs:
$\begin{matrix} & 1\prec 2\prec 5\prec 3\prec 4,\ \ \ \ 1\prec 2\prec 3\prec 5\prec 4 \\ & 1\prec 2\prec 5\prec 4\prec 3,\ \ \ \ 1\prec 2\prec 4\prec 5\prec 3 \\ & 1\prec 2\prec 3\prec 4\prec 5,\ \ \ \ 2\prec 1\prec 5\prec 3\prec 4 \\ & 2\prec 1\prec 3\prec 5\prec 4,\ \ \ \ 2\prec 1\prec 5\prec 4\prec 3 \\ & 2\prec 1\prec 4\prec 5\prec 3,\ \ \ \ 2\prec 1\prec 3\prec 4\prec 5 \\ \end{matrix}$
According to the method of determining the minimum and maximum maintenance delay times of a single-aircraft, the delay time interval $\left[ T_{j}^{\min },T_{j}^{\max } \right]$ of every aircraft is canceled. The worst and best dominance of each aircraft is shown in the order of maintenance of the upper and lower bounds, and the result is shown in Figure 3.
Figure 3
Figure 3.
Minimum and maximum maintenance delay time of each aircraft
According to the results shown in Figure 3, we choose the dominant maintenance program $\text{2}\prec \text{1}\prec \text{3}\prec \text{4}\prec \text{5}$ as the starting point of the branch, and e eventually get the dominant maintenance programs $\text{1}\prec \text{2}\prec \text{5}\prec \text{3}\prec \text{4}$ and $\text{2}\prec \text{1}\prec \text{5}\prec \text{3}\prec \text{4}$. According to the idea of interference management, the deviation between the adjusted maintenance program and the initial maintenance program should be made as small as possible, and the initial maintenance program is $\text{1}\prec \text{2}\prec \text{3}\prec \text{4}$, so we choose $\text{1}\prec \text{2}\prec \text{5}\prec \text{3}\prec \text{4}$ as the best maintenance program. According to the maintenance program, it is known that only aircraft 4 does not complete the maintenance task before the latest maintenance time. The maintenance time of aircraft 4 exceeded its latest maintenance of three minutes, that is, $\max \left( T_{j}^{\max } \right)=T_{4}^{\max }=3$.
If according to the classic FCFS algorithm to re-develop the maintenance program, the program is $\text{1}\prec \text{2}\prec \text{3}\prec \text{4}\prec 5$, the calculation shows that aircraft 5 was finally repaired and exceeded the maximum maintenance delay time of 20 minutes. The maintenance results of these two programs are shown in Figure 4.
Figure 4
Figure 4.
Comparison of the results of two programs
7. Conclusions
When the number of support stations for serving the combat fleet maintenance is limited, the new faulty aircraft will disturb the existing maintenance program of the support station. In response to this problem, this paper presents a new maintenance support emergency strategy based on the Allen relationship, which provides ideas and methods for effectively solving the disturbances in the process of fleet maintenance.
The main advantages of this method are summarized as follows:
(1) Under the assumption that the support station is in good condition, according to the determination of the Allen time relationship, the Allen’s relationship between the disturbed faulty aircraft and each of the original planned aircraft can be determined. At the same time, the corresponding parameters are defined.
(2) The program proposed in this paper can deal with the disturbance of the fault plane systematically so that situations that can only be handled locally can be avoided.
(3) Although this paper presents a single support station of single aircraft disturbance as an example, this method can be extended to more complex situations, and it has a wide range of applications.
(4) The algorithm has some advantages, including relatively simple processing, less demand for data, lower computational cost, and high usability.
This paper applied the Allen relationship to resource scheduling and reorganization of the assurance plan. Better results indicate that this method can be used to optimize the safeguard time when disturbances occur in the security process. Perturbation events are more frequent in the scheduling of complex systems, and applying this method can improve the operation, scheduling, and anti-interference ability of complex systems.
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We investigate a joint user association and interference management problem in two-tier downlink heterogeneous networks where pico base stations (BSs) are densely deployed in areas of high traffic demand. We employ macro almost blanking subframes (ABSs) to mitigate cross-tier interference. To manage co-tier interference among picocells, we introduce pico operation mode (POM): During each POM, a distinct subset of picocells serve users simultaneously; different POMs operate at different times at different durations. We formulate the problem as maximizing the network utility under proportional fairness with the optimization over user association and resource partitioning (RP) (including the macro ABS and the amount of time allocated to each POM). As an exhaustive search over all possible POMs in the optimization may become computationally infeasible, we propose a method of preselecting favorable POMs to reduce the dimension of the optimization. With the selected POMs, we explicitly examine the structure of the optimal solutions and show that 1) the optimal user association is load-aware and can be determined by rate bias values of each BS, and 2) all the POMs and the macro BSs have balanced load in the sense that the ratios of the number of associated users to the allocated time are the same. Based on the analysis, an alternating algorithm is developed to obtain the RP and the biases. We demonstrate through numerical examples that 1) the proposed POM selection method does not incur significant performance loss, compared with the case where all possible POMs are considered; 2) the alternating algorithm provides near-optimal cell association and RP solutions; 3) by applying the proposed framework, significant network performance improvement can be achieved with dense pico deployments, compared with baselines without co-tier interference management and baselines with sparse pico deployments.
A Robust Approach for the Single Machine Scheduling Problem
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DOI:10.1007/s10951-007-0010-3
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This paper describes a robust approach for the single machine scheduling problem 1| r i | L max . The method is said to be robust since it characterizes a large set of optimal solutions allowing to switch from one solution to another, without any performance loss, in order to face potential disruptions which occur during the schedule execution. It is based on a dominance theorem that characterizes a set of dominant sequences, using the interval structure defined by the relative order of the release and the due dates of jobs. The performance of a set of dominant sequences can be determined in polynomial time by computing the most favorable and the most unfavorable sequences associated with each job, with regard to the lateness criterion. A branch and bound procedure is proposed which modifies the interval structure of the problem in order to tighten the dominant set of sequences so that only the optimal sequences are conserved.
Branch and Bound Algorithm for Optimal Sensor Network Design
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A Branch-and-Bound Algorithm for Globally Optimal Calibration of a Camera-and-Rotation-Sensor System,”in
DOI:10.1109/ICCV.2009.5459343
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[Cited within: 1]
We propose a branch-and-bound algorithm to obtain the globally optimal relative rotation between a camera and the rotation sensor attached to it. Compared to previous methods, our approach directly minimizes the image space error related to the measurements which is very natural for camera-based systems. Our algorithm is based on the observation that we may evaluate the residual when the rotation matrix is known. We propose a feasibility test algorithm for the branch-and-bound to efficiently reduce the search volume of the rotation domain. Experimental results are provided using synthetic and real data sets.
