\r\n\trescue missions especially in difficult to access areas. Precise control of aerial robotic systems with multiple degrees of freedom arms is a challenging task due to the fact that the translational and rotational dynamics of the aerial vehicle are strongly coupled with the dynamics of the manipulator. In order to accomplish complex missions in presence of uncertainties in the environment, to achieve better maneuverability and precise 3D position and attitude control, nonlinear control techniques have been found effective. Swarm robotics (multiple robot working together) is another exciting application of the aerial robotics. This book intends to provide a wide range of readers in applied mathematics and various engineering disciplines an excellent survey of recent studies of aerial robotic systems.
",isbn:"978-1-78984-567-9",printIsbn:"978-1-78984-566-2",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"53805f091c3107536edd2579c9987649",bookSignature:"Dr. Mahmut Reyhanoglu and Dr. Geert De Cubber",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/7792.jpg",keywords:"Nonlinear Dynamics, Lagrangian Formulation, Visual Tracking, Way Point, Lyapunov Technique, Backstepping, Surveilance, Emergency Response, Crop Monitoring, Sliding Mode Observer, Nonlinear Filter, Coordinated Control, Leader-Follower",numberOfDownloads:1023,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"December 11th 2018",dateEndSecondStepPublish:"January 15th 2019",dateEndThirdStepPublish:"March 16th 2019",dateEndFourthStepPublish:"May 21st 2019",dateEndFifthStepPublish:"July 20th 2019",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,editors:[{id:"15068",title:"Dr.",name:"Mahmut",middleName:null,surname:"Reyhanoglu",slug:"mahmut-reyhanoglu",fullName:"Mahmut Reyhanoglu",profilePictureURL:"https://mts.intechopen.com/storage/users/15068/images/system/15068.png",biography:"Mahmut Reyhanoglu is presently the Glaxo Wellcome Distinguished Professor of Engineering at the University of North Carolina at Asheville, North Carolina, USA. His extensive research makes use of advanced mathematical techniques and models that arise from fundamental physical principles. His major research interests are in the areas of nonlinear dynamical systems and control theory, with particular emphasis on applications to mechatronics and aerospace systems. He has edited 3 books, and authored/co-authored several book chapters and over 130 peer-reviewed journal/proceedings papers. He served on the IEEE Transactions on Automatic Control Editorial Board and on the IEEE Control Systems Society Conference Editorial Board as an Associate Editor. He also served as International Program Committee member for several conferences and as a member of AIAA Guidance, Navigation, and Control Technical Committee. He is currently serving as an editor of International Journal of Aerospace Engineering.",institutionString:"University of North Carolina",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"3",totalChapterViews:"0",totalEditedBooks:"3",institution:{name:"University of North Carolina at Asheville",institutionURL:null,country:{name:"United States of America"}}}],coeditorOne:{id:"81429",title:"Dr.",name:"Geert",middleName:null,surname:"De Cubber",slug:"geert-de-cubber",fullName:"Geert De Cubber",profilePictureURL:"https://mts.intechopen.com/storage/users/81429/images/system/81429.png",biography:"Geert De Cubber works at the Unmanned Vehicle Centre of the Belgian Royal Military Academy, where he is leading the research activities of the research group on robotics for high-risk applications. The specialization of this research unit is the development of robots for applications like search and rescue and humanitarian demining. Geert’s main task is to apply computer vision techniques to mobile robots, rendering these robots able to perceive, analyze, and – to some degree – understand their environment.\nGeert was the coordinator of the EU-FP7-ICARUS projectnwhich was a large-scale EU-project dealing with the development of unmanned tools (aerial, ground and marine robots) which can assist search and rescue workers to save human survivors after a major crisis (earthquake, tsunami, typhoon, shipwreck, etc.).",institutionString:"Royal Military Academy",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"242",title:"Aerial Robotics",slug:"aerial-robotics"}],chapters:[{id:"67435",title:"A System for Continuous Underground Site Mapping and Exploration",slug:"a-system-for-continuous-underground-site-mapping-and-exploration",totalDownloads:136,totalCrossrefCites:0,authors:[null]},{id:"67297",title:"Decentralised Scalable Search for a Hazardous Source in Turbulent Conditions",slug:"decentralised-scalable-search-for-a-hazardous-source-in-turbulent-conditions",totalDownloads:137,totalCrossrefCites:0,authors:[null]},{id:"67003",title:"Vision-Based Autonomous Control Schemes for Quadrotor Unmanned Aerial Vehicle",slug:"vision-based-autonomous-control-schemes-for-quadrotor-unmanned-aerial-vehicle",totalDownloads:162,totalCrossrefCites:0,authors:[null]},{id:"67705",title:"Advanced UAVs Nonlinear Control Systems and Applications",slug:"advanced-uavs-nonlinear-control-systems-and-applications",totalDownloads:312,totalCrossrefCites:0,authors:[null]},{id:"68391",title:"Robotic Search and Rescue through In-Pipe Movement",slug:"robotic-search-and-rescue-through-in-pipe-movement",totalDownloads:280,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"194666",firstName:"Nina",lastName:"Kalinic Babic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/194666/images/4750_n.jpg",email:"nina@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. 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Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"19327",title:"The Prevention and Treatment of Penile Prosthesis Infections",doi:"10.5772/24859",slug:"the-prevention-and-treatment-of-penile-prosthesis-infections",body:'Since they introduction in the treatment of erectile dysfunction (ED), phosphodiesterase type 5 (PDE-5) inhibitors have achieved widespread acceptance. Today PDE-5 inhibitors are considered as first-line oral pharmacotherapy in the management of ED (Hatzimouratidis et al., 2010). However, penile implants are still a popular choice, especially in patients who have failed to achieve erections by chemical enhancement, who prefer a permanent solution to their condition or in those who have considerable scar tissue in the penis resulting in erection deformalities (Mulcahy 1999). Despite its invasiveness, penile prostheses provide high satisfaction rates (Montague & Angermeier 2001).
The types of prosthesis most commonly implanted are the two-piece and the three-piece inflatable device, and the soft and malleable prosthesis. In the last few years the three-piece inflatable device has been used for preference, as it improves the erection with the most acceptable functional and cosmetical results (Minervini et al., 2006; Bettocchi et al., 2008).
Engineering changes and designs revisions have reduced the mechanical malfunctions associated with inflatable penis prostheses to less than 5 % (Carson et al., 2000; Carson 2004). As penile prostheses are now expected to function for an average of 8-12 years post implantation, infection has become a more significant problem. The incidence of infection has been reported to range from 0.5 to 17.7% (Quesada & Light, 1993; Wilson & Delk, 1995) usually about 1-3 % in case of primary implantation, and about 10-13 % in case of revision or reimplantation (Abouassaly et al., 2004).
The traditional treatment of penile prosthesis infection is systematic and local antibiotics application with the complete removal of the device followed by reinsertion within 2–12 months. However, removal of the device can lead to corporeal fibrosis, making dilation of the corporeal bodies difficult and reinsertion of a new device more complicated (Brant et al., 1996; Mulcahy, 1999).
To reduce the risk of device associated infections and to avoid the difficulties associated with late reinsertions many modifications have been developed such as antibiotic or hydrophilic coated devices and immediate replacement of the infected prosthesis (salvage techniques).
The aim of this chapter is to summarize the different methods of prevention and treatment of penile prosthesis infections.
Staphylococcus species, especially Staphylococcus epidermidis are the most common infecting pathogens, isolated from 35 to 56% of infected penile prostheses patients (Carson, 2003). Gram-negative enteric bacteria including Proteus mirabilis, Pseudomonas aeruginosa, Escherichia coli and Serratia marcenses may also be pathogens, accounting for 20 % of infections (Abouassaly et al., 2004). Gram-negative bacteria can combine with anaerobic organisms in severe infections, such as Bacteroides species, and lead to gangrene of the penis. Fungi, mycobacteria and Neisseria gonorrhea have also been reported as etiological agents in penile prosthesis infections (Carson, 1989; Abouassaly et al., 2004).
Penile prostheses get infected predominantly secondary to bacterial seeding at the time of surgery. Prosthetic materials attract bacteria and support subsequent biofilm formation. In a multicenter study culture positive bacteria were found in 70% of patients with clinically uninfected penile prostheses during revision surgery for mechanical malfunction. Saphylococcus species were cultured in 90 % of the cases (Henry et al., 2004), which have an enhanced ability to produce glycocalyx biofilm.
Penile prosthesis infections can be divided into clinically apparent and subclinical infections. Clinical infections present with, penile pain, induration, erythema, fever, purulent drainage from the wound and extrusion. Subclinical infections most often manifest by chronic prosthesis-associated pain.
Known risk factors for penile prosthesis infection include urinary tract infection, infections elsewhere in the body and hematogenous spread (Carson & Robertson, 1988; Little & Rhodus, 1992). There is an increased incidence of infection among patients undergoing primary implantation with penile reconstruction or secondary prosthesis revision compared to primary implantation alone, probably due to the increased duration of surgery (Quesada & Light, 1993; Jarow, 1996). The role of diabetes mellitus and spinal chord injury, as risk factors of penile prosthesis infection are contradictory (Jarow, 1996; Cakan et al., 2003).
Because in most cases bacterial contamination occurs at the time of surgery, it is essential to use appropriate preoperative preparations. Short preoperative hospital stays have been documented to maintain low virulence (Carson, 2003). It is important to eliminate skin infections and to have sterile urine prior to surgery. Washing the genital region with strong soap in the days before the procedure, preoperative shaving and an aggressive scrub of the operating area is recommended to decrease the risk of infection (Mulcahy, 1999; Gomelsky & Dmochowski, 2003).
During surgery adequate sterile technique, short operating time, minimal tissue devitalization along with effective wound closure can all decrease the rate of perioperative infections (Scott, 1987).
Athough the effectiveness of prophylactic perioperative antibiotics for implantation of penile prosthesis has never been proven by prospective studies, their use has become established and favored by most urologists. Antibiotics should be administered 1-2 hours prior to surgery and continued for 36-48 hours postoperative. Most common pathogenic organisms most likely to produce infections must be targeted when choosing prophylactic antibiotics. Therefore traditional prophylaxis include a parenteral aminoglycoside for Gram-negative and a first- or second generation cephalosporin or vancomycin for Gram-positive organisms coverage (Schwartz et al., 1996; Naber et al., 2001). Schwarz et al found in a randomized prospective trial of 20 patients that oral fluoroquinolone (ofloxacin) administered 2 hours before surgery achieved significantly higher intracavernosal levels and was more cost-effective than the combination of gentamicin and cefazolin (Schwartz et al., 1996). To estimate the safety and efficacy of this prophylaxis modality, further studies with similar settings, but bigger sample size should be performed.
Early efforts in device impregnation focused on coating catheters with antibiotics. In 1995 Raad et al reported that in in vitro studies catheters coated with a combination of rifampin and minocycline provided significantly better inhibitory activities against S. epidermidis, S. aureus and E. faecalis than catheters coated with either drug alone or vancomycin (Raad et al., 1995). After additional in vitro and in vivo studies in 2001 the US Food and Drug Administration approved the use of penile prosthesis coated with a combination of rifampin and minocycline called InhibiZone. The concentrations of the antibiotics, while adequate for decreasing colonization, provided only minimal serum levels of the agents. Coated inflatable penile prostheses are implanted in a fashion similar to those without antibiotic treatment except that the devices are not soaked prior to implantation (Carson, 2004).
In a retrospective study Carson et al reported 61,7% decrease in perioperative infection with InhibiZone compared to controls at 1 year post infection (Carson, 2004). The same group recently published their long-term clinical outcomes of almost 40,000 implants. There were significantly less revisions due to infections in the impregnated compared to the non-impregnated group at up to 7.7 years of follow-up (1.1% vs 2.5%, respectively)(Carson et al., 2011). In a subset of diabetic patients in the same series, the rate of infection-related revisions was significantly lower in the impregnated group compared to the controls at 7 years (1,62 % vs 4,24 %)(Mulcahy & Carson, 2011).
In 2007 Wilson et al. began a prospective randomized study comparing the infection rate of rifampin and minocycline coated implants with implants without impregnation (Wilson et al., 2007). After it became evident that the infection rate was less with the impregnated group they abandoned the other arm because of ethical considerations and compared they results with the previously published series of the same surgical team with noncoated implants (Wilson & Delk, 1995; Wilson et al. 1998). The use of antibiotic coated inflatable penile prosthesis resulted in a statistically significant reduction in the infection rates compared with the historical data in nondiabetic virgin implant patients (p=0,0024), diabetic virgin implant patients (p=0,0141) and in revision patients in whom washout with antiseptic solutions was used (p= 0,0095). Revision without washout had the same infection rate (10%) as with noncoated implants.
In 2002 a hydrophilic penile prosthesis coating was developed which has been shown to decrease bacterial adherence in vitro and in animal models (Rajpurkar et al., 2004). This coating absorbs surgeon chosen intraoperative antibiotics that can elute into surrounding tissues over 24-72 h to further decrease infection (Hellstrom et al., 2003).
Mansouri and colleagues compared the spectrum and durability, both in vitro and in vivo of the hydrophilic coated prosthesis dipped in vancomycin and the InhibiZone implants, and found that the antibiotic pre-impregnated prosthesis had a broader spectrum in vitro and a more durable antimicrobial activity in vitro and in an animal model than implants dipped in vancomycin (Mansouri et al. 2009).
Clinical data on the hydrophilic coated inflatable penile prosthesis is limited. Wolter et al. presented their one-year experience with the device (Titan, Mentor Corporation, Santa Barbara, CA) (Wolter & Hellstrom 2004), the infection rate for 2357 coated penile prostheses was 1,06 % compared to 2,07 % in 482 uncoated penile prostheses implanted over the same time period. Although preliminary data using this device shows promise, long-term follow-up and prospective studies are not yet available.
Subclinical infections may be more common than clinically apparent infections. These infections are difficult to diagnose and even more challenging to treat. Parsons et al. recommend initial trial of oral antibiotic therapy using long-term antibiotics (ciprofloxacin 500mg twice daily) (Parsons et al., 1993). Following initiation of antibiotics, pain suppression should suggest continuing antibiotics for 10–12 weeks. The authors reported a 60% success rate with conservative treatment of subclinical penile prosthesis infections. The use of oral cephalosporins (cefalexin and cephradine) has also been suggested for 10-12 weeks, although success rates are lower at 25-30% (Choong & Whitfield, 2000; Carson, 2003). If pain fails to resolve or rapidly returns after antibiotics, however, surgical intervention is appropriate. Parsons et al. have reported 90% success rate in treating these prostheses with an exchange protocol including systemic antibiotics for 24–48 h using vancomycin. The suspected infected prosthesis is then removed and a combination of vancomycin and protamine was used for antibiotic irrigation prior to reimplantation of a new prosthesis. Patients are maintained on vancomycin and parental antibiotics for 1 week (Parsons et al., 1993; Carson, 2003).
In case of clinically apparent infection surgical intervention along with antibiotics is of critical importance. The traditional treatment consists of the immediate removal of all the components followed by delayed reimplantation 2-12 months later (Gomelsky & Dmochowski, 2003; Mulcahy, 2003). The advantage of this solution is that the new implant is scheduled only when the infection has completely cleared. However, removal of the device along with inflammation from the infectious process leads to corporeal fibrosis and scarring, which almost always results in penile shortening and may make dilation of the corporeal bodies very difficult, resulting reinsertion of a new device more complicated and sometimes impossible (Brant et al., 1996; Mulcahy, 1999).
A salvage protocol was instituted in 1991 to avoid difficult reinsertion and maintain as much penile length as possible. The salvage procedure involves removing all parts of the infected prosthesis, washing the wound, and replacing the device at the same procedure. Mulcahy et al. recommend a sequence of irrigating solutions including kanamycin and bacitracin in normal saline followed by half-strength hydrogen peroxide, half-strength povidone-iodine solution, pressurized normal saline containing vancomycin and gentamicin, half-strength povidone-iodine, half-strength hydrogen peroxide, and finally another kanamycin/bacitracin solution (Mulcahy et al., 1995). Gloves, instruments, gowns, and drapes are changed and a new inflatable penile prosthesis is immediately implanted. Postoperatively, patients are treated with antibiotics (2x500 mg ciprofloxacin) for 4-6 weeks. Antibiotics can be modified based on culture and sensitivity results. The reported success rate of the salvage procedure is 84-91% (Brant et al., 1996; Mulcahy, 2003). To avoid complications of late reinsertion the salvage protocol is a treatment alternative of traditional delayed reimplantation, although in patients with life-threatening systemic conditions such as septicemia, or diabetic ketoacidosis, or in whom necrotizing infections with death of penile skin is occurring salvage procedure is not recommended (Brant et al., 1996; Mulcahy, 1999).
The delayed salvage procedure consists of placement of a drainage tube after removal of the prosthesis; antibiotic solution is irrigated through the drain and a new prosthesis is placed about 3 days later. However, Knoll et al could not find a statistically significant difference between immediate and delayed salvage procedure (Knoll, 1998), while there are the additional cost of the second surgical procedure.
Prospective studies and long-term follow up are needed to make stronger recommendations about the different methods in the prevention or treatment of penile prostheses infections, especially about the hydrophilic coated penile prosthesis.
The efforts to apply more effective methods of prevention and the new developments of prosthesis coatings resulted a significant reduction of the infectious complications of penile prosthesis implantation. Further improvements of surgical procedures and prosthesis materials and coatings can lead to further decrease of the infection rates in the future.
Research in multi-robot systems is motivated by several notions; namely, some motivation can be put as [1]:
It is complex for one single robot system to fulfill complex tasks. Instead, more than one system would simplify the solution.
Tasks are generally distributed in nature.
Multiple limited-resource robot systems are more efficient to deal with than a single powerful robot system.
Speed of the task process increases through parallelism in multiple robot systems.
Robustness increases as redundancy is introduced in multiple systems.
Until recently, the number of real-life implementations of multi-robot systems is relatively small. The reason is the complexity associated with the field. Also, the related technologies are relatively new. Emergence of autonomous driving vehicle technology and market can push the boundaries in the field. As technology develops, new venues for application will open for mainstream use rather than only in research and development labs. Due to its promising applicability, autonomous cars and vehicles (or various intelligent transportation systems in general) sit at the forefront [2, 3]. To name a few, benefits include reducing congestions [4], increasing road safety [5], and, of course, self-driving cars [6]. Another application in civil environments is related to safety and security like rescue missions of searching for missing people in areas hard for humans to operate in [7] or searching for dangerous materials or bombs [8] in an evacuated building. Also, another area of application of multi-robot systems is in military area; research was done heavily in the fields of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) [9, 10].
Many approaches are developed to tackle the issue of multiple robot systems. Under the inspiration of biological systems and the need of technologies, many problems are defined as cooperative motions. Cooperative motion is discussed in [11, 12, 13, 14, 15, 16]. Optimization in both time and energy has been tackled in the literature [17, 18, 19, 20]. There is an opportunity to incorporate concept of time/energy optimization into the paradigm of multi-robot systems.
This paper investigates the solution of a time-energy optimal control problem for multiple mobile robots; namely, the paper is to study the problem as a nonlinear programming (NLP) problem. The main idea of the solution used here is to utilize distributed optimization techniques to solve the overall optimization problem. Solving for optimal time and energy of more than one robot system adds more burden on the problem; robot interaction with each other is added to the problem. This paper will focus more on the distributed aspect of the problem; more details about the numerical optimal control problem formulation can be found in [21]. In [21], the problem of controlling the motion of a single mobile robot is solved using the direct method of numerical optimal control (see [22]); this showed great flexibility in incorporating physical constraints and nonlinear dynamics of the system.
The rest of this section will define the global problem formulation. Discussion about distributed optimization and associated algorithm is presented in Section 2. Section 3 will apply the method on the multi-robot problem. Application to wheeled mobile robots and simulation examples are discussed in Section 4 followed by the conclusion.
We can present the discrete time global optimization (numerical optimal control) problem for L robots as follows:
with
System behavior is governed by the nonlinear dynamic system in
The above optimization problem can be viewed as having two sets of control variables; the first set resembles the discretized system inputs,
with
Here in this section, the concept of distributed optimization is explored. This area tackles optimization problems with distributed nature. Consider the following optimization problem:
with
Observing the global problem in (2), we can see that it is just equivalent to the combination of all the subproblems; it is easy to see that solving for each subproblem (3) individually will result in the solution for the whole global problem. Now, however, consider the following problem:
You see clearly that the above problem cannot be trivially separated into some subproblems due to the constraint
Decomposition in mathematics is the concept of breaking a mathematical problem into smaller subproblems that can be solved independently while not violating the original problem. Primary works of [23, 24] discuss multiple aspects of optimization in general while exploring specific classes as well; these works are excellent resources for reading and understanding. Viewing the applications of distributed optimization will convey the impression that they, however different, are all mostly very similar theoretically. Terms of networked, distributed, decentralized, cooperative, and the like are becoming all corresponding to somewhat similar problems. Other works related to this area and the area of multi-agent systems can be found in [25, 26, 27, 28, 29].
Before going further, we discuss a method used in solving distributed optimization problem which will help us in solving the problem of this paper. This method is called subgradient methods [30]. These methods are similar to the popular optimization algorithms using gradient descent. However, they are extended to escape function differentiation. The works [31, 32, 33] also explore the method in the perspective of multi-agent systems.
Consider the typical problem:
This typical problem can be solved using any gradient descent method. At iteration
with
However, for the subgradient method [33], we will have a definition of
with
The subgradient method is a simple first-order algorithm to minimize a possibly nondifferentiable function. The above definition escapes the requirement of a differentiated objective function. It is defined as finding any vector that makes the optimization algorithm go to better value in a first-order optimality sense. Of course, when a gradient
Now observe the following constrained optimization problem:
Let
The vector
The dual optimization problem is the pair of two optimization problems, namely, a maximization in
Now, an algorithm for solving the dual problem utilizing subgradient method is discussed. Let us define
The above definition is to clarify the minimum attained at any value of
Now, it is obvious from (14) that at iteration
The projection operator
Recall the problem of combination of
As mentioned, the constraints
Put in mind that
then we can apply the primal-dual update from (13) and (15) at an iteration
We can see that the above pair of updates can easily be distributed; after the relaxing of the constraints, the primal problem can be separated. The facility of subgradients lets us propose that any iteration
The function
Now, in this section we can apply the previous discussion into the problem of optimizing the motion of multiple robots. Recall the global optimization problem of motion of
You can see that the problem above is just the combination of
The above definition is just to reduce the notation of robot input sequence. If we have, for example, two robot inputs
with objectives
which are subject to the set of individual robot
Returning back to the primal-dual problem pair in Section 2, we can establish the algorithm updates according to the defined updates in (19). At each iteration
The minimizer update
In this brief section, elaboration is put forth about how to practically use the algorithm. The ultimate goal is to optimize primal problem with no collision violation, i.e., reaching optimal dual (maximum) solution. At each global iteration, we only need to improve the primal problem values for the updated extra cost of the interaction constraint,
We can also distribute the stopping decision to individual robots by observing the change in individual objective values.
With condition (25) on its own, we cannot always be satisfying the collision requirement. So, this condition can be accompanied by a condition on the collision constraint violation. For all robots, elements of the complete constraint vector
Specific values of
Figure 1 shows the individual robot system considered here. Robot state includes
Wheeled mobile robot.
Here, we will discuss the formulation and the structure of the coupling constraints. The robots can be designed to perform any cooperative strategy in their motion. Here, we only consider the global goal of optimizing the motion of each robot in time and energy while avoiding colliding with each other during the motion. Let us define the coupling constraint vector across the discrete time indexes as
For the ith robot, it tries to avoid colliding with the rest of
We define the collision avoidance by constraining motion of other robots to be outside a safety circle region around each
So, we can define each element of the constraint vector as
The radius of the safety region is chosen as
You can follow the whole distributed algorithm for the time-energy optimization of multi-robot system with collision avoidance in the flowchart in Figure 2. View the flowchart as the process for each individual robot (subproblem). We implement the algorithm in Figure 2. For the primal minimizer update in (24), the nonlinear programming (NLP) function of
Distributed algorithm flowchart to optimize multi-robot motion.
Example 1. In this example, exploration of the behavior of the algorithm is shown. The problem has the following desired values of initial and final positions and orientations for the three robots:
Here, robot 1 has equal objective weights of 5 on both time and energy, robot 2 has weights of 10 on energy and 1 on time, and robot 3 has 10 on time and 1 on energy. The maximum number of internal NLP iterations (primal update) is set to only 10. The step size is set to
Figure 3 contains the objective (time-energy) value evolution throughout iterations. You can see the stable convergence as the algorithm progresses. Figure 4 shows each of the robots’ safety clearances during the optimized motion. In Figure 5, snapshots of motion of the three robots at different time instants are depicted. This illustrates the collision avoidance attained throughout the optimized motion. Observe also how different are the speeds of each robot because of objective weights; note from Figure 4 that each robot has a different final time for their motion. The algorithm has shown good performance at eliminating collision constraint violations. Figures 4 and 5 show an instant (around t = 12) where robots 1 and 3 violate collision distance with very small value, but no collision occurs. This is because the maximum number of iterations of the algorithm is exhausted. This indicates the possibility for the motion to be optimized even more if collision constraints were relaxed or if more algorithm iterations were allowed. Initialization of the algorithm also plays a role in algorithm evolution.
The time-energy objective values throughout global iterations (Example 1).
Collision avoidance (Example 1).
Snapshots of optimized motions at different instants (Example 1).
Example 2. This example illustrates more the satisfaction of the objectives. In this example the safety circle radius is put as
After applying our approach, you can see the resulting optimized motions in Figure 6. In Figure 6, errors in x- and y-coordinates and orientation of each robot are shown with respect to time. It is clear that errors of zero are achieved. In Figure 7 for each robot, constraint evaluations, i.e., safety clearance, are displayed for the other two robots throughout time. You can see that robots come close to each other sometimes but without violating the safety distance. This result is attained maybe because of special structure of initial and final positions and orientations. That could have given flexibility for the algorithm. Optimized trajectories of the three robots: each row of plots shows x-coordinate error, y-coordinate error, and orientation error, respectively; each column of plots show robots 1, 2, and 3 errors, respectively (Example 2). Collision avoidance (Example 2).Figure 6.
Figure 7.
The paper investigated the time-energy minimization onto the multi-robot case. A global objective function is formulated as the sum of individual robot objectives in time and energy. Constraints are divided into two sets, namely, robot-individual constraints and robots’ interaction constraints. The problem is decomposed into L subproblems with L being the number of robot systems. The subproblems are coupled with each other by the collision avoidance information. Applying a distributed algorithm solved the problem iteratively. The overall output gives optimized motions for all robots in time and energy while adhering and not colliding with each other. We applied our approach to the case of three wheeled mobile robots: we generated in parallel for each robot an optimized control input trajectory.
An extension to this study is to generate optimized motion trajectories and apply them experimentally. A possible area for experimentation is full-scale autonomous vehicles. Issues related to communication and distributing information during the parallel algorithm will need to be incorporated and investigated. Also, aspects of state estimation and localization of the robot system will come into the place which were not considered in this work. A possible other investigation is to distribute the problem further onto the time variable
The authors would like to thank King Fahd University of Petroleum and Minerals supporting this work.
"Open access contributes to scientific excellence and integrity. It opens up research results to wider analysis. It allows research results to be reused for new discoveries. And it enables the multi-disciplinary research that is needed to solve global 21st century problems. Open access connects science with society. It allows the public to engage with research. To go behind the headlines. And look at the scientific evidence. And it enables policy makers to draw on innovative solutions to societal challenges".
\n\nCarlos Moedas, the European Commissioner for Research Science and Innovation at the STM Annual Frankfurt Conference, October 2016.
",metaTitle:"About Open Access",metaDescription:"Open access contributes to scientific excellence and integrity. It opens up research results to wider analysis. It allows research results to be reused for new discoveries. And it enables the multi-disciplinary research that is needed to solve global 21st century problems. Open access connects science with society. It allows the public to engage with research. To go behind the headlines. And look at the scientific evidence. And it enables policy makers to draw on innovative solutions to societal challenges.\n\nCarlos Moedas, the European Commissioner for Research Science and Innovation at the STM Annual Frankfurt Conference, October 2016.",metaKeywords:null,canonicalURL:"about-open-access",contentRaw:'[{"type":"htmlEditorComponent","content":"The Open Access publishing movement started in the early 2000s when academic leaders from around the world participated in the formation of the Budapest Initiative. They developed recommendations for an Open Access publishing process, “which has worked for the past decade to provide the public with unrestricted, free access to scholarly research—much of which is publicly funded. Making the research publicly available to everyone—free of charge and without most copyright and licensing restrictions—will accelerate scientific research efforts and allow authors to reach a larger number of readers” (reference: http://www.budapestopenaccessinitiative.org)
\\n\\nIntechOpen’s co-founders, both scientists themselves, created the company while undertaking research in robotics at Vienna University. Their goal was to spread research freely “for scientists, by scientists’ to the rest of the world via the Open Access publishing model. The company soon became a signatory of the Budapest Initiative, which currently has more than 1000 supporting organizations worldwide, ranging from universities to funders.
\\n\\nAt IntechOpen today, we are still as committed to working with organizations and people who care about scientific discovery, to putting the academic needs of the scientific community first, and to providing an Open Access environment where scientists can maximize their contribution to scientific advancement. By opening up access to the world’s scientific research articles and book chapters, we aim to facilitate greater opportunity for collaboration, scientific discovery and progress. We subscribe wholeheartedly to the Open Access definition:
\\n\\n“By “open access” to [peer-reviewed research literature], we mean its free availability on the public internet, permitting any users to read, download, copy, distribute, print, search, or link to the full texts of these articles, crawl them for indexing, pass them as data to software, or use them for any other lawful purpose, without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. The only constraint on reproduction and distribution, and the only role for copyright in this domain, should be to give authors control over the integrity of their work and the right to be properly acknowledged and cited” (reference: http://www.budapestopenaccessinitiative.org)
\\n\\nOAI-PMH
\\n\\nAs a firm believer in the wider dissemination of knowledge, IntechOpen supports the Open Access Initiative Protocol for Metadata Harvesting (OAI-PMH Version 2.0). Read more
\\n\\nLicense
\\n\\nBook chapters published in edited volumes are distributed under the Creative Commons Attribution 3.0 Unported License (CC BY 3.0). IntechOpen upholds a very flexible Copyright Policy. There is no copyright transfer to the publisher and Authors retain exclusive copyright to their work. All Monographs/Compacts are distributed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). Read more
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\\n\\nThe Open Access publishing model employed by IntechOpen eliminates subscription charges and pay-per-view fees, enabling readers to access research at no cost. In order to sustain operations and keep our publications freely accessible we levy an Open Access Publishing Fee for manuscripts, which helps us cover the costs of editorial work and the production of books. Read more
\\n\\nDigital Archiving Policy
\\n\\nIntechOpen is committed to ensuring the long-term preservation and the availability of all scholarly research we publish. We employ a variety of means to enable us to deliver on our commitments to the scientific community. Apart from preservation by the Croatian National Library (for publications prior to April 18, 2018) and the British Library (for publications after April 18, 2018), our entire catalogue is preserved in the CLOCKSS archive.
\\n"}]'},components:[{type:"htmlEditorComponent",content:'The Open Access publishing movement started in the early 2000s when academic leaders from around the world participated in the formation of the Budapest Initiative. They developed recommendations for an Open Access publishing process, “which has worked for the past decade to provide the public with unrestricted, free access to scholarly research—much of which is publicly funded. Making the research publicly available to everyone—free of charge and without most copyright and licensing restrictions—will accelerate scientific research efforts and allow authors to reach a larger number of readers” (reference: http://www.budapestopenaccessinitiative.org)
\n\nIntechOpen’s co-founders, both scientists themselves, created the company while undertaking research in robotics at Vienna University. Their goal was to spread research freely “for scientists, by scientists’ to the rest of the world via the Open Access publishing model. The company soon became a signatory of the Budapest Initiative, which currently has more than 1000 supporting organizations worldwide, ranging from universities to funders.
\n\nAt IntechOpen today, we are still as committed to working with organizations and people who care about scientific discovery, to putting the academic needs of the scientific community first, and to providing an Open Access environment where scientists can maximize their contribution to scientific advancement. By opening up access to the world’s scientific research articles and book chapters, we aim to facilitate greater opportunity for collaboration, scientific discovery and progress. We subscribe wholeheartedly to the Open Access definition:
\n\n“By “open access” to [peer-reviewed research literature], we mean its free availability on the public internet, permitting any users to read, download, copy, distribute, print, search, or link to the full texts of these articles, crawl them for indexing, pass them as data to software, or use them for any other lawful purpose, without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. The only constraint on reproduction and distribution, and the only role for copyright in this domain, should be to give authors control over the integrity of their work and the right to be properly acknowledged and cited” (reference: http://www.budapestopenaccessinitiative.org)
\n\nOAI-PMH
\n\nAs a firm believer in the wider dissemination of knowledge, IntechOpen supports the Open Access Initiative Protocol for Metadata Harvesting (OAI-PMH Version 2.0). Read more
\n\nLicense
\n\nBook chapters published in edited volumes are distributed under the Creative Commons Attribution 3.0 Unported License (CC BY 3.0). IntechOpen upholds a very flexible Copyright Policy. There is no copyright transfer to the publisher and Authors retain exclusive copyright to their work. All Monographs/Compacts are distributed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). Read more
\n\nPeer Review Policies
\n\nAll scientific works are Peer Reviewed prior to publishing. Read more
\n\nOA Publishing Fees
\n\nThe Open Access publishing model employed by IntechOpen eliminates subscription charges and pay-per-view fees, enabling readers to access research at no cost. In order to sustain operations and keep our publications freely accessible we levy an Open Access Publishing Fee for manuscripts, which helps us cover the costs of editorial work and the production of books. Read more
\n\nDigital Archiving Policy
\n\nIntechOpen is committed to ensuring the long-term preservation and the availability of all scholarly research we publish. We employ a variety of means to enable us to deliver on our commitments to the scientific community. Apart from preservation by the Croatian National Library (for publications prior to April 18, 2018) and the British Library (for publications after April 18, 2018), our entire catalogue is preserved in the CLOCKSS archive.
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