Hybrid spiral-dynamic bacteria-chemotaxis algorithm with application to control two-wheeled machines
- K. M. Goher^{1}Email author,
- A. M. Almeshal^{3},
- S. A. Agouri^{2},
- A. N. K. Nasir^{2},
- M. O. Tokhi^{2},
- M. R. Alenezi^{3},
- T. Al Zanki^{3} and
- S. O. Fadlallah^{4}
Received: 27 October 2016
Accepted: 5 May 2017
Published: 16 June 2017
Abstract
This paper presents the implementation of the hybrid spiral-dynamic bacteria-chemotaxis (HSDBC) approach to control two different configurations of a two-wheeled vehicle. The HSDBC is a combination of bacterial chemotaxis used in bacterial forging algorithm (BFA) and the spiral-dynamic algorithm (SDA). BFA provides a good exploration strategy due to the chemotaxis approach. However, it endures an oscillation problem near the end of the search process when using a large step size. Conversely; for a small step size, it affords better exploitation and accuracy with slower convergence. SDA provides better stability when approaching an optimum point and has faster convergence speed. This may cause the search agents to get trapped into local optima which results in low accurate solution. HSDBC exploits the chemotactic strategy of BFA and fitness accuracy and convergence speed of SDA so as to overcome the problems associated with both the SDA and BFA algorithms alone. The HSDBC thus developed is evaluated in optimizing the performance and energy consumption of two highly nonlinear platforms, namely single and double inverted pendulum-like vehicles with an extended rod. Comparative results with BFA and SDA show that the proposed algorithm is able to result in better performance of the highly nonlinear systems.
Keywords
Background
Optimization algorithms play a dominant role in solving real problems [38, 58]. Bacterial foraging algorithm (BFA) [42] and spiral-dynamics algorithm (SDA) [50, 51] are well-known optimization techniques in solving real-world problems. Evolutionary algorithms (EA) have been used extensively in literature: soft computing techniques [46], particle swarm optimization [53, 55], incremental encoding [13], neural stochastic multi-scale optimization [9], multi-objective optimization [12, 23], multi-criteria optimization [43] and fuzzy logic and genetic programming [48].
Nasir et al. [33, 34, 36] proposed linear and nonlinear adaptive BFA where the bacteria step size is varied based on the combination of bacteria and iteration index. Chen and Lin [14], Farhat and El-Hawary [18] and Huang and Lin [22] utilized index and total number of chemotaxis to vary bacteria step size within a specified range. Niu et al. [39], Yan et al. [57] and Xu et al. [56] varied the step size within a user-defined range using combination of index and total number of iterations. Supriyono and Tokhi [49] developed various versions of BFA based on linear and nonlinear mathematical formulations to establish relationship between bacteria step size and their current fitness value. This relationship enables bacteria to have different step sizes in similar iteration as well as through the whole operation. There are other adaptive approaches considered the variation of the step size based on fitness value [16, 28, 29, 44, 45, 54]. Nasir et al. [30–32] proposed adaptive spiral-dynamic algorithm (ASDA) to establish relationship between spiral radius (r) and fitness value of each search point. They introduced schemes to make variation in the spiral radius within a specific range, enabling each search point to have different spiral radius in moving from one location to another location. Moreover, the movement step of each search agent was made with respect to its fitness value at the current location. As a result of the variation, there was improvement to the performance mainly on the accuracy of the final solution.
Hybrid optimization techniques
Hybrid approach is the combination of two or more algorithms aimed to retain the advantages and eliminate the weaknesses of the original algorithms. This includes the synergization between different groups such as bio-inspired, nature-inspired, etc. Biswas et al. [10, 11] proposed hybrid BFA-PSO where a chemotactic strategy of bacteria was designed to represent exploitation part of the algorithm, while the exploration of optimum location was accomplished by PSO. The same approach using a constant step size was implemented by Korani [26], where the PSO operator was used to determine new direction of bacteria motion. Ghaffar et al. [19] adopted a modified PSO operator to determine new direction of bacteria to avoid local optima solution. Biswas et al. [11] proposed chemotactic differential evolution algorithm where adaptive chemotactic strategy of bacteria has been used to improve fitness accuracy of classical differential evolution (DE). Sinha et al. [47] implemented the same approach on an electric power system. Kim et al. [24] and Kim [25] used GA and BFA to tune a PID controller for automatic voltage regulation. Panigrahi and Ravikumar [40] and Hooshmand et al. [21] incorporated Nelder–Mead method into bacteria chemotaxis phase to enhance the search strategy and improve bacteria location. Other hybrid approaches involving BFA [41, 59] used bee colony algorithm and Tabu search.
Limitations of BFA and SDA
BFA is a well-known bio-inspired algorithm. It has a comparable or better performance compared to other types of optimization algorithm [17]. Therefore, it has been adopted by many researchers worldwide to solve real-world problems in many areas [52]. However, BFA has a slow convergence speed and longer computation time. Due to this issue, the application of original BFA in online and offline tuning for solving a complex real-world problem is unsatisfactory [15]. On the other hand, SDA is a relatively new and a simple algorithm developed inspired from natural spiral phenomena on earth. It has a relatively fast convergence speed which can complement the drawback of BFA performance. Previous study showed that SDA has a similar or better performance compared to other differential evolutionary (DE) and particle swarm optimization (PSO) algorithms [50, 51]. However, SDA has a premature convergence issue where it hardly provides an optimal solution for complex problems.
Hybrid spiral-dynamic bacteria-chemotaxis
A hybrid bacteria-chemotaxis spiral-dynamic algorithm (HSDBC) has been proposed by Nasir et al. [30–32] to synergize the chemotactic strategy of bacteria and ASDA. The chemotaxis phase in BFA was designed such that it represents exploration stage and placed at the first phase of the algorithm, while the ASDA as the exploitation stage and was placed at the second phase of the algorithm. The combination simplified the BFA algorithm and greatly reduced the total computation time of BFA. Moreover, comparison with original algorithms concluded that it improved the accuracy of the final solution and had the capability to avoid the local optima problem. HSDBC is a new variant of hybrid-type BFA-SDA algorithm developed to solve the issues aforementioned above. Our previous study showed that the algorithm outperformed both BFA and SDA algorithms in terms of accuracy in finding a global optima solution. Compared to BFA, the total computation time has been significantly reduced and its convergence speed has been considerably increased [31, 37].
HSDBC algorithm parameters
θ _{tumble} | Bacteria angular displacement on x _{ i } − x _{ j } plane around the origin for tumbling |
θ _{swim} | Bacteria angular displacement on x _{ i } − x _{ j } plane around the origin for swimming |
r _{tumble} | Spiral radius from bacteria tumble |
r _{swim} | Spiral radius for bacteria swim |
m | Number of search points |
k _{max} | Maximum iteration number |
N _{sw} | Maximum number of swim |
x _{ i } (k) | Bacteria position |
R ^{ n } | n × n matrix |
Contribution overview and paper organization
Establishing the optimal control strategy for nonlinear dynamic systems, specifically inverted pendulum-based systems, has been and still remains a field of interest for a countless number of research studies due to their various promising real-life applications including personal transport systems and wheelchairs. This paper presents an extended study of the proposed algorithm in solving complex problem of two-wheeled inverted pendulum systems. We will implement HSDBC algorithm to control two different configurations of two-wheeled machines. A detailed simulation study of the HSDBC algorithm using several unimodal and multimodal benchmark functions can be found in the work of Nasir and Tokhi [37]. A hybrid fuzzy-like PD and I controller is designed and implemented on the two systems.
This paper is organized as follows: “Background” section introduces both ASDA and ABFA optimization algorithms, along with an explanation of the principle of HSDBC algorithm. In order to test and validate the proposed HSDBC algorithm on real dynamic systems, two case studies are considered in the study and are introduced in “Methods” section. “Case study I: single IP with an extended rod” section describes in details the first case study that involves a single inverted pendulum (IP) system. A double IP system with an extended rod is considered as the second case study and is presented in “Case study II: double IP with an extended rod” section. The results of the investigation are presented at the end of each of the previously mentioned sections, sections “Case study I: single IP with an extended rod” and “Case study II: double IP with an extended rod”. At last, the paper is concluded in “Conclusion” section.
Methods
An inverted pendulum as a typical multi-input multi-output system has the characteristics of nonlinear, multivariable and close coupling Luo et al. [27]. The uniqueness and wide application of technology derived from this unstable system has drawn interest of many researchers including Akesson et al. [2], Askari et al. [5] and Balan et al. [6, 7]. There are various applications of IP configuration including design of walking gaits, wheelchairs, and personal transport systems.
The system considered in this paper is a two-wheeled machine (TWM) with an extendable rod as described by Goher et al. [20] and verified by Almeshal et al. [3, 4]. This system stabilizes it extendable intermediate body (IB) by controlling the wheel movements in a desired manner. A TWM is designed such that either the center of mass of the robot is above or below the axle joining two wheels. Statically unstable TWM have evoked a lot of interest in present decade [8]. Two case studies are used to test and validate the developed algorithm; single IP and double IP with an extended rod. For consistency, the two systems are considered to move along an inclined surface. The results of the simulation are shown in a comparative manner with three different optimization algorithms; BFA, SDA and HSDBC.
Case study I: single IP with an extended rod
System description
Mathematical modeling of the single IP with an extended rod
Detailed explanations of the constant parameters appearing in Eqs. (4)–(9) are formulated in “Appendix 2”.
Control strategy
Fuzzy rule base
ê | e | ||||
---|---|---|---|---|---|
NB | NS | Z | PS | PB | |
NB | NB | NB | NB | NS | Z |
NS | NB | NB | NS | Z | PS |
Z | NB | NS | Z | PS | PB |
PS | NS | Z | PS | PB | PB |
PB | Z | PS | PB | PB | PB |
Constrained optimization
Boundary limits of the controller gain parameters
Gain parameters | Minimum value | Maximum value |
---|---|---|
Kp _{ 1 } | 4 | 5 |
Kd _{ 1 } | 3 | 4 |
Ki _{ 1 } | 0.4 | 0.8 |
Kp _{ 2 } | 4 | 5 |
Kd _{ 2 } | 3 | 4 |
Ki _{ 2 } | 1 | 1.3 |
Kp _{ 3 } | 10 | 13 |
Kd _{ 3 } | 15 | 20 |
Ki _{ 3 } | 2 | 3 |
Objective functions
BFA parameters
P | S | Nc | Ns | Nre | Ned | Ped | Sr |
---|---|---|---|---|---|---|---|
9 | 40 | 10 | 6 | 2 | 2 | 0.25 | S/2 |
SDA parameters
P | R | Theta | Initial points | Iterations |
---|---|---|---|---|
9 | 0.9 | π/4 | 5 | 90 |
HSDBC parameters
P | R | Rzw | Ns | Theta | Initial points | Iterations |
---|---|---|---|---|---|---|
9 | 0.95 | 0.55 | 2 | π/4 | 5 | 90 |
Optimized gain values
Parameter | BFA | SDA | HSDBC | |
---|---|---|---|---|
Loop 1 | Kp _{ 1 } | 4.2287 | 4.0000 | 4.0003 |
Kd _{ 1 } | 3.0064 | 3.1065 | 3.0089 | |
Ki _{ 1 } | 0.7380 | 0.6773 | 0.7267 | |
Loop 2 | Kp _{ 2 } | 4.5638 | 4.3183 | 4.7770 |
Kd _{ 2 } | 3.2615 | 3.6085 | 3.4461 | |
Ki _{ 2 } | 1.0322 | 1.2380 | 1.1306 | |
Loop 3 | Kp _{ 3 } | 11.4488 | 10.7368 | 11.3992 |
Kd _{ 3 } | 19.3417 | 17.1030 | 18.0021 | |
Ki _{ 3 } | 2.6508 | 2.0113 | 2.6529 |
Cost functions
Minimum cost function value | BFA | SDA | HSDBC |
---|---|---|---|
J | 0.922 | 0.8804 | 0.8517 |
Simulation results
Case study II: double IP with an extended rod
Control strategy
Minimization of the objective function J is used to find the optimal controller gain parameters that result in the minimum control loop errors in the stability region of the system.
Constrained optimization
Boundary limits of the controller gain parameters
Parameter | Lower | Upper | |
---|---|---|---|
Loop 1 | Kp _{ 1 } | 1.5 | 2.4 |
Kd _{ 1 } | 0.5 | 1 | |
Ki _{ 1 } | 0.9 | 1.4 | |
Loop 2 | Kp _{ 2 } | 5 | 6.5 |
Kd _{ 2 } | 2.5 | 4 | |
Ki _{ 2 } | 1.5 | 2 | |
Loop 3 | Kp _{ 3 } | 8 | 12 |
Kd _{ 3 } | 7.5 | 9 | |
Ki _{ 3 } | 0 | 0.5 | |
Loop 4 | Kp _{ 4 } | 8 | 10 |
Kd _{ 4 } | 5 | 8 | |
Ki _{ 4 } | 0 | 0.5 | |
Loop 5 | Kp _{ 5 } | 30 | 50 |
Kd _{ 5 } | 10 | 20 | |
Ki _{ 5 } | 1 | 10 |
Results and discussion
BFA parameters
P | S | Nc | Ns | Nre | Ned | Ped | Sr |
---|---|---|---|---|---|---|---|
15 | 20 | 14 | 6 | 2 | 2 | 0.25 | S/2 |
SDA parameters
P | R | Theta | Initial points | Iterations |
---|---|---|---|---|
15 | 0.95 | π/4 | 10 | 150 |
HSDBC parameters
P | R | Rzw | Ns | Theta | Initial points | Iterations |
---|---|---|---|---|---|---|
15 | 0.95 | 0.55 | 2 | π/4 | 10 | 150 |
Optimized gain values
Parameter | BFA | SDA | HSDBC | |
---|---|---|---|---|
Loop 1 | Kp _{ 1 } | 2.0729 | 2.3452 | 2.1566 |
Kd _{ 1 } | 0.8572 | 0.8714 | 0.8095 | |
Ki _{ 1 } | 1.3925 | 1.2778 | 1.2026 | |
Loop 2 | Kp _{ 2 } | 6.0155 | 5.1504 | 5.1530 |
Kd _{ 2 } | 2.8185 | 3.1264 | 2.6917 | |
Ki _{ 2 } | 1.6390 | 1.9794 | 1.8754 | |
Loop 3 | Kp _{ 3 } | 8.7514 | 11.3330 | 11.4514 |
Kd _{ 3 } | 8.1889 | 8.3229 | 8.9946 | |
Ki _{ 3 } | 0.2449 | 0.2731 | 0.3771 | |
Loop 4 | Kp _{ 4 } | 8.9718 | 9.8522 | 9.9903 |
Kd _{ 4 } | 5.1315 | 6.7829 | 6.6239 | |
Ki _{ 4 } | 0.0071 | 0.0532 | 0.0410 | |
Loop 5 | Kp _{ 5 } | 49.9646 | 36.5230 | 36.6753 |
Kd _{ 5 } | 13.6834 | 14.2519 | 14.3583 | |
Ki _{ 5 } | 4.0408 | 5.3567 | 5.4203 |
Cost functions
Minimum cost function | BFA | SDA | HSDBC |
---|---|---|---|
J | 0.3684 | 0.3685 | 0.3682 |
Conclusions
A novel hybrid spiral-dynamics bacteria-chemotaxis (HSDBC) optimization algorithm has been proposed. Chemotactic strategy of bacteria through spiral tumble and swim actions of bacteria is adopted to improve exploration strategy of SDA. Moreover, spiral radius and angular displacement of spiral model is made adaptive to enhance the movement of bacteria within feasible region. Incorporating these two schemes have successfully saved the SDA from getting trapped into local optima point and provides faster convergence. The proposed algorithm has been utilized to optimize the performance of two different IP platforms; single and double IP with a new configuration of an extended intermediate body. Simulation results have shown that the proposed hybrid algorithm outperformed its predecessor algorithms (BFA and SDA) in terms of increased convergence speed and better fitness accuracy. Furthermore, implementation of the HSDBC yielded significant saving in the energy consumption of the two tested platforms.
Future work will consider investigating standard PID tuning methods, such as Ziegler–Nichols method, and evaluating and comparing their performance with the HSDBC algorithm.
Declarations
Authors’ contributions
KG initiated the concept and developed the system of two-wheeled machine with extended rods. AN developed the HSDBC algorithm. AA and SA contributed to the modeling and simulation of the system. OT was overseeing the entire research and he gave technical insights in the control part. MA and TA helped with the editing of the final draft. Final editing of the manuscript is managed by KG. All authors read and approved the final manuscript.
Acknowledgements
The authors of this paper would like to thank Lincoln University in New Zealand by offering the funding support for this publication.
Competing interests
The authors declare that they have no competing interests.
Funding
This research is originally funded by a governmental PhD scholarships and research grants from various countries including New Zealand, Malaysia, Kuwait and the UK.
Publisher’s Note
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Authors’ Affiliations
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