A Taguchi-Based Heterogeneous Parallel Metaheuristic ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 4, AUGUST 2015

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A Taguchi-Based Heterogeneous Parallel Metaheuristic ACO-PSO and Its FPGA Realization to Optimal Polar-Space Locomotion Control of Four-Wheeled Redundant Mobile Robots Hsu-Chih Huang, Member, IEEE

Abstract—This paper presents an efficient Taguchi-based heterogeneous parallel metaheuristic ant colony optimization (ACO)-particle swarm optimization (PSO), called THPACOPSO, and its field-programmable gate array (FPGA) realization to optimal polar-space locomotion control of four-wheeled redundant mobile robots. This THPACOPSO consists of one slave Taguchi-based ACO (TACO) and one slave Taguchi-based PSO (TPSO) along with a master communication operator in one chip using system-on-a-programmable chip (SoPC) methodology. This approach takes the advantages of the Taguchi quality method, ACO, PSO, parallel processing, and FPGA technique, thereby obtaining better population diversity, avoiding premature convergence and keeping parallelism. Experimental results and comparative works are conducted to present effective optimization and high accuracy of the proposed FPGA-based THPACOPSO locomotion controller for four-wheeled Swedish redundant mobile robots. Index Terms—Field-programmable gate array (FPGA), metaheuristic, redundant control, system-on-a-programmable chip (SoPC).

I. I NTRODUCTION

M

ETAHEURISTIC algorithms are widely recognized as one of the efficient and practical approaches for many complex combinatorial optimization problems that are not tractable by deterministic methods [1]–[4]. This is due to the fact that most real-world optimizations are highly nonlinear and multimodal with various complex constraints. In this rapidly growing field of metaheuristics research, ant colony optimization (ACO) and particle swarm optimization (PSO) are currently gaining popularity for their increasing applications in solving engineering problems of diverse domains. They have been widely applied to many aspects in real-world application and have shown great success by exploiting their strong optimization ability [5]–[8]. In order to expedite the processing time and increase the searching diversity in sequential algorithms, the parallel ACOs

Manuscript received September 15, 2014; revised March 13, 2015; accepted May 18, 2015. Date of publication May 31, 2015; date of current version July 31, 2015. This work was supported by the Ministry of Science and Technology, Taiwan, under Grant MOST 103-2221-E-197-028. Paper no. TII-14-1031. The author is with the Department of Electrical Engineering, National Ilan University, Yilan 26047, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TII.2015.2440173

and parallel PSOs have been proposed to address the optimization problems in recent years [9]–[11]. Although the parallel or distributed metaheuristics have been proven useful in solving complex combinatorial problems, these individual algorithms have their strengths and restrictions. To date, there are several modified ACOs and PSOs proposed to improve the system performance [12]–[14]. However, the control parameters in these parallel and hybrid metaheuristic researches are usually set by an inefficient trial-and-error approach [9]–[14] and there has no attempt to develop a heterogeneous parallel metaheuristic algorithm using Taguchi quality method. Moreover, these personal computer (PC)-based metaheuristic algorithms [9]–[14] suffer from their high cost, so that they are not suitable for real-time embedded robotics applications. These disadvantages of the conventional metaheuristic paradigms can be circumvented by using the heterogeneous parallel processing, Taguchi method, and system-ona-programmable chip (SoPC) hardware/software codesign in field-programmable gate array (FPGA) chip. Taguchi quality method is a statistical method developed by Genichi Taguchi to improve the product quality and cost in industrial design [15]–[17]. On the other hand, FPGA-based metaheuristics is a new challenging field in computational intelligence. This computing transition from general purpose processors to embedded systems has been emerging as a new paradigm in terms of time-to-market, cost, implementation flexibility, and long-term maintenance [18]–[21]. To the authors’ best understanding, there has no attempt to designing intelligent FPGA-based Taguchi-based heterogeneous parallel metaheuristic ACO-PSO (THPACOPSO) optimal redundant controllers for four-wheeled mobile robots. Mobile robots with Swedish omnidirectional driving capability are superior to those with differential wheels in terms of dexterity and locomotion mechanism [22]–[24]. The fourwheeled mobile robots are expected to have better effective floor traction, even though at the expense of increased mechanical and control complexities. However, there exists a redundant control problem because more than three wheels provide redundancy [22]. Mathematical modeling and control of redundant fourwheeled omnidirectional mobile robots have been investigated by several researchers [25]–[28]. Overall, these studies applied more complex mechanisms of the mobile robots to resolve the

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Fig. 1. Architecture of the proposed metaheuristic THPACOPSO algorithm.

redundant problem in Cartesian space. These researches neither cope with the polar-space redundant control problem of four-wheeled mobile robots, nor address the control parameter optimization problem. Polar-space locomotion controllers of mobile robots have been proven superior to the Cartesian controllers in tracking the special trajectories, such as Archimedes’ spiral, rose curves, and Limacon of Pascal [29]–[33]. These trajectories are easily described in polar coordinates, but they are more complex in Cartesian space. The polar-space tracking problems for nonholonomic mobile robots have been investigated by several researchers [29]–[31]. There have been few studies related to holonomic three-wheeled polar-space omnidirectional mobile robot control [32], [33], in which the control parameters are obtained by a trial-and-error approach. However, to the authors’ best understanding, the polar-space redundant control problem of four-wheeled mobile robots remains open. The objective of this paper is to develop a pragmatic metaheuristic THPACOPSO to optimal polar-space locomotion control of four-wheeled redundant mobile robots to achieve trajectory tracking. This paper is organized as follows. In Section II, the THPACOPSO algorithm is proposed. Section III elaborates the FPGA realization to optimal polar-space THPACOPSO locomotion control of fourwheeled redundant mobile robots to achieve trajectory tracking. Section IV conducts experimental results and comparative works to show the performance and merit of the proposed methods. Section V concludes this paper. II. TAGUCHI -BASED H ETEROGENEOUS PARALLEL M ETAHEURISTIC ACO-PSO Fig. 1 presents the architecture of the proposed THPACOPSO which is a parallel processing strategy using both ACO and PSO with Taguchi-based control parameter tuners. This THPACOPSO consists of one Taguchi-based ACO (TACO) slave processor and one Taguchi-based PSO (TPSO) processor along with a communication master operator. The independent TACO and TPSO works in parallel for searching the optimal solutions more efficiently by using parallel architecture and Taguchi orthogonal array method. This heterogeneous parallel model is employed to not only circumvents the premature convergence problem in conventional

metaheuristics, but also expedite the computing in searching optimal solutions. In the proposed THPACOPSO, the two Taguchi-qualified slave processors concurrently executed with one master processor that aims at sharing information about the TACO and TPSO searching experiences. More precisely, the slave TACO and TPSO processor independently evolve new generations in parallel, and master processor is responsible for collecting and evaluating all the best solutions in slave processors. The master processor serves as a communication operator that is designed to allow the two slave processors to share their best solutions. As shown in Fig. 1, the searching spaces for both slaves are independent, and the solutions evolve separately for a certain number of generations, called communication step. After the communication step, the master processor finds the parallel global best (Gbest ), and then send it to each slave processor. The worst solution in each slave processor is then replaced by the received parallel global best (Gbest ). Compared with conventional hybrid and parallel computing methods, this THPACOPSO not only increases the searching diversity, but also decreases the probability to get stuck into a local optimum. The heterogeneous distributed metaheurictic algorithm employs more searching space and faster computing speed to search for a global optimum in the optimization problems. A. Taguchi-Based ACO TACO is a standard ACO tuned with Taguchi quality method using the orthogonal array and signal-to-noise ratio (SNR) techniques. Compared to conventional ACOs, this method has fewer experiments to obtain optimal ACO parameters in solving multiobjective optimization problems. 1) ACO for Solving Optimization Problems: The main idea of ACO is to model an optimization problem as the search for a minimum cost path in a graph consisting of nodes and edges. Each ant constructs a solution by making a sequence of local decisions which are guided by pheromone information and some problem-dependent heuristic information [5], [6]. The solutions to optimization problems can be expressed in terms of feasible paths on the ACO graph. The ACO aims at finding the best path with minimum cost and its corresponding optimal solution in a multiobjective optimization problem. If ant k is currently located at node i, it selects the next node j ∈ Nik , based on the transition probability ⎧ ⎨  [τij (t)]α [ηij ]β , if j ∈ Nik k [τil (t)]α [ηil ]β l∈N k (1) ϕ ij (t) = ⎩0, i if j ∈ / N k. i

τij is the pheromone trails on edge (i, j), Nik is the set of feasible nodes connected to node i, with respect to ant k. α and β are positive constants. The heuristic information ηij is a problem-dependent function to be minimized, given by ηij =

1 dij

(2)

where dij is the cost between the nodes i and j. The pheromone update rule is given by τij (t + 1) = (1 − ρ)τij (t) + Δτij (t)

(3)

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TABLE I ACO C ONTROL PARAMETERS AND THE L EVELS

with Δτij (t) =

m 

Δτ k ij (t)

(4)

k=1

where Δτ k ij (t) is the amount of pheromone deposited by ant k on link (i, j) at time step t, m denotes the number of ants, and 0 < ρ < 1 is the pheromone decay parameter. Δτ k ij (t) is given by Q , if kth ant uses edge(i, j) in its tour k Δτ ij (t) = Lk 0, otherwise

TABLE II TAGUCHI L25 (56 ) O RTHOGONAL A RRAY

where Q is a positive constant and Lk is the tour cost of the kth ant. This algorithm will terminate either when the maximum number of iteration is reached or an acceptable solution is found. 2) ACO Tuned With Taguchi Method: This section is devoted to developing a qualified ACO tuned with Taguchi method; this computing method resolves the control parameter optimization problem in the ACO algorithms. Taguchi method is an experimental method that can reduce the disturbance caused by randomness and can help determine the optimal parameters in the ACO. An orthogonal array is a fractional factorial matrix that provides a balanced comparison of levels of parameters or factors. This approach can provide an efficient way to obtain the optimal ACO parameters. SNR is used in Taguchi method to measure the quality of each experiment in the orthogonal array. This paper adopts the small-the-better characteristic and the SNR is given by SNR = −10 log

Nt 1  yi 2 Nt i=1

(5)

where Nt represents the number of repeated experiments and yi is the loss function for the ith experiment. With the orthogonal array and SNR, the control parameters in ACO can be properly set to obtain optimal performance only using a few experiments rather than the unreliable hand-tuned approach. In TACO, there are six control parameters to be optimized using Taguchi quality method, including three positive constants α, β, Q, number of generation (NACO ), number of artificial ants (m), and decay parameter (ρ). Table I lists the ACO control factors and their five levels, and Table II presents the L25 (56 ) orthogonal array of the TACO in this paper. The full factorial design method requires 56 = 15 625 experiments, while the Taguchi method needs only 25 experiments to obtain the approximate optimal values. This qualified TACO serves as a slave processor in the proposed metaheuristic THPACOPSO algorithm. B. Taguchi-Based PSO 1) PSO for Solving Optimization Problems: In PSO computing, each particle represents a feasible solution of the optimization problem and is evaluated by the problem-dependent fitness function. The particles are moving through a multidimensional search space, where the position of each particle is adjusted according to its own experience and that of its

neighbors [7], [8]. The position of the particle is changed by adding a velocity vi (t) to the current position, given by xi (t + 1) = xi (t) + vi (t + 1)

(6)

where xi (t) is the position of particle i in the search space at discrete time step t. The best position reached by the single particle is called gbest and the best location found by the rest of the swarm is called pbest . Each particle i renews its velocity using pbest and gbest , expressed by vij (t + 1) = wvij (t) + c1 ϕ1 (pbest − xij (t)) + c2 ϕ2 (gbest − xij (t))

(7)

where vij (t) is the velocity of particle i in dimension j at time step t, xij (t) is the position of particle i in j at time step t, c1 and c2 are positive acceleration constants. ϕ1 and ϕ2 are uniform random numbers with the range [0, 1], and w is called inertia weight. 2) PSO Tuned With Taguchi Quality Method: TPSO is one of the slave processors in the proposed THPACOPSO parallel paradigm. Similar to TACO, this TPSO is a PSO soft computing tuned with Taguchi method using orthogonal array and

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TABLE III PSO C ONTROL PARAMETERS AND T HEIR L EVELS

where

⎡

− sin(θ(t)) ⎢ − cos(θ(t)) P (θ(t)) = ⎢ ⎣ sin(θ(t)) cos(θ(t))

cos(θ(t)) − sin(θ(t)) − cos(θ(t)) sin(θ(t))

⎤ L L⎥ ⎥. L⎦ L

(9)

R denotes the radius of each motorized wheel; L represents the distance from the wheel’s center to the geometric center of the mobile robot; υi (t) and ωi (t), i = 1, 2, 3, 4, respectively, denote the linear and angular velocities of each Swedish wheel; [x(t) y(t) θ(t)]T represents the pose of the redundant mobile robot in Cartesian coordinates. In this redundant robotic system, the matrix P (θ(t)) in (9) is not square because there exists a redundancy. Although the matrix P (θ(t)) is singular for any θ, its left inverse matrix can be found by P # (θ(t))P (θ(t)) = I3 , where I3 is the 3 × 3 identity matrix. This pseudoinverse matrix is expressed by ⎡ − sin(θ(t)) − cos(θ(t)) sin(θ(t)) cos(θ(t)) ⎤ Fig. 2. Structure and geometry of the four-wheeled Swedish redundant mobile robot.

SNR techniques. There are six control parameters in the conventional PSOs, including the number of generations (NPSO ), swarm size (Ss ), neighborhood size (Ns ), and three positive constants w, c1, c2. Table III lists the control parameters and their five levels. Furthermore, the slave TPSO processor adopts the Taguchi L25 (56 ) orthogonal array in Table II to determine the qualified control parameters by using only 25 experiments. C. Communication Module In the proposed master–slave metaheuristic THPACOPSO computing, each slave processor is regarded as an independent agent. Each slave processor independently evolves new solutions in parallel, and the master processor is employed for collecting and evaluating the best solutions in slave processors to find the parallel global best (Gbest ), and then send it to each slave processor. Therefore, the worst solution in each slave processor is replaced by the received parallel global best (Gbest ). The master processor and two slave processors communicate with each other in a communication interval.

⎢ P # (θ(t)) = ⎢ ⎣

2

cos(θ(t)) 2 1 4L

2

2

2

− sin(θ(t)) − cos(θ(t)) sin(θ(t)) 2 2 2 1 4L

1 4L

1 4L

⎥ ⎥. ⎦ (10)

To express the kinematic model (8) in polar coordinates, we have the following transformation:

r(t) = x2 (t) + y 2 (t), x(t) = r(t) cos ϕ(t), y(t) = r(t) sin ϕ(t) (11) where r denotes the polar radius and ϕ represents the polar angle in polar coordinates. With the time derivative of (11) r(t) ˙ =

x(t)x(t) ˙ + y(t)y(t) ˙ x(t)x(t) ˙ + y(t)y(t) ˙

= r(t) x2 (t) + y 2 (t)

x(t) ˙ = r(t) ˙ cos ϕ(t) − r(t)ϕ(t) ˙ sin ϕ(t)

(12)

y(t) ˙ = r(t) ˙ sin ϕ(t) + r(t)ϕ(t) ˙ cos ϕ(t) one can combine (10), (11), and (12) to obtain the kinematic model of the Swedish four-wheeled mobile robot in polar coordinates. Thus, it follows from (12) that the time derivative of the polar radius is given by x(t)x(t) ˙ + y(t)y(t) ˙ r(t) x(t)r(t) ˙ cos ϕ(t) + y(t)r(t) ˙ sin ϕ(t) = r(t) = x(t) ˙ cos ϕ(t) + y(t) ˙ sin ϕ(t)  1 1 sin(ϕ(t) − θ(t))− cos(ϕ(t) − θ(t)) = 2 2

r(t) ˙ = III. FPGA-BASED THPACOPSO FOR R EDUNDANT C ONTROL OF M OBILE ROBOTS A. Kinematic Model in Polar Space Fig. 2 depicts the structure and geometry of the four-wheeled Swedish omnidirectional driving configuration with respect to a world frame in Cartesian coordinates. In what follows describes the kinematic model of this kind of redundant robot, where θ represents the vehicle orientation. The Cartesian-space kinematic model of the four-wheeled Swedish mobile robot is presented in [22]; the relationship between the wheel velocity vector and the vehicle velocity vector can be given by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Rω1 (t) V1 (t) x(t) ˙ ⎢ V2 (t) ⎥ ⎢ Rω2 (t) ⎥ ⎥ ⎢ ⎥ ⎣ ˙ ⎦ (8) υ(t) = ⎢ ⎣ V3 (t) ⎦ = ⎣ Rω3 (t) ⎦ = P (θ(t)) y(t) ˙ θ(t) V4 (t) Rω4 (t)

⎡

⎤ Rω1 (t) ⎢ Rω2 (t) ⎥ 1 1 ⎥ × sin(ϕ(t) − θ(t)) cos(ϕ(t) − θ(t)) ⎢ ⎣ Rω3 (t) ⎦ . 2 2 Rω4 (t) (13) Using the equality 

y(t) ˙ cos ϕ(t) − x(t) ˙ sin ϕ(t) = r(t)(sin2 ϕ(t) + cos2 ϕ(t))ϕ(t) ˙ = r(t)ϕ(t) ˙

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and (12) yields 1 (y(t) ˙ cos ϕ(t) − x(t) ˙ sin ϕ(t)) r(t)  1 1 1 = cos(ϕ(t) − θ(t)) sin(ϕ(t) − θ(t)) r(t) 2 2 ⎡ ⎤  Rω1 (t) ⎢ Rω2 (t) ⎥ 1 1 ⎥ cos(ϕ(t) − θ(t))− sin(ϕ(t) − θ(t)) ⎢ ⎣ Rω3 (t) ⎦ . 2 2 Rω4 (t) (14) Moreover, from (10), it is easy to obtain ⎡ ⎤ Rω1 (t) ⎥  ⎢ ˙ = 1 1 1 1 ⎢ Rω2 (t) ⎥ . θ(t) (15) 4L 4L 4L 4L ⎣ Rω (t) ⎦ 3 Rω4 (t)

ϕ(t) ˙ =

Combining (13), (14), and (15) gives the kinematic model of the four-wheeled Swedish redundant mobile robot in polar coordinates as follows: ⎡ ⎤ ⎡ ⎤ Rω1 (t) r(t) ˙ ⎢ Rω2 (t) ⎥ ⎥ ⎣ ϕ(t) ˙ ⎦ = T # (r(t), ϕ(t) − θ(t)) ⎢ (16) ⎣ Rω3 (t) ⎦ ˙ θ(t) Rω4 (t) where T # (r(t), ϕ(t) − θ(t)) is defined at the bottom of this page. The pseudoinverse matrix of T # (r(t), ϕ(t) − θ(t)) can be found as below T (r(t), ϕ(t) − θ(t)) ⎡ ⎤ sin(ϕ(t) − θ(t)) r(t) cos(ϕ(t) − θ(t)) L ⎢ − cos(ϕ(t) − θ(t)) r(t) sin(ϕ(t) − θ(t)) L ⎥ ⎢ ⎥ =⎢ ⎥. ⎣ − sin(ϕ(t) − θ(t)) −r(t) cos(ϕ(t) − θ(t)) L ⎦ cos(ϕ(t) − θ(t)) −r(t) sin(ϕ(t) − θ(t)) L Note that T # (θ(t))T (θ(t)) = I3 , where I3 is the 3 × 3 identity matrix. B. Pseudoinverse Tracking Control in Polar Coordinates With the polar-space kinematic model of the four-wheeled Swedish redundant mobile robot in (16), this section is devoted to developing a kinematic locomotion controller to achieve trajectory tracking for the Swedish wheeled mobile robot in Fig. 2. The trajectory control goal is to find the controlled velocity vec T tor V1 V2 V3 V4 to steer the redundant mobile robot from  T any starting pose r0 ϕ0 θ0 to track any given time-varying, T  smooth, and differentiable trajectory rd (t) ϕd (t) θd (t) . ⎡ ⎢ T # (r(t), ϕ(t) − θ(t)) = ⎢ ⎣

1 2

sin(ϕ(t) − θ(t))

1 2r(t)

cos(ϕ(t) − θ(t)) 1 4L

The current pose of the Swedish wheeled mobile robot is rep T resented by r(t) ϕ(t) θ(t) and the tracking error vector of the redundant robot is expressed by ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ r(t) rd (t) re (t) ⎣ ϕe (t) ⎦ = ⎣ ϕ(t) ⎦ − ⎣ ϕd (t) ⎦ . (17) θ(t) θe (t) θd (t) Thus, one obtains ⎤ ⎡ ⎡ ⎡ ⎤ ⎤ Rω1 (t) r˙e (t) r˙d (t) ⎥ ⎢ (t) Rω 2 ⎥ ⎣ ⎣ ϕ˙ e (t) ⎦ = T # (r(t), ϕ(t) − θ(t)) ⎢ ⎦ ⎣ Rω3 (t) ⎦ − ϕ˙ d (t) . ˙θe (t) ˙θd (t) Rω4 (t) (18) The control aim is therefore  to find a set of control T output ω1 (t) ω2 (t) ω3 (t) ω4 (t) that makes the closedloop system asymptotically stable. In doing so, the following pseudoinverse redundant control law is proposed; the gain matrices KP and KI are symmetric and positive definite, meaning that KP = diag{Kp1 , Kp2 , Kp3 } = KPT > 0, KI = diag{Ki1 , Ki2 , Ki3 } = KIT > 0 ⎡ ⎤ ω1 (t) ⎢ ω2 (t) ⎥ 1 ⎢ ⎥ ⎣ ω3 (t) ⎦ = R T (r(t), θ(t) − ϕ(t)) ω4 (t) ⎛ ⎞ ⎤ ⎡ t ⎡ ⎤ ⎡ ⎤ re (τ )dτ 0 r˙d (t) ⎟ re (t) ⎜ ⎥ ⎢ ⎣ ϕe (t) ⎦ − KI ⎢ t ϕe (τ )dτ ⎥ + ⎣ ϕ˙ d (t) ⎦⎟ . −K ×⎜ p ⎝ ⎠ ⎦ ⎣ 0 θe (t) t θ˙d (t) θ (τ )dτ 0 e (19) Substituting (19) into (18) leads to the underlying closedloop error system governed by ⎤ ⎡ t ⎡ ⎤ ⎡ ⎤ r (τ )dτ 0 e r˙e (t) re (t) ⎥ ⎢ ⎣ ϕ˙ e (t) ⎦ = −Kp ⎣ ϕe (t) ⎦ − KI ⎢ t ϕe (τ )dτ ⎥ (20) ⎦ ⎣ 0 θe (t) t θ˙e (t) θ (τ )dτ 0 e where the globally asymptotical stability of the closed-loop error system (20) can be easily proven by selecting the quadratic Lyapunov function ⎤ ⎡ re (t)   1 re (t) ϕe (t) θe (t) ⎣ ϕe (t) ⎦ V (t) = 2 θe (t)   t t 1 t + re (τ )dτ 0 ϕe (τ )dτ 0 θe (τ )dτ 0 2 ⎤ ⎡ t r (τ )dτ 0 e ⎥ ⎢t ⎥ (21) × KI ⎢ ⎣ 0 ϕe (τ )dτ ⎦ . t θ (τ )dτ 0 e

− 12 cos(ϕ(t) − θ(t)) 1 2r(t)

− 21 sin(ϕ(t) − θ(t))

1 2

cos(ϕ(t) − θ(t))

⎤

⎥ 1 1 sin(ϕ(t) − θ(t)) − 2r(t) cos(ϕ(t) − θ(t)) − 2r(t) sin(ϕ(t) − θ(t)) ⎥ ⎦. 1 4L

1 4L

1 4L

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Hence, for the robot’s polar-space kinematic model (16) with the desired smooth and differentiable trajectory T  rd (t) ϕd (t) θd (t) and the trajectory tracking control law (19), the Swedish mobile robot can be steered to exactly follow the trajectory in the sense of globally asymptotical stability, i.e., r(t) → rd (t), ϕ(t) → ϕd (t), and θ(t) → θd (t) as t → ∞. C. Optimal THPACOPSO Locomotion Controller Although the polar-space locomotion controller for the fourwheeled Swedish redundant mobile robots was synthesized in (19) using the pseudoinverse control method, the two control matrices KP and KI were not optimally set to ensure system stability and obtain optimal performance. This section aims at developing an efficient THPACOPSO optimal controller for the four-wheeled Swedish mobile robots. The optimal control parameters KP = diag{kp1 , kp2 , kp3 } and KI = diag{ki1 , ki2 , ki3 } are evolved via the THPACOPSO process to achieve trajectory tracking. In the proposed metaheuristic THPACOPSO for solving the polar-space optimal redundant control problem of mobile robots, the ACO path of the TACO slave processor is defined by Path = {kp1 , kp2 , kp3 , ki1 , ki2 , ki3 }, and the particle in the slave TPSO processor is expressed by Particle = {kp1 , kp2 , kp3 , ki1 , ki2 , ki3 }. The optimal solutions (control parameters) with minimum fitness value KP = diag{k ∗ p1 , k ∗ p2 , k ∗ p3 } and KI = diag{k ∗ i1 , k ∗ i2 , k ∗ i3 } are evolved by using the THPACOPSO evolution process described in Section II to solve the robotic optimal redundant problem. The fitness function of the proposed THPACOPSO in polar coordinates can be defined by the integral square error (ISE)  t  2  re (τ ) + ϕe 2 (τ ) + θe 2 (τ ) dτ (22) ISE = 0

where re and ϕe are the position errors in polar coordinates and θe is the error of vehicle’s orientation. The SNR of the TACO and TPSO in the proposed THPACOPSO redundant controller is defined by SNR = −10 log(ISE).

(23)

D. FPGA Implementation Fig. 3 depicts the architecture of the Altera FPGA implementation for the proposed polar-space THPACOPSO redundant locomotion controller using SoPC technology and hardware/ software codesign technique. There are four very high speed integrated circuit (VHSIC) hardware description language (VHDL) hardware intellectual property (IP) cores developed for the four-wheeled mobile robot, including quadratureencoder-pulse (QEP), pulse width modulation (PWM), communication module, and digital filter. The QEPs aim at calculating the posture of the mobile robot from the encoders mounted on the dc motors. The PWM is a technique used to encode a message into a pulsing signal for steering the dc motors. These components are interconnected by means of the interconnection network.  T of The current position and orientation r(t) ϕ(t) θ(t) the Swedish redundant mobile robot in polar coordinates can

Fig. 3. FPGA realization of the proposed THPACOPSO redundant locomotion controller.

be directly dead-reckoned by the embedded central processing unit (CPU) via the obtained QEP signals from the motor encoders. With this QEP information, the proposed polar-space THPACOPSO optimal locomotion controller determine the outT  to the PWM module, put ω = ω1 (t) ω2 (t) ω3 (t) ω4 (t) which conforms the pulse duration and then steers the four DC brushless motors on the four Swedish omnidirectional wheels to achieve trajectory tracking. IV. E XPERIMENTAL R ESULTS AND D ISCUSSIONS The aims of this section are to examine the effectiveness and performance of the proposed polar-space THPACOPSO optimal redundant control law (19) to the four-wheeled Swedish omnidirectional mobile robot. This heterogeneous parallelization strategy increases the searching diversity for this optimal redundant problem. A. System Architecture of the Experimental Swedish FourWheeled Mobile Robot Fig. 4 presents the picture of the experimental mobile robot. The four Swedish omnidirectional wheels are driven by four dc brushless servomotors with four mounted rotary encoders to provide QEP information and then achieve dead-reckoning of the redundant mobile robot. The proposed metaheuristic THPACOPSO tuning method and polar-space control law are realized using C/C++ code in the embedded processors incorporating with the robotics custom logic IPs. In this study, the FPGA chip integrated the embedded processors, real-time operating system (RTOS), lightweight IP (lwIP), and VHDL-based IPs for performing the THPACOPSO optimal control law of the Swedish mobile robot. B. Polar-Space Limacon of Pascal Trajectory Tracking This experimental result was conducted to explore how the proposed embedded THPACOPSO polar-space locomotion controller (19) steers the Swedish mobile robot to track the Limacon of Pascal trajectory expressed by rd (t) = 2 + 1.5 cos ϕd (t) (m), ϕd (t) = 0.2t (rad), and θd (t) = π/4 (rad).

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Fig. 6. Evolutions of performance index for the THPACOPSO and conventional Taguchi controllers. Fig. 4. Picture of the experimental four-wheeled redundant mobile robot.

Fig. 7. Evolutions of performance index for the FPGA-based THPACOPSO, hybrid, and parallel controllers.

V. C ONCLUSION

Fig. 5. Experimental result of the Limacon of Pascal trajectory tracking.

Fig. 5 depicts the experimental results in which the initial poses are inside and outside the reference Limacon of Pascal trajectory. Moreover, Fig. 6 depicts the evolutions of performance index of the proposed THPACOPSO and conventional Taguchibased paradigms to achieve this trajectory tracking. These results indicate that the proposed THPACOPSO polar-space locomotion controller (19) is capable of successfully steering the Swedish mobile robot to track this special trajectory. In order to exhibit the merit of the proposed polar-space THPACOPSO tuning approach over conventional hybrid and parallel approaches, Fig. 7 presents the evolutions of performance index for the proposed SoPC-based THPACOPSO optimal redundant controller and the two conventional controllers to achieve this Limacon of Pascal trajectory tracking. As shown in Fig. 7, the proposed FPGA-based THPACOPSO computational intelligence converges to the optimal solution with better performance index.

This paper has presented an efficient THPACOPSO and its FPGA realization to optimal polar-space locomotion control of four-wheeled redundant mobile robots. Based on the polar-space kinematic model, the THPACOPSO optimal locomotion controller has been synthesized. Through experimental results, the proposed THPACOPSO-based optimal locomotion controller has been shown to achieve trajectory tracking successfully. These results indicate that the proposed FPGA-based THPACOPSO polar-space redundant controller outperforms the conventional controllers. R EFERENCES [1] G. Tambouratzis, “Using an ant colony metaheuristic to optimize automatic word segmentation for ancient Greek,” IEEE Trans. Evol. Comput., vol. 13, no. 4, pp. 742–753, Aug. 2009. [2] X. Zhang, S. Hu, D. Chen, and X. Li, “Fast covariance matching with fuzzy genetic algorithm,” IEEE Trans. Ind. Informat., vol. 8, no. 1, pp. 148–157, Feb. 2012. [3] H. C. Huang, “Intelligent motion control for four-wheeled holonomic mobile robots using FPGA-based artificial immune system algorithm,” Adv. Mech. Eng., vol. 2013, pp. 1–11, 2013. [4] S. W. Lin and S. C. Chen, “Parameter tuning, feature selection and weight assignment of features for case-based reasoning by artificial immune system,” Appl. Soft Comput., vol. 11, pp. 5042–5052, 2011. [5] C. H. Hsu and C. F. Jung, “Multi-objective continuous-ant-colonyoptimized FC for robot wall-following control,” IEEE Comput. Intell. Mag., vol. 8, no. 3, pp. 28–40, Aug. 2013.

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Hsu-Chih Huang (S’08–M’09) received the M.S. degree in system on a programmable chip and embedded systems from the Institute of Biomedical Engineering, National Cheng-Kung University, Tainan, Taiwan, and the Ph.D. degree in electrical engineering from National Chung-Hsing University, Taichung, Taiwan, in 1999 and 2009, respectively. Currently, he is an Associate Professor with the Department of Electrical Engineering, National Ilan University, Yilan, Taiwan. His research interests include intelligent control, mobile robots, embedded systems, SoPC, and nonlinear control.