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Southern Taiwan University Using GPC to predict the target position(2/5) (III) We calculated the distances between the former and current positions of the target, and we determined the speed V 0 of the target by using the sampling time T0. The equation below shows the calculations involved. d0 GF x G x 2 GF y G y 2 V0 d 0 / T0 1 DECISION AND CONTROL LAB Southern Taiwan University Using GPC to predict the target position(3/5) (IV) After calculating the velocity V 0 of the target, we designed a GPC by using V 0 , the direction of the target and the sampling time T0 , which was then used to find the subsequent position of the target GL x , GL y , as calculated by means of the equation below and illustrated in Figure 4. d G GL 2 G GL 2 0 x x y y GFy G y G y GL y GFx G x G x GLx 2 DECISION AND CONTROL LAB Southern Taiwan University Subsequent position of the target Fig.4. Subsequent position of the target. 3 DECISION AND CONTROL LAB Southern Taiwan University Using GPC to predict the target position(4/5) (V) We then calculated the time T1 needed for the robot to reach the target at its central speed V c , based on the distance (d ) that it had to cover to reach its current position. The equation below shows the calculations involved ( V L is the speed of the left wheel of the Robot, V R is the speed of the right wheel). VL VR Vc 2 T1 d / Vc 4 DECISION AND CONTROL LAB Southern Taiwan University Using GPC to predict the target position(5/5) (VI) If T1 is larger than T0 , then the robot followed to the next position of the target; if T1 is smaller than T0 , the robot proceeded to the current position of the target. By repeating steps I to VI, the target could be reached in less time. 5 DECISION AND CONTROL LAB Southern Taiwan University Motion fuzzy controller structure(1/4) In this part, we start with a design for a fuzzy logic controller (FLC) aimed at producing the velocities of the right and left wheels of the robot. Two input parameters of an FLC are distance（d）and angle（ψ）. d Robot goal Figure 5. Relationship between d and ψ 6 DECISION AND CONTROL LAB Southern Taiwan University Motion fuzzy controller structure(2/4) We represented d, ψ, VLL and VRR as e1, e2, y1 and y2, respectively. Afterwards, we set the values of variables e3 and e4 ( e1 = e3 and e2 = e4 ). We designed two fuzzy controllers to control the velocity of the right and left wheels of the robot. In the first fuzzy controller, e1 and e2 are used as the input variables and y1 as the output variable. In the other fuzzy controller, e3 and e4 are used as the input variables and y2 as the output variable. 7 DECISION AND CONTROL LAB Southern Taiwan University fuzzy rule table The fuzzy rules on which were based these fuzzy controllers are described in Table 1 and Table 2. Table 1. Fuzzy rule base of the Table 2. Fuzzy rule base of the right- left-wheel velocity fuzzy controller wheel velocity fuzzy controller e1 e3 y1 y2 NB NM NS Z PS PM PB NB NM NS Z PS PM PB MS NM NB NS NS NS Z Z NS NS NS NB NM NM NB y1( 3,3) y1( 2,3) y1( 1,3) y1( 0,3) y1(1,3) y1( 2,3) y1(3,3) NB y 2 ( 3,3) y 2 ( 2,3) y 2 ( 1,3) y 2 ( 0 ,3) y 2 (1,3) y2 ( 2 , 3) y 2 ( 3, 3 ) Z NM NM NM NS Z PM PM Z NS NM NM NM Z NM y1( 3, 2 ) y1( 2, 2 ) y1( 1, 2) y1( 0, 2 ) y1(1, 2 ) y1( 2, 2 ) y1(3, 2 ) NM y2 ( 3, 2 ) y2 ( 2, 2 ) y2 ( 1, 2) y2 ( 0 , 2 ) y2 (1, 2 ) y2 ( 2 , 2 ) y 2 ( 3, 2 ) Z NM NM Z NS Z PM PM Z NS Z NM NM Z NS y1( 3,1) y1( 2,1) y1( 1,1) y1( 0,1) y1(1,1) y1( 2,1) y1(3,1) NS y2 ( 3,1) y2 ( 2,1) y2 ( 1,1) y2 ( 0,1) y2 (1,1) y2 ( 2,1) y2 (3,1) PM Z NM PM Z PM PS e4 PS PM Z PM NM Z PM e2 Z y1( 3, 0 ) y1( 2, 0 ) y1( -1, 0 ) y1( 0, 0 ) y1(1, 0 ) y1( 2, 0 ) y1(3, 0 ) Z y2 ( 3, 0 ) y2 ( 2, 0 ) y2 ( -1, 0 ) y2 ( 0 , 0 ) y2 (1, 0 ) y2 ( 2 , 0 ) y 2 ( 3, 0 ) PS PM Z PS PM PS PB PB PS PM PS Z PM PS PS y1( 3, 1) y1( 2, -1) y1( 1, 1) y1( 0, 1) y1(1, 1) y1( 2, 1) y1(3, 1) PS y2 ( 3, 1) y2 ( 2, -1) y2 ( 1, 1) y2 ( 0, 1) y2 (1, 1) y2 ( 2, 1) y2 ( 3, 1) PM Z NM PB PM PS PB PB PS PM PB NM PS PM PM y1( 3, 2 ) y1( 2,-2) y1( 1, 2 ) y1( 0, 2 ) y1(1, 2 ) y1( 2, 2 ) y1(3, 2 ) PM y2 ( 3, 2 ) y2 ( 2,-2) y2 ( 1, 2 ) y2 ( 0, 2 ) y2 (1, 2 ) y2 ( 2, 2 ) y2 ( 3, 2 ) Z NM NS PB PM PS PB PB PM PB NS NM Z PB PB PS y1( 3, 3) y1( 2, -3) y1( 1, 3) y1( 0, 3) y1(1, 3) y1( 2, 3) y1(3, 3) y2 ( 3, 3) y2(-2,-3) y2 ( 1, 3) y2 ( 0, 3) y2 (1, 3) y2 ( 2, 3) y2 ( 3, 3) 8 DECISION AND CONTROL LAB Southern Taiwan University Motion fuzzy controller structure(3/4) Ry1 j1 , j2 : IF e1 is A1, j1 AND e2 is A 2, j2 , THEN y1 is y1 j1 , j2 j1 , j2 3, 2, 1,0,1, 2,3 Ry2 j3 , j4 : IF e3 is A 3, j3 AND e4 is A 4, j4 , THEN y2 is y2 j3 , j3 j3 , j4 3, 2, 1,0,1, 2,3 The following term sets were used to describe the fuzzy sets of each input and output fuzzy variables: T ei NB, NM , NS , Z , PS , PS , PM , PB , i 1,2,3,4 Ai ,3 , Ai ,2 , Ai ,1 , Ai ,0 , Ai ,1 , Ai ,2 , Ai ,3 T ym NB, NM , NS , Z , PS , PM , PB , m 1,2 y m,3 , y m,2 , y m,1 , y m,0 , y m,1 , y m,2 , y m,3 9 DECISION AND CONTROL LAB Southern Taiwan University membership function μAi(xi) NB NM NS Z PS PM PB 1 A(i,-3) A(i,-2) A(i,-1) A(i,0) A(i,1) A(i,2) A(i,3) μym(ym) NB NM NS Z PS PM PB y(m,-3) y(m,-2) y(m,-1) y(m,0) y(m,1) y(m,2) y(m,3) 1 a(i,-3) a(i,-2) a(i,-1) 0 0 0 a(i,1) a(i,2) a(i,3) xi b(m,-3) b(m,-2) b(m,-1) 0 b(m,1) b(m,2) b(m,3) ym (a) (b) Figure 6. Membership functions: (a) the fuzzy sets for ei; (b) the fuzzy sets for ym 10 DECISION AND CONTROL LAB Southern Taiwan University Motion fuzzy controller structure(4/4) Based on the weighted average method, the final outputs of these fuzzy controllers can be described by following equations. 3 3 3 3 y1 w y j1 , j2 1 j1 , j2 y2 w j3 3 j4 3 y j3 , j4 1 j3 , j4 j1 3 j2 3 where w(j , 1 j2) and w(j , 3 j4) were determined according the following equations. w j1 , j2 min A1, j e1 , A 2, j e2 1 2 w j3 , j4 min A3, j e3 , A 4, j e4 3 4 min e1 , A e2 min e3 , A e4 3 3 3 3 A1, j 2, j2 A 3, j 4, j4 1 3 j1 3 j2 3 j3 3 j4 3 11 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(1/7) The main function of the Ant Colony Algorithm is to solve problems in identifying the optimal path to be taken, which is similar to the problem faced by the robot. To this end, we devised the state equation for the robot. The moving velocity of the robot was calculated according to v r . The moving acceleration of the robot was calculated according to a r . 12 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(2/7) We defined the mathematical model of the equation of the robot’s movements as follows: r D r l x Vl _ x y V l_y m V r _ x n V r _ y q r l cos w r sin l p r r cos s r r sin r a r l al 13 DECISION AND CONTROL LAB Southern Taiwan University The state equation Having defined the state variables, x1 x, x2 y, x3 m, x4 n, x5 Vl _ x , x6 Vl _ y , x7 Vr _ x , x8 Vr _ y , x9 r , x10 l We were able to determine the state equation: x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 T x5 x6 x7 x8 r al cos r al sin r ar cos r ar sin ar al T Consequently, we identified the optimal vector of the velocity of the left wheel as GVl x5 x6 , and that of the right wheel as GVr x7 x8 14 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(3/7) After we set the fuzzy membership functions, we revised them by using possible questions contained in the ant colony algorithm. First, we set the membership functions a , a , a , a , a , a , a , a , a , a , a , a , 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 4,1 4,2 4,3 b , b , b and b , b , b , where a and b are the range of 1,1 1,2 1,3 2,1 2,2 2,3 the membership function of the fuzzy controller. 15 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(4/7) Therefore, the ants have to choose the optimal route among these three domains. We can explain this by means of Figure 7. ai ,2 Optimal solution ai ,3 ai ,1 Ant colony Figure 7. The form of fuzzy rules changed into route one 16 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(5/7) If we consider bi t as the number of ants at time t in rule i, the ants have to go to four rules, which constitute the optimal solution. As a result, we have i=1,2,3,4. However, in terms of the computations involved, i is used to express n the domain, m b t represents the total number of ants, i i 1 and dij is the geometric distance from rule i to rule j, which is called the density value of the trace at time t+1. 17 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(6/7) This creates the possibility of choosing among targets, which implies that it is possible for the ant to reach the next rule under the influence of visibility and pheromones. This possibility of choice is expressed by following equation. ij t ij if j N ik k pij t k ij t ij jNi 0 others We applied this equation to our system, and the main function of this system is to calculate the probability of a certain route being chosen by the robot. 18 DECISION AND CONTROL LAB Southern Taiwan University ACA-based FLC design method(7/7) The variable ij t 1 stands for the value of pheromones at time t+1 along the route from i to j. This is expressed in following equations. ij t 1 ij t ij t , t 1 m ij t , t 1 ij t , t 1 k k 1 19 DECISION AND CONTROL LAB Southern Taiwan University ACA used in obstacle avoidance(1/4) The ACA uses an adaptive pheromone updating strategy to ensure that the robot reaches the target in the shortest time and follows the best obstacle-avoidance path, as illustrated in Figure 8. (b x ,b y ) (R x1 , R y1 ) (R x ,R y) Figure 8. Obstacle-avoidance path 20 DECISION AND CONTROL LAB Southern Taiwan University ACA used in obstacle avoidance(2/4) Step 1: Parameter Initialization. At search time N=0, set a predetermined search time of NC. Generate m initial solutions at random. Posit that there are s initial solutions following path (i, j), the total length of which is L1 , L2 ,..., Ls . Finally, initialize the pheromones of path (i, j) by means of following equation, where Q is a constant. s Q ij (0) k 1 Lk 21 DECISION AND CONTROL LAB Southern Taiwan University ACA used in obstacle avoidance(3/4) Step 2: Iterative process. Calculate the distribution range of the ant colony concentration at starting point i according to following equation. s (i ) e ( r 1) 1 max s (i ) r m s (i ) ( al ) 2 l 1 r Then, calculate the probability of the path choice according to following equation. ij ij (t ) j Ak pij (t ) rAk ir (t )ir (t ) k a 0 other 22 DECISION AND CONTROL LAB Southern Taiwan University ACA used in obstacle avoidance(4/4) Step 3: Update the pheromone concentration for the path according to following equation. ij (t 1) 1 ij (t ) ij Step 4: Repeat Steps 2 and 3 until the ant reaches its target point. Step 5: Stop the iterative search when one in m ants has already completed its search for the path length and has exceeded the best path length of the previous iteration. Step 6: Make N=N+1, place the ant at the starting point, reset the target point at N<NC, and repeat Step 2. Otherwise, output the best path and stop the Algorithm. 23 DECISION AND CONTROL LAB Southern Taiwan University Simulation results(1/6) Figure 9. Use of SVM-FLC and ACA-FLC to control the speed of the robot 24 DECISION AND CONTROL LAB Southern Taiwan University Simulation results(2/6) Figure 10. Before using the GPC to Figure 11. Using the GPC to predict the next target position (x, y predict the next target position (x, coordinates: inch) y coordinates: inch) 25 DECISION AND CONTROL LAB Southern Taiwan University Simulation results(3/6) Figure 12. Simulation of obstacle- Figure 13. Simulation of obstacle- avoidance path of soccer robot by avoidance path of soccer robot by using MATLAB (x, y coordinates: using FIRA simulation inch) 26 DECISION AND CONTROL LAB Southern Taiwan University Simulation results(4/6) Figure 14. Using ACA-FLC to Figure 15. Using ACA-FLC to seek the path of the robot, in the seek the path of the robot, in the context of a MATLAB simulation context of a FIRA simulation (x, y coordinates: inch) 27 DECISION AND CONTROL LAB Southern Taiwan University Simulation results(5/6) Figure 16. Moving trick 28 DECISION AND CONTROL LAB Southern Taiwan University Simulation results(6/6) (a) (b) (c) Figure 17. Membership functions of (a) x1 and x3, (b) x2 and x4, and (c) y1 and y2, as determined by the proposed ACA-FLC method 29 DECISION AND CONTROL LAB Southern Taiwan University Conclusions The results of the experiment presented above show that the method we propose can be effectively applied to a wheeled robot, and the generalized predictive control function we designed can clarify the position of the target at the next sampling time. Also, we used the fuzzy ant colony algorithm to reduce the time required by a robot moving at top velocities to successfully find a path to its optimal target. 30 DECISION AND CONTROL LAB Southern Taiwan University Thanks for your attention! 31 DECISION AND CONTROL LAB

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posted: | 9/12/2012 |

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