Robotics 2010 Current and future challenges Part 9 docx

where µ denotes the friction coefficient, and F N denotes the normal force applied on the inter- nal surface of the pipe by the robot’s wheels. Therefore, the resisting torque due to the internal friction can be obtained from the following equation: T f = ΓµbF N + K f 1 ˙ φ + K f 2 ˙ θ (13) One should note that in (13) : 1. ΓµbF N models the Coulomb friction applied to the hub acting on the wheels. 2. K f 1 ˙ φ and K f 2 ˙ θ model the viscous friction on the hub and the wheels, respectively. From (2), the angular velocities of the hub and the wheels, namely ˙ φ and ˙ θ are related. There- fore one can write; T f = ΓµbF N + K f ˙ φ (14) where : K f = K f 1 + b + r rC δ K f 2 (15) The hydrodynamic drag force induced by the flow on the robot, projected onto the generalized coordinate φ, can be expressed as follows: T D = bS δ ρC d A 2 ( (b + r) ˙ φS δ + ν ) 2 (16) where ρ, A, ν and C d are as listed in Table 1. One should note that in (16): 1. The effect of the rotational motion of the robot on the drag coefficient is not considered, therefore, the drag coefficient is assumed to remain constant as the robot moves. 2. Drag force on the wheels is negligible. By substituting (14) and (16) in (11), the generalized force Q will be computed as: Q = T m − ΓµbF N − K f ˙ φ − bS δ ρC d A 2 ( (b + r) ˙ θS δ + ν ) 2 (17) Using (17) and substituting T and V from (6) and (9) into (10), the following closed form solution in form of a nonlinear 2 nd -order differential equation for the wheels motion (and correspondingly the robot motion) can be obtained: ¨ φ = T m − f ( ˙ φ, ν ) − a 1 a 2 + a 3 + I B (18) where: ⎧      ⎨      ⎩ f ( ˙ φ, ν ) = K f ˙ φ + bS δ ρC d A 2 ( (b + r) ˙ φS δ + ν ) 2 a 1 = ΓµbF N + (M m + M h + Γm)(b + r)g tan(δ) a 2 = (M m + M h + Γm + Γ I WX r 2 ) ( (b + r)tan(δ) ) 2 a 3 = (m + I WZ r 2 )Γb 2 (19) From (18), one can realize that the motion of the robot can be controlled by changing parame- ters such as the wheel inclination, δ the normal force exerted on the pipe wall via the wheels, F N , and the torque applied to the wheels actuators, T m . The only control input that can vary on the fly in our design is the motor torque, namely T m . How to manipulate this torque in order to maintain a constant speed of motion when the robot is subjected to flow disturbances (i.e., variation in the flow speed, ν) will be discussed in section 5. where T Motor , T Hull and T AW denote kinetic energies of the motor, hull and the angled wheels, respectively, and Γ denotes the number of angled (active) wheels. In (4), the kinetic energy of the passive straight wheels is disregarded. T Motor , T Hull and T AW can be calculated as: T Motor = 1 2 M m ˙ z 2 T Hull = 1 2 M h ˙ z 2 + 1 2 I B ˙ φ 2 (5) T AW =  (mr 2 + I WZ )  bC δ b+r  2 + (mr 2 + I WX )S 2 δ  ˙ θ 2 In (5), S δ and C δ represent the short form of sin(δ) and cos(δ) , respectively. Considering (1) and (5) the total kinetic energy of the system can be written as: T = 1 2   (b + r) S δ C δ  2 α M + Γb 2 α m + I B  ˙ φ 2 (6) where:  α M = (M m + M h + Γm + Γ I WX r 2 ) α m = (m + I WZ r 2 ) (7) An infinitesimal change in the potential energy of the robot due to the gravity when moving in a vertical pipe can be calculated as: dV = (M m + N h + Γm)gdz (8) After substituting eqn. (1) in (8) one gets: dV = (M m + M h + Γm)(b + r)gdφ tan(δ) (9) Considering the angle of rotation of the hull φ as the only generalized coordinate in the La- grange formulation, one can write: d dt  ∂L ∂ ˙ φ i  − ∂L ∂φ = Q (10) The generalized force Q applied on the robot moving inside the pipe is given by: Q = T m − T f − T D (11) where the right hand side of the above equation represents the non-potential generalized torques such as the electromechanical torque generated by the motor, T m , the resisting torques due to the friction between the wheels and their axles T f , and the resisting torque due to hy- drodynamic drag force posed on the system T D all projected onto the generalized coordinate, φ. Friction plays a significant role in creating the motion of the robot. Insufficient friction at the point-of-contact between the wheels and the pipe wall leads to wheel slippage. The slippage constraint of a wheel is expressed as (using Coulomb friction law): F T ≤ µF N (12) where µ denotes the friction coefficient, and F N denotes the normal force applied on the inter- nal surface of the pipe by the robot’s wheels. Therefore, the resisting torque due to the internal friction can be obtained from the following equation: T f = ΓµbF N + K f 1 ˙ φ + K f 2 ˙ θ (13) One should note that in (13) : 1. ΓµbF N models the Coulomb friction applied to the hub acting on the wheels. 2. K f 1 ˙ φ and K f 2 ˙ θ model the viscous friction on the hub and the wheels, respectively. From (2), the angular velocities of the hub and the wheels, namely ˙ φ and ˙ θ are related. There- fore one can write; T f = ΓµbF N + K f ˙ φ (14) where : K f = K f 1 + b + r rC δ K f 2 (15) The hydrodynamic drag force induced by the flow on the robot, projected onto the generalized coordinate φ, can be expressed as follows: T D = bS δ ρC d A 2 ( (b + r) ˙ φS δ + ν ) 2 (16) where ρ, A, ν and C d are as listed in Table 1. One should note that in (16): 1. The effect of the rotational motion of the robot on the drag coefficient is not considered, therefore, the drag coefficient is assumed to remain constant as the robot moves. 2. Drag force on the wheels is negligible. By substituting (14) and (16) in (11), the generalized force Q will be computed as: Q = T m − ΓµbF N − K f ˙ φ − bS δ ρC d A 2 ( (b + r) ˙ θS δ + ν ) 2 (17) Using (17) and substituting T and V from (6) and (9) into (10), the following closed form solution in form of a nonlinear 2 nd -order differential equation for the wheels motion (and correspondingly the robot motion) can be obtained: ¨ φ = T m − f ( ˙ φ, ν ) − a 1 a 2 + a 3 + I B (18) where: ⎧      ⎨      ⎩ f ( ˙ φ, ν ) = K f ˙ φ + bS δ ρC d A 2 ( (b + r) ˙ φS δ + ν ) 2 a 1 = ΓµbF N + (M m + M h + Γm)(b + r)g tan(δ) a 2 = (M m + M h + Γm + Γ I WX r 2 ) ( (b + r)tan(δ) ) 2 a 3 = (m + I WZ r 2 )Γb 2 (19) From (18), one can realize that the motion of the robot can be controlled by changing parame- ters such as the wheel inclination, δ the normal force exerted on the pipe wall via the wheels, F N , and the torque applied to the wheels actuators, T m . The only control input that can vary on the fly in our design is the motor torque, namely T m . How to manipulate this torque in order to maintain a constant speed of motion when the robot is subjected to flow disturbances (i.e., variation in the flow speed, ν) will be discussed in section 5. where T Motor , T Hull and T AW denote kinetic energies of the motor, hull and the angled wheels, respectively, and Γ denotes the number of angled (active) wheels. In (4), the kinetic energy of the passive straight wheels is disregarded. T Motor , T Hull and T AW can be calculated as: T Motor = 1 2 M m ˙ z 2 T Hull = 1 2 M h ˙ z 2 + 1 2 I B ˙ φ 2 (5) T AW =  (mr 2 + I WZ )  bC δ b+r  2 + (mr 2 + I WX )S 2 δ  ˙ θ 2 In (5), S δ and C δ represent the short form of sin(δ) and cos(δ) , respectively. Considering (1) and (5) the total kinetic energy of the system can be written as: T = 1 2   (b + r) S δ C δ  2 α M + Γb 2 α m + I B  ˙ φ 2 (6) where:  α M = (M m + M h + Γm + Γ I WX r 2 ) α m = (m + I WZ r 2 ) (7) An infinitesimal change in the potential energy of the robot due to the gravity when moving in a vertical pipe can be calculated as: dV = (M m + N h + Γm)gdz (8) After substituting eqn. (1) in (8) one gets: dV = (M m + M h + Γm)(b + r)gdφ tan(δ) (9) Considering the angle of rotation of the hull φ as the only generalized coordinate in the La- grange formulation, one can write: d dt  ∂L ∂ ˙ φ i  − ∂L ∂φ = Q (10) The generalized force Q applied on the robot moving inside the pipe is given by: Q = T m − T f − T D (11) where the right hand side of the above equation represents the non-potential generalized torques such as the electromechanical torque generated by the motor, T m , the resisting torques due to the friction between the wheels and their axles T f , and the resisting torque due to hy- drodynamic drag force posed on the system T D all projected onto the generalized coordinate, φ. Friction plays a significant role in creating the motion of the robot. Insufficient friction at the point-of-contact between the wheels and the pipe wall leads to wheel slippage. The slippage constraint of a wheel is expressed as (using Coulomb friction law): F T ≤ µF N (12) ∙ The accommodation of plant dynamics; The AI applications in the design and implementation of automatic control systems have been broadly described as ”intelligent control”. Such decision-making is inevitably autonomous and should result in improved overall performance over time. In this context, a neural- network-based fuzzy logic control strategy has been adopted in our system. The rational for this selection is that a precise linear dynamic model of our pipe crawler cannot be obtained. FLC’s incorporate heuristic control knowledge in the form of ”IF-THEN” rules and are a con- venient choice when a precise linear dynamic model of the system to be controlled cannot be easily obtained. Furthermore, FLC’s have also shown a good degree of robustness in face of large variability and uncertainty in the system parameters (Wang, 1994),(Dimeo & Lee, 1995). An ANN can learn fuzzy rules from I/O data, incorporate prior knowledge of fuzzy rules, fine tune the membership functions and act as a self learning fuzzy controller by automatically generat- ing the fuzzy rules needed (Jang, 1993). This capability of the NN was utilized to form an FL-based controller based on data obtained via Human-In-The-Loop (HITL) simulator. 5.2.1 Structure of the FLC The rule-base of the proposed FLC contains rules of first order TSK type (Takagi & Sugeno, 1985). In our proposed FLC the two inputs to the controller are error in linear velocity of the robot e (t) and the rate of change in the error ˙ e(t) as follows: { e (t) = ˙ Z set − ˙ z (t); ˙ e (t) = − ¨ z (t); (21) where ” ˙ Z set ” is the set-point in velocity. The controller output is the voltage applied to the DC motor of the hub, namely v (t). The rationale for this selection of the input variables is that, intuitively speaking, human makes a decision about the value of v (t) based on a visual feedback (detailed under human-in-the-loop simulator) of the change of the velocity of the robot (i.e. e (t)) and the rate of this change (i.e. ˙ e(t)). This FLC adjusts the control variable, namely the input voltage provided to the hub’s actuator in order to maintain a constant speed in the robot when subjected to flow disturbances. The structure of ANFIS model implemented is based on : ∙ A first order TSK fuzzy model where the consequent part of the fuzzy IF-THEN rules is first order in terms of the premise parameters; ∙ To performs fuzzy ”AND”, algebraic ”minimum” is manipulated as the T-norm ; ∙ To performs fuzzy ”OR”, algebraic ”maximum” is manipulated as the T-norm ; ∙ Three sets of product-of-two-sigmoidal MF’s on each input were implemented. These MF’s are depicted in Fig. 4 and are represented by : f (x; q) = 1 1 + e −a 1 (x−c 1 ) × 1 1 + e −a 2 (x−c 2 ) (22) where q = [a 1 , a 2 ,c 1 ,c 2 ]. 4.3 Motor Dynamics The dynamics of a permanent magnet DC motor is represented by : T m = K m i a di a dt = − R L i a (t) − e b L + 1 L v app (t) (20) e b (t) = K b ˙ φ (t) where T m is the mechanical torque generated by the motor, e b is the back EMF of the motor and i a is the armature current. Here v app is the input voltage (i.e., the control variable) and i a denotes the armature current. In (20) it is assumed that the DC motor is not geared (i.e., direct drive). 5. Controller Design The primary objective of a controller is to provide appropriate inputs to a plant to obtain some desired output. In this research, the controller strives to balance hydrodynamic forces exerted on the robot due to the flow disturbances while maintaining a constant speed for the robot. Two sets of disturbance models in the form of step and also sinusoidal changes in flow velocity were generated randomly in a simulated environment. The controller tracks the response of the system to its user defined velocity set-point ˙ Z set and sends a correction command in terms of the input voltage provided to the DC motor actuators. We compare the behavior of two controllers in this research: a conventional PID controller and a fuzzy logic controller (FLC) trained using adaptive network-based fuzzy inference system (ANFIS) algorithm. ANFIS generates a fuzzy inference system (FIS) that is in essence a complete fuzzy model based on data obtained from an operator through real-time HITL virtual reality simulator to tune the parameters of the FLC. More specifically parameters that define the membership functions on the inputs to the system and those that define the output of our system. 5.1 Servomechanism Problem The servomechanism problem is one the most elementary problems in the field of automatic control, where it is desired to design a controller for the plant which satisfies the following two criteria for the system while maintaining closed-loop stability: 1.Regulation : The outputs are independent of the disturbances affecting the system. 2.Tracking :The outputs asymptotically track a referenced input signal applied to the system. The controller’s objective is to maintain a constant linear speed in robot’s motion in the pres- ence of disturbances. In general, robot’s motion can be regulated by either changing the nor- mal force F N exerted on the pipe’s wall via robot’s wheels, changing active wheels’ inclination angle δ offline, or by changing the input voltage provided to the DC motor on fly. The latter is adopted as the control variable. 5.2 Fuzzy Logic Control : An Overview Recently, researchers have been exploiting Artificial Intelligence (AI) techniques to address the following two major issues where conventional control techniques still require improvement: ∙ Accuracy of nonlinear system modeling; ∙ The accommodation of plant dynamics; The AI applications in the design and implementation of automatic control systems have been broadly described as ”intelligent control”. Such decision-making is inevitably autonomous and should result in improved overall performance over time. In this context, a neural- network-based fuzzy logic control strategy has been adopted in our system. The rational for this selection is that a precise linear dynamic model of our pipe crawler cannot be obtained. FLC’s incorporate heuristic control knowledge in the form of ”IF-THEN” rules and are a con- venient choice when a precise linear dynamic model of the system to be controlled cannot be easily obtained. Furthermore, FLC’s have also shown a good degree of robustness in face of large variability and uncertainty in the system parameters (Wang, 1994),(Dimeo & Lee, 1995). An ANN can learn fuzzy rules from I/O data, incorporate prior knowledge of fuzzy rules, fine tune the membership functions and act as a self learning fuzzy controller by automatically generat- ing the fuzzy rules needed (Jang, 1993). This capability of the NN was utilized to form an FL-based controller based on data obtained via Human-In-The-Loop (HITL) simulator. 5.2.1 Structure of the FLC The rule-base of the proposed FLC contains rules of first order TSK type (Takagi & Sugeno, 1985). In our proposed FLC the two inputs to the controller are error in linear velocity of the robot e (t) and the rate of change in the error ˙ e(t) as follows: { e (t) = ˙ Z set − ˙ z (t); ˙ e (t) = − ¨ z (t); (21) where ” ˙ Z set ” is the set-point in velocity. The controller output is the voltage applied to the DC motor of the hub, namely v (t). The rationale for this selection of the input variables is that, intuitively speaking, human makes a decision about the value of v (t) based on a visual feedback (detailed under human-in-the-loop simulator) of the change of the velocity of the robot (i.e. e (t)) and the rate of this change (i.e. ˙ e(t)). This FLC adjusts the control variable, namely the input voltage provided to the hub’s actuator in order to maintain a constant speed in the robot when subjected to flow disturbances. The structure of ANFIS model implemented is based on : ∙ A first order TSK fuzzy model where the consequent part of the fuzzy IF-THEN rules is first order in terms of the premise parameters; ∙ To performs fuzzy ”AND”, algebraic ”minimum” is manipulated as the T-norm ; ∙ To performs fuzzy ”OR”, algebraic ”maximum” is manipulated as the T-norm ; ∙ Three sets of product-of-two-sigmoidal MF’s on each input were implemented. These MF’s are depicted in Fig. 4 and are represented by : f (x; q) = 1 1 + e −a 1 (x−c 1 ) × 1 1 + e −a 2 (x−c 2 ) (22) where q = [a 1 , a 2 ,c 1 ,c 2 ]. 4.3 Motor Dynamics The dynamics of a permanent magnet DC motor is represented by : T m = K m i a di a dt = − R L i a (t) − e b L + 1 L v app (t) (20) e b (t) = K b ˙ φ (t) where T m is the mechanical torque generated by the motor, e b is the back EMF of the motor and i a is the armature current. Here v app is the input voltage (i.e., the control variable) and i a denotes the armature current. In (20) it is assumed that the DC motor is not geared (i.e., direct drive). 5. Controller Design The primary objective of a controller is to provide appropriate inputs to a plant to obtain some desired output. In this research, the controller strives to balance hydrodynamic forces exerted on the robot due to the flow disturbances while maintaining a constant speed for the robot. Two sets of disturbance models in the form of step and also sinusoidal changes in flow velocity were generated randomly in a simulated environment. The controller tracks the response of the system to its user defined velocity set-point ˙ Z set and sends a correction command in terms of the input voltage provided to the DC motor actuators. We compare the behavior of two controllers in this research: a conventional PID controller and a fuzzy logic controller (FLC) trained using adaptive network-based fuzzy inference system (ANFIS) algorithm. ANFIS generates a fuzzy inference system (FIS) that is in essence a complete fuzzy model based on data obtained from an operator through real-time HITL virtual reality simulator to tune the parameters of the FLC. More specifically parameters that define the membership functions on the inputs to the system and those that define the output of our system. 5.1 Servomechanism Problem The servomechanism problem is one the most elementary problems in the field of automatic control, where it is desired to design a controller for the plant which satisfies the following two criteria for the system while maintaining closed-loop stability: 1.Regulation : The outputs are independent of the disturbances affecting the system. 2.Tracking :The outputs asymptotically track a referenced input signal applied to the system. The controller’s objective is to maintain a constant linear speed in robot’s motion in the pres- ence of disturbances. In general, robot’s motion can be regulated by either changing the nor- mal force F N exerted on the pipe’s wall via robot’s wheels, changing active wheels’ inclination angle δ offline, or by changing the input voltage provided to the DC motor on fly. The latter is adopted as the control variable. 5.2 Fuzzy Logic Control : An Overview Recently, researchers have been exploiting Artificial Intelligence (AI) techniques to address the following two major issues where conventional control techniques still require improvement: ∙ Accuracy of nonlinear system modeling; . Fig. 6. FLC-based closed-loop system. 5.2.3 Acquiring Real-Time Data The simulink model used for this purpose is depicted in Fig. 8. The disturbance in form of flow velocity and also the open-loop control signal in form of voltage (controlled by the trainee subject as explained below) are applied to the simulated system and the required data for training ANFIS (i.e. applied voltage v (t), error e(t) and the rate of change of error ˙ e(t)) are captured and saved for manipulation in ANFIS. Also, the scope is the aforementioned HMI as in Fig. 7. A joystick was used as the haptic device to control the voltage applied to the DC motor actuator in the simulation environment and also experiment. The operator can continuously monitor the robot motion in real-time to correct its course of motion by varying the voltage provided to the motor.The objective is to make ˙ z (t) follow ˙ Z set closely and consequently minimize the error. Following the above procedure, we asked our trainee to accomplish the control task in the presence of step flow disturbance. The trainees go through a few trials in order to become an expert and the data provided by them can be used for training our ANFIS. The data acquisition time was set at 40s for the trainee to have enough time, between each of the four jumps in the flow velocity, to bring the system back to its set-point. 5.3 ANFIS Architecture Here we elaborate on the ANFIS structure adopted in the proposed servomechanism control problem. As explained previously (see section 5.2.1) there are three MF’s on each input which yield a rule base with nine fuzzy if-then rules of first order TSK type (Turing, 1950). Rule #i : IF e (t h ) is A j1 and ˙ e(t h ) is A j2 THEN v i = p i e(t h ) + q i ˙ e (t h ) + r i where i = {1, .,9} is the rule number, {e(t h ), ˙ e(t h )} are the numerical values of the error inputs at sampling time t h and A jk ’s are linguistic variables ( i.e. { NEGATIVE , ZERO , POS- ITIVE } ). Also j = {1,2, 3} is the node number and k = 1,2 is the indicator of the input (”1” referring to a linguistic variable on ”e” and ”2” referring to a linguistic variable on ” ˙ e”) . The corresponding equivalent ANFIS structure is shown in Fig. 9. The node functions in each layer are of the same family. . Fig. 4. Membership functions on the two inputs of the system : error and the rate of change in error before tuning. . Fig. 5. Closed-loop system of the HITL simulator. 5.2.2 Human-In-the-Loop Simulator (HITL) A real-time virtual reality HITL simulator was designed. Data acquired via this simulator was employed for training the ANFIS. The operator learns to control the velocity of the pipe crawler when subjected to flow disturbances, in the Human-Machine Interface (HMI) de- signed for this purpose. Fig. 5 shows the closed-loop system modeled in the HITL simulator. In this research we replace the ”human operator” of the closed-loop with a stand-alone FLC whose parameters are tuned using the data acquired from the human operator, as depicted in Fig. 6 The disturbance on the system is simulated in the form of step changes in the flow velocity in the pipe. A snapshot of the HMI is given in Fig. 7. In this figure, ˙ z (t) and ˙ Z set are depicted on top with a solid and a dashed line, respectively. The randomly generated flow disturbance (used for training) is also shown at the bottom of the figure. We will show through simulation that the controller tuned based on this type of disturbance is capable of rejecting different disturbances such as sinusoidal as well. . Fig. 6. FLC-based closed-loop system. 5.2.3 Acquiring Real-Time Data The simulink model used for this purpose is depicted in Fig. 8. The disturbance in form of flow velocity and also the open-loop control signal in form of voltage (controlled by the trainee subject as explained below) are applied to the simulated system and the required data for training ANFIS (i.e. applied voltage v (t), error e(t) and the rate of change of error ˙ e(t)) are captured and saved for manipulation in ANFIS. Also, the scope is the aforementioned HMI as in Fig. 7. A joystick was used as the haptic device to control the voltage applied to the DC motor actuator in the simulation environment and also experiment. The operator can continuously monitor the robot motion in real-time to correct its course of motion by varying the voltage provided to the motor.The objective is to make ˙ z (t) follow ˙ Z set closely and consequently minimize the error. Following the above procedure, we asked our trainee to accomplish the control task in the presence of step flow disturbance. The trainees go through a few trials in order to become an expert and the data provided by them can be used for training our ANFIS. The data acquisition time was set at 40s for the trainee to have enough time, between each of the four jumps in the flow velocity, to bring the system back to its set-point. 5.3 ANFIS Architecture Here we elaborate on the ANFIS structure adopted in the proposed servomechanism control problem. As explained previously (see section 5.2.1) there are three MF’s on each input which yield a rule base with nine fuzzy if-then rules of first order TSK type (Turing, 1950). Rule #i : IF e (t h ) is A j1 and ˙ e(t h ) is A j2 THEN v i = p i e(t h ) + q i ˙ e (t h ) + r i where i = {1, .,9} is the rule number, {e(t h ), ˙ e(t h )} are the numerical values of the error inputs at sampling time t h and A jk ’s are linguistic variables ( i.e. { NEGATIVE , ZERO , POS- ITIVE } ). Also j = {1,2, 3} is the node number and k = 1,2 is the indicator of the input (”1” referring to a linguistic variable on ”e” and ”2” referring to a linguistic variable on ” ˙ e”) . The corresponding equivalent ANFIS structure is shown in Fig. 9. The node functions in each layer are of the same family. . Fig. 4. Membership functions on the two inputs of the system : error and the rate of change in error before tuning. . Fig. 5. Closed-loop system of the HITL simulator. 5.2.2 Human-In-the-Loop Simulator (HITL) A real-time virtual reality HITL simulator was designed. Data acquired via this simulator was employed for training the ANFIS. The operator learns to control the velocity of the pipe crawler when subjected to flow disturbances, in the Human-Machine Interface (HMI) de- signed for this purpose. Fig. 5 shows the closed-loop system modeled in the HITL simulator. In this research we replace the ”human operator” of the closed-loop with a stand-alone FLC whose parameters are tuned using the data acquired from the human operator, as depicted in Fig. 6 The disturbance on the system is simulated in the form of step changes in the flow velocity in the pipe. A snapshot of the HMI is given in Fig. 7. In this figure, ˙ z (t) and ˙ Z set are depicted on top with a solid and a dashed line, respectively. The randomly generated flow disturbance (used for training) is also shown at the bottom of the figure. We will show through simulation that the controller tuned based on this type of disturbance is capable of rejecting different disturbances such as sinusoidal as well. . Fig. 8. Simulink model used for data acquisition. . Fig. 9. The ANFIS structure adopted in this work. where T m,p is the m-th component of the p-th target output vector, and O L m,p is the m-th compo- nent of actual output vector produced by the presentation of the p-th input vector. Therefore, the overall error measure is equal to E = ΣE p and the derivative of the overall error measure E with respect to the premise parametes α is: ∂E ∂α = P ∑ p=1 ∂E p ∂α (28) The updated formula for the premise parameters α is : ∆α = −η ∂E ∂α (29) where: η = k √ ∑ α ( ∂E ∂α ) 2 (30) is the learning rate for α and k is the step size and can be varied to change the speed of con- vergence. . Fig. 7. A snapshot of the HMI used in this paper. 5.3.1 Hybrid Learning Rule The architecture of ANFIS shows that the output can be expressed as: (Ghafari et al., 2006): output = F( ⃗ I, S) (23) where ⃗ I is the set of input variables S in the set of parameters. There will exist an identity function H such that the composite of H ∘ F is linear in some of the elements of consequent parameters S, then these elements can be identified by the Least Squared Estimation (LSE). More formally, if the parameter set S can be decomposed into two sets as: S = S 1 ⊕ S 2 (24) where ⊕ represents direct sum, such that H ∘ F is linear in the elements of S 2 , then upon applying H to (23), we have: H (output) = H ∘ F( ⃗ I, S) (25) which is linear in the elements of S 2 . Hence, given values of premise parameters S 1 , we can plug P training data into (25) and obtain a matrix equation : AX = B (26) where X is a vector of unknown parameters in S 2 , and A and B are the set of inputs and outputs, respectively. Let ∣S 2 ∣=M, then the dimensions of A, X and B are P × M, M × 1 and P × 1, respectively. As the number of training data P is usually greater than the number of linear parameters M, a least squared estimate is used to seek X. On the other hand, the error measure for the p-th (1 ≤ p ≤ P) training data can be defined as the sum of squared errors: E p = #(L ) ∑ m=1 (T m,p − O L m,p ) 2 (27) . Fig. 8. Simulink model used for data acquisition. . Fig. 9. The ANFIS structure adopted in this work. where T m,p is the m-th component of the p-th target output vector, and O L m,p is the m-th compo- nent of actual output vector produced by the presentation of the p-th input vector. Therefore, the overall error measure is equal to E = ΣE p and the derivative of the overall error measure E with respect to the premise parametes α is: ∂E ∂α = P ∑ p=1 ∂E p ∂α (28) The updated formula for the premise parameters α is : ∆α = −η ∂E ∂α (29) where: η = k √ ∑ α ( ∂E ∂α ) 2 (30) is the learning rate for α and k is the step size and can be varied to change the speed of con- vergence. . Fig. 7. A snapshot of the HMI used in this paper. 5.3.1 Hybrid Learning Rule The architecture of ANFIS shows that the output can be expressed as: (Ghafari et al., 2006): output = F( ⃗ I, S) (23) where ⃗ I is the set of input variables S in the set of parameters. There will exist an identity function H such that the composite of H ∘ F is linear in some of the elements of consequent parameters S, then these elements can be identified by the Least Squared Estimation (LSE). More formally, if the parameter set S can be decomposed into two sets as: S = S 1 ⊕ S 2 (24) where ⊕ represents direct sum, such that H ∘ F is linear in the elements of S 2 , then upon applying H to (23), we have: H (output) = H ∘ F( ⃗ I, S) (25) which is linear in the elements of S 2 . Hence, given values of premise parameters S 1 , we can plug P training data into (25) and obtain a matrix equation : AX = B (26) where X is a vector of unknown parameters in S 2 , and A and B are the set of inputs and outputs, respectively. Let ∣S 2 ∣=M, then the dimensions of A, X and B are P × M, M × 1 and P × 1, respectively. As the number of training data P is usually greater than the number of linear parameters M, a least squared estimate is used to seek X. On the other hand, the error measure for the p-th (1 ≤ p ≤ P) training data can be defined as the sum of squared errors: E p = #(L ) ∑ m=1 (T m,p − O L m,p ) 2 (27) 6. Simulation and Experimental Results 6.1 Simulation Results MATLAB VR2008a together with SIMULINK, the Fuzzy Logic Toolbox and WinCon V5.0 from Quanser (Quanser, 2009) were used for real-time simulation of our proposed system. The control objective was to maintain a pre-set constant linear speed ˙ Z set while moving the robot inside a vertical pipe in the presence of hydrodynamic forces due to flow. The SIMULINK model of the feedback-loop with the proposed FLC is shown in Fig. 10. . Fig. 10. Closed-loop system using stand-alone FLC used in simulation. 6.1.1 External Disturbance Models Two flow disturbance models were used in the simulation environment : (1) step changes and (2) sinusoidal changes in flow velocity as depicted on top of Fig. 11. A variety of simulations were conducted based on the classical PID and also the stand-alone . Fig. 11. Flow disturbance models used in simulation. intelligent controller (FLC based on ANFIS), both of which were tested in a closed-loop system in the presence of the two aforementioned disturbance models and ˙ Z set m s = {0.10,0.15, 0.30}. 6.1.2 PID Control The tests were carried out with a classical PID controller of the form : u (t) = K p e(t) + K d de dt + K I ∫ t 0 e(τ)dτ + u 0 (32) 5.3.2 Hybrid Learning Algorithm Given the values of the premise parameters, the overall output of the proposed type-3 ANFIS structure can be expressed as a linear combination of the consequent parameters, i.e. the output v can be expressed as : v = 9 ∑ i=1  w i v i ∑ 9 i =1 w i  (31) = 9 ∑ i=1  ( ¯ w i e)p i  + 9 ∑ i=1  ( ¯ w i ˙ e )q i  + 9 ∑ i=1  ( ¯ w i )r i  which is linear in terms of the consequent parameters {p i ,q i ,r i }. a) Forward Pass : In the forward pass of the hybrid learning algorithm, the node out- puts go forward till layer 4 where the consequent parameters are identified by the Least Square Estimate (LSE) from (26). a) Backward Pass : In the backward pass, the error rates of each node output propa- gate from the output end toward the first layer, where now the premise parameters are updated by the gradient descent using (29). Table 2 summarizes the activities in each path. This hybrid learning algorithm is shown to efficiently obtain the optimal premise and consequent parameters during the learning process. Forward Pass Backward Pass Premise Parameters Fixed Gradient Descent Consequent Parameters LSE Fixed Signals Node Outputs Error Rates Table 2. The hybrid learning procedure for ANFIS in two passes (Jang, 1993) 5.3.3 Tuning the FLC using ANFIS In order to tune parameters of both the linguistic variables’ membership functions µ A jk (x k ) (i.e. the set {a 1 , a 2 ,c 1 ,c 2 } as in (22)) and the parameters of the rules’ consequents (i.e. {p i ,q i ,r i } for each rule i) we used the acquired data (see section 5.2.3) based on {e(t), ˙ e(t)} as inputs to the controller and the DC motor voltage v (t) as the output of the system. In other words, using ANFIS, the objective is to find a relationship between the inputs and output of the controller of the form v (t) = k 1 e(t) + k 2 ˙ e (t) + k 3 for each rule i. One can readily conclude by referring to (31) that : k 1 = 9 ∑ i=1 w i p i ; k 2 = 9 ∑ i=1 w i q i ; k 3 = 9 ∑ i=1 w i r i For this purpose, each trainee accomplishes the control task for 4000 time steps or 40 seconds in each trial (sampling time was set at δt = 0.01sec). From each training run, 2000 data points were randomly selected to tune the FLC using ANFIS. After having been trained, ANFIS was tested with the remaining 2000 sampled data for verification. [...]... parameters a2 -222.4 -222.4 -222.4 - 89. 33 - 89. 33 - 89. 33 -222.4 -222.4 -222.4 - 89. 33 - 89. 33 - 89. 33 c2 0.11 0.18 0.24 -0.05 0.12 0. 29 0.10 0.15 0.24 -0.03 0.12 0. 29 Fig 22 Control surface of the FLC using three product-of-sigmoidal MF’s, four views i 1 2 3 4 5 6 7 8 9 pi 2.46 -0.1 29 -2.274 -1.602 10.36 3.308 8.068 8 .91 7 2.43 qi 2. 195 1.607 2. 49 2.343 2 .96 2 -1.373 1 .93 2 1.853 0.072 Table 4 The final value... ( 198 5) Fuzzy identification of systems and its applications to modeling and control, IEEE transactions on systems, man, and cybernetics 15(1): 116–132 Turing, A ( 195 0) Computing machinery and intelligence, Mind 59( 236): 433–460 Wang, L ( 199 3) Stable adaptive fuzzy control of nonlinear systems, IEEE Transactions on Fuzzy Systems 1(2): 146–155 Wang, L ( 199 4) Adaptive fuzzy systems and control: design and. .. Mobile and Autonomous Systems, Vol 10, pp 25–26 ˙ Hunt, K., Sbarbaro, D., Zbikowski, R & Gawthrop, P ( 199 2) Neural networks for control systems: a survey, Automatica (Journal of IFAC) 28(6): 1083–1112 Jang, J ( 199 3) ANFIS: Adaptive-network-based fuzzy inference system, IEEE transactions on systems, man, and cybernetics Koji, K ( 199 9) Underwater inspection robot–AIRIS 21, Nuclear Engineering and Design(Switzerland)... River, NJ, USA Wang, L & Mendel, J ( 199 2) Fuzzy basis functions, universal approximation, and orthogonalleast-squares learning, IEEE Transactions on Neural Networks 3(5): 807–814 Zadeh, L ( 197 3) Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man, and Cybernetics 3: 28–44 Ziegler, J & Nichols, N ( 199 3) Optimum settings for automatic controllers,... function Fig 21 MF’s on the two inputs of the system : error and the rate of change in error, after tuning INITIAL VALUES FINAL VALUES MF µ A11 (e) µ A21 (e) µ A31 (e) ˙ µ A12 (e) ˙ µ A22 (e) ˙ µ A32 (e) µ A11 (e) µ A21 (e) µ A31 (e) ˙ µ A12 (e) ˙ µ A22 (e) ˙ µ A32 (e) a1 222.4 222.4 222.4 89. 33 89. 33 89. 33 222.4 222.4 222.4 89. 33 89. 33 89. 33 c1 0.041 0.108 0.1756 -0.2185 -0.05 0.12 0.04 0.12 0.15... Nuclear Engineering and Design(Switzerland) 188(3): 367–371 Kulpate, C (2006) Design and development of a stereo scanner for condition assessment of water pipes, Technical Report UR/CSIR-01-2006, University of Regina Lee, C ( 199 0) Fuzzy logic in control systems: fuzzy logic controller II, IEEE Transactions on systems, man and cybernetics 20(2): 4 19 435 Mamdani, E et al ( 197 4) Application of fuzzy algorithms... ANFIS The inputs to the FLC were the error, e(t) and the rate of change of ˙ error, e(t) (see section 5.2.1) and are presented in Fig 19 The error tolerance in ANFIS was set at 10−6 and was reached after 97 epochs on average The trend in epochs is depicted in Fig 20 Also the modified MFs and the pertaining control surface after tuning are shown in Figs 21 and 22, respectively It is noteworthy that as the... using Hybrid Neuro-Fuzzy ANFIS Approach, IEEE International Conference on Robotics and Biomimetics, 2006 ROBIO’06, pp 733–738 Glass, S., Levesque, M., Engels, G., Klahn, F & Fairbrother, D ( 199 9) UNDER-WATER ROBOTIC TOOLS FOR NUCLEAR VESSEL AND PIPE EXAMINATION, Framatome ANP, Lynchburg Virginia Griffiths, G (2003) Technology and applications of autonomous underwater vehicles, CRC Press Grigg, N (2006)... International Conference on Mechatronics and Machine Vision in Practice,(M 2 VIP97), pp 142–147 Quanser (20 09) Consulting co., http://www.quanser.com Ratanasawanya, C., Binsirawanich, P., Yazdanjo, M., Mehrandezh, M., Poozesh, S., Paranjape, R & Najjaran, H (2006) Design and Development of a Hardware-in-the Loop Simulation System for a Submersible Pipe Inspecting Robot, Electrical and Computer Engineering, 2006... system analysis, design, and technology, IEEE transactions on control systems technology 13(4): 5 59 576 Bradbeer, R., Harrold, S., Luk, B., Li, Y., Yeung, L & Ho, H (2000) A mobile robot for inspection of liquid filled pipes, Workshop on Service Automation and Robotics, City University of Hong Kong Fig 33 The proposed HIL simulation system Chaudhuri, T., Hamey, L & Bell, R ( 199 6) Historical Perspective:” . p i q i r i 1 2.46 2. 195 2.582 2 -0.1 29 1.607 2.728 3 -2.274 2. 49 2.8 49 4 -1.602 2.343 3.03 5 10.36 2 .96 2 0 .93 16 6 3.308 -1.373 2.846 7 8.068 1 .93 2 1.277 8 8 .91 7 1.853 1.116 9 2.43 0.072 2. 497 Table 4 p i q i r i 1 2.46 2. 195 2.582 2 -0.1 29 1.607 2.728 3 -2.274 2. 49 2.8 49 4 -1.602 2.343 3.03 5 10.36 2 .96 2 0 .93 16 6 3.308 -1.373 2.846 7 8.068 1 .93 2 1.277 8 8 .91 7 1.853 1.116 9 2.43 0.072 2. 497 Table 4 222.4 0.1756 -222.4 0.24 VALUES µ A 12 ( ˙ e ) 89. 33 -0.2185 - 89. 33 -0.05 µ A 22 ( ˙ e ) 89. 33 -0.05 - 89. 33 0.12 µ A 32 ( ˙ e ) 89. 33 0.12 - 89. 33 0. 29 µ A 11 (e) 222.4 0.04 -222.4 0.10 µ A 21 (e)