A MIMO Fuzzy Controller for tracking: Robot control. Carl Crane Professor, Mechanical Engineering University of Florida Gainesville, FL 32611

Transcription

1 A MIMO Fuzzy Controller for tracking: Robot control Paul A.C. Mason Assistant Professor, Kevin Walchko Research Assistant Carl Crane Professor, Mechanical Engineering University of Floria Gainesville, 36 Abstract In the formulation of most practical robot control problems the analytical moel use in the controller esign may neglect certain nonlinearities or system ynamics in orer to simplify the esign. Furthermore, unknown loa ynamics can also effect the response of the system. For many moel-base controllers, these iscrepancies can lea to poor performance or even instability. This is especially true of Multi-Input Multi-Output (MIMO) systems. Therefore, it is important to evelop robust MIMO controllers that can account for such iscrepancies, an uncertainties. To aress these issues, this work escribes the evelopment an implementation of a MIMO hybri controller for a two-link robot arm. The presente MIMO formulation is generic in nature an can reaily be integrate into existing applications with minimal effort. The hybri structure of the controller takes avantage of classical proportional-erivative control logic while maintaining a significant egree of robustness, performance an portability. A two-link robot simulation stuy is use to illustrate the merits of the propose scheme.. Introuction

2 Robots have revolutionize the manufacturing, meical, an entertainment inustries. In manufacturing, robots are use in a wie varity of applications, such as material hanling, spraying painting, spot weling, inspection, assembly, etc. Current control research in the area of robotics examines ways to improve the robustness of the position an or force control for tracking an regulation. Avancements in technology have le to the evelopment of complex multi-functional systems such as multi-link kinematic systems (Scharf 985), autonomous vehicles Asit (95), etc. In the case of robotic system, formulating an analytical escription of the ynamics or kinematics requires a significant amount of analysis an insight. There are many inherent ifficulties in efining the nonlinear amping in the joints (Smaili an Sannah 994) an formulating the inverse or forwar kinematics. In many cases it is impractical to formulate an accurate analytical expression of these ynamics. In these situations intuitive information can be use to esign a traitional controller in an a hoc manner, or synthesis of a non-moel base intelligent controller. In aition these controllers must account for uncertainty an isturbance. The problems associate with esigning a controller in this environment are compoune with issues associate with a Multiple Input Multi-Output (MIMO) system. Some of the most common an intuitive robot control techniques such as those evelope by Arimoto an Miyazaki (983) utilize the PID schemes. As system complexity increases, more robust schemes such as output feeback (Wang et al. 989) are require to achieve the esire performance. This is especially true when uncertainty is taken into consieration. For example, Lin an Brant (998) escribe an optimal control approach base on the work of Lin (99) for robust control of a robot manipulators. This work extens optimal controllers to account for uncertainties an isturbances. Like sliing moe control (Slotine 984), uncertainty bouns are require to esign the controller. In aition, the esign process can become complicate an a significant

3 amount of effort is require to implement the controller. The problem of esigning MIMO control laws for systems with a significant amount of nonlinearities in the ynamics is the primary motivation for investigating the evelopment of MIMO control schemes. The scheme presente in this paper is straightforwar, easy to implement, an utilizes the esigner s intuition. This work eals with the evelopment of intelligent MIMO controllers that utilize the esigner s heuristic knowlege as well as any mathematical escription of the system to provie a reasonable controlle response. The propose controller combines the traitional Proportional-Derivative () control law with a controller in a MIMO framework to create a scheme that can be integrate into a variety of existing systems with minimal effort.. Benefits of logic for complex systems Fuzzy logic is one of the most promising intelligent control schemes for complex, nonlinear systems. In the literature, Single-Input Single-Output (SISO) logic controllers have been successfully applie to numerous applications in the military (Gonsalves an Caglayan 95), inustry (Bartolini 985), an research (Lee, et. al. 94). Fuzzy logic allows operator experience an system knowlege to be encoe into the controller through membership functions an rules rather than complex mathematical moels. For this reason, it lens itself to the control of complex time-varying SISO systems, which are not always practical to moel accurately. Since many systems in real-worl applications are MIMO, it is essential that a MIMO version be formulate to control systems such as multi-link robots. To make this scheme practical, issues associate with MIMO control, accuracy, an implementation must be aresse.. Theory

4 . Control Law There is an immense knowlege base associate with classical controls. Any textbook on classical controls can provie etaile theory on the esign, analysis, an implementation of these controllers (Ogata 97). Classical controllers can be integrate into most existing linear SISO system. As the system complexity increases the esign process becomes ifficult an the performance may be egrae. The classical controller is characterize by the following time-omain expression: u = K e c p + K e t () where e is the error between the reference value an the actual value.. Fuzzy Logic The power of logic is its tolerance for ambiguity an its ability to increase robustness. It interprets controller information in a linguistic manner via the membership functions. In the linguistic omain the input/output relationship is efine by a set of rules. These rules an the membership functions are use to prouce the appropriate control effect. Membership functions convert crisp inputs into linguistic variables or visa versa. A given input value can belong to one or several membership functions, an the egree to which it belongs to each membership function is given by the value between zero an one. If-then rules are evaluate in parallel for the fuzzifie inputs. Each rule evaluation forms a output set. The iniviual output sets are aggregate to form a final output set, which is referee to as efuzzification. The Center-of-gravity (COG) is the most common efuzzification metho (Kosko 997). The entire process is epicte graphically in Figure.

5 Crisp Inputs Fuzzification Rule Evaluation Output Fuzzy Sets : if then w Aggregation : if then w wm Deffuzification n m: if then Crisp Output Figure. The Fuzzy Inference Process..3 Hybri Structure The motivation behin eveloping a hybri scheme, which is a combination of various types of controllers, is to take avantage of the attributes of various controllers while accounting for their weaknesses. Recent research in intelligent controls has incorporate classical control techniques into control structure, stability analysis (Wang, et. al. 96, Spooner an Passino 96, Petroff et al. 98), an controller esign methoology (Walchko et al. 98). These intelligent controllers are referre to as hybris. One of the most use hybris is the PID (Brehm an Kulip 94, Misir 96). Due to their structure, hybris provie a more efine control structure over control while increasing the accuracy an portability of the controller. An example of a hybri structure is given in Figure.

6 e Error (e) e t Fuzzy u Figure. The hybri structure..4 MIMO evelopment In this section the MIMO controller is formulate by extening the SISO concepts to the vector omain. This formulation resolves the control vector into a irection, magnitue an position-erivative ratio. By reucing the control vector into these components, the MIMO problem of solving for the appropriate control effort is reuce to a scalar one. In the stanar SISO system the hybri controller uses the error an its erivative as the inputs to the inference system to etermine the corresponing scalar output control effort u c. u c ( e, e& ) In the MIMO case, the error terms are vectors. If the system can be assume uncouple, three SISO can be utilize to achieve the esire response. If the system is couple the require rules an membership functions significantly increase as the number of states an outputs increase, which makes the control esign impractical. In this work a SISO inference system prouces a magnitue, K mag, an a position-erivative ratio, T. Given e an e& the control law is u c = K mag ( e + T e& ) ()

7 Since, the magnitue is a positive number, all negative MF s are remove an the complexity of inference system is significantly reuce. In orer to exten the SISO controller to a MIMO framework, the control law is resolve into unit vectors, a position-erivative ratio an a magnitue. u c ) ) ) ~ = K pkunitse + K K& ~ ) ~ unitse & = + = ) K ) K p e Ke & K e + e & ~ K p u c = K ) ( e + T e) ) & where K p, K are controller gain, e ) is the unit vector of the error an K, an K & units represents the units of the error (meters) an it s time erivative (meters/sec). These constants are conversion units factors. For example K units = N, therefore, the output of K K eˆ.is torque. p units u = K mag v ( e + T e) D v & (3) Note that equation () is the same as the SISO controller. The MIMO controller uses the norm of the error an it s erivative in the inference system to etermine a magnitue an a position-erivative ratio. [ K, T ] = norm( e ), norm( e& ) mag ( ) (4) The T term is crucial to ensure a ampe response. Table -3 contains the simplifie rules for this MIMO robotic application.

8 Table Magnitue rules for error e BP MP P SP Z BP MP P SP Z Table Magnitue rules for err_ot e& BP MP P SP Z BP MP P SP Z Table 3 T rules for error an it s erivative e e& BP MP P SP Z BP MP P SP Z BP BP SP SP SP.5 Robot Moel Description In orer to illustrate this concept, a simple a two-link robot is use in the simulation stuy. The moel of the robot is obtaine from Slotine an Li (99). Figure 3 contains a picture of the robot. This two-link robot system utilizes quaternion angles for computation stability. The moel of the system is given below. H H H H q&& hq& + q&& hq& hq& hq& q& g τ + = q& g τ Where = [ q q ] T are the joint angles, [ ] T q τ = τ τ are the input torques an

9 g H H H h = m l l g = m l = H c = m l c = m l c = m l c c sin g sin g cos + l + m + I = m l l c [ l + l + l l cos( q )] cos c c ( q) + mlc + I + l ( q ) ( q ) + mg[ lc cos( q + q) + l cos( q )] ( q + q ) Figure 3. Robot system.5 an Feeback Linearization controllers Two common types of controllers use in robotic control are the classical scheme an the Linearization feeback metho. These methos are moel epenent an require some insight to tune the controllers. Since the robotic system is a MIMO system the controller is efine as. u = K e + K P D e& (5) where K, K are scalar constants. This allows for a simple intuitive esign. p

10 The Feeback Linearization () controller is a conceptually simple scheme. However, implementation an esign can be involve an time consuming. The basic concept behin the LF controller is to replace the system ynamics with the esire ynamics. This is achieve by efining the control torque as τ H = τ H H H v v hq& + hq& hq& hq& q& g + q& g (6) Where Y q q ~ = && & q~ [ v ] T Y = v an q ~ = q q. Plugging v into equation 6 prouces the tracking error q ~ & + λ q~ & + λ q~ =, which converges to zero if is a positive value. The rate of convergence is a function of the moel accuracy an lamba. In aition, this metho is very sensitive to saturation an loa isturbances. 3. Results The simulations stuies consists of two parts. The first simulation examines the performance of a MIMO hybri controller in ieal conitions. This will be compare against a stanar controller an a feeback linearization () scheme. In the next stuy, an unknown loa isturbance is ae to the system an the performances of the controllers are compare. In the first case stuy all the ynamics are assume known an the performance of the three controllers are compare. The position errors of the first an secon state are shown in Figure 3-4. All three controller prouce about the same response time. However, the scheme was a little

11 faster position response in the secon link. In the velocity errors the controller response was slightly more ampe. Upon further inspection one can see that the an the schemes prouce a larger steay state error than the scheme (Figure 5). In aition, the control effort of the an the are larger than the scheme (Figure 6). Overall the scheme out performs the other controllers. It shoul be note that the scheme can be altere to prouce a faster, ampe response by altering the rules. However, this was not one to show how a simple set of rules can prouce a response that is better than a moel base controller..5 Position error of the first link Position error (ra) time 8 Velocity error of the first link Velocity error (ra/sec) time Figure 3. Position an velocity errors of the first link

12 .5 Position error of the secon link Position error (ra) time sec 4 Velocity error of the secon link Velocity error(ra/sec) Figure 4. Position an velocity errors of the secon link Steay state errors of the first link. Position error (ra) Steay state errors of the secon link Velocity error (ra/sec) Figure 5. Position steay state errors

13 5 First Control effort control effort (N-m) Secon Control effort control effort (N-m) Figure 6. Control effort In the secon stuy a isturbance, which is a combination of a ranom an a eterministic source, is ae to the system (Figure ) an the performance of the three controllers are compare. The position errors of the first an secon state are shown in Figure 7-8. Again, all three controller prouces about the same response time. However, the effects of the isturbance egrae the accuracy of the controllers. This is especially true when a eterministic portion of the isturbance. The was able to account for the stochastic an eterministic isturbance better than the of the schemes. This is especially true for the steay state response of the an scheme (Figure 9). Finally, the control effort of all controller was about the same.

14 .5 Position error of the first link with isturbance Position error (ra) Velocity error of the first link with isturbance Velocity error(ra/sec) Figure 7. Position an velocity errors of the first link with isturbance.5 Position error of the secon link with isturbance Position error (ra) Velocity error of the secon link with isturbance Velocity error (ra/sec) Figure 8. Position an velocity errors of the secon link with isturbance

15 Steay state errors of the first link with isturbance Position error Steay state errors of the secon link with isturbance. Velocity error Figure 9. Position steay state errors with isturbance 5 First Control effort with isturbance control effort (N-m) Secon Control effort with isturbance control effort (N-m) Figure. Control effort with istrubance

16 istrubance 5 5 magnitue Figure. Distrubance 4. Conclusions This paper escribe the evelopment an implementation of a MIMO controller, which can compensates for unmoele isturbances in a robotic application. Base on the results, the MIMO outperforme the an the controllers. This is ue to the fact that these controllers are fixe gain controllers an can not easily compensate for uncertainty in the ynamics. In aition, the scheme require the same or less control effort to achieve a reasonable response. This is significant point in real applications. One of the main avantages the scheme has is its portability. Unlike the fixe gain schemes, the controller can be easily porte to another robot with minimal effort. Furthermore, the

17 scheme is more robust to ranom an eterministic isturbances. In summary, this work is a crucial step in the evelopment of a more robust MIMO technique, which can be implemente, in various MIMO robotic applications. 5. References Asit, P., Crane, C., Gamble, D., Rigeway, S., an Tosunoglu, S., 995, Design, Fabrication, an Control of a Robotic Wall-Crawler for Remote Inspection Operations, IEEE st International Conference on Inustrial Electronics, Control, an Instrumentation IECON proceeings, pp. -5. Arimoto, S., an Miyazaki, F., 983, Stability an robustness of PID feeback control for robot manipulators of sensory capabilities, st Int. Symp. Robot Res. proceeings, pp Bartolini, G., Casalino, G., Davoli, F., Mastretta, M., an Morten, E., 985, Development of Performance Aaptive Fuzzy Controllers with Application to Continuous Casting Plants, Inustrial Applications of Fuzzy Control, Eite by Sugeno, M., Elsevier Science Publishers, North Hollan, pp.??-??. Brehm, T., an Kulip, S., R., 995, Proportional Fuzzy Logic Controller: Classical Proportional-Plus-Derivative Like Response, IEEE International Conference on Systems, Man, an Cybernetics, Vol. 3, pp Ewars, C., an Spurgeon, S. K., 998, Sliing Moe Control Theory an Applications, Taylor & Francis Lt., Lonon. Gonsalves, P., G., an Caglayan, A., K., 995, Fuzzy Logic PID Controller for Missile Terminal Guiance, IEEE International Symposium on Intelligent Control - Proceeings, pp

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19 ientification of the joint for esign with most effective amping enhancement, Proceeings of the 994 ASMEW esign technical Conference- 3 Biennial Mechanisms conference, Minnesota, pp. -9. Smart, D. W., an Wiens, G. J., 995, Analysis of Gravitational Effects on Two Link, Flexible Multiboy Systems, ASME 6th Biennial Conference on Mechanical Vibration an Noise, DE-Vol 84-, No. 3A, pp Spooner, J., an Passino, K. M., 996, Stable Aaptive Control Using Fuzzy Systems an Neural Networks, IEEE Transactions on Fuzzy Systems, Vol. 4, No. 3, pp Walchko, K., Petroff, N., Mason, P., an Reisinger, K., 998, "Development of Moular SISO Fuzzy Logic Controller," Proceeings of the ANNIE 98 Smart Engineering System Design, pp Wang, W., Lu, S., an Hsu, C., 989 Experiments on the position Control of a One-link Flexible Robot Arm, IEEE Transactions on Robots an Automation, Vol. 5, No. 3, pp