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MechatronicSystems,Simulation,ModellingandControl110 time between rising edges of P E ) and T I (the time between rising edge of P E and trailing edge of P I ) are measured by the unit, as shown in Fig. 10. The phase difference is calculated from 2 180 C I C T T T . (2) This value is measured as average in averaging factor N a cycles of pulse signal P E . Thus, the operating frequency is updated every N a cycles of the driving signals. The updated operating frequency f n+1 is given by rpnn Kff 1 , (3) where f n is the operating frequency before update, r is the admittance phase at resonance, is the calculated admittance phase from eq. (2) (at a frequency of f n ), K p is a proportional feedback gain. To stabilize the tracing, K p should be selected as following inequality is satisfied. Fig. 7. Voltage/current detecting unit. Fig. 8. Control unit with a microcomputer. In Hall Element Out Amp/LPF Comp. P E P I A E A I Micro Computer SH-7045F PS/2 Keyboard LCD Display COM Port EEPROM 24C16 P E A E A I P I MTU AD Con DDS S K p 2 , (4) where S is the slope of the admittance phase vs. frequency curve at resonanse frequency. The updated frequency is transmitted to the DDS. Repeating this routine, the operating frequency can approach resonance frequency of transducer. 4. Application for Ultrasonic Dental Scaler 4.1 Ultrasonic dental scaler Ultrasonic dental scaler is an equipment to remove dental calculi from teeth. the scaler consists of a hand piece as shown in Fig. 10 and a driver circuit to excite vibration. A Langevin type ultrasonic transducer is mounted in the hand piece. the structure of the transducer is shown in Fig. 11. Piezoelectric elements are clamped by a tail block and a hone block. A tip is attached on the top of the horn. The blocks and the tip are made of stainless steel. The transducer vibrates longitudinally at first-order resonance frequency. One vibration node is located in the middle. To support the node, the transducer is bound by a silicon rubber. To carry out the following experiments, a sample scaler was fabricated.Frequency response of the electric charactorristics of the transducer was observed with no mechanical load and input voltage of 20 V p-p . The result is shown in Fig. 12. From this result, the resonance Fig. 9. Measurement of cycle and phase diference. Fig. 10. Example of ultrasonic dental scalar hand piece. Fig. 11. Structure of transducer for ultrasonic dental scalar. P E P I T C T I Tip Hand pieceHand piece Tip HornTail block PZT Rubber supporter Tip ResonanceFrequencyTracingSystemforLangevinTypeUltrasonicTransducers 111 time between rising edges of P E ) and T I (the time between rising edge of P E and trailing edge of P I ) are measured by the unit, as shown in Fig. 10. The phase difference is calculated from 2 180 C I C T T T . (2) This value is measured as average in averaging factor N a cycles of pulse signal P E . Thus, the operating frequency is updated every N a cycles of the driving signals. The updated operating frequency f n+1 is given by rpnn Kff 1 , (3) where f n is the operating frequency before update, r is the admittance phase at resonance, is the calculated admittance phase from eq. (2) (at a frequency of f n ), K p is a proportional feedback gain. To stabilize the tracing, K p should be selected as following inequality is satisfied. Fig. 7. Voltage/current detecting unit. Fig. 8. Control unit with a microcomputer. In Hall Element Out Amp/LPF Comp. P E P I A E A I Micro Computer SH-7045F PS/2 Keyboard LCD Display COM Port EEPROM 24C16 P E A E A I P I MTU AD Con DDS S K p 2 , (4) where S is the slope of the admittance phase vs. frequency curve at resonanse frequency. The updated frequency is transmitted to the DDS. Repeating this routine, the operating frequency can approach resonance frequency of transducer. 4. Application for Ultrasonic Dental Scaler 4.1 Ultrasonic dental scaler Ultrasonic dental scaler is an equipment to remove dental calculi from teeth. the scaler consists of a hand piece as shown in Fig. 10 and a driver circuit to excite vibration. A Langevin type ultrasonic transducer is mounted in the hand piece. the structure of the transducer is shown in Fig. 11. Piezoelectric elements are clamped by a tail block and a hone block. A tip is attached on the top of the horn. The blocks and the tip are made of stainless steel. The transducer vibrates longitudinally at first-order resonance frequency. One vibration node is located in the middle. To support the node, the transducer is bound by a silicon rubber. To carry out the following experiments, a sample scaler was fabricated.Frequency response of the electric charactorristics of the transducer was observed with no mechanical load and input voltage of 20 V p-p . The result is shown in Fig. 12. From this result, the resonance Fig. 9. Measurement of cycle and phase diference. Fig. 10. Example of ultrasonic dental scalar hand piece. Fig. 11. Structure of transducer for ultrasonic dental scalar. P E P I T C T I Tip Hand pieceHand piece Tip HornTail block PZT Rubber supporter Tip MechatronicSystems,Simulation,ModellingandControl112 frequency was 31.93 kHz, admittance phase coincided with 0 at the resonance frequency, electorical Q factor was 330 and the admittance phase response had a slope of -1 [deg/Hz] in the neighborhood of the resonanse frequency. 4.2 Tracing test Dental calculi are removed by contact with the tip. The applied voltage is adjusted according to condition of the calculi. Temparature rises due to high applied voltage. Therefore, during the operation, the resonance frequency of the transducer is shifted with the changes of contact condition, temperature and amplitude of applied voltage. The oscillating frequency was fixed in the conventional driving circuit. Consequently, vibration amplitude was reduced due to the shift. The resonance frequency tracing system was apllied to the ultrasonic dental scaler. Fig. 12. Electric frequency response of the transducer for ultrasonic dental scalar. Fig. 13. Step responses of the resonance frequency tracing system with the transducer for ultrasonic dental scaler. 0 4 8 12 -90 0 90 31.7 31.8 31.9 32 32.1 Current [mA] Admittance phase [deg] Frequency [kHz] Applied voltage: 20V p-p 31.7 31.8 31.9 32 Time [ms] Frequency [kHz] K P = 1 / 2 K P = 1 / 4 K P = 1 / 16 K P = 1 / 8 Applied voltage: 20V p-p 0 40 80 120 160 The transducer was driven by the tracing system, where averaging factor N a was set to 8. To evaluate the system characteristic, step responses of the oscillating frequency were observed in the same condition as the measurement of the electric frequency response. In this measurement, initial operating frequency was 31.70 kHz. the frequency was differed from the resonance frequency (31.93 kHz). At a time of 0 sec, the tracing was started. Namely, the terget frecuency was changed, as a step input, to 31.93 kHz from 31.7 kHz. The transient response of the oscillating frequency was observed. The oscillating frequency was measured by a modulation domain analyzer in real time. Figure 13 shows the measurement results of the responces. With each K p , the oscillating frequency in steady state was 31.93 kHz. the frequency coincided with the resonance frequency. A settling time was 40 ms with K p of 1/4. The settling time was evaluated from the time settled within ±2 % of steady state value. The response speed is enough for the application to the dental scaler. Contact load does not change faster than the response speed since the scaler is wielded by human. The temperature and the amplitude of applied voltage also do not change so fast in normal operation. 4.3 Dental diagnosis When the transducer is contacted with an object, the natural frequency of the transdcer is shifted. A value of the shift depends on stiffness and damping factor of the object (Nishimura et. al, 1994). The contact model can be discribed as shown in Fig. 14. In this model, the natural angular frequency of the transducer with contact is presented as 2 2 2 1 m C K l AE m C C , (5) where m is the equivalent mass of the transducer, A is the section area of the transducer, E is the elastic modulus of the material of the transducer, l is the half length of the transducer, K c is the stiffness of the object and C c is the damping coefficient of the object. Equation (5) indicates that the combination factor of the damping factor and the stiffness can be estimated from the natural frequency shift. The shift can be observed by the proposed resonance frequency tracing system in real time. If the correlation between the combination factor and the material properties is known, the damping factor or the stiffness of unknown material can be predicted. For known materials, the local stiffness on the contacting point can be estimated if the damping factor is assumed to be constant and known. Geometry also can be evaluated from the estimated stiffness. For a dental health diagnosis, the stiffness Fig. 14. Contact model of the transducer. 2l Support point Transducer Object K C C C ResonanceFrequencyTracingSystemforLangevinTypeUltrasonicTransducers 113 frequency was 31.93 kHz, admittance phase coincided with 0 at the resonance frequency, electorical Q factor was 330 and the admittance phase response had a slope of -1 [deg/Hz] in the neighborhood of the resonanse frequency. 4.2 Tracing test Dental calculi are removed by contact with the tip. The applied voltage is adjusted according to condition of the calculi. Temparature rises due to high applied voltage. Therefore, during the operation, the resonance frequency of the transducer is shifted with the changes of contact condition, temperature and amplitude of applied voltage. The oscillating frequency was fixed in the conventional driving circuit. Consequently, vibration amplitude was reduced due to the shift. The resonance frequency tracing system was apllied to the ultrasonic dental scaler. Fig. 12. Electric frequency response of the transducer for ultrasonic dental scalar. Fig. 13. Step responses of the resonance frequency tracing system with the transducer for ultrasonic dental scaler. 0 4 8 12 -90 0 90 31.7 31.8 31.9 32 32.1 Current [mA] Admittance phase [deg] Frequency [kHz] Applied voltage: 20V p-p 31.7 31.8 31.9 32 Time [ms] Frequency [kHz] K P = 1 / 2 K P = 1 / 4 K P = 1 / 16 K P = 1 / 8 Applied voltage: 20V p-p 0 40 80 120 160 The transducer was driven by the tracing system, where averaging factor N a was set to 8. To evaluate the system characteristic, step responses of the oscillating frequency were observed in the same condition as the measurement of the electric frequency response. In this measurement, initial operating frequency was 31.70 kHz. the frequency was differed from the resonance frequency (31.93 kHz). At a time of 0 sec, the tracing was started. Namely, the terget frecuency was changed, as a step input, to 31.93 kHz from 31.7 kHz. The transient response of the oscillating frequency was observed. The oscillating frequency was measured by a modulation domain analyzer in real time. Figure 13 shows the measurement results of the responces. With each K p , the oscillating frequency in steady state was 31.93 kHz. the frequency coincided with the resonance frequency. A settling time was 40 ms with K p of 1/4. The settling time was evaluated from the time settled within ±2 % of steady state value. The response speed is enough for the application to the dental scaler. Contact load does not change faster than the response speed since the scaler is wielded by human. The temperature and the amplitude of applied voltage also do not change so fast in normal operation. 4.3 Dental diagnosis When the transducer is contacted with an object, the natural frequency of the transdcer is shifted. A value of the shift depends on stiffness and damping factor of the object (Nishimura et. al, 1994). The contact model can be discribed as shown in Fig. 14. In this model, the natural angular frequency of the transducer with contact is presented as 2 2 2 1 m C K l AE m C C , (5) where m is the equivalent mass of the transducer, A is the section area of the transducer, E is the elastic modulus of the material of the transducer, l is the half length of the transducer, K c is the stiffness of the object and C c is the damping coefficient of the object. Equation (5) indicates that the combination factor of the damping factor and the stiffness can be estimated from the natural frequency shift. The shift can be observed by the proposed resonance frequency tracing system in real time. If the correlation between the combination factor and the material properties is known, the damping factor or the stiffness of unknown material can be predicted. For known materials, the local stiffness on the contacting point can be estimated if the damping factor is assumed to be constant and known. Geometry also can be evaluated from the estimated stiffness. For a dental health diagnosis, the stiffness Fig. 14. Contact model of the transducer. 2l Support point Transducer Object K C C C MechatronicSystems,Simulation,ModellingandControl114 estimation can be applied. To discuss the possibility of the diagnosis, the frequency shifts were measured using the experimental apparatus as shown in Fig. 15. A sample was supported by an aluminum disk through a silicon rubber sheet. The transducer was fed by a z-stage and contacted with the sample. The contact load was measured by load cells under the aluminum disk. This measuring configuration was used in the following experiments. The combination factor were observed in various materials. The natural frequency shifts in contact with various materials were measured with the change of contact load. The shape and size of the sample was rectangular solid and 20 mm x 20 mm x 5 mm except the LiNbO 3 sample. the size of the LiNbO 3 sample was 20 mm x 20 mm x 1 mm. The results are plotted in Fig. 16. the natural frequency of the transducer decreased with the increase of contact load in the case of soft material with high damping factor such as rubber. The natural frequency did not change so much in the case of silicon rubber. The natural frequency increased in the case of other materials. Comparing steel (SS400) and aluminum, stiffness of steel is higher than that of aluminum. Frequency shift of LiNbO 3 is larger than that of steel Fig. 15. Experimental apparatus for measurement of the frequency shifts with contact. Fig. 16. Measurement of natural frequency shifts with the change of contact load in contact with various materials. Aluminum disk Silicon rubber sheet Sample Load cell Contact load Transducer Supporting part -300 -200 -100 0 100 200 300 400 500 0 1 2 3 4 5 6 LiNbO 3 Steel (SS400) Aluminium Acrylic Silicon rubber Rubber f [Hz] Contact load [N] Applied voltage: 20V p-p within 4 N though stiffness of LiNbO 3 is approximately same as that of steel. This means that mechanical Q factor of LiNbO 3 is higher than that of steel, namely, damping factor of LiNbO 3 is lower. Frequency shift of LiNbO 3 was saturated above 5 N. The reason can be considered that effect of the silicon rubber sheet appeared in the measuring result due to enough acoustic connection between the transducer and the LiNbO 3 . The geometry was evaluated from local stiffness. The frequency shifts in contact with aluminum blocks were measured with the change of contact load. The sample of the aluminum block is shown in Fig. 17 (a). Three samples were used in the following experiments. One of the samples had no hole, another had thickness t = 5 mm and the other had the thickness t = 1 mm. Measured frequency shifts are shown in Fig. 17 (b). The frequency shifts tended to be small with decrease of thickness t. These results show that the Fig. 17. Measurement of natural frequency shifts with the change of contact load in contact with aluminum blocks. Fig. 18. Measurement of natural frequency shifts with the change of contact load in contact with teeth. (a) f 20 f 10 t 20 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 No hole 5mm 1mm f [Hz] Contact load [N] (b) Applied voltage: 20V p-p Contact pointContact point A B A B (a) 0 100 200 300 400 500 0 1 2 3 4 5 6 f [Hz] Contact load [N] Non-damaged (B) (b) Damaged (A) Applied voltage: 20 V p-p ResonanceFrequencyTracingSystemforLangevinTypeUltrasonicTransducers 115 estimation can be applied. To discuss the possibility of the diagnosis, the frequency shifts were measured using the experimental apparatus as shown in Fig. 15. A sample was supported by an aluminum disk through a silicon rubber sheet. The transducer was fed by a z-stage and contacted with the sample. The contact load was measured by load cells under the aluminum disk. This measuring configuration was used in the following experiments. The combination factor were observed in various materials. The natural frequency shifts in contact with various materials were measured with the change of contact load. The shape and size of the sample was rectangular solid and 20 mm x 20 mm x 5 mm except the LiNbO 3 sample. the size of the LiNbO 3 sample was 20 mm x 20 mm x 1 mm. The results are plotted in Fig. 16. the natural frequency of the transducer decreased with the increase of contact load in the case of soft material with high damping factor such as rubber. The natural frequency did not change so much in the case of silicon rubber. The natural frequency increased in the case of other materials. Comparing steel (SS400) and aluminum, stiffness of steel is higher than that of aluminum. Frequency shift of LiNbO 3 is larger than that of steel Fig. 15. Experimental apparatus for measurement of the frequency shifts with contact. Fig. 16. Measurement of natural frequency shifts with the change of contact load in contact with various materials. Aluminum disk Silicon rubber sheet Sample Load cell Contact load Transducer Supporting part -300 -200 -100 0 100 200 300 400 500 0 1 2 3 4 5 6 LiNbO 3 Steel (SS400) Aluminium Acrylic Silicon rubber Rubber f [Hz] Contact load [N] Applied voltage: 20V p-p within 4 N though stiffness of LiNbO 3 is approximately same as that of steel. This means that mechanical Q factor of LiNbO 3 is higher than that of steel, namely, damping factor of LiNbO 3 is lower. Frequency shift of LiNbO 3 was saturated above 5 N. The reason can be considered that effect of the silicon rubber sheet appeared in the measuring result due to enough acoustic connection between the transducer and the LiNbO 3 . The geometry was evaluated from local stiffness. The frequency shifts in contact with aluminum blocks were measured with the change of contact load. The sample of the aluminum block is shown in Fig. 17 (a). Three samples were used in the following experiments. One of the samples had no hole, another had thickness t = 5 mm and the other had the thickness t = 1 mm. Measured frequency shifts are shown in Fig. 17 (b). The frequency shifts tended to be small with decrease of thickness t. These results show that the Fig. 17. Measurement of natural frequency shifts with the change of contact load in contact with aluminum blocks. Fig. 18. Measurement of natural frequency shifts with the change of contact load in contact with teeth. (a) f 20 f 10 t 20 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 No hole 5mm 1mm f [Hz] Contact load [N] (b) Applied voltage: 20V p-p Contact pointContact point A B A B (a) 0 100 200 300 400 500 0 1 2 3 4 5 6 f [Hz] Contact load [N] Non-damaged (B) (b) Damaged (A) Applied voltage: 20 V p-p MechatronicSystems,Simulation,ModellingandControl116 hollow in the contacted object can be investigated from the frequency shift even though there is no difference in outward aspect. Such elastic parameters estimation and the hollow investigation were applied for diagnosis of dental health. The natural frequency shifts in contact with real teeth were also measured on trial. Figure 18 (a) shows the teeth samples. Sample A is damaged by dental caries and B is not damaged. The plotted points in the picture indicate contact points. To simulate real environment, the teeth were supported by silicon rubber. Measured frequency shifts are shown in Fig. 18 (b). It can be seen that the natural frequency shift of the damaged tooth is smaller than that of healthy tooth. Difference of resonance frequency shifts was observed. To conclude the possibility of dental health diagnosis, a large number of experimental results were required. Collecting such scientific date is our future work. 5. Conclusions A resonance frequency tracing system for Langevin type ultrasonic transducers was built up. The system configuration and the method of tracing were presented. The system does not included a loop filter. This point provided easiness in the contoller design and availability for various transducers. The system was applied to an ultrasonic dental scaler. The traceability of the system with a transducer for the scaler was evaluated from step responses of the oscillating frequency. The settling time was 40 ms. Natural frequency shifts under tip contact with various object, materials and geometries were observed. The shift measurement was applied to diagnosis of dental health. Possibility of the diagnosis was shown. 6. References Ide, M. (1968). Design and Analysis of Ultorasonic Wave Constant Velocity Control Oscillator, Journal of the Institute of Electrical Engineers of Japan, Vol.88-11, No.962, pp.2080-2088. Si, F. & Ide, M. (1995). Measurement on Specium Acousitic Impedamce in Ultrsonic Plastic Welding, Japanese Journal of applied physics, Vol.34, No.5B, pp.2740-2744. Shimizu, H., Saito, S. (1978). Methods for Automatically Tracking the Transducer Resonance by Rectified-Voltage Feedback to VCO, IEICE Technical Report, Vol.US78, No.173, pp.7-13. Hayashi, S. (1991). On the tracking of resonance and antiresonance of a piezoelectric resonator, IEEE Transactions on. Ultrasonic, Ferroelectrics and Frequency Control, Vol.38, No.3, pp.231-236. Hayashi, S. (1992). On the tracking of resonance and antiresonance of a piezoelectric resonator. II. Accurate models of the phase locked loop, IEEE Transactions on. Ultrasonic, Ferroelectrics and Frequency Control, Vol.39, No.6, pp.787-790. Aoyagi, R. & Yoshida, T. (2005), Unified Analysis of Frequency Equations of an Ultrasonic Vibrator for the Elastic Sensor, Ultrasonic Technology, Vol.17, No.1, pp. 27-32. Nishimura, K. et al., (1994), Directional Dependency of Sensitivity of Vibrating Touch sensor, Proceedings of Japan Society of Precision Engineering Spring Conference, pp. 765- 766. NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 117 NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM A.Traslosheros,L.Angel,J.M.Sebastián,F.Roberti,R.CarelliandR.Vaca X New visual Servoing control strategies in tracking tasks using a PKM i A. Traslosheros, ii L. Angel, i J. M. Sebastián, iii F. Roberti, iii R. Carelli and i R. Vaca i DISAM, Universidad Politécnica de Madrid, Madrid, España, ii Facultad de Ingeniera Electrónica Universidad Pontificia Bolivariana Bucaramanga, Colombia, iii Instituto de Automática, Universidad Nacional de San Juan, San Juan, Argentina 1. Introduction Vision allows a robotic system to obtain a lot of information on the surrounding environment to be used for motion planning and control. When the control is based on feedback of visual information is called Visual Servoing. Visual Servoing is a powerful tool which allows a robot to increase its interaction capabilities and tasks complexity. In this chapter we describe the architecture of the Robotenis system in order to design two different control strategies to carry out tracking tasks. Robotenis is an experimental stage that is formed of a parallel robot and vision equipment. The system was designed to test joint control and Visual Servoing algorithms and the main objective is to carry out tasks in three dimensions and dynamical environments. As a result the mechanical system is able to interact with objects which move close to 2m=s . The general architecture of control strategies is composed by two intertwined control loops: The internal loop is faster and considers the information from the joins, its sample time is 0:5ms . Second loop represents the visual Servoing system and it is an external loop to the first mentioned. The second loop represents the main study purpose, it is based in the prediction of the object velocity that is obtained from visual information and its sample time is 8:3ms . The robot workspace analysis plays an important role in Visual Servoing tasks, by this analysis is possible to bound the movements that the robot is able to reach. In this article the robot jacobian is obtained by two methods. First method uses velocity vector-loop equations and the second is calculated from the time derivate of the kinematical model of the robot. First jacobian requires calculating angles from the kinematic model. Second jacobian instead, depends on physical parameters of the robot and can be calculated directly. Jacobians are calculated from two different kinematic models, the first one determines the angles each element of the robot. Fist jacobian is used in the graphic simulator of the system due to the information that can be obtained from it. Second jacobian is used to determine off-line the work space of the robot and it is used in the joint and visual controller of the robot (in real time). The work space of the robot is calculated from the condition number of the jacobian (this is a topic that is not studied in article). The dynamic model of the mechanical system is based on Lagrange multipliers, and it uses forearms and end effector platform of non-negligible inertias for the 8 MechatronicSystems,Simulation,ModellingandControl118 development of control strategies. By means of obtaining the dynamic model, a nonlinear feed forward and a PD control is been applied to control the actuated joints. High requirements are required to the robot. Although requirements were taken into account in the design of the system, additional protection is added by means of a trajectory planner. the trajectory planner was specially designed to guarantee soft trajectories and protect the system from exceeding its Maximum capabilities. Stability analysis, system delays and saturation components has been taken into account and although we do not present real results, we present two cases: Static and dynamic. In previous works (Sebastián, et al. 2007) we present some results when the static case is considered. The present chapter is organized as follows. After this introduction, a brief background is exposed. In the third section of this chapter several aspects in the kinematic model, robot jacobians, inverse dynamic and trajectory planner are described. The objective in this section is to describe the elements that are considered in the joint controller. In the fourth section the visual controller is described, a typical control law in visual Servoing is designed for the system: Position Based Visual Servoing. Two cases are described: static and dynamic. When the visual information is used to control a mechanical system, usually that information has to be filtered and estimated (position and velocity). In this section we analyze two critical aspects in the Visual Servoing area: the stability of the control law and the influence of the estimated errors of the visual information in the error of the system. Throughout this section, the error influence on the system behaviour is analyzed and bounded. 2. Background Vision systems are becoming more and more frequently used in robotics applications. The visual information makes possible to know about the position and orientation of the objects that are presented in the scene and the description of the environment and this is achieved with a relative good precision. Although the above advantages, the integration of visual systems in dynamical works presents many topics which are not solved correctly yet. Thus many important investigation centers (Oda, Ito and Shibata 2009) (Kragic and I. 2005) are motivated to investigate about this field, such as in the Tokyo University ( (Morikawa, et al. 2007), (Kaneko, et al. 2005) and (Senoo, Namiki and Ishikawa 2004) ) where fast tracking (up to 6m=s and 58m=s 2 ) strategies in visual servoing are developed. In order to study and implementing the different strategies of visual servoing, the computer vision group of the UPM (Polytechnic University of Madrid) decided to design the Robotenis vision-robot system. Robotenis system was designed in order to study and design visual servoing controllers and to carry out visual robot tasks, specially, those involved in tracking where dynamic environments are considered. The accomplishment of robotic tasks involving dynamical environments requires lightweight yet stiff structures, actuators allowing for high acceleration and high speed, fast sensor signal processing, and sophisticated control schemes which take into account the highly nonlinear robot dynamics. Motivated by the above reasons we proposed to design and built a high-speed parallel robot equipped with a vision system. a) Fi g T h th e ev a s ys R o th e re a o n pr e S y ha m e se l se l 3. B a ac q ef f th i co n th e g . 1. Robotenis s y h e Robotenis S y st e development o f a luate the level s tem in applica t o botenis S y stem i s e vision s y stem a so n s that motiv a n the performanc e e cision of the m o stem have been p s been optimize d e thod solved t w l ectin g the actua t l ected. Robotenis de s a sicall y , the Rob o q uisition s y stem . f ector speed is 4 m i s article reside s n siderin g static a e camera and t h y stem and its env i em was created t f a tool in order of inte g ration b e t ions with hi g h s inspired b y the is based in one a te us the choic e e of the s y stem, o vements. The ki n p resented b y A n d from the view o w o difficulties: d t ors. In additio n s cription o tenis platform . The parallel ro b m =s. The visual s s in tracking a a nd d y namic cas e h e ball is const a i ronment: Robot, t akin g into accou n to use in visual s e tween a hi g h-s p temporar y requ i DELTA robot ( C camera allocate d e of the robot is a especiall y with r n ematic anal y sis a ng el, et al. (An g e l f both kinematic s d eterminin g the n , the vision s y st e (Fig. 1.a) is for m b ot is based on a sy stem is based o black pin g pon g e . Static case con s a nt. D y namic ca s b) c) camera, back g r o n t mainly two p u s ervoin g researc h p eed parallel ma i rements. The m C lavel 1988) (Sta m d at the end eff e a consequence of r e g ard to velocit y a nd the optimal d l , et al. 2005). Th e s and d y namics r dime n sions of t e m and the con t m ed b y a paral l a DELTA robot a o n a camera in h a g ball. Visual c s iders that the d e s e considers th a o und, ball and pa d u rposes. The firs t h . The second o n nipulator and a m echanical struc t m per and Tsai 19 9 e ctor of the rob o the hi g h requir e y , acceleration a n d esi g n of the Ro b e structure of th e espectivel y . The d t he parallel rob o t rol hardware w a l el robot a n d a a nd its maximu m a nd and its ob j e c ontrol is desi gn e sired distance b e a t the desired d i d dle. t one is n e is to vision t ure of 9 7) and o t. The e ments n d the b otenis e robot d esi g n o t and a s also visual m end- c tive in n ed by e tween i stance NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 119 development of control strategies. By means of obtaining the dynamic model, a nonlinear feed forward and a PD control is been applied to control the actuated joints. High requirements are required to the robot. Although requirements were taken into account in the design of the system, additional protection is added by means of a trajectory planner. the trajectory planner was specially designed to guarantee soft trajectories and protect the system from exceeding its Maximum capabilities. Stability analysis, system delays and saturation components has been taken into account and although we do not present real results, we present two cases: Static and dynamic. In previous works (Sebastián, et al. 2007) we present some results when the static case is considered. The present chapter is organized as follows. After this introduction, a brief background is exposed. In the third section of this chapter several aspects in the kinematic model, robot jacobians, inverse dynamic and trajectory planner are described. The objective in this section is to describe the elements that are considered in the joint controller. In the fourth section the visual controller is described, a typical control law in visual Servoing is designed for the system: Position Based Visual Servoing. Two cases are described: static and dynamic. When the visual information is used to control a mechanical system, usually that information has to be filtered and estimated (position and velocity). In this section we analyze two critical aspects in the Visual Servoing area: the stability of the control law and the influence of the estimated errors of the visual information in the error of the system. Throughout this section, the error influence on the system behaviour is analyzed and bounded. 2. Background Vision systems are becoming more and more frequently used in robotics applications. The visual information makes possible to know about the position and orientation of the objects that are presented in the scene and the description of the environment and this is achieved with a relative good precision. Although the above advantages, the integration of visual systems in dynamical works presents many topics which are not solved correctly yet. Thus many important investigation centers (Oda, Ito and Shibata 2009) (Kragic and I. 2005) are motivated to investigate about this field, such as in the Tokyo University ( (Morikawa, et al. 2007), (Kaneko, et al. 2005) and (Senoo, Namiki and Ishikawa 2004) ) where fast tracking (up to 6m=s and 58m=s 2 ) strategies in visual servoing are developed. In order to study and implementing the different strategies of visual servoing, the computer vision group of the UPM (Polytechnic University of Madrid) decided to design the Robotenis vision-robot system. Robotenis system was designed in order to study and design visual servoing controllers and to carry out visual robot tasks, specially, those involved in tracking where dynamic environments are considered. The accomplishment of robotic tasks involving dynamical environments requires lightweight yet stiff structures, actuators allowing for high acceleration and high speed, fast sensor signal processing, and sophisticated control schemes which take into account the highly nonlinear robot dynamics. Motivated by the above reasons we proposed to design and built a high-speed parallel robot equipped with a vision system. a) Fi g T h th e ev a s ys R o th e re a o n pr e S y ha m e se l se l 3. B a ac q ef f th i co n th e g . 1. Robotenis s y h e Robotenis S y st e development o f a luate the level s tem in applicat o botenis S y stem i s e vision s y stem a so n s that motiv a n the performanc e e cision of the m o stem have been p s been optimize d e thod solved t w l ectin g the actua t l ected. Robotenis de s a sicall y , the Rob o q uisition system. f ector speed is 4 m i s article reside s n sidering static a e camera and t h y stem and its env i em was created t f a tool in order of inte g ration b e t ions with high s inspired b y the is based in one a te us the choic e e of the s y stem, o vements. The ki n p resented b y A n d from the view o w o difficulties: d t ors. In additio n s cription o tenis platform . The parallel ro b m =s. The visual s s in tracking a a nd dynamic cas e h e ball is const a i ronment: Robot, t akin g into accou n to use in visual s e tween a hi g h-s p temporary requ i DELTA robot ( C camera allocate d e of the robot is a especiall y with r n ematic anal y sis a ng el, et al. (An g e l f both kinematic s d eterminin g the n , the vision s y st e (Fig. 1.a) is for m b ot is based on a sy stem is based o black pin g pon g e . Static case con s a nt. D y namic ca s b) c) camera, back g r o n t mainly two p u s ervoin g researc h p eed parallel ma i rements. The m C lavel 1988) (Sta m d at the end eff e a consequence of r e g ard to velocit y a nd the optimal d l , et al. 2005). Th e s and d y namics r dime n sions of t e m and the con t m ed b y a paral l a DELTA robot a o n a camera in h a g ball. Visual c s iders that the d e s e considers th a o und, ball and pa d u rposes. The firs t h . The second o n nipulator and a m echanical struct m per and Tsai 19 9 e ctor of the rob o the hi g h requir e y , acceleration a n d esi g n of the Ro b e structure of th e espectivel y . The d t he parallel rob o t rol hardware w a l el robot a n d a a nd its maximu m a nd and its ob j e c ontrol is desi gn e sired distance b e a t the desired d i d dle. t one is n e is to vision t ure of 9 7) and o t. The e ments n d the b otenis e robot d esi g n o t and a s also visual m end- c tive in n ed by e tween i stance [...]... 1 1 0 0 x 31 21 m 2 z Py a s( 2 2 ) s( 3 2 ) 0 0 1 2 s( 2 3 ) s( 3 3 ) 0 0 m 3z P 1 3 z (27) Where m i z s 1i 2i s 3i m i x c 1i 2i s 3i c i c 3i s i m i y c 1i 2i s 3i s i c 3i c i (28) Note that the right and left part of the eq (27)... ( 13) : 1 2tan i 2 s i 1 1 tan 2 i 2 and 1 1 tan 2 i 2 c i 1 1 tan 2 i 2 (15) And we can obtain the following second order equation: 1 1 M i Fi tan 2 2 i 2Ei tan 2 i M i Fi 0 And the angle can be finally obtained as: (16) 124 Mechatronic Systems, Simulation, Modelling and Control Where ��� , ��� and. .. identify a group of direct kinematic singularities when the last column in the three rows is cero, this is: s( 1 1 2 1 ) s( 3 1 ) s( 1 2 2 2 ) s( 3 2 ) s( 1 3 2 3 ) s( 3 3 )0 When 1i 2i 0 or a) i 1, 2, 3 or when b) 3i 0 or i 1, 2, 3 (30 ) ... member of the first and third row of the eq (2) by using trigonometric identities and making Ψ� � �� sin���� � sin���� � and Υ� � �� � �� cos��� � sin��� �: (5) 122 Mechatronic Systems, Simulation, Modelling and Control i c( 1i ) i s( 1i )C ix (6) i s( 1i ) i c( 1i )C iz Note that from (6) we can obtain: C C i ix s( 1i ) i iz 2 2 i i and C C ... s c 1i 2i 3i bi b c 3i s 3i s 1i 2i 0 and 1i 1i 0 (26) All of them are expressed in the � �coordinate system Substituting equations in (26) into (25) and after operating and simplifying we have: New visual Servoing control strategies in tracking tasks using a PKM m 1x m 2 x m 3x m 1y m2 y m 3y 127 m1z P ... represents the inverse and direct Jacobians or , see respectively An inverse kinematic singularity occurs when Fig 5 a) and b) On the other hand direct kinematic singularities occur when rows of the left matrix become linearly dependent The above is: k 1 m 1 k 2 m 2 k 3 m 3 0 Where k 1 , k 2 , k 3 and not all are cero (29) Equation (29) is not as clear as the right part of the equation...120 Mechatronic Systems, Simulation, Modelling and Control between the ball and the camera can be changed at any time Image processing is conveniently simplified using a black ball on white background The ball is moved through a stick (Fig 1.c) and the ball velocity is close to 2m=s The visual system of the Robotenis platform is formed by a camera located at the end effector (Fig 1.b) and a frame... blocked The third type of singularity it is a combined singularity and can occurs in parallel robots with special architecture or under especial considerations Sometimes singularities can be 126 Mechatronic Systems, Simulation, Modelling and Control identified from the Jacobian almost directly but sometimes Jacobian elements are really complex and singularities are difficult to identify Singularities can... derivative of the eq (19) in the following equation J x x J q q where J x f x, q x and J q f x, q q (20) Note that �� and �� are the time derivate of � and � respectely The direct and the inverse Jacobian can be obtained as the following equations x J Dq and q J I x where J D J x 1J q and J I J q 1J x (21) A robot singularity occurs when the determinant of the Jacobian... degrees of freedom and is not possible to control We will see that the Jacobian relates the actuators velocity with the end effector velocity and singularities occur when the Jacobian rank drops New visual Servoing control strategies in tracking tasks using a PKM 121 Nowadays there are excellent references to study in depth parallel robots, (Tsai 1999), (Merlet 2006) and recently (Bonev and Gosselin 2009) . equations in (26) into (25) and after operating and simplifying we have: ( ) ( ) 0 0 1 1 1 1 1 2 1 3 1 0 ( ) ( ) 0 2 2 2 22 3 2 1 2 0 0 ( ) ( ) 2 3 3 3 1 3 3 3 3 m m m P s s x y z x m m m P. equations in (26) into (25) and after operating and simplifying we have: ( ) ( ) 0 0 1 1 1 1 1 2 1 3 1 0 ( ) ( ) 0 2 2 2 22 3 2 1 2 0 0 ( ) ( ) 2 3 3 3 1 3 3 3 3 m m m P s s x y z x m m m P. ) ( ) 0 1 1 21 3 1 1 2 22 3 2 1 3 2 3 3 3 s s s s s s When 0 1 2i i or 1,2,3i or when 0 3i or 1,2,3i (30 ) a) b) NewvisualServoing control strategiesintrackingtasksusingaPKM
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