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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 MechatronicSystems,Simulation,ModellingandControl120 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 grabber (SONY XC-HR50 and Matrox Meteor 2-MC/4 respectively) The motion system is formed by AC brushless servomotors, Ac drivers (Unidrive) and gearbox. Fig. 2. Cad model and sketch of the robot that it is seen from the side of the i-arm In section 3.1 3.1 Robotenis kinematical models A parallel robot consists of a fixed platform that it is connected to an end effector platform by means of legs. These legs often are actuated by prismatic or rotating joints and they are connected to the platforms through passive joints that often are spherical or universal. In the Robotenis system the joints are actuated by rotating joints and connexions to end effector are by means of passive spherical joints. If we applied the Grüble criterion to the Robotenis robot, we could note that the robot has 9 DOF (this is due to the spherical joints and the chains configurations) but in fact the robot has 3 translational DOF and 6 passive DOF. Important differences with serial manipulators are that in parallel robots any two chains form a closed loop and that the actuators often are in the fixed platform. Above means that parallel robots have high structural stiffness since the end effector is supported in several points at the same time. Other important characteristic of this kind of robots is that they are able to reach high accelerations and forces, this is due to the position of the actuators in the fixed platform and that the end effector is not so heavy in comparison to serial robots. Although the above advantages, parallel robots have important drawbacks: the work space is generally reduced because of collisions between mechanical components and that singularities are not clear to identify. In singularities points the robot gains or losses 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. Nowadays there are excellent references to study in depth parallel robots, (Tsai 1999), (Merlet 2006) and recently (Bonev and Gosselin 2009). For the position analysis of the robot of the Robotenis system two models are presented in order to obtain two different robot jacobians. As was introduced, the first jacobian is utilized in the Robotenis graphic simulator and second jacobian is utilized in real time tasks. Considers the Fig. 2, in this model we consider two reference systems. In the coordinate system are represented the absolute coordinates of the system and the position of the end effector of the robot. In the local coordinate system (allocated in each point ) the position and coordinates ( ) of the i-arm are considered. The first kinematic model is calculated from Fig. 2 where the loop-closure equation for each limb is: A B B C O P PC O A i i i i x y z i x y z i (1) Expressing (note that and the eq. (1) in the coordinate system attached to each limb is possible to obtain: 1 3 1 2 3 1 3 1 2 a c b s c C i i i i i x C b c i y i C a s bs s i z i i i i (2) Where and are related by c s 0 s c 0 0 0 0 0 1 P C h H i i x ix i i P C y i i iy P C z iz (3) In order to calculate the inverse kinematics, from the second row in eq. (2), we have: 1 c 3 C i y i b i (4) can be obtained by summing the squares of and of the eq. (2). 2 2 2 2 2 2 c 3 2 C C C a b a b s i x i y i z i i → 2 2 2 2 2 1 c 2 2 3 C C C a b ix iy iz i a b s i (5) By expanding left member of the first and third row of the eq. (2) by using trigonometric identities and making and : NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 121 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 grabber (SONY XC-HR50 and Matrox Meteor 2-MC/4 respectively) The motion system is formed by AC brushless servomotors, Ac drivers (Unidrive) and gearbox. Fig. 2. Cad model and sketch of the robot that it is seen from the side of the i-arm In section 3.1 3.1 Robotenis kinematical models A parallel robot consists of a fixed platform that it is connected to an end effector platform by means of legs. These legs often are actuated by prismatic or rotating joints and they are connected to the platforms through passive joints that often are spherical or universal. In the Robotenis system the joints are actuated by rotating joints and connexions to end effector are by means of passive spherical joints. If we applied the Grüble criterion to the Robotenis robot, we could note that the robot has 9 DOF (this is due to the spherical joints and the chains configurations) but in fact the robot has 3 translational DOF and 6 passive DOF. Important differences with serial manipulators are that in parallel robots any two chains form a closed loop and that the actuators often are in the fixed platform. Above means that parallel robots have high structural stiffness since the end effector is supported in several points at the same time. Other important characteristic of this kind of robots is that they are able to reach high accelerations and forces, this is due to the position of the actuators in the fixed platform and that the end effector is not so heavy in comparison to serial robots. Although the above advantages, parallel robots have important drawbacks: the work space is generally reduced because of collisions between mechanical components and that singularities are not clear to identify. In singularities points the robot gains or losses 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. Nowadays there are excellent references to study in depth parallel robots, (Tsai 1999), (Merlet 2006) and recently (Bonev and Gosselin 2009). For the position analysis of the robot of the Robotenis system two models are presented in order to obtain two different robot jacobians. As was introduced, the first jacobian is utilized in the Robotenis graphic simulator and second jacobian is utilized in real time tasks. Considers the Fig. 2, in this model we consider two reference systems. In the coordinate system are represented the absolute coordinates of the system and the position of the end effector of the robot. In the local coordinate system (allocated in each point ) the position and coordinates ( ) of the i-arm are considered. The first kinematic model is calculated from Fig. 2 where the loop-closure equation for each limb is: A B B C O P PC O A i i i i x y z i x y z i (1) Expressing (note that and the eq. (1) in the coordinate system attached to each limb is possible to obtain: 1 3 1 2 3 1 3 1 2 a c b s c C i i i i i x C b c i y i C a s bs s i z i i i i (2) Where and are related by c s 0 s c 0 0 0 0 0 1 P C h H i i x ix i i P C y i i iy P C z iz (3) In order to calculate the inverse kinematics, from the second row in eq. (2), we have: 1 c 3 C i y i b i (4) can be obtained by summing the squares of and of the eq. (2). 2 2 2 2 2 2 c 3 2 C C C a b a b s i x i y i z i i → 2 2 2 2 2 1 c 2 2 3 C C C a b ix iy iz i a b s i (5) By expanding left member of the first and third row of the eq. (2) by using trigonometric identities and making and : MechatronicSystems,Simulation,ModellingandControl122 c( ) s( ) 1 1 s( ) ( ) 1 1 C i i i i ix c C i i i i iz (6) Note that from (6) we can obtain: ( ) 1 2 2 C C i iz i ix s i i i and ( ) 1 2 2 C C i ix i iz c i i i (7) Equations in (7) can be related to obtain ߠ ଵ as: 1 tan 1 C C i iz i ix i C C i ix i iz (8) In the use of above angles we have to consider that the “Z” axis that is attached to the center of the fixed platform it is negative in the space that the end effector of the robot will be operated. Taking into account the above consideration, angles are calculated as: 1 c 3 C i y i b 2 2 2 2 2 1 c 2 2 3 C C C a b ix iy iz i a b s i 1 tan 1 C C i iz i ix i C C i ix i iz (9) Second kinematic model is obtained from Fig. 3. Fig. 3. Sketch of the robot taking into account an absolute coordinate reference system. If we consider only one absolute coordinate system in Fig. 3, note that the segment ܤ ܥ is the radius of a sphere that has its center in the point ܤ and its surface in the pointܥ , (all points in the absolute coordinate system). Thus sphere equation as: P θ 1i a b B i 0 xyz X A i Y ø i H i C i h i Z 2 2 2 2 0C B C B C B b i ix ix iy iy iz iz (10) From the Fig. 3 is possible to obtain the point B i = O x y z B i in the absolute coordinate system. c c c s s O x y z H a B i i i i x B H a i y i i i B a i z i where µ i = µ 1i (11) Replacing eq. (11) in eq. (10) and expanding it the constraint equationࢣ is obtained: 2 2 2 2 2 2 2 c 2 s 2 c 2 c c 2 s 2 c s 0 H a b C C C H C H C H a a C i i i x i y i z i i x i i i y i i i i x i i a C a C i z i i y i i (12) In order to simplify, above can be regrouped, thus for the i-limb: s c 0E F M i i i i i (13) Where: 2 c s 2 2 2 2 2 2 2 2 c s F a C C H E a C i i x i i y i i i i z M b a C C C H H C C i i x i y i z i i i x i i y i (14) The following trigonometric identities can be replaced into eq. (13): 1 2tan 2 s 1 2 1 tan 2 i i i and 1 2 1 tan 2 c 1 2 1 tan 2 i i i (15) And we can obtain the following second order equation: 1 1 2 tan 2 tan 0 2 2 M F E M F i i i i i i i (16) And the angle ߠ can be finally obtained as: NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 123 c( ) s( ) 1 1 s( ) ( ) 1 1 C i i i i ix c C i i i i iz (6) Note that from (6) we can obtain: ( ) 1 2 2 C C i iz i ix s i i i and ( ) 1 2 2 C C i ix i iz c i i i (7) Equations in (7) can be related to obtain ߠ ଵ as: 1 tan 1 C C i iz i ix i C C i ix i iz (8) In the use of above angles we have to consider that the “Z” axis that is attached to the center of the fixed platform it is negative in the space that the end effector of the robot will be operated. Taking into account the above consideration, angles are calculated as: 1 c 3 C i y i b 2 2 2 2 2 1 c 2 2 3 C C C a b ix iy iz i a b s i 1 tan 1 C C i iz i ix i C C i ix i iz (9) Second kinematic model is obtained from Fig. 3. Fig. 3. Sketch of the robot taking into account an absolute coordinate reference system. If we consider only one absolute coordinate system in Fig. 3, note that the segment ܤ ܥ is the radius of a sphere that has its center in the point ܤ and its surface in the pointܥ , (all points in the absolute coordinate system). Thus sphere equation as: P θ 1i a b B i 0 xyz X A i Y ø i H i C i h i Z 2 2 2 2 0C B C B C B b i ix ix iy iy iz iz (10) From the Fig. 3 is possible to obtain the point B i = O x y z B i in the absolute coordinate system. c c c s s O x y z H a B i i i i x B H a i y i i i B a i z i where µ i = µ 1i (11) Replacing eq. (11) in eq. (10) and expanding it the constraint equationࢣ is obtained: 2 2 2 2 2 2 2 c 2 s 2 c 2 c c 2 s 2 c s 0 H a b C C C H C H C H a a C i i i x i y i z i i x i i i y i i i i x i i a C a C i z i i y i i (12) In order to simplify, above can be regrouped, thus for the i-limb: s c 0E F M i i i i i (13) Where: 2 c s 2 2 2 2 2 2 2 2 c s F a C C H E a C i i x i i y i i i i z M b a C C C H H C C i i x i y i z i i i x i i y i (14) The following trigonometric identities can be replaced into eq. (13): 1 2tan 2 s 1 2 1 tan 2 i i i and 1 2 1 tan 2 c 1 2 1 tan 2 i i i (15) And we can obtain the following second order equation: 1 1 2 tan 2 tan 0 2 2 M F E M F i i i i i i i (16) And the angle ߠ can be finally obtained as: [...]... 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 ) 128 Mechatronic Systems, Simulation, Modelling and Control c) d) Fig 5 a) Inverse kinematic singularities if b) Inverse kinematic singularities where... 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)... MF 2 i Considering that i can be replaced in (35 ) and 2 M 2 F 2 2 i 1E 1 E M 1 E F 1 2 E E 1 2 M M 1 2 F F 1 1 2 2 (35 ) i (36 ) 130 Mechatronic Systems, Simulation, Modelling and Control On the other hand we know that: Fi 2 aC i x c i 2 aC i y s i M i ... 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 ... constant and (40) and the time derivate C ix C iy Ciz P x P y Pz (41) New visual Servoing control strategies in tracking tasks using a PKM 131 Substituting the above equation in (40) and the expanding the equation, finally the inverse Jacobian of the robot is given by: D 1 1x 4 D 2 2x D 3 3x D 1y D2 y D3y... (generalized and redundant coordinates) Lagrangian equations of the first type can be expressed: k i d L L Q j i q q dt j i 1 q j j j 1, 2 , , n Where is the constraint equation, is the Lagrangian multiplier, constraint equation, is the number of coordinates (Note that ( 43) is the number of 132 Mechatronic Systems, Simulation, Modelling and Control and in our... the body in the center of mass, is the moment of inertia and is the body's angular velocity Thus the total kinetic energy of the robot is (mobile platform, 3 input links and 3 input shafts, and 3 parallelogram links): 1 2 2 2 2 2 2 2 2 2 2 2 2 K m p p x p y p z m a a 2 I 1 2 3 m b a 2 1 2 3 3m b p x p y p z 2 (47) ),... above to the Robotenis system, we have that , and can define the full system and can be chosen as generalized coordinates moreover to simplify the Lagrange expression and to solve the Lagrangian by means of Lagrange multipliers we choose 3 additional redundant coordinates , and Thus the generalized coordinates are: and External forces and position, velocity and acceleration of the end effector (mobile... 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...124 Mechatronic Systems, Simulation, Modelling and Control Where ��� , ��� and ��� are: E E 2 F 2 M 2 i i i i M i Fi i 2 t a n 1 si ci C P c i x x i C P s i y y i C P 0 i z z 0 (17) 0 h i 0 0 1 0 (18) 3. 2 Robot Jacobians In robotics, . 2 1 3 1 12 2 2 32 1 3 23 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) Mechatronic Systems, Simulation, Modelling and Control1 28 c). 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
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