Mechatronic Systems, Simulation, Modeling and Control 2012 Part 4 pptx

MechatronicSystems,Simulation,ModellingandControl134         = 2 i P h H ac c x i i P x         ǡ ൌͳǡʹǡ͵       =2 i P h H ac s y i i P y               ǡ ൌͳǡʹǡ͵     = 2 i P as z i P z      ǡ ൌͳǡʹǡ͵         1 = 2 1 1 1 1 1 a c P c P h H s P c x y z                         ߲߁ ଶ ߲ߠ ଶ ൌ ߲߁ ଷ ߲ߠ ଷ ൌͲ         2 = 2 2 2 2 2 2 a c P c P h H s P c x y z                         ߲߁ ଵ ߲ߠ ଵ ൌ ߲߁ ଷ ߲ߠ ଷ ൌͲ         3 = 2 3 3 3 3 3 a c P c P h H s P c x y z                         ߲߁ ଵ ߲ߠ ଵ ൌ ߲߁ ଶ ߲ߠ ଶ ൌͲ Once we have the derivatives above, they are substituted into equation (44) and the Lagrangian multipliers are calculated. Thus for݆ൌͳǡʹǡ͵.                                                             2 1 1 1 2 2 2 3 3 3 3 P h H a c c P h H a c c P h H a c c x x x m m p F p b x P x                                                                         2 1 1 1 2 2 2 3 3 3 3 P h H a c s P h H a c s P h H a c s y y y m m p F p b y P y                                                2 3 3 1 1 2 2 3 3 P a s P a s P a s m m p g m m F z z z p b z p b P z (50) Note that ܨ ௉௫ ǡܨ ௉௬ and ܨ ௉௭ are the componentsሺܳ ௝ ǡ݆ൌͳǡʹǡ͵ሻ of an external force that is applied on the mobile platform. Once that the Lagrange multipliers are calculated the (45) is solved (where݆ൌͶǡͷǡ͸) and for the actuator torquesሺ߬ ௖௞ ൌܳ ௞ାଷ ǡ݇ൌͳǡʹǡ͵ሻ.               2 2 2 1 1 1 1 1 1 1 1 m a I m a m m g a c a P c P s h H s P c c a b a b x y z                                              2 2 2 2 2 2 2 2 2 2 2 m a I m a m m g a c a P c P s h H s P c c a b a b x y z                                              2 2 2 3 3 3 3 3 3 3 3 m a I m a m m g a c a P c P s h H s P c c a b a b x y z                                (51) Fig. 7. Basic architecture of the control system of the Robotenis platform. The results above are used in real time to control each joint independently. The joint controller is based in a classical computed-torque controller plus a PD controller (Santibañez and Kelly 2001). The objective of the computed-torque controller is to Feedback a signal that cancels the effects of gravity, friction, the manipulator inertia tensor, and Coriolis and centrifugal force, see in Fig. 7. 3.4 Trajectory planner The structure of the visual controller of the Robotenis system is called dynamic position- based on a look-and-move structure (Corke 1993). The above structure is formed of two intertwined control loops: the first is faster and makes use of joints feedback, the second is external to the first one and makes use of the visual information feedback, see in Fig. 7. Once that the visual control loop analyzes the visual information then, this is sent to the joint controller as a reference. In other words, in tracking tasks the desired next position is calculated in the visual controller and the joint controller forces to the robot to reach it. Two control loops are incorporated in the Robotenis system: the joint loop is calculated each 0.5 ms; at this point dynamical model, kinematical model and PD action are retrofitted. The external loop is calculated each 8.33 ms and it was mentioned that uses the visual data. As the internal loop is faster than the external, a trajectory planner is designed in order to accomplish different objectives: The first objective is to make smooth trajectories in order to avoid abrupt movements of the robot elements. The trajectory planner has to guarantee that the positions and its following 3 derivates are continuous curves (velocity, acceleration and jerk). The second objective is to guarantee that the actuators of the robot are not saturated and that the robot specifications are not exceeded, the robot limits are: MVS= maximum allowed velocity, MAS= maximum allowed acceleration and MJS= maximum allowed jerk (maximum capabilities of the robot are taken from the end effector). In the Robotenis system maximum capabilities are:     ,     , and    .      Robot Vision system Visual controller Inverse and direct Kinematics PD Joint Controller End effector Trajectory planner Inverse Dynamics                                       NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 135         = 2 i P h H ac c x i i P x         ǡ ൌͳǡʹǡ͵       =2 i P h H ac s y i i P y               ǡ ൌͳǡʹǡ͵     = 2 i P as z i P z      ǡ ൌͳǡʹǡ͵         1 = 2 1 1 1 1 1 a c P c P h H s P c x y z                         ߲߁ ଶ ߲ߠ ଶ ൌ ߲߁ ଷ ߲ߠ ଷ ൌͲ         2 = 2 2 2 2 2 2 a c P c P h H s P c x y z                         ߲߁ ଵ ߲ߠ ଵ ൌ ߲߁ ଷ ߲ߠ ଷ ൌͲ         3 = 2 3 3 3 3 3 a c P c P h H s P c x y z                         ߲߁ ଵ ߲ߠ ଵ ൌ ߲߁ ଶ ߲ߠ ଶ ൌͲ Once we have the derivatives above, they are substituted into equation (44) and the Lagrangian multipliers are calculated. Thus for݆ൌͳǡʹǡ͵.                                                             2 1 1 1 2 2 2 3 3 3 3 P h H a c c P h H a c c P h H a c c x x x m m p F p b x P x                                                                         2 1 1 1 2 2 2 3 3 3 3 P h H a c s P h H a c s P h H a c s y y y m m p F p b y P y                                                2 3 3 1 1 2 2 3 3 P a s P a s P a s m m p g m m F z z z p b z p b P z (50) Note that ܨ ௉௫ ǡܨ ௉௬ and ܨ ௉௭ are the componentsሺܳ ௝ ǡ݆ൌͳǡʹǡ͵ሻ of an external force that is applied on the mobile platform. Once that the Lagrange multipliers are calculated the (45) is solved (where݆ൌͶǡͷǡ͸) and for the actuator torquesሺ߬ ௖௞ ൌܳ ௞ାଷ ǡ݇ൌͳǡʹǡ͵ሻ.               2 2 2 1 1 1 1 1 1 1 1 m a I m a m m g a c a P c P s h H s P c c a b a b x y z                                              2 2 2 2 2 2 2 2 2 2 2 m a I m a m m g a c a P c P s h H s P c c a b a b x y z                                              2 2 2 3 3 3 3 3 3 3 3 m a I m a m m g a c a P c P s h H s P c c a b a b x y z                                (51) Fig. 7. Basic architecture of the control system of the Robotenis platform. The results above are used in real time to control each joint independently. The joint controller is based in a classical computed-torque controller plus a PD controller (Santibañez and Kelly 2001). The objective of the computed-torque controller is to Feedback a signal that cancels the effects of gravity, friction, the manipulator inertia tensor, and Coriolis and centrifugal force, see in Fig. 7. 3.4 Trajectory planner The structure of the visual controller of the Robotenis system is called dynamic position- based on a look-and-move structure (Corke 1993). The above structure is formed of two intertwined control loops: the first is faster and makes use of joints feedback, the second is external to the first one and makes use of the visual information feedback, see in Fig. 7. Once that the visual control loop analyzes the visual information then, this is sent to the joint controller as a reference. In other words, in tracking tasks the desired next position is calculated in the visual controller and the joint controller forces to the robot to reach it. Two control loops are incorporated in the Robotenis system: the joint loop is calculated each 0.5 ms; at this point dynamical model, kinematical model and PD action are retrofitted. The external loop is calculated each 8.33 ms and it was mentioned that uses the visual data. As the internal loop is faster than the external, a trajectory planner is designed in order to accomplish different objectives: The first objective is to make smooth trajectories in order to avoid abrupt movements of the robot elements. The trajectory planner has to guarantee that the positions and its following 3 derivates are continuous curves (velocity, acceleration and jerk). The second objective is to guarantee that the actuators of the robot are not saturated and that the robot specifications are not exceeded, the robot limits are: MVS= maximum allowed velocity, MAS= maximum allowed acceleration and MJS= maximum allowed jerk (maximum capabilities of the robot are taken from the end effector). In the Robotenis system maximum capabilities are:     ,     , and    .      Robot Vision system Visual controller Inverse and direct Kinematics PD Joint Controller End effector Trajectory planner Inverse Dynamics                                       MechatronicSystems,Simulation,ModellingandControl136 Fig. 8. Flowchart of the Trajectory planner. And the third objective is to guarantee that the robot is in prepared to receive the next reference, in this point the trajectory planner imposes a cero jerk and acceleration at the end of each trajectory. In order to design the trajectory planner it has to be considered the system constrains, the maximum jerk and maximum acceleration. As a result we have that the jerk can be characterized by:       s n s n m a x 3 k j e j e T        (52) Where   is the maximum allowed jerk, ,            ,  is the real time clock,   and   represent the initial and final time of the trajectory. Supposing that the initial and final acceleration are cero and by considering that the acceleration can be obtained from the integral of the eq. (52) and that if      then       we have:       m a x 1 c o s T j a       (53) By supposing that the initial velocity (  ) is different of cero, the velocity can be obtained from the convenient integral of the eq. (53). A new   is defined and the end effector moves towards the new   (considering the maximum capabilities of the robot) 1 no yes   is used in the trajectory planner and   is reached.                                                  1                                          2 s i n m a x T j v v i                (54) Finally, supposing ݌ ௜ as the initial position and integrating the eq. (54) to obtain the position:     3 2 s 1 m a x 2 2 2 T j c o p p T v i i                      (55) We can see that the final position ݌ ௙ is not defined in the eq. (55). ݌ ௙ is obtained by calculating not to exceed the maximum jerk and the maximum acceleration. From eq. (53) the maximum acceleration can be calculated as:     m a x m a x 1 s m a x 1 m a x 2 2 T j a a c o j T           (56) The final position of the eq. (55) is reached when߬ൌͳ, thus substituting eq. (56) in eq. (55) when߬ൌͳ, we have: 4 m a x 2 a p p T v f i i T          (57) By means of the eq. (57) ܽ ௠௔௫ can be calculated but in order to take into account the maximum capabilities of the robot. Maximum capabilities of the robot are the maximum speed, acceleration and jerk. By substituting the eq. (56) in (54) and operating, we can obtain the maximum velocity in terms of the maximum acceleration and the initial velocity.   2 s i n m a x m a x m a x 2 1 T j T a v v v i i                   (58) Once we calculate a m a x from eq. (57) the next is comparing the maximum capabilities from equations (56) and (58). If maximum capabilities are exceeded, then the final position of the robot is calculated from the maximum capabilities and the sign of a m a x (note that in this case the robot will not reach the desired final position). See the Fig. 8. The time history of sample trajectories is described in the Fig. 9 (in order to plot in the same chart, all curves are normalized). This figure describes when the necessary acceleration to achieve a target, is bigger than the maximum allowed. It can be observed that the fifth target position ( 83:3ms ) is not reached but the psychical characteristics of the robot actuators are not exceeded. NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 137 Fig. 8. Flowchart of the Trajectory planner. And the third objective is to guarantee that the robot is in prepared to receive the next reference, in this point the trajectory planner imposes a cero jerk and acceleration at the end of each trajectory. In order to design the trajectory planner it has to be considered the system constrains, the maximum jerk and maximum acceleration. As a result we have that the jerk can be characterized by:       s n s n m a x 3 k j e j e T        (52) Where   is the maximum allowed jerk, ,            ,  is the real time clock,   and   represent the initial and final time of the trajectory. Supposing that the initial and final acceleration are cero and by considering that the acceleration can be obtained from the integral of the eq. (52) and that if      then       we have:       m a x 1 c o s T j a       (53) By supposing that the initial velocity (  ) is different of cero, the velocity can be obtained from the convenient integral of the eq. (53). A new   is defined and the end effector moves towards the new   (considering the maximum capabilities of the robot) 1 no yes   is used in the trajectory planner and   is reached.                                                  1                                          2 s i n m a x T j v v i                (54) Finally, supposing ݌ ௜ as the initial position and integrating the eq. (54) to obtain the position:     3 2 s 1 m a x 2 2 2 T j c o p p T v i i                      (55) We can see that the final position ݌ ௙ is not defined in the eq. (55). ݌ ௙ is obtained by calculating not to exceed the maximum jerk and the maximum acceleration. From eq. (53) the maximum acceleration can be calculated as:     m a x m a x 1 s m a x 1 m a x 2 2 T j a a c o j T           (56) The final position of the eq. (55) is reached when߬ൌͳ, thus substituting eq. (56) in eq. (55) when߬ൌͳ, we have: 4 m a x 2 a p p T v f i i T          (57) By means of the eq. (57) ܽ ௠௔௫ can be calculated but in order to take into account the maximum capabilities of the robot. Maximum capabilities of the robot are the maximum speed, acceleration and jerk. By substituting the eq. (56) in (54) and operating, we can obtain the maximum velocity in terms of the maximum acceleration and the initial velocity.   2 s i n m a x m a x m a x 2 1 T j T a v v v i i                   (58) Once we calculate a m a x from eq. (57) the next is comparing the maximum capabilities from equations (56) and (58). If maximum capabilities are exceeded, then the final position of the robot is calculated from the maximum capabilities and the sign of a m a x (note that in this case the robot will not reach the desired final position). See the Fig. 8. The time history of sample trajectories is described in the Fig. 9 (in order to plot in the same chart, all curves are normalized). This figure describes when the necessary acceleration to achieve a target, is bigger than the maximum allowed. It can be observed that the fifth target position ( 83:3ms ) is not reached but the psychical characteristics of the robot actuators are not exceeded. MechatronicSystems,Simulation,ModellingandControl138 Fi g n o 4. C o w o re s ca m co o Tr a ki n is c an its Al H u ba 20 0 ob p o co n ca s ef f g . 9. Example of t o t reached (whe n Description o f o ordinated s y ste m o rd coordinate s y s pectivel y . Othe r m era coordinate o rdinate syste m a nsformation m a n ematical model c alculated b y m e d b y means of t h calculation requ i thou g h there ar u tchinson, Visua l sed in position 0 6). Schematic c o tained thou g h t h o sition ( n siderin g when s e). Once the er r f ector. B y means o t he time respons e t = 83:3ms ) but r f the visual co n m s are shown in y stem, the end-e f r notations defi n system, w p e rep r m . w p e is obta i a trices are w R e and e R c is obtai e ans of the mass c h e diameter of th e i res sub-pixel pr e e advanced co n l servo control. I (Chaumette and o ntrol can be ap p h e difference bet w ). In the prese n desired position r or is obtained, t o f the tra j ector y p e of the tra j ector y r obot capabilitie s n troller. the Fig. 10 and a f fector robot s y s n ed are: c p b rep r r esents the positi ined by mean s , w R c and e R ned from the ca m c enter of the pro j e ball ( ). Dia m e cisio n technique s n trollers that ha v I Advanced app r Hutchinson, Vi s p reciated in the F w een the referen c n t article the co n is fixed (static c t he controller ca l p lanner and the J y planner, note t h s are not exceede d a re § w , § e , and § tem and the ca m r esents the posi t on of the robot e n s of the direc R c where w R e i m era calibratio n . j ection of the bal l m eter of the ball is s . v e been propos e r oaches 2007), t h s ual servo contr o F ig. 7 and Fig. 1 c e position ( n trol si g nal is o c ase) and when i l culates the desi r J acobian matrix, a h at the target pos i d . § c which repres e m era coordinate s t ion of the ball n d effector in th e t kinematical m i s calculated fr o The position of t h l on the image ( principally criti c e d b y (Chaumet t h e controller sele c o l. I. Basic appr o 0, the error func ) and the me a btained as a re s i t is variable (d y r ed velocit y of t h a ll the j oint moti o i tion is e nt the sy stem in the e word m odel. o m the h e ball ) c al and t e and c ted is o aches c tion is a sured s ult of y namic h e end o ns are calculated. Signals are supposed as known in the instant ݇ܶ, where ܶ is the sample time in the visual control loop (in order to simplify we suppose ݇ܶ as ݇). Fig. 10. Coordinated systems that are considered in the controller. 4.1 Static case In the Fig. 7 can be observed that the position error can be expressed as follows:   * ( ) c c e k p p k b b   (59) In this section ݌ ௖ ௕ כ ሺ ݇ ሻ is the desired position of the ball in the camera coordinate system and in this section is considered as constant and known. ݌ ௖ ௕ ሺ ݇ ሻ is the position of the ball in the camera coordinate system. Thus, by considering the position in the word coordinate system:       ( ) c c w w p k R p k p k b w b c   (60) If (60) is substituted into (59) then we obtain: * ( ) ( ( ) ( ) ) c c w w e k p R p k p k b w b c    (61) The system (robot) is supposed stable and in order to guarantee that the error will decrease exponentially we choose: c  C Y ( , , ) b b b X Y Z c b p w b p w c p C X C Z w  w Z w X w Y NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 139 Fi g n o 4. C o w o re s ca m co o Tr a ki n is c an its Al H u ba 20 0 ob p o co n ca s ef f g . 9. Example of t o t reached (whe n Description o f o ordinated s y ste m o rd coordinate s y s pectivel y . Othe r m era coordinate o rdinate s y ste m a nsformation m a n ematical model c alculated b y m e d b y means of t h calculation requ i thou g h there ar u tchinson, Visua l sed in position 0 6). Schematic c o tained thou g h t h o sition ( n siderin g when s e). Once the er r f ector. B y means o t he time respons e t = 83:3ms ) but r f the visual co n m s are shown in y stem, the end-e f r notations defi n system, w p e rep r m . w p e is obta i a trices are w R e and e R c is obtai e ans of the mass c h e diameter of th e i res sub-pixel pr e e advanced co n l servo control. I (Chaumette and o ntrol can be ap p h e difference bet w ). In the prese n desired position r or is obtained, t o f the tra j ector y p e of the tra j ector y r obot capabilitie s n troller. the Fig. 10 and a f fector robot s y s n ed are: c p b rep r r esents the positi i ned b y mean s , w R c and e R ned from the ca m c enter of the pro j e ball ( ). Dia m e cisio n technique s n trollers that ha v I Advanced app r Hutchinson, Vi s p reciated in the F w een the referen c n t article the co n is fixed (static c t he controller ca l p lanner and the J y planner, note t h s are not exceede d a re § w , § e , and § tem and the ca m r esents the posi t on of the robot e n s of the direc R c where w R e i m era calibratio n . j ection of the bal l m eter of the ball is s . v e been propos e r oaches 2007), t h s ual servo contr o F i g . 7 and Fig. 1 c e position ( n trol si g nal is o c ase) and when i l culates the desi r J acobian matrix, a h at the target pos i d . § c which repres e m era coordinate s t ion of the ball n d effector in th e t kinematical m i s calculated fr o The position of t h l on the image ( principally criti c e d b y (Chaumet t h e controller sele c o l. I. Basic appr o 0, the error fun c ) and the me a btained as a re s i t is variable (d y r ed velocit y of t h a ll the j oint moti o i tion is e nt the sy stem in the e word m odel. o m the h e ball ) c al and t e and c ted is o aches c tion is a sured s ult of y namic h e end o ns are calculated. Signals are supposed as known in the instant ݇ܶ, where ܶ is the sample time in the visual control loop (in order to simplify we suppose ݇ܶ as ݇). Fig. 10. Coordinated systems that are considered in the controller. 4.1 Static case In the Fig. 7 can be observed that the position error can be expressed as follows:   * ( ) c c e k p p k b b   (59) In this section ݌ ௖ ௕ כ ሺ ݇ ሻ is the desired position of the ball in the camera coordinate system and in this section is considered as constant and known. ݌ ௖ ௕ ሺ ݇ ሻ is the position of the ball in the camera coordinate system. Thus, by considering the position in the word coordinate system:       ( ) c c w w p k R p k p k b w b c   (60) If (60) is substituted into (59) then we obtain: * ( ) ( ( ) ( ) ) c c w w e k p R p k p k b w b c    (61) The system (robot) is supposed stable and in order to guarantee that the error will decrease exponentially we choose: c  C Y ( , , ) b b b X Y Z c b p w b p w c p C X C Z w  w Z w X w Y MechatronicSystems,Simulation,ModellingandControl140     0e k e k w h e r e      (62) Deriving (60) and supposing that     and    are constant, we obtain:       ( ) c w w e k R v k v k w b c     (63) Substituting (61)and (63)into (62), we obtain:       * ( ) w w c T c c v k v k R p p k c b w b b     (64) Where    and    represent the camera and ball velocities (in the word coordinate system) respectively. Since                the control law can be expressed as:       *w c T c c u k v k R p p k b w b b           (65) The equation (65) is composed by two components: a component which predicts the position of the ball (       ) and the other contains the tracking error (            ).The ideal control scheme (65) requires a perfect knowledge of all its components, which is not possible, a more realistic approach consist in generalizing the previous control as       * ˆ ˆ ˆ ( ) w w c T c c u k v k v k R p p k c b w b b            (66) Where, the estimated variables are represented by the carets. A fundamental aspect in the performing of the visual controller is the adjustment of, therefore  will be calculated in the less number of sample time periods and will consider the system limitations. This algorithm is based in the future positions of the camera and the ball; this lets to the robot reaching the control objective (            ). By supposing “n” small, the future position (in the  instant) of the ball and the camera in the word coordinate system are:     ˆ ˆ ˆ w w w p k n p v k n T b b b    (67)     w w w p k n p v k n T c c c    (68) Where  is the visual sample time period. As was mentioned, the control objective is to reach the target position in the shorter time as be possible. By taking into account eq. (61), the estimated value    and by considering that the error is cero  in the instant, we have:     * ˆ 0 c c w w p R p k n p k n b w b c            (69) Substituting (67) and (68) into (69), we obtain (70).         * ˆ ˆ c c w w w w p R p k v k n T p k v k n T b w b b c c           (70) Taking into account that the estimate of the velocity of the ball is:       ˆ ˆ ( ) c c w w p k R p k p k b w b c   (71) Then the control law can be expressed as:         1 * ˆ ˆ ˆ w w c T c c u k v k v k R p p k c b w b b nT           (72) If (66) and (72) are compared, we can obtain the λ parameter as: 1 nT   (73) The equation (73) gives a criterion for adjust  as a function of the number of samples required  for reaching the control target. The visual control architecture proposed above does not consider the physical limitations of the system such as delays and the maximum operation of the components. If we consider that the visual information (       ) has a delay of 2 sampling times () with respect to the joint information, then at an instant, the future position of the ball can be:         ˆ ˆ ˆ w w w p k n p k r v k r T n r b b b       (74) The future position of the camera in the word coordinate system is given by (68). Using the (74) is possible to adjust the  for the control law by considering the following aspect: -The wished velocity of the end effector is represented by (72). In physical systems the maximal velocity is necessary to be limited. In our system the maximal velocity of each joint is taken into account to calculate. Value of depends of the instant position of the end effector. Therefore through the robot jacobian is possible to know the velocity that requires moving each joint and the value of  is adjusted to me more constrained joint (maximal velocity of the joint). 4.2 Dynamic case Static case is useful when the distance between the ball and the camera must be fixed but in future tasks it is desirable that this distance change in real time. In this section, in order to carry out above task a dynamic visual controller is designed. This controller receives two parameters as are the target position and the target velocity. By means of above parameters the robot can be able to carry out several tasks as are: catching, touching or hitting objects NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 141     0e k e k w h e r e       (62) Deriving (60) and supposing that     and    are constant, we obtain:       ( ) c w w e k R v k v k w b c     (63) Substituting (61)and (63)into (62), we obtain:       * ( ) w w c T c c v k v k R p p k c b w b b     (64) Where    and    represent the camera and ball velocities (in the word coordinate system) respectively. Since                the control law can be expressed as:       *w c T c c u k v k R p p k b w b b           (65) The equation (65) is composed by two components: a component which predicts the position of the ball (       ) and the other contains the tracking error (            ).The ideal control scheme (65) requires a perfect knowledge of all its components, which is not possible, a more realistic approach consist in generalizing the previous control as       * ˆ ˆ ˆ ( ) w w c T c c u k v k v k R p p k c b w b b            (66) Where, the estimated variables are represented by the carets. A fundamental aspect in the performing of the visual controller is the adjustment of, therefore  will be calculated in the less number of sample time periods and will consider the system limitations. This algorithm is based in the future positions of the camera and the ball; this lets to the robot reaching the control objective (            ). By supposing “n” small, the future position (in the  instant) of the ball and the camera in the word coordinate system are:     ˆ ˆ ˆ w w w p k n p v k n T b b b    (67)     w w w p k n p v k n T c c c    (68) Where  is the visual sample time period. As was mentioned, the control objective is to reach the target position in the shorter time as be possible. By taking into account eq. (61), the estimated value    and by considering that the error is cero  in the instant, we have:     * ˆ 0 c c w w p R p k n p k n b w b c            (69) Substituting (67) and (68) into (69), we obtain (70).         * ˆ ˆ c c w w w w p R p k v k n T p k v k n T b w b b c c           (70) Taking into account that the estimate of the velocity of the ball is:       ˆ ˆ ( ) c c w w p k R p k p k b w b c   (71) Then the control law can be expressed as:         1 * ˆ ˆ ˆ w w c T c c u k v k v k R p p k c b w b b nT           (72) If (66) and (72) are compared, we can obtain the λ parameter as: 1 nT   (73) The equation (73) gives a criterion for adjust  as a function of the number of samples required  for reaching the control target. The visual control architecture proposed above does not consider the physical limitations of the system such as delays and the maximum operation of the components. If we consider that the visual information (       ) has a delay of 2 sampling times () with respect to the joint information, then at an instant, the future position of the ball can be:         ˆ ˆ ˆ w w w p k n p k r v k r T n r b b b       (74) The future position of the camera in the word coordinate system is given by (68). Using the (74) is possible to adjust the  for the control law by considering the following aspect: -The wished velocity of the end effector is represented by (72). In physical systems the maximal velocity is necessary to be limited. In our system the maximal velocity of each joint is taken into account to calculate. Value of depends of the instant position of the end effector. Therefore through the robot jacobian is possible to know the velocity that requires moving each joint and the value of  is adjusted to me more constrained joint (maximal velocity of the joint). 4.2 Dynamic case Static case is useful when the distance between the ball and the camera must be fixed but in future tasks it is desirable that this distance change in real time. In this section, in order to carry out above task a dynamic visual controller is designed. This controller receives two parameters as are the target position and the target velocity. By means of above parameters the robot can be able to carry out several tasks as are: catching, touching or hitting objects MechatronicSystems,Simulation,ModellingandControl142 that are static or while are moving. In this article the principal objective is the robot hits the ball in a specific point and with a specific velocity. In this section     is no constant and        is considered instead,        is the relative target velocity between the ball and the camera and the error between the target and measured position is expressed as:       *c c e k p k p k b b   (75) Substituting (60) in (75) and supposing that only    is constant, we obtain its derivate as:           *c c w w e k v k R v k v k b w b c     (76) Where        is considered as the target velocity to carry out the task. By following a similar analysis that in the static case, our control law would be:               * * ˆ ˆ ˆ w w c T c c c u k v k v k R p k p k v k c b w b b b             (77) Where       and       are estimated and are the position and the velocity of the ball. Just as to the static case, from the eq. (61) is calculated if the error is cero in .         * ˆ 0 c c w w p k n R p k n p k n b w b c       (78) Substituting (67) and (68) in (78) and taking into account the approximation:       * * *c c c p k n p k nT v k b b b    (79) Is possible to obtain:               * * ˆ ˆ 0 c c c w w w w p k n T v k R p k nT v k p k n T v k b b w b b c c       (80) Taking into account the eq. (71), the control law can be obtained as:               1 * * ˆ ˆ ˆ w w c T c c c u k v k v k R p k p k v k c b w b b b n T            (81) From eq. (77) it can be observed that  can be    where “” is “small enough”. 4.3 Stability and errors influence By means of Lyapunov analysis is possible to probe the system stability; it can be demonstrated that the error converges to zero if ideal conditions are considered; otherwise it can be probed that the error will be bounded under the influence of the estimation errors and non modelled dynamics. We choose a Lyapunov function as:     1 2 T V e k e k (82)     T V e k e k   (83) If the error behavior is described by the eq. (62) then ( ) ( ) 0 T V e k e k      (84) The eq. (84) implies      when  and this is only true if           . Note that above is not true due to estimations (              ) and dynamics that are not modelled. Above errors are expressed in    and is more realistic to consider (           ):         ˆ w w u k v k v k k c c     (85) By considering the estimated velocity of the ball (    ) in eq. (76) and substituting the eq. (85) is possible to obtain:           *c c w w e k v k R v k v k k b w b c             (86) Note that estimate errors are already included in. Consequently the value of       is:             * *w w c T c c c v k v k R p k p k v k c b w b b b            (87) Substituting eq. (87) in (86):                   * * *c c w w c T c c c e k v k R v k v k R p k p k v k k b w b b w b b b                       (88) Operating in order to simplify:                   * * *c c c c c c e k v k v k p k p k R k R k e k b b b b w w             (89) Taking into account the Lyapunov function in eq. (82):             T T T c V e k e k e k e k e k R k w        (90) Thus, by considering   we have that the following condition has to be satisfied: NewvisualServoingcontrolstrategiesintrackingtasksusingaPKM 143 that are static or while are moving. In this article the principal objective is the robot hits the ball in a specific point and with a specific velocity. In this section     is no constant and        is considered instead,        is the relative target velocity between the ball and the camera and the error between the target and measured position is expressed as:       *c c e k p k p k b b   (75) Substituting (60) in (75) and supposing that only    is constant, we obtain its derivate as:           *c c w w e k v k R v k v k b w b c     (76) Where        is considered as the target velocity to carry out the task. By following a similar analysis that in the static case, our control law would be:               * * ˆ ˆ ˆ w w c T c c c u k v k v k R p k p k v k c b w b b b             (77) Where       and       are estimated and are the position and the velocity of the ball. Just as to the static case, from the eq. (61) is calculated if the error is cero in .         * ˆ 0 c c w w p k n R p k n p k n b w b c       (78) Substituting (67) and (68) in (78) and taking into account the approximation:       * * *c c c p k n p k nT v k b b b    (79) Is possible to obtain:               * * ˆ ˆ 0 c c c w w w w p k n T v k R p k nT v k p k n T v k b b w b b c c       (80) Taking into account the eq. (71), the control law can be obtained as:               1 * * ˆ ˆ ˆ w w c T c c c u k v k v k R p k p k v k c b w b b b n T            (81) From eq. (77) it can be observed that  can be    where “” is “small enough”. 4.3 Stability and errors influence By means of Lyapunov analysis is possible to probe the system stability; it can be demonstrated that the error converges to zero if ideal conditions are considered; otherwise it can be probed that the error will be bounded under the influence of the estimation errors and non modelled dynamics. We choose a Lyapunov function as:     1 2 T V e k e k (82)     T V e k e k   (83) If the error behavior is described by the eq. (62) then ( ) ( ) 0 T V e k e k     (84) The eq. (84) implies      when  and this is only true if           . Note that above is not true due to estimations (              ) and dynamics that are not modelled. Above errors are expressed in    and is more realistic to consider (           ):         ˆ w w u k v k v k k c c     (85) By considering the estimated velocity of the ball (    ) in eq. (76) and substituting the eq. (85) is possible to obtain:           *c c w w e k v k R v k v k k b w b c             (86) Note that estimate errors are already included in. Consequently the value of       is:             * *w w c T c c c v k v k R p k p k v k c b w b b b            (87) Substituting eq. (87) in (86):                   * * *c c w w c T c c c e k v k R v k v k R p k p k v k k b w b b w b b b                       (88) Operating in order to simplify:                   * * *c c c c c c e k v k v k p k p k R k R k e k b b b b w w             (89) Taking into account the Lyapunov function in eq. (82):             T T T c V e k e k e k e k e k R k w        (90) Thus, by considering   we have that the following condition has to be satisfied: [...]... ( x)) ¨ + p3 ( x M3 − r1 ( x)) + (2x p4 tan x p3 − p3 + p9 ) (e1 − r1 ( x)) p10 x p4 cos x p3 p4 (3) − x M4 + e1 − x p2 p1 sin x p1 − p2 cos x p1 + + p10 ¨ (e − r1 ( x)) p5 cos x p3 + p6 sin x p3 + p7 x p4 p4 cos x p3 1 p p10 x − 10 ( p3 − p9 ) tan x p3 p4 cos x p3 p4 p4 (3) e1 p10 tan x p3 {( p3 − p9 ) x M4 − k1 x M1 − k2 x M2 − k3 x M3 − k4 x M4 } p4 p10 p (4) − x x + k5 x M5 + k6 x M6 + k7 x M7... M7 + p10 ¨ tan x p3 (e1 − r1 ( x)) p4 ( 14) 152 Mechatronic Systems, Simulation, Modelling and Control • Step 3 Further differentiating ( 14) gives rise to (3) e2 ¨ ¨ ∂ e (3) ∂ e2 { f ( x ) + G ( x ) u } + 2 e1 ¨ ∂x ∂ e1 p10 (3) = tan x p3 − x p2 p1 sin x p1 − p2 cos x p1 + p3 ( x M3 − r1 ( x)) − x M4 + e1 p4 p10 ¨ x (e − r1 ( x)) − p2 x p6 + x M8 − 9 p4 cos x p3 p4 1 = + p10 sin x p3 ( p3 − p9 ), 0,... Masahide Ito, and Masaaki Shibata “Vision-based motion control for robotic systems.” Vol 4 no 2 Edited by Hoboken John Wiley February 2009 Santibañez, Victor, and Rafael Kelly “PD control with feedforward compensation for robot manipulators: analysis and experimentation.” Robotica (Cambridge University Press) 19, no 1 (2001): 11-19  146 Mechatronic Systems, Simulation, Modelling and Control Sebastián,... uncertainties and external disturbances The resultant experimental data show that the method constructed with the inclusion of these additional terms produces the best control performance  148 Mechatronic Systems, Simulation, Modelling and Control 2 System Description Consider the tandem rotor model helicopter of Quanser Consulting, Inc shown in Figs 1 and 2 The helicopter body is mounted at the end of an arm and. .. Robots and Systems, 2005 21 34- 2139 Bonev, Ilian A., and Clément Gosselin Fundamentals of Parallel Robots Edited by Springer 2009 Chaumette, F., and S Hutchinson “Visual servo control I Basic approaches.” (Robotics & Automation Magazine, IEEE) 13, no 4 (December 2006): 82-90 Chaumette, F., and S Hutchinson “Visual servo control II Advanced approaches.” (Robotics & Automation Magazine, IEEE) 14, no 1... 144 Mechatronic Systems, Simulation, Modelling and Control e   Above means that if the error is bigger that tend to cero, finally the error is bounded by e (91) then the error will decrease but it will not   (92) By considering that errors from the estimation of the position and velocity are bigger that errors from the system dynamics, then can be obtained if we replace (77) and (87)... DE=DF, g [m/s2 ]: gravitational acceleration, 150 Mechatronic Systems, Simulation, Modelling and Control Jε , Jθ , Jφ [kg·m2 ]: Moment of inertia about the elevation, pitch and travel axes, ηε , ηθ , ηφ [kg·m2 /s]: Coefficient of viscous friction about the elevation, pitch and travel axes The forces of the front and rear rotors are assumed to be Ff =Km Vf and Fb =Km Vb [N], respectively, where Km [N/V]... elements are used in the visual controller and the robot controller has to satisfied the visual controller requirements Thanks to the joint controller the robot is supposed stable and its response is considered faster than the visual system Two cases are presented in this paper: the static case that is exposed in other works and some results and videos are shown, the another controller is called the dynamic... Following Control for a 3-DOF Model Helicopter 153 where ˙ ¯ e1 = −σ12 e1 − σ11 e1 (3) ¯ ¨ e2 = −σ 24 e2 − σ23 e2 − σ22 e˙2 − σ21 e2 r1 ( x) = − p1 cos x p1 − p2 sin x p1 − p3 x p2 + x M3 − p1 sin x p1 − p2 cos x p1 r2 ( x ) = − + p10 p9 p10 tan x p3 + x p4 p4 cos x p3 p4 p10 x tan x p3 p1 cos x p1 + p2 sin x p1 p4 p2 x p2 p p3 p10 x + 10 tan x p3 p3 p9 − p1 sin x p1 + p2 cos x p1 p4 cos x p3 p4 p4 ( x... Roberti, and R Carelli “Parallel robot high speed objec tracking.” Chap 3, by Image Analysis and recognition, edited by Aurélio Campilho Mohamed Kamel, 295-306 Springer, 2007 Senoo, T., A Namiki, and M Ishikawa “High-speed batting using a multi-jointed manipulator.” Vol 2 Robotics and Automation, 20 04 Proceedings ICRA ' 04 20 04 IEEE International Conference on, 20 04 1191- 1196 Stamper, Richard Eugene, and . p 7 x p4  +  p 10 p 4 cos x p3 x p4 − p 10 p 4 ( p 3 − p 9 ) tan x p3  e (3) 1 + p 10 p 4 tan x p3 { ( p 3 − p 9 ) x M4 − k 1 x M1 − k 2 x M2 − k 3 x M3 − k 4 x M4 } − p 10 p 4 cos x p3 x p4 x M4 +. p 7 x p4  +  p 10 p 4 cos x p3 x p4 − p 10 p 4 ( p 3 − p 9 ) tan x p3  e (3) 1 + p 10 p 4 tan x p3 { ( p 3 − p 9 ) x M4 − k 1 x M1 − k 2 x M2 − k 3 x M3 − k 4 x M4 } − p 10 p 4 cos x p3 x p4 x M4 +. obtain ¨ e 2 = −p 9 x p6 + x M7 + p 10 p 4 tan x p3 ( ¨ e 1 − r 1 (x)) ( 14) Mechatronic Systems, Simulation, Modelling and Control1 52 • Step 3 Further differentiating ( 14) gives rise to e (3) 2 = ∂ ¨ e 2 ∂x { f