AdvancesinRobotManipulators472 Although the actuating forces can be decreased by varying the orientations of the moving platform, the required maximum leg lengths increase as indicated in Figs. 8(a), (b). It means that a larger task space is necessary to accommodate the planned singularity-free path. (2) Time optimum For this problem, the travel time f t is to be determined. Based on the singularity-free path planning algorithm, the planned trajectory is shown in Fig.9 (i.e. the line for μ =1) with the corresponding minimal travel time f t =5.85 sec. Fig. 6. Variations of determinant along corresponding planned paths (a) (b) Fig. 7. Actuating forces along planned paths with (a) constant orientation and (b) varied orientations (3) Energy efficiency Fig. 9 also shows the minimal-energy trajectory with the corresponding travel time f t =20.01 sec (i.e. the line for μ =0). Compared with the time optimal trajectory planning, reduction in the travel time is at the expense of a greater consumed energy, a poorer fitness value, and a larger force. (4) Mixed cost function The cost function is defined as 1   t T f t G μ λ Δt μ dt 0 ( ) + ( )= - f l (36) The optimal singular free trajectories for μ =0.3, 0.6 and 0.8 with the corresponding determined travel time f t =7.758, 6.083 and 6.075 sec are also respectively shown in Fig. 9. (a) (b) Fig. 8. Leg lengths along planned paths with (a) constant orientation and (b) varied orientations 5. Conclusions In this chapter, a numerical technique is presented to determine the singularity-free trajectories of a parallel robot manipulator. The required closed-form dynamic equations for the parallel manipulator with a completely general architecture and inertia distribution are OntheOptimalSingularity-FreeTrajectoryPlanningofParallelRobotManipulators 473 Although the actuating forces can be decreased by varying the orientations of the moving platform, the required maximum leg lengths increase as indicated in Figs. 8(a), (b). It means that a larger task space is necessary to accommodate the planned singularity-free path. (2) Time optimum For this problem, the travel time f t is to be determined. Based on the singularity-free path planning algorithm, the planned trajectory is shown in Fig.9 (i.e. the line for μ =1) with the corresponding minimal travel time f t =5.85 sec. Fig. 6. Variations of determinant along corresponding planned paths (a) (b) Fig. 7. Actuating forces along planned paths with (a) constant orientation and (b) varied orientations (3) Energy efficiency Fig. 9 also shows the minimal-energy trajectory with the corresponding travel time f t =20.01 sec (i.e. the line for μ =0). Compared with the time optimal trajectory planning, reduction in the travel time is at the expense of a greater consumed energy, a poorer fitness value, and a larger force. (4) Mixed cost function The cost function is defined as 1   t T f t G μ λ Δt μ dt 0 ( ) + ( )= - f l (36) The optimal singular free trajectories for μ =0.3, 0.6 and 0.8 with the corresponding determined travel time f t =7.758, 6.083 and 6.075 sec are also respectively shown in Fig. 9. (a) (b) Fig. 8. Leg lengths along planned paths with (a) constant orientation and (b) varied orientations 5. Conclusions In this chapter, a numerical technique is presented to determine the singularity-free trajectories of a parallel robot manipulator. The required closed-form dynamic equations for the parallel manipulator with a completely general architecture and inertia distribution are AdvancesinRobotManipulators474 developed systematically using the new structured Boltzmann-Hamel-d’Alembert approach, and some fundamental structural properties of the dynamics of parallel manipulators are validated in a straight proof. In order to plan a singularity-free trajectory subject to some kinematic and dynamic constraints, the parametric path representation is used to convert a planned trajectory into the determination of a set of undetermined control points in the workspace. With a highly nonlinear expression for the constrained optimal problem, the PSOA needing no differentiation is applied to solve for the optimal control points, and then the corresponding trajectories are generated. The numerical results have confirmed that the obtained singularity-free trajectories are feasible for the minimum actuating force problem, time optimal problem, energy efficient problem and mixed optimization problem. The generic nature of the solution strategy presented in this chapter makes it suitable for the trajectory planning of many other configurations in the parallel robot manipulator domain and suggests its viability as a problem solver for optimization problems in a wide variety of research and application fields. Fig. 9. Planned paths for time optimum, energy efficiency and mixed cost function 6. References Bhattacharya, S.; Hatwal, H. & Ghosh, A. (1998). Comparison of an exact and an approximate method of singularity avoidance in platform type parallel manipulator. Mech. Mach. Theory, 33, 965-974. Chen, C.T. (2003). A Lagrangian formulation in terms of quasi-coordinates for the inverse dynamics of the general 6-6 Stewart platform manipulator. JSME International J. Series C, 46, 1084-1090. Chen, C.T. & Chi, H.W. (2008). Singularity-free trajectory planning of platform-type parallel manipulators for minimum actuating effort and reactions. Robotica, 26(3), 357-370. Chen, C.T. & Liao, T.T. (2008). Optimal path programming of the SPM using the Boltzmann- Hamel-d’Alembert dynamics formulation model. Adv Robot, 22(6), 705–730. Dasgupta, B. & Mruthyunjaya, T.S. (1998). Singularity-free planning for the Stewart platform manipulator. Mech. Mach. Theory, 33, 771-725. Dasgupta, B. & Mruthyunjaya, T.S. (1998). Closed-form dynamic equations of the general Stewart platform through the Newton-Euler approach. Mech. Mach. Theory, 33, 993- 1012. Dasgupta, B. & Mruthyunjaya, T.S. (1998). A Newton-Euler formulation for the Inverse dynamics of the Stewart platform manipulator. Mech. Mach. Theory, 33, 1135-1152. Do, W. & Yang, D. (1998). Inverse dynamic analysis and simulation of a platform type of robot. J. Robot. Syst., 5, 209-227. Dash, A.K.; Chen, I.; Yeo, S. & Yang, G. (2005). Workspace generation and planning singularity-free path for parallel manipulators. Mech. Mach. Theory, 40, 776-805. Kennedy, J. & Eberhart, R. (1995). Particle swarm optimization, in Proc. of IEEE Int. Conf. on Neural Network, pp. 1942-1948, Australia. Lebret, G.; Liu, K. & Lewis, F.L. (1993). Dynamic analysis and control of a Stewart platform manipulator. J. Robot. Syst., 10, 629-655. Liu, M.J.; Li, C.X. & Li, C.N. (2000). Dynamics analysis of the Gough-Stewart platform manipulator. IEEE Trans. Robot. Automat., 16, 94-98. Nakamura, Y. & Ghodoussi, M. (1989). Dynamics computation of closed-link robot mechanisms with nonredundant and redundant actuators, IEEE Transactions on Robotics and Automation, 5, 294-302. Nakamura, Y. & Yamane, K. (2000). Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Trans. Robotics and Automation, 16, 124-134. Pang, H. & Shahinpoor, M. (1994). Inverse dynamics of a parallel manipulator. J. Robot. Syst., 11, 693-702. Sen, S.; Dasgupta, B. & Bhattacharya, S. (2003). Variational approach for singularity-free path-planning of parallel manipulators. Mech. Mach. Theory, 38, 1165-1183. Tsai, L.W. (2000). Solving the inverse dynamics of a Stewart-Gough manipulator by the principle of virtual work. Trans. ASME J. Mech. Design, 122, 3-9. Wang, S.C.; Hikita, H.; Kubo, H.; Zhao, Y.S.; Huang, Z. & Ifukube, T. (2003). Kinematics and dynamics of a 6 degree-of-freedom fully parallel manipulator with elastic joints. Mech. Mach. Theory, 38, 439-461. Wang, J. & Gosselin, C.M. (1998). A new approach for the dynamic analysis of parallel manipulators. Multibody System Dynamics, 2, 317-334. Zhang, C. & Song, S. (1993). An efficient method for inverse dynamics of manipulators based on the virtual work principle. J. Robot. Syst., 10, 605-627. Acknowledgments This work is supported by the National Science Council of the ROC under the Grant NSC 98-2221-E-212 -026 and NSC 98-2221-E-269-015. Appendix ■ Dynamic equation of general parallel robot manipulators   p p p p M x + D x + G = f (A1) where   T T ( ) p M C M J M J C 1 1 1 2 1 1 = T p C M C 1 1 OntheOptimalSingularity-FreeTrajectoryPlanningofParallelRobotManipulators 475 developed systematically using the new structured Boltzmann-Hamel-d’Alembert approach, and some fundamental structural properties of the dynamics of parallel manipulators are validated in a straight proof. In order to plan a singularity-free trajectory subject to some kinematic and dynamic constraints, the parametric path representation is used to convert a planned trajectory into the determination of a set of undetermined control points in the workspace. With a highly nonlinear expression for the constrained optimal problem, the PSOA needing no differentiation is applied to solve for the optimal control points, and then the corresponding trajectories are generated. The numerical results have confirmed that the obtained singularity-free trajectories are feasible for the minimum actuating force problem, time optimal problem, energy efficient problem and mixed optimization problem. The generic nature of the solution strategy presented in this chapter makes it suitable for the trajectory planning of many other configurations in the parallel robot manipulator domain and suggests its viability as a problem solver for optimization problems in a wide variety of research and application fields. Fig. 9. Planned paths for time optimum, energy efficiency and mixed cost function 6. References Bhattacharya, S.; Hatwal, H. & Ghosh, A. (1998). Comparison of an exact and an approximate method of singularity avoidance in platform type parallel manipulator. Mech. Mach. Theory, 33, 965-974. Chen, C.T. (2003). A Lagrangian formulation in terms of quasi-coordinates for the inverse dynamics of the general 6-6 Stewart platform manipulator. JSME International J. Series C, 46, 1084-1090. Chen, C.T. & Chi, H.W. (2008). Singularity-free trajectory planning of platform-type parallel manipulators for minimum actuating effort and reactions. Robotica, 26(3), 357-370. Chen, C.T. & Liao, T.T. (2008). Optimal path programming of the SPM using the Boltzmann- Hamel-d’Alembert dynamics formulation model. Adv Robot, 22(6), 705–730. Dasgupta, B. & Mruthyunjaya, T.S. (1998). Singularity-free planning for the Stewart platform manipulator. Mech. Mach. Theory, 33, 771-725. Dasgupta, B. & Mruthyunjaya, T.S. (1998). Closed-form dynamic equations of the general Stewart platform through the Newton-Euler approach. Mech. Mach. Theory, 33, 993- 1012. Dasgupta, B. & Mruthyunjaya, T.S. (1998). A Newton-Euler formulation for the Inverse dynamics of the Stewart platform manipulator. Mech. Mach. Theory, 33, 1135-1152. Do, W. & Yang, D. (1998). Inverse dynamic analysis and simulation of a platform type of robot. J. Robot. Syst., 5, 209-227. Dash, A.K.; Chen, I.; Yeo, S. & Yang, G. (2005). Workspace generation and planning singularity-free path for parallel manipulators. Mech. Mach. Theory, 40, 776-805. Kennedy, J. & Eberhart, R. (1995). Particle swarm optimization, in Proc. of IEEE Int. Conf. on Neural Network, pp. 1942-1948, Australia. Lebret, G.; Liu, K. & Lewis, F.L. (1993). Dynamic analysis and control of a Stewart platform manipulator. J. Robot. Syst., 10, 629-655. Liu, M.J.; Li, C.X. & Li, C.N. (2000). Dynamics analysis of the Gough-Stewart platform manipulator. IEEE Trans. Robot. Automat., 16, 94-98. Nakamura, Y. & Ghodoussi, M. (1989). Dynamics computation of closed-link robot mechanisms with nonredundant and redundant actuators, IEEE Transactions on Robotics and Automation, 5, 294-302. Nakamura, Y. & Yamane, K. (2000). Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Trans. Robotics and Automation, 16, 124-134. Pang, H. & Shahinpoor, M. (1994). Inverse dynamics of a parallel manipulator. J. Robot. Syst., 11, 693-702. Sen, S.; Dasgupta, B. & Bhattacharya, S. (2003). Variational approach for singularity-free path-planning of parallel manipulators. Mech. Mach. Theory, 38, 1165-1183. Tsai, L.W. (2000). Solving the inverse dynamics of a Stewart-Gough manipulator by the principle of virtual work. Trans. ASME J. Mech. Design, 122, 3-9. Wang, S.C.; Hikita, H.; Kubo, H.; Zhao, Y.S.; Huang, Z. & Ifukube, T. (2003). Kinematics and dynamics of a 6 degree-of-freedom fully parallel manipulator with elastic joints. Mech. Mach. Theory, 38, 439-461. Wang, J. & Gosselin, C.M. (1998). A new approach for the dynamic analysis of parallel manipulators. Multibody System Dynamics, 2, 317-334. Zhang, C. & Song, S. (1993). An efficient method for inverse dynamics of manipulators based on the virtual work principle. J. Robot. Syst., 10, 605-627. Acknowledgments This work is supported by the National Science Council of the ROC under the Grant NSC 98-2221-E-212 -026 and NSC 98-2221-E-269-015. Appendix ■ Dynamic equation of general parallel robot manipulators   p p p p M x + D x + G = f (A1) where   T T ( ) p M C M J M J C 1 1 1 2 1 1 = T p C M C 1 1 AdvancesinRobotManipulators476          T T T T ( ) ( p D C M J M J C D J D J J M J C 1 1 1 2 1 1 1 1 2 1 1 2 1 1 ) =      T T ( ) p C M J M J C D C 1 1 1 2 1 1 1   T T ( ) p G C G J G 1 1 1 2 ( ) T T p f C Jacob f 1 The actuating forces vector      1 6 T f f f In (A1), the velocity transformation matrix, C 1 , is defined as         B I C C 3 3 1 0 0 (A2) where B C is the angular velocity transformation of the moving platform. In addition, J 1 and J 2 are the sub-matrices appropriately partitioned while developing the equations of motion, and are defined as   ( ) ( ) ( ) ( ) ( ) ( ) T T T T 1 1 1 6 6 6 1 6                                          s s b s s b s s b s s b T T T T T T T T l l l l 24 24 1 1 6 6 1 1 1 1 N N N B N N N B N N N B N N N B R R R R R R J = J J I R R R R R R 1 1 6 6 1 2 1 1 1 1 6 6 6 6 (A3) in which n n I is an n n  unitary matrix such that    24 24 J Ι 2 . ■ Inertia matrix, Coriolis and centrifugal terms, gravity vector        M M M 11 1 22 0 0 ,        M M M M M 33 34 2 4 3 44 (A4) where   B m M I 11 3 3  B M I 22 12 62 , ,= m m     M diag 33 12 62 T T T T m m      1 12 6 62 s d s d, ,M diag 34    12 62 m m      12 1 62 6 d s d s, ,M diag 43      , , 1 644 M diag I I and   2 2 1 2 1 T T m l m l     2 2 2      s s d d s 1 i i i i 2 i i i i i i i i I I I , 1, ,6i         D D 1 22 0 0 0 ,        D D D D 34 2 4 3 44 0 (A5) where   T B B D I ω 22 ( ) , , ( )               T T m l m l 12 11 62 61 D diag ω s s d ω s s d 34 1 1 1 12 6 6 6 62               T T T T m l m l 12 11 62 61 ( ) , , ( )D diag s d s ω s d s ω 43 1 12 1 1 6 62 6 6   44  1 6 , , D dia g h h and  i i i i i           T T T T T m l l m l 2 1 1 2 2 1 2 ( ) ( ) ( ) i i i i i i i i i i h s d s I I ω ω d s 1 2 , 1, ,6i  The tildes over the matrix-vector products in D 22 and i h denote a skew-symmetric matrix formed from the matrix-vector product.        G G G 11 1 21 ,        G G G 31 2 4 1 (A6) where   B mG g 11  0  G 21 3 1 12 62        Τ T T m m N N G g Rs g Rs 31 1 1 6 6 l l                  G T T T T T m m m m m m 11 12 11 12 61 62 61 62 ( ) ( ) N N g R d s d g R d s d 41 1 11 i 12 6 61 i 62 OntheOptimalSingularity-FreeTrajectoryPlanningofParallelRobotManipulators 477          T T T T ( ) ( p D C M J M J C D J D J J M J C 1 1 1 2 1 1 1 1 2 1 1 2 1 1 ) =      T T ( ) p C M J M J C D C 1 1 1 2 1 1 1   T T ( ) p G C G J G 1 1 1 2 ( ) T T p f C Jacob f 1 The actuating forces vector      1 6 T f f f In (A1), the velocity transformation matrix, C 1 , is defined as         B I C C 3 3 1 0 0 (A2) where B C is the angular velocity transformation of the moving platform. In addition, J 1 and J 2 are the sub-matrices appropriately partitioned while developing the equations of motion, and are defined as   ( ) ( ) ( ) ( ) ( ) ( ) T T T T 1 1 1 6 6 6 1 6                                          s s b s s b s s b s s b T T T T T T T T l l l l 24 24 1 1 6 6 1 1 1 1 N N N B N N N B N N N B N N N B R R R R R R J = J J I R R R R R R 1 1 6 6 1 2 1 1 1 1 6 6 6 6 (A3) in which n n I is an n n  unitary matrix such that    24 24 J Ι 2 . ■ Inertia matrix, Coriolis and centrifugal terms, gravity vector        M M M 11 1 22 0 0 ,        M M M M M 33 34 2 4 3 44 (A4) where   B m M I 11 3 3  B M I 22 12 62 , ,= m m     M diag 33 12 62 T T T T m m      1 12 6 62 s d s d, ,M diag 34    12 62 m m      12 1 62 6 d s d s, ,M diag 43      , , 1 644 M diag I I and   2 2 1 2 1 T T m l m l     2 2 2      s s d d s 1 i i i i 2 i i i i i i i i I I I , 1, ,6i         D D 1 22 0 0 0 ,        D D D D 34 2 4 3 44 0 (A5) where   T B B D I ω 22 ( ) , , ( )               T T m l m l 12 11 62 61 D diag ω s s d ω s s d 34 1 1 1 12 6 6 6 62               T T T T m l m l 12 11 62 61 ( ) , , ( )D diag s d s ω s d s ω 43 1 12 1 1 6 62 6 6   44  1 6 , , D dia g h h and  i i i i i           T T T T T m l l m l 2 1 1 2 2 1 2 ( ) ( ) ( ) i i i i i i i i i i h s d s I I ω ω d s 1 2 , 1, ,6i  The tildes over the matrix-vector products in D 22 and i h denote a skew-symmetric matrix formed from the matrix-vector product.        G G G 11 1 21 ,        G G G 31 2 4 1 (A6) where   B mG g 11  0  G 21 3 1 12 62        Τ T T m m N N G g Rs g Rs 31 1 1 6 6 l l                  G T T T T T m m m m m m 11 12 11 12 61 62 61 62 ( ) ( ) N N g R d s d g R d s d 41 1 11 i 12 6 61 i 62 AdvancesinRobotManipulators478 Programming-by-DemonstrationofReachingMotionsusingaNext-State-Planner 479 Programming-by-Demonstration of Reaching Motions using a Next- State-Planner AlexanderSkoglund,BoykoIlievandRainerPalm 0 Programming-by-Demonstration of Reaching Motions using a Next-State-Planner Alexander Skoglund, Boyko Iliev and Rainer Palm ¨ Orebro University Sweden 1. Introduction Programming-by-Demonstration (PbD) is a central research topic in robotics since it is an im- portant part of human-robot interaction. A key scientific challenge in PbD is to make robots capable of imitating a human. PbD means to instruct a robot how to perform a novel task by observing a human demonstrator performing it. Current research has demonstrated that PbD is a promising approach for effective task learning which greatly simplifies the programming process (Calinon et al., 2007), (Pardowitz et al., 2007), (Skoglund et al., 2007) and (Takamatsu et al., 2007). In this chapter a method for imitation learning is presented, based on fuzzy mod- eling and a next-state-planner in a PbD framework. For recent and comprehensive overviews of PbD, (also called “Imitation Learning” or “Learning from Demostration”) see (Argall et al., 2009), (Billard et al., 2008) or (Bandera et al., 2007). What might occur as a straightforward idea to copy human motion trajectories using a simple teaching-playback method, it turns out to be unrealistic for several reasons. As pointed out by Nehaniv & Dautenhahn (2002), there is significant difference in morphology between body of the robot and the robot, in imitation learning known as the correspondence problem. Fur- ther complicating the picture, the initial location of the human demonstrator and the robot in relation to task (i.e., object) might force the robot, into unreachable sections of the workspace or singular arm configurations. Moreover, in a grasping scenario it will not be possible to reproduce the motions of the human hand since there so far do not exist any robotic hand that can match the human hand in terms of functionality and sensing. In this chapter we will demonstrate that the robot can generate an appropriate reaching motion towards the target object, provided that a robotic hand with autonomous grasping capabilities is used to execute the grasp. In the approach we present here the robot observes a human first demonstrating the envi- ronment of the task (i.e., objects of interest) and the the actual task. This knowledge, i.e., grasp-related object properties, hand-object relational trajectories, and coordination of reach- and-grasp motions is encoded and generalized in terms of hand-state space trajectories. The hand-state components are defined such that they are invariant with respect to perception, and includes the mapping between the human and robot hand, i.e., the correspondence. To 24 AdvancesinRobotManipulators480 enable a next-state-planner (NSP) to perform reaching motions from an arbitrary robot config- uration to the target object, the hand-state representation of the task is then embedded into the planner. An NSP plans only one step ahead from its current state, which contrasts to traditional robotic approaches where the entire trajectory is planned in advance. In this chapter we use the term “next-state-planner”, as defined by Shadmehr & Wise (2005), for two reasons. Firstly, because it emphasizes on the next–state planning ability, the alternative term being “dynamic sys- tem” which is very general. Secondly, the NSP also act as a controller which is an appropriate name, but “next-state-planner” is chosen because the controller in the context of an industrial manipulator refers to the low level control system. In this chapter the term planner is more appropriate. Ijspeert et al. (2002) were one of the first researchers to use an NSP approach in imitation learning. A humanoid robot learned to imitate a demonstrated motion pattern by encoding the trajectory in an autonomous dynamical system with internal dynamic variables that shaped a “landscape” used for both point attractors and limit cycle attractors. To address the above mention problem of singular arm configurations Hersch & Billard (2008) consid- ered a combined controller with two controllers running in parallel; one controller acts in joint space, while the other one acts in Cartesian space. This was applied in a reaching task for controlling a humanoid’s reaching motion, where the robot performed smooth motion while avoiding configurations near the joint limits. In a reaching while avoiding an obstacle task, Iossifidis & Schöner (2006) used attractor dynamics, where the target object acts as a point attractor on the end effctor. Both the end-effector and the redundant elbow joint avoided the obstacle as the arm reaches for an object. The way to combine the demonstrated path with the robots own plan distinguishes our use of the NSP from from previous work (Hersch & Billard, 2008), (Ijspeert et al., 2002) and (Ios- sifidis & Schöner, 2006). Another difference is the use of the hand state space for PbD; most approaches for motion planning in the literature uses joint space (Ijspeert et al., 2002) while some other approaches use the Cartesian space. To illustrate the approach we describe two scenarios where human demonstrations of goal- directed reach-to-grasp motions are reproduced by a robot. Specifically, the generation of reaching and grasping motions in pick-and-place tasks is addressed as well as the ability to learn from self observation. In the experiments we test how well the skills perform the demon- strated task, how well they generalize over the workspace and how skills can be adapted from self execution. The contributions of the work are as follows: 1. We introduce a novel next-state-planner based on a fuzzy modeling approach to encode human and robot trajectories. 2. We apply the hand-state concept (Oztop & Arbib, 2002) to encode motions in hand-state trajectories and apply this in PbD. 3. The combination of the NSP and the hand-state approach provides a tool to address the correspondence problem resulting from the different morphology of the human and the robot. The experiments shows how the robot can generalize and use the demonstration despite the fundamentally difference in morphology. 4. We present a performance metric for the NSP, which enables the robot to evaluate its performance and to adapt its actions to fit its own morphology instead of following the demonstration. 2. Learning from Human Demonstration In PbD the idea is that the robot programmer (here called demonstrator) shows the robot what to do and from this demonstration an executable robot program is created. We assume the demonstrator to be aware of the particular restrictions of the robot. Given this assumption, the demonstrator shows the task by performing it in a way that seems to be feasible for the robot. In this work the approach is entirely based on proprioceptive information, i.e., we consider only the body language of the demonstrator. Furthermore, interpretation of human demonstrations is done under two assumptions: firstly, the type of tasks and grasps that can be demonstrated are a priori known by the robot; secondly, we consider only demonstrations of power grasps (e.g., cylindrical and spherical grasps) which can be mapped to–and executed by–the robotic hand. 2.1 Interpretation of Demonstrations in Hand-State Space To create the associations between human and robot reaching/grasping we employ the hand- state hypothesis from the Mirror Neuron System (MNS) model of (Oztop & Arbib, 2002). The aim is to resemble the functionality of the MNS to enable a robot to interpret human goal- directed reaching motions in the same way as its own motions. Following the ideas behind the MNS-model, both human and robot motions are represented in hand-state space. A hand- state trajectory encodes a goal-directed motion of the hand during reaching and grasping. Thus, the hand-state space is common for the demonstrator and the robot and preserves the necessary execution information. Hence, a particular demonstration can be converted into the corresponding robot trajectory and experience from multiple demonstrations is used to con- trol/improve the execution of new skills. Thus, when the robot execute the encoded hand- state trajectory of a reach and grasp motion, it has to move its own end-effector so that it follows a hand-state trajectory similar to the demonstrated one. If such a motion is success- fully executed by the robot, a new robot skill is acquired. Seen from a robot perspective, human demonstrations are interpreted as follows. If hand motions with respect to a potential target object are associated with a particular grasp type denoted G i , it is assumed that there must be a target object that matches this observed grasp type. In other words, the object has certain grasp-related features which affords this particular type of grasp (Oztop & Arbib, 2002). The position of the object can either be retrieved by a vision system, or it can be estimated from the grasp type and the hand pose, given some other motion capturing device (e.g., magnetic trackers). A subset of suitable object affordances is identified a priori and learned from a set of training data for each grasp type G i . In this way, the robot can associate observed grasp types G i with their respective affordances A i . According to Oztop & Arbib (2002), the hand-state must contain components describing both the hand configuration and its spatial relation with respect to the affordances of the target object. Thus, the general definition of the hand-state is in the form: H =  h 1 , h 2 , . . . h k−1 , h k , . . . h p  (1) where h 1 . . . h k−1 are hand-specific components which describe the motion of the fingers during a reach-to-grasp motion. The remaining components h k . . . h p describe the motion of the hand in relation to the target object. Thus, a hand-state trajectory contains a record of both the reaching and the grasping motions as well as their synchronization in time and space. The hand-state representation in Eqn. 1 is invariant with respect to the actual location and orientation of the target object. Thus, demonstrations of object-reaching motions at different locations and initial conditions can be represented in a common domain. This is both the [...]... for using the joint space as state space is the highly non-linear relationship between joint space and hand-state space (i.e., a Cartesian space): two neighboring points in joint space are neighboring in Cartesian space but not the other way around This means that action selection is better done in joint space since the same action is more likely to be suitable for two neighboring points than in hand-state... 11(2): 113 127 Vijayakumar, S., D’Souza, A & Schaal, S (2005) Incremental online learning in high dimensions, Neural Computation 17(12): 2602–2634 502 Advances in Robot Manipulators Robot Arms with 3D Vision Capabilities 503 25 0 Robot Arms with 3D Vision Capabilities Theodor Borangiu and Alexandru Dumitrache Politehnica University of Bucharest Romania 1 Introduction The use of industrial robots in production... representation in Eqn 1 is invariant with respect to the actual location and orientation of the target object Thus, demonstrations of object-reaching motions at different locations and initial conditions can be represented in a common domain This is both the 482 Advances in Robot Manipulators strength and weakness of the hand-state approach Since the origin of the hand-state space is in the target object,... original motions H6 and H11 were selected since they performed best In addition H20 was selected, because it had the lowest positive Q-value 6 Conclusions and Future Work In this article, we present a method for programming-by-demonstration of reaching motions which enables robotic grasping To allow the robot to interpret both the human motions and Programming-by-Demonstration of Reaching Motions using... Human-like Machines, Itech, Vienna, Austria, chapter Robot Learning by Active Imitation, pp 519–535 Billard, A., Calinon, S., Dillmann, R & Schaal, S (2008) Springer handbook of Robotics, Springer, chapter Robot Programming by Demonstration, pp 137 1 139 4 Billard, A., Epars, Y., Calinon, S., Schaal, S & Cheng, G (2004) Discovering optimal imitation strategies, Robotics and Autonomous Systems 47(2–3): 69–77... for trajectory formation and postural control, in W A Hershberger (ed.), Volitonal Action, Elsevier Science Publishing B.V (North-Holland), pp 253–297 500 Advances in Robot Manipulators Calinon, S., Guenter, F & Billard, A (2007) On learning, representing, and generalizing a task in a humanoid robot, IEEE Transactions on Systems, Man and Cybernetics, Part B 37(2): 286–298 Charusta, K., Dimitrov, D.,... Journal of Intelligent Service Robotics 2(1): 23–30 Tegin, J., Wikander, J & Iliev, B (2008) A sub e1000 robot hand for grasping – design, simulation and evaluation, International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, Coimbra, Portugal Ude, A (1993) Trajectory generation from noisy positions of object features for teaching robot paths, Robotics and... the robot have to be evaluated to enable the robot to improve its performance Therefore, we use the state-action value concept from reinforcement learning to evaluate each action (skill) in a point of the state space (joint configuration) The objective is to assign a metric to each skill to determine its performance, and include this is a reinforcement learning framework In contrast to most other reinforcement... matching is also used in mobile robotics, where comparing two succesive images taken with the same camera leads to estimation of the motion of the robot For robotic applications, which require 6-DOF part localization in 3D space, stereo vision is considered also a mature technology: (Hardin, 2008) and (Iversen, 2006) 3D part localization is also possible even with a single 2D camera, using a trained... reconstruction system An application involving a 6-DOF robot arm and a profile scanning device is presented in this section The structure of the system can be seen in Fig 3(a) The main components are: • Short range laser probe, able to measure distances between 100 and 200 mm with 30 µm accuracy, using one laser beam and two CCD cameras; 506 Advances in Robot Manipulators • 6-DOF vertical robot arm, with 650 mm reach . research topic in robotics since it is an im- portant part of human -robot interaction. A key scientific challenge in PbD is to make robots capable of imitating a human. PbD means to instruct a robot how. 6 61 i 62 Advances in Robot Manipulators4 78 Programming-by-DemonstrationofReachingMotionsusingaNext-State-Planner 479 Programming-by-Demonstration of Reaching Motions using a Next- State-Planner AlexanderSkoglund,BoykoIlievandRainerPalm 0 Programming-by-Demonstration of. running in parallel; one controller acts in joint space, while the other one acts in Cartesian space. This was applied in a reaching task for controlling a humanoid’s reaching motion, where the robot