Evolutionary Robotics Part 13 potx

Frontiers in Evolutionary Robotics 472 Actuators are placed in A and C. Attaching to each link a vector, on the OABPO respectively OCDPO, we can write successively the relations: DPCDOCOPBPABOAOP ++=++= ; (4) Based on the above relations, the coordinates of the point P have the following forms: 4231 4231 sinsinsinsin coscos 2 coscos 2 qLqlqLqly qLql d qLql d x P P +=+= ++−=++= (5) In this part, kinematics of a planar micro parallel robot articulated with revolute type joints has been formulated to solve direct kinematics problem, where the position, velocity and acceleration of the micro parallel robot end-effector are required for a given set of joint position, velocity and acceleration. The Direct Kinematic Problem (DKP) of micro parallel robot is an important research direction of mechanics, which is also the most basic task of mechanic movement analysis and the base such as mechanism velocity, mechanism acceleration, force analysis, error analysis, workspace analysis, dynamical analysis and mechanical integration. For this kind of micro parallel robot solving DKP is easy. Coordinates of point P in the case when values of joint angles are known 1 q and 2 q are obtained from relations: C BCDD x P 2 4 2 −±− = , DB DBP yy xxxA − −− = )( y P (6) where: )( 2 1 222222 BPDPBBDD LLyxyxA +−−−+−= AyyyALyxyyB DBDDPDDDB )(2)()( 22222 −−+−+−= 22 )()( DBDB xxyyC −+−= )(2)(2))((2 2 DBDBDDBDBD xxAyyxxxyyyD −−−−−−= (7) 2 cos 2 ql d x D +−= 2 sin qly D = 1 cos 2 ql d x B += 1 sin qly B = The speed of the point P is obtained differentiating the relations (1). Thus results: Evolving Behavior Coordination for Mobile Robots using Distributed Finite-State Automata 473 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅ 3 1 ω ω B y x A J V V J (8) where sincos sincos 44 33 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = qLqL qLqL J A (9) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −⋅ −⋅ = )sin(0 0)sin( 42 31 qqLl qqLl J B (10) or ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 3 1 ω ω J V V y x (11) where ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−− −−− − == − )sin(cos)sin(cos )sin(sin)sin(sin )sin( 423314 423314 34 1 qqqqq qqqqqq qq L JJJ AB ϕ (12) and J represents the Jacobian matrix. Acceleration of the point P is obtained by differentiating of relation (8), as it yields: 3 1 3 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ω ω ε ε J dt d J A A y x (13) -8 -6 -4 -2 0 2 4 6 8 -5 0 5 10 15 y x O -8 -6 -4 -2 0 2 4 6 8 -5 0 5 10 15 y x O Figure 6. The two forward kinematic models: (a) the up-configuration and (b) the down- configuration Based on the inverse kinematics analysis are determined the motion lows of the actuator links function of the kinematics parameters of point P. Frontiers in Evolutionary Robotics 474 The values of joint angles i q , (i = 1…4) knowing the coordinates x P , y P of point P, may be computed with the following relations: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −−+− = AC ACBB arctgq i )( 2 222 1 σ (14) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −+ = 2 2 222 3 ) M N arctg P PNM arctgq ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −−+− = ef efBB arctgq i )( 2 222 2 σ 1-or 1= , )( 2 i 222 4 σ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +−±− = EF EFbb arctgq where ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−= 2 2 d xlA P (15) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−= 2 2 d xLa P 22 2 2 2 Lly d xC PP −++ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= λ LM 2= μ LN 2= Lyb P 2−= 22 2 2 2 Lly d xc PP −−+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +−= 2 2 d xle P 22 2 2 2 Lly d xf PP −++ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += lyB P 2−= Evolving Behavior Coordination for Mobile Robots using Distributed Finite-State Automata 475 22 2 2 2 Lly d xF PP +−+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +−= 2 2 d xLE P 22 μλ −−=P dqlql +⋅−⋅= 21 coscos λ 21 sinsin qlql ⋅−⋅= μ From Eq. (14), one can see that there are four solutions for the inverse kinematics problem of the 2-dof micro parallel robot. These four inverse kinematics models correspond to four types of working modes (see Fig. 7). -8 -6 -4 -2 0 2 4 6 8 -5 0 5 10 15 y x O -8 -6 -4 -2 0 2 4 6 8 -5 0 5 10 15 y x O a) b) -8 -6 -4 -2 0 2 4 6 8 -5 0 5 10 15 y x O -8 -6 -4 -2 0 2 4 6 8 -5 0 5 10 15 y x O c) d) Figure 7. The four inverse kinematics models: (a)”+−“ model; (b)” −+“ model; (b)” −−“ model; (d)”++“ model Frontiers in Evolutionary Robotics 476 Figure 8. Graphical User Interface for solving the inverse kinematics problem of 2 DOF micro parallel robot Figure 9. Robot configuration for micro parallel robot x P =-15 mm y P =100 mm Figure 10. Robot configuration for micro parallel robot x P =-30 mm y P =120 mm Evolving Behavior Coordination for Mobile Robots using Distributed Finite-State Automata 477 Figure 11. Robot configuration for micro parallel robot x P =40 mm y P =95 mm Figure 12. Robot configuration for micro parallel robot x P =0 mm y P =130 mm 3.3 Singularities analysis of the planar 2-dof micro parallel robot In the followings, vector v is used to denote the actuated joint coordinates of the manipulator, representing the vector of kinematic input. Moreover, vector u denotes the Cartesian coordinates of the manipulator gripper, representing the kinematic output. The velocity equations of the micro parallel robot can be rewritten as: 0vBuA =+ && (16) Frontiers in Evolutionary Robotics 478 Where [] T q,qv 21 &&& = , [] T PP y,xu &&& = and where A and B are square matrices of dimension 2, called Jacobian matrices, with 2 the number of degrees of freedom of the micro parallel robot. Referring to Eq. (13), (Gosselin and Angeles, 1990), has defined three types of singularities which occur in parallel kinematics machines. (I) The first type of singularity occurs when det(B)=0. These configurations correspond to a set of points defining the outer and internal boundaries of the workspace of the micro parallel robot. (II) The second type of singularity occurs when det(A)=0. This kind of singularity corresponds to a set of points within the workspace of the micro parallel robot. (III) The third kind of singularity when the positioning equations degenerate. This kind of singularity is also referred to as an architecture singularity (Stan, 2003). This occurs when the five points ABCDP are collinear. -5 0 5 10 15 -5 0 5 10 15 y x O -5 0 5 10 15 -5 0 5 10 15 y x O a) b) -5 0 5 10 15 -5 0 5 10 15 y x O -5 0 5 10 15 -5 0 5 10 15 y x O c) d) Figure 13. Some configurations of singularities: (a) the configuration when l b and l c are completely extended (b) both legs are completely extended; (c) the second leg is completely extended and (d) the first leg is completely extended In this chapter, it will be used to analyze the second type of singularity of the 2-dof micro parallel robot introduced above in order to find the singular configuration with this type of micro parallel robot. For the first type of singularity, the singular configurations can be obtained by computing the boundary of the workspace of the micro parallel robot. Evolving Behavior Coordination for Mobile Robots using Distributed Finite-State Automata 479 Furthermore, it is assumed that the third type of singularity is avoided by a proper choice of the kinematic parameters. For this micro parallel robot, we can use the angular velocities of links l c and l b as the output vector. Matrix A is then written as: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅−⋅ ⋅−⋅ = )sin()(ins )cos()cos( 3b4c 3b4c qlql qlql A (17) From Eq. (17), one then obtains: ).sin()det( 34bc qqllA −⋅⋅= (18) From Eq. (18), it is clear that when ,,2,1,0,qq 34 ±±=+= nn π then .0)Adet( = In other words if the two links l c and l b are along the same line, the micro parallel robot is in a configuration which corresponds to be second type of singularity. Figure 14. Examples of architectural singular configurations of the R RRRR micro parallel robot 3.4 Optimal design of the planar 2-dof micro parallel robot The performance index chosen corresponds to the workspace of the micro parallel robot. Workspace is defined as the region that the output point P can reach if q 1 and q 2 changes from 2π without the consideration of interference between links and the singularities. There Frontiers in Evolutionary Robotics 480 were identified five types of workspace shapes for the 2-dof micro parallel robot as it can be seen in Figs. 15-20. Each workspace is symmetric about the x and y axes. Workspace was determined using a program made in MATLAB™. Analysis, visualization of workspace is an important aspect of performance analysis. A numerical algorithm to generate reachable workspace of parallel manipulators is introduced. Figure 15. The GUI for calculus of workspace for the planar 2 DOF micro parallel robot Figure 16. Workspace of the 2 DOF micro parallel robot [...]... architecture for evolutionary controllers based on neural networks, which allows the selective introduction of bias knowledge in the neural controller during the evolutionary process The architecture allows the introduction of external knowledge on selected stages of the evolutionary process, affecting only selected parts of the controller that need to accommodate that information The evolutionary controller... chapter ends with a discussion of the results, the conclusions and future lines of application 2 Evolutionary robotics for complex robots Evolutionary robotics is a framework for the automatic creation of autonomous robots It reproduces on robots selective reproduction of the fittest, that means, it uses evolutionary algorithms to develop the control program of an autonomous robot (Nolfi & Floreano,... this stage task After 1000 generations the maximum fitness reached is 1650 We performed the evolutionary process ten times, obtaining a mean fitness value after 1000 generations of 1570 All ten evolutionary processes were able to generate the garbage collector behavior 502 Frontiers in Evolutionary Robotics Third evolutionary stage Figure 9: At the final stage, two additional sensors were included (in... and Testing, Robotics – AQTR 2006 (THETA 15), May 25-28 2006, Cluj-Napoca, Romania, IEEE Catalog number: 06EX1370, ISBN: 1-4244-0360-X, pp 278-283 Stan, S and Lăpuşan, C., (2006) Workspace analysis of a 2 dof mini parallel robot, The 8th National Symposium with International Participation COMPUTER AIDED DESIGN PRASIC'06, Braşov, 9 - 10th November 2006, pag 175-180, ISBN (10)973-653-824-0; (13) 978-973-635-824-1... controller for the robot using only artificial evolution 492 Frontiers in Evolutionary Robotics We will show how our method applies to general robots Additional results will show its application to a complex Aibo robot The rest of the chapter is divided as follows: section two provides a description of the problem of evolutionary robotics that we try to solve, including a review of existing solutions... Catalonia Spain 1 Introduction Evolutionary robotics is a good method for the generation of controllers for autonomous robots However, up to date, evolutionary methods do not achieve the generation of behaviors for complex robots with a fixed body structure composed of lots of sensors and actuators For such cases, no satisfactory results exist due to the large search space that the evolutionary algorithm has... problems of evolutionary robotics is the creation of behaviors for complex robots We measure the complexity of a robot by its number of devices, that is, the number of sensors and the number of actuators that it has, which require a coordination in order to achieve a task, or generate a behavior As this number increases, more difficult will be to generate an evolutionary controller for it At present, evolutionary. .. beginning of the evolutionary process with a minimum fitness value that could guide the evolutionary path towards the final solution (Nolfi & Floreano, 1998) Progressive Design through Staged Evolution 493 Up to date, there is no general method which avoids such problems, even if partial solutions have been proposed For instance, incremental evolution (or incremental learning) is an evolutionary method... at each evolutionary stage on a simplified task, we keep reduced the searching space the evolutionary algorithm has to face Second, by evolving new modules keeping the functionality obtained in previous stages, the likelihood of obtaining a bootstrap is reduced, since the evolution of the newly added modules depart from a previous stable solution which can be scored, and by hence, directs the evolutionary. .. each other for the generation of the global robot task This is accomplished through an evolutionary process using a neuro -evolutionary algorithm Due to the fact that the evolutionary process has to Progressive Design through Staged Evolution 497 evolve different neural networks for different roles on a common task, a co -evolutionary algorithm is required, that is, the simultaneous evolution of several . Advanced robotics, 9(4):443-461. Merlet, J. P., (1995). Determination of the orientation workspace of parallel manipulators. Journal of intelligent and robotic systems, 13: 143–160. Frontiers in Evolutionary. lows of the actuator links function of the kinematics parameters of point P. Frontiers in Evolutionary Robotics 474 The values of joint angles i q , (i = 1…4) knowing the coordinates x P ,. kinematics models: (a)”+−“ model; (b)” −+“ model; (b)” −−“ model; (d)”++“ model Frontiers in Evolutionary Robotics 476 Figure 8. Graphical User Interface for solving the inverse kinematics