ÉCOLE POLYTECHNIQUE DE MONTRÉAL

Transcription

1 ÉCOLE POLYTECHNIQUE DE MONTRÉAL MODELIZATION OF A 3-PSP 3-DOF PARALLEL MANIPULATOR USED AS FLIGHT SIMULATOR MOVING SEAT. MASTER IN ENGINEERING PROJET III MEC693 SUBMITTED TO: Luc Baron Ph.D. Mechanical Engineering Department SUBMITTED BY: Jaime Pacheco ( ) April 23, 213.

2 Table of Contents 1. Introduction 1 2. Manipulator Description Inverse Kinematic Model Direct Kinematic Model 9 3. Jacobian Matrix Analysis Workspace analysis Conclusion References 22 I

3 Figure List Fig. 1. Flight simulator moving seat using a 3-PSP parallel manipulator... 2 Fig. 2. Kinematic diagram of the 3-PSP parallel manipulator Fig. 3. Tilt-and-torsion angle representation [12]... 4 Fig. 4. Vectors in 3-PSP parallel manipulator Fig. 5. Kinematic simulation of the 3-PSP parallel manipulator performed in Catia V Fig. 6. Parasite displacement for different values of and for a 3-PSP parallel manipulator with Fig. 7. Manipulator simplification used to obtain direct kinematic equations. 9 Fig. 8. Result of the numerical simulation using jacobian matrix Fig. 9. Workspace in coordinated system for a 3-PSP parallel manipulator with. Ratio Fig. 1. Workspace in coordinated system for a 3-PSP parallel manipulator with and. Ratio Fig. 11. Workspace in coordinated system for a 3-PSP parallel manipulator with and. Ratio Fig. 12. Workspace for a 3-PSP parallel Manipulator in coordinated system Fig. 13. Workspace in coordinated system for a 3-PSP parallel manipulator with and whit (a) and Ratio, (b) and Ratio,and (c) and Ratio Fig. 14. Reference point position for the flight simulator moving seat workspace Fig. 15. Workspace in coordinated system ; (a), (b)and(c); and (d) in coordinate system for the flight simulator moving seat Fig. 16. Relation between the maximum reachable value and the ratio II

4 1. Introduction In recent years the use of parallel manipulator has been increased due they have a number of advantages, when compared to traditional serial arms, such as high accuracy, high rigidity, low inertia, high load-to weight ratio and good dynamic performance [1]. The actual research is focused over the applications like high-speed machine tools, motion simulators, micro motion manipulators, force /torque sensor, etc [2]. However the parallel manipulator has some disadvantage such as small workspace, complex forward kinematics, and a complicated set of universal and spherical joints. Initially the attention was concentrated in parallel manipulators with six degrees of freedom (DOF) [3]. For flight simulators, the most used mechanism is the Steward Platform, a 6-DOF fully parallel manipulator. However this mechanism is very expensive and complex, and it is best suited for full trained simulators. Actually, the research increasingly focuses on parallel manipulators with a limited-dof because there are many applications that don t need all DOF, and specifically, flight simulator with a limited-dof can be used in the first stages of pilot training [4]. Furthermore, these mechanisms have a lot of advantages in terms of simplicity in construction and control, and a reduced cost [5]. Especially the 3-DOF spatial parallel manipulators (SPM) have been considered to be the most useful SPM for hybrid machine tool [5,6], e.g., the 3-PRS (Prismatic-Revolute-Spherical) type. They are used primarily as a plug and play module for numerical control machine for the fabrication of large components in the aircraft and automobile industries [7]. However, some parallel manipulator presents complications due a coupled orientation and position for the end-effector [8]. In the literature, the most studied configuration for limited-dof SPM is three equal kinematical chains used as legs that join two platforms and has 3-DOF. This configuration is very useful from the point of view of design and construction, and moreover, that permits to interchange the role of the base and moving platform because the mechanism is topologically symmetric [9]. One of these manipulators is the 3-PSP (prismatic-spherical-prismatic) 3-DOF, which is studied in this paper for an application as flight simulator moving seat shown in Fig. 1. In this case all passive joints are kept in the base platform in order to ensure the security of the pilot, obtaining a configuration that is upside-down relative to those studied in the literature [2,7,8]. The 3-DOF SPM are grouped in function of the DOF used by their end effector as: (1) translational, (2) Rotational and (3) coupled rotationaltranslational. The most famous full translational mechanism is the Delta robot presented By Clavel [1] that have been quite studied in the literature. There is also a lot of research in fully rotational 3-DOF parallel manipulator like The Agile Eye presented by Gosselin & Hamel[11]. The 3-PSP is in the third group, because position is coupled with the orientation. 1

5 For this report, two rotational DOF ( ) and one transnational (z) DOF are studied, and displacements in X and Y are considered as parasitic motion depended on and. First, in section 2, a description of the 3-PSP manipulator and the orientation representation Tilt and Torsion [12] is done. A geometric method is use to solve the forward kinematic problem and a simplification of the mechanism is used to solve the direct kinematic problem. Next, the jacobian matrix is obtained from the forward equations and then a numerical simulation of the position, speed and accelerations is performed. Finally in section 4, the workspace is analyzed using an iterative technique. Fig. 1. Flight simulator moving seat using a 3-PSP parallel manipulator. 2

6 2. Manipulator Description The 3-PSP parallel manipulator has a Moving platform, three legs and a fixed base. The moving platform is an equilateral triangle inscribed into a circumference with radius r. There are three legs and each one has an actuated prismatic joint rigidly attached to the mobile platform and normal to it. The connection points between legs and the moving platform are the vertex of the triangle identified as. These points define the plane containing the mobile platform which will be named { }. The fixed base has three non-actuated prismatic joints that are coplanar and equally spaced at 12. A spherical joint makes the connection between the active prismatic joint of the leg and the passive one on the fixed base. The plane containing the fixed base and the three passive prismatic joints will be called and it is defined by the points, and. These points correspond to the center of the spherical joints. Thus, we have three kinematic chains, Prismatic- Spherical-Prismatic, in this mechanism and 3-DOF. B3 Plane { } Mobile Platform Active prismatic joint W P U V B2 Passive prismatic joint A3 B1 Z O X Y 12 Plane { }. Fixed base A2 A1 Fig. 2. Kinematic diagram of the 3-PSP parallel manipulator. As shown in Fig. 2, the fixed coordinated frame denoted as { } has its origin at the intersection point of the axis of displacement of the three passive prismatic joints in the fixed base. The -axis is normal to the base platform and oriented toward moving platform, and the -axis is orientated 3

7 toward point. The origin of the moving coordinate frame { } is placed in the center of the moving platform, the -Axis is oriented toward point and the -axis is normal to the moving platform. The relation of moving coordinate frame { } with respect to base coordinate frame { } is defined by rotation matrix [2]. [ ] 1) In this case an orientation representation called till-and-torsion [12] is used, which is represented by three angles. Figure 3(a) shows The angle, called Tilt, that is a rotation around axis placed onto original plane. The orientation of axis is done by angle, called azimuth, which is the angle between the projection of axis onto the fixed plane and the fixed axis. The third angle is shown by Fig. 3(b) and is a rotation about the and is called Torsion. For 3-PSP the angle is always zero. In order to avoid the singularity at, we set the ranges of azimuth and tilt angles, respectively, and. Fig. 3. Tilt-and-torsion angle representation [12]. 4

8 2.1 Inverse Kinematic Model The unitary vector, in the coordinate frame { } prismatic joints can be written as: for the direction of passive (2) where. Equally, the position vector of point in the coordinate frame { } is defined by: (3) where is the radius of the moving platform. The vector expressed in the coordinate frame { } is given as: The position vector of the origin of frame { }, i.e. point, can be expressed in the coordinate frame { } as: (4) The vector from origin point of the frame { } to the point is defined by: (5) Let be defined as a unit vector normal to the moving platform in the coordinate frame { } and having the same positive direction of the displacement axis of active prismatic joints: (6) Thus, vector expressed in coordinate frame { } is: (7) 5

9 Mobile Platform Active prismatic joint W P U V bi Bi mi p Fixed base Passive prismatic joint ki O ai Ai Fig. 4. Vectors in 3-PSP parallel manipulator. As shown in Fig. 4, vectors and are coplanar, therefore we must have: (8) (9) where is the distance from point to point, is the distance from point to point and is a unit vector defined in (2). The inverse kinematic model of the 3-PSP parallel manipulator can be obtained from (8). The displacements of the active prismatic joints are done as function of the generalized coordinates and : (1) 6

10 (11) (12) The 3-PSP have 1-DOF in translation, and 2-DOF in rotation and. An important feature of the 3-PSP parallel manipulator is the parasitic displacement in and, which can be calculated from (9): (13) (14) The invers kinematic equation was verified using a kinematical simulation performed in Catia V5 as shown in Fig. 5. Active prismatic joint axis Moving platform. Passive prismatic joint axis Fixed base Plan { Fig. 5. Kinematic simulation of the 3-PSP parallel manipulator performed in Catia V5. 7

11 Y(mm) Y(mm) Y(mm) Y(mm) Parasitic displacement for z = 2 Workspace limit Parasitic displacement for z = (a) (b) Workspace limit 4 Parasitic displacement for z = Parasitic displacement for z = Workspace limit Workspace limit (c) (d) Fig. 6. Parasite displacement for different values of and for a 3-PSP parallel manipulator with. Figure 6 shows the parasite displacement for various values of z and. We can see how the capacity of the 3-PSP Parallel manipulator to rise different position and and orientation change as function of z. In this case the manipulator finds its maximum capability at. 8

12 2.2 Direct Kinematic Model In order to simplify the calculation of the direct kinematic equations is necessary to clarify that the value of generalized coordinate is always the minimal distance between the point and the fixed plane { }; and is the angle between moving plane { } and the fixed plan { }. The plans { } and { } has been defined in section 2. Thus, it is possible to invert the manipulator placing the original moving platform as base plane and introducing a new coordinate frame { } where axis is normal to the { } plane and axis is oriented toward point, as shown in Fig.7. Furthermore, we ignore the passive prismatic joins for this calculation because the only information needed is the position of the center of the spherical joints defined as point A2 Plane{ }. A3 n1 n2 B2 A1 W P V U B3 B1 Plane { } Pl Fig. 7. Manipulator simplification used to obtain direct kinematic equations. The position vector of the point, in coordinated frame { } can be written as: (15) 9

13 The position vector of the origin point written as: in the coordinate frame { }, can be (15) To define the plane { } formed by points, and, it is necessary set two vectors as: (16) (17) To obtain the equation of the plane { } the cross product between (16) and (17) is computed: (18) [ ] where is a vector normal to plane { }. Thus, the equation of plane { } is: (19) From (19) we can calculate the value of the generalized coordinate, that is the distance from plane { } to point. (2) Calculating the angle between the normal vectors to each plane, obtained, as follow: can be ( ) (21) 1

14 where is the unit vector of the -axis in the coordinate frame { }. Developing (21), the direct kinematic equation of can be written as: ( ) (22) From (1) and (11), the direct equation of can be computed and written as: (23) (24) (25) 3. Jacobian Matrix Analysis The velocities and accelerations of the active prismatic joins of the 3-PSP parallel manipulator can be obtained by direct differentiation of the inverse kinematics equations (1),(11) and (12); and can be expressed as follow: (26) (27) Equation (26) can be rewritten in matrix form as: (28) where is the serial jacobian matrix, is the parallel jacobian matrix, and. 11

15 From (26) and (28), the parallel jacobian matrix can be written as: ( ) (28) [ ( ) ] In this case the serial jacobian matrix is a 3x3 identity matrix. The jacobian matrix was validated using the resolved motion rate (RMR) method. The RMR can be expressed as: [ ] (28) Using this method and knowing the initial value of and, and applying small increments of the generalized coordinates, the final position of the active prismatic joints can be obtained. These increments must be as smallest as possible to ensure the accurate of the calculation. A numerical simulation using a MatLab Script have been performed applying a constant velocity for the coordinates and. The values used in the simulation are: The results obtained are shown in fig.8. As we can see, constant velocities in and, develop a great variation in linear speed and acceleration in the active prismatic joints. 12

16 Fig. 8. Result of the numerical simulation using jacobian matrix. 13

17 4. Workspace analysis To obtain the workspace of the 3-PSP parallel manipulator, an iterative method has been performed. The algorithm has three nested loops. In the outermost, z value increases from zero to a maximum value, which corresponds to the maximum value of displacement of the three active prismatic joints ( ). The second loop sweeps from to. Finally, the inner loop sweeps from zero to the point at which the limits of the prismatic joints is achieved. At that point, the value of the three generalized coordinates and, and the parasite displacements coordinates, and are stored. With this list of points, the external surface of the manipulator workspace can be obtained. Fig. 9 shows the workspace in the coordinate system for a manipulator with. Algorithm to obtain the external point of the workspace using Invers Kinematic solution. for z= TO 1, step size=.1 for = - TO ; step size=.1 =; for = TO ; step size=.1 Computing Invers kinematic of the 3-PSP having as results if Point inside of workspace. else Point outside of workspace Save de previous point as a limit element of the workspace end end end end An alternative method using the direct kinematic solution is performed. In this case three nested loop increase de value of from to, keeping the active prismatic joints displacements inside their joints limits. Thus, all the points obtained by using the algorithm are inside the workspace. Algorithm to obtain the workspace using Direct Kinematic solution. for = z= TO 1, step size=.1 for = z= TO 1, step size=.1 for = z= TO 1, step size=.1 Computing direct kinematic of the 3-PSP having as results end end end 14

18 Y (mm) Y (mm) Z (mm) Y (mm) (a) (b) (c) Fig. 9. Workspace in coordinated system for a 3-PSP parallel manipulator with. Ratio. Figure 9(a) shows the volume that forms the workspace. This begins with a point in. As can be seen in the Fig. 9(b) and 9(c), the cross sections begin to increase until reaching approximately, at this point the cross section begins to decrease again until a point at z = 1mm. Additionally, the work space is symmetrical every 12. An important factor 15

19 Y (mm) Y (mm) Z (mm) that we have found is the ratio between the radius of the moving platform and the maximum displacement of the active prismatic joints,. When this ratio is changed, the form of the work space is modified. As can be seen in Fig. 1, when the ratio is reduced to.5, the z value where the maximal extension of the workspace is reached increase front 56mm to 64mm. Additionally, making a comparison between Fig. 9, 1 and 11, we can see that when this ratio increases, the upper working space becomes convex approaching a spherical section and the workspace volume become bigger, when the ratio is reduced, the workspace becomes more concave Y (mm) -2-4 (a) (b) (c) Fig. 1. Workspace in coordinated system for a 3-PSP parallel manipulator with and. Ratio. 16

20 Y (mm) Y (mm) Z (mm) Y (mm) (a) mm mm -4 mm (b) (c) Fig. 11. Workspace in coordinated system for a 3-PSP parallel manipulator with and. Ratio. Fig. 12 shows the workspace in the coordinate system, for a 3-PSP parallel manipulator with. In this case, only the surface that indicate the upper and lower boundaries is shown. The lateral limit are determined by the range defined for and in section 2., i.e., and. Thus, the workspace is all the points between the upper and lower boundaries. 17

21 Fig. 12. Workspace for a 3-PSP parallel Manipulator in coordinated system. Making a comparison between the fig. 13(a), 13(b) and 13(c) is easy to determine that for a larger ratio, we obtain a greater orientation capacity of manipulator in. The Relation between the maximum attainable value of and the ratio is shown in Fig. 16. So far, we have studied the case in which the displacement of the active prismatic joints is equal to the maximum length of the leg. This case does not apply to our flight simulator in which the actuators only have a maximum displacement of 25 mm and the benchmark for the workspace is located in the center of the driver's head, at a height of 1255 mm about the fixed base, as shown in Fig. 14. To get this workspace, we use the algorithm based on the direct kinematics and taking a displacement between 153mm and 1555mm for the prismatic joint and a mobile base with 382 mm radius. Figure 15 (a), (b) and (c) show the workspace in the coordinate system, and Fig. 15 (d) shows it in the system. As we can see, the result achieved is very different from that obtained when considered complete displacement actuators. In addition, analyzing Fig. 15 (d) we can observe that the orientation in theta capacity is extremely low, reaching only a maximum of.44 rad. This because the ratio between the and is very small. 18

22 Fig. 13. Workspace in coordinated system for a 3-PSP parallel manipulator with and whit (a) and Ratio, (b) and Ratio,and (c) and Ratio. 19

23 1255mm Fig. 14. Reference point position for the flight simulator moving seat workspace. (a) (b) (c) (d) Fig. 15. Workspace in coordinated system ; (a), (b)and(c); and (d) in coordinate system for the flight simulator moving seat. 2

24 (rad) q i max/r Fig. 16. Relation between the maximum reachable value and the ratio. 5. Conclusion 1. A new architecture of the 3-PSP parallel manipulator 3-DOF was introduced using the orientation representation Tilt and Torsion. 2. The inverse kinematic model was obtained with a geometric method and verified using a kinematic simulation performed in Catia V5. 3. To resolve the direct kinematic problem, a simplification of the 3-PSP parallel manipulator mechanism with a geometric approach was used to develop the equation for and. The Angle was obtained from inverse kinematic equations. 4. The Parallel jacobian matrix was obtained by direct differentiation of the inverse kinematic equations and validated using Resolved Motion Rate method. 5. Finally, the robot reachable workspace was determinate. The variations for different configuration were analyzed, introducing the ratio as an index of the ability of orientation in for the 3-PSP manipulator. The difference in the workspace between full and partial displacement actuator for the active prismatic joints was established. 21

25 6. References [1] Liu X. Optimal kinematic design of a three translational Dofs Parallel Manipulator. Robotica, 26,24: [2] Hao Q, guan L, wang J, wang L. Dynamic feedfoward control of a novel 3-PSP 3-DOF parallel Manipulator. Chinese Journal of Mechanical Engineering,29, 2: 1-8. [3] Harib K, Ullah A, Sharif M, Hammami A. A hexapod-based machine tool with hybrid structure: kinematic analysis and trajectory planning. International journal of machine tools & manufacture, 27,47: [4] Pouliot N, Gosselin C, Nahon M. Motion simulation capabilities of three-degree-of-freedom flight simulators. AIAA Journal Aircraft, 1998,35:9 17. [5] Sun T, Song Y, LI Y, LIU L. Dimensional synthesis of 3-DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix. CHINA Technological Sciences, 21,53: [6] Chen C, Huang Y, Han X, Cheng X, Guo S. Study and Analysis of the 3-PRS Parallel Mechanism. IEEE International conference on mechatronics and automation, 29: [7] Rezaei A, Akbarzadeh A, Aahmoodi P, Akbarzadeh M. Position, Jacobian and workspace Analysis of a 3-PSP spatial parallel manipulator. Robotics and Computer-integrated manufactured, 213, 29: [8] HAO Q, GUAN L, LIU X, WANG L. Dynamic analysis of a novel 3-PSP 3-DOF parallel manipulator. ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots, 29: [9] Di Gregorio R, Parenti-Castelli V. Position Analysis in analytical form of the 3-PSP Mechanism. Journal of Mechanical Design, 21, 123: [1] Clavel R. DELTA, a fast robot with parallel geometry. Proceedings of the 18th International Symposium on Industrial Robots, 1988: 91-1 [11] Gosselin C, Hamel J. The Agile Eye - a high-performance 3- degree-of-freedom camera-orienting device. IEEE international conference on robotics and automation proceedings, 1994: [12] Bonev I, Gosselin C. Singularity locy of spherical parallel mechanism. International Conference on Robotics and Automation, 25: