robotics Designing the Mechanisms for Automated Machinery Part 11 pps

9.2 Dynamics of Manipulators 315 • Taking parts or blanks from a feeder tray or conveyor and transferring them to a machine, e.g., an indexing table, another conveyor, chuck, vise, etc.; • Taking parts from chucks, vises, etc., and transferring them for further handling; • Assembly operations, which include taking parts from trays, hoppers, etc., and putting them into some other assembly positions (e.g., in electronic circuit assembly); • Taking tools from magazines or hoppers and putting them into spindles, or vice versa; bringing the proper tool into the correct position at the right moment; • Moving welding electrodes or burners in accordance with the welding trace; moving and activating resistance-point welding heads; • Moving dye or paint sprayers; • Functioning as X-Ycoordinate tables for different purposes, for instance, in all manufacturing stages of integrated-circuit wafer production; • Operating in hostile environments such as radioactive, hot or extremely cold, or poisonous atmospheres (devices acting in outer space and under water may also be included in this category). 9.2 Dynamics of Manipulators We have already mentioned that manipulators usually work in concert, creating automatic production machines of several levels of flexibility. We have also empha- sized the importance of time, when the productivity of a process is under considera- tion. Thus, we begin the discussion of manipulator design problems with a consideration of their dynamics. We consider this problem on the basis of some examples, and try to show the approach for answering a nontrivial question: What are the unconstrained trajectories of the links, for a manipulator with multiple degrees of freedom, to bring a gripper from point A to point B (arbitrarily chosen) in minimal time? The trajectory providing this condition is the so-called optimal-time trajectory. The example under discussion presents a cylindrical manipulator with three degrees of freedom, as in Figure 9.1. Here FIGURE 9.1 Layout of cylindrical-type manipulator. TEAM LRN 316 Manipulators the gripper with the grasped part is represented by sliding mass m. The distance of this mass from the axis of rotation is r(t), which is a function of time. So also is the angle of rotation <j)(t), the azimuth. The coordinate z(£) describes the height of the manipulator's arm. The angular motion is caused by torque T, and the radial motion by force E The variables are Neglecting frictional forces, the motion equations are Here,/ 0 = moment of inertia of the unloaded arm (without the mass m), and the control constraints are Here M X and u 2 are control functions describing the behavior of the force and torque. In this case the z(r) degree of freedom is not considered, because it does not influence the azimuth and radial displacements of the mass m. The motion of the manipulator is considered from an arbitrary initial position A (at standstill): to an arbitrary final position B (at standstill) at time r: in such a way as to complete this motion in the shortest time. Let us suppose that the system shown in Figure 9.1 is described by the following data TEAM LRN 9.2 Dynamics of Manipulators 317 Computation of Equations (9.1) with control constraints (9.2) and boundary con- ditions (9.3) and (9.4) brings us to the results presented in Figure 9.2. Here, movement of the arm and mass from point A to point B took about 1.2 seconds, and their posi- tion was calculated every 0.1 second. To achieve this minimal time, both the arm and the mass m along the arm were moved, so that the force changes direction twice during the operation while the torque T changes once (line c). The control functions obtained for this case have the following form: The changes in the force occur at about 0.4 second and 0.8 second, while the torque changes once in the middle of the operating time at about 0.6 second. It is interesting to note that when the mass is not moved during this operation and stays at the end of the arm, the operating time (when the same T max is applied) equals 1.5 seconds (line d). Thus, 20% of the time can be saved by introducing optimal dynamic control of the manipulator. The calculations for this example (and later ones in this section) are taken from the paper "Time-Optimal Motions of Robots in Assembly Tasks" by H. E Geering, L. Guzzella, S. A. R. Hepner, and C. H. Onder, presented at the 24th IEEE Conference on Decision and Control, Fort Lauderdale, Florida, December 11-13,1985. Obviously, the solution presented in Figure 9.2 for the manipulator's optimal behav- ior is not the only possibility. For instance, when the value F max is considerably greater than in the above case, the mass m will rapidly reach the axis 0 and its further move- ment (if the design of the manipulator permits it) will not be desired (to minimize the moment of inertia of the system). This leads to the necessity of keeping the control function u^ = 0 for a certain time. The next example we consider is presented in Figure 9.3. This kind of device is often called a serpent-like manipulator. It consists of arm 1 and lever 2. The location of FIGURE 9.2 Optimal-time trajectory of the gripper providing fastest transfer from point A to point B. TEAM LRN 318 Manipulators FIGURE 9.3 Serpent-like manipulator. gripper 3 can be spatially defined by coordinate z and two angles 0 and if/. In this case also, vertical displacement z(t) does not influence the dynamics of the mechanism in the horizontal plane. (We do not consider here the remaining degrees of freedom of the gripper.) Such manipulators are convenient for automated assembly. The variables describing the motion of this system are The movement of the manipulator is provided by applying torques 7^ and T ¥ ; thus, the control variables are We denote the parameters as follows: /i, / 2 = lengths of links 1 and 2, respectively; r 2 = distance from the joint to the center of mass of the second link; m 2 , m 3 = masses of link 2 and gripper 3; /! = moment of inertia of link 1 with respect to the first axis; / 2 = moment of inertia of link 2 with respect to the second axis. Then, TEAM LRN 9.2 Dynamics of Manipulators 319 FIGURE 9.3a) General view of a cylinder-type (serpent-like) robot with hydraulic drives. This device is used in teaching and is produced by Cybernetic Application Company (Partway Trading Estate Andover Hants SP103LF, England). We can now rewrite the motion equations in the following form: The control constraints are Note that, because of changing moments of inertia, even constant forces or torques applied to the links will cause variable accelerations. TEAM LRN 320 Manipulators The computation was carried out for the following set of data: Figure 9.4 shows the optimal trajectory traced by the gripper of the manipulator for equidistant time moments when the boundary conditions are The time needed to complete this transfer is 1.25 seconds and the torques to carry out this minimal (for the given circumstances) time have to behave in the following manner: The meaning of these functions is simple. Arm 1 is accelerated by the maximum value of torque r^, max half its way (here, until angle 0 reaches ;r/2); afterwards arm 1 is decel- erated by the negative torque -T^, max until it stops. Obviously, this is true when friction can be ignored. Link 2 begins its movement, being accelerated due to torque T rmax for 0.278 second. Then it is decelerated by torque -T ¥max until 0.625 second has elapsed, and then again accelerated by torque T vmyx . After 0.974 second the link is decelerated by negative torque -r rmax until it comes to a complete stop after a total of 1.25 seconds. FIGURE 9.4 Optimal-time trajectory C of the gripper shown in Figure 9.3, providing fastest travel from point A to point B for link 1; rotational angle of 0 = n. TEAM LRN 9.2 Dynamics of Manipulators 321 It is interesting (and important for better understanding of the subject) to compare these results with those for a simple arc-like trajectory connecting points A and B, made by straightened links 1 and 2 so that the length of the manipulator is constant and equals ^ + 1 2 . To calculate the time needed for carrying out the transfer of mass m 3 from point A to point B under these conditions, we have to estimate the moment of inertia/of the moving masses. This value, obviously, is described in the following form: Applying to this mass a torque r^ max we obtain an angular acceleration a Considering the system as frictionless, we can assume that for half the way, nJ2, it is accelerated and for the other half, decelerated. Thus, the acceleration time ^ equals which gives, for the whole motion time T, The previous mechanism gives a 17% time saving (although the more complex manip- ulator is also more expensive). The mode of solution (the shape of the optimal trajectory) depends to a certain extent on the boundary conditions. The examples presented in Figures 9.5, 9.6, and 9.7 illustrate this statement. FIGURE 9.5 Optimal-time trajectory C of the gripper providing fastest travel from point A to point B for link 1; rotational angle of <j> = 1. TEAM LRN 322 Manipulators FIGURE 9.6 Optimal-time trajectory C of the gripper providing fastest travel from point A to point B for link 1; rotational angle of <j> = 1. FIGURE 9.7 Optimal-time trajectory C of the gripper providing fastest travel from point A to point B for link 1; rotational angle of 0 = 0.76. TEAM LRN 9.2 Dynamics of Manipulators 323 For the conditions: we have, for example, a motion mode shown in Figure 9.5. The transfer time T= 1.085 seconds and the control functions have the following forms: In Figure 9.6 we see another path of motion of the links for the same conditions (9.19) and here the control functions are Note that in Figure 9.5, link 1 does not pass the maximum angle, while in the trajec- tory shown in Figure 9.6, link 1 passes this angle a little and then returns. By decreasing the maximum angle 0 B , we obtain another very interesting mode ensuring optimal motion time for this manipulator. Indeed, for the result is as shown in Figure 9.7. Link 2 here moves in only one direction, creating a loop-like trajectory of the gripper when it is transferred from point A to point B. The control functions in this case are TEAM LRN 324 Manipulators Comparing these results (the time needed to travel from point A to point B for the examples shown in Figures 9.5, 9.6, and 9.7) with the time T" calculated for the con- ditions (9.16), (9.17), and (9.18) (i.e., links 1 and 2 move as a solid body and y/ = 0), we obtain the following numbers: Figure T optimal T'fory = 0 Time saving 9.5 1.085 sec 1.17 sec ~ 13 % 9.6 1.085 sec 1.17 sec ~ 13% 9.7 0.9755 sec 1.02 sec ~ 10 % The ideal motion described by the Equation Sets (9.1) and (9.2) does not take into account the facts that: the links are elastic, the joints between the links have back- lashes, no kinds of drives can develop maximum torque values instantly, the drives (gears, belts, chains, etc.) are elastic, there is friction and other kinds of resistance to the motion, or there may be mechanical obstacles in the way of the gripper or the links, all of which do not permit achieving the optimal motion modes. Thus, real conditions may be "hostile" and the minimum time values obtained by using the approach con- sidered here may differ when all the above factors affect the motion. However, an optimum in the choice of the manipulator's links-motion modes does exist, and it is worthwhile to have analyzed it. Note: The mathematical description here is given only to show the reader what kind of analytical tools are necessary even for a relatively simple—two-degrees-of-freedom system—dynamic analysis of a manipulator. We do not show here the solution proce- dure but send those who are interested to corresponding references given in the text and Recommended Readings. Another point relevant to the above discussion is that, in Cartesian manipulators (see Chapter 1), such an optimum does not exist. In Cartesian devices the minimum time simply corresponds with the shortest distance. Therefore, if the coordinates of points A and B are X A , Y A , Z A and X B , Y B , Z B , respectively, as shown in Figure 9.8, the dis- tance AB equals, obviously, Physically, the shortest trajectory between the two points is the diagonal of the paral- lelepiped having sides (X B - X A ), (Y B - Y A ), and (Z A - Z H ). Thus, the resulting force Facting along the diagonal must accelerate the mass half of the way and decelerate it during the other half. Thus, the forces along each coordinate cause the corresponding accelerations Here, a x , a Y > &z = accelerations along the corresponding coordinates, F x , F Y , F z = force components along the corresponding coordinates, m x , m Y , m z =the accelerated masses corresponding to the force component. TEAM LRN [...]... located on the base and transmissions transferring the motion to the corresponding links them To calculate the coordinates of point A (the gripper or the part the manipulator deals with), one has to know the angles lt 2, etc., between the links caused by the cylinders (or any other drive) In Figure 9.13 we show the calculation scheme Thus, we obtain for the coordinates of point A the following... 1) is 6 The sum of the torques Tfor all n drives (n = the number of degrees of freedom) gives an indication of the power the system consumes for both approaches, and this sum can be expressed for the model in Figure 9 .11 as: and for the model in Figure 9.12 as We derive two conclusions from this last assumption: • These sums of torques depend on the values (kl} • The ratio T^JP^I describes the average... in Figure 9.12 this dependence has the form: We will now illustrate an approach for evaluating the optimum choice of drive location: on the joints or on the bases We make the comparison for the worst case when the links are stretched in a straight line (in this case, obviously, the torques the drives must develop are maximal) for the two models given in Figure 9 .11 and Figure 9.12 We assume: 1 All... write for (9.35), TEAM LRN 9.3 Kinematics of Manipulators 333 The weights Pm are described in terms of the above assumptions in the following form: Here, C= number of combinations For the second model (Figure 9.12) we assume, in addition: 5 That the weights Pt of each link together with the kinematic elements of transmission are proportional to the torque the link develops Thus, Then the weight Pm of the. .. 9.9, for example, gripper 8 driven by cylinder 9 constitutes an additional degree of freedom; The resultant displacement of the gripper does not depend on the sequence in which the drives are actuated; The power or force that every drive develops depends on the place it occupies in the kinematic chain of the device The closer the drive is to the base, the more powerful it must be to carry all the links... weights P0 and lengths / 2 For the model in Figure 9 .11 the weight of each driving motor Pd is directly proportional to the torque T developed by it in the form 3 The weight Pm of link number m together with the drive can be applied to the left joint 4 The energy W or work consumed or expended by the whole system can be estimated as Here, A0m = rotation of a link at joint m, for m = 1, 2, , n We assume... (9.46a) in the following form: and here are the inertial coefficients of the dynamic system, and they correspondingly are TEAM LRN 344 Manipulators Stiffness can be introduced either as generalized forces or in the form of potential energy where r is the number of the corresponding generalized coordinate In this example r = 1,2 Completing the procedure of writing the Lagrange equation we obtain the following... expressions: (These expressions are written for the assumption that the lengths of all links equal /.) The point is that, to obtain the desired position of point A, we have to find a suitable set of angles 01; 02» — 0n» and control the corresponding drives so as to form these angles FIGURE 9.13 TEAM LRN Kinematics calculation scheme for the design shown in Figure 9 .11 330 Manipulators The design considered... simple 1:1 ratio between the angles 0 and iff, while the second approach suffers from the mutual influences of the angles ijs on TEAM LRN 9.3 Kinematics of Manipulators 335 the angles 0 (see Equations (9.33-36)) On the other hand, the second approach is preferable from the point of view of the inertial forces, torques, and powers that the whole system consumes How can we combine these two advantages in... to these vibrations we observe a considerable increase of the dynamic loads acting on the system After the positioning of the arm is completed (in the absence of external forces) free vibrations of the mechanism occur We now show how to estimate the parameters of these vibrations These parameters are the natural frequency and the main shapes of the vibrations Our explanation is based on an example given . obtained for this case have the following form: The changes in the force occur at about 0.4 second and 0.8 second, while the torque changes once in the middle of the operating. - Z H ). Thus, the resulting force Facting along the diagonal must accelerate the mass half of the way and decelerate it during the other half. Thus, the forces along each. the drives are actuated; • The power or force that every drive develops depends on the place it occupies in the kinematic chain of the device. The closer the drive is to the