robotics Designing the Mechanisms for Automated Machinery Part 12 pot

9.3 Kinematics of Manipulators 345 The general solution of the dynamic equations then is The parameters B u , B u , M lt M 2 are determined through the initial conditions: Thus, the free vibrations of the arm close to the average position consist of two oscil- lation processes with two different and usually not commensurable frequencies k^ and k 2i which, in turn, means that in general these oscillations are not of a periodic nature. For further analyses it is convenient to use the so-called main or normal coordinates z m which are defined in the following form: For any initial conditions these new variables change periodically, monoharmonically: where the constants D m and y/ m are determined by the initial conditions. An example follows. Supposing m l = m 2 = m and Cj = c 2 = c; we rewrite (9.46c) in the form For q = 0 this becomes For the natural frequencies in this case we obtain TEAM LRN 346 Manipulators For the natural frequencies in this case we obtain The coefficients describing the shape of the oscillations are correspondingly It is possible to obtain an approximate estimation for the initial deformations appear- ing in the manipulator under discussion which determine the amplitudes of the free oscillations after the motors are stopped. For this purpose we write kinetostatical equa- tions of acting torques Q l and Q 2 . This is done with an assumption that the angle q = 0 and that angular acceleration before the stop was e Q = constant. From (9.46m) follows for the initial amplitudes cor- respondingly Some simple dependences for evaluating the behavior of a robot's arm can be helpful. We denote: 0max = maximal value of the oscillations amplitude; / = moment of inertia with respect to the rotating shaft of the vibrating body; </max - maximal angle the arm travels; t = travelling time; k = frequency of the oscillations. Then we can use the following approximations: An illustration of the process of vibrations calculated for Case (9.461) is given in Figure 9.23a). TEAM LRN 9.3 Kinematics of Manipulators 347 FIGURE 9.23b) Oscillations of the two-freedom degree manipulator. fil = Plot[Cos[.3561]+.454 Cos[3.027 t] Exp[ 15 t],{t,0,20}] Robots with parallel connections of links We consider here some aspects of the principles used in the device called the Stewart platform (SP) in regard to manipulators. We begin with a description of the main prop- erties of this device. The definition of SP is: two rigid basic bodies connected by six rods (noncomplanar and noncolinear) with variable lengths. Such a structure is shown in Figure 9.23c) and possesses the following main properties: 1. This system has six degrees of freedom; 2. One-to-one unambiguous connection between the lengths of the rods and the mutual positions of the two bodies; 3. Two main problems may be formulated in this regard: a. The length of the rods is given—what is the relative position of the bodies? (FKT—Forward Kinematic Transform.) b. The relative position of the bodies is given—what is the length of the rods needed for this situation? (IKT—Inverse Kinematic Transform.) FIGURE 9.23c) Layout of the Stewart platform: 1) upper body; 2) lower body; 3) variable rods. TEAM LRN 348 Manipulators 4. The length of the rods is changed in parallel, i.e., each rod is controlled inde- pendently relative to the base, which makes this kinematic solution preferable because of the lack of cumbersome kinematic chains. The vectors appearing in Figure 9.23d) are denned as follows: Q = vector between a certain point in the basic coordinate system and another certain point on the platform. This vector represents one of the six rods con- necting the two bodies comprising the mechanism; R 0 = vector denning the begining of the coordinates of the platform relative to the basic coordinate system; R = vector defining the position of the point belonging to the platform (the point the rod is fastened to the platform) in the basic coordinate system; p = vector of the same point in the coordinate system of the platform; b = vector of the point in the basic coordinate system (the lower fastening point of the rod); and K= cosine matrix. Then the connection between all these vectors looks as follows: Now we define K: Here: 5 = sin and C = cos of a corresponding angle; 0 = rotation angle relative to the axis x (roll); 6 - rotation angle relative to the axis y (pitch); iff = rotation angle relative to the axis z (azimuth). Now it is possible to solve the IKT which is the real case in most applications of this kind of mechanism-as-a-manipulator (robot) for manipulating parts for processing or other purposes. We then need to calculate the length of the rods /, (i=J, ,6) knowing FIGURE 9.23d) Coordinates describing the position of the platform relative to the base. TEAM LRN 9.3 Kinematics of Manipulators 349 the desired position of the platform. This is done, for instance, in the following way using the above-shown Expressions (9.44q): In the photograph shown in Figure 9.23e) there is an embodiment of a Stewart plat- form where the rods are pneumatic cylinders. Controlling the positions of the piston rods correspondingly to the calculated values /,-, we obtain the desired location and orientation of the platform. (By the way, the FKT is more unpleasant. The expressions we get in this case are nonlinear and the equations have a number of formal solutions, which makes the pro- cedure of finding the practical one complicated enough.) The structure of the SP has a wide potential of creative possibilities in theoretical, design, and application domains. For instance, the micro domain of applications opens interesting theoretical and design alternatives. For small displacements of parts, espe- cially when we deal with dimensions of the order of 10" 4 ,10" 7 m, the description of the movement can be simplified. The cosine matrix in this case is Another field for new SP applications occurs when combinations of these devices are investigated. One such idea given in Figure 9.23f), interesting for the robotics field, is to create a "trunk"-like structure by using a series of SP for robotics applications. FIGURE 9.23e) Embodiment of a Stewart platform built in the Mechanical English Department of Ben-Gurion University (Israel). TEAM LRN 350 Manipulators FIGURE 9.23f) Idea of a "trunk" made of Stewart-platform-like elements. This idea belongs to Dr. A. Sh. Kiliskor. 9.4 Grippers In previous sections we have discussed the kinematics and dynamics of manipu- lators. Now let us consider the tool that manipulators mainly use—the gripper. To manipulate, one needs to grip and hold the object being manipulated. Grippers of various natures exist. For instance, ferromagnetic parts can be held by electromag- netic grippers. This gripping device has no moving parts (no degrees of freedom and no drives). It is easily controlled by switching the current in the coil of the electro- magnet on or off. However, its use is limited to the parts' magnetic properties, and magnetic forces are sometimes not strong enough. When relatively large sheets are handled, vacuum suction cups are used; for instance, for feeding aluminum, brass, steel, etc., sheets into stamps for producing car body parts. Glass sheets are also handled in this way, and some printing presses use suction cups for gripping paper sheets and introducing them into the press. Obviously, the surface of the sheet must be smooth enough to provide reliability of gripping (to seal the suction cup and prevent leakage of air and loss of vacuum). Here, also, no degrees of freedom are needed for gripping. The vacuum is switched on or off by an automatically controlled valve. (We illustrated the use of such suction cups in the example shown in Figure 2.10.) Grippers essentially replace the human hand. If the gripping abilities of a mechan- ical five-finger "hand" are denoted as 100%, then a four-finger hand has 99% of its ability, a three-finger hand about 90%, and a two-finger hand 40%. We consider here some designs of two-fingered grippers. In the gripper shown in Figure 9.24, piston rod 1 moves two symmetrically attached connecting links 2 which in turn move gripping levers 3, which have jaws 4. (Cylinder 5 can obviously be replaced by any other drive: electromagnet, cable wound on a drum driven by a motor, etc.) The jaws shown here are suitable for gripping cylindrical bodies having a certain range of diameters. Attempts to handle other shapes or sizes of parts may lead to asymmet- rical gripping by this device, because the angular displacements of jaws may not be parallel. To avoid skewing in the jaws, solutions like those shown in Figure 9.25a) or b) are used. In Case a) a simple cylinder 1 with piston 2 and jaws 3 ensures parallel TEAM LRN 9.4 Grippers 351 FIGURE 9.24 Design of a simple mechanical gripper. FIGURE 9.25 Grippers with translational jaw motion. displacement of the latter. In case b) a linkage as in Figure 9.24, but with the addition of connecting rods 6 and links 7 with attached jaws 4, provides the movement needed. These additional elements create parallelograms which provide the transitional move- ment of the jaws. Various other mechanical designs of grippers are possible. For instance, Figure 9.26 shows possible solutions a) and b) with angular movement of jaws 1, while cases c) and d) provide parallel displacement of jaws 1. In all cases the gripper is driven by rod 2. All the cases presented in Figure 9.26 possess rectilinear kinematic pairs 3. Intro- duction of higher-degree kinematic pairs are shown in Figure 9.27. In case a) cam 1 fastened on rod 2 moves levers 3 to which jaws 4 are attached. Spring 5 ensures the contact between the levers and the cam. In case b) the situation is reversed: cams 1 are fastened onto levers 3 and rod 2 actuates the cams, thus moving jaws 4. Spring 5 closes the kinematic chain. In case c), which is analogous to case b), springs 5 also play the role of joints. In case d) the higher-degree kinematic pair is a gear set. Rack 1 (moved by rod 2) is engaged with gear sector 3 with jaws 4 attached to them. Cases a) to d) have dealt with angular displacement of jaws. In case e) we see how the addition of parallelograms 5 (as in the example in Figure 9.25b)) to the mechanism shown in Figure 9.27d) makes the motion of the jaws translational. The last two cases do not need springs, since the chain is closed kinematically. TEAM LRN 352 Manipulators FIGURE 9.26 Designs of grippers using low-degree kinematic pairs. FIGURE 9.27 Designs of grippers using high-degree kinematic pairs. TEAM LRN 9.4 Grippers 353 To describe these mechanisms quantitatively we use the relationships between: 1. Forces F G which the jaws develop, and the force F d which the driving rod applies; and 2. The displacements S d of the driving rod and the jaws of the gripper S G . Figure 9.28 illustrates these parameters and graphically shows the functions S G (S d ) and F G /Fd=flSj for a gripper. This discussion of grippers has been influenced by the paper by J. Volmer, "Tech- nische Hochschule Karl-Marx-Stadt, DDR, Mechanism fur Greifer von Handhaberg- eraten," Proceedings of the Fifth World Congress on Theory of Machines and Mechanisms, 1979, ASME. We should note that the examples of mechanical grippers discussed above permit a certain degree of flexibility in the dimensions of parts the gripper can deal with. This property allows using these grippers for measuring. For instance, by remembering the values of S d by which the driving rod moves to grip the parts, the system can compare the dimensions of the gripped parts. When the manipulated parts are relatively small and must be positioned accurately, miniaturization of the gripper is required. A solution of the type shown in Figure 9.29 can be recommended, for example, in assembly of electronic circuits. Here, the gripper FIGURE 9.28 Characteristics of a mechanical gripper. TEAM LRN 354 Manipulators is a one-piece tool made of elastic material that can bend and surround the gripped part, of diameter d, to create frictional force to hold the part, and then to release it when it is fastened on the circuit board. The overlap h = Q.2d serves this purpose. Three-fingered grippers are also available (or can be designed for special purposes). Figure 9.30 shows a concept of a three-fingered gripper. Part a) presents a general view and part b) shows a side view. Here, 1 is the base of the gripper and 2 the driving rod, which is connected by joints and links to fingers 3. When rod 2 moves right, the fingers open, and when it moves left, they close. This gripper (as well as some considered earlier) can grip a body from both the outside and the inside. (Such grippers are pro- duced by Mecanotron Corporation, South Plainfield, New Jersey, U.S.A.) One of the most serious problems that appears in manipulators equipped with dif- ferent sorts of grippers is control of the grasping force the gripper develops. Obviously, there must be some difference between grasping a metal blank, a wine glass, or an egg, even when all these objects are the same size. This difference is expressed in the dif- ferent amounts of force needed to hold the objects and (what is more important) the limited pressure allowed to be applied to some objects. Figure 9.31 shows a possible solution for handling tender, delicate objects. Here, hand 1 is provided with two elastic FIGURE 9.30 Three-fingered gripper. FIGURE 9.31 A soft gripper for grasping delicate objects. TEAM LRN [...]... direct the support 4 of cutter 3 The cutter develops force P at the cutting point Decomposition of this force yields its three components Px, Py, and Pz Together with the weight G of the moving part, these forces cause the guides to react with forces A, B, and Cin the Z-Fplane and frictional forces fA, fB, and fc along the X-axis (when movement occurs) Statics equations permit finding the reactive forces... design for guides where the frictional force is nearly linearly dependent on the speed (complete lubricational friction) Thus, The motion equation for the mass driven by force F takes the following form instead of (9.52): (Here the deformation of the rod shown in Figure 9.40 is neglected.) For initial conditions the solution is (similar equations were solved in Chapter 3) This expression indicates the. .. of the guides and are shown in Figure 9.36 The obtained pressure values are average values, and the real local pressure might not be uniformly distributed along the guides The allowed maximum pressures depend on the materials the guides are made of and their surfaces, and are about 300 N/cm 2 for slow-moving systems to 5 N/cm 2 for fast-running sliders Obviously, the lower the pressure, the less the. .. Frictional force FF appearing in a slide pair depends on the speed of relative motion x, as shown in Figure 9.39 This means that, when the speed is close to 0, the frictional force FST is higher than it is at faster speeds Thus, Here F is the driving force, and Fdin is the frictional force at the final sliding speed This can be analyzed further with the help of Figure 9.40 Mass M of the slider is driven by force... Shields 6 and 7 keep the guides clean Rolling guides have much lower friction than sliding guides, and therefore the Fsr values are much smaller However, these guides employ more matching surfaces: between the housing and the rolling elements, and between the rolling elements and the moving part In addition, deviations in the shapes and dimensions of the rolling elements affect the precision, and such... LRN deflections For heavier loads, the area of contact between the guide and the moving 9.5 Guides 359 part must be larger To prevent excess wear, the guides must apply low pressure to the moving part, which also entails a certain width of the guide and length of the support (to create the required contact area) It is important to mention that, above all, wear of the guides depends on the maintenance... solution has the form By substituting this solution into Equation (9.53), we obtain the following expressions for a and CD: Under the initial conditions (when t = 0) the displacement x = XST, and speed x = 0 So we obtain for the coefficients A and B Thus, finally, the solution is For instance, for M- 100 kg, c = 104 N/cm, FST= 100 N and a = 1 Nsec/m, we find from (9.50) that and from (9.55) that The ratio... grasping force can be controlled, and the object handled by the gripper with a light or heavy touch For simpler grippers (as in Figures 9.24, 9.28, and 9.30), force-sensitive jaws can be made as shown in Figure 9.33 Here, part 1 is grasped by jaws 2 which develop grasping force FG The force is measured by sensor 3 located in base 4 which connects the gripper with drive rod 5 The latter moves rack 6 and the. .. kinematics of the gripper Force Fd, which is developed by rod 5, determines grasping force FG Sensor 3 enables the desired ratio FG/Fd to be achieved The sensor can be made so as to measure more than one force, say, three projections offerees and torques relative to a coordinate axis These devices help to control the grasping force; however, its value must be predetermined (before using the gripper) and the. .. with the floor or anything else Let us try to develop an algorithm that permits describing the position of the vehicle relative to some immovable coordinate system X, Y (with initial point 0) through the number of revolutions of the wheels and, say, the angle of the steering fork (We assume that no slippage between the wheel and the floor occurs.) Thus, by continually calculating these two values, the . defining the position of the point belonging to the platform (the point the rod is fastened to the platform) in the basic coordinate system; p = vector of the same point in the coordinate. be formulated in this regard: a. The length of the rods is given—what is the relative position of the bodies? (FKT—Forward Kinematic Transform.) b. The relative position of the . system of the platform; b = vector of the point in the basic coordinate system (the lower fastening point of the rod); and K= cosine matrix. Then the connection between all these vectors