Robotics 2 E Part 7 pdf

170 Kinematics and Control of Automatic Machines FIGURE 4.65 Oscillation amplitudes al of the mass M versus ratio k 2 /m. (Example). FIGURE 4.66 Vibration amplitudes of mass M versus ratio k 2 /m and time during 5 seconds of the process (Example). FIGURE 4.67 Layout of a DD device. 4.7 Electrically Controlled Vibration Dampers 171 FIGURE 4.68 Photograph of one of the DDs used in our experiments. FIGURE 4.69 Layout of an active damper. Force P changes depending upon the free vibrations of the mass m. FIGURE 4.70 Oscillation amplitude versus frequency of P and time during the first 5 seconds. A "valley" of almost zero amplitudes at freo^iency about co = 18 I/sec is clearly seen. 172 Kinematics and Control of Automatic Machines FIGURE 4.71 Active damping force generator. 1) Oscillating mass; 2) Core of the magnet; 3) Coil of the magnet; 4) Armature. FIGURE 4.72 Comparison of the free oscillation of mass M (computation) without (damping takes about 20 sec) and with actuation of the AD (damping takes about 10 sec). force is applied to the mass M. Obviously, the bigger the mass of the armature 4, the bigger the force. The core 4 is fastened to the arm of the manipulator (or any other object). An example of a comparison of the vibrations damping processes is shown in Figure 4.72. One process, taking about 20 seconds, is calculated for a usual system, without any artificial damping means, while the other, taking about 11 seconds, is the result of AD use. A special control system that carries out all signal transformations must be used for this method. Its general layout for one control channel is shown in Figure 4.73. The accelerometer and the active damper are placed on the end of a robot's arm. The signal FIGURE 4.73 Layout of the proposed AD system. Exercises 173 from each accelerometer is doubly integrated and amplified. Thereafter, the obtained power signal enters the active damper where it generates the force P(t) required for damping. This latter idea of electrically controlled damping being designed to suit different and various mechanical systems (including manipulators) seems to be a very fruitful means for increasing accuracy of automatic manufacturing machines. The main advan- tage of this idea is the possibility to interact between the mechanics and the control electronics or computer. This kind of interaction recently has been given the name mechatronics. Exercise 4E-1 For the mechanisms shown in Figure 4E-1 a) and b), write the motion functions y = n(.x) and y f = n'(jt), respectively. For case a) calculate the speed y and the acceleration y of link 2 when x = 0.05 m, x = 0.1 m/sec, x = 0, and L = 0.15 m, and the force acting on link 1 to overcome force F= 5N acting on link 2. For case b) calculate the speed y and the acceleration y of link 3 when 0 = 30°, 0 = 5 rad/sec, 0 = 0, AO = 0.2 m and ACIAB = 2. Exercise 4E-2 A cam mechanism is shown in Figure 4E-2. The radius of the initial dwelling circle is r 0 = 0.08 m. The follower moves along a line passing through the camshaft center O (i.e., e = 0). The law of motion of the follower y(0) is given by: FIGURE 4E-1a) 174 Kinematics and Control of Automatic Machines FIGURE 4E-1b) FIGURE 4E-2 During rotation for 9 = 45°, the cam's profile completes the displacement of the fol- lower for a distance h. Calculate the maximum allowed value h which provides the condition where the pressure angle a does not exceed the permitted value a max = 20°; calculate the profile angle 0* at which the pressure angle becomes worse. 5 Feedback Sensors Referring to Figure 1.5 we see that, beginning from level 8, feedbacks are introduced into the design of an automatic machine or robot. These serve to control the machine or process, assuring automatic correction response of the system when conditions change. Sensors are the principal elements of a feedback system. This chapter pre- sents a brief review of the most important feedback domains and sensors appropriate to them. The sensors can be divided into two main groups: analog and digital. To the first group belong those sensors that respond to changes in the measured value by changing some other physical value in their output, say, voltage, resistance, pressure, etc. In contrast, digital sensors transform the measured value into a sequence of elec- trical pulses. Information is carried encoded as the amount of pulses (say, the higher the number of pulses, the larger the measured dimension), as the frequency of pulses, or as some other pulse-duration parameter. The amplitude of the pulses usually has no importance in information transmission. 5.1 Linear and Angular Displacement Sensors The most common task of a feedback is to gather information about the real loca- tions of robot or machine links using, for example, sensors that respond to displace- ment or changes in location. There are several kinds of these sensors, some of which will be considered here. Electrical sensors The simplest displacement sensor is a potentiometer: a variable electrical resistor in which the slide arm is mechanically connected to the moving link. Thus, the resis- tance changes in accordance with the displacement. The electrical displacement or location sensors are usually a part of an electrical bridge, the layout of which is shown 175 176 Feedback Sensors in Figure 5.la). When a constant voltage V 0 is introduced, the off-balance voltage AV can be expressed as follows: There are several methods to use these bridges. For instance, keeping the resis- tances R-L, R 2 , and R± constant so that R l = R 2 = R^ = R and using the resistance R 3 as a sensor, i.e., a variable resistor responding to changes in the measured value, we can rewrite Expression (5.1) as Substituting here R 3 = R + Aft, where AR is a small change of the resistance, so as AR«R we obtain, from (5.2), In the simplest case, the displacement (or the measurement of some dimension) is transformed directly into displacement of the slide arm of the resistor. Thus, as follows from Relation (5.3), the change in the output voltage AV across the bridge's diagonally opposite pair of terminals a-a is directly proportional to the displacement (for small displacements). However, it is possible to increase the sensitivity of the bridge by using a so-called differential layout, as shown in Figure 5.1 b. For this case, by sub- stituting the following in Expression (5.1), we obtain FIGURE 5.1 Layout of an electrical measurement bridge: a) Common circuit; b) Differential circuit. 5.1 Linear and Angular Displacement Sensors 177 This concept of a bridge feedback can be realized in a design such as that shown in Figure 5.2. This layout is called a compensating bridge. Here resistors R^ and R 3 are variable. The slide arm of resistor R l indicates the location of cutter support 1 driven by motor 2 via screw drive 3. The slide arm of resistor R 3 is connected to feeler 4 which traces the program template 5 (master cam) fastened onto carrier 6, driven by motor 7 via screw drive 8. Thus, when resistance R 3 changes its value due to the template's displacement, the balance of the bridge is disturbed and voltage AV" occurs on the output of the circuit. This voltage is amplified by amplifier 9 and actuates motor 2, which moves the cutter so as to change the value of resistance R 1 until the imbalance of the bridge vanishes. Thus, motor 2 compensates for the disturbances in the circuit caused by motor 7. From Expression (5.1), by substituting J? x = R + AR and R 3 = R-AR while R 2 = R 4 = R, we obtain Assuming AR«R this can be rewritten as The accuracy of such sensors is not high, about 0.5%, and absolute values of about 0.25 mm can be measured. When the resistors have a circular form, angular displace- ments can be measured. Sometimes a sensor that gives a functional dependence between the rotation and output voltage is required. Figure 5.3 gives an example. Here, bases 1 are wound with high resistance wire 2 so that subsequent winds touch one another. Arm 3 is able to rotate around center 0. The function this device provides is Figure 5.4 shows a rotating resistance sensor that produces a trapezoidal relation between the angle and the output voltage. Here 1 is a resistor, 2 is a conductor, and 3 is a slide FIGURE 5.2 Electrical bridge used for feedback in tracking machine. 178 Feedback Sensors FIGURE 5.3 Resistance sensor for measuring angular displacements with a harmonic relation between the measured angle and the output voltage. arm. The resistance wire must be wound uniformly to provide linearity during the appropriate rotation intervals. The angles 2a 0 are made of high-conductivity material. Much higher sensitivity can be achieved by using variable-induction sensors (also called variable-reluctance pick-ups). The layout of the simplest of this kind of sensor is shown in Figure 5.5. It consists of a core 1, coils 2, and armature 3. The coils are fed by alternating current with a constant frequency CD. The alternating-current resistance Z in this case can be expressed in the form where R = ohmic resistance, and X L = inductive reactance. The latter is described as where L = inductance of the system. For the layout in Figure 5.5 this parameter is described by the following formula: FIGURE 5.4 Resistance sensor for measuring angular displacements with a trapezoidal relation between the measured angle and the output voltage. 5.1 Linear and Angular Displacement Sensors 179 FIGURE 5.5 Layout of an induction displacement sensor. where // = magnetic permeability, Q = cross-sectional area of the core (Q = a • h), a,h = the dimensions of the cross section of the magnetic circuit, W= the number of winds, 8 = the width of the gap. We assume here (to make the formula simple) that the cross-sectional areas of the core and armature are equal, as are the materials of which they are made. Obviously, the gap can be represented as the following sum: where § 0 = initial gap and x = the measured displacement. Substituting (5.11) into (5.10) and the latter into (5.9), we see that (5.8) is a func- tion of jc. A more complicated design for an induction sensor is shown in Figure 5.6. This device consists of housing 1, made of ferromagnetic material with a high magnetic permeability, which constitutes the core of the sensor. Two coils 2 and 3 generate the FIGURE 5.6 Differential induction sensor for displacement measurements. Cross-sectional view. [...]... continuous measurement of the thickness of a metal strip FIGURE 5.14 Pneumatic device for hole-diameter control 186 Feedback Sensors device measures the difference between the initial distance (between the nozzles) and the distance as affected by either the thickness of the strip or the diameter of the holes This method is useful for controlling the center distance between two openings Figure 5.15 presents... pressure sensors, a different approach than that described above must be used In devices like those shown in Figures 5.35-5. 37, the force affects an elastic element and is balanced by the elasticity of the system via either a special spring or the membrane itself The higher the measured forces, pressures, or acceleration, the less accurate are the measured values However, for small forces a more effective... At) These temperatures are registered by thermoresistors (or other temperature sensors) 2 and 3, respectively, which together with constant resistors R create a bridge Warming the gas by a certain temperature increment (here At) requires different quantities of heat energy introduced by the heater for different mass flow rates of the gas The voltage AV that appears when the bridge is out of balance is... can be amplified, and when integrated the displacement x is obtained, and when differentiated the acceleration x is obtained Vibrations of 20 Hz to 500 Hz can be measured Often the piezoelectric effect is used for measuring acceleration In Figure 5 .28 such a piezoelectric transducer is represented Housing 1 is threaded to connect it, by means of its thread 2, to the object being measured A piezoelement... speed measurement, especially when a digital readout is desired An analogous kind of speed sensor, of an electrical nature, is shown in Figure 5 .21 This device consists of a permanent magnet 1 fastened to the moving element and an immovable coil 2 When relative movement occurs between these two elements, an electromotive force (EMF) appears in the coil This EMF can easily be transformed into voltage,... increase the precision of measurement, devices without mechanically moving parts can be introduced In Figure 5 .24 a thermal flow-rate sensor is presented The pipe section serves as a housing for the device and is provided with a heater 1 Before reaching the heater (say, from the left side), the flowing liquid or gas has a temperature tQ, and after it passes the heater its temperature rises to the value... principle of a Michelson interferometer that can be applied for accurate displacement determination in industrial systems where machine elements must move with high precision Interference results from the algebraic addition of the individual components of two or more light beams If two of the light beams are of the same frequency, the extent of their interference will depend on the phase shift between them... latter beam serves as a reference to which the beam reflected from moving mirror 3 is compared (mirror 4 is strictly immobile) Because of the interference due to the phase shift occurring between these two beams, the detector obtains (and processes) information about the movement of mirror 3 (and element 1) It is easy to see that the beam striking mirror 3 traverses the thickness of the splitter three... compensate for the influence of temperatures on the readout, another strain gauge 2 of the same type and material is glued perpendicular to the first By connecting them in opposite arms of the measurement bridge, only those changes in resistance that are due to stresses will be detected These devices are characterized by a sensitivity coefficient k, which is defined in the following way: FIGURE 5.30... the lower ends of which are immersed in the mercury The tubes contain contacts located at different levels When the mercury level in tube 5 reaches its contact, the coil of relay R2 is energized and the contacts of this relay are actuated The normally open contacts become closed and the normally closed, open Thus, lamp 11 is lit When the pressure increases further and the mercury level reaches the . Feedback Sensors device measures the difference between the initial distance (between the nozzles) and the distance as affected by either the thickness of the strip or the . permanent magnet 1 fastened to the moving element and an immovable coil 2. When relative movement occurs between these two elements, an electromotive force (EMF) appears in the . reflected and enters detector 7. The other beam is reflected by mirror 4 and part of it is transmitted by the splitter to the detector. This latter beam serves as a reference