Chapter 2 Actuators and Drive Systems

Transcription

1 Introduction to obotics, H. Harry Asada 1 Chapter Actuators and Drive Systes Actuators are one of the key coponents contained in a robotic syste. A robot has any degrees of freedo, each of which is a servoed joint generating desired otion. We begin with basic actuator characteristics and drive aplifiers to understand behavior of servoed joints. Most of today s robotic systes are powered by electric servootors. Therefore, we focus on electroechanical actuators..1 DC Motors Figure -1 illustrates the construction of a DC servootor, consisting of a stator, a rotor, and a coutation echanis. The stator consists of peranent agnets, creating a agnetic field in the air gap between the rotor and the stator. The rotor has several windings arranged syetrically around the otor shaft. An electric current applied to the otor is delivered to individual windings through the brush-coutation echanis, as shown in the figure. As the rotor rotates the polarity of the current flowing to the individual windings is altered. This allows the rotor to rotate continually. Stator Winding otor Windings N i a Brush Inertia Load S Bearings Shaft Angle θ Coutator Figure.1.1 Construction of DC otor Let τ be the torque created at the air gap, and i the current flowing to the rotor windings. The torque is in general proportional to the current, and is given by Figure by MIT OCW. τ = i (.1.1) K t K t where the proportionality constant is called the torque constant, one of the key paraeters describing the characteristics of a DC otor. The torque constant is deterined by the strength of the agnetic field, the nuber of turns of the windings, the effective area of the air gap, the radius of the rotor, and other paraeters associated with aterials properties. In an attept to derive other characteristics of a DC otor, let us first consider an idealized energy transducer having no power loss in converting electric power into echanical Departent of Mechanical Engineering

2 Introduction to obotics, H. Harry Asada power. Let E be the voltage applied to the idealized transducer. The electric power is then given by E i, which ust be equivalent to the echanical power: P in = E i = τ ω (.1.) where ω is the angular velocity of the otor rotor. Substituting eq.(1) into eq.() and dividing both sides by i yield the second fundaental relationship of a DC otor: E = ω (.1.3) K t The above expression dictates that the voltage across the idealized power transducer is proportional to the angular velocity and that the proportionality constant is the sae as the torque constant given by eq.(1). This voltage E is called the back ef (electro-otive force) generated at the air gap, and the proportionality constant is often called the back ef constant. Note that, based on eq.(1), the unit of the torque constant is N/A in the etric syste, whereas the one of the back ef constant is V/rad/s based on eq.(). Exercise.1 Show that the two units, N/A and V/rad/s, are identical. The actual DC otor is not a loss-less transducer, having resistance at the rotor windings and the coutation echanis. Furtherore, windings ay exhibit soe inductance, which stores energy. Figure.1. shows the scheatic of the electric circuit, including the windings resistance and inductance L. Fro the figure, di u = i + L + E (.1.4) dt where u is the voltage applied to the arature of the otor. L u i E ω τ Figure.1. Electric circuit of arature Cobining eqs.(1), (3) and (4), we can obtain the actual relationship aong the applied voltage u, the rotor angular velocity ω, and the otor torque τ. Kt u dτ Kt = τ + Te + ω (.1.5) dt Departent of Mechanical Engineering

3 Introduction to obotics, H. Harry Asada 3 L where tie constant T e =, called the otor reactance, is often negligibly sall. Neglecting this second ter, the above equation reduces to an algebraic relationship: τ Kt Kt = u ω (.1.6) This is called the torque-speed characteristic. Note that the otor torque increases in proportion to the applied voltage, but the net torque reduces as the angular velocity increases. Figure.1.3 illustrates the torque-speed characteristics. The negative slope of the straight lines, K t, iplies that the voltage-controlled DC otor has an inherent daping in its echanical behavior. The power dissipated in the DC otor is given by P dis = i = τ (.1.7) Kt fro eq.(1). Taking the square root of both sides yields P dis τ =, K = (.1.8) K K t where the paraeter K is called the otor constant. The otor constant represents how effectively electric power is converted to torque. The larger the otor constant becoes, the larger the output torque is generated with less power dissipation. A DC otor with ore powerful agnets, thicker winding wires, and a larger rotor diaeter has a larger otor constant. A otor with a larger otor constant, however, has a larger daping, as the negative slope of the torquespeed characteristics becoes steeper, as illustrated in Figure.1.3. τ axτ u P out -K 1 axω axω ω Figure.1.3 Torque-speed characteristics and output power Departent of Mechanical Engineering

4 Introduction to obotics, H. Harry Asada 4 Taking into account the internal power dissipation, the net output power of the DC otor is given by P out Kt = τ ω = ( u K ω) ω (.1.9) This net output power is a parabolic function of the angular velocity, as illustrated in Figure.1.3. It should be noted that the net output power becoes axiu in the iddle point of the velocity axis, i.e. 50 % of the axiu angular velocity for a given arature voltage u. This iplies that the otor is operated ost effectively at 50 % of the axiu speed. As the speed departs fro this iddle point, the net output power decreases, and it vanishes at the zero speed as well as at the axiu speed. Therefore, it is iportant to select the otor and gearing cobination so that the axiu of power transfer be achieved.. Dynaics of Single-Axis Drive Systes DC otors and other types of actuators are used to drive individual axes of a robotic syste. Figure..1 shows a scheatic diagra of a single-axis drive syste consisting of a DC otor, a gear head, and ar links 1. An electric otor, such as a DC otor, produces a relatively sall torque and rotates at a high speed, whereas a robotic joint axis in general rotates slowly, and needs a high torque to bear the. In other words, the ipedance of the actuator: Z torque angular velocity τ ω = = (..1) is uch saller than that of the. Ar Links Gearing τ ω DC Motor Joint Axis Figure..1 Joint axis drive syste 1 Although a robotic syste has ultiple axes driven by ultiple actuators having dynaic interactions, we consider behavior of an independent single axis in this section, assuing that all the other axes are fixed. Departent of Mechanical Engineering

5 Introduction to obotics, H. Harry Asada 5 To fill the gap we need a gear reducer, as shown in Figure..1. Let r > 1 be a gear reduction ratio (If d 1 and d are diaeters of the two gears, the gear reduction ratio is r = d / d 1). The torque and angular velocity are changed to: τ 1 = r τ, ω = ω (..) r where τ and ω are the torque and angular velocity at the joint axis, as shown in the figure. Note that the gear reducer of gear ratio r increases the ipedance r ties larger than that of the otor axis Z : Z = r (..3) Z Let I be the inertia of the otor rotor. Fro the free body diagra of the otor rotor, 1 I ω = τ τ (..4) r 1 where r τ is the torque acting on the otor shaft fro the joint axis through the gears, and ω is the tie rate of change of angular velocity, i.e. the angular acceleration. Let be the inertia of the ar link about the joint axis, and b the daping coefficient of the bearings supporting the joint axis. Considering the free body diagra of the ar link and joint axis yields I ω = τ bω (..5) l I l Eliinating τ fro the above two equations and using eq.(.1.6) and (..) yields Iω + Bω = k u (..6) where I, B, k are the effective inertia, daping, and input gain reflected to the joint axis: I = I + r (..7) l I K B = b + r (..8) Kt k = r (..9) Note that the effective inertia of the otor rotor is r ties larger than the original value when reflected to the joint axis. Likewise, the otor constant becoes r ties larger when reflected to the joint axis. The gear ratio of a robotic syste is typically 0 ~ 100, which eans that the effective inertia and daping becoes 400 ~ 10,000 ties larger than those of the otor itself. For fast dynaic response, the inertia of the otor rotor ust be sall. This is a crucial requireent as the gear ratio gets larger, like robotics applications. There are two ways of I Departent of Mechanical Engineering

6 Introduction to obotics, H. Harry Asada 6 reducing the rotor inertia in otor design. One is to reduce the diaeter and ake the rotor longer, as shown in Figure..-(a). The other is to ake the otor rotor very thin, like a pancake, as shown in Figure..-(b). (a) Long and slender (b) Pancake Figure.. Two ways of reducing the otor rotor inertia Most robots use the long and slender otors as Figure (a), and soe heavy-duty robots use the pancake type otor. Figure..3 shows a pancake otor by Mavilor Motors, Inc. Figure reoved for copyright reasons. Figure..3 Pancake DC otor Exercise - Assuing that the angular velocity of a joint axis is approxiately zero, obtain the optial gear ratio r in eq.(7) that axiizes the acceleration of the joint axis. Departent of Mechanical Engineering

7 Introduction to obotics, H. Harry Asada 7.3 Power Electronics Perforance of servootors used for robotics applications highly depends on electric power aplifiers and control electronics, broadly tered power electronics. Power electronics has shown rapid progress in the last two decades, as seiconductors becae faster, ore powerful, and ore efficient. In this section we will briefly suarize power electronics relevant to robotic syste developent..3.1 Pulse width odulation (PWM) In any robotics applications, actuators ust be controlled precisely so that desired otions of ars and legs ay be attained. This requires a power aplifier to drive a desired level of voltage (or current indirectly) to the otor arature, as discussed in the previous section. Use of a linear aplifier (like an operational aplifier), however, is power-inefficient and ipractical, since it entails a large aount of power loss. Consider a siple circuit consisting of a single transistor for controlling the arature voltage, as shown in Figure.3.1. Let V be the supply voltage connected to one end of the otor arature. The other end of the arature is connected to the collector of the transistor. As the base voltage varies the eitter-collector voltage varies, and thereby the voltage drop across the otor arature, denoted u in the figure, varies accordingly. Let i be the collector current flowing through the transistor. Then the power loss that is dissipated at the transistor is given by P loss 1 = ( V u) i = ( V u) u (.3.1) where is the arature resistance. Figure.3. plots the internal power loss at the transistor against the arature voltage. The power loss becoes the largest in the iddle, where half the supply voltage V/ acts on the arature. This large heat loss is not only wasteful but also harful, burning the transistor in the worst case scenario. Therefore, this type of linear power aplifier is seldo used except for driving very sall otors. u P loss V Worst range i V CE 0 V/ V u Figure.3.1 Analogue power aplifier for driving the arature voltage Figure.3. Power loss at the transistor vs. the arature voltage. An alternative is to control the voltage via - switching. Pulse Width Modulation, or PWM for short, is the ost coonly used ethod for varying the average voltage to the otor. In Figure.3. it is clear that the heat loss is zero when the arature voltage is either 0 or V. This eans that the transistor is copletely shutting down the current () or copletely Departent of Mechanical Engineering

8 Introduction to obotics, H. Harry Asada 8 aditting the current (). For all arature voltages other than these coplete - states, soe fraction of power is dissipated in the transistor. Pulse Width Modulation (PWM ) is a technique to control an effective arature voltage by using the - switching alone. It varies the ratio of tie length of the coplete state to the coplete state. Figure.3.3 illustrates PWM signals. A single cycle of and states is called the PWM period, whereas the percentage of the state in a single period is called duty rate. The first PWM signal is of 60% duty, and the second one is 5 %. If the supply voltage is V=10 volts, the average voltage is 6 volts and.5 volts, respectively. The PWM period is set to be uch shorter than the tie constant associated with the echanical otion. The PWM frequency, that is the reciprocal to the PWM period, is usually ~ 0 khz, whereas the bandwidth of a otion control syste is at ost 100 Hz. Therefore, the discrete switching does not influence the echanical otion in ost cases. 60% PWM T PWM PWM Period 5% PWM Figure.3.3 Pulse width odulation As odeled in eq.(.1.4), the actual rotor windings have soe inductance L. If the electric tie constant T e is uch larger than the PWM period, the actual current flowing to the otor arature is a sooth curve, as illustrated in Figure.3.4-(a). In other words, the inductance works as a low-pass filter, filtering out the sharp - profile of the input voltage. In contrast, if the electric tie constant is too sall, copared to the PWM period, the current profile becoes zigzag, following the rectangular voltage profile, as shown in Figure.3.4-(b). As a result, unwanted high frequency vibrations are generated at the otor rotor. This happens for soe types of pancake otors with low inductance and low rotor inertia. (a) T e >> T PWM (b) T << e T PWM Figure.3.4 Current to the otor is soothed due to inductance.3. PWM switching characteristics As the PWM frequency increases, the current driven to the otor becoes soother, and the nonlinearity due to discrete switching disappears. Furtherore, high PWM frequencies cause no audible noise of switching. The noise disappears as the switching frequency becoes higher Departent of Mechanical Engineering

9 Introduction to obotics, H. Harry Asada 9 than the huan audible range, say 15 khz. Therefore, a higher PWM frequency is in general desirable. However, it causes a few adverse effects. As the PWM frequency increases: The heat loss increases and the transistor ay over-heat, Harful large voltage spikes and noise are generated, and adio frequency interference and electroagnetic interference becoe proinent. The first adverse effect is the ost critical one, which liits the capacity of a PWM aplifier. Although no power loss occurs at the switching transistor when it is copletely or, a significant aount of loss is caused during transition. As the transistor state is switched fro to or vise versa, the transistor in Figure.3.1 goes through interediate states, which entail heat loss, as shown in Figure.3.. Since it takes soe finite tie for a seiconductor to ake a transition, every tie it is switched, a certain aount of power is dissipated. As the PWM frequency increases, ore power loss and, ore iportantly, ore heat generation occur. Figure.3.5 illustrates the turn-on and turn-off transitions of a switching transistor. When turned on, the collector current I c increases and the voltage V ce decreases. The product of these two values provides the switching power loss as shown by broken lines in the figure. Note that turn-off takes a longer tie, hence it causes ore heat loss. Transistor Voltage Current Power loss Switching power loss I c V ce µ s 1.0 µ s Turn- Turn- Tie Figure.3.5 Transient responses of transistor current and voltage and associated power loss during turn-on and turn-off state transitions Fro Figure.3.5 it is clear that a switching transistor having fast turn-on and turn-off characteristics is desirable, since it causes less power loss and heat generation. Power MOSFETs (Metal-Oxide-Seiconductor Field-Effect Transistors) have very fast switching characteristics, enabling 15 ~ 100 khz of switching frequencies. For relatively sall otors, MOSFETs are widely used in industry due to their fast switching characteristics. For larger otors, IGBTs (Insulated Gate Bipolar Transistor) are the rational choice because of their larger capacity and relatively fast response. As the switching speed increases, the heat loss becoes saller. However, fast switching causes other probles. Consider eq.(.1.4) again, the dynaic equation of the arature: Departent of Mechanical Engineering

10 Introduction to obotics, H. Harry Asada 10 di u = i + L + E (.1.4) dt High speed switching eans that the tie derivative of current i is large. This generates a large di inductance-induced kickback voltage L that often daages switching seiconductors. As dt illustrated in Figure.3.6-(a), a large spike is induced when turning on the seiconductor. To get rid of this proble a free-wheeling-diode (FWD) is inserted across the otor arature, as shown in Figure.3.6-(b). As the voltage across the arature exceeds a threshold level, FWD kicks in to bypass the current so that the voltage ay be claped, as shown in figure (c). Power supply Without FWD: Spikes of V DS V DS V DS FWD Arature With FWD: The spike is claped. (a) V DS Gate Drive Switching transistor, MOSFET V DS (c) (b) Figure.3.6 Voltage spike induced by inductance (a), free-wheeling diode (b), and the claped spike with FWD (c) High speed PWM switching also generates Electroagnetic Interference (EMI), particularly when the wires between the PWM aplifier and the otor get longer. Furtherore, high speed PWM switching ay incur adio-frequency Interference (FI). Since the PWM wavefors are square, significant FI can be generated. Whenever PWM switching edges are faster than 10 µ s, FI is induced to soe extent. An effective ethod for reducing EMI and FI is to put the PWM aplifier inside the otor body. This otor architecture, called Integrated Motor or Sart Motor, allows confining EMI and FI within the otor body by iniizing the wire length between the otor arature and the power transistors..3.3 The H-bridge and bipolar PWM aplifiers In ost robotics applications, bi-directional control of otor speed is necessary. This requires a PWM aplifier to be bipolar, allowing for both forward and backward rotations. The architecture described in the previous section needs to be extended to eet this bipolar requireent. The H-Bridge architecture is coonly used for bipolar PWM aplifiers. As shown in Figure.3.7, the H-Bridge architecture resebles the letter H in the arrangeent of switching transistors around the otor arature. Switching transistors A and B are pulled up to the supply voltage V, whereas transistors C and D are connected to ground. Cobinations of these four switching transistors provide a variety of operations. In figure (i), gates A and D are, and B and C are. This gate cobination delivers a current to the arature in the Departent of Mechanical Engineering

11 Introduction to obotics, H. Harry Asada 11 forward direction. When the gate states are reversed, as shown in figure (ii), the direction of current is reversed. Furtherore, the otor coasts off when all the gates are turned, since the arature is totally isolated or disconnected as shown in figure (iii). On the other hand, the arature windings are shortened, when both gates C and D are turned and A and B are turned. See figure (iv). This shortened circuit provides a braking effect, when the otor rotor is rotating. +V +V Gate A + _ Gate B Gate A _ + Gate B Gate C Aratur + e Gate C Arature Gate D Gate D (i) Forward otion +V (ii) everse otion +V Gate A Gate B Gate A Gate B Gate C Arature Gate D Gate C Arature Gate D (iii) The otor arature coasts off (iv) The otor windings are shortened causing a braking effect. Figure.3.7 H-bridge and four quadrant control It should be noted that there is a fundaental danger in the H-bridge circuit. A direct short circuit can occur if the top and botto switches connected to the sae arature terinal are turned on at the sae tie. A catastrophic failure results when one of the switching transistors on the sae vertical line in Figure.3.7 fails to turn off before the other turns on. Most of H-bridge power stages coercially available have several protection echaniss to prevent the direct short circuit..4 obot Controls and PWM Aplifiers of the.1 Laboratory DC otors and PWM aplifiers, the two ost iportant coponents involved in a robot power train, have been described. Now we are ready to introduce the specific drive syste and controls to be used for building robots for the design project. This ter we will use controllers and drives produced by Innovation First, Inc. The syste consists of bipolar PWM aplifiers, a PIC-based on-board robot controller with a wireless ode, a stationary controller hooked up to a laptop coputer. Potentioeters are used for Departent of Mechanical Engineering

12 Introduction to obotics, H. Harry Asada 1 easuring the angular displaceent of joint axes. They are connected to A/D converter ports of the on-board controller for position feedback control. Additional sensors can be hooked up to the on-board controllers. A C-language based developent environent is available for the syste. Figure.4.1 Bipolar PWM aplifier with a built-in cooling fun, IFI, Inc. Courtesy of IFI obotics. Used with perission. Wireless Counication Operator Interface (Stationary) obot Controller (On-Board) Figure.4. On-board and stationary controllers, IFI.Inc. Courtesy of IFI obotics. Used with perission. Departent of Mechanical Engineering

13 Introduction to obotics, H. Harry Asada 13 Courtesy of IFI obotics. Used with perission. Figure.4.3 Control syste operation diagra, IFI, Inc..5 Optical Shaft Encoders The servoechanis described in the previous section is based on analogue feedback technology, using a potentioeter and a tachoeter generator. These analogue feedbacks, although siple, are no longer used in industrial robots and other industrial applications, due to liited reliability and perforance. A potentioeter, for exaple, is poor in reliability, resolution, accuracy, and signal to noise ratio. The output tap of the variable resistance slides on a track of resistive aterial, aking a echanical contact all the tie. This slide contact causes not only electric noise but also wear of the contacting surfaces. The resolution and S/N ratio of the sensor are also liited by the echanical contact. Furtherore, linearity depends on the unifority of the resistive aterial coated on the substrate, and that is a liiting factor of a potentioeter s accuracy. Today s industrial standard is optical shaft encoders, having no sliding contact. This will be discussed next. Departent of Mechanical Engineering

14 Introduction to obotics, H. Harry Asada Basic principle An optical encoder consists of a rotating disk with grids, light sources, photodetectors, and electronic circuits. As shown in Figure.5.1, a pattern of alternating opaque and translucent grids is printed on the rotating disk. A pair of light source and photodetector is placed on both sides of the rotating disk. As an opaque grid coes in, the light bea is blocked, while it is transitted through the disk, when the translucent part coes in. The light bea is then detected by the photodetector. The disk is coupled to a otor shaft or a robot joint to easure. As it rotates, an alternating - signal is obtained with the photodetector. The nuber of grids passing through the optical eleents represents the distance traveled. Opaque Translucent Light source: LED Disk with grid pattern Photodetector Figure.5.1 Basic construction of optical shaft encoder This optical shaft encoder has no echanical coponent aking a slide contact, and has no coponent wear. An optical circuit is not disturbed by electric noise, and the photodetector output is a digital signal, which is ore stable than an analogue signal. These ake an optical shaft encoder reliable and robust; it is a suitable choice as a feedback sensor for servootors..5. Position easureent One proble with the above optical encoder design is that the direction of rotation cannot be distinguished fro the single photodetector output. The photodetector output is the sae for both clockwise and counter-clockwise rotations. There is no indication as to which way the disk is rotating. Counting the pulse nuber erely gives the total distance the shaft has rotated back and forth. To easure the angular position, the direction of rotation ust be distinguished. One way of obtaining the directional inforation is to add another pair of light source/photodetector and a second track of opaque/translucent grids with 90 degrees of phase difference fro the first track. Figure.5. illustrates a double track pattern and resultant output signals for clockwise and counter-clockwise rotations. Note that track A leads track B by 90 degrees for clockwise rotation and that track B leads track A for counter-clockwise rotation. By detecting the phase angle the direction of rotation can be distinguished, and this can be done easily with an up-down counter. By siply feeding both A phase and B phase encoder signals to an up-down counter, the direction of rotation is first detected, and the nuber of rising edges and falling edges of both signals is counted in such a way that the counter adds the incoing edge nuber for clockwise rotation and subtract the edge nubers for counter-clockwise rotation. The up-down counter indicates the cuulative nuber of edges, that is, the angular position of the otor. The output of the up-down counter is binary n-bit signals ready to be sent to a digital controller without A/D conversion. Departent of Mechanical Engineering

15 Introduction to obotics, H. Harry Asada 15 Track A Track B Photodetectors A Clockwise rotation A Counter-clockwise rotation B +90 o B -90 o Figure.5. Double track encode for detection of the direction of rotation It should be noted that this type of encoder requires initialization of the counter prior to actual easureent. Usually a robot is brought to a hoe position and the up-down counters are set to the initial state corresponding to the hoe position. This type of encoder is referred to as an increental encoder, since A-phase and B-phase signals provide relative displaceents fro an initial point. Whenever the power supply is shut down, the initialization ust be perfored for increental encoders. A Phase B Phase Up-Down Counter Most sensitive bit n-bit parallel Clear/Initialization Least sensitive bit Figure.5.3 Up-down counter for an increental encoder An absolute encoder provides an n-bit absolute position as well as the direction of rotation without use of an up-down counter and initialization. As shown in Figure.5.4, the rotating disk has n-tracks of opaque-translucent grid patterns and n pairs of light sources and photodetectors. The n-tracks of grid patterns differ in the nuber of grids; the innerost track has only 1= 0 pair of opaque and translucent slits, the second track has = 1 pairs, and the i-th track has i-1 pairs. The n outputs fro the photodetectors directly indicate the n-bit absolute position of the rotating disk. In general, absolute encoders are ore coplex and expensive than increental encoders. In case of power failure, increental encoders need a laborious Departent of Mechanical Engineering

16 Introduction to obotics, H. Harry Asada 16 initialization procedure for recovery. For quick recovery as well as for safety, absolute encoders are often needed in industrial applications. Sorry for this poor drawing. I did in haste. Figure.5.4 Absolute encoder.5.3 Velocity estiate Velocity feedback is needed for iproving accuracy of speed control as well as for copensating for syste dynaics. A salient feature of optical encoders is that velocity inforation can be obtained along with position easureent. Without use of a dedicated tachoeter generator, velocity easureent can be attained by siply processing pulse sequences generated by an optical encoder. Figure.5.5 shows a pulse sequence coing fro an optical encoder. Each pulse indicates a rising edge or a falling edge of phase A & B signals. Therefore, the density of this pulse train, i.e. the pulse frequency, is approxiately proportional to the angular velocity of the rotating shaft. The pulse density can be easured by counting the nuber of incoing pulses in every fixed period, say T=10 s, as shown in the figure. This can be done with another up-down counter that counts A phase and B phase pulses. Counting continues only for the fixed sapling period T, and the result is sent to a controller at the end of every sapling period. Then the counter is cleared to re-start counting for the next period. As the sapling period gets shorter, the velocity easureent is updated ore frequently, and the delay of velocity feedback gets shorter. However, if the sapling period is too short, discretization error becoes proinent. The proble is ore critical when the angular velocity is very sall. Not any pulses are generated, and just a few pulses can be counted for a very short period. As the sapling period gets longer, the discretization error becoes saller, but the tie delay ay cause instability of the control syste. Counter clear Sapling period T = Pulse counting interval Figure.5.5 Velocity estiate based on pulse frequency easureent For siplicity only an increental encoder is considered. Departent of Mechanical Engineering

17 Introduction to obotics, H. Harry Asada 17 An effective ethod for resolving these conflicting requireents is to use a dual ode velocity easureent. Instead of counting the nuber of pulses, the interval of adjacent pulses is easured at low speed. The reciprocal to the pulse interval gives the angular velocity. As shown in Figure.5.6, the tie interval can be easured by counting clock pulses. The resolution of this pulse interval easureent is uch higher than that of encoder pulse counting in a lower speed range. In contrast, the resolution gets worse at high speed, since the adjacent pulse interval becoes sall. Therefore, these two ethods suppleent to each other. The dual ode velocity easureent uses both counters and switches the depending on the speed. Low speed counter High speed counter Figure.5.6 Dual ode velocity easureent.6 Brushless DC Motors The DC otor described in the previous section is the siplest, yet efficient otor aong various actuators applied to robotic systes. Traditional DC otors, however, are liited in reliability and robustness due to wear of the brush and coutation echanis. In industrial applications where a high level of reliability and robustness is required, DC otors have been replaced by brushless otors and other types of otors having no echanical coutator. Since brushless otors, or AC synchronous otors, are increasingly used in robotic systes and other autoation systes, this section briefly describes its principle and drive ethods. Stator otor Stator Peranent agnet Magnetic flux Peranent agnet a) Conventional DC otor b) Brushless DC otor Figure by MIT OCW. Figure.6.1 Construction of brushless DC otor and conventional DC otor Departent of Mechanical Engineering

18 Introduction to obotics, H. Harry Asada 18 In the brushless otor, the coutation of currents is perfored with an electronic switching syste. Figure.6.1 shows the construction of a brushless otor, copared with a traditional DC otor. In the brushless otor, the rotor and the stator are swapped. Unlike the traditional DC otor, the stator of the brushless otor consists of windings, whereas the rotor coprises peranent agnets. The coutation is accoplished by easuring the rotor position using a position sensor. Depending on the rotor position, currents are delivered to the corresponding windings though electronic switching circuits. The principle of torque generation reains the sae, and the torque-speed characteristics and other properties are ostly preserved. Therefore, the brushless otor is highly efficient with added reliability. A drawback of this brushless otor design is that the torque ay change discontinuously when switches are turned on and off as the rotor position changes. In the traditional DC otor this torque ripple is reduced by siply increasing the coutator segents and dividing the windings to any segents. For the brushless otor, however, it is expensive to increase the nuber of electronic switching circuits. Instead, in the brushless otor the currents flowing into individual windings are varied continuously so that the torque ripple be iniu. A coon construction of the windings is that of a three-phase windings, as shown in Figure.6.. Let I A, I B and I C be individual currents flowing into the three windings shown in the figure. These three currents are varies such that: I I I A B C = I = I = I O O O sinθ sin( θ + π ) 3 4 sin( θ + π ) 3 (.6.1) where I O is the scalar agnitude of desired current, and θ is the rotor position. The torque generated is the suation of the three torques generated at the three windings. Taking into account the angle between the agnetic field and the force generated at each air gap, we obtain 4 τ = k O[ I A sinθ + I B sin( θ + π ) + IC sin( θ + π )] (.6.) 3 3 where k 0 is a proportionality constant. Substituting eq.(1) into eq.() yields τ = koio (.6.3) 3 The above expression indicates a linear relationship between the output torque and the scalar agnitude of the three currents. The torque-current characteristics of a brushless otor are apparently the sae as the traditional DC otor. Departent of Mechanical Engineering

19 Introduction to obotics, H. Harry Asada 19 eference input (analogue) esolver to digital converter esolver otor ead OM only eory OM OM Windings Multiplying D/A converter M/DA M/DA M/DA Triangular Wave Pulse width odulator PWM PWM PWM Power aps Figure.6. Brushless DC otor and drive aplifier Figure by MIT OCW. Departent of Mechanical Engineering