Peter I. Corke - Pumaservo - The Unimation Puma servo system.rar (779752), страница 7
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vm and vs are directlymeasurable and were fed to an FFT analyzer which computed the transfer function. Thesystem exhibited good coherence, and the low frequency gain was used to estimate Rm sinceRs is known. The resistance values for the base and wrist motors determined in this fashionare summarized in Table 3.4 along with manufacturer's data for the `similar' Kawasaki Pumamotors. The experiments give the complete motor circuit resistance including contributions dueto the long umbilical cable, internal robot wiring, and connectors.3.3 Current loopThe Unimate current loop is implemented in analog electronics on the so called analog servoboard, one per axis, within the controller backplane. A block diagram of the Unimate currentloop is shown in Figure 3.5. The transfer function of the block marked compensator was determined from analysis of schematics [20] and assuming ideal op-amp behaviour.
The compensatorincludes an integrator, adding an open-loop pole at the origin in order to achieve Type 1 systemcharacteristics in current following. The remaining dynamics are at frequencies well beyond therange of interest, and will be ignored in subsequent modeling. The voltage gain of the blockmarked power amplier, k, has been measured as approximately 50.Current feedback is from a shunt resistor in series with the motor. The shunts are 0:2 and0:39 for the base and wrist joints respectively.
The high forward path gain, dominated by thecompensator stage, results in the closed-loop gain, Ki , being governed by the feedback path(3.11)Ki = VIm = 6:06 1 RIdsThe measured frequency response of the joint 6 current loop is shown in Figure 3.6, and themagnitude and phase conrm the analytic model above. The response is at to 400 Hz whichis consistent with Armstrong's [4] observation that current steps to the motor settle in lessthan 500 s. The use of a current source to drive the motor eectively eliminates the motor'selectrical pole from the closed-loop transfer function.The measured current loop transconductance of each axis is summarized in Table 3.5.
Thegains are consistently higher than predicted by (3.11), but no more than expected given thetolerance of components used in the various gain stages3. The maximum current in Table3.5 is estimated from the measured transconductance and the maximum current loop drive ofvid = 10 V.3The circuitry uses only standard precision, 10%, resistors.31VIm6:06RSCurrent FBShuntRAVId++k:710 )(10 ): 105 (9(0)(2:6106 )51 4+7Compensator+-Vm+-Power amp1Lm s+RmMotor ImpedanceImEb(0)105 (107 )Voltage FB5Figure 3.5: Block diagram of motor current loop.
Eb is the motor back EMF, Vm motor terminalvoltage, and RA the amplier's output impedance.-2Mag (dB)-4-6-8-10-12110231010410Freq (Hz)Phase (deg)-200-250-300110231010410Freq (Hz)Figure 3.6: Measured joint 6 current loop frequency response Im=VId .3.4 Combined motor and current-loop dynamicsThe measured transfer function between motor current and motor velocity for joint 6 is givenin Figure 3.7, along with a tted transfer function. The transfer function corresponds to alinearization of the non-linear model of Figure 3.3 for a particular level of excitation.
The ttedmodel ism = 24:4 rad=s=V(3.12)V(31:4)Idwhich has a pole at 5Hz. From Figure 3.3 and (3.11) the model response is given bym = KmKiVId Jeff s + Beff(3.13)where Jeff and Beff are the eective inertia and viscous friction due to motor and transmission.From the measured break frequency and Armstrong's inertia value [2] of 33 10 6 kgm2 the32Joint Ki (A=V) immax (A)1-0.8838.832-0.8778.773-0.8748.744-0.4424.425-0.4424.426-0.4494.49Table 3.5: Measured current loop transconductances and estimated maximum current. Themeasurement is a lumped gain from the DAC terminal voltage to current loop output.3530ModelMag (dB)252015105 -110011010210Freq (Hz)Figure 3.7: Measured joint 6 motor and current loop transfer function, m =VId , with ttedmodel response.estimated eective viscous friction isBeff = 33 10 6 31:4 = 1:04 10 3 Nms=radm(3.14)Substituting these friction and inertia values into equation (3.13) givesm =KmKi6VId (33 10 )s + (1:04 10 3)(3.15)Equating the numerator with (3.12), and using known Ki from Table 3.5 leads to an estimateof torque constant of 0:060.The estimated eective viscous friction is signicantly greater than the measured value of37 10 6 Nms=rad from Table 3.1.
This is a consequence of the linear identication techniqueused which yields the `small-signal' response. The eective viscous friction includes a substantialcontribution due to Coulomb friction particularly at low speed. Coulomb friction may belinearized using sinusoidal or random input describing functions giving an eective viscousdamping ofBeff = B + pkC(3.16)2_where _ is the RMS velocity, and k = 1:27 (sinusoidal) or k = 1:13 (random).33Joint Iloop fuse1 120562 200973 11052422105221062110Table 3.6: Maximum torque (load referenced) at current loop and fuse current limits. Alltorques in Nm.The large-signal response may be measured by step response tests, and these indicate amuch higher gain | approximately 200 rad=s=V. From (3.13) the DC gain isKm Ki(3.17)Beffwhich leads to an estimate of eective viscous friction of 126 10 6 Nms=rad.
As expected atthe higher speed, this estimate is closer to the measured viscous friction coecient of Table 3.1.3.4.1 Current and torque limitsMaximum motor current provides an upper bound on motor torque. When the motor is stationary, the current is limited only by the armature resistance(3.18)imax = Rvamwhere va is the maximum amplier voltage which is 40V for the Puma. From the armatureresistance data given in Table 3.4 it can be shown that the maximum current given by (3.18)is substantially greater than the limit imposed by the current loop itself.
Maximum current isthus a function of the current loop, not armature resistance. The sustained current is limitedfurther by the fuses, or circuit breakers, which are rated at 4A and 2A for the base and wristmotors respectively. The fuse limits are around half the maximum achievable by the currentloop.Using maximum current data from Table 3.5, and known motor torque constants, the maximum joint torques can be computed.
Table 3.6 shows these maxima for the case of currentlimits due to the current loop or fuse.3.4.2 Back EMF and amplier voltage saturationA robot manipulator with geared transmission necessarily has high motor speeds, and thushigh back EMF. This results in signicant dynamic eects due to reduced motor torque andamplier voltage saturation. Such eects are particularly signicant with the Puma robot whichhas a relatively low maximum amplier voltage.
Figure 3.8 shows these eects very clearly |maximum current demand is applied but the actual motor current falls o rapidly as motor speedrises. Combining (3.9) and (3.6), and ignoring motor inductance since steady state velocity andcurrent are assumed, the voltage constraint can be written as_ m + RT j vajK(3.19)Kmwhere RT is the total circuit resistance comprising armature resistance, Rm and amplier outputimpedance Ra.
Rising back EMF also diminishes the torque available from the actuator during34Shaft velocity (radm/s)Shaft angle (radm)0-50-100-15000.2Time (s)0.42000qd_max = -439.5-200-400-60000.2Time (s)0.4Motor current (A)20-2I_mean = -0.2877-4I_max = -4.312-600.2Time (s)0.4Figure 3.8: Measured motor and current loop response for a step current demand of VId = 10.Motor angle, velocity and current versus time are shown.
q_max is the steady state velocity, andImean is the steady state current. The nite rise-time on the current step response is due to theanti-aliasing lter used. The sampling interval is 5ms.acceleration_ m ) Kmavail = (va K(3.20)RTHowever during deceleration, back EMF works to increase motor current which is then limitedby the current loop or fuse. When the frictional torque equals the available torque_ = (va K_ m ) KmC + B(3.21)RTthe joint velocity limit due to amplier voltage saturation(3.22)_vsat = vaRKmB +RKT2CTmis attained.
Armature resistance, RT , and friction, serve to lower this value below that due toback EMF alone. The corresponding motor current isa + Km Civsat = Bv(3.23)RT B + Km2Experiments were conducted in which the maximum current demand was applied to eachcurrent loop, and the motor position and current history recorded, as shown in Figure 3.8. Theinitial current peak is approximately the maximum current expected from Table 3.5, but falls35Joint123456Measured Estimated_vsat ivsat _vsat ivsat120 2.1 149 3.2163 0.26 165 1.3129 0.42 152 1.7406 0.56 534 0.84366 0.39 516 0.75440 0.29 577 0.51Table 3.7: Comparison of experimental and estimated velocity limits due to back EMF andamplier voltage saturation.
Velocity and current limits are estimated by (3.22) and (3.23)respectively. Velocity in radm/s and current in Amps.o due to voltage saturation as motor speed rises. Without this eect the motor speed wouldultimately be limited by friction. Table 3.7 compares computed maximum joint velocities andcurrents with the results of experiments similar to that leading to Figure 3.8. The estimates for_vsat are generally higher than the measured values, perhaps due to neglecting the amplier output resistance (which has not been measured).
Measurements for joints 2 and 3 are complicatedby the eect of gravity which eectively adds a conguration dependent torque term to (3.20).To counter this, gravity torque was computed and the corresponding current subtracted fromthat measured. At voltage saturation, the current is generally around 30% of the maximumcurrent achievable by the power amplier.3.4.3 MATLAB simulationA complete SIMULINK model of the motor and current loop is shown in Figure 3.9.
Thismodel is a sti non-linear system and is thus very slow to simulate. The high-order poles dueto motor inductance and the current loop compensator are around 300 times faster than thedominant mechanical pole of the motor. Non-linearities are introduced by voltage saturation andCoulomb friction. The reduced order model, Figure 3.10, has similar non-linear characteristicsbut does not have the high-order poles, is much faster to integrate, and is to be preferred forsimulation purposes. Increasing the value of Rm above that in Table 3.4 allows the model tomore accurately predict the saturation speed and current. This could be considered as allowingfor the currently unmodeled amplier output impedance.3.5 Velocity loopLike the current loop, the Unimate velocity loop is implemented in analog electronics on theanalog servo board.
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