Peter I. Corke - Pumaservo - The Unimation Puma servo system.rar (779752), страница 6
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Experimental points and lines of best t are shown.Joints+C+B+sCBB10.569 0.435 1.46e-3 -0.588 -0.395 -1.49e-3 1.48e-320.141 0.126 0.928e-3 -95.1e-3 -70.9e-3 -0.705e-3 0.817e-330.164 0.105 1.78e-3 -0.158 -0.132 -0.972e-3 1.38e-34 14.7e-3 11.2e-3 64.4e-6 -21.8e-3 -16.9e-3 -77.9e-6 71.2e-65 5.72e-3 9.26e-3 93.4e-6 -13.1e-3 -14.5e-3 -71.8e-6 82.6e-66 5.44e-3 3.96e-3 40.3e-6 -9.21e-3 -10.5e-3 -33.1e-6 36.7e-6Table 3.1: Measured friction parameters | motor referenced (Nm and Nms/rad). Positive andnegative joint velocity are indicated by the superscripts.
The column B is the mean of B + andB .The average stiction over 20 trials was taken. The standard deviation was very high for joint1, around 16% of the mean, compared to 5% of the mean for the wrist joints.3.2 MotorThe Puma robot uses two dierent size of motor | one for the base axes (joints 1-3) andanother for the wrist axes (joints 4-6). Data on these motors is dicult to nd, and the motorsthemselves are unlabeled. There is speculation about the manufacturer and model in [1], and itis strongly rumored that the motors are `specials' manufactured by Electrocraft for Unimation.It is conceivable that dierent types of motor have been used by Unimation over the years.Tarn and Bejczy et al.
[17, 19] have published several papers on manipulator control basedon the full dynamic model and cite sources of motor parameter data as Tarn et al. [18] andGoor [8]. The former has no attribution for motor parameter data quoted, while the latterquotes \manufacturer's specications" for the base motors only. The source of Tarn's data forthe wrist motors [19] is not given.
Kawasaki manufacture the Puma 560 under licence, anddata on the motors used in that machine was provided by the local distributor. Those motorsare Tamagawa TN3053N for the base, and TN3052N for the wrist. However some of theseparameters appear dierent to those quoted by Tarn and Goor.27τ distImKmτmτ+-1J eff s + BarmaturedynamicsΩm1GΩlgearboxτCCoulombfrictionFigure 3.3: Block diagram of motor mechanical dynamics. dist represents disturbance torquedue to load forces or unmodeled dynamics.A complete block diagram of the motor system dynamics is shown in Figure 3.3 and assumesa rigid transmission. The motor torque constant, Km , is a gain that relates motor current toarmature torquem = Km im(3.3)and is followed by a rst-order stage representing the armature dynamics(3.4)m = J s + Beffwhere m is motor velocity, Jeff the eective inertia due to the armature and link, and B theviscous friction due to motor and transmission.
The so-called mechanical pole is given bypm = JB(3.5)effCoulomb friction, c , described by (3.2), is a non-linear function of velocity that opposes thearmature torque. The friction and inertia parameters are lumped values representing the motoritself and the transmission mechanism. Finally, there is a reduction gear to drive the manipulatorlink.An equivalent circuit for the servo motor is given in Figure 3.4. This shows motor impedancecomprising resistance, Rm, due to the armature winding and brushes, and inductance, Lm , dueto the armature winding. Rs is the shunt resistor which provides the current feedback signal tothe current loop.
The electrical dynamics are embodied in the relationship for motor terminalvoltageVm = sKm + sLm Im + (Rs + Rm)Im + Ec(3.6)which has components due to back EMF, inductance, resistance and contact potential dierencerespectively. The latter is a small constant voltage drop, typically around 1 to 1.5V [9], whichwill be ignored here. The so-called electrical pole is given bype = RL m(3.7)m28vmimEcLmRmEbvsRsFigure 3.4: Schematic of motor electrical model.Parameter Armstrong Tarn Kawasaki PreferredJm1291e-6 198-6200e-6200e-6Jm2409e-6 203e-6200e-6200e-6Jm3299e-6 202e-6200e-6200e-6Jm435e-6 18.3e-620e-633e-6Jm535e-6 18.3e-620e-633e-6Jm633e-6 18.3e-620e-633e-6Table 3.2: Comparison of motor inertia values from several sources | motor referenced ( kg m2).3.2.1 InertiaThere are two components of inertia `seen' by the motor.
One is due to the the rigid-body linkdynamics `reected' through the gear system, and the other due to the rotating armature. Thetotal inertia sets the upper bound on acceleration, and also aects the location of the mechanicalpole by (3.5).Several sources of armature inertia data are compared in Table 3.2. Armstrong's [2] values(load referenced and including transmission inertia) were divided by G2i , from Table A.2, togive the values tabulated. These estimates are based on total inertia measured at the joint withestimated link inertia subtracted and is subject to greatest error where link inertia is high. Fromknowledge of motor similarity the value for motor 2 seems anomalous.
Values given by Tarn [18]are based on an unknown source of data for armature inertia, but also include an estimate forthe inertia of the shafts and gears of the transmission system for the base axes. These inertiacontributions are generally less than 2% of the total and could practically be ignored. Thevery dierent estimates of armature inertia given in the literature may reect dierent modelsof motor used in the robots concerned.
The preferred values for the base axes are based on aconsensus of manufacturer values rather than Armstrong, due to the clearly anomalous valueof one of his base motor inertia estimates. Inertia of the drive shaft, exible coupling and gearwill be more signicant for the wrist axes. Frequency response measurements in Section 3.4 areconsistent with the higher values of Armstrong and therefore these are taken as the preferredvalues.Change in link inertia with conguration has a signicant eect on the dynamics of the axiscontrol loop and creates a signicant challenge for control design if it is to achieve stability andperformance over the entire conguration space.29Parameter Armstrong Paul [16] CSIROKm0.1890.255 0.223Km0.2190.220 0.226Km0.2020.239 0.240Km0.0750.078 0.069Km0.0660.070 0.072Km0.0660.079 0.066123456Table 3.3: Measured motor torque constants - motor referenced (Nm/A).3.2.2 Torque constantFor a permanent magnet DC motor the torque and back EMF are given by [9] = 2Z im = Km imEb = 2Z _ = Km_(3.8)(3.9)where is the magnetic ux due to the eld, Z the number of armature windings, and themotor shaft angle.
If a consistent set of units is used, such as SI, then the torque constant inNm/A and back-EMF constant in Vs/rad will have the same numerical value.Armature reaction is the weakening of the ux density of the permanent magnet eld, by theMMF (magneto-motive force measured in Ampere-turns) due to armature current. This couldpotentially cause the torque constant to decrease as armature current is increased.
Howeveraccording to Kenjo [9] the ux density increases at one end of the pole and decreases at theother, maintaining the average ux density. Should the ux density become too low at one endof the pole, permanent de-magnetization can occur. A frequent cause of de-magnetization isover-current at starting or during deceleration. A reduction of ux density leads to reductionof torque constant by (3.8).Table 3.3 compares measurements of torque constant of the robot used in this work, withthose obtained by other researchers for other Puma 560 robots.
The values in the column headed`CSIRO' were obtained by a colleague using the common technique of applying known loads torobot joints in position control mode and measuring the current required to resist that load.Values in the column headed `Armstrong' were computed from the maximum torque and currentdata in [2]. Tarn [17] gives the torque constant for the base axis motors as 0.259 Nm=A (apparently from the manufacturer's specication).
The Kawasaki data indicate torque constants of0.253 and 0.095 Nm=A for base and wrist motors respectively.Considerable variation is seen in Table 3.3, but all values for the base motor are less thanTarn's value. This may be due to loss of motor magnetization in the robots investigated, compared to the \as new" condition of the servo motors. Another source of error is the measurementprocedure itself | since the joint is not moving the load torque is resisted by both the motorand the stiction mechanism. Lloyd [12] used an elaborate procedure based on the work involvedin raising and lowering a mass so as to cancel out the eects of friction.3.2.3 Armature impedanceFigure 3.4 shows the motor armature impedance Rm + sLm , where Rm is the resistance dueto the armature winding and brushes, and Lm is inductance due to the armature winding. Forthe Puma 560 armature inductance is low (around 1 mH [19]) and of little signicance since themotor is driven by a current source.Armature resistance, Rm, is signicant in determining the maximum achievable joint velocityby (3.22) but is dicult to measure directly.
Resistance measurement of a static motor exhibits30Axis Expt. Kawasaki TarnBase2.11.6 1.6Wrist6.73.83Table 3.4: Comparison of measured and manufacturer's values of armature resistance ().Experimental results obtained using (3.10).a strong motor position dependence due to brush and commutation eects. A conventionallocked-rotor test also suers this eect and introduces the mechanical problem of locking themotor shaft without removing the motor from the robot. Measurements on a moving motormust allow for the eect of back EMF. Combining (3.6) and (3.3) and ignoring inductance wecan writevm = Rm + Rs + Km2 _(3.10)vsRsRsmwhere the second term represents the eect of back EMF, which may be minimized by increasingthe torque load on the motor, and reducing the rotational speed.
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