WHAT A SINGLE JOINT IS MADE OF RA

Transcription

1 Anthropomorphic robotics WHAT A SINGLE JOINT IS MADE OF

2 Notation d F ( mv) mx Since links are physical objects with mass dt J J f i i J = moment of inertia F r F r

3 Moment of inertia Around an axis m3 m1 m2 i1 J N mr 2 i i density J volume 2 rdv

4 Parallel axis theorem J J Mr 2 c Through the center of gravity

5 Example y x z J 0 x Mass M, M l y l / l /2 Ml J rdv xdx x l /2 3 l /2 12 Ml l l J M M yl/2

6 Sum of J y c x c z a x a b L M J x a b 12 M 2 2 Jy ( a c ) 12 M 2 2 Jz ( b c ) ( b ) M a J top x ( a c ) M top ( L ) 12 2 e.g. top J hand x J top x J side x J bottom x

7 Experimental estimation of J object torsional spring Use a photodiode and a computer to measure the frequency Requires calibration from known J f 1 2 K J

8 Experimental estimation of J object h center of gravity f 1 2 Mgh J

9 Work and power E const if Fext 0 W K s2 2 Fds W E, E energy 1 2 s1 P mv 2 kinetic energy dw Power P dt Fv

10 Rotational case E const if ext 0 W K 2 d W E, E energy J 2 P kinetic energy dw Power P dt

11 Single joint model end-point link/load motor reduction gears

12 Motor Let s imagine for now that it is something that generates a given torque

13 Mechanical transmission Gears Belts Lead screws Cables Cams etc.

14 Gears 1 2 N1 N2 Distance traveled is the same: r1 1 r2 2 Because the size of teeth is the same: N1 N2 r r 1 2

15 Furthermore r r N r N r No loss of energy

16 Combining N1 r N r # of teeth Inverse relationship between speed and torque output input 2 1 N 2 N 1 TR N 1 N2 mechanical parameter

17 Equivalent J N J J N N N N N J J J N N J TR J J as seen from the motor

18 In reality TR Where is the efficiency of the mechanism (from 0 to 1) is related to power, speed ratio doesn t change is also the ratio of input power vs. power at the output t

19 For example input 20% output

20 Example

21 Motion conversion Start with N N 1 Design TR, more torque (usually) TR 1 N N 2 1 J J

22 Viscous friction Easy: viscous B 2 2 eq _ viscous TR viscous TR B Beq 1 TR B2 2 Beq TR B2 Coulomb friction: eq TR F sgn( ) c 2

23 Lead screw Rotary to linear motion conversion (P=pitch in #of turns/mm or inches) x mass/load motor [ rad] 2 Px 2 Px E E 1 1 M v J 2 2 M J load 2 (2 P) 2 2 rot lin load

24 Harmonic drives From the harmonic drive website de

25 Gearhead (for real) Standard (serial) Planetary

26 Example Designing i the single joint Given: J J J TR 2 max eq max eq max load max Then taking into account some more realistic components: max J load TR 2 max

27 Example (continued) max J load TR 2 max P given max max get P motor power, from catalog This guarantees that the motor can still deliver maximum torque at maximum speed

28 More on real world components Efficiency Eccentricity Backlash Vibrations To get better results during design mechanical systems can be simulated

29 Control of a single joint microprocessor actuator mechanics sensor reference amplifier

30 Components Digital microprocessor: Microcontroller, processor + special interfaces Amplifier (drives the motor) Turns control signals into power signals Actuator E.g. electric motor Mechanics/load The robot! Sensors For intelligence

31 Actuators Various types: AC, DC, stepper, etc. DC Brushless With brushes We ll have a look at the DC with brushes, simple to control, widely used in robotics

32 DC-brushless

33 DC with brushes

34 Modeling the DC motor Speed-torque and torque-current t relationships are linear

35 In particular No-load speed current-torquetorque speed-torque stall torque

36 Real numbers!

37 Electrical diagram Ra La V E arm R g l + E g () t K E

38 Meaning of components R a V arm R l E g L a Armature resistance (including brushes) Armature voltage Losses due to magnetic field Back EMF produced by the rotation of the armature in the field Coil inductance

39 We can write V R I L I () t K arm a a a a E for R R l a which is the case at the frequency of interest, and we also have KI T a

40 On torque and current F il B F qv B lorentz

41 Thus for many coils

42 Back to motor modeling ( J J ) () t B () t M L f gr J M J J L f gr Torque generated Inertia of the motor Inertia of the load Friction Gravity

43 Furthermore V R I L I () t K arm a a a a E K I ( J J ) () t B () t T M L f gr a

44 Consequently R a K E Varm I L L a a La Ia a K T B f gr J M J L J M J L JM JL A linear system of two equations (differential) Q: can you write a transfer function from these equations? Q: can you transform the equations into a block diagram?

45 By Laplace-transforming Varm() s () s KE Varm() s Ra Ia () s La Ia () s s() s KE Ia () s R L s K I Varm() s () s KE K ( J J ) () s sb() s R L s T T M L f gr a a a a a

46 and finally () s KT La JT 2 V () s s [( R J L B) L J ] s( K K R B) L J arm a T a a T T E a a T Considering gravity and friction as additional inputs

47 Block diagram f gr V arm - R a I - 1 a 1 K sl T B sj T a K E

48 Analysis tools Control: determine V Va so to move the motor as desired Root locus Pole placement Frequency response Etc.

49 First block diagram * - A V Km (1 s ) (1 s ) a m 1 s ˆ K p A K K m p Hopen _ loop 1 s 1 s s a m

50 Root locus A K K m p open _ loop m p 1sa 1sm s H K AK K j 1 a 1 m

51 Changing K small K t higher K t

52 Let s add something, second diagram * - - A V Km (1 s ) (1 s ) a m 1 s ˆ K g K p H open _ loop AK ( K sk ) m p g (1 s )(1 s ) s a m

53 Analysis H open _ loop AK K m p K g (1 s ) K (1 s )(1 s ) s a m p K AKm Kp Z feedback K g K p

54 Root locus (case 1) j 1 1 a m K K p g K p K g 1 m

55 Root locus (case 2) j 1 K p 1 K a g m K p K g 1 m

56 Overall f gr * A Varm 1 a I s a - R a sl a K T BsJ T s K E K g K p

57 Error and performance d M () s s KT ( R sl )( B sj ) K K a a T E T closed loop (position) 1 () s () s s 1 () s () s s 1 1 ( s) K p s () s A M () s 1 s a A 1 M ( s ) K 1 s a closed loop (velocity) g

58 finally lim sh () s lim ht () s0 t 1 d s () s d lim s ( s) lim s s s0 s s0 1 1 ( sk ) p s d K p For zero error K must be 1 or the control structure must be different

59 Same line of reasoning final gr Ra AK K T p Final value due to friction and gravity R K R gr a gr a max p T p AKTmax AK K K p min gr R AK T a max

60 PID controller * - K p sk d K i s V arm - R a 1 sl a f gr Ia 1 K T BsJ T K E - 1 s

61 PID controller We now know why we need the proportionalp We also know why we need the derivative Finally, we add the integral Integrates the error, in practice needs to be limited

62 Interpreting the PID Proportional: to go where required, linked to the steady-state error Derivative: damping Integral: to reduce the steady-state error

63 Global view microprocessor actuator mechanics sensor reference amplifier OK OK

64 About the amplifiers Linear amplifiers H type T type PWM (switching) amplifiers

65 Let s consider the linear as a starting point V cc

66 H-type The motor doesn t have a reference to ground (floating) It s difficult to get feedback signals (e.g. to measure the current flowing through the motor)

67 T-type V cc V cc

68 On the T-type Bipolar DC supply Dead band (around zero) Need to avoid simultaneous conduction (short circuit)

69 Things not shown Transistor protection (currents flowing back from the motor) Power dissipation and heat sink Cooling Sudden stop due to obstacles High currents current limits and timeouts

70 T-type V cc I c R transisor Vcc R motor V cc

71 PWM amplifiers P V I ce c same current

72 PWM signal P V I ce c Transistors either on or off When off, current is very low, little power too When on, V is low, working point close to (or in) saturation, power dissipation is low

73 Comparison 12W for a 6A current using a switching amplifier 72W for a corresponding linear amplifier

74 Why does it work? () s KT LaJT 2 V () s s [( R J L B) L J ] s( K K R B) L J arm a T a a T T E a a T In practice the motor transfer function is a lowpass filter T s with f s f e ( f s 100 f e ) Switching frequency must be high enough (s=switching switching, e=electric electric pole)

75 PWM signal V cc T s V cc T s T switching

76 Feedback in servo amplifiers V cc - V in + VV cc Voltage feedback amplifier

77 Operating characteristic Vout RL R1 AV in RL R 2 R R 2 1 I L

78 We ve already seen this V I V in A arm 1 s R sl 1 a - a a Ia 1 K T BsJ T K E () s K L J A V s s R J L B L J s K K R B L J s T a T v 2 in () [( a T a ) a T ] ( T E a ) a T (1 a)

79 Current feedback - V in + V out

80 Current feedback V out R R L 1 RL R 2 AV in R R 2 1 I L

81 Motor driven by a current amplifier V in A 1s a Ia 1 K T BsJ T () s KTAi V () s ( sj B)(1 s ) in T a

82 Back to the global view microprocessor actuator mechanics sensor reference amplifier OK OK OK

83 Sensors Potentiometers Encoders Tachometers Inertial sensors Strain gauges g Hall-effect sensors and many more

84 Potentiometer - r thin resistive film V out r V R cc V cc V out + Simple but noisy Requires A/D conversion Absolute position (good!)

85 Note TR turns Encoder Motor Gearbox N TR N N 2 (most of the time) N N N N N 1 turn The resolution of the sensor multiplied by TR

86 Encoder Absolute Incremental

87 Absolute encoder phototransistors LEDs transparent motor x x x x x x x x x x x x motor shaft 13 bits required for degrees opaque

88 Incremental encoder Disk single track instead of multiple No absolute position Usually an index marks the beginning gof a turn

89 Incremental encoder output TTL t t Sensitive to the amount of light collected The direction of motion is not measured

90 Two-channel encoder 2 channels 90 degrees apart (quadrature signals) allow measuring the direction of motion A B t t

91 Moreover There are differential encoders Taking the difference of two sensors 180 degrees apart Typically A, B, Index channel A, B, Index (differential) A counter is used to compute the position from an incremental encoder

92 Increasing resolution Counting UP and DOWN edges X2 or X4 circuits

93 Absolute position A potentiometer and incremental encoder can be used simultaneously: the pot for the absolute reference, and the encoder because of good resolution and robustness to noise

94 Analog locking Use digital encoder as much as possible Get to zero error or so using the digital signal When close to zeroing the error: Switch to analog: use the analog signal coming from the photodetector (roughly sinusoidal/triangular) Much higher resolution, precise positioning

95 Tachometer Use a DC motor The moving coils in the magnetic field will get an induced EMF In practice is better to design a special purpose DC motor for measuring velocity Ripple: typ. 3%

96 As already seen e e ripple

97 Measuring speed with digital encoders Frequency to voltage converters Costly (additional electronics) Much better: in software Take the derivative (for free!) vkt ( ) pkt ( ) p(( k1) T) T

98 Inertial sensors Accelerometers: position sensor K K Ma 2 Kx a 2Kx M a

99 Gyroscopes Quartz forks driver (causing oscillations) amp V F F 2m V

100 Strain gauges Principle: deformation R (resistance) Example: conductive paint (Al, Cu) The paint covers a deformable non-conducting substrate L R L, Aconst R A conductivity

101 Reading from a strain gauge V cc strain gauge R 2 a b R 1 balance V cc RR R R V V f ( R ) 1 2 g b ab 0 ab g

102 Hall-effect sensors + - V out charges B F F qv B lorentz if electrons

103 Example Measuring angles (magnetic encoders)

104 Back to the global view microprocessor actuator mechanics sensor reference amplifier OK OK OK OK

105 Microprocessors Special DSPs for motion control Some are barely yprogrammable (the control law is fixed) Others are general purpose and they are mixed mode (analog and digital in a single chip)

106 Example DSP 16 bit ALU and instruction set PWM generator (simply attach this to either T or H amplifier) A/D conversion CAN bus, Serial ports, digital I/O Encoder counters Flash memory and RAM on-board Enough of all these to control two motors (either brush- or brushless)