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Matrix • Order Number of row or columns • Rank of Matrix: Order of largest non-zero determinant. A matrix whose order exceeds its rank is singular • Matrix Operations – Addition/Subtraction A and B must be same Order AB C Ci,j Ai,j Bi,j – Multiplication • By Scalar • By Matrix or Vector 02/11/2002 AB C Ci,j Ai B j Robotics 1 Copyright Martin P. Aalund, Ph.D. Inverse of a matrix • Matrix of Cofactors divided by the determinate a11 a12 a 21 a22 a a32 A1 31 . . . . an1 an 2 02/11/2002 1 . . . a1n A11 A . . . a2 n 12 . . . . 1 A13 . . . . A . . . . . . . . . ann A1n A21 . . . A22 . . . A23 . . . . . . . . . . . A2 n . . . Robotics 1 Copyright Martin P. Aalund, Ph.D. An1 An 2 . . . Ann Cofactors and Determinants • Cofactor of Matrix Ajk 1 j k M jk • Determinate of Matrix n n k 1 k 1 A aik Aik akj Akj • We could use any row or any column 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Matrices • Inverse of a diagonal Matrix a 0 1 / a 0 0 b 0 1 / d • Inverse of a symmetrical matrix is symmetrical • Inverse of an anti-symmetrical matrix is anti-symmetrical • Inverse of the product of matrices is equal to the reordered product of the inverses. AB1 B1A1 • Normal Matrix A A • Orthogonal Matrix A A • Other Identities A AT T 1 T A 1 1 A I 1 I 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Definitions • Actuator: A motor or transducer that converts energy (Electrical, Hydraulic, or Pneumatic Etc..) into motion. • Transducer: A device for converting one form of energy to another. An example would be a microphone. It converts acoustic energy(Sound) to electrical energy. • A/D: Analog to Digital converter. Converts an analog voltage to a digital value. Used to interface sensors to a computer. Also written (ATOD). • D/A: Digital to Analog converter. Converts a digital value to an analog voltage. Often connected to the input of a control system or amp. Also written DTOA. 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Definitions Continued • Repeatability: How well a robot can return to the same point. • Accuracy: How well a robot can move to an arbitrary point in space • Precision: The smallest increment with which a robot can be positioned. • Resolution: Sensor Increment Desired Position Precision Accuracy Ajacent Position 02/11/2002 Actual Position Robotics 1 Copyright Martin P. Aalund, Ph.D. Definitions Continued • As upper limits the precision is equal to the resolution and the Accuracy is 1/2 the precision. • Most robots repeatability, accuracy and precision changes throughout its workspace. Type Catesian Cylindircal Spherical SCARA Articulated 02/11/2002 Hozizontal Uniform Decreases Radially Decreases Radially Varies Varies Vertical Uniform Uniform Decreases Radially Uniform Varies Robotics 1 Copyright Martin P. Aalund, Ph.D. Type of Robot Actuation • Direct Drive • Geared – Belts, Gears, Harmonic Drives, Cycloidal Cam Backlash Power Density Speed Friction/Stiction Disturbances Inetia Changes Process Feedback Noise Reliability P:ostion Sensor Weight 02/11/2002 Direct Drive None Poor High Low Seen Directly Seen Directly Fealt Directly Low Very Good Coaxial Heavy Gear Reduced Can be Significant Can be Very Good Suficient for Most Applications Can be High Divided by Gear Ration Divided by Gear Ration Squared Is Masked by Stiction in Geartrain Can be Loud Good to Poor Can Take Advantage of Gear Ratio Low Robotics 1 Copyright Martin P. Aalund, Ph.D. SCARA (IBM/Sankyo) 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Type of Actuation Actuation Type Hydraulics Pneumatics Electrical Torque/Force Density Speed Positional Repeatability Control Very High Slow Poor Variable Valve Medium Fast Limited 2 position or PWM Low Fast Good Simple Type Good Linear Hydraulics Some Rotation Leak, Control Good Linear Life of Seals, Some Rotation Control Good Linear Power Density Good Rotation • Electrical Most Popular • Hydraulic used mainly in welding and underwater activities. • Pneumatics used for gripping and detented motion 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Cons Motor • Brush DC – Brushes may wear out. Perceived as a reliability issue. Brushes produce dust. • Brushless DC – Require a full H-Bridge and a sensor for comutation • AC Induction – Requi • Stepper • Reluctance 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Electric Motor Types Commutation Power Density Mechanical High Brushless DC 6 Step Electrical Halls High Brushless DC Sinusoidal Sensor Based None Motor Type Brush DC Stepper Reluctance Sensor Based Inductance Sensor Based 02/11/2002 Field Stator Torque Ripple Magnets Low Magnets 3 Windings Medium High Magnets 3 Windings Very Low Medium Magnets or Iron Medium Low Medium Low Fields Rotor DC Windings Iron Induced N Windings 3 Windings 3 Windings Thermal Poor Windings 4 on Rotor Transistors Good Windings 6 on Stator Transistors Good Windings 6 on Stator Transistors High Fair Low Fair Very Low Fair Robotics 1 Copyright Martin P. Aalund, Ph.D. Amp Type Reliabilty N Transistors 6 Transistors 6 Transistors Speed Brushes Medium None High None High None Low None Medium None Medium Position Sensing • Sense at Joint – Don’t Worry about Deflection or Backlash • Sense at Motor – Low Cost Sensor • Sense at End-Effector – Limited View – Cost 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Sensor Comparison • Incremental Encoders and Resolvers are Most Popular Type Encoder Incremental Pseudo Ambsolute Encoder Absolute Sinusoidal Encoders Typical Resolution Incremental Bits or Absolute Resolver Inductosyn Capacitive Inductive 02/11/2002 Signal Types Signals Homing Required Cost Electrical Imunity Number of Wires Electrical Interface 12 Incremental Digital A, B, Index Yes $ Very Good 4-8 None 20+ Apsolute Digutal A, B, or Sin Cos Coded Ref Small Motion $$ Good 6-10 Yes 16 Absolute Digital Binary No $$$ Very Good 20+ None 20+ Absolute Sin, Cos, 485 No $$ Good 6-10 Interpolator 14 Absolute Sin,Cos No $$$ Good 6 R/D Converter 24 Incremental Sin, Cos Yes $$$$ Fair 6 Amp +Converter 24 Either Sin, Cos Yes/No $$ Fair 6-10 Converter 24 Either Sin, Cos Yes/No $$ Fair 6-10 Converter Digital and Analog Analog Volts RMS Analog millivolts Analog millivolts Analog millivolts Robotics 1 Copyright Martin P. Aalund, Ph.D. Encoder • • • Generally Have two picks up that are 90 degrees out of Phase (A and B) This allows you to determine the direction of rotation and thus count up or down By using the rising and falling edges of both A and B we can get 4 times the number of slots. May have one or more index marks for homing. 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Absolute Encoder • Gray Code Vs Binary • Gray Code only changes by one bit per transition. • At least one sensor per track. 11 10 11 10 01 00 00 01 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Resolver • • • • • Uses an AC signal to excite the rotor winding. Stator has two windings at 90 degrees to each other. As the rotor turns the coupling to the two windings will change Can have multiple poles, but lose absolute capability. Converters usually are analog and can be expensive, $200 for 14-16 bits. Cosine Reference Sin 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Inductosyn • • • • • • • Similar to a resolver but made in two planes. The Inductosyn has many pole pairs, 50 + The output will repeat ones for each pole pair. Each cycle can be decoded to 14+ bits Require very precise alignment, and high quality amplifiers. Expensive Analog Encoders offer similar solution at a lower cost. 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Analog Encoder. • Uses a pattern and a matched diffraction grate to transmit light at different amounts as a function of rotation • Optical sensors generate voltages proportional to the light hitting them. • These voltages are digitize and used to produce absolute position values for a cycle. • Encoders can be designed to produce multiple cycles per revolution. For example a disk can have 2048 cycles and each cycle can be decoded to 10 bits to result in 22 bits of position information. • Multiple tracks can be place on a disk One track with many cycles can be used to obtain fine resolution One tack can be used to determine which cycle of the fine track the encoder is in. Similar to an Hour, Minute and second hand on a clock. • Requires additional Electronics to decode. • LEDs require relatively high power. 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Sin/Cosine 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Reflective Sin/Cosine 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Capacitive and Inductive Sensors • • • • • Operate similar to analog encoders. Patterns are placed on the rotor and stator. Rotor and stator can be made of low cost materials Technology similar to printed circuit board fabrication. Capable of very low power operation. This would allow for battery baked operation. • Resolution similar to analog encoders and Inductosyns • Electronics utilize Digital to Analog converters and DSPs or PLDs. 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D. Potentiometer • • • • • • • • • Apply Voltage across Resistive Element Uses a Brush Sliding on a Resistive Element Brush Acts as a Voltage Divider Low Cost Noisy Varies with Temperature and Time Contact will Wear May produce particles New Laser Trimmed Films Show Promise for Linear Applications 02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
