Encoder applications. I Most common use case: Combination with motors

Transcription

1 3.5 Rotation / Motion - Encoder applications Intelligent Robotics Encoder applications I Most common use case: Combination with motors I Used to measure relative rotation angle, rotational direction and rotational speed I Knowledge about connected transmission and wheels allows to determine the distance 107

2 3.5.1 Rotation / Motion - Encoder applications - Self-localization of mobile robots Intelligent Robotics Self-localization of mobile robots I In most cases, the motors used in mobile robotic systems are equipped with incremental encoders I Using knowledge about the transmission and the wheel diameter and circumference, the position of the moving robot can be determined I A global coordinate system has to be referenced for this purpose I This basic procedure for the localization of mobile robots is called dead reckoning I The relative position of the mobile robot is determined using the history of accumulated measurement values from the incremental encoders 108

3 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Dead reckoning I The simplest case of dead reckoning for mobile robots can be set up using a di erential drive I On a di erential drive, the two wheels of a robot are located on asharedaxis I Wheel speeds can be controlled and adjusted separately I The center of the robot is located in the middle of the "connection" between the two wheels I If wheel speeds are equal, the robot moves forward or backward I If wheel speeds di er, the robot moves along a circular path 109

4 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Dead reckoning (cont.) I The center of the circular path which the robot moves along is necessarily a point on the shared axis of the wheels I This point is called the instantaneous center of curvature (ICC) I Variation of the wheel speeds changes the location of the ICC 110

5 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Dead reckoning (cont.) I Let! be the angular velocity of the rotation of the robot around the instantaneous center of curvature I Let l be the distance between the two wheels I Let R be the distance between the center of the robot and the ICC With the above definitions the velocities of the wheels (v r and v l ) are determined through: Wheel velocity v r =! (R + l/2) v l =! (R l/2) 111

6 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Dead reckoning (cont.) I!, R, v l and v r are time-dependent terms I At each point in time! and R can be calculated as follows: Angular velocity and radius! = v r v l l R = l 2 vl + v r v r v l 112

7 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Dead reckoning (cont.) If v l = v r : I Equation for the radius is not solvable I Denominator equals zero I Radius is e ectively infinite I Robot drives straight ahead If v l = v r : I Numerator of the equation for the radius becomes zero I The robot is turning on the spot 113

8 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Forward kinematics I While driving, the robot changes its position (x, y) and orientation ( ) in relation to a global or world coordinate system I The triple (x, y, ) determined by position and orientation is called the pose of the robot I The angle is the angle in relation to the x-axis of the global coordinate system 114

9 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Forward kinematics (cont.) I The calculation of the pose which is achieved for given wheel velocities v r and v l is called forward kinematics I In this context the ICC is calculated as follows: Instantaneous center of curvature x R sin( ) ICC = y + R cos( ) 115

10 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Forward kinematics (cont.) Knowing the ICC, the pose (x 0, y 0, 0 ) of the robot can be determined at the time of t = t 0 + t (if v r and v l remain constant): 2 x cos(! t) sin(! t) 0 x 4y 0 5 = 4sin(! t) cos(! t) 05 4y ICC x ICC x ICC y ICC y 5! t I Through integration the pose of the robot can be determined for any point in time t starting from an initial situation (x 0, y 0, 0 ) at t = 0 I Wheel velocities v l (t) and v r (t) must be known 116

11 3.5.2 Rotation / Motion - Encoder applications - Dead reckoning Intelligent Robotics Forward kinematics (cont.) I In practice, use of incremental encoders allows to determine wheel velocities v l and v r at any given time I This procedure is carried out periodically, using an interval period of t I Thus, integration turns into summation (accumulation) I It is assumed that the speeds remain constant during I General problem: Measurement errors accumulate as well! t 117

12 3.5.3 Rotation / Motion - Encoder applications - Odometry Intelligent Robotics Odometry I The process of calculating the pose of a robot from wheel speeds is called odometry I Errors in orientation have a specifically high impact on the deviation of the estimated pose from the real one I Nevertheless, odometry is used in all established mobile robot systems: I Odometry is combined with absolute pose measurements I Using landmarks for absolute pose determination, a precise odometry may help reducing the number of landmarks needed I Sometimes odometry is the only available source of data 118

13 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics Causes for deviations Systematic deviation I Varying wheel diameters I Actual wheel diameter di ers from expected diameter I Actual wheel distance di ers from expected distance I Wheels are not on the same axis I Finite resolution of the encoders I Finite sampling rate of the encoders 119

14 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics Causes for deviations (cont.) Random deviation I Uneven ground I Unexpected objects on the ground I Spinning wheels: I Slippery ground I Excessive acceleration I Skidding I Internal/external forces I No dedicated contact point to the ground 120

15 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics Error propagation I Deviation estimation procedures exist I Only systematic deviation is considered, since random deviation is unbounded 121

16 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics Measurement of odometry deviation I Errors due to unmatched wheel diameters: E d = D r /D l I Errors due to uncertainty about the wheel distance E b = b actual /b nominal 122

17 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) I Error values: x, y, x = x abs y = y abs = abs x calc y calc calc 123

18 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) (cont.) 124

19 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) (cont.) 125

20 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) (cont.) 126

21 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) (cont.) I The square is repeatedly circled in clockwise and counter-clockwise directions I Resulting end coordinates usually form a cluster 127

22 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) (cont.) I Cluster centres are calculated as follows: nx x c.g.,cw/ccw = 1 n y c.g.,cw/ccw = 1 n i=1 nx i=1 xi,cw/ccw yi,cw/ccw 128

23 3.5.4 Rotation / Motion - Encoder applications - Odometry deviation Intelligent Robotics "Square Path" experiment (UMBmark) (cont.) I The absolute deviation of the centres is calculated as: q r c.g,cw = (x c.g.,cw ) 2 +(y c.g.,cw ) 2 r c.g,ccw = q (x c.g.,ccw ) 2 +(y c.g.,ccw ) 2 I The bigger one of the two values is a measure for the accuracy of the odometry I In practice, the absolute deviation is important - therefore, it is chosen instead of the average of the cluster I Orientation errors lead to deviation of the position, therefore they are not considered 129

24 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Reduction of odometry deviation I Robots with a narrow wheel base are prone to orientation errors I Free-turning wheels may cause spinning of the drive wheels (depending on the weight load of the free-turning wheels) I Reduction of the contact area to the ground I Quality of the drive system (di erential drive, synchro-drive, etc.) I Application of additional, non-driven "encoder wheels" I Odometry calibration 130

25 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry calibration I Errors of type A (E b ) vs. errors of type B (E d ) 131

26 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry calibration (cont.) I Assumption: I E d and E b are the dominant error sources I A wrong wheel distance (E b ) causes errors on rotation only I Unmatched wheel diameters (Ed ) are only causing errors during straight driving I Eb causes only errors of type A I E d causes only errors of type B 132

27 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry calibration (cont.) I Error of type A: is the deviation from the 90 rotation: = x c.g.,cw + x c.g.,ccw 4L 180 I Error of type B: is the incrementally grown deviation of the orientation at the end of each straight line: = x c.g.,cw x c.g.,ccw 4L

28 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry calibration (cont.) I The radius of the curved path can be determined through: R = L/2 sin /2 I Knowing the radius, the relation between the wheel diameters can be determined: E d = D R = R + b/2 D L R b/2 134

29 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry calibration (cont.) I Due to the wheel base b being directly proportional to the circumvolution, the following equation applies: b actual 90 = b nominal 90 ) b actual = b nominal I As a consequence: E b =

30 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry calibration (cont.) I This calibration procedure was shown to yield a reduction of the systematic error by approximately a factor of 10x I E d and E b can be used as compensation factors in the control software 136

31 3.5.5 Rotation / Motion - Encoder applications - Reduction of odometry deviation Intelligent Robotics Odometry after calibration 137

32 3.5.6 Rotation / Motion - Encoder applications - Tachometer Intelligent Robotics Tachometer I Tachometer vs. Speedometer I Tachometer = counts revolutions I Speedometer = indicates instantaneous speed I Angular change is divided by elapsed time ) angular velocity I Accurate tachometers use incremental encoders or resolvers I Cheaper alternatives count rotations (e.g. bicycle computer) 138

33 3.6 Rotation / Motion - Gyroscope Intelligent Robotics Gyroscope I A gyroscope is a "direction keeper" I An alternative to a magnetic compass I Most commonly used sensor in navigation I Used in outer space applications (position/orientation) I Types of gyroscopes: I Mechanical I Semiconductor (MEMS) I

34 3.6.1 Rotation / Motion - Gyroscope - Mechanical gyroscope Intelligent Robotics Mechanical gyroscope I Solid disc rotating around an axis I Rotation axis (spin axis) is located in a frame I This frame can rotate around one (or two) axes 140

35 3.6.1 Rotation / Motion - Gyroscope - Mechanical gyroscope Intelligent Robotics Mechanical gyroscope (cont.) Two properties: 1. Spin axis of a free gyroscope stays fixed in relation to a global coordinate system 2. A gyroscope outputs a torque proportional to the angular speed of a rotation around an axis perpendicular to the axis of rotation due to precession 141

36 3.6.2 Rotation / Motion - Gyroscope - Semiconductor gyroscope Intelligent Robotics Semiconductor gyroscope I Micro-Electro-Mechanical System (MEMS) in silicone I Manufactures using surface or bulk micromechanic processes I Various implementations exist 142

37 3.6.2 Rotation / Motion - Gyroscope - Semiconductor gyroscope Intelligent Robotics Semiconductor gyroscope (cont.) I Vibrating/Oscillating structure I Secondary oscillation due to external forces I Capacitive evaluation 143

38 3.6.2 Rotation / Motion - Gyroscope - Semiconductor gyroscope Intelligent Robotics Semiconductor gyroscope (cont.) 144

39 3.7 Rotation / Motion - Literature Intelligent Robotics Literature list [1] Jacob Fraden. Handbook of Modern Sensors: Physics, Designs, and Applications, chapter 8, pages Springer New York, 4. edition,