Rigid Body Transformations

Transcription

1 F1/10 th Racing Rigid Body Transformations (Or How Different sensors see the same world) By, Paritosh Kelkar

2 Mapping the surroundings Specify destination and generate path to goal The colored cells represent a potential that is used to plan paths Rviz is used to specify different goals to the robot

3 Why should you watch these lectures Following the wall here blindly is going to be really hard You will need a very complicated Route definition file

4 Why should you watch these lectures Simultaneous Localization and Planning Planning

5 Lets begin

6 Scope of the Lecture PART 1 The concept of frames and transforms (different views of the same world) Why is this important to us The Homogenous Transformation Matrix PART 2 How ROS deals with these frames, conventions in ROS

7 Frames of Reference

8 Part 1

9 Transformations and Frames: Heads up 1. w.r.t = with respect to 2. Map frame where are you w.r.t the map co-oridnates from origin

10 Transformations and Frames: Heads up The Sensor frame how does the world look w.r.t the sensor Does this tell you anything about where obstacles are in the map? Does this tell you anything about where we are in the map? We must link frames together Transformations

11 Transformations and Frames The frame of reference in which the measurement is taken Z Δz X Distance measurements returned by LIDAR

12 Transformations and Frames The frame of reference in which the measurement is taken Δz X Z Y The scan Values from the LIDAR will not tell us how far away are the obstacles. We must take care of the offsets Δx

13 Transformations and Frames Δz X Z Y Note: Axes X,Y,Z of Frames of Reference are orthogonal(90 o ) to each other. X,Y,Z represent the axes along the 3 dimensions.

14 Transformations and Frames Car frame X Z Y Y Map frame Y laser frame Between frames there will exist transformations that convert measurements from one frame to another Important Point: Note what the transformation means w.r.t frames

15 Transformations and Frames Between frames there will exist transformations that convert measurements from one frame to another Z Y laser frame Y There should exist a relationship Between these frames Car frame X Y Y Map frame Y Transform from car to laser Y Transform from map to car

16 A world without frames and transformations

17 The actual motion of the car

18 Rigid Body Transforms: An Aside What s with it being Rigid? The distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. Play-Doh: Obviously not a rigid body

19 Rigid Body Transforms Y A Z A X A

20 Rigid Body Transforms Y A Z A X A

21 Rigid Body Transforms θ Y A Z A X A

22 Rigid Body Transforms θ Y A A d B Z A X A

23 Rigid Body Transforms B p θ Y A A d B Z A X A

24 Rigid Body Transforms B p A p θ Y A A d B Z A X A

25 Rigid Body Transforms What we need is Point p with respect to Frame A, given its pose in Frame B B p A p θ Y A A d B Z A X A

26 Rigid Body Transforms Special type of matrices called Rotation matrices X B = R 11 X A + R 21 Y A + R 31 Z A Y A Y B = R 12 X A + R 22 Y A + R 32 Z A Z B = R 13 X A + R 23 Y A + R 33 Z A Z A X A θ

27 Rigid Body Transforms Special type of matrices called Rotation matrices X B = R 11 X A + R 21 Y A + R 31 Z A Y B = R 12 X A + R 22 Y A + R 32 Z A Z B = R 13 X A + R 23 Y A + R 33 Z A Y A Z A A R B = R 11 R 12 R 13 R 21 R 22 R 23 R 31 R 32 R 33 Takes points in frame B and represents their orientation in frame A θ X A

28 Rigid Body Transforms: Rotation Matrices X B = R 11 X A + R 21 Y A + R 31 Z A (0,5,0) A p =? Y B = R 12 X A + R 22 Y A + R 32 Z A Y A B p Sine component Z B = R 13 X A + R 23 Y A + R 33 Z A Z A θ Cosine component XA X B = cos(θ) X A +sin(θ) Y A +0 Z A R 11 R 21 R 31

29 Rigid Body Transforms: Rotation Matrices A R B = Cos(θ) R 12 R 13 Sin(θ) R 22 R 23 0 R 32 R 33 X A Y A Z A A C S 0 RB S C X B Y B Z B ) C θ = Cos(θ ) S θ = Sin(θ

30 We have the Rotation Matrix, so now what? p R p A A B B R = For example θ We now have the point P as referenced in frame A Τ = π 6 Known Known A p = (-2.5,4.3,0) (-2.5,4.3,0) Y A A p Z A X A

31 Important point to remember The rotation matrix will take care of perspectives of orientation, what about displacement? Y A Y A Z A Z A X A X A Origins of both the frames are at the same location A d B

32 Rigid Body Transforms: And We are back to the Future A A B A B p R p d B Y B Y A Z B X B Z A X A

33 Rigid Body Transforms What we need is Point p with respect to Frame A, given its pose in Frame B p H p B p A A B B p A p θ Y A Z A X A A d B A H = Homogenous transformations that transforms B measurements in Frame B to those in Frame A A C S 0 RB S C R d A A A B B H B 0 1

34 Part 2

35 Map frame Map Frame

36 Map frame: Importance Position with respect to map MAP FRAME

37 Map Frame: Properties Used as a long term reference Dependence on localization engine (Adaptive Monte Carlo Localization AMCL used in our system more about this in later lectures) Localization engine - responsible for providing pose w.r.t map Frame Authority

38 Map Frame: ROS The tf package tracks multiple 3D coordinate frames - maintains a tree structure b/w frames access relationship b/w any 2 frames at any point of time ROS REP(ROS Enhancement Proposals) 105 describes the various frames involved Normal hierarchy world_frame Has no parent A tf tree is a structure that maintains relations between the linked frames. Note: Tf = transformer class map Child of world frame

39 Odom Frame

40 Odom frame: Calculation Frame in which odometry is measured Odometry is used by some robots, whether they be legged or wheeled, to estimate (not determine) their position relative to a starting location -Wikipedia Source: eg: Wheel encoders. Count wheel ticks

41 Odom Frame: Calculation Difference in count of ticks of wheels orientation Integrating the commanded velocities/accelarations Integrating values from IMU

42 Odom Frame: Uncertainty Initial Position Error can accumulate leading to a drift in values Incorrect diameter used? Slippage? Dead Reckoning Notice how the uncertainty increases

43 Odom Frame: Properties Continuous actual data from actuators/motors Evolves in a smooth manner, without discrete jumps Short term ; accurate local reference

44 Odom Frame: ROS General ROS frame hierarchy world_frame map Note that if the frame is connected in the tf tree, we can obtain a representation of that frame with any other frame in the tree odom Tf tree

45 Base_link and fixed frames attached to the robot

46 Base link: What is it Attached to the robot itself base_footprint; base_link; base_stabilized

47 Base link: Properties Odom -> base link transform provided by Odometry source Map -> base_link transform provided by localization component

48 Fixed Frames: Source Where do we get the relationships between the fixed frame on the car Frame for various hardware components(sensors) Robot description provides the transformations Urdf file Look up the tutorial related to this lecture

49 Base_link Frame: ROS General ROS frame hierarchy world_frame map odom base_link Tf tree

50 ROS.W.T.F Its actually a tool just very cleverly named Host of tf debugging tools provided by ROS Look at tutorial for further details $ rosrun tf view_frames $ rosrun tf view_monitor $ roswtf

51 In Conclusion Rigid Body Transformations the concept and the importance in robotic systems We now know how to correlate measurements from different sensors The upcoming lecture SLAM Simultaneous Localization and Mapping

52 Why do you have to remember all of this stuff Again, you are developing the platform in this framework Don t you want to know how you could get maps of your surroundings? what we just covered are building blocks of the upcoming topics

53 Upcoming Lectures We will go into detail about the packages that we use for mapping and localizing

54 Map frame: Properties Discontinuity Y Z Map frame (0,0,0) X

55 Map frame: Properties New sensor reading gives us new information Discontinuity Y Z Map frame (0,0,0) X Jump in position, i.e, not continuous (2,0,0)

56 Map Frame: Properties

57 Map Frame: Why Discontinuity is a Problem What pose coordinates will the controller act on?

58 Transformations and Frames The frame of reference in which the measurement is taken Z Δz X Y Δx

59 Odom Frame: Uncertainty Initial placement of odom and map frames Final placement of odom and map frames after robot has moved some distance