Fundamental problems in mobile robotics

Transcription

1 ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino

2 Mobile & Service Robotics Kinematics

3 Fundamental problems in mobile robotics Locomotion: how the robot moves in the environment Perception: how the robot perceives the environment Mapping: how to build the map of the environment Localization: where is the robot wrtthe map Representation: how the robot organizes the knowledge about the environment Path planning/action planning: what the robot shall do to go from here to there; what are the actions to be performed to complete a specified task Supervision and control: how are the command to actuators generated to perform simple or complex tasks. How to generate tasks Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW 3

4 Kinematics Locomotion depends on the robot kinematics Almost all mobile robots are underactuated, i.e., they have a number of actuators (and the relative control signals) that is less than the number of degrees of freedom of the structure Various wheels arrangements exist Each one has a different kinematic model We will study two wheeled structures Differential drive robots Bicycle-like robots A brief introduction to quadcopters Basilio Bona -DAUIN-PoliTo ROBOTICS 01PEEQW 4

5 Underactuated robots A car has two actuators, thrust and steer, and moves in a 2D space (3dof) A fixed-wing aircraft has four actuators, forward thrust, ailerons, elevator and rudder), and moves in a 3D space (6dof) A helicopter has four actuators: thrust vector magnitude and direction from the main rotor, and yaw moment from the tail rotor, and moves in a 3D space (6dof) A quadcopterhas four actuators, the four rotor thrusts, and moves in a 3D space (6dof) A ship has two propellers on parallel axes and one rudder, and moves in a 2D space (3dof) Basilio Bona -DAUIN-PoliTo ROBOTICS 01PEEQW 5

6 Kinematics β() t the steering angle is the angle between the velocity vector and the steered wheel direction is equivalent to the angle between the normal to the velocity vector and the steered wheel rotation axis we indicate with robot pose the position and the orientation (with respect to some give inertial axis) ( θ) p() t = x y c s 0 θ θ 0 R = s c 0 m θ θ T 6

7 Instantaneous Curvature Centre (ICC) If an ICC exists, the wheel motion occurs without slippage ICC Basilio Bona -DAUIN-PoliTo ROBOTICS 01PEEQW 7

8 ICC If an ICC does not exist, the wheel motion occurs with slippage? wheels slip Basilio Bona -DAUIN-PoliTo ROBOTICS 01PEEQW 8

9 Kinematics ω W R W j W δ α i W j r d y R r ω r α v=rϕɺ i r R 0 x Basilio Bona -DAUIN-PoliTo ROBOTICS 01PEEQW 9

10 Kinematics Fixed wheel The kinematic model represents a rover frame with a generic active non-steering wheel, located at a given position, with a local orientation d = αδ, ( dcosα dsinα 0) given constants T The kinematic equations describe the relations and constraints between the wheel angular velocity and the angular velocity of the frame Linear velocity of the wheel at the contact point with the plane Angular velocity of the wheel Angular velocity of the robot v= ω ω W r ( v v 0) = = x y T ( ω ω 0) Wx Wy ( 0 0 ω ) rz T T 10

11 Kinematic constraints Hypothesis ω ω r W = θɺ = ϕɺ Constraints: To avoid slippage the tangential velocity at the wheel contact point, due to the rover rotation around its center, must be equal to the advance velocity of the wheel v = rϕɺ In the wheel reference frame we have ω ω r W v= = = ( 0 0 1) ( 0 1 0) ( 1 0 0) T T T θɺ rϕɺ T ϕɺ 11

12 Differential Drive (DD) Rover Left wheel, active, non steering Passive support (castor, omniwheel) Right wheel, active, non steering 12

13 Differential Drive Rover j r Left wheel l 2 r l 2 k r i r Right wheel 13

14 Differential Drive Kinematics ICC θ ɺ () t v l () t R r v () t R 0 v () r t 14

15 Differential Drive Kinematics ICC v() l t θ ɺ () t v () t v () r t R 0 15

16 Differential Drive Kinematics ICC C i Consider the generic time instant ρ i v l v v r 16

17 Differential Drive Kinematics v v r v l R r C i after v > v r l v > v l r Turn left Turn right v l v v r R r before 17

18 Differential Drive Kinematics C i δθ t θ 18

19 Differential Drive Kinematics v k C ik ρ ik j rk δθ k δθ k θ k j 0 c ik ρ j ik rk θ k t k+1 v k 0 t rk R 0 i 0 19

20 Differential Drive Kinematics C ik δθ k B AB= t s lk s k A δθ 2 k srk 20

21 Differential Drive Kinematics 21

22 Differential Drive Kinematics Considering the sampled time equations and assuming constant values during the time interval ICC k C x = = C y ρ sinθ xk k k k + ρ cosθ yk k k k ω v ρ k k k v vl = ; δθ = k l v + v rk lk = 2 v ( v v ) k l + rk lk = = ω 2( v v ) rk k rk lk k rk lk s s l ω( ρ + l/2) = k k v ω( ρ l/2) = v k k l rk k 22

23 Differential Drive Kinematics t xt () = cos θτ ( )( vτ)d τ; yt () = sin θτ ( )( vτ)d τ; 0 0 t θ() t = ωτ ( )dτ 0 Approximation when sampling period is small d ω = θ ωt θ θ δθ + 1 dt k k k k k t x cos( δθ) sin( δθ) 0 x C C k+ 1 k k k xk xk y sin( δθ) cos( δθ) 0 y C C k+ 1 = k k k yk + yk θ θ δθ k+ 1 k k (A) 23

24 Differential Drive Kinematics EULER APPROXIMATION x = x + vt cos θ + 1 y = y + vtsinθ k k k k k+ 1 k k k θ = θ + ω k+ 1 k k T 24

25 Differential Drive Kinematics RUNGE-KUTTA APPROXIMATION 1 x = x + vtcos θ ωt k+ 1 k k + k k 2 1 y = y + vt sin θ ωt k+ 1 k k + k k 2 θ = θ + ωt k+ 1 k k 25

26 Differential Drive Kinematics ω x = x + ( sinθ sinθ) EXACT INTEGRATION k k+ 1 k k+ 1 k vk ωk = ( cosθ cosθ ) (B) k+ 1 k k+ 1 k vk y y θ = θ + ωt k+ 1 k k (A) and (B) are the same 26

27 Differential Drive Kinematics 27

28 Path Planning 28

29 Path Planning free space obstacles final pose 2D final pose initial pose initial pose 29

30 Path Planning? 3D 30

31 Path Planning A robot path from pose A to pose B can always be decomposed into a series of arcs of different radii arcs of zero radius represent turn-in-place maneuvers arcs of infinite radius represents straight line maneuvers 31

32 Path Planning Driving a vehicle to a goal (no orientation) relative to frame R A x, y f f R A θ (0,0) (0, ρ ) 32

33 Path Planning Driving a vehicle to a goal (with orientation) relative to frame R A (0, r ) 2y R B θ 2 R B x, y f1 f1 x, y f2 f2 R A θ 1 (0, 0) (0, ρ) 1 33

34 Path Planning 34

35 Path Planning 35

36 Unicycle Unicycle is the simplest example of wheeled robot. It consists of a single actuated steering wheel ωis the steering velocity x cosθ 0 ɺ sin 0 yɺ = θ v+ θɺ 0 1 ω y R r θ v r R 0 x states/coordinates T x y θ control commands 36

37 Unicycle motion 37

38 Unicycle-like robot (polar coordinates) y θ δ ρ α v r ρ α 2 2 = x + y = atan2(, yx) θ R r θ δ= θ+ α= atan2(, yx) x Basilio Bona - DAUIN - PoliTo 38 ROBOTICS 01PEEQW

39 Bicycle-like robot xɺ = vcosθ yɺ = vsinθ v θɺ = tanφ L R r NON HOLONOMIC CONSTRAINT xɺ sinθ yɺcosθ = 0 Lθ ɺ v φ steering angle y f y L θ orientation R 0 x x f 39

40 Bicycle equations 1 1) 2) 1 ) 40

41 Bicycle equations 2 41

42 Bicycle equations 3 42

43 Bicycle equations 5 43

44 Bicycle vs unicycle motion 44

45 Quadcopters 45

46 Quadcopter model F 4 N 3 N 1 F 3 R B N 4 F 1 i B F 2 j B k B d N 2 46

47 Quadcopter equations 47

48 Quadcopter equations 48

49 Quadcopter equations 49

50 Quadcopter equations 50

51 Odometry Odometry is the estimation of the successive robot poses based on the wheel motion. Wheel angles are measured and used for pose computation Odometric errors increase with the distance covered, and are due to many causes, e.g. Imperfect knowledge of the wheels geometry Unknown contact points: in the ideal DD robot there are two geometric contact points between wheels and ground. In real robots the wheels are several centimeters wide and the actual contact points are undefined Slippage of the wheel wrtthe terrain Errors can be compensated sensing the environment around the robot and comparing it with known data (maps, etc.) 51

52 Odometry Errors Map Desired path Estimated path based only on odometry 52

53 Other important problems that must be addressed 1. Path planning: the definition of an optimal geometric path in the environment 2. Mapping: how to build environment maps (geometrical vs semantic) 3. Localization: where am I? 4. Simultaneous Localization and Mapping (SLAM): build a map and at the same time localize the robot in the map while moving 53

54 Path Planning Path planning: it computes which route to take, from the initial to the final pose, based on the current internal representation of the terrain and considering a cost function (minimum time, minimum energy, maximum comfort, etc) Complete path generation: this path is created as a sequence of successive moves from the initial pose Motion planning: is the execution of this theoretical route, by translating the plan from the internal representation to the physical movement of the wheels Next move selection: at each step, the algorithm must decide which way to move next. An efficient dynamic implementation is required 54

55 Path Planning Where am I going? Mission planning What s the best way there? Path planning Where have I been? Map making Where am I? Localization Cartographer Navigation Mission Planner de eliberative How am I going to get there? Behaviors Behaviors Behaviors Behaviors reactive Introduction to AI Robotics (MIT Press), copyright Robin Murphy

56 Path Planning Fixed obstacles Mobile obstacles B Velocity constraints Acceleration constraints A 56

57 Path Planning k=2 k= N k=1 j 0 k=0 R 0 k 0 i 0 Which commands shall I give to the rover? 57

58 Path Planning B Final pose Path B Path A A Initial pose 58

59 Motion Planning B Move 3:rotation Final pose Move 2: go straight A Move 1: rotation Initial pose 59

60 Non-holonomic constraints Non-holonomic constraints limit the possible incremental movements in the configuration space of the robot Robots with differential drive move on a circular trajectory and cannot move sideways Omni-wheel robots can move sideways 60

61 Holonomic vs. Non-Holonomic Non-holonomic constraints reduce the control space with respect to the current configuration (e.g., moving sideways is impossible) Holonomic constraints reduce the configuration space 61

62 Academic year 2010/11 by Prof. Alessandro De Luca Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW 62

63 Academic year 2010/11 by Prof. Alessandro De Luca Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW 63

64 Academic year 2010/11 by Prof. Alessandro De Luca Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW 64

65 Academic year 2010/11 by Prof. Alessandro De Luca Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW 65