CSE4421/5324: Introduction to Robotics

Transcription

1 CSE442/5324: Introduction to Robotics

2 Contact Information Burton Ma Lassonde 246 EECS442/5324 lectures Monday, Wednesday, Friday :3-2:3PM (SLH C) Lab Thursday 2:3-2:3, Prism 4 Lab 2 Thursday 2:3-4:3, Prism 4 (web site not comlete yet) 2

3 General Course Information introduces the basic concets of robotic maniulators and autonomous systems. After a review of some fundamental mathematics the course examines the mechanics and dynamics of robot arms, mobile robots, their sensors and algorithms for controlling them. 3

4 Textbook no required textbook first 6 weeks of course uses notation consistent with Robot Modeling and Control by MW Song, S Hutchinson, M Vidyasagar 4

5 Assessment labs/assignments 6 x 5% midterm, 3% exam, 4% 5

6 6 Introduction to maniulator kinematics

7 Robotic Maniulators a robotic maniulator is a kinematic chain i.e. an assembly of airs of rigid bodies that can move resect to one another via a mechanical constraint the rigid bodies are called links the mechanical constraints are called joints 7 Symbolic Reresentation of Maniulators

8 A5 Robotic Arm link 2 link 3 8 Symbolic Reresentation of Maniulators

9 Joints most maniulator joints are one of two tyes. revolute (or rotary) like a hinge 2. rismatic (or linear) like a iston our convention: joint i connects link i to link i when joint i is actuated, link i moves 9 Symbolic Reresentation of Maniulators

10 Joint Variables revolute and rismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable q i : joint variable for joint i. revolute q i = i : angle of rotation of link i relative to link i 2. rismatic q i = d i : dislacement of link i relative to link i Symbolic Reresentation of Maniulators

11 Revolute Joint Variable revolute like a hinge allows relative rotation about a fixed axis between two links axis of rotation is the z axis by convention joint variable q i = i : angle of rotation of link i relative to link i link i link i i joint i Symbolic Reresentation of Maniulators

12 Prismatic Joint Variable rismatic like a iston allows relative translation along a fixed axis between two links axis of translation is the z axis by convention joint variable q i = d i : dislacement of link i relative to link i link i link i d i joint i 2 Symbolic Reresentation of Maniulators

13 Common Maniulator Arrangments most industrial maniulators have six or fewer joints the first three joints are the arm the remaining joints are the wrist it is common to describe such maniulators using the joints of the arm R: revolute joint P: rismatic joint 3 Common Maniulator Arrangements

14 Articulated Maniulator RRR (first three joints are all revolute) joint axes z : waist z : shoulder (erendicular to z ) z 2 : elbow (arallel to z ) z z z shoulder forearm elbow waist 4 Common Maniulator Arrangements

15 Sherical Maniulator RRP Stanford arm htt://infolab.stanford.edu/ub/voy/museum/ictures/dislay/robots/img_244armfrontpeekingout.jpg z z d 3 2 shoulder z 2 waist 5 Common Maniulator Arrangements

16 SCARA Maniulator RRP Selective Comliant Articulated Robot for Assembly htt:// z z 2 z 2 d 3 6 Common Maniulator Arrangements

17 Parallel Robots all of the receding examles are examles of serial chains base (link ) is connected to link by a joint link is connected to link 2 by a joint link 2 is connected to link 3 by a joint... and so on a arallel robot is formed by connecting two or more serial chains htts:// 7 Parallel Robots

18 Forward Kinematics given the joint variables and dimensions of the links what is the osition and orientation of the end effector? a 2 2 a 8 Forward Kinematics

19 Forward Kinematics choose the base coordinate frame of the robot we want (x, y) to be exressed in this frame (x, y)? a 2 y 2 a x 9 Forward Kinematics

20 Forward Kinematics notice that link moves in a circle centered on the base frame origin (x, y)? a 2 y 2 a ( a cos, a sin ) x 2 Forward Kinematics

21 Forward Kinematics choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame (x, y)? y a 2 2 y x a ( a cos, a sin ) x 2 Forward Kinematics

22 Forward Kinematics notice that link 2 moves in a circle centered on frame (x, y)? y a 2 ( a 2 cos ( + 2 ), y 2 x a 2 sin ( + 2 ) ) a x ( a cos, a sin ) 22 Forward Kinematics

23 Forward Kinematics because the base frame and frame have the same orientation, we can sum the coordinates to find the osition of the end effector in the base frame (a cos + a 2 cos ( + 2 ), a sin + a 2 sin ( + 2 ) ) y a 2 ( a 2 cos ( + 2 ), y 2 x a 2 sin ( + 2 ) ) a x ( a cos, a sin ) 23 Forward Kinematics

24 Forward Kinematics we also want the orientation of frame 2 with resect to the base frame x 2 and y 2 exressed in terms of x and y y 2 x 2 a 2 y 2 a x 24 Forward Kinematics

25 Forward Kinematics without roof I claim: x 2 = (cos ( + 2 ), sin ( + 2 ) ) y 2 = (-sin ( + 2 ), cos ( + 2 ) ) y 2 a 2 x 2 y 2 a x 25 Forward Kinematics

26 Forward Kinematics find,, and exressed in frame y 2 y y x 2 (x, y)? d x 2 x 26

27 Forward Kinematics find,, and exressed in frame y 3 27 x

28 Inverse Kinematics given the osition (and ossibly the orientation) of the end effector, and the dimensions of the links, what are the joint variables? y 2 (x, y) x 2 a 2 y 2? a? x 28 Inverse Kinematics

29 Inverse Kinematics harder than forward kinematics because there is often more than one ossible solution (x, y) a 2 y a x 29 Inverse Kinematics

30 Inverse Kinematics law of cosines b 2 a 2 a aa2 cos( 2) x y (x, y) y b 2? a 2 a x 3 Inverse Kinematics

31 Inverse Kinematics 3 Inverse Kinematics ) cos( a a a a y x ) cos( ) cos( 2 2 and we have the trigonometric identity cos C a a a a y x therefore, We could take the inverse cosine, but this gives only one of the two solutions.

32 Inverse Kinematics Instead, use the two trigonometric identities: 2 2 sin cos 2 tan sin cos to obtain 2 tan C C which yields both solutions for 2. In many rogramming languages you would use the four quadrant inverse tangent function atan2 c2 = (x*x + y*y a*a a2*a2) / (2*a*a2); s2 = sqrt( c2*c2); theta2 = atan2(s2, c2); theta22 = atan2(-s2, c2); 32 Inverse Kinematics

33 Inverse Kinematics Exercise for the student: show that tan y x tan a a2 sin2 a cos Inverse Kinematics

34 34 Satial Descritions

35 Points and Vectors oint : a location in sace vector : magnitude (length) and direction between two oints q v 35

36 Coordinate Frames choosing a frame (a oint and two erendicular vectors of unit length) allows us to assign coordinates ŷ o ˆx q q v q 2 36

37 Coordinate Frames the coordinates change deending on the choice of frame o ˆx ŷ q q v q 2 37

38 Dot Product 38 the dot roduct of two vectors u n u u u 2 v n v v v 2 v u u v u v u v v u T n n 2 2 u v cos u cos v u v u

39 Vector Projection and Rejection u rojection of u on v rejection of u from v v uv u v vv uv v vv if u and v are unit vectors (have magnitude equal to ) then the rojection becomes uˆ vˆ vˆ 39

40 Translation ŷ ŷ o ˆx o ˆx suose we are given o exressed in {} o 3 4

41 Translation ŷ ŷ o ˆx d o ˆx the location of {} exressed in {} d o o 3 3 4

42 Translation d j i. the translation vector can be interreted as the location of frame {j} exressed in frame {i} 42

43 Translation 2 a oint exressed in frame {} ŷ ŷ o ˆx d o ˆx exressed in {} d

44 Translation 2 d j i 2. the translation vector can be interreted as a coordinate transformation of a oint from frame {j} to frame {i} 44

45 Translation 3 ŷ d q 2 o ˆx q exressed in {} q d

46 Translation 3 3. the translation vector d can be interreted as an oerator that takes a oint and moves it to a new oint in the same frame 46

47 Rotation suose that frame {} is rotated relative to frame {} ŷ ŷ ˆx o o ˆx sin cos 47

48 Rotation the orientation of frame {} exressed in {} ŷ ŷ ˆx o o ˆx R x x x y y y x y 48

49 Rotation R j i. the rotation matrix can be interreted as the orientation of frame {j} exressed in frame {i} 49

50 Rotation 2 exressed in {} ŷ ŷ ˆx o o ˆx R cos sin sin cos 5

51 Rotation 2 R j i 2. the rotation matrix can be interreted as a coordinate transformation of a oint from frame {j} to frame {i} 5

52 Rotation 3 q exressed in {} ŷ q o ˆx q R cos sin sin cos 52

53 Rotation 3 3. the rotation matrix R can be interreted as an oerator that takes a oint and moves it to a new oint in the same frame 53

54 Proerties of Rotation Matrices R T = R - the columns of R are mutually orthogonal each column of R is a unit vector det R = (the determinant is equal to ) 54

55 Rotation and Translation ŷ ŷ ˆx o ˆx o 55

56 Rotations in 3D 56 z z z y z x y z y y y x x z x y x x R

57 57 Rotations

58 Proerties of Rotation Matrices R T = R - the columns of R are mutually orthogonal each column of R is a unit vector det R = (the determinant is equal to ) 58

59 Rotations in 3D 59 z z z y z x y z y y y x x z x y x x R ˆx ŷ o o ẑ ˆx ŷ ẑ

60 Rotation About z-axis zˆ z ˆ +'ve rotation ŷ ŷ ˆx ˆx 6

61 Rotation About x-axis ẑ ẑ ŷ ŷ +'ve rotation ˆx ˆx 6

62 Rotation About y-axis ẑ ẑ yˆ y ˆ +'ve rotation ˆx ˆx 62

63 Relative Orientation Examle ẑ ˆx 45 ˆx ŷ yˆ z ˆ 63

64 Successive Rotations in Moving Frames. Suose you erform a rotation in frame {} to obtain {}. 2. Then you erform a rotation in frame {} to obtain {2}. What is the orientation of {2} relative to {}? ẑ ẑ zˆ z ˆ2 ẑ 2 ẑ ˆx ˆx ŷ 2 ˆx ŷ 2 ˆx R R y, yˆ y ˆ ˆx 2 R R ŷ 2 z, ˆx 2 R 2? ŷ 64

65 Successive Rotations in a Fixed Frame. Suose you erform a rotation in frame {} to obtain {}. 2. Then you rotate {} in frame {} to obtain {2}. What is the orientation of {2} relative to {}? ẑ ẑ ẑ ẑ ẑ 2 ẑ ˆx ˆx ŷ ˆx ŷ 2 ˆx R R y, yˆ y ˆ ˆx R R z, ŷ ˆx 2 R 2? ŷ 65

66 Comosition of Rotations. Given a fixed frame {} and a current frame {} and if {2} is obtained by a rotation R in the current frame {} then use ostmulitlication to obtain: 2. Given a fixed frame {} and a frame {} and R R 2 and R 2 R R if {2} is obtained by a rotation R in the fixed frame {} then use remultilication to obtain: R 2 RR 2 R 66

67 Rotation About a Unit Axis 67 c v k s k v k k s k v k k s k v k k c v k s k v k k s k v k k s k v k k c v k R z x z y y z x x z y y z y x y z x z y x x k 2 2 2, ˆx ŷ ẑ k x k y k z kˆ cos sin cos v s c

68 68 Rigid Body Transformations

69 Homogeneous Reresentation translation reresented by a vector d vector addition rotation reresented by a matrix R matrix-matrix and matrix-vector multilication convenient to have a uniform reresentation of translation and rotation obviously vector addition will not work for rotation can we use matrix multilication to reresent translation? 69

70 Homogeneous Reresentation 7 consider moving a oint by a translation vector d z z y y x x z y x z y x d d d d d d d z z y y x x z y x d d d? not ossible as matrix-vector multilication always leaves the origin unchanged

71 Homogeneous Reresentation 7 consider an augmented vector h and an augmented matrix D z y x d d d D z y x h z z y y x x z y x z y x h d d d d d d D

72 Homogeneous Reresentation 72 the augmented form of a rotation matrix R 3x3 R 3x3 R 3x3 3x3 R R R z y x h

73 Rigid Body Transformations in 3D d {} {} d 2 {2} 73

74 Rigid Body Transformations in 3D suose {} is a rotated and translated relative to {} what is the ose (the orientation and osition) of {} exressed in {}? T? d {} {} 74

75 Rigid Body Transformations in 3D suose we use the moving frame interretation (ostmultily transformation matrices). translate in {} to get { } 2. and then rotate in { } to get {} D ' ' D R ' d { } {} d { } {} Ste {} 75 Ste 2

76 Rigid Body Transformations in 3D suose we use the fixed frame interretation (remultily transformation matrices). rotate in {} to get { } R 2. and then translate in {} in to get {} DR {} { } d {} Ste {} 76 Ste 2 { }

77 Rigid Body Transformations in 3D 77 both interretations yield the same transformation 3x3 3x3 d R R d DR T

78 Homogeneous Reresentation every rigid-body transformation can be reresented as a rotation followed by a translation in the same frame as a 4x4 matrix T R d where R is a 3x3 rotation matrix and d is a 3x translation vector 78

79 Homogeneous Reresentation in some frame i oints P i i vectors V i i v 79

80 Inverse Transformation 8 the inverse of a transformation undoes the original transformation if then d R T d R R T T T

81 Transform Equations {} T 2 {2} {} T 4 T 2 T 3 3 T 4 {4} {3} 8

82 Transform Equations give exressions for: T 2 3 T 4 82

83 Transform Equations {} T {} T 2 2 T 3 {3} {2} 83

84 Transform Equations how can you find T T 2 2 T 3 T 3 84