1. Kinematics of Robot Manipulator
Introduction to ROBOTICS
Kinematics of Robot Manipulator

2. Outline Review Robot Manipulators Kinematics Robot Configuration
Robot Specification
Number of Axes, DOF
Precision, Repeatability
Kinematics
Preliminary
World frame, joint frame, end-effector frame
Rotation Matrix, composite rotation matrix
Homogeneous Matrix
Direct kinematics
Denavit-Hartenberg Representation
Examples
Inverse kinematics

3. Review What is a robot? Why use robots?
By general agreement a robot is:
A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human.
To qualify as a robot, a machine must be able to:
1) Sensing and perception: get information from its surroundings
2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects
3) Re-programmable: can do different things
4) Function autonomously and/or interact with human beings
Why use robots?
Perform 4A tasks in 4D environments
4A: Automation, Augmentation, Assistance, Autonomous
4D: Dangerous, Dirty, Dull, Difficult

4. Manipulators Robot arms, industrial robot
Rigid bodies (links) connected by joints
Joints: revolute or prismatic
Drive: electric or hydraulic
End-effector (tool) mounted on a flange or plate secured to the wrist joint of robot

5. Robotic Manipulators a robotic manipulator is a kinematic chain
i.e. an assembly of pairs of rigid bodies that can move respect to one another via a mechanical constraint
the rigid bodies are called links
the mechanical constraints are called joints

6. A150 Robotic Arm
link 3
link 2

7. Joints most manipulator joints are one of two types
revolute (or rotary)
like a hinge
allows relative rotation about a fixed axis between two links
axis of rotation is the z axis by convention
prismatic (or linear)
like a piston
allows relative translation along a fixed axis between two links
axis of translation is the z axis by convention
our convention: joint i connects link i – 1 to link i
when joint i is actuated, link i moves

8. Joint Variables
revolute and prismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable
qi : joint variable for joint i
revolute
qi = qi : angle of rotation of link i relative to link i – 1
prismatic
qi = di : displacement of link i relative to link i – 1

9. Revolute Joint Variable
qi = qi : angle of rotation of link i relative to link i – 1
link i qi
link i – 1

10. Prismatic Joint Variable
qi = di : displacement of link i relative to link i – 1
link i – 1
link i di

11. Common Manipulator Arrangments
most industrial manipulators have six or fewer joints
the first three joints are the arm
the remaining joints are the wrist
it is common to describe such manipulators using the joints of the arm
R: revolute joint
P: prismatic joint

12. Articulated Manipulator
RRR (first three joints are all revolute)
joint axes
z0 : waist
z1 : shoulder (perpendicular to z0)
z2 : elbow (parallel to z1) z0 z1 z2 q2 q3
shoulder
forearm
elbow q1
waist

13. Spherical Manipulator
RRP
Stanford arm z0 z1 d3 q2
shoulder z2 q1
waist

14. SCARA Manipulator
RRP
Selective Compliant Articulated Robot for Assembly z1 z2 z0 q2 d3 q1

15. Manipulators Robot Configuration: Cartesian: PPP Cylindrical: RPP
Spherical: RRP
Hand coordinate:
n: normal vector; s: sliding vector;
a: approach vector, normal to the
tool mounting plate
SCARA: RRP
(Selective Compliance Assembly Robot Arm)
Articulated: RRR

16. Manipulators Motion Control Methods Point to point control
a sequence of discrete points
spot welding, pick-and-place, loading & unloading
Continuous path control
follow a prescribed path, controlled-path motion
Spray painting, Arc welding, Gluing

17. Manipulators Robot Specifications Number of Axes
Major axes, (1-3) => Position the wrist
Minor axes, (4-6) => Orient the tool
Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration
Degree of Freedom (DOF)
Workspace
Payload (load capacity)
Precision v.s. Repeatability
Precision (validity) is defined as how accurately a specified point can be reached. It is a function of the resolution of the actuators, as well as its feedback devices. Most industrial robots can have precision of inch or better.
Repeatability (variability) is how accurately the same position can be reached if the motion is repeated many times. The robot may not reach the same point every time, but will be within a certain radius from the desired point. The radius of a circle that is formed by this repeated motion is called repeatability. It is much more important than precision. If a robot is not precise, it will generally show a consistent error, which can be predicted and thus corrected through programming. However, if the error is random, it cannot be predicted and thus cannot be eliminated. Repeatability defines the extent of this random error. Manufactures must specify repeatability in conjunction with the number of tests, the applied payload during the tests, and the orientation of the arm. Have repeatability in the inch range.
Depending on their configuration and the size of their links and wrist joints, robots can reach a collection of points called a workspace.
Payload is the weight a robot can carry and still remain within its other specification.
how accurately a specified point can be reached
how accurately the same position can be reached if the motion is repeated many times
Which one is more important?

18. What is Kinematics Forward kinematics Given joint variables
End-effector position and orientation, -Formula?

19. What is Kinematics Inverse kinematics End effector position
and orientation
Joint variables -Formula?

20. Example 1

21. Example II
given the joint variables and dimensions of the links what is the position and orientation of the end effector? a2 q2 a1 q1

22. Forward Kinematics choose the base coordinate frame of the robot
we want (x, y) to be expressed in this frame
(x, y) ? a2 q2 y0 a1 q1 x0

23. Forward Kinematics
notice that link 1 moves in a circle centered on the base frame origin
(x, y) ? a2 q2 y0 a1 q1
( a1 cos q1 , a1 sin q1 ) x0

24. Forward Kinematics
choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame
(x, y) ? y1 a2 q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0

25. Forward Kinematics
notice that link 2 moves in a circle centered on frame 1
(x, y) ? y1 a2
( a2 cos (q1 + q2),
a2 sin (q1 + q2) ) q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0

26. Forward Kinematics
because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame
(a1 cos q1 + a2 cos (q1 + q2),
a1 sin q1 + a2 sin (q1 + q2) ) y1 a2
( a2 cos (q1 + q2),
a2 sin (q1 + q2) ) q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0

27. Forward Kinematics
we also want the orientation of frame 2 with respect to the base frame
x2 and y2 expressed in terms of x0 and y0 y2 x2 a2 q2 y0 q1 a1 q1 x0

28. Forward Kinematics x2 = (cos (q1 + q2), sin (q1 + q2) ) y2 x2
y2 = (-sin (q1 + q2),
cos (q1 + q2) ) a2 q2 y0 q1 a1 q1 x0

29. Inverse Kinematics
given the position (and possibly the orientation) of the end effector, and the dimensions of the links, what are the joint variables? y2 x2
(x, y) a2
q2 ? y0 a1
q1 ? x0

30. Inverse Kinematics
harder than forward kinematics because there is often more than one possible solution
(x, y) a2 y0 a1 x0

31. Inverse Kinematics
law of cosines
(x, y) a2 b
q2 ? y0 a1 x0

32. Inverse Kinematics and we have the trigonometric identity therefore,
We could take the inverse cosine, but this gives only one of the two solutions.

33. Inverse Kinematics Instead, use the two trigonometric identities:
to obtain
which yields both solutions for q2 . In many programming languages you would use the
four quadrant inverse tangent function atan2
c2 = (x*x + y*y – a1*a1 – a2*a2) / (2*a1*a2);
s2 = sqrt(1 – c2*c2);
theta21 = atan2(s2, c2);
theta22 = atan2(-s2, c2);

34. Inverse Kinematics

35. Preliminary Robot Reference Frames World frame Joint frame Tool frame
WF is a universal coordinate frame.
JF is used to specify movements of each individual joint of the robot.
HF specifies movements of the robot’s hand relative to a frame attached to the hand T P W R

36. Points and Vectors point : a location in space
vector : magnitude (length) and direction between two points

37. Coordinate Frames
choosing a frame (a point and two perpendicular vectors of unit length) allows us to assign coordinates

38. Coordinate Frames
the coordinates change depending on the choice of frame

39. Preliminary Coordinate Transformation Reference coordinate frame OXYZ
Body-attached frame O’uvw
O, O’
Point represented in OXYZ:
Point represented in O’uvw:
Two frames coincide ==>

40. Preliminary Properties: Dot Product
Let and be arbitrary vectors in and be the angle from to , then
Properties of orthonormal coordinate frame
Mutually perpendicular
Unit vectors

41. Preliminary Coordinate Transformation Rotation only
How to relate the coordinate in these two frames?

42. Preliminary Basic Rotation
, , and represent the projections of onto OX, OY, OZ axes, respectively
Since

43. Preliminary
Basic Rotation Matrix
Rotation about x-axis with

44. Preliminary
Is it True?
Rotation about x axis with

45. Basic Rotation Matrices
Rotation about x-axis with
Rotation about y-axis with
Rotation about z-axis with

46. Preliminary Basic Rotation Matrix
Obtain the coordinate of from the coordinate of
Dot products are commutative!
<== 3X3 identity matrix

47. Properties of Rotation Matrices
RT = R-1
the columns of R are mutually orthogonal
each column of R is a unit vector
det R = 1 (the determinant is equal to 1)
4/23/2017

48. Example
A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.

49. Example
A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis.

50. Composite Rotation Matrix
A sequence of finite rotations
matrix multiplications do not commute
rules:
if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix
if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

51. Example Find the rotation matrix for the following operations:
Pre-multiply if rotate about the OXYZ axes
Post-multiply if rotate about the OUVW axes

52. Translation
suppose we are given o1 expressed in {0}

53. Translation 1
the location of {1} expressed in {0}

54. Translation 1
the translation vector can be interpreted as the location of frame {j} expressed in frame {i}

55. Translation 2
a point expressed
in frame {1}
p1 expressed in {0}

56. Translation 2
the translation vector can be interpreted as a coordinate transformation of a point from frame {j} to frame {i}

57. Translation 3
q0 expressed in {0}

58. Translation 3
the translation vector can be interpreted as an operator that takes a point and moves it to a new point in the same frame

59. Coordinate Transformations
position vector of P in {B} is transformed to position vector of P in {A}
description of {B} as seen from an observer in {A}
Rotation of {B} with respect to {A}
Translation of the origin of {B} with respect to origin of {A}

60. Coordinate Transformations
Two Special Cases
1. Translation only
Axes of {B} and {A} are parallel
2. Rotation only
Origins of {B} and {A} are coincident

61. Homogeneous Representation
Coordinate transformation from {B} to {A}
Homogeneous transformation matrix
Rotation matrix
Position vector
Scaling

62. Homogeneous Transformation
Special cases
1. Translation
2. Rotation

63. Example
Translation along Z-axis with h:
O, O’
O, O’ h

64. Example
Rotation about the X-axis by

65. Homogeneous Transformation
Composite Homogeneous Transformation Matrix
Rules:
Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre-multiplication
Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication

66. Example
Find the homogeneous transformation matrix (T) for the following operations:

67. Homogeneous Representation
A frame in space (Geometric Interpretation)
(z’)
(y’)
(X’)
Principal axis n w.r.t. the reference coordinate system

68. Homogeneous Transformation
Translation

69. Homogeneous Transformation
Composite Homogeneous Transformation Matrix ?
Transformation matrix for adjacent coordinate frames
Chain product of successive coordinate transformation matrices

70. Example
For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5
Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?

71. Orientation Representation
Rotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body.
Any easy way?
Euler Angles Representation

72. Orientation Representation
Euler Angles Representation ( , , )
Many different types
Description of Euler angle representations
Euler Angle I Euler Angle II Roll-Pitch-Yaw
Sequence about OZ axis about OZ axis about OX axis
of about OU axis about OV axis about OY axis
Rotations about OW axis about OW axis about OZ axis

73. Euler Angle I, Animated
w'= z
w'"=
w" f
v'"
v "  v' y
u'"  u'
=u" x

74. Orientation Representation
Euler Angle I

75. Euler Angle I
Resultant eulerian rotation matrix:

76. Euler Angle II, Animated
w'= z
w"'=
w" 
v"'  v'
=v"  y
u"'
u"
Note the opposite (clockwise) sense of the third rotation, f. u' x

77. Orientation Representation
Matrix with Euler Angle II
Quiz: How to get this matrix ?

78. Orientation Representation
Description of Roll Pitch Yaw Z Y X
Quiz: How to get rotation matrix ?

79. Inverse Transformation
the inverse of a transformation undoes the original transformation if
then

80. Thank you!
Homework 1 is posted on the web.
Next class: kinematics II