1. Matrix Methods in Kinematics
Rigid Body Rotation Matrices
Rigid Body – points have same relative position
Displacement = Rotation and Translation
Rotation about
1. Right hand Cartesian axes (x,y,z)
2. Arbitrary Axis
3. Euler Angles

2. Matrix Methods in Kinematics
Rotate about Z
Components in the fixed system
x-y
Rotation matrix

3. Matrix Methods in Kinematics
Rotation about three Cartesian axes y z v2 v2 v1 v2 y x z
X,Y,Z axes fixed in space x

4. Matrix Methods in Kinematics
Plane rotation (2D)
Spatial Rotation

5. Matrix Methods in Kinematics

6. Matrix Methods in Kinematicsy u
Rotation about axis u
Rotate u to z and then back again v1 x z

7. Matrix Methods in Kinematics
ux,uy,uz are dir cos of unit vector along u
pretty useful
u is fixed in space

8. Matrix Methods in Kinematics
Euler Angles
Each rotation about axis
That depends preceding rotation
Moving reference frame 1 2 3

9. Matrix Methods in Kinematics

10. Matrix Methods in Kinematics
Rigid Body Displacement Matrix 2d Cartesian Rotation p1 q1
Fixed x-y
May know p1,p
Find position q

11. Matrix Methods in Kinematics
Now a displacement matrix
3x3 planar (2D)

12. Matrix Methods in Kinematics
Spatial (3D) Rigid Body Displacement
Replace
Cartesian
Axis
Euler
Using Cartesian
new
original
4x4 matrix (3D)

13. Matrix Methods in Kinematics
Example Displacement of a point
Moving with a rigid body

14. Matrix Methods in Kinematics
3x3 Displacement matrix

15. Matrix Methods in Kinematics
Finite Rotation Pole – plane rotation about p0
With new position vectors
p1=p2=p0

16. Original displacement matrix
Matrix Methods in Kinematics
Previously
p1 and p2 in
D13 and D23
Displacement matrix D now written as:
q2 is the same point
Original displacement matrix

17. Matrix Methods in Kinematics

18. Matrix Methods in Kinematics
Screw displacement matrix y q u q1
p=p1+su s p1 z x
Screw displacement matrix

19. Matrix Methods in Kinematics
HW #3 Salute z q
shoulder
30◦ p y x
p1=elbow
q1=tip of finger
Use Euler angles, [D]

20. Matrix Methods in Kinematics
HW #3 Salute
Y,x’ x” Y
Changed axes
notation 2
X,y’,y” X q
shoulder 1
30◦
Z,z’,z” p Z
p1=elbow
Y,x’ x”
q1=tip of finger
X,y’ y”
z-x-z rotation 3
Z,z’ z”

21. Matrix Methods in Kinematics
Finding the Displacement Matrix by Inversion y B1
Known points C1 C2 A1 A2 B2 x

22. Matrix Methods in Kinematics
Displacement Matrix by Inversion

23. Matrix Methods in Kinematicsq q1 d a x y
2D Planar motion z
D=d*a-1

24. Matrix Methods in Kinematics
Finding the inverse of A (by hand – the long way)
adj - adjoint
ith row=ith column
α=co factor
Mij= minor of A

25. Matrix Methods in Kinematics
Using MATLAB
inv(a)
ans =
d =
a =
>> e=d*inv(a)
e =
Displacement matrix

26. Matrix Methods in Kinematics
Coordinate transformations – Vector
Rigid body motion in terms of axes fixed in the body
x’,y’ moves with the body p
2 co-ord systems
moving
1 fixed co-ord system
fixed

27. Matrix Methods in Kinematics
Transformation [T] (between 2 co-ord sys) = inverse rotation[R] (one fixed system)

28. Matrix Methods in Kinematics
Coordinate transformations – Point p
x’,y’ initially coincident with x,y
[T]=[R]-1
Transformation Matrix is inverse of Displacement Matrix

29. Matrix Methods in Kinematics
Increase rotation by p

30. Matrix Methods in Kinematics
Hartenberg-Denavit Notation (J.ASME 1955)
Co-ordinate trans for axes fixed in a rigid body (x1,y1,z1) and second set
(x2,y2,z2) fixed in a second adjacent body (kinematic chain)

31. Matrix Methods in Kinematics

32. Matrix Methods in Kinematics
a1=Perpendicular distance between
z1 and z2
(may not be physical link length)
α1= twist angle z1 into z2 (along a1)
θ1= screws x1 into x2 (along S1)
S1 (d1)= distance from axes x1 to x2

33. Matrix Methods in Kinematics
4 motions – 2 rotations, 2 translations
Hartenberg-Denavit matrix

34. Matrix Methods in Kinematics
Forward Kinematics – determine the position of the end effector
given the joint variables (angles/extensions)
Inverse Kinematics – what are joint variables for a desired end effector position

35. Matrix Methods in Kinematics
2D Planar Elbow
z (rotation) axes are all parallel
ai= _ dist between z (z0,z1,z2)
ai = link length
αi=angle between Z, αi =0
Si=di=dist between origins along z,
di =0
θi=rotation angles
Link ai αi di θi 1 a1 θ1 2 a2 θ2
Base frame
x0 direction is arbitrary

36. Matrix Methods in Kinematics
H-D Matrix

37. Matrix Methods in Kinematics

38. Matrix Methods in Kinematics
We end up with

39. Matrix Methods in Kinematics
Forward Kinematics
Rotation matrix of 2 to 0
Translation of 2 to 0

40. Matrix Methods in Kinematics
a1=a2=10 in.
θ1=θ2

41. Matrix Methods in Kinematics
Example: 3 Link cylindrical manipulator d3 L3 J2 d2 L2 L1
O1 at J1 J1

42. Matrix Methods in Kinematics
Example: 3 Link cylindrical manipulator
Prismatic
joint
(Video) d3 L3 O2 O3 J2
Prismatic
joint d2 L2 L1 J1 O0 O1
Link ai αi
di (Si) θi 1 d1
θ1 (var) 2
-90
d2 (var) 3
d3 (var)

43. Matrix Methods in Kinematics
H-D Matrix
Link ai αi
di (Si) θi 1 d1
θ1 (var) 2
-90
d2 (var) 3
d3 (var)

44. Matrix Methods in Kinematics

45. Matrix Methods in KinematicsJ1 J2 O0 L1 L2 L3 d3 d2 O3 O2 O1

46. Matrix Methods in Kinematics
Spherical Wrist
z3-z4
plane d6
Link ai αi di θi 4
-90 θ4 5
+90 θ5 6 d6 θ6
H-D Matrix

47. Matrix Methods in Kinematics

48. Matrix Methods in Kinematics

49. Matrix Methods in Kinematics
Cylindrical Manipulator with spherical wrist
Slide 46
Seems like it
should be this direction
a=approach
s=sliding
n=normal

50. Matrix Methods in Kinematics

51. Matrix Methods in Kinematics

52. Matrix Methods in Kinematics
Puma 560 Manipulator

53. Matrix Methods in Kinematics
Puma 560 Manipulator

54. Matrix Methods in Kinematics
Puma Manipulator
HW #4
Determine T Matrix
10 in.

55. Matrix Methods in Kinematics

56. Matrix Methods in Kinematics
Co-Ordinate System for offset slider using H-D notation S&R fig 3.11
a1=Perpendicular distance between
z axes
(may not be physical link length)
α1= twist angle zn into zn+1 (along a1)
θ1= screws xn into xn+1 (along S1)
S1 = distance from axes xn to xn+1

57. Matrix Methods in Kinematics

58. Matrix Methods in Kinematics
Assume θ4 defined
Solve for other parameters, equate elements of [A] and [I]

59. Matrix Methods in Kinematics

60. Matrix Methods in Kinematics
Inverse Kinematics- finding required angles for given position
Cincinnati Milacron T3

61. Matrix Methods in Kinematics
Kinematic Model i αi ai si θi 1
+90 θ1
0+/- 120 2 a2 θ2
45+/-45 3 a3 θ3
-60+/-60 4
-90 a4 θ4
0+/120 5 θ5
90+/-120 6 l θ6
0+/-180 HD
notation

62. Matrix Methods in Kinematics
Forward Kinematics, find AH for given angles θ
Inverse Kinematics, For given AH, find explicit solutions for θs
No general approach or closed form solutions available
However, analytic solutions developed for manipulators with
6 DOF (7-1)-2(6)
3 consecutive intersecting axes or
3 consecutive parallel axes
Called simple manipulators

63. Matrix Methods in Kinematics

64. Matrix Methods in Kinematics

65. Matrix Methods in Kinematics
A2,A3,A4 are parallel, so multiply
Combining using cosθ23=cos(θ2+θ3), sinθ23=sin(θ2+θ3)

66. Matrix Methods in Kinematics
A5 and A6 are intersecting (Zs) so multiply them….
then

67. Matrix Methods in Kinematics
A5 and A6 are intersecting (Zs) so multiply them….
Call this R.H.S.

68. Matrix Methods in Kinematics
Bring A1 to other side..call this L.H.S
A1 is othogonal matrix A-1=AT

69. Matrix Methods in Kinematics
Look at RHS and LHS for elements with one variable
e.g. L33 =R33, L34=R34
0+/-120
2 values – branch cases

70. Matrix Methods in Kinematics
Since θ1 known, can find θ5 (previous equations) 2 solutions
Also all elements of LHS are now known ( AT1AH)
Now pick 2 other elements with single unknown (θ6)
R31 and R32
Are known

71. Matrix Methods in Kinematics
Use ATan2 for correct quadrant
May have a singular point, sθ5=0 y
(1,1) x
(-1,-1)

72. Matrix Methods in Kinematics
Tracking which element pairs selected 1 2 3

73. Matrix Methods in Kinematics
Use L13=R13, L23=R23
Sθ5=0 singularity

74. Matrix Methods in Kinematics
known
Eliminates θ23

75. Matrix Methods in Kinematics

76. Matrix Methods in Kinematics
So far
Θ1= 2 values
Θ5= 2 values
Θ6= 1 value (ATan2 recognizes correct quadrant)
Θ234 = 1 value (same with ATan2)
Θ2= 2tan-1(t) 2 values of t (roots of second order eqn)
ex. Tanα = 1 α = 45 or 225
tan(α/2) = 1 α=90
tan(α/2) = 0.7
α = 2tan-1(0.7) 2 x 35 = 70
2 X 215 = =70
1 solution

77. Matrix Methods in Kinematics
From slide 74
Complete solution
Θ1= 2 solutions
Θ5= 2 values
Θ6= 1 value
Θ2= 2tan-1(t) 2 values for t
Θ3=θ23-θ2
Θ4=θ234-θ23

78. Matrix Methods in Kinematics
Another example
Adept One Robot

79. Matrix Methods in Kinematics
Kinematic Model

80. Matrix Methods in Kinematics
In Base frame

81. Matrix Methods in Kinematics
In Base frame
Base (0,0,0) p3

82. Matrix Methods in Kinematics
For x and y
Typically have these transformation equations

83. Matrix Methods in Kinematics
From the position equations 1)
Squaring and adding, eliminates θ12
get
Rotation about z by θ1 2)

84. Matrix Methods in Kinematics2)
Solution of 2) in the form

85. Matrix Methods in Kinematics
Alternate form:
Eq 1) multiply each by p1 and p2, add and subtract
Reverse arguments and other stuff
Substituting for θ1

86. Matrix Methods in Kinematics
Equal length links

87. Matrix Methods in Kinematics
For angular position of the effector, from picture

88. Matrix Methods in Kinematics
Summary a1=a2 case

89. Matrix Methods in Kinematics
Because σ1 is +/-

90. Matrix Methods in Kinematics
All possible solutions
For real solutions
Limitations on angles
Vector length squared

91. Matrix Methods in Kinematics
Solution approach for slide 83

92. Matrix Methods in Kinematics

93. Matrix Methods in Kinematics

94. Matrix Methods in Kinematics

95. Matrix Methods in Kinematics