1. ROTATIONS & TRANSLATIONS
In robotics, the kinematic descriptions of ma­nipulators and their assigned tasks are utilized to set up the fundamental equations for dynamics and control.
 Because the links and arms in a robotic system are modeled as rigid bodies, the properties of rigid body displacement takes a central place in robotics.
 We define a rigid body as a collection of particles such that the distance between any two of them remains fixed, regardless of any motion of the body or forces exerted on the body.
 Thus, if p and q are any two points on a rigid body then, as the body moves, or as time passes, p and q must satisfy
||p(t) – q(t)|| = ||p(0) – q(0)|| = constant, for all time t
(where ||.|| means “norm” or euclidean distance between two points)
That is, a rigid motion of an object is a continuous movement of all the particles of the object such that the distance between any two particles remains fixed at all times.
 In general, a rigid displacement may consist of both rotation and translation.
Vector and matrix algebra are utilized to develop a systematic and generalized approach to describe and represent the location of the arms of a robot with respect to a global fixed reference frame G.

2. KINEMATICS (FORWARD KINEMATICS)
Since the arms of a robot may rotate or translate with respect to each other, body- attached coordinate frames A, B, C, … or B1, B2, B3, … will be established along with the joint axis for each link to find their relative configurations, and within the reference frame G.
 The position of one link B relative to another link A is defined kinematically by a coordinate transformation Axb (ATb or ARb) between reference frames attached to the link.
 The direct kinematics problem is reduced to finding a transformation matrix Gxb (GTb) that relates the body attached local coordinate frame B to the global reference coordinate frame G.
 A 3 x 3 rotation matrix is uti­lized to describe the rotational operations of the local frame with respect to the global frame.
 The homogeneous coordinates are then used to represent position vectors and directional vectors in a three-dimensional space.

3. Elementary Rotations in 2D …
x1 = r cos a
y1 = r sin a
r is magnitude
Rotation of coordinates
Rotation of vector
x2 = r cos (a - q) = x1 cos q + y1 sin q
y2 = r sin (a - q) = - x1 sin q + y1 cos q
x2 = r cos (a + q) = x1 cos q - y1 sin q
y2 = r sin (a + q) = x1 sin q + y1 cos q
Using matrix notation:
Using matrix notation: x2 y2 x1 y1
cos q
-sin q
sin q = x2 y2 x1 y1
cos q
sin q
- sin q =
Notice that the determinant of the “rotation” matrices is 1, they are called unimodular matrices.

4. Example of Elementary Rotations in 2D
Rotation of coordinates
Rotation of vector
What are the new coordinates of point P(3,7)
if the coordinate axes are rotated 23o CCW?
What are the new coordinates of point P(3,7)
if it is rotated 23o CCW while the frame is fixed?
Firstly we find the rotation angle in radians:
23o = (23/180) =
sin( ) =
cos( ) =
Therefore:
Firstly we find the rotation angle in radians:
23o = (23/180) =
sin( ) =
cos( ) =
Therefore: 3 7 x2 y2 x2 y2 3 7 = = x2 y2 x2 y2 = =
Therefore: P( , )
Therefore: P( , )
As a reference note that P in polar coordinates is P(3,7) = P( , o) and 23o o = o

5. Rotation a used in the textbook …
Equations as used in the textbook
x1 = r cos a
y1 = r sin a
r is magnitude t t x2 y2
cos q
- sin q
sin q x1 y1 = t t x1 y1
cos q
- sin q
sin q x2 y2 =
cos q
sin q
- sin q x2 y2
Rotation of coordinates = x1 y1
x2 = r cos (a - q) = x1 cos q + y1 sin q
y2 = r sin (a - q) = - x1 sin q + y1 cos q x2 y2 = x1 y1
Xxx
(Textbook’s notation!)
Using matrix notation:
x2 = r cos (a - q) = x1 cos q + y1 sin q
y2 = r sin (a - q) = - x1 sin q + y1 cos q x2 y2 x1 y1
cos q
-sin q
sin q =
We get the same equations for x2 and y2!

6. Rotation in the opposite direction in 2D …X Y x y a q P y2 y1 x1 x2 r
x1 = r cos a
y1 = r sin a
r is magnitude
Rotation of coordinates
Rotation of coordinates in the opposite direction
x2 = r cos (a - q) = x1 cos q + y1 sin q
y2 = r sin (a - q) = - x1 sin q + y1 cos q
x2 = r cos (a + q) = x1 cos q - y1 sin q
y2 = r sin (a + q) = x1 sin q + y1 cos q
Using matrix notation:
Using matrix notation: x2 y2 x1 y1
cos q
-sin q
sin q = x2 y2 x1 y1
cos q
sin q
- sin q =
Rotating the coordinates in the opposite direction only causes to change q to -q in the matrix.

7. The Rotations vectors are SO(2)  R2x2…
Let’s study some of the special properties of the rotation matrices given below:
cos q
-sin q
sin q
cos q
sin q
- sin q = A
and = B
First, let’s notice that det(A) = det(B) = (cos q)2 + (sin q)2 = 1, that is, the length of a vector is unchanged after applying these matrices to it as a rotation matrix.
Next, for the 1st matrix call its 1st column vector v1 = (cos q)i – (sin q)j and its 2nd column vector v2 = (sin q)i + (cos q)j. We can verify that |v1| = 1 and |v2| =1, that is, they are unit vectors, both are equal to (cos q)2 + (sin q)2 = 1.
Can verify the same for the 2nd matrix calling v3 = (cos q)i + (sin q)j and v2 = - (sin q)i + (cos q)j, both column vector are unit vectors since |v3| =1 and |v4| =1, same reason, both are equal to (cos q)2 + (sin q)2 = 1.
Now, let’s calculate the dot product v1 . v2 = [(cos q)i – (sin q)j] . [(sin q)i + (cos q)j] = (cos q)(sin q) - (sin q)(cos q) = 0. That is, vectors v1 and v2 are orthogonal to each other. Similarly, v3 . v4 = [(cos q)i + (sin q)j] . [- (sin q)i + (cos q)j] = - (cos q)(sin q) + (sin q)(cos q) = 0. That is, vectors v3 and v4 are also orthogonal to each other.
Moreover, notice that B = (A-transpose) = At and vice-versa A = Bt.
Lastly, notice that A*B = I, the unit matrix! that is: A = B-1 and B = A-1, they are inverse of each other!

8. Elementary Sequence of Rotations in 2D …x1 y1 x y
cos a
-sin a
sin a =
= Ra X
X1 = x2 y2 x1 y1
cos b
-sin b
sin b =
= Rb X1
X2 = x2 y2
cos b
-sin b
sin b =
= Rb Ra X
X2 = x y
cos a
-sin a
sin a
cos(a + b)
-sin(a + b)
sin(a + b)
cos(a +b)

9. Exercise on Sequence of Rotations in 2D …
Evaluate the rotation matrix to perform the following set of
consecutive rotations: a 22 degrees in CCW followed by a 49
degrees also in CCW and lastly an 8 degrees rotation in CW
direction. Do it using a single rotation formula and compare that
Result with the three consecutive rotations in sequence.
cos(a + b + g)
-sin(a + b + g)
sin(a + b + g)
R = Rg Rb Ra =
Total rotation: 22+49–8 = 63 deg = 63*pi/180 = Rad
R = [ ; ]
cos(22)
-sin(22)
sin(22)
cos(49)
-sin(49)
sin(49)
cos(8)
-sin(8)
sin(8)
Ra=
Rb =
Rg =
[ ]
[ ]
R = =

10. Elementary Translations in 2D & 3D …x z y X Y y1 Z Y z1 y z2 P P y1 y2 l k x x2 y2 k h x1 X x2
x2 = x1 – h
y2 = y1 – k h x1 x1 y1 1 h k = x2 y2 h k l = x2 y2 z2 1 x1 y1 z1
x2 = x1 – h
y2 = y1 – k
z2 = z1 – l

11. Rotations in 3D about the z-axis … an example …
In 3D rotations about the axes are well defined.
The 3x3 rotation matrix Ra through an angle a about
the z-axis is: Y Z x
P(x, y, z) X y z v v1 =
= Ra X
X1 = x1 y1 z1
cos a
sin a
- sin a 1 x y z
Example: Assume the frame performs a positive, that is, a 450)
CCW (counter-clockwise) rotation about the z-axis given that
point P has coordinates P(1,1,1). The new P would be
P(1.4142, 0, 1).,What about if P(3.5, 4, 1.5)? Then the new P
would be P( , , ). Use Octave/Matlab.
(In this case: sin(450) = cos(450) = 1/  2). x1 y1 z1
1/ 2
1/2) 1 1
X1 =
 2 =
- 1/ 2
1/2 = 1

12. Rotations in 3D about the coordinate axes & the location of the “– sine” term
The 3x3 positive rotation matrix Rz,a through an angle a about the z-axis takes the Y axis into negative X:
Rz,a =
cos a
sin a
- sin a 1
The 3x3 positive rotation matrix Rx,a through an angle a about the x-axis takes the Z axis into negative Y:
Rx,a =
cos a
sin a
- sin a 1
The 3x3 positive rotation matrix Ry,a through an angle a about the y-axis takes X axis into negative Z:
Ry,a =
cos a
sin a
- sin a 1

13. Several Rotations plus (see below) Sequence of Rotations are not commutative …
a. original
b. /2 about x-axis
c.  about x-axis
d. −/2 about x-axis
e. /2 about y-axis
f. /2 about z-axis Z X Y
Series of rotations about different axes
1. original
2. /2 about x-axis
2’. /𝟐 about y-axis
3. /2 about y-axis
3’. /2 about x-axis Z X Y
1’. original
The order in which the rotations are executed makes a difference: “rotations are not commutative”