Chapter 2 Mathematical Analysis for Kinematics

Table of Contents Mathematical background Representation of coordinates Rotation matrix Multiple rotations Homogeneous Transformation Denavit-Hartenberg Coordinates Forward kinematics with HT

2.1 Mathematical Background To bring the end-effector to a desired posture, the end-effector coordinates need to be represented with respect to the reference frame. Cartesian coordinates, cylindrical coordinates, and spherical coordinates are generally used. In this chapter, only Cartesian coordinates are handled.

Representations and Inner Product Types Representations Examples Scalar Small Italic x, y Vector Small Gothic Matrix, Set Capital X, Y 대문자 [Table 2-1] Representations

Properties of Inner Product [Definition 1] Property of inner product: (2.1) Prove this property using x =[2 3 4]T and y = [3 1 2]T

2.2 Representation of Coordinates Position: With respect to the reference coordinates {A}, a given position a can be represented as Orientation: {B} w.r.t. {A} (2.2) (2.3)

2.3 Rotation Matrix [Fig. 2-1] Two coordinates for the same vector

Vector Representation The vector can be represented in {0} frame and {1} frame as follows: are unit vectors along axes, respectively. (2.4)

Transformation (2.5)

Transformation Matrix This can be represented in a matrix equation as where This 3 X 3 matrix transforms from coordinates to .. (2.6)

Rotation Matrix As the same way, (2.7) This can be represented in a matrix equation as where (2.8)

Orthogonal Matrix Matrix is the inverse of . Since , it is easy to show that This matrix is named as orthogonal since the inverse is the same as the transpose. Also the columns are unit and orthogonal to each other. Therefore, is called as orthonormal matrix . (2.9)

Properties of Rotation Matrix When the frames are determined by the right hand rule, Simply, rotation matrices have unit determinant and they are orthogonal. (2.10)

Basic Rotation Matrix [Fig. 2-2] rotation about X axis.

Basic Rotation Matrix From [Fig. 2-2], we know that Therefore, (2.11) (2.12)

Basic Rotation Matrix For y and z axes, the rotation matrices are determined as (2.13)

Composition of Rotation Matrices The same vector is represented w.r.t. the different frames as, (2.14) Be careful not to change the multiplication order.

Composition of Rotation Matrices Instead of , when we rotate about the fixed axis,  (2.15)

Composition of Rotation Matrices Rotate about y axis, and rotate about z axis, the rotation matrix R becomes (2.16)

Properties of Rotation Matrix (2.17)

Fixed Axis Rotation Rotate about the fixed axis, and rotate about fixed axis, the resulting rotation matrix is different.

Rotation about Fixed Axis P [Fig. 2-3] rotation about y and z axis.

Rotation around a Fixed Axis About axis, rotate,  about axis, rotate  In this case, is a fixed frame. The multiplication sequence will be reversed in this case. When , , and are the same vector , there is the relations:

Second Rotation The second rotation along z axis is done w.r.t. the fixed frame, not w.r.t. the current frame, . Therefore, . implies the rotation about axis.

Undo the Previous Rotation To solve this problem, can be matched to . By rotating about y axis through - , Now the rotation about z axis will be done through . The inverse operation will be followed by rotating about y axis. That is, (2.18)

Reverse Order Finally the rotation w.r.t. the fixed frame results the reverse order multiplication of the rotation matrices. (2.19)

Example 2.2 1) can be represented in {A} as using For fixed frame, {A} and rotated frame {B} which is rotated about Through , 1) can be represented in {A} as using (2.20)

Sol) As follows: (2.21)

Sol) 2) If , represent w.r.t {B}? sol) Since , (2.22)

2.4 Composition of Rotation Any frames can be represented w.r.t. a reference frame by consecutive rotations. The total rotation matrix is composed of basic rotation matrices. Note that AB ≠ BA.

Euler Angles Rotation matrix for a wrist manipulator which has Roll –Pitch – Roll type can be represented by Euler Angles. PUMA-560 is a typical example of wrist manipulator with Euler angles. The relation between two frames can be represented by three angles about three axes, respectively.

Euler Angles [Fig 2-5] Z-Y-Z Euler Angle

Euler Angles

Z-Y-Z Euler Angles (2.23)

Roll, Pitch, Yaw Angles Roll-Pitch-Yaw rotation: Rotation about x axis, about y axis, and about z axis, will complete and rotational transformation. [Fig. 2-6] Wrist of Cinccinati Milacron

Roll, Pitch, Yaw Angles Aviation system needs orientation w.r.t., the absolute coordinates. R-P-Y angles are suitable for the system. [Fig. 2-7] Roll-Pitch-Yaw Angles

Roll, Pitch, and Yaw

Roll, Pitch, and Yaw (2.24)

Example 2.5 [Fig. 2-8] Yaw-Pitch-Roll Rotation

Sol ) 1) 2) 3) sol) Yaw, Pitch, and Roll from the right. (2.25)

Rotation about an Arbitrary Axis [Fig 2-9] Rotation about arbitrary axis k.

Rotation about an Arbitrary Axis is a directional unit vector in frame. Let’s obtain the rotation matrix, , which is rotated through about k axis.

General Rotation (2.26)

General Rotation (2.27) (2.28)

2.5 Homogeneous Transformation Homogeneous matrix is defined as A combination of a position vector , and orientation matrix, as, (2.29)

Homogeneous Transformations [Fig. 2-10] Coordinates transformation

Homogeneous Transformations vector can be represented in frame {0}, as This can be represented as a homogeneous matrix as, (2.30) (2.31a)

Homogeneous Transformations Transforming in frame {2} into frame {1}, we have If we transform, in frame {2} into frame {0}, (2.32b)

Homogeneous Transformations (2.33) (2.34)

Homogeneous Transformations Since R is orthogonal, (2.35)

2.6 Denavit-Hartenberg Coordinates There are four parameters in D-H representation. These constraints need to be satisfied: (DH1): Axis x1 is orthogonal to axis z0. (DH2) : Axis x1 intersects axis z0 .

Denavit-Hartenberg Frame [Fig. 2-11] Denavit-Hartenberg frame

Denavit-Hartenberg Frame How to obtain ? Frame {i-1} match  to frame{i}. : Rotate about , to make parallel to : Move along to make is on the line of : Move along to make same as : Rotate about , to make same as

Denavit-Hartenberg Frame Homogeneous matrix, , is represented as (2.36)

Assignments of Frames It starts from 0,…,n satisfying DH1 & DH2 conditions [Fig. 2-12] Assignments of frame

Alpha Ⅱ Robotic Arm [Fig. 2-13] Alpha ⅡRobotic arm & Link coordinates

SCARA Robot (Adept One) [Fig. 2-14] SCARA Robot (Adept One) & Link coordinates

2.7 Forward Kinematics Forward kinemaitcs with D-H representation Step 1: Locations of joint axes are determined with suitable labels. Step 2: Assign the reference frame. The origin on the axis...

Repeat step 3: Assign oi at the intersection of zi and zi-1. Steps 3 – 5 are repeated. step 3: Assign oi at the intersection of zi and zi-1. If two are parallel, assign on zi. step 4: Assign xi at oi vertical to both of zi and zi-1. step 5: Assign yi following right screw rule. step 6: Assign frames of the end-effector.

Repeat rotation about translation along Step 7 : Draw the link parameter table rotation about translation along Step 8 : Obtain the homogeneous transformation matrix

<Example1> Planar Robot link 1 a1 2 a2 [Table 2-2] Link parameter of Planar Robot variable [Fig. 2-15] Frames of Planar Robot

Homogeneous Matrix. (2.37) (2.38)

Multiplication (2.39)

<Example 2> SCARA Robot link 1 a1 2 a2 180° 3 4 [Table 2-3] Link parameter of SCARA Robot [Fig. 2-16] Frames of SCARA Robot variable

Homogeneous Matrix (2.40) (2.41)

Homogeneous Matrix (2.42) (2.43)

Resulting Matrix (2.44)

<Example 3> 2 links Cartesian Manipulator 1 -90° 2 [Table 2-4] Link parameter of Cartesian manipulator variable [Fig. 2-17] Two link Cartesian manipulator.

Homogeneous Matrix (2.45) (2.46)

Multiplication (2.47)

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