50176176-6-Robot-Kinematics-and-Dynamics-1.pdf

8/16/2019 50176176-6-Robot-Kinematics-and-Dynamics-1.pdf

1/45

6 Basic Coordinates

Rotation and Translation in basic of Robot Coordinates, Homogeneous Transform

Euler Angles, Denavit-Hartenberg Notation

Affiliated Professor : Jaeyoung Lee

E-mail : [email protected]

. – -


2/45

한국 IT직업전문학교

To ics

Basic Coordinates

-

Forward and Inverse Kinematics of Robot Manipulators

Mani ulator Jacobian

Review of Lagrangian Dynamics

Lagrangian Dynamics of Robot Manipulators

Newton-Euler Equations of Robot Manipuators Forward and Inverse Dynamics

Robot Trajectory Planning

Robot Control

로봇 학 론 2


3/45


Basic Coordinates

Operators for Translation and Rotation

- -

로봇 학 론 3


4/45


Basic Coordinate

Coordinate Frames

- -

로봇 학 론 4


5/45


Basic Coordinate

Different View Based Upon Different Coordinates

- -

로봇 학 론 5


6/45


Basic Coordinate

Rotation and Translation Combined

-

- A single rotation and a single translation after another pair oftranslation and rotation:

- Scheme shown above to represent transform operations is not agood way of simplifying notation and computations.

- homogeneous transform

로봇 학 론 6


7/45


8/45


Basic Coordinate

Rotation Matrix R

- Notin that x A A and z A re resent the com onents of alon the x and z _ _ _axes of coordinate A.

- Therefore,

로봇 학 론 8


9/45


10/45


Basic Coordinate

Homogeneous Transform

- In homo eneous transforms translation and orientations have identical characteristics.

- Position vector [x y z]T is represented by an augmented vector [x y z 1]T .

-

로봇 학 론 10


11/45


Basic Coordinate

- Rotation is represented by a matrix:

로봇 학 론 11


12/45


Basic Coordinate

- If translation and rotation are combined

로봇 학 론 12


13/45


Basic Coordinate

- If translation and rotation are combined

Therefore,

- Inverse of homogeneous transform

로봇 학 론 13


14/45


Basic Coordinate

More On Rotation : Euler Angles

- Man different sets of Euler an les exist

- Z-Y-Z, Z-Y-X, and etc,

- Example : Z-Y-X : Rotate around z-axis, and then around y-axis, and finally- .

- Rotational Transformation (Post-multiplication)

로봇 학 론 14


15/45


16/45


Basic Coordinate

More On Rotation : Euler Angles

- X-Y-Z Fixed An les Pitch-Yaw-Roll

- Note

- -

로봇 학 론 16


17/45


Basic Coordinate

Equivalent Angle-Axis

- Re resented b a unit vector that indicates the axis of rotation and an an le to rotate.

- If the axis of rotation is k = [K_x K_y K_z]^T,

로봇 학 론 17


18/45


Basic Coordinate

Denavit-Hartenberg Notation

- ConventionsJoint i moves link i

Frame i is fixed at (attached to) link i and located at the distal joint, which meansthat frame i is located at joint i+1.

로봇 학 론 18

Z-axes are aligned with joint axes.


19/45


Basic Coordinate

4 Link Parameters

- _ends of Link n.

- Link Twist(alpha_n): "twist" angle of two z-axes at both ends of Link n- _ .

- Link Offset (d_n): distance from the origins of Frame n-1 to Frame n inthe direction of Z_n-1-axis.

- Joint Ang e(T etat_n): ang e rom x_n-1-axis to x_n -axis in t e irection

of z_n-1-axis.

로봇 학 론 19


20/45


Basic Coordinate

Miscellaneous Conventions

- - – - , _ , , _1Ⅹz_n.

- Positive direction of z-axis can be either.

로봇 학 론 20


21/45


Basic Coordinate

Miscellaneous Conventions

- .

- Z-axis of frame 0 is aligned with z-axis of joint 1.

- For a prismatic joint, z-axis is aligned with the sliding direction of theo nt. In t s case, n o set s t e o nt var a e.

- Homogeneous transform :

- Homogeneous transform can be obtained directly by observing thecoor inates in most cases.

로봇 학 론 21


22/45


Basic Coordinate

Example :

로봇 학 론 22


23/45


Basic Coordinate

Example :

Note that T^0_1 can be directly determined

로봇 학 론 23


24/45


Kinematics & Inverse Kinematics

Forward Kinematics

- For a given set of joint displacements, the end-effector position andorientation can be calculated. (Forward Kinematics)

Inverse Kinematics

- For a given set of end-effector position

and orientation, joint displacements arecomputed.

-

however, we may not able to solve for

solutions analytically.

로봇 학 론 24


25/45



Kinematics & Inverse Kinematics for a 2-D Revolute-Jointed Manipulator

Forward Kinematics

로봇 학 론 25


26/45




orwar nemat cs

로봇 학 론 26


27/45




n - ector os t on End-Effector Orientation

로봇 학 론 27


28/45




- For a given tip position [x y]^T, the joint displacement [theta_1 theta_2]^Tis to be determined.

Geometric Method(Using cosine rule)

로봇 학 론 28


29/45




로봇 학 론 29


30/45




로봇 학 론 30

IT


31/45




Sign: "+", if -p < theta_2 < 0, i.e., at "elbow up“ configuration "-", otherwise.

로봇 학 론 31

IT


32/45




- -right-hand-side: a matrix whose elements are functions of joint angles

로봇 학 론 32

IT


33/45




로봇 학 론 33

IT


34/45




Let

Then

로봇 학 론 34


35/45

IT


36/45




where

Therefore,

로봇 학 론 36



37/45


Kinematic Decoupling

kinematics is simplified.

Three axes z4 , z5 , and z6.

The position of the wrist center, w, is determinedby q1, q2 , and q3 only.

로봇 학 론 37



38/45


Kinematic Decoupling

-position can be computed:

w ere

-be computed.

- The orientation matrix R^0_3 can be computed after q1, q2 , and q3 aree erm ne .

- The rest of joint variables, q 4, q5 , and q6 , can be computed from R_6^3,

which in turn can be com uted b

로봇 학 론 38



39/45


SCARA Robot

로봇 학 론 39



40/45


SCARA Robot

- Inverse Kinematics of SCARA Robot

로봇 학 론 40



41/45


SCARA Robot

-

로봇 학 론 41



42/45


SCARA Robot

- upon a horizontal plane, which forms a 2-link manipulator,

- Again, similarly to a 2-link manipulator,

- and

로봇 학 론 42



43/45


SCARA Robot

- Solvabilit : A mani ulator is defined to be solvable if all the sets of oint variables for a given position and orientation can be determined by an algorithm. Manymanipulators have multiple solutions for a single configuration. For example,PUMA 560 has 8 solutions.

- or space : e vo ume o space t at t e en -e ector o a man pu ator can reac .Within the workspace, inverse kinematics solutions exist.

- Closed Form Solution : Inverse kinematic solutions can be grouped into closed form.

algebraic expressions; whereas numerical solutions are iterative. Numericalsolutions are computationally more expensive and slower.

- Pieper's work : A general 6 d.o.f. manipulator does not have a closed form solution.If the three consecutive axies intersect at a point, a closed form solution exists. Formost commercial manipulators, the last consecutive axes intersect at a point.

- Notes: Even if inverse kinematic solutions are found, they may not be physically.

로봇 학 론 43



44/45

HW 6

139p, 연습문제 1

로봇 학 론 44



45/45

REFERENCE

기초로봇 공학

-

Introduction to Robotics, b J, J. Crai- Ch 2. Spatial descriptions and transformations

- Ch 3. Manipulator kinematics

- Ch 4. Inverse manipulator kinematics

로봇 학 론 45

Date post:	06-Jul-2018
Category:	Documents
Upload:	cosorandrei-alexandru
View:	214 times
Download:	0 times

50176176-6-Robot-Kinematics-and-Dynamics-1.pdf

Documents