View
238
Download
1
Category
Preview:
Citation preview
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 1/21
1/21ROBOT KINEMATICS
Vaclav Hlavac
Czech Technical University, Faculty of Electrical EngineeringDepartment of Cybernetics, Center for Machine Perception
121 35 Praha 2, Karlovo nam. 13, Czech Republic
hlavac@fel.cvut.cz, http://cmp.felk.cvut.cz
LECTURE PLAN
1. Kinematics, what is?
2. Open, closed kinematic mechanisms.3. Sequence of joint transformations (matrix multiplications)
4. Direct vs. inverse kinematic task.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 2/21
2/21KINEMATICS
KINEMATICS – the analytical study of the geometry of
motion of a mechanism:
• with respect to a fixed reference co-ordinate system,
• without regard to the forces or moments that cause
the motion.
In order to control and programme a robot we must have
knowledge of both its spatial arrangement and a means of reference to the environment.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 3/21
3/21OPEN CHAIN MANIPULATOR KINEMATICS
Mechanics of a manipulator can be
represented as a kinematic chainof rigid bodies (links) connected by
revolute or prismatic joints.
One end of the chain is constrainedto a base, while an end effector is
mounted to the other end of the
chain.
The resulting motion is obtained
by composition of the elementary
motions of each link with respect to
the previous one.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 4/21
4/21CLOSED KINEMATIC CHAIN
Much more difficult.
Even analysis has to take into account statics, constraints
from other links, etc.
Synthesis of closed kinematic mechanisms is very difficult.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 5/21
5/21KINEMATICS vs. DIFFERENTIAL KINEMATICS
Kinematics describes the analytical relationship between
the joint positions and the end-effector position and
orientation.
Differential kinematics describes the analytical relationship
between the joint motion and the end-effector motion in
terms of velocities.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 6/21
6/21COORDINATE FRAMES
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 7/21
7/21TWO FRAMES KINEMATIC RELATIONSHIP
There is a kinematic relationship between two frames,
basically a translation and a rotation.
This relationship is represented by a 4 × 4 homogeneous
transformation matrix.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 8/21
8/21HOMOGENEOUS TRANSFORMATION
r 1 r 3
r 4 r 5 r 6
r 7 r 8 r 9
r 2
000 1
Dx
Dy
Dz
3x3 rotation matrix 3x1 translation
global scale1x3 perspective
Rotation matrix R is orthogonal ⇔ RTR = I ⇒ 3 independent
entries, e.g., Euler angles.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 9/21
9/21TWO BASIC JOINTS
Revolute Prismatic
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 10/21
10/21OPEN KINEMATIC CHAIN
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 11/21
11/21DIRECT vs. INVERSE KINEMATICS
In manipulator robotics, there are two kinematic tasks:
Direct (also forward) kinematics – Given are joint relations
(rotations, translations) for the robot arm. Task: What is
the orientation and position of the end effector?
Inverse kinematics – Given is desired end effector position and
orientation. Task: What are the joint rotations and
orientations to achieve this?
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 12/21
12/21DIRECT KINEMATICS
One joint: xi = Axi−1.
Chain of joints: xn−1 = An−1 An−2 . . . A1 A0 x0.
Easy to compute (matrix multiplication).
Unique solution.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 13/21
13/21INVERSE KINEMATICS
For a kinematic mechanism, the inverse kinematic problem
is difficult to solve.
The robot controller must solve a set of non-linear
simultaneous algebraic equations.
Source of problems:
• Non-linear equations (sin, cos in rotation matrices).
•
The existence of multiple solutions.• The possible non-existence of a solution.
• Singularities.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 14/21
14/21
INVERSE KINEMATICS, SIMPLIFICATIONS
Divide and conquer strategy. Decouple the problem into
independent subproblems.
The spherical wrist. Positioning of the wrist + positioning
within the wrist.
Design conventions, e.g. Denavit-Hartenberg systematicframe assignment.
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 15/21
15/21
MANIPULATOR KINEMATIC (1)
Cartesian Gantry
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 16/21
16/21
MANIPULATOR KINEMATIC (2)
Cylindrical Sphere
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 17/21
17/21
MANIPULATOR KINEMATIC (3)
SCARA Anthropomorphic
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 18/21
18/21
KINEMATICS → DYNAMICS, CONTROL
Kinematics is only the first step towards robot control !
Cartesian Space Joint Space Actuator Space
zy
x
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 19/21
19/21CLOSED PARALLEL CHAIN
Hexamod
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 20/21
20/21
REAL HEXAMOD (1)
7/28/2019 11 Kinematics Robot
http://slidepdf.com/reader/full/11-kinematics-robot 21/21
21/21
REAL HEXAMOD (2)
Recommended