Post on 04-Apr-2018
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ROBOT KINEMATICS
Vclav Hlav
Czech Technical University in Prague
Faculty of Electrical Engineering, Department of Cybernetics
Center for Machine Perceptionhttp://cmp.felk.cvut.cz/hlavac, hlavac@fel.cvut.cz
Outline of the talk:1. Kinematics, what is?
2. Open, closed kinematic mechanisms.
3. Sequence of joint transformations (matrix multiplications).
4. Direct vs. inverse kinematic task.
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2/31Kinematics
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. Knowledge of both
robot (mechanism) spatial arrangement
and a means of reference to the environmentis needed in order to control and programme a robot.
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3/31Open chain manipulator kinematics
Mechanics of a manipulator can be represented
as a kinematic chain of rigid bodies (links) con-
nected by revolute or prismatic joints.
One end of the chain is constrained to a base,
while an end effector is mounted to the otherend of the chain.
The resulting motion is obtained by composition
of the elementary motions of each link with re-
spect to the previous one.
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4/31Closed 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.
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5/31Direct vs. inverse kinematics
In an open chain kinematic manipulator robotics, there are two kinematic tasks:
1. Direct (also forward) kinematicsGiven: joint relations (rotations, translations) for the robot arm.
Task: What is the orientation and position of the end effector?
2. Inverse kinematics
Given: the desired end effector position and orientation.Task: What are the joint rotations and orientations to achieve this?
In a more general case ofclose kinematic chain mechanisms, a more general statement is needed:
1. Direct kinematics
Given: the geometric structure of the manipulator and the values of a number of jointpositions equal to the number of degrees of freedom of the mechanism.
Task: Find a relative position and orientation of any two designed joints.
2. Inverse kinematics
Given: a relative position and orientation of any two designed joints.
Task: Find values of all joints position and orientations.
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6/31Kinematics 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.
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7/31Kinematics dynamics, control
Kinematics is only the first step towards robot control !
Cartesian Space Joint Space Actuator Space
zy
x
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8/31Coordinate frames
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9/31Manipulator description
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10/31Configuration parameters
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11/31Generalized coordinates
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12/31Generalized coordinates (2)
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13/31End-effector configuration parameters
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14/31Operational (joints) coordinates
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15/31Joint coordinates joint space
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16/31Manipulator redundancy
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17/31Two 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.
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18/31Homogeneous transformation
r1 r3
r4 r5 r6
r7 r8 r9
r2
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.
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19/31Two basic types of joints
Revolute Prismatic
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20/31Kinematic open chain
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21/31Direct vs. inverse kinematics, a reminder
In an open chain kinematic manipulator robotics, there are two kinematic tasks:
1. Direct (also forward) kinematics
Given: joint relations (rotations, translations) for the robot arm.
Task: What is the orientation and position of the end effector?
2. Inverse kinematics
Given: the desired end effector position and orientation.
Task: What are the joint rotations and orientations to achieve this?
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22/31Direct kinematics
One joint: xi = Axi1.
Chain of joints: xn1 = An1 An2 . . . A1 A0 x0.
Easy to compute (matrix multiplication).
Unique solution.
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23/31Inverse kinematics
For an open chain kinematic mechanism (a robot), the inverse kinematicproblem 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.
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24/31Inverse 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 systematic frame assignment.
(1)
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25/31Manipulator kinematic (1)
Cartesian Gantry
M i l ki i (2)
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26/31Manipulator kinematic (2)
Cylindrical Sphere
M i l ki i (3)
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27/31Manipulator kinematic (3)
SCARA Anthropomorphic
M th d l i th i ki ti t k
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28/31Methods solving the inverse kinematics task
1. Closed-form solutions. Relevant for industrial manipulators.
Algebraic methods.
Geometric methods.
2. Numerical methods.
Symbolic elimination methods: involve analytical manipulations toeliminate variables from a system of nonlinear equations to reduce it to
a smaller set of equations.
Continuation methods: involve tracking a solution path from a start
system with known solutions to a target system. Iterative methods: are in general based on Newton-Raphson metod
using 1st order approximation of the original algebraic equation. They
converge in a single solution (from several possible) based on the initial
guess.
Cl d ll l h i
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29/31Closed parallel chain
Hexamod
Real hexamod (1)
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30/31Real hexamod (1)
Real hexamod (2)
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31/31Real hexamod (2)
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