020 Kinema Tics

7/30/2019 020 Kinema Tics

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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, [email protected]

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.


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


20/31

20/31Kinematic open chain


21/31

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?


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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Cylindrical Sphere

M i l ki i (3)


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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)


30/31

30/31Real hexamod (1)

Real hexamod (2)


31/31

31/31Real hexamod (2)
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020 Kinema Tics

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