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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.

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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.

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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.

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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.

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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.

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6/21COORDINATE FRAMES

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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.

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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.

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9/21TWO BASIC JOINTS

Revolute Prismatic

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10/21OPEN KINEMATIC CHAIN

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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?

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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.

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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.

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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.

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15/21

MANIPULATOR KINEMATIC (1)

Cartesian Gantry

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16/21

MANIPULATOR KINEMATIC (2)

Cylindrical Sphere

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17/21

MANIPULATOR KINEMATIC (3)

SCARA Anthropomorphic

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18/21

KINEMATICS → DYNAMICS, CONTROL

Kinematics is only the first step towards robot control !

Cartesian Space Joint Space Actuator Space

zy

x

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19/21CLOSED PARALLEL CHAIN

Hexamod

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20/21

REAL HEXAMOD (1)

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21/21

REAL HEXAMOD (2)