C02,C03 – 2009.01.22,27,29 Advanced Robotics for Autonomous Manipulation

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Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering. C02,C03 – 2009.01.22,27,29 Advanced Robotics for Autonomous Manipulation. Giacomo Marani Autonomous Systems Laboratory, University of Hawaii. http://www2.hawaii.edu/~marani. 1.

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Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering

C02,C03 – 2009.01.22,27,29

Giacomo Marani

Autonomous Systems Laboratory, University of Hawaii

http://www2.hawaii.edu/~marani

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ME696 - Advanced Robotics – C02

Contents

1. Vectors deriv.

2. Angular velocity

3. Derivative for P.

4. Generalized Vel.

5. Derivative for R

6. Joint Kinematics

7. Simple kin. Joint

• Summary
• Vectors derivatives
• Angular velocity
• Derivative for points
• Generalized velocity
• Derivative of orientation matrix
• Joint kinematics
• Simple kinematic joint

Kinematics – Part A

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ME696 - Advanced Robotics – C02

• Vector Derivatives
• Time derivative of geometrical vector , computed w.r.t. frame <a>:
• (2.1)
• Same time derivative but in the different reference frame <b>:
• In general:

k

j

Oa

k

< a >

i

< b >

Ob

i

j

Vector derivative

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ME696 - Advanced Robotics – C02

• Vector Derivatives
• Proof:
• (2.2)
• Hence the result (very important):

k

j

Oa

k

< a >

i

< b >

Ob

i

j

Vector derivative

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ME696 - Advanced Robotics – C02

• Vector Derivatives
• If we project the (2.1) over the frame <b> we have:
• FIRST derive THEN project (not allowed the reverse)
• Meaning: An observer integral with <a>sees the change of the components over <b> of . These components change independently from the place of the observer.
• This the definition of derivative of algebraic vector.

k

j

Oa

k

< a >

i

< b >

Ob

i

j

Vector derivative

5

ka

kb

q

jb

ja

ME696 - Advanced Robotics – C02

ia

ib

• Angular Velocity
• Since the rotation matrix between <a> and <b> is time dependent, we can define Angular Velocityof the frame <a>w.r.t. the frame <b> the vector b/a which, at any instant, gives the following information:
• Its versor indicates the axis around which, in the considered time instant, an observer integral with <a> may suppose that <b> is rotating;
• The component (magnitude) along its versor indicates the effective instantaneous angular velocity (rad/sec.)
• To the vector Angular Velocity we can associate the following differential form:
• The above relationship does not coincide with any exact differential.

w(t)

Angular Velocity

6

ka

kb

q

jb

ja

ME696 - Advanced Robotics – C02

ia

ib

• Angular Velocity
• We want not to write in a different form the (2.2):
• We need Poisson formulae:
• Thus we have:
• (2.3)
• If  is constant: (rigid body)

w(t)

Angular Velocity

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ME696 - Advanced Robotics – C02

• Angular Velocity
• Properties:
• b/a = - a/b
• Given n frames, the angular velocity of <k> w.r.t. <h> if given by adding the successive ang. Velocities encounteredalong any path.
• In this example:

Angular Velocity

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ME696 - Advanced Robotics – C02

• Time derivative for points in space
• We define:
• “velocity of P computed w.r.t. the frame <a>”:
• “velocity of P computed w.r.t. the frame <b>”:
• It is possible to proof that:
• where vp/b is the velocity of the origin of the frame <b> w.r.t <a>

Angular Velocity

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ME696 - Advanced Robotics – C02

• Time derivative for points in space
• Proof:
• We define vb/a the velocity of the origin on the frame <b> w.r.t. <a>:
• Using the (2.3) with the opportune indexes we have:

Angular Velocity

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ME696 - Advanced Robotics – C02

Generalized velocity

In order to completely describe the relative motion between 2 frames we organize the angular velocity and the velocity of the origin within a vector called Generalized Velocity :

We can project the G.V. in any frame:

This definition is valid forany point integral with the frame <b>:

where

Angular Velocity

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ka

kb

q

jb

ja

ME696 - Advanced Robotics – C02

ia

ib

Derivative of the orientation matrix

Problem: we want to compute the relationship between the derivative of the orientation matrix and the angular velocity:

Remember that:

Deriving w.r.t. time:

w(t)

Derivative of the Orientation matrix

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ka

kb

q

jb

ja

ME696 - Advanced Robotics – C02

ia

ib

Derivative of the orientation matrix

Finally:

(2.4)

Remembering the transformation of the cross-prod operator:

the previous equation becomes:

(2.5)

The (2.4) and (2.5) are very useful in computing the time evolution of the orientation matrix:

w(t)

Derivative of the Orientation matrix

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ME696 - Advanced Robotics – C02

Group definition

A group is a set, G, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation, such as the addition. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:

Closure. For all a, b in G, the result of the operation a • b is also in G.

Associativity. For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds.

Identity element. There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds.

Inverse element. For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation a • b = b • a may not always be true.

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Rotation Group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space.

Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group.

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Joint Kinematics

In general, the set of all the relative positions between two free bodies constitutes a group that may be represented by the matrix:

SO(3) is the Special Euclidian group.

Kinematics in G can be represented as an object belonging to its Lie algebra:

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Joint Kinematics

The joint can be characterized by a relationship that involves the generalized velocity of the frame <b> w.r.t. <a>:

(2.6)

where q is the “configuration”. This means:

If the distribution q è integrable, the constraint is Holonomic.

In case that the axis are integral with at least one body, the matrix A is constant. Will name this kind of joints as Simple Kinematic Joints.

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Simple Kinematic Joint s

In this case, the solution of the (2.6) is given by:

where the column of H creates a base for the kernel of A and r is the number of degreesof freedom of the joint:

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Simple Kinematic Joint s

H is the Joint Matrix. Often p is known as quasivelocity.

Examples of joint matrices:

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Parameterization of Simple Kinematic Joint s

In general, the joint configuration is defined by the previous differential equation:

which can be re-written as:

We can now integrate the above equation, obtaining the evolution of the transformation matrix T.

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Parameterization of Simple Kinematic Joint s

Example:

r=1

H1 is the direction of the rotation axis, hence:

H2 is the direction of the translation, so we have:

which, integrated, gives:

If H has more columns:

Kinematics of the joints

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ME696 - Advanced Robotics – C02

Example: spherical joint

Example:

Kinematics of the joints

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ka

kb

jb

ja

ME696 - Advanced Robotics – C02

ia

ib

Example: spherical joint

Finally:

Kinematics of the joints

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