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机器人学 kinematics2

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commercial use

肈

薁2.4 Kinematic Analysis of a Particle in 3D Space

膂

the coordinates of P with respect to

O ?

xyz

膆Investigation into the motion of a particle is a simplest yet the most fundamental aspect in kinematics. In this section, we deal with the kinematic equations of a particle P moving in 3D space by

肀Takingderivatives once and twice with respect time gives the velocity andacceleration of point P

using vector notation. Three coordinate systems will be introduced to describe the curvilinear motion of a particle: They are
?

腿

x i

y j

z k

(1b)

螇

?r?

?x?i

?y?j

?z?k

(1c)

羃Rectangular(Cartesian) Coordinate System (

)

节Obviously, this is the form which we are very familiar with.蒁Projectile Problem(A classical problem for particle kinematics)

薂Cylindrical Coordinate System(

???

)

聿Spherical Coordinate System(

???

羅Rectangular (Cartesian) Coordinate System (

)

袁An object located at origin of a Cartesian

system with its

axis upward and

axis

肂Consider a particle P moving in 3D

horizontal as shown, is thrown through the air

with an initial velocity

and an angle

space, in the rectangular coordinate

about horizontal at

, determine the

system

O ?

xyz

as shown in the

trajectory function of the object.

figure, its position vector

can be

given as

薆Solution:
薆Initial conditions

螁

羃

x i

y j

z k

(1a)

the unit vectors of three

羂Position:

orthogonal axes. Note that they are constant since all axes have

莈Velocity:

x ?0

cos?0

y ?0

sin?0

蕿Acceleration:

?x?

?y?

( g--the gravitational acceleration)

蚆?

?x?

cont

0cos?

莃?

?y?

y
?y 0

d y

y?v 0

sin?0

d y

t?0

gdt

肀One more integration leads to

Substituting	t ?		x	into the above equation finally results
莇		v	0cos?0

袀Defineer,e?and ezas three orthogonal unit vectors in the

senseof positive r, ?and z

directions,respectively. The position ?r

vector rof Pcanthen be expressed ?e?

bye???e???

芀r?rer?zez (2a) Oy

er??er

袅Inorder to achieve the velocity of P, er ??

letus rotate er,e?and ezabout ?er ??

aninfinitesimal time interval
zwith an small angle ??during

?t. ?

Thenthe differentiation of rhas x

theform

螆Cylindrical Coordinate System (

???

)

羆

r?e

z?e

(2b)

螃In the cylindrical coordinates, the

芁Since

keeps unchanged,

. While the small changes in

position of a particle P can be

and

occur as follows

located by

蚈Thus

薈

-----the radial distance from the

-axis

羈

lim

?t?0

lim

?t?0

?t e?

??e?

膆

?---- the angle from the

-axis

to the radial line

肅This will also lead to

袆

---the vertical distance from the

蚂and thereby

x ?

plane.

蒀

d r

lim
??0

r e

r??e?

z e

(2c)

膀

r 1?e?

1 ?1

r e 3 r

??3

羂

,???

蚇Differentiating Eq.(2c) finally results in the acceleration of P

??3

r 3?e?

3 ?3

e?1

,??

膅

?r?

?r?e

r e

?r??e??r???e??r??e?

??r??e??r???e??r??2

??r????2 r???e???z?e z

?z?e

(2d)

?r?e

?r?

?r??e?

r??2?e r

?z?e

莅Velocity Polygon

r 1

,??

,???

芅y

肃Sample Problem

r 1

薁x

袈Determine

r 3

r?3

and

??3

of a RRPR planar mechanism

芅y

薀x

e?3

as shown provided that

r 1

r 3

and

??1

have been

,??

,???

r 3

?r?3

r 3

?r?3

known.

蒆Solution:

r ?

,??

,???

膅The polar coordinate system may be used for this problem. The

general formulae 2(a), 2(c) and 2(d) are reduced to

芈y

肇x

r r ?

r?e?

??r?

r??2

?r???

r??

蝿VelocityAnalysis

肆Setp1: Establish the polar coordinate system			r 1	??1	with				e	r	1
and	e?1	being as the unit vectors. The velocity of A in				r 1	?	?1	can

be expressed by

薅Evaluated in

r 1

finally results in

蒅Since

r 1

is a constant,

r 1?

. Thus

薁Keep in mind that

r 1

is a constant such that

莂Step 2: Establish another polar coordinate system

r 3

with

蚈Evaluated in

r 3

and

e?3

being as the unit vectors. The velocity of point A in

芅The acceleration constraint condition gives

r 3

can be expressed by

膁

r 3?e?

3?3

--- the velocity of a fictitious point B on link 3, which is

肃Projecting the left hand term onto

e?3

and

momentarily coincident with the point A.

蝿r e 3 r 3 --- the sliding velocity of A relative to B along the direction

of e r 3 .

蒈Step 3: The velocity of the point A evaluated by using these two

systems SHOULD BE THE SAME. Thus

螇Projecting

r A?

r 1?e?

1 ?1

onto

e?3

and

, respectively, leads

莀SphericalCoordinate System

Whena radial distance and two angles are used to specify the

positionof a particle as radar measurement, spherical coordinates

r,?,?can be used as shown in the figure. However, since the

derivationof the acceleration is quite complicated, we only give the

resultshere.

r 3?3 2

?r 3???

r 3??

?e?3

袃??

?r 1

cos

sin

r 3

??3

r 1?e??

` 1?1

r e 3

螂Acceleration Analysis

??3

袄Acceleration analysis can be implemented in the same way.

r 1??

` 1
2

薈Acceleration Polygon