kinematics part i

MEAM 535

University of Pennsylvania 1

Kinematics

MEAM 535


Scalar and Vector Functionsv(q1, q2,...,qn) is a vector function v of n variables.

In any reference frame {A}, we can find three independent vectors a1, a2, and a3 that are basis vectors.

The vector function v(q1, q2,...,qn) can be expressed as a linear combination of the three vectors:v(q1, q2,...,qn) = v1(q1, q2,...,qn) a1 +

v2(q1, q2,...,qn) a2 + v3(q1, q2,...,qn) a3

The three coefficients are the three scalar functions v1, v2, and v3. They are called components and these three functions are unique once the vectors a1, a2, and a3 are specified.

φ(q1, q2,...,qn) is a scalar function of nvariables

φ(q1, q2,...,qn) is independent of reference frames – scalar invariant

a1

a2

a3

e1

e2

e3

NB: No assumptions of orthogonality!

MEAM 535


Reference Frames

A robotic arm is a system of rigid bodies (reference frames) A, B, and C. D is the inertial or the laboratory reference frame that is considered fixed.

Components of vectors depend on the reference triad

q1

q2

q3

C

D

A B

p

[ ] [ ]

p e e ev e e e

p v

= + += + +

=

=

p p pv v v

ppp

vvv

E E

1 1 2 2 3 3

1 1 2 2 3 3

1

2

3

1

2

3

,,

,

a1

a2

a3

e1

e2

e3

[ ] [ ]E Ap p≠

MEAM 535


Standard Reference TriadThree vectors rigidly attached to a reference frame satisfying the equations below constitute a standard reference triad or simply a reference triad.

Projection rule

Composition rule

a1

a2

a3

e1

e2

e3

213

132

321,,

,1,0

aaaaaaaaa

aa

=×=×=×

=≠

=⋅jiji

ji

ii u=⋅eu

∑=++==

3

1332211

iiiuuuu eeeeu

( ) ii

i eeuu ∑=

⋅=3

1

MEAM 535


Transformation of VectorsComponents in {A} and in {E} are related by:

e1

e2

e3

a1

a2

a3

[ ] ( ) [ ]∑ ⋅=j

jA

jiiE paep

[ ] [ ]

[ ] ( )

( ) [ ]

( ) [ ]

Ei

ii

Aj

jj

Aj j i

ii

j

i ji j

Aj i

i jA

jji

i

p e p a

p a e e

e a p e

e a p e

∑ ∑

∑∑

∑

∑∑

=

= ⋅

= ⋅

= ⋅

,

[ ] [ ] [ ]

[ ]paeaeaeaeaeaeaeaeae

pRp

A

AA

EE

⋅⋅⋅⋅⋅⋅⋅⋅⋅

=

=

332313

322212

312111

Rotation matrix that transforms components in {A} into components in {E}

MEAM 535


Rotation Matrices

MEAM 535


Properties of Rotation Matricesξ3Orthogonal

Matrix times its transpose equals 1

Special orthogonalDeterminant is +1

Closed under multiplication

Composition ARC = ARB × BRC

The inverse of a rotation matrix is also a rotation matrix

Composition and inverse operations are “continuous functions”

The set of all rotations is a Lie Group, SO(3)

SO(3) can be parameterized by 3 coordinates

ξ1

ξ2

MEAM 535


Example: “Simple” RotationsRotation about the x-axis through θ

( )

θθθ−θ=θ

cossin0sincos0001

,xRot

x

y

zθ

e1

e2

e3

θ

a2

a3

⋅⋅⋅⋅⋅⋅⋅⋅⋅

332313

322212

312111

aeaeaeaeaeaeaeaeae

MEAM 535


Example: RotationRotation about the y-axis through θ

e2

e3

e1

θ

a3

a1

( )

θθ−

θθ=θ

cos0sin010

sin0cos,yRot ( )

θθθ−θ

=θ1000cossin0sincos

,zRot

e2

e3

e1

θ

a2

a1

⋅⋅⋅⋅⋅⋅⋅⋅⋅

332313

322212

312111

aeaeaeaeaeaeaeaeae

Rotation about the z-axis through θ

MEAM 535


Composition of Three Rotations

a1

a2

a3

b1

b3

b2

ψ

c1

c3

c2φ

d1

d3

d2

θ

ARD = ARB × BRC × CRD

ARD = Rot(z, φ) × Rot(y, θ) × Rot(z, ψ)

MEAM 535


Euler Angles: Parameterization of Rotation MatricesSequence of three rotations about body-fixed axes

Rot(z, φ)Rot(y, θ)Rot(z, ψ)

Three Euler Anglesφ, θ, and ψParameterize rotations

Noteθ= 0 is a special (singular) case

R = Rot(z, φ) × Rot(y, θ) × Rot(z, ψ)

MEAM 535


Determination of Euler Angles

R = Rot(z, φ) × Rot(y, θ) × Rot(z, ψ)

:= R

− ( )cos φ ( )cos θ ( )cos ψ ( )sin φ ( )sin ψ − − ( )cos φ ( )cos θ ( )sin ψ ( )sin φ ( )cos ψ ( )cos φ ( )sin θ

+ ( )sin φ ( )cos θ ( )cos ψ ( )cos φ ( )sin ψ − + ( )sin φ ( )cos θ ( )sin ψ ( )cos φ ( )cos ψ ( )sin φ ( )sin θ

− ( )sin θ ( )cos ψ ( )sin θ ( )sin ψ ( )cos θ

R ,1 1 R ,1 2 R ,1 3

R ,2 1 R ,2 2 R ,2 3

R ,3 1 R ,3 2 R ,3 3

θ= cos33Rψθ−= cossin13R

ψθ= sinsin23R

φθ=φθ=

sinsincossin

23

13

RR

MEAM 535


Position, Velocity, Angular Velocity and Acceleration Vectors

MEAM 535


Position, Velocity and Acceleration Vectorsp is a position vector of P in A

Emanates from a point fixed to AEnds up at P

AvP is the velocity of P in A

AaP is the acceleration of P in A

P

O

p

A O'

p'dtdA

PA pv =

( )dt

d PAAPA va =

What if a different position vector were chosen?

MEAM 535


Velocity of P in A is independent of choice of “origin” in A!

p is a position vector of P in AEmanates from a point fixed to AEnds up at P

AvP is the velocity of P in A

P

O

p

A O'

p'r

( )dt

ddtd

A

APA

rp

pv

+=

′=

MEAM 535


Example

q1

q2

q3

D

A

B C

p

b1

b2

b3

2q∂∂p

1. Are the following equal?

2q

A

∂∂p

2q

B

∂∂p

2q

C

∂∂p

dtdA p

2. If motor (joint) rates are given, calculate

Q

A robotic arm is a system of rigid bodies (reference frames) B, C, and D. A is the inertial or the laboratory reference frame that is considered fixed.

33

22

11

qq

qq

qqdt

d AAAA&&&

∂∂

+∂∂

+∂∂

=pppp

3. Find the velocity of Q in A

MEAM 535


Angular Velocity (Kane)The angular velocity of B in A, denoted by AωB, is defined as:

Defined in terms of a reference triad attached to BIndependent of reference triad attached to AGeneralizes to three dimensionsYields simple results for derivatives of vectors

+

+

=ω 2

131

323

21 ... bbbbbbbbb

dtd

dtd

dtd AAA

BA

A

Bb1

b2

θ

a1

a2

θ−θ=θ+θ=

sincossincos

122

211

aabaab

Example

MEAM 535


Differentiation of vectors1. Vector fixed to B

r = r1 b1 + r2 b2 + r3 b3

Composition and Projection rule

A

BP Qr

b1

b2

b3

dtdr

dtdr

dtdr

dtd AAAA

33

22

11

bbbr++=

331

221

1111 bbbbbbbbbb

⋅+

⋅+

⋅=

dtd

dtd

dtd

dtd AAAA

⋅−

dtdA

31

bb

MEAM 535


Differentiation of vectors (cont’d)

121

313

232

11 ... bbbbbbbbbbb ×

+

+

=×ω

dtd

dtd

dtd AAA

BA

313

2211 bbbbbbb

⋅−

⋅=

dtd

dtd

dtd AAA

BP Qr

b1

b2

b3

Important Result 1

iBAi

A

dtd bb

×ω=

Important Result 2

r

bbb

bbbr

×ω=

×ω+×ω+×ω=

++=

BA

BABABA

AAAA

rrrdtdr

dtdr

dtdr

dtd

332211

33

22

11

A

MEAM 535


Velocity of P (attached to B) in A when A and B have a common point O

A

B

O

Pp=r

b1

b2

b3ppv ×ω== BA

APA

dtd

Choose p to be a position vector of P in A

MEAM 535


Differentiation of vectors2. Vector not fixed to B

rr

bbbr

bbbbbbr

×ω+=

×ω+×ω+×ω+=

+++++=

BAB

BABABAB

AAAA

dtd

rrrdtd

dtdr

dtdr

dtdr

dtdr

dtdr

dtdr

dtd

332211

33

22

113

32

21

1

BO

rb1

b2

b3

rrr×ω+= BA

BA

dtd

dtd

P

Ar can be any vector

MEAM 535


Simple Angular Velocity

Angular velocity of B in Ais along a2 as seen in Ais along b2 as seen in B

A rigid body B has a simple angular velocity in A, when there exists a unit vector k whose orientation (as seen) in both A and in B is constant (independent of time).

q1

q2

q3

C

D

A Ba1

a2

a3 b1b2b3

In each frame, the angular velocity has a constant direction (magnitude may change)

MEAM 535


Simple Angular Velocity

Axis 1

Axis 2

P

A

Axis 3

q1 q2

q3

B

C

D

How about DωC ?

However, DωC is not a simple angular velocity. The motion of Crelative to D is such that there is no vector fixed in D that also remains fixed in C.

DωA, AωB and CωB are simple angular velocities.

MEAM 535


Addition Theorem for Angular VelocitiesLet A, B, and C be three rigid bodies. The addition theorem for angular velocities states:

ProofLet r be fixed to C.

( )( ) r

rr

rrr

rrr

×ω+ω=

×ω+ω+=

×ω+×ω+=

×ω+=

BACB

BACBC

BACBC

BABA

dtd

dtd

dtd

dtd

rr×ω= CA

A

dtd

And,

CBBACA ω+ω=ω

A

B

C r

MEAM 535


Angular velocities can be found by adding up “simple” angular velocities

Axis 1

Axis 2

P

A

Axis 3

q1 q2

q3

B

C

D

DωC is not a simple angular velocity. The motion of C relative to D is such that there is no vector fixed in D that also remains fixed in C.

But,

CBBAADCD ω+ω+ω=ω

DωA, AωB and CωB are simple angular velocities.

MEAM 535


ExampleThe rolling (and sliding) disk on a horizontal plane

A circular disk C of radius R is in contact with a horizontal plane (not shown in the figure)at the point P. The point P is attached to the disk. The plane is the x-y plane. It is rigidly attached to the earth. The standard reference triad bi is chosen so that b1 is along the direction of progression of the disk (parallel to the tangent to the disk at P), b2 is parallel to the plane of the disk, and b3 is normal to the disk. Note that this triad is not fixed to the disk. Call the earth-fixed reference frame A and choose the standard reference triad ax, ay, and azin an obvious fashion along the x, y, and z axes shown in the figure.

x

y

z

P

q2

q3q1

q5

q4

C

b1

b2

b3

C*

MEAM 535


Reference Triads

Rotate triad A about z through q1followed by rotation about x by 90 deg to get ERotate triad E about -x through q2 to get BRotate triad B about z through q3 to get C (not shown)

xP

q3q1

q5

q4

C

b1

b2

b3

C*

Locus of thepoint of contact Q on the plane A

e1

e2

e3

a1

a2

a3

Imagine B to be a virtual body that is attached to C*

z

Imagine E to be a virtual body that is attached to Qq2

y

MEAM 535


TransformationsTwo coordinate transformations

ai in terms of ei

ei in terms of bi

x

y

z

P

q2

q3q1

q5

q4

C

b1

b2

b3

C*

e1

e2

e3

a1

a2

a3

MEAM 535


Reference Triads

Rotate triad A about z through q1followed by rotation about x by 90 deg to get ERotate triad E about -x through q2 to get BRotate triad B about z through q3 to get C (not shown)

xP

q3q1

q5

q4

C

b1

b2

b3

C*

Locus of thepoint of contact Q on the plane A

e1

e2

e3

a1

a2

a3

Imagine B to be a virtual body that is attached to C*

z

Imagine E to be a virtual body that is attached to Qq2

y

MEAM 535


TransformationsTwo coordinate transformations

ai in terms of ei

ei in terms of bi

x

y

z

P

q2

q3q1

q5

q4

C

b1

b2

b3

C*

e1

e2

e3

a1

a2

a3

MEAM 535


Angular Velocity: ComponentsAωC = u1 b1 + u2 b2 + u3 b3

ui are the components of the angular velocity of the disk with respect to the reference triad B

AωC = ux ax + uy ay + uz azuα are the components of the angular velocity of the disk with respect to the reference triad A

=

5

4

3

2

1

5

4

3

2

1

qqqqq

uuuuu

&

&

&

&

&

X

=

5

4

3

2

1

5

4

3

2

1

uuuuu

qqqqq

Y

&

&

&

&

&

MEAM 535


Angular AccelerationThe angular acceleration of B in A, denoted by AαB, is defined as the first time-derivative in A of the angular velocity of B in A:

Addition theorem for angular accelerations?

( )BAA

BAdtd

ω=α

kinematics part i

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