kinematics part i
TRANSCRIPT
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Kinematics
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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!
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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≠
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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
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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}
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Rotation Matrices
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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
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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
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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
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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)
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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)
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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
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Position, Velocity, Angular Velocity and Acceleration Vectors
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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?
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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
+=
′=
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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
qqdt
d AAAA&&&
∂∂
+∂∂
+∂∂
=pppp
3. Find the velocity of Q in A
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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
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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
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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
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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
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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
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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)
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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.
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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
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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.
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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*
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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
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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
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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
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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
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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
&
&
&
&
&
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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