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finite order CT LTI system H with input f = f (t), output
g = g(t), and state x = x(t) has the form
x˙ (t) = Ax(t) + Bf (t),
g(t) = Cx(t) + Df (t),
(2.1)
(2.2)
where A, B, C, D are constant matrices with real entries.
H =
A B
C D
�
�
frequently serves as a shortcut notation. Given an input f = f (t), the outpu... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
(kT ),
x[k] = x(kT ), and g[k] = g(kT ), equations have the form
x[k + 1] = Ax[k] + Bf [k],
g[k] = Cx[k] + Df [k],
(2.5)
(2.6)
The transfer matrix (transfer function in the case when both f and g are scalar) of the
system is defined for all complex z such that zI − A is invertible by
H(z) = D + C(zI − A)−1B.
2.... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
∈� = sup �max(H(ej�)).
�→[−�,�]
The H2 norm ∈H∈2 of a finite order stable CT LTI system H with D = 0 is defined
by the integral
∈H∈2 =
2
�
0
trace(h(t)h(t)≥)dt =
�
1
2� −�
trace(H(jσ)H(jσ)≥)dσ,
where
h(t) = Ce AtB
is the impulse response matrix of H.
In the discrete time case (where D ≥= 0 is allowed), t... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
zero initial conditions”
(whatever this means), we expect the “energy” of the output to be bounded by the energy
of the input times the L2 gain squared. Since non-zero initial conditions can produce non
zero output even for zero input, the actual definition says that the difference between the
energies must be bounde... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
0
�
0
|g T (t)|2dt
0
�
=
1
2� −�
�
1
2� −�
=
|˜T
g (jσ)|2dσ =
|H(jσ)f˜T (jσ)|2dσ
�
�
1
2� −�
�
∈H∈2
�|fT (jσ)|2dσ = ∈H∈2
�
|fT (t)|2dt = ∈H∈2
�
0
In the case of non-zero initial conditions the total system response is given by g =
g0 + g1, where g0 is the zero state response and g1 is zero ... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
��nity gain of a stable state space model is defined as the supremum of the
amplitude of its time domain response to an input signal of unit energy.
Theorem 2.2 L2-to-L-Infinity gain of a stable LTI system with a scalar output equals
its H2 norm.
Proof Consider the continuous-time case (the DT case is similar).
To s... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
��ect in integration: if
γ =
t2
t1
h(t)f (t)dt,
8
where h = h(t) is a row vector of appropriate length, then
Eγ = 0, E|γ|2 =
t2
t1
|h(t)|2dt.
Combining this information with the definition of H2 norm, one can conclude that, for
a stable LTI system, the asymptotic (as t � →) variance of white noi... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
gain of the nonlinear feedback system on the left of Figure 2.2 based on
the L2 gain properties of � and G.
Theorem 2.3 If G and � separately have L2 gains less than one then the feedback in
terconnection on Figure 2.2, left, has L2 gain (from w2 to z2) less than 1.
9
Proof Consider the signals consistent with th... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
) =
−�e
s − 1
−
δ
1
,
1 + 0.5δ 1 + 0.5δ s
W (s) = 0.042
1 + s
,
10 + s
and �(s) is known to have H-Infinity norm not larger than one. The corresponding
feedback design diagram are shown below.
1
s+1
Transfer Fcn
1
Out1
1
In1
g
Gain
3
Out3
3
In3
P0
LTI System
1/d
Gain1
2
Out2
2
In2
0.042*... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
t2
t1
E
�
�
�
�
E
�
�
�
�
�
T
where the constant may depend on the input/output pair (f, g). A similar definition can
be given in the discrete time case. When dealing with randomized signals and systems,
the L2 gain bounding condition can be replaced by its “expected value” version
E|g(t)|2dt � const + π2
T
... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
and
2
T
E|w2(t)|2dt � const +
E|z2(t)|2dt,
0
0
for all T . Then
lim sup
T ��
T
0
1
T
E|z1(t)|2dt � ∈G11∈2 +
2
2
∈G12∈2 · ∈G21∈2 .
2
1 − ∈G22∈2
2
12
2.3.4 Example: analysis via H2 small gain
Consider a simple example which can be easily solved analytically with or without us... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
[k], w1[k] = v1[k], w2[k] = v[k]y[k].
Then z and w are related through an LTI transformation with transfer matrix
G = G(z) =
1
z−a
1
z−a
�
b
z−a
b
z−a
.
�
Assuming |a| < 1,
∈G11∈2 = ∈G21∈2 =
2
2
1
1 − a2 , ∈G12∈2 = ∈G22∈2 =
2
2
b2
.
1 − a2
By the assumptions made about vi[k], signal w = [w1; w2] is unco... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/5bca558808b8b652b14b2be6128d8e7d_lec2_6245_2004.pdf |
8.871 Lecture 3
M. Padi
November 14, 2004
Let’s use dimensional analysis to make some general remarks about the tensions of various branes. We’ll
start with the M2- and M5-branes. Note that the tension of an object that fills 2 space dimensions must
have units of [L−3] = [L2] . The only length scale we have is lp, ... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/5bcdff3a3071c28befd43ad3ef4fc350_lec3.pdf |
theory. We use the fact that the M2-brane compactified
over one of its dimensions becomes an F1-string in Type IIA. Thus we can write
TF 1 = =
1 R
l3
l2
p
s
.
(3)
An M2 which does not wrap the M theory circle becomes a D2 brane in Type IIA and we can write a relation
between the tensions:
TD2 =
=
.
(4)
1
gs... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/5bcdff3a3071c28befd43ad3ef4fc350_lec3.pdf |
the D3-brane is self-dual and the D5-brane becomes the
NS5-brane. So now we have the following equalities:
s
TD1 =
TF 1 =
1
gsl2
s
1
l2
s
= T � =
F 1
= T � =
D1
1
l�2
s
1
g� l�2
s s
From these equations we can derive the following relation between ls and l� :s
�2 = gsl2 .
s
s
l
References
(8)
(9)
(10)
... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/5bcdff3a3071c28befd43ad3ef4fc350_lec3.pdf |
Frequency response: Resonance, Bandwidth, Q factor
Resonance.
Let’s continue the exploration of the frequency response of RLC circuits by investigating
the series RLC circuit shown on Figure 1.
I
R
C
Vs
R
+
VR
-
The magnitude of the transfer function when the output is taken across the resistor is
Figure 1
H
(
)
... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
)
)
p t
( )
=
=
V
R
V I
R R
cos(
ω
t I
)
R
t
ω
2
cos (
6.071/22.071 Spring 2006, Chaniotakis and Cory
cos(
t
ω
)
)
(1.4)
2
And the average power becomes
P
(
)
ω =
=
1
2
1
2
V I
R R
2
I R
R
(1.5)
Notice that this power is a function of frequency since the amplitudes
frequency depende... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
2
⎞
⎟
⎠
+
1
2
ω
0
6.071/22.071 Spring 2006, Chaniotakis and Cory
(1.8)
(1.9)
(1.10)
4
The bandwidth is the difference between the half power frequencies
Bandwidth B
=
=
−
1ω ω
2
(1.11)
By multiplying Equation (1.9) with Equation (1.10) we can show that
0ω is the geometric ... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
(1.15)
At the resonance frequency where
ω ω=
0
=
1
LC
the energy stored in the circuit
becomes
SE
=
2
CA
1
2
6.071/22.071 Spring 2006, Chaniotakis and Cory
(1.16)
5
The energy dissipated per period is equal to the average resistive power dissipated times
the... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
of the capacitor and the inductor act as an open
circuit. Therefore at the resonance the total current flows through the resistor.
and the impedance seen by the source is purely
= 0
1
2
LCω−
If we look at the current flowing through the resistor as a function of frequency we obtain
according to the current divider... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
1
=
1
RC
Q
=
ω
0
B
P
=
ω
0
RC
=
R
L
ω
0
(1.23)
(1.24)
(1.25)
(1.26)
(1.27)
6.071/22.071 Spring 2006, Chaniotakis and Cory
8
Summary of the properties of RLC resonant circuits.
Series
I
R
C
Parallel
Circuit
Vs
R
+
VR
-
I s(t)
IR(t)
R
L
C
Transfer function
H
(
)
ω
≡
... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
ω ω=
2
1
=
R
L
PB
−
ω ω=
2
1
=
1
RC
Q factor
Q
=
0
L
ω ω
0
=
R
B
S
=
1
RC
ω
0
Q
=
ω
0
B
P
=
ω
0
RC
=
R
L
ω
0
6.071/22.071 Spring 2006, Chaniotakis and Cory
9
Example:
A very useful circuit for rejecting noise at a certain frequency such as the i... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
above values for L and C is shown on Figure 7
for various values of R.
Figure 7
Since the capacitor and the inductor are in parallel the bandwidth for this circuit is
B
=
1
RC
(1.32)
If we require a bandwidth of 5 Hz, the resistor R=212Ω. In this case the pot of the
transfer function is shown on Figure 8.
Figure... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/5bcec4bfba5f2e99754b77509e9e7ab4_resonance_qfactr.pdf |
15.083J/6.859J Integer Optimization
Lecture 3: Methods to enhance formulations
1 Outline
• Polyhedral review
• Methods to generate valid inequalities
• Methods to generate facet defining inequalities
Slide 1
2 Polyhedral review
2.1 Dimension of polyhedra
• Definition: The vectors x1 , . . . , xk ∈ (cid:3)n
uni... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
be a valid inequality for a polyhedron P , and let F = {x ∈
P | f (cid:2) x = g}. Then, F is called a face of P and we say that f (cid:2) x ≥ g
represents F . A face is called proper if F = Ø(cid:6)
, P .
• A face F of P represented by the inequality f
(cid:2) x ≥ g, is called a facet of
P if dim(F ) = dim(P ) − 1. We... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
3 Methods to generate valid inequalities
3.1 Rounding
• Choose u = (u1, . . . , um)(cid:2) ≥ 0; Multiply ith constraint with ui and sum:
Slide 5
Slide 6
Slide 7
(cid:2)
n
(u
(cid:2)Aj )xj ≤ u
(cid:2)b.
j=1
• Since (cid:5)u (cid:2)Aj (cid:6) ≤ u (cid:2)Aj and xj ≥ 0:
(cid:2) (cid:3)
n
(cid:4)
(cid:2)Aj (... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
�E(U )
xe ≤ 1 |U |.
2
(cid:2)
xe ≤
e∈E(U )
(cid:9)
(cid:10)
1 |U | =
2
|U | − 1 ,
2
3.2 Superadditivity
3.2.1 Definition
A function F : D ⊂ (cid:3) (cid:8)n → (cid:3) is superadditive if for a1, a2 ∈ D,: a1 + a2 ∈ D :
Slide 9
F (a1) + F (a2) ≤ F (a1 + a2),
It is nondecreasing if
F (a1) ≤ F (a2), if a1 ≤ a... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
F (Aj )xj ≤
F (Aj xj ).
Slide 12
j=1
j=1
3
By superadditivity,
⎛
n
(cid:2)
F (Aj xj ) ≤ F
⎝
n
(cid:2)
Aj xj
⎞
⎠ = F (Ax).
j=1
j=1
Since Ax ≤ b and F is nondecreasing
F (Ax) ≤ F (b).
3.3 Modular arithmetic
(cid:6)
n
S = x ∈ Z+
(cid:8)
,
aj xj = a0
(cid:7) n
(cid:2)
(cid:7) (cid:7) (cid:7)
j... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
:6), and thus the following
j=1
(cid:5)aj (cid:6)xj ≤ a0. Since x ∈ Z,
(cid:5)
n
j=1
inequality is valid for S
(cid:2)
n
(aj − (cid:5)aj (cid:6))xj ≥ a0 − (cid:5)a0(cid:6).
j=1
3.4 Disjunctions
3.4.1 Proposition
(cid:5)n
If the inequality
(cid:5)
n
j=1 cj xj ≤ d is valid for S2 ⊂ (cid:3) , then the inequ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
, (−x1 + x2) − (x2 − 2) ≤ 1.
α = 2, −x1 + x2 ≤ 1 is valid.
3.5 Mixed integer rounding
3.5.1 Proposition
• For v ∈ (cid:3), f (v) = v − (cid:11)v(cid:12), v+ = max {0, v}.
• X = {(x, y) ∈ Z × (cid:3)+ | x − y ≤ b}
y
3
5
2
2
3
2
1
1
2
x − y =
3
2
x − 2y = 2
1
3
2
2
3
4
x
• The inequality x − 1
1−f (b)... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
)
3.5.3 Theorem
(cid:7)
(cid:7)
(cid:5)n Aj xj ≤ b,
S = x ∈ Z n (cid:7)
+ (cid:7)
(cid:17)
(cid:11)u(cid:2)Aj (cid:12) +
inequality
j=1
(cid:5)
n
j=1
j = 1, . . . , n . For every u ∈ Qm the
� b)]+
u
A
[f (u�
(
f
−
)
j
b
u
)
(
f
−
1
�
xj ≤ (cid:11)u(cid:2)b(cid:12) is valid for conv(S).
(cid:18)
+
(cid... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
(i, r(i)) ∈(cid:10) E. x i with
r(i) = 1, and zero elsewhere is in S, and satisfies inequality (*) with
x i
i = 1, x i
equality.
• e1, . . . , ek, x k+1 , . . . , x n are linearly independent; hence, affinely independent.
Conversely, since U is not maximal, there is a node i /∈ U such that U ∪ {i} is a clique,
xj ≤ 1 (∗... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
{3, 5}, {3, 6}, {4, 6} satisfy it with equality. Not facet,
since there are no other stable sets that satisfy (1) with equality.
(cid:12)
• (1) is facet defining for S ∩ x ∈ {0, 1}6 | x1 = 0 .
• Consider ax1 + x2 + x3 + x4 + x5 + x6 ≤ 2, a > 0.
• Select a in order for (1) to be still valid, and to define a facet for S... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
for S.
• If S1 (cid:10)
1x
aj xj s.t. x ∈ S1 .
Z =
= Ø, then a 1 +
(cid:5) n
(cid:5) n
j=2
j=2
j x
a j ≤ a0 (3) is valid for S for any a1 ≤ a
0 − Z
,
• If a1 = a0 − Z and (2) defines a face of dimension k of conv(S0), then (3) gives
a face of dimension k + 1 of conv(S).
4.2.2 Geometry
7
Slide 26
Slide 27
Slide... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
b} nonempty?
2
Integer points in lattices
• B = [b1 , . . . , bd
] ∈ Rn×d , b1 , . . . , bd are linearly independent.
n
L = L(B) = {y ∈ R
| y = Bv, v ∈ Z d}
is called the lattice generated by B. B is called a basis of L(B).
• bi = ei, i = 1, . . . , n ei is the i-th unit vector, then L(e1, . . . , en) = Z .
•... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
unimodular, then B = BU is a basis of the lattice L.
• If B and B are bases of L, then there exists a unimodular matrix U such that
B = BU .
• If B and B are bases of L, then |det(B)| = |det(B)|.
2.3 Proof
• For all x ∈ L: x = Bv with v ∈ Z d .
• det(U ) = ±1, and det(U −1) = 1/ det(U ) = ±1.
• x = BU U −1 v. ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
2.4 Convex Body Theorem
Let L be a lattice in R and let A ∈ R be a convex set such that vol(A) >
2ndet(L) and A is symmetric around the origin, i.e., z ∈ A if and only if −z ∈ A.
Then A contains a non-zero lattice point.
n
n
2.5
Integer normal form
• A ∈ Z m×n of full row rank is in integer normal form, if it is ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
1,1 ≥ a1,2 ≥ . . . ≥ a1,n ≥ 0.
• Since A is of full row rank, a1,1 > 0. Let k = max{i : a1,i > 0}. If k = 1, then
we have transformed A into a matrix of the form (1). Otherwise, k ≥ 2 and by
applying k − 1 operations (c) we transform A to
�
�
�
�
a1,1 A2, . . . , Ak−1 −
a1,2
�
a1,k−1 Ak, Ak, Ak+1, . . . ,... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
a
unimodular matrix U 2 = I −2I j,j , that is an identity matrix except that element
(j, j) is −1. det(U 2) = −1.
(iii) Adding f ∈ Z times column k to column j, corresponds to multiplying
matrix A by a unimodular matrix U 3 = I + f I k,j . Since det(U 3) = 1, U 3 is
unimodular.
• Performing two elementary column ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
B, 0] = AU . Let b ∈ Z m and S = {x ∈ Z n | Ax = b}.
(a) The set S is nonempty if and only if B −1b ∈ Z
m
(b) If S (cid:6)= ∅, every solution of S is of the form
.
Slide 10
x = U 1B −1b + U 2z, z ∈ Z n−m ,
where U 1, U 2: U = [U 1, U 2].
(c) L = {x ∈ Z n | Ax = 0} is a lattice and the column vectors of U 2 const... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
| Ax = 0}. By setting b = 0 in part (b) we obtain that
L = {x ∈ Z n | x = U 2z, z ∈ Z n−m}.
Thus, by definition, L is a lattice with basis U 2.
2.10 Example
• Is S = {x ∈ Z 3 | Ax = b} is nonempty
�
�
A =
3
4
6
5
1
5
Slide 12
and b =
�
�
.
3
2
• Integer normal form: [B, 0] = AU , with
�
[B, 0] =
1
5... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
0,
y(cid:2)A ∈ Z and y ∈ Z
.
(cid:2)b /
m
2.12 Proof
• Assume that S = ∅. If there exists y ∈ Qm, such that y (cid:2)A ∈ Z and y ∈ Z,
(cid:2)b /
m
(cid:6)
then y (cid:2)Ax = y (cid:2)
b with y
(cid:2)Ax ∈ Z and y (cid:2)b /
∈ Z.
• Conversely, if S = ∅, then by previous theorem, u = B −1b ∈ Z/
m
, that is t... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
. Bz ≥ −x 0
z ∈ Z n−m .
6
MIT OpenCourseWare
http://ocw.mit.edu
15.083J / 6.859J Integer Programming and Combinatorial Optimization
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
4 SCETI LAGRANGIAN
where
Wn =
(cid:88)
(cid:88) −g)k
(
k p
erm
k!
(cid:32)
n¯
· An(q1) · · · n¯ · An(qk)
[n¯ · q1][n¯ · (q1 + q2)] · · · [n¯ · (cid:80)
k
i=1 qi]
(cid:33)
.
(3.34)
Here Wn is the momentum space version of a Wilson line built from collinear An gluon fields. In position
space the corresponding Wilson li... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
separate large (label) momenta from small (residual) momenta.
4.1 SCET Quark Lagrangian
Lets construct the leading order SCET collinear quark Lagrangian. This desired properties that this
Lagrangian must satisfy include
• Yielding the proper spin structure of the collinear propagator
• Contain both collinear quark... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
ultrasoft gluon with the collinear quark. The LO collinear
quark propagator must be smart enough to give the correct leading order result without further expansions,
irrespective of whether it later emits a collinear gluon or ultrasoft gluon.
We will achieve the desired collinear Lagrangian in several steps.
4.1.1 ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(4.4)
since ξnPn = 0 and ϕ¯n¯Pn¯ = 0. These simiplifications leave us with the Lagrangian
¯
L = ξˆ
n
n/
2
in · D ξˆn + ϕ¯iD/ ⊥ ξˆn + ξˆ
n
n iD/ ⊥ϕn¯ + ϕn¯
n/
2
in¯ · Dϕn¯ .
(4.5)
So far this is just QCD written in terms of the ξˆn and ϕn¯ fields. However, the field ϕn¯ corresponds to the
spinor components which were ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
= 0
n/¯ ˆ
iD/ ⊥ ξn ,
2
where the second line is obtained by multiplying the first by /¯n/2 on the left, and the plus sign in the last
= −i / ¯ Plugging this result back into our Lagrangian, two terms cancel,
line comes from using /¯
and the other two terms give the Lagrangian for the ξˆn field
niD/ ⊥
D⊥n/.
(cid:18... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
. Separate the collinear and usoft momentum components with a multipole expansion.
We then can expand in the fields and momenta and keep the leading pieces.
4.1.2 Step 2: Separate collinear and ultrasoft gauge fields
n
µ
n ∼ (λ2 , 1, λ) ∼ pn
Recall that Aµ
and Aµ ∼ (λ2, λ2, λ2) ∼ kµ Since k2
the ultrasoft gluons en... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
for our leading order analysis here (they are
given below in Eq.()). The interpretation of Aµ
n will also prove useful
us as a background field to ξn and Aµ
23
4.1 SCET Quark Lagrangian
4 SCETI LAGRANGIAN
when we derive the collinear gluon lagrangian and when we later consider gauge transformations in the
theory.... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
= −
(pn
1
us)(pn + k−
+
⊥
us) + (pn
+ k−
1
us) + p
− + + k+
pn (pn
=
+ k⊥ )2
us
⊥
2k⊥ · pus n
−
⊥ 2
n
[pn (pn
− + + k+
us) + p
⊥ 2]2
n
+ . . . .
(4.11)
By power counting, we see that the first term scales as λ−2 and the second term scales as λ−1 . Although
the first term dominates the second, we need to... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
3-point Feynman rule where the small momentum k is
ignored relative to the large momenta p1 and p2, and that total momentum is not conserved at the vertex.
For the next order term we get
(cid:90)
(cid:90)
¯
dx ψ(x)x(i∂Aus)(0)ψ(x) =
dp1 dp2 dk δ(cid:48)(p1 − p2) k ψ(p1)Aus(k)ψ(p2).
¯
(4.13)
Here the Feynman rule invo... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
Fourier transform ξ˜n(p) of the quark part of our field we have
(cid:90)
˜ξn(p) =
d4x eip·x ˆξn(x).
(4.14)
Now to separate momentum scales, we define our momentum pµ to be a sum of a large momentum
components pµ called the label momentum and a small momentum pµ
r called the residual momentum.
c
µ
µ
µ
p = p + p
r
c ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
integrand is homogeneous in λ.
⊥
⊥
+
µ
In practice the grid picture is a bit misleading, since actually the boxes are infinite and with momentum
components (pc, pr) we are really dealing with a product of continuous spaces R3 × R4/I where I are a
group of relations that remove redundancy order by order in λ. (I incl... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(cid:96),q(cid:96) δ4(pr − qr)(2π)4.
(4.17)
(4.18)
(4.19)
Every collinear field carries both label and residual momenta, they are both conserved at all vertices,
but Feynman rules may depend on only one or the other of these components. For example, what was
previously a nonconservation of momenta for an interaction b... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
n,p£ (x)
26
k,uspplr)(,pplr)(+kus
4.1 SCET Quark Lagrangian
4 SCETI LAGRANGIAN
• Interactions with ultrasoft gluons or quarks leave the label momenta of collinear fields conserved.
• Interactions with collinear gluons or quarks will change label momenta.
• The label n for the collinear direction is preserv... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(x) in terms of label momenta
ˆξn(x) =
(cid:88)
e−ip(cid:96)·x ξn,p(cid:96)(x)
p
(cid:96)=0(cid:54)
= e−iP·x
(cid:88)
ξn,p(cid:96)(x)
p(cid:96)(cid:54)=0
P ξn(x) .
≡ e−i x
(4.25)
In the last line we defined ξn(x) =
. Since the label operator allows us to encode the phase
factor involving label momenta as an operator... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
) ,
(cid:96)
ψ+ −→ ˆξ+(x) =
n
ψ− −→ ˆξn
−(x) =
(cid:88)
pl(cid:54)=0
(cid:88)
p
(cid:96)(cid:54)=0
27
(4.27)
(4.28)
(cid:54)
4.1 SCET Quark Lagrangian
4 SCETI LAGRANGIAN
where both have a θ(p0) = θ(¯n · pc). Because of charge conjugation symmetry it is convenient to combine
the particle and anti-particle... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
For collinear gluons, we proceed analogously to find
ˆAµ
n =
(cid:88)
q(cid:96)(cid:54)=0
e−iq(cid:96)·xAµ
n,q = e−iP·
(cid:96)
xAµ
n(x)
where
Aµ
n(x) =
(cid:88)
q
(cid:96)=0(cid:54)
Aµ
n, q .
(cid:96)
Since the gluon field Aµ
n = An T A where An (x) is real we also have
µA
µA
[AµA
n,q£
(x)] ∗ = An,−q£
µA
(x) .
(... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(x).
(4.35)
(4.35)
In the last line, we can suppress the exponent if we assume that label momenta are always conserved.
Effectively, by introducing the label operator we have replaced the ordinary derivative operation by
i∂µφˆn(x) → (P µ + i∂µ)φn(x).
(4.36)
28
4.1 SCET Quark Lagrangian
4 SCETI LAGRANGIAN... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
2 terms that can be dropped in our leading order analysis
(but later on we will see are required by gauge symmetry when considering power suppressed operators).
Keeping only the lowest order terms, we have the following lagrangian
L(0) = e− ·P ¯ξn
nξ
ix
(cid:16)
in · D + iD/ n
1
⊥ in¯ · Dn
iD/ n
(cid:17) n/¯
⊥ 2
ξn ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
hard scale through the inverse P ∼ λ0 . The fact that there is locality
except at the hard scale is a key feature of SCETI. Some attempts to tweak the formalism described
here, in order to simplify SCET, lead to actions that are non-local at the small scale ∼ λ2 because
they integrate out some onshell particles, whi... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
0 that will be
discussed later on. But this form does make it more clear why the collinear particles share many of
the properties of the full QCD Lagrangian (for example, we have the same renormalization properties
for the gauge coupling).
The computation of the propagator from L is also greatly simplified without t... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
Lagrangian L
In position space the
defining equations for a Wilson line are W (x, x) = 1 and its equation of motion, which we can transform
to momentum space
(0)
nξ .
in¯ · DxW (x, −∞) = 0
(position space)
⇓ Fourier Transform
in¯ · DnWn = (P + gn¯ · An)Wn = 0 .
(4.45)
30
k,uspplr)(,pplr)(+kus,pplr)(,pplr)(+,qq... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
�n · An for the Wilson line Wn. Inverting these results
also gives useful operator identities
also gives useful operator identities
1
1
in¯ · Dn
i¯n · Dn
1
1
= W † Wn ,
= W †
Wn ,
n
n
P
P
1
1
P
P
= Wn
= Wn
1
1
W † .
W †
in¯ · D n .
n
in¯ · Dn
n
The first relation allows us to rewrite L as
(0)
nξ
L(0) = e
nξ
−i... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
c
-n
'
"
Gauge Fixing Term
'
-n
Ghost Term
1
2
"
-n
'
Gauge Kinetic Term
"
(4.52)
where Gµν = [Dµ, Dν ]. Expanding the covariant derivative as we did in the quark sector we keep only
the leading order terms. For a covariant derivative acting on collinear fields the leading order terms are
i
g
iDµ → iDµ = ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
γ
nµ + µ p/⊥
n¯·p
p/(cid:48) γ⊥
+ ⊥ µ
n¯·p (cid:48) − /(cid:48) p/⊥ ⊥
p
n¯
n¯·p n¯·p (cid:48) µ
(cid:21)
¯
n/
2
= ig2 T A T B
n¯·(p−q)
(cid:20)
⊥γν
µ
γ
⊥ − µ
⊥p/⊥
¯n·p ¯nν − p (cid:48)/⊥γ⊥
¯n·p (cid:48) ¯nµ + p (cid:48)/⊥p/⊥
n¯·p n¯·p (cid:48) µn¯ν
n¯
ν
(cid:21)
¯
n/
2
+ ig2 T B T A
n¯·(q+p(cid:48))
⊥γµ
ν
⊥ − γ p/⊥
¯n·... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
2
P + P µ +
⊥
µ
n¯
2
n¯
in · ∂ + gn · Aus.
2
The resulting leading order collinear gluon Lagrangian is then
L(0)
ng =
1
2g2
(cid:8)
Tr ([i µ, i
D D
µ
(cid:9)
])2
+ τ Tr ([i µ , An µ])2 + 2Tr cn[i us, [i µ, cn]]
Dµ D
Dus
(cid:9)
(cid:8)
(cid:8)
(4.54)
(cid:9)
.
(4.55)
For the Langrangian with only ultrasoft quarks ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
ark and gluon collinear modes in the n direction, and quark and gluon ultrasoft modes,
L(0)
= L
(0) + L(0) + L(0)
us
ng
nξ
.
(4.57)
32
μ , Appɂμ , Appɂμ , Aν , Bq 4.4 Feynman Rules for Collinear Quarks and Gluons
4 SCETI LAGRANGIAN
=
−
i
n¯ ·q n·k + q2 + i0
⊥
(cid:18)
µν − (1 − τ )
g
µqν
q
n¯ ·q n·k + q2
⊥
(c... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
the collinear quark and gluon
Lagrangians. We do not show the purely ultrasoft interactions which are identical to those of QCD, nor
do we show the purely collinear gluon interactions which are also identical to those of QCD.
The Feynman rules that follow from the leading order collinear quark Lagrangian are shown i... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
, k)b, νc, λa, μq2q1a, μb, νc, λd, ρa, μb, νc, λd, ρ 4.5 Rules for Combining Label and Residual Momenta in Amplitudes
4 SCETI LAGRANGIAN
Instead of this, we need to use a Continuum EFT picture where the EFT modes have propagators that
extend over all momenta, but integrands which obtain their key contribution from t... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(pc, pr) ∈ Rd−1 × Rd . The upshot is that in the simplest
cases the residual momentum can simply be dropped or absorbed into a label momentum in the same
direction (making it continuous), while in the most complicated cases the formalism leads to so-called 0
bin subtractions for collinear integrands. These subtracti... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
n¯ · pc
−
and pc⊥, and a single residual momentum p . This amounts to picking βµ above to contain the full pr
and pµ components. The onshell condition for the collinear particles is then simply p − p − pp 2 = 0.
All propagators for intermediate collinear and ultrasoft lines are then simply determined by momentum
cons... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
a qr that this will be
equal to an integration over all of the qµ momentum space, since it does not depend on how we divide the
momentum into the two components. For notational convenience we denote the label space integration as
a sum rather than an integral. In d-dimensions we have
(cid:90)
(cid:90)
(cid:88)
ddqr... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
step we use the fact that F is constant throughout each box in the grid picture of Fig. 5 so its
the same with the first two arguments shifted by residual momenta. (In the continuum EFT picture its
the same property, F does not depend on residual momenta in these components.) In the final equality we
then combined the... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
will be dependence on the residual momentum components from higher
order terms in the multipole expansion of the collinear propagators. If these terms correspond to the
momentum components q− and q⊥ that do not appear inside any ultrasoft propagators then the resulting
integration is zero
r
r
2) :
(cid:88) (cid:90... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(cid:90)
(cid:90)
(cid:90)
d k F (q−, q⊥, q , k ) ,
d
+
µ
(4.62)
ddqr
ddkr F (q−, q
r , kµ
⊥, +
q
(cid:96)
(cid:96)
(cid:90)
d
r ) = d q
q(cid:96)
which in general is nonzero. This integrand corresponds to a mixed two-loop diagram with one loop
momentum with collinear scaling and one with ultrasoft scaling.
4After im... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
easy to determine what the set of restrictions are for any diagram, since we have
one such condition for every collinear propagator. At leading order in λ only the zero-bin subtractions
corresponding to collinear gluon propagators can give non-zero contributions since operators containing
an ultrasoft quark together... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
∼ λ2, expanding, and keeping the
dominant and any sub-dominant scaling terms up to those that are the same order in λ as the original
loop integration. If the original integrand F ∼ λ−4, then this corresponds to keeping just the terms up to
F 0 ∼ λ−8, which is often the leading term. (Together with the standard scal... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
of zero-bin subtractions that avoids the use of notation like q£ = 0 is to note that in
a theory with both collinear and ultrasoft modes, each collinear propagator is actually a distribution, like a generalized +
function, that induces these subtraction terms. The fact that we drop higher order terms in the λ expansi... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
Introduction to Robotics, H. Harry Asada
1
Chapter 6
Statics
Robots physically interact with the environment through mechanical contacts. Mating
work pieces in a robotic assembly line, manipulating an object with a multi-fingered hand, and
negotiating a rough terrain through leg locomotion are just a few examples ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
the mass centroid Ci is denoted mig, where
mi is the mass of link i and g is the 3x1 vector representing the acceleration of gravity. The
balance of linear forces is then given by
f
i
,1
i
−
f
− +
ii
1,
+
mi
g
=
0
,
i
(cid:34)=
,1
,
n
(6.1.1)
Note that all the vectors are defined with respect to the base coordinate... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Figure 6.1.1 Free body diagram of the i-th link
The force
,1−f
i
i
and moment
N
i
,1−
i
are called the coupling force and moment between
. These are
the adjacent links i and i-1. For i=1, the coupling force and moment are
interpreted as the reaction force and moment applied to the base link to which the arm mec... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
accommodate in order to
perform a given task. Thus, we specify this pair of coupling force and moment, and solve the
, +nnN
simultaneous equations. For convenience we combine the force
1
to define the following six-dimensional vector:
, are the force and moment that the end-effecter applies to the
and the moment ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
. One is to
separate the net force or moment generating energy from the constraint forces and moments
irrelevant to energy. Second, we need to find independent displacement variables that are
geometrically admissible satisfying kinematic relations among the links.
Figure 6.2.1 shows the actuator torques and the cou... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
−f
i
i
,1−N
i
i
Revolute Joint
Oi-1
Link i
Actuator i+1
Joint i+1
Prismatic
Joint
1, +iiN
Link i+1
Joint i
1+iτ
Link i-1
1, +iif
Actuator i
Oi
z
O
x
y
1−ib
ib
Figure 6.2.1 Joint torques as components of coupling force and moment
We combine all the joint torques from joint 1 through joint n to define th... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
�F
ℜ∈τ
are given by
16 x
1×n
τ
= T
FJ
⋅
(6.2.4)
where J is the 6 x n Jacobian matrix relating infinitesimal joint displacements dq to infinitesimal
end-effecter displacements dp:
d
qJp
⋅=
d
(6.2.5)
Note that the joint torques in the above expression do not account for gravity and friction.
They are the net torqu... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Introduction to Robotics, H. Harry Asada
6
δ
Work
=
q
q
+
δτδτ
1
⋅
⋅
1
2
=
T
T
pFqτ
−
δ
δ
+
(cid:34)
+
q
⋅
δτ
n
n
2
−
f
nn
,
T
1
+
⋅
δ
Nx
−
e
T
1
+
nn
,
φ
⋅
δ
e
(6.2.6)
vanishes for arbitrary virtual displacements that conform to geometric
According to the principle of virtual work, the linkage system is in equilib... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
y
2ττ=τ
(
1
T . Assume no friction.
needed for pushing the surface with an endpoint
(=F
)
,
The Jacobian matrix relating the end-effecter coordinates
x
e and
2θ has been obtained in the previous chapter:
1θ and
displacements
ey
to the joint
J
=
−
(cid:65)
⎛
⎜⎜
⎝
(cid:65)
sin
1
cos
(cid:65)
θ
−
1
(cid:65)
θ
+
1... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Introduction to Robotics, H. Harry Asada
7
y
Endpoint
Force
F
⎛
⎞
x
⎜
⎟
⎜
⎟
F
⎝
⎠
y
2τ
2θ
1τ
1θ
O
x
Figure 6.2.3 Two-dof articulated robot pushing the environment surface
6.3 Duality of Differential Kinematics and Statics
We have found that the equivalent joint torques are related to the endpoint force by the... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
, generating a mapping from the m-dimensional vector
space Vm, associated with the Cartesian coordinates of the end-effecter, to the n-dimensional
vector space Vn, associated with the joint coordinates. Therefore the joint torques τ are always
determined uniquely for any arbitrary endpoint force F. However, for give... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
figure. Also, space S3, i.e., the orthogonal complement of R(JT) is identical to N(J). What this
implies is that, in the direction in which joint velocities do not cause any end-effecter velocity, the
joint torques cannot be balanced with any endpoint force. In order to maintain a stationary
configuration, the joint... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
A
Link 4
4δθ
x
δ
⎛
e
⎜⎜
y
δ
⎝
e
⎞
⎟⎟
⎠
Joint 4
Endpoint
Force
F
⎞
x
⎟
⎟
F
⎠
y
⎛
⎜
⎜
⎝
Link 0
1τ
3τ
Joint 1
Joint 3
3δθ
Link 3
x
Figure 6.4.1 Five-bar-link robot exerting endpoint force
We begin by revisiting the five-bar-link planar robot shown in Figure 6.4.1. This robot
has two degrees of freedom, compris... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
-effecter are
uniquely determined by the virtual displacements of Joints 1 and 3. In fact, the former is related
to the latter via the Jacobian matrix:
J
=
⎛
⎜
⎜
⎜
⎜
⎝
1
x
x
∂
∂
e
e
θθ
∂
∂
3
y
y
∂
∂
e
θθ
∂
∂
3
1
e
⎞
⎟
⎟
⎟
⎟
⎠
(6.4.3)
Using this Jacobian,
Department of Mechanical Engineering
Massachusetts Institut... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
.
(6.4.7)
Then the equivalent joint torques
by
τ
= T
FJ
⋅
τ
1n
×ℜ∈
to bear an arbitrary endpoint force
F
1m
×ℜ∈
is given
(6.4.8)
Note that the joint coordinates associated with the active joints are not necessarily
generalized coordinates that uniquely locate the system. For example, the arm configuration of ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
e
11
(6.5.1)
Note that only three virtual displacements of the four joint angles are independent. There exists a
differential relationship between one of the joints, say
kinematic constraint. Let us write it as
4θ , and the other three due to the
δθ
4
= c
J δ
q
⋅
(6.5.2)
(
δθ
1
δθ
3
T
)
δ =q
where
constraint du... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Introduction to Robotics, H. Harry Asada
12
Exercise 6.2
Define geometric parameters needed in Figure 6.5.1, and obtain the two Jacobian
matrices associated with the two-fingered hand holding an object. Furthermore, obtain the grasp
force using the Jacobian matrices and the joint torques.
Department of Mechanical ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Lecture 9 (On-Line Video)
The Hydrogen Atom
Today’s Program:
1. Angular momentum, classical and quantum mechanical.
2. The Hydrogen atom semi-classical approach.
3. The Hydrogen atom quantum mechanical approach.
4. Eigenfunctions and eigenvalues common to Hˆ , Lˆ2 and Lˆ
z .
5. The radial dependence.
6. Energy... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
�
2
p2
e
2 r
Where:
m m
e
mme
p
p me is a reduced mass.
Semiclassical Bohr model that yielded correct energy (but produced generally incorrect results)
relied on the conservation of energy, angular momentum and the balance of forces.
1
E v
2
2
e2
r
Energy
2
v
r
e2
2 Balanc... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
rˆ
1, pˆ
2, pˆ
1, rˆ
2
2
pˆ1
2m1
2
pˆ2
2m2
Vˆ
2 rˆ
rˆ
1
Let’s perform the following coordinate transform – break our system into a relative motion of
electron and a nucleus the motion of the center of mass:
ˆ
R
CM
2
rˆ
1 m2rˆ
m1
m1 m2
ˆ
rˆ ˆ
r1 r2
ˆ
CM p1 pˆ
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
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