lecture 12: dynamics: euler-lagrange equations two-link cartesian manipulator for this system we...
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Lecture 12: Dynamics: Euler-Lagrange Equations
• Examples
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 1/17
Lecture 12: Dynamics: Euler-Lagrange Equations
• Examples
• Properties of Equations of Motion
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 1/17
Example: Two-Link Cartesian Manipulator
For this system we need• to solve forward kinematics problem;• to compute manipulator Jacobian;• to compute kinetic and potential energies and the
Euler-Lagrange equationsc©Anton Shiriaev. 5EL158: Lecture 12 – p. 2/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
are
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 3/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = 0, d = q1, a = 0, α = −π
2
T 12 : θ = 0, d = q2, a = 0, α = 0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 3/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = 0, d = q1, a = 0, α = −π
2
T 12 : θ = 0, d = q2, a = 0, α = 0
The kinetic energy of the system is
K = 12
[m1v2
c1 + ωT
1I1ω1
]+ 1
2
[m2v2
c2 + ωT
2I2ω2
]
and
vc1 =[
J(1)v1 J
(2)v1
][
q1
q2
]
= J(1)v1 q1 + J
(2)v1 q2
vc2 =[
J(1)v2 J
(2)v2
][
q1
q2
]
= J(1)v2 q1 + J
(2)v2 q2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 3/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = 0, d = q1, a = 0, α = −π
2
T 12 : θ = 0, d = q2, a = 0, α = 0
The kinetic energy of the system is
K = 12
[m1v2
c1 + ωT
1I1ω1
]+ 1
2
[m2v2
c2 + ωT
2I2ω2
]
and
ω1 =[
J(1)ω1
J(2)ω1
][
q1
q2
]
= J(1)ω1
q1 + J(2)ω1
q2
ω2 =[
J(1)ω2
J(2)ω2
][
q1
q2
]
= J(1)ω2
q1 + J(2)ω2
q2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 3/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = 0, d = q1, a = 0, α = −π
2
T 12 : θ = 0, d = q2, a = 0, α = 0
To compute the Jacobian we can use the DH-frames, i.e
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
J(i)ω =
0, for prismatic joint
z0i−1, for revolute joint
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 3/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = 0, d = q1, a = 0, α = −π
2
T 12 : θ = 0, d = q2, a = 0, α = 0
To compute the Jacobian we can use the DH-frames, i.e
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
⇒ Jv1 =[~z00, 0
]=
0
0
1
,
0
0
0
, Jv2 =
[~z00, ~z0
1
]=
0
0
1
,
0
1
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 3/17
Forward Kinematics and Jacobian (Cont’d)
To sum up:
• Angular velocities ω1 and ω2 of both links are zeros
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 4/17
Forward Kinematics and Jacobian (Cont’d)
To sum up:
• Angular velocities ω1 and ω2 of both links are zeros
• Linear velocities of centers of mass are
vc1 =[
J(1)v1 , J
(2)v1
][
q1
q2
]
=
0
0
1
q1 +
0
0
0
q2
vc2 =[
J(1)v2 , J
(2)v2
][
q1
q2
]
=
0
0
1
q1 +
0
1
0
q2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 4/17
Forward Kinematics and Jacobian (Cont’d)
To sum up:
• Angular velocities ω1 and ω2 of both links are zeros
• Linear velocities of centers of mass are
vc1 =[
J(1)v1 , J
(2)v1
][
q1
q2
]
=
0
0
1
q1 +
0
0
0
q2
vc2 =[
J(1)v2 , J
(2)v2
][
q1
q2
]
=
0
0
1
q1 +
0
1
0
q2
• The kinetic energy is
K = 12m1v2
c1+12m2v2
c2 = 12
[
q1
q2
]T
[
m1 + m2 0
0 m2
] [
q1
q2
]
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 4/17
Potential Energy (PE) for Two-Link Cartesian Manipulator
Observations• PE is independent of the second link position;• It depends on the height of center of mass of robot;
• P = g · (m1 + m2) · q1 + Const
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 5/17
Euler-Lagrange Equations for 2-Link Cartesian Manipulator
Given the kinetic K and potential P energies, the dynamics are
d
dt
[∂(K − P)
∂q
]
−∂(K − P)
∂q= τ
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 6/17
Euler-Lagrange Equations for 2-Link Cartesian Manipulator
Given the kinetic K and potential P energies, the dynamics are
d
dt
[∂(K − P)
∂q
]
−∂(K − P)
∂q= τ
With kinetic and potential energies
K = 12
[
q1
q2
]T
[
m1 + m2 0
0 m2
] [
q1
q2
]
, P = g (m1 + m2) q1+C
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 6/17
Euler-Lagrange Equations for 2-Link Cartesian Manipulator
Given the kinetic K and potential P energies, the dynamics are
d
dt
[∂(K − P)
∂q
]
−∂(K − P)
∂q= τ
With kinetic and potential energies
K = 12
[
q1
q2
]T
[
m1 + m2 0
0 m2
] [
q1
q2
]
, P = g (m1 + m2) q1+C
the Euler-Lagrange equations are
(m1 + m2)q1 + g(m1 + m2) = τ1
m2q2 = τ2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 6/17
Example: Planar Elbow Manipulator
For this system we need• to compute forward kinematics and manipulator Jacobian;• to compute kinetic and potential energies and the
Euler-Lagrange equationsc©Anton Shiriaev. 5EL158: Lecture 12 – p. 7/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
are
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 8/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = q1, d = 0, a = l1, α = 0
T 12 : θ = q2, d = 0, a = l2, α = 0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 8/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = q1, d = 0, a = l1, α = 0
T 12 : θ = q2, d = 0, a = l2, α = 0
The kinetic energy of the system is
K = 12
[m1v2
c1 + ωT
1I1ω1
]+ 1
2
[m2v2
c2 + ωT
2I2ω2
]
and
vc1 =[
J(1)v1 J
(2)v1
][
q1
q2
]
= J(1)v1 q1 + J
(2)v1 q2
vc2 =[
J(1)v2 J
(2)v2
][
q1
q2
]
= J(1)v2 q1 + J
(2)v2 q2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 8/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = q1, d = 0, a = l1, α = 0
T 12 : θ = q2, d = 0, a = l2, α = 0
The kinetic energy of the system is
K = 12
[m1v2
c1 + ωT
1I1ω1
]+ 1
2
[m2v2
c2 + ωT
2I2ω2
]
and
ω1 =[
J(1)ω1
J(2)ω1
][
q1
q2
]
= J(1)ω1
q1 + J(2)ω1
q2
ω2 =[
J(1)ω2
J(2)ω2
][
q1
q2
]
= J(1)ω2
q1 + J(2)ω2
q2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 8/17
Forward Kinematics and Jacobian
DH parameters for computing homogeneous transformations
T (qi) = Rotz,θ · Transz,d · Transx,a · Rotx,α
areT 0
1 : θ = q1, d = 0, a = l1, α = 0
T 12 : θ = q2, d = 0, a = l2, α = 0
To compute the Jacobian we can use the DH-frames, i.e
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
J(i)ω =
0, for prismatic joint
z0i−1, for revolute joint
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 8/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
gives
J(1)v1 = ~z0 × (~oc1
− ~o0) =
0
0
1
×
lc1 cos q1
lc1 sin q1
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 9/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
gives
J(1)v1 = ~z0 × (~oc1
− ~o0) =
0
0
1
×
lc1cos q1
lc1sin q1
0
=
−lc1sin q1
lc1cos q1
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 9/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
gives
J(1)v1 = ~z0 × (~oc1
− ~o0) =
0
0
1
×
lc1cos q1
lc1sin q1
0
=
−lc1sin q1
lc1cos q1
0
J(1)v2 = ~z0 × (~oc2 − ~o0)
=
0
0
1
×
l1 cos q1
l1 sin q1
0
+
lc2cos(q1 + q2)
lc2sin(q1 + q2)
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 9/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
gives
J(1)v1 = ~z0 × (~oc1
− ~o0) =
0
0
1
×
lc1cos q1
lc1sin q1
0
=
−lc1sin q1
lc1cos q1
0
J(1)v2 = ~z0 × (~oc2 − ~o0) =
−l1 sin q1 − lc2sin(q1 + q2)
l1 cos q1 + lc2cos(q1 + q2)
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 9/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
gives
J(1)v1 = ~z0 × (~oc1
− ~o0) =
0
0
1
×
lc1cos q1
lc1sin q1
0
=
−lc1sin q1
lc1cos q1
0
J(1)v2 = ~z0 × (~oc2 − ~o0) =
−l1 sin q1 − lc2sin(q1 + q2)
l1 cos q1 + lc2cos(q1 + q2)
0
J(2)v2 = ~z1 × (~oc2 − ~o1) =
0
0
1
×
lc2cos(q1 + q2)
lc2sin(q1 + q2)
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 9/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)v =
z0i−1, for prismatic joint
z0i−1 ×
[
o0c − o0
i−1
]
, for revolute joint
gives
J(1)v1 = ~z0 × (~oc1
− ~o0) =
0
0
1
×
lc1cos q1
lc1sin q1
0
=
−lc1sin q1
lc1cos q1
0
J(1)v2 = ~z0 × (~oc2 − ~o0) =
−l1 sin q1 − lc2sin(q1 + q2)
l1 cos q1 + lc2cos(q1 + q2)
0
J(2)v2 = ~z1 × (~oc2 − ~o1) =
−lc2sin(q1 + q2)
lc2cos(q1 + q2)
0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 9/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)ω =
0, for prismatic joint
z0i−1, for revolute joint
gives
J(1)ω1
= ~z0 =
0
0
1
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 10/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)ω =
0, for prismatic joint
z0i−1, for revolute joint
gives
J(1)ω1
= ~z0 =
0
0
1
J(1)ω2
= ~z0 =
0
0
1
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 10/17
Forward Kinematics and Jacobian (Cont’d)
The formula
J(i)ω =
0, for prismatic joint
z0i−1, for revolute joint
gives
J(1)ω1
= ~z0 =
0
0
1
J(1)ω2
= ~z0 =
0
0
1
J(2)ω2
= ~z1 =
0
0
1
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 10/17
Forward Kinematics and Jacobian (Cont’d)
To sum up, the kinetic energy K is
K = 12
[m1v2
c1 + ωT
1I1ω1
]+ 1
2
[m2v2
c2 + ωT
2I2ω2
]
= 12
[
m1
(
J(1)v1
q1
)2+ I1
(
J(1)ω1
q1
)2]
+
+12
[
m2
(
J(1)v2
q1 + J(2)v2
q2
)2+ I2
(
J(1)ω2
q1 + J(2)ω2
q2
)2]
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 11/17
Forward Kinematics and Jacobian (Cont’d)
To sum up, the kinetic energy K is
K = 12
[m1v2
c1 + ωT
1I1ω1
]+ 1
2
[m2v2
c2 + ωT
2I2ω2
]
= 12
[
m1
(
J(1)v1
q1
)2+ I1
(
J(1)ω1
q1
)2]
+
+12
[
m2
(
J(1)v2
q1 + J(2)v2
q2
)2+ I2
(
J(1)ω2
q1 + J(2)ω2
q2
)2]
= 12
q1
q2
T
d11 d12
d12 d22
q1
q2
with
d11 = m1l2c1
+ m2
(l21 + l2c2
+ 2l1lc2cos q2
)+ I1 + I2
d12 = m2
(l2c2
+ l1lc2cos q2
)+ I2
d22 = m2l2c2
+ I2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 11/17
Potential Energy (PE) for Two-Link Elbow Manipulator
• PE of the 1st link is P1 = m1gyc1= m1glc1
sin q1
• PE of the 2nd link isP2 = m1gyc2
= m2g (l1 sin q1 + lc2sin(q1 + q2))
• Total PE is P1 + P2
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 12/17
Lecture 12: Dynamics: Euler-Lagrange Equations
• Examples
• Properties of Equations of Motion
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 13/17
Passivity Relation
Given a mechanical system
d
dt
[∂L
∂q
]
−∂L
∂q= τ ⇔ D(q)q + C(q, q)q + g(q) = τ
withL = 1
2qT D(q)q − P (q)
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 14/17
Passivity Relation
Given a mechanical system
d
dt
[∂L
∂q
]
−∂L
∂q= τ ⇔ D(q)q + C(q, q)q + g(q) = τ
withL = 1
2qT D(q)q − P (q)
Its energy is given by
H = 12qT D(q)q + P (q)
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 14/17
Passivity Relation
Given a mechanical system
d
dt
[∂L
∂q
]
−∂L
∂q= τ ⇔ D(q)q + C(q, q)q + g(q) = τ
withL = 1
2qT D(q)q − P (q)
Its energy is given by
H = 12qT D(q)q + P (q)
What will happen with ddt
H?
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 14/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
= qT D(q)q + 12qT d
dt[D(q)] q + qT
∂P
∂q
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
= qT D(q)q + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT [τ − C(q, q)q − g(q)] + 12qT d
dt[D(q)] q + qT
∂P
∂q
Here we use the Euler-Lagrange equations
D(q)q + C(q, q)q + g(q) = τ
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
= qT D(q)q + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT [τ − C(q, q)q − g(q)] + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT τ + qT
(12
ddt
[D(q)] − C(q, q))
q + qT
(∂P
∂q− g(q)
)
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
= qT D(q)q + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT [τ − C(q, q)q − g(q)] + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT τ + qT
(12
ddt
[D(q)] − C(q, q))
q + qT
(∂P
∂q− g(q)
)
︸ ︷︷ ︸
= 0
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
= qT D(q)q + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT [τ − C(q, q)q − g(q)] + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT τ + qT
(12
ddt
[D(q)] − C(q, q))
︸ ︷︷ ︸
= 0
q
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
Differentiating H along a solution of the system, we have
ddt
H = ddt
[12qT D(q)q + P (q)
]
= 12qT D(q)q + 1
2qT D(q)q + 1
2qT d
dt[D(q)] q + qT
∂P
∂q
= qT D(q)q + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT [τ − C(q, q)q − g(q)] + 12qT d
dt[D(q)] q + qT
∂P
∂q
= qT τ + qT
(12
ddt
[D(q)] − C(q, q))
︸ ︷︷ ︸
= 0
q
= qT τ
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 15/17
Passivity Relation (Cont’d)
The differential relationddt
H = qT τ
can be integrated, so that
∫ T
0
ddt
H(q(t), q(t))dt = H(q(T ), q(T )) − H(q(0), q(0))
=
∫ T
0q(t)T τ (t)dt
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 16/17
Passivity Relation (Cont’d)
The differential relationddt
H = qT τ
can be integrated, so that
∫ T
0
ddt
H(q(t), q(t))dt = H(q(T ), q(T )) − H(q(0), q(0))
=
∫ T
0q(t)T τ (t)dt
⇒∫ T
0 q(t)τ (t)dt ≥ −H(q(0), q(0))
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 16/17
Passivity Relation (Cont’d)
The differential relationddt
H = qT τ
can be integrated, so that
∫ T
0
ddt
H(q(t), q(t))dt = H(q(T ), q(T )) − H(q(0), q(0))
=
∫ T
0q(t)T τ (t)dt
⇒∫ T
0 q(t)τ (t)dt ≥ −H(q(0), q(0))
These relations are called• passivity (dissipativity) relation• passivity (dissipativity) relation in the integral form
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 16/17
Skew Symmetry of D(q) − C(q, q)
To check that
N = ddt
[D(q)] − 2C(q, q), NT = −N
look at (k, j)th-component
ddt
dkj − 2ckj =
n∑
i=1
∂dkj
∂qi
qi −
n∑
i=1
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]
qi
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 17/17
Skew Symmetry of D(q) − C(q, q)
To check that
N = ddt
[D(q)] − 2C(q, q), NT = −N
look at (k, j)th-component
ddt
dkj − 2ckj =
n∑
i=1
∂dkj
∂qi
qi −
n∑
i=1
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]
qi
=
n∑
i=1
{∂dkj
∂qi
−
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]}
qi
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 17/17
Skew Symmetry of D(q) − C(q, q)
To check that
N = ddt
[D(q)] − 2C(q, q), NT = −N
look at (k, j)th-component
ddt
dkj − 2ckj =
n∑
i=1
∂dkj
∂qi
qi −
n∑
i=1
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]
qi
=
n∑
i=1
{∂dkj
∂qi
−
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]}
qi
=n∑
i=1
{∂dij
∂qk
−∂dki
∂qj
}
qi
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 17/17
Skew Symmetry of D(q) − C(q, q)
To check that
N = ddt
[D(q)] − 2C(q, q), NT = −N
look at (k, j)th-component
ddt
dkj − 2ckj =
n∑
i=1
∂dkj
∂qi
qi −
n∑
i=1
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]
qi
=
n∑
i=1
{∂dkj
∂qi
−
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]}
qi
=n∑
i=1
{∂dij
∂qk
−∂dki
∂qj
}
qi =n∑
i=1
{∂dji
∂qk
−∂dki
∂qj
}
qi
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 17/17
Skew Symmetry of D(q) − C(q, q)
To check that
N = ddt
[D(q)] − 2C(q, q), NT = −N
look at (k, j)th-component
ddt
dkj − 2ckj =
n∑
i=1
∂dkj
∂qi
qi −
n∑
i=1
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]
qi
=
n∑
i=1
{∂dkj
∂qi
−
[∂dkj
∂qi
+∂dki
∂qj
−∂dij
∂qk
]}
qi
=n∑
i=1
{∂dij
∂qk
−∂dki
∂qj
}
qi =n∑
i=1
{∂dji
∂qk
−∂dki
∂qj
}
qi
⇒ nkj = −njk
c©Anton Shiriaev. 5EL158: Lecture 12 – p. 17/17