Lecture 12: Dynamics: Euler-Lagrange Equations Two-Link Cartesian Manipulator For this system we need • to solve forward kinematics problem; • to compute manipulator Jacobian;

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