Interconnected rigid bodies: Dynamic modelling - …banavar/resources/notes/sneha... · Rigid body dynamics Symmetry breaking potentials Rolling constraints Interconnected rigid bodies:

Rigid body dynamics Symmetry breaking potentials Rolling constraints

Interconnected rigid bodies: Dynamic modelling

Sneha Gajbhiye and Ravi N Banavar([email protected]) 1

1Systems and Control Engineering,IIT Bombay, India

Research Symposium, ISRO-IISc Space Technology Cell,IISc, Bengaluru, February 22 - 27, 2016.

February 22, 2016

February, 2016 STC-IISc Workshop


Outline

1 Rigid body dynamics

2 Symmetry breaking potentials

3 Rolling constraints



Outline






Rotational Rigid body

x3X3

x2

X2

x1

X1

p

• xp

(t) 2 R3- position of a particle p in thebody in the spatial coordinate system

• Xp

- position of a particle p in bodyreference frame

• Configuration space of the body at timet is determined by a rotation matrix R(t)

Obody

R�! Ospatial

• the map X 7! x = RX is called thebody � to� space map

• Rotation matrices belong to a set ofspecial orthogonal group and have afollowing property

•SO(n) = {R 2 GL(n,R)|RRT = I

n

, det(R) = 1}• Configuration space Q of a rotational rigid body is SO(3)



Rotational and translational rigid body

x1

x2

x3

X1

X2

X3

r

Ospatial

Obody

Configurationof rigid body

(Obody

�Ospatial

) 2 R3

[X1 | X2 | X3 ] 2 SO(3)

• Q = SO(3)⇥ R3 for a single rigid body

• For k rigid bodies,

Qf

= (SO(3)⇥ R3)⇥ · · ·⇥ (SO(3)⇥ R3)| {z }kcopies

• This is a mechanical system without constraints.



Rotational and translational rigid body

x1

x2

x3

X1

X2

X3

r

Ospatial

Obody

Configurationof rigid body

(Obody

�Ospatial

) 2 R3

[X1 | X2 | X3 ] 2 SO(3)

• Q = SO(3)⇥ R3 for a single rigid body

• For k rigid bodies,

Qf

= (SO(3)⇥ R3)⇥ · · ·⇥ (SO(3)⇥ R3)| {z }kcopies

• This is a mechanical system without constraints.



Interconnected rigid bodies

• Many mechanical systems consist of rigid bodies which areinterconnected.

• An interconnected mechanical system is a collection of rigid bodiesrestricted to move on a submanifold Q of Q

f

. The manifold Q is calleda configuration manifold.

• Coordinates of Q are denoted by (q1, · · · , qn) are called “generalizedcoordinates”.

• And for i 2 (1, · · · , k), ⇧i

: Q ! SO(3)⇥ R3 gives the configuration ofith body.






f



• And for i 2 (1, · · · , k), ⇧i







f



• And for i 2 (1, · · · , k), ⇧i







f



• And for i 2 (1, · · · , k), ⇧i







f



• And for i 2 (1, · · · , k), ⇧i




Example:Two link manipulator

• Q = SO(2)⇥ SO(2)

• Coordinates (✓1, ✓2)

• ⇧1(✓1, ✓2) = (R1, r1) and

• ⇧1(✓1, ✓2) = (R1, r1), where

R1 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 , R2 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 ,

r1 = l1R1s1, r2 = l1R1s1 + (l1 + l2)R2s1




• Q = SO(2)⇥ SO(2)


• ⇧1(✓1, ✓2) = (R1, r1) and

• ⇧1(✓1, ✓2) = (R1, r1), where

R1 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 , R2 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 ,

r1 = l1R1s1, r2 = l1R1s1 + (l1 + l2)R2s1




• Q = SO(2)⇥ SO(2)


• ⇧1(✓1, ✓2) = (R1, r1) and

• ⇧1(✓1, ✓2) = (R1, r1), where

R1 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 , R2 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 ,

r1 = l1R1s1, r2 = l1R1s1 + (l1 + l2)R2s1




• Q = SO(2)⇥ SO(2)


• ⇧1(✓1, ✓2) = (R1, r1) and

• ⇧1(✓1, ✓2) = (R1, r1), where

R1 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 , R2 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 ,

r1 = l1R1s1, r2 = l1R1s1 + (l1 + l2)R2s1




• Q = SO(2)⇥ SO(2)


• ⇧1(✓1, ✓2) = (R1, r1) and

• ⇧1(✓1, ✓2) = (R1, r1), where

R1 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 , R2 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 ,

r1 = l1R1s1, r2 = l1R1s1 + (l1 + l2)R2s1




• Q = SO(2)⇥ SO(2)


• ⇧1(✓1, ✓2) = (R1, r1) and

• ⇧1(✓1, ✓2) = (R1, r1), where

R1 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 , R2 =

2

4cos ✓1 � sin ✓1 0sin ✓1 cos ✓1 00 0 1

3

5 ,

r1 = l1R1s1, r2 = l1R1s1 + (l1 + l2)R2s1



Lagrangian Mechanics

• The set of all possible configurations of a mechanical system is asmooth manifold Q and velocities is the tangent bundle TQ.

• The Lagrangian is a map L : TQ �! R• The equations of motion on TQ are given by the principle of leastaction applied to a Lagrangian function L.

Symmetry

• The Lagrangian function is invariant under a Lie group action

L(g · q) = L(q) 8q 2 TQ, 8g 2 G,

where G is a Lie group.

• Dynamics on a reduced space

• Reconstruction





• The Lagrangian is a map L : TQ �! R

• The equations of motion on TQ are given by the principle of leastaction applied to a Lagrangian function L.

Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,



• Reconstruction






Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,



• Reconstruction






Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,



• Reconstruction






Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,



• Reconstruction






Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,



• Reconstruction



Lagrangian

• Rigid body B undergoing motion t 7! R(t)

• Rigid body B with ⇢(X) as the density of the body

• Mass : m =RB ⇢(X) d3X

• Lagrangian of the rigid body is equal to rotational kinetic energy

L(R, R) = Krot

=12

Z

B⇢(X)kxk2 d3X =

12

Z

B⇢(X)k ˙RXk2 d3X

=12

Z

B⇢(X)kRXk2 d3X

• Spatial angular velocity: t 7! b!(t) = R(t)R�1(t)

• Body angular velocity: t 7! b⌦(t) = R�1(t)R(t)



Lagrangian





L(R, R) = Krot

=12

Z

B⇢(X)kxk2 d3X =

12

Z

B⇢(X)k ˙RXk2 d3X

=12

Z

B⇢(X)kRXk2 d3X





Lagrangian





L(R, R) = Krot

=12

Z

B⇢(X)kxk2 d3X =

12

Z

B⇢(X)k ˙RXk2 d3X

=12

Z

B⇢(X)kRXk2 d3X





Lagrangian





L(R, R) = Krot

=12

Z

B⇢(X)kxk2 d3X =

12

Z

B⇢(X)k ˙RXk2 d3X

=12

Z

B⇢(X)kRXk2 d3X





• The tangent vector (R, R) translated to so(3) (which is equal toTI

SO(3)), by the tangent lift of left/right multiplication by R�1

TR

LR

�1

(R, R) = (I, R�1R), TR

LR

�1

(R, R) = (I, RR�1)

• Both b! and b⌦ lie in so(3) ) define !,⌦ 2 R3 by the rule2

40 �c bc 0 �a�b a 0

3

5 (a, b, c)

• Adjoint map

ad :so(3)⇥ so(3) ! so(3)

(bx, by) 7! adbxby = (x⇥ y)^

• Dual of adjoint map

ad⇤ :so(3)⇥ so

⇤(3) ! so

⇤(3)

(bx, ✓) 7! ad⇤bx✓ = �(x⇥ ✓)^





TR

LR

�1

(R, R) = (I, R�1R), TR

LR

�1

(R, R) = (I, RR�1)


40 �c bc 0 �a�b a 0

3

5 (a, b, c)

• Adjoint map

ad :so(3)⇥ so(3) ! so(3)



ad⇤ :so(3)⇥ so

⇤(3) ! so

⇤(3)

(bx, ✓) 7! ad⇤bx✓ = �(x⇥ ✓)^





TR

LR

�1

(R, R) = (I, R�1R), TR

LR

�1

(R, R) = (I, RR�1)


40 �c bc 0 �a�b a 0

3

5 (a, b, c)

• Adjoint map

ad :so(3)⇥ so(3) ! so(3)



ad⇤ :so(3)⇥ so

⇤(3) ! so

⇤(3)

(bx, ✓) 7! ad⇤bx✓ = �(x⇥ ✓)^





TR

LR

�1

(R, R) = (I, R�1R), TR

LR

�1

(R, R) = (I, RR�1)


40 �c bc 0 �a�b a 0

3

5 (a, b, c)

• Adjoint map

ad :so(3)⇥ so(3) ! so(3)



ad⇤ :so(3)⇥ so

⇤(3) ! so

⇤(3)

(bx, ✓) 7! ad⇤bx✓ = �(x⇥ ✓)^



LagrangianL : TQ ! RL(R, R)

Group action

SO(3)⇥ SO(3)L�! SO(3)

R 7! LR

�1

R = R�1R

Reduced LagrangianL(TL

R

�1

(R, R))= L(R�1R,R�1R)

= l(I, ⌦)

•

Krot

=12

Z

B⇢(X)kR�1RXk2 d3X,

=12

Z

B⇢(X)kb⌦Xk2 d3X =

12

Z

B⇢(X)k bX⌦k2 d3X,

=12⌦T

✓Z

B⇢(X) bXT bX d3X

◆⌦

• Identity concerning hat map bXT bX = kXk2I = XTX, looks remarkablylike a moment of inertia tensor

I =✓Z

B⇢(X)kXk2I = XTX d3X

◆

• Reduced Lagrangian l(⌦) = 12⌦T I⌦




Group action

SO(3)⇥ SO(3)L�! SO(3)

R 7! LR

�1

R = R�1R


R

�1

(R, R))= L(R�1R,R�1R)

= l(I, ⌦)•

Krot

=12

Z


=12

Z


12

Z


=12⌦T

✓Z

B⇢(X) bXT bX d3X

◆⌦


I =✓Z


◆





Group action

SO(3)⇥ SO(3)L�! SO(3)

R 7! LR

�1

R = R�1R


R

�1

(R, R))= L(R�1R,R�1R)

= l(I, ⌦)•

Krot

=12

Z


=12

Z


12

Z


=12⌦T

✓Z

B⇢(X) bXT bX d3X

◆⌦


I =✓Z


◆





Group action

SO(3)⇥ SO(3)L�! SO(3)

R 7! LR

�1

R = R�1R


R

�1

(R, R))= L(R�1R,R�1R)

= l(I, ⌦)•

Krot

=12

Z


=12

Z


12

Z


=12⌦T

✓Z

B⇢(X) bXT bX d3X

◆⌦


I =✓Z


◆




Hamilton’s variational principle

�

Zt

1

t

0

l(⌦) dt = 0,

�R = 0 at t = t0 and t = t1; �⌦ restricted to be of the form

�b⌦ = b⌘ + adb⌦b⌘

where b⌘ = R�1�R is a variation vanishing at endpoints ⌘(t0) = ⌘(t1) = 0



Reduced Variational principle

�

Zb

a

l(⌦(t)) dt = 0,

)Z

b

a

h @l@⌦

, �⌦idt = 0

)Z

b

a

h @l@⌦

, ⌘ + ad⌦⌘i dt =Z

b

a

h @l@⌦

, ⌘i+ h @l@⌦

, ad⌦⌘i dt = 0

Zb

a

h� d

dt

✓@l

@⌦

◆+ ad⇤⌦

✓@l

@⌦

◆, ⌘i dt = 0

• The resulting dynamics is Euler-Poincare Equation

d

dt

@l

@⌦� ad⇤⌦

@l

@⌦= 0

ad⇤⇠

: so⇤(3) �! so

⇤(3) is dual of adjoint action on so(3)

• Calculating the dynamics: I⌦ = I⌦⇥ ⌦

• Reconstruction : g(t) = g(t)⇠ =) R = Rb⌦




�

Zb

a

l(⌦(t)) dt = 0,

)Z

b

a

h @l@⌦

, �⌦idt = 0

)Z

b

a

h @l@⌦

, ⌘ + ad⌦⌘i dt =Z

b

a

h @l@⌦

, ⌘i+ h @l@⌦

, ad⌦⌘i dt = 0

Zb

a

h� d

dt

✓@l

@⌦

◆+ ad⇤⌦

✓@l

@⌦

◆, ⌘i dt = 0


d

dt

@l

@⌦� ad⇤⌦

@l

@⌦= 0

ad⇤⇠

: so⇤(3) �! so







�

Zb

a

l(⌦(t)) dt = 0,

)Z

b

a

h @l@⌦

, �⌦idt = 0

)Z

b

a

h @l@⌦

, ⌘ + ad⌦⌘i dt =Z

b

a

h @l@⌦

, ⌘i+ h @l@⌦

, ad⌦⌘i dt = 0

Zb

a

h� d

dt

✓@l

@⌦

◆+ ad⇤⌦

✓@l

@⌦

◆, ⌘i dt = 0


d

dt

@l

@⌦� ad⇤⌦

@l

@⌦= 0

ad⇤⇠

: so⇤(3) �! so






Outline






Breaking symmetry

The full Lie group symmetry is sometimes broken. For instance in arotating top the gravitational potential breaks the symmetry of theLagrangian L.

X k g

Figure: Heavy top example

• Q = SO(3)

• Group : G = SO(3)

• Symmetric group:G

k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).

• The Lagrangian function’s G-invariance is now expressed with anadvected parameter. (the terminology “advected” finds its source influid modeling as invariants of a flow 1).

1D. D. Holm et al: Geometric Mechanics and Symmetry, Oxford Texts, 2009.February, 2016 STC-IISc Workshop


Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.

• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Breaking symmetry


X k g


• Q = SO(3)



k

= {R 2 SO(3)|RT k = k}.• where k = (0, 0, 1).




Lagrangian reduction

• We can take advantage of full broken symmetry as follows

• Start with extended space Q = SO(3)⇥R3 so that k can be consideredas the new coordinate v 2 R3

• Define L = T (SO(3))⇥ R3 ! R on Q

• Such thatL

ext

(R, R, v, v)|v=k

= L(R, R) (1)

• Motion is determined by L(via Hamilton’s principle) corresponds tothe motion determined by L

ext

, with added constraint.







• Such thatL

ext

(R, R, v, v)|v=k

= L(R, R) (1)


ext








• Such thatL

ext

(R, R, v, v)|v=k

= L(R, R) (1)


ext








• Such thatL

ext

(R, R, v, v)|v=k

= L(R, R) (1)


ext








• Such thatL

ext

(R, R, v, v)|v=k

= L(R, R) (1)


ext




Symmetry breaking potential and the associated machinery

• Lagrangian L : TSO(3)⇥ R3 ! R

L(R, k, R) = l(R�1R,R�1k,R�1R)

= l(e,�, b⌦)

• Lagrangian: L(R, R) = 12

RB ⇢(X)kRXk d3X +mglhk,RX i

l =12hIb⌦, b⌦i+mglhR�1

k,�i = 12hIb⌦, b⌦i+mgh�,�i


�

Zt

1

t

0

l(⌦,�) dt = 0,

�R = 0 at t = t0 and t = t1

�b⌦ = ⌘ + ad⌦⌘ �� = �b⌘�

b⌘ = R�1�R is a variation vanishing at endpoints ⌘(t0) = ⌘(t1) = 0





L(R, k, R) = l(R�1R,R�1k,R�1R)

= l(e,�, b⌦)






�

Zt

1

t

0

l(⌦,�) dt = 0,

�R = 0 at t = t0 and t = t1

�b⌦ = ⌘ + ad⌦⌘ �� = �b⌘�






L(R, k, R) = l(R�1R,R�1k,R�1R)

= l(e,�, b⌦)






�

Zt

1

t

0

l(⌦,�) dt = 0,

�R = 0 at t = t0 and t = t1

�b⌦ = ⌘ + ad⌦⌘ �� = �b⌘�






L(R, k, R) = l(R�1R,R�1k,R�1R)

= l(e,�, b⌦)






�

Zt

1

t

0

l(⌦,�) dt = 0,

�R = 0 at t = t0 and t = t1

�b⌦ = ⌘ + ad⌦⌘ �� = �b⌘�




The dynamic equations

•d

dt

@l

@⌦� ⌦⇥ @l

@⌦= �⇥ @l

@�

• Carrying out the di↵erentials we get

I⌦ = I⌦⇥ ⌦+mg�⇥ �, Euler-Poincare equation

� = �⇥ ⌦. Advection dynamics

�(t) = R�1(t)k.




•d

dt

@l

@⌦� ⌦⇥ @l

@⌦= �⇥ @l

@�




�(t) = R�1(t)k.




•d

dt

@l

@⌦� ⌦⇥ @l

@⌦= �⇥ @l

@�




�(t) = R�1(t)k.



Heavy top with two rotors

Figure: Heavy top with two rotors, eachconsists of two rigidly coupled disk.

• Q = SO(3)⇥ R3

| {z }group space

⇥ S1 ⇥ S1

| {z }shape space

• ✓ = (✓1, ✓2) 2 S1 ⇥ S1

• Lie group G = SO(3)

L(R, k, ✓, R, ✓) = l(R�1R,R�1k, ✓, R�1R, ✓)

= l(e,�, ✓, b⌦, ✓)





• Q = SO(3)⇥ R3

| {z }group space

⇥ S1 ⇥ S1

| {z }shape space

• ✓ = (✓1, ✓2) 2 S1 ⇥ S1


L(R, k, ✓, R, ✓) = l(R�1R,R�1k, ✓, R�1R, ✓)

= l(e,�, ✓, b⌦, ✓)





• Q = SO(3)⇥ R3

| {z }group space

⇥ S1 ⇥ S1

| {z }shape space

• ✓ = (✓1, ✓2) 2 S1 ⇥ S1


L(R, k, ✓, R, ✓) = l(R�1R,R�1k, ✓, R�1R, ✓)

= l(e,�, ✓, b⌦, ✓)





• Q = SO(3)⇥ R3

| {z }group space

⇥ S1 ⇥ S1

| {z }shape space

• ✓ = (✓1, ✓2) 2 S1 ⇥ S1


L(R, k, ✓, R, ✓) = l(R�1R,R�1k, ✓, R�1R, ✓)

= l(e,�, ✓, b⌦, ✓)





• Q = SO(3)⇥ R3

| {z }group space

⇥ S1 ⇥ S1

| {z }shape space

• ✓ = (✓1, ✓2) 2 S1 ⇥ S1


L(R, k, ✓, R, ✓) = l(R�1R,R�1k, ✓, R�1R, ✓)

= l(e,�, ✓, b⌦, ✓)



Hamilton’s Variational Principle

�

Zt

1

t

0

l(⌦,�, ✓, ✓) dt = 0,

�R = 0 and �✓ = 0 at t = t0 and t = t1

�b⌦ = ⌘ + ad⌦⌘ �� = �b⌘�


The dynamics Equation

d

dt

@l

@⌦� ⌦⇥ @l

@⌦= �⇥ @l

@�Euler-Poincare equation

d

dt

@l

@✓� @l

@✓= ⌧. Euler-Lagrange equation



Hamilton’s Variational Principle

�

Zt

1

t

0

l(⌦,�, ✓, ✓) dt = 0,

�R = 0 and �✓ = 0 at t = t0 and t = t1

�b⌦ = ⌘ + ad⌦⌘ �� = �b⌘�


The dynamics Equation

d

dt

@l

@⌦� ⌦⇥ @l

@⌦= �⇥ @l

@�Euler-Poincare equation

d

dt

@l

@✓� @l

@✓= ⌧. Euler-Lagrange equation



Outline






A spherical robot

Configuration space

Q = R2 ⇥ SO(3)⇥ S1 ⇥ S1 ⇥ S1



A spherical robot

Configuration space

Q = R2 ⇥ SO(3)⇥ S1 ⇥ S1 ⇥ S1



Rolling constraint

C

O

x

r

Figure: A rolling sphere with velocity of thecenter being x

VC

= 0.

x = xe1 + ye2

x = VC

+ (!s

)I ⇥ re3

x = (!s

)I ⇥ re3.



Lagrangian mechanics


• The Lagrangian is a map L : TQ �! R• A distribution of velocities is a linear subspace D

q

⇢ Tq

Q at everypoint q 2 Q (appears in the context of nonholonomic systems.)


Structure properties

• System is independent of certain base variable Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,


• The equation of motion independent of base body configuration,Reduction

• Dynamic equation is termed as the Euler-Poincare equation.






q

⇢ Tq






L(g · q) = L(q) 8q 2 TQ, 8g 2 G,









q

⇢ Tq




• System is independent of certain base variable

Symmetry


L(g · q) = L(q) 8q 2 TQ, 8g 2 G,









q

⇢ Tq






L(g · q) = L(q) 8q 2 TQ, 8g 2 G,









q

⇢ Tq






L(g · q) = L(q) 8q 2 TQ, 8g 2 G,









q

⇢ Tq






L(g · q) = L(q) 8q 2 TQ, 8g 2 G,


• The equation of motion independent of base body configuration,

Reduction







q

⇢ Tq






L(g · q) = L(q) 8q 2 TQ, 8g 2 G,









q

⇢ Tq






L(g · q) = L(q) 8q 2 TQ, 8g 2 G,



• Dynamic equation is termed as the Euler-Poincare equation.February, 2016 STC-IISc Workshop


Spherical Robot (Contd.)

• Q = (SO(3)⇥ R2)⇥ S where S = S1 ⇥ S1 ⇥ S1

• Lagrangian of the system with 3-rotors is given by

L =12m

T

kxk2 + 12!T

s

(Is

+ J)!s

+12

⇣⇥J⇥+ 2!T

s

J⇥⌘

• Rolling constraint x = (!s

)I ⇥ re3

Symmetry

• Restriction of R3 to horizontal subspace R2 breaks the symmetry of thesystem.

• Left group action, G = SO(3)n R3 on manifold Q.

• L and distribution D is invariant with respect to subgroup Ge

3

of G

Ge

3

= {(Rs

, b) 2 G = SO(3)⇥R3|RT

s

e3 = e3, hb, e3i = constant} = SO(2)nR2.




• Q = (SO(3)⇥ R2)⇥ S where S = S1 ⇥ S1 ⇥ S1


L =12m

T

kxk2 + 12!T

s

(Is

+ J)!s

+12

⇣⇥J⇥+ 2!T

s

J⇥⌘


)I ⇥ re3

Symmetry




3

of G

Ge

3

= {(Rs

, b) 2 G = SO(3)⇥R3|RT

s





• Q = (SO(3)⇥ R2)⇥ S where S = S1 ⇥ S1 ⇥ S1


L =12m

T

kxk2 + 12!T

s

(Is

+ J)!s

+12

⇣⇥J⇥+ 2!T

s

J⇥⌘


)I ⇥ re3

Symmetry




3

of G

Ge

3

= {(Rs

, b) 2 G = SO(3)⇥R3|RT

s




Reduction

• Start with extended configuration Q = SO(3)⇥ R3 ⇥ S1 ⇥ S1 ⇥ S1 andthe action is with respect to G = SO(3)n R3.

L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = L(Rs

, p, Rs

, X, R↵

, R'

, R↵

, R'

) |p=e

3

• The system reduced from TQ to se(3)⇥O⇥TS1 ⇥TS1 ⇥TS1 by groupaction. On se(3)⇥O ⇥ TS1 ⇥ TS1 we define a reduced Lagrangian

L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

)

• From this, the reduced equations of motion are obtained onso(3)⇥O ⇥ TS1 ⇥ TS1 ⇥ TS1 primarily in terms of the reducedconstrained Lagrangian

l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

) = lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).



Reduction


L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = L(Rs

, p, Rs

, X, R↵

, R'

, R↵

, R'

) |p=e

3


L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

)


l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

) = lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).



Reduction


L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = L(Rs

, p, Rs

, X, R↵

, R'

, R↵

, R'

) |p=e

3


L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

)


l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

) = lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).



Reduction


L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = L(Rs

, p, Rs

, X, R↵

, R'

, R↵

, R'

) |p=e

3


L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

)


l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

) = lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).




�

Zt

1

t

0

l(!s

s

, Y ,�,↵,', ↵, ') dt = 0,

�Rs

= 0, �↵ = 0 and �' = 0 at t = t0 and t = t1

�b!s

s

= ⌘ + ad!

ss⌘ �Y = r ad

!

ss⌘�+ r

d

dt(b⌘�) �� = �b⌘�

b⌘ = RT

s

�Rs

is a variation vanishing at endpoints ⌘(t0) = ⌘(t1) = 0




�

Zt

1

t

0

l(!s

s

, Y ,�,↵,', ↵, ') dt = 0,

�Rs

= 0, �↵ = 0 and �' = 0 at t = t0 and t = t1

�b!s

s

= ⌘ + ad!

ss⌘ �Y = r ad

!

ss⌘�+ r

d

dt(b⌘�) �� = �b⌘�

b⌘ = RT

s

�Rs




Dynamics of spherical robot

d

dt

✓@l

c

@!s

s

◆�

✓@l

c

@!s

s

◆⇥ !s

s

= �✓

@l

@Y⇥ �

◆+

✓@l

@�⇥ �

◆,

d

dt

✓@l

@⇥

◆� @l

@⇥= 0,

� = � !!

s

s

⇥�.

The governing equation is calculated as

M(�)

"!!

s

s

⇥

#=

"ad⇤

!

ss

⇣@lc@!

ss

⌘

0

#+

0u

�.



yoke

shell

pendulum

plane

Figure: Spherical Robot with internal pendulum mechanism

Configuration space

Q = SO(3)⇥ R2

| {z }group space

⇥ S1 ⇥ S1

| {z }shape space



o y

z

x

(xc

, yc

)

xs

zs

ys

'

yoke

Sphere

pendulum

↵

Figure: Coordinate frames for the system

Rolling Constraints

• x- velocity of the center of the sphere

x = (!s

)I ⇥ re3



Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp

i| {z }P.E.ofpendulum

+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum

Rolling constraint:!rs

= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry


• Symmetry group Ge

3

= {(Rs

, b) 2 SO(3)n R3|RT

s

e3 = e3, hb, e3i =constant} = SO(2)n R2.

• The gravitational field breaks the symmetry of the system.




Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp


+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum


= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry



3

= {(Rs

, b) 2 SO(3)n R3|RT

s






Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp


+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum


= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry



3

= {(Rs

, b) 2 SO(3)n R3|RT

s






Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp


+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum


= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry



3

= {(Rs

, b) 2 SO(3)n R3|RT

s






Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp


+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum


= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry



3

= {(Rs

, b) 2 SO(3)n R3|RT

s






Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp


+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum


= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry



3

= {(Rs

, b) 2 SO(3)n R3|RT

s






Lagrangian

L =12m

s

k!rs

k2 + 12hI !

!s

s

,!!

s

s

i| {z }

K.E.ofsphere

+mp

glhe3, Rs

R↵

R'

kp


+12m

p

k!rs

+ [Rs

!!

s

s

+Rs

R↵

(!!

↵

)Y +Rs

R↵

R'

(!!

'

)P ]⇥Rs

R↵

R'

kp

k2| {z }

K.E.ofpendulum


= (!!

s

)I ⇥ re3 =) !rs

= (b!s

)Ire3

Symmetry



3

= {(Rs

, b) 2 SO(3)n R3|RT

s






Reduced Lagrangian

L : T (SO(3)⇥ R2 ⇥ S1 ⇥ S1) �! R

l : se(3)⇥O ⇥ S1 ⇥ S1 �! R

lc

: so(3)⇥O ⇥ S1 ⇥ S1 �! R

L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

),

= lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).


�

Zt

1

t

0

l(!s

s

, Y ,�,↵,', ↵, ') dt = 0,

�Rs

= 0, �↵ = 0 and �' = 0 at t = t0 and t = t1

�b!s

s

= ⌘ + ad!

ss⌘ �Y = r ad

!

ss⌘�+ r

d

dt(b⌘�) �� = �b⌘�

b⌘ = RT

s

�Rs




Reduced Lagrangian

L : T (SO(3)⇥ R2 ⇥ S1 ⇥ S1) �! R

l : se(3)⇥O ⇥ S1 ⇥ S1 �! R

lc

: so(3)⇥O ⇥ S1 ⇥ S1 �! R

L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

),

= lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).


�

Zt

1

t

0

l(!s

s

, Y ,�,↵,', ↵, ') dt = 0,

�Rs

= 0, �↵ = 0 and �' = 0 at t = t0 and t = t1

�b!s

s

= ⌘ + ad!

ss⌘ �Y = r ad

!

ss⌘�+ r

d

dt(b⌘�) �� = �b⌘�

b⌘ = RT

s

�Rs




Reduced Lagrangian

L : T (SO(3)⇥ R2 ⇥ S1 ⇥ S1) �! R

l : se(3)⇥O ⇥ S1 ⇥ S1 �! R

lc

: so(3)⇥O ⇥ S1 ⇥ S1 �! R

L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

),

= lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).


�

Zt

1

t

0

l(!s

s

, Y ,�,↵,', ↵, ') dt = 0,

�Rs

= 0, �↵ = 0 and �' = 0 at t = t0 and t = t1

�b!s

s

= ⌘ + ad!

ss⌘ �Y = r ad

!

ss⌘�+ r

d

dt(b⌘�) �� = �b⌘�

b⌘ = RT

s

�Rs




Reduced Lagrangian

L : T (SO(3)⇥ R2 ⇥ S1 ⇥ S1) �! R

l : se(3)⇥O ⇥ S1 ⇥ S1 �! R

lc

: so(3)⇥O ⇥ S1 ⇥ S1 �! R

L(Rs

, e3, Rs

, X, R↵

, R'

, R↵

, R'

) = l(e,RT

s

Rs

, RT

s

X, RT

s

e3, R↵

, R'

, R↵

, R'

),

= l(b!s

s

, Y ,�, R↵

, R'

, R↵

, R'

),

= lc

(b!s

s

, b!s

s

�,�, R↵

, R'

, R↵

, R'

).


�

Zt

1

t

0

l(!s

s

, Y ,�,↵,', ↵, ') dt = 0,

�Rs

= 0, �↵ = 0 and �' = 0 at t = t0 and t = t1

�b!s

s

= ⌘ + ad!

ss⌘ �Y = r ad

!

ss⌘�+ r

d

dt(b⌘�) �� = �b⌘�

b⌘ = RT

s

�Rs

is a variation vanishing at endpoints ⌘(t0) = ⌘(t1) = 0February, 2016 STC-IISc Workshop


The dynamic equation

• d

dt

⇣@lc@!

ss

⌘� ad⇤

!

ss

⇣@lc@!

ss

⌘= �

⇣@l

@Y

⇥ �⌘+

�@l

@�⇥ �

�,

d

dt

�@l

@↵

�� @l

@↵

= 0,d

dt

⇣@l

@'

⌘� @l

@'

= 0,

� = � !!

s

s

⇥�.

• Computing the derivatives, we get

M(�,↵,')

2

64

!!

s

s

↵'

3

75 =� d

dt(M(�,↵,'))

2

4!!

s

s

↵'

3

5+

2

64ad⇤

!

ss

⇣@lc@!

ss

⌘

@T (�,↵,')@↵

@T (�,↵,')@'

3

75

+

2

64

@l

��⇥ �

� @V (�,↵,')@↵

� @V (�,↵,')@'

3

75+

2

4� �

@l

@Y

�⇥ �00

3

5+

2

40⌧↵

⌧'

3

5 .




• d

dt

⇣@lc@!

ss

⌘� ad⇤

!

ss

⇣@lc@!

ss

⌘= �

⇣@l

@Y

⇥ �⌘+

�@l

@�⇥ �

�,

d

dt

�@l

@↵

�� @l

@↵

= 0,d

dt

⇣@l

@'

⌘� @l

@'

= 0,

� = � !!

s

s

⇥�.


M(�,↵,')

2

64

!!

s

s

↵'

3

75 =� d

dt(M(�,↵,'))

2

4!!

s

s

↵'

3

5+

2

64ad⇤

!

ss

⇣@lc@!

ss

⌘

@T (�,↵,')@↵

@T (�,↵,')@'

3

75

+

2

64

@l

��⇥ �

� @V (�,↵,')@↵

� @V (�,↵,')@'

3

75+

2

4� �

@l

@Y

�⇥ �00

3

5+

2

40⌧↵

⌧'

3

5 .




• d

dt

⇣@lc@!

ss

⌘� ad⇤

!

ss

⇣@lc@!

ss

⌘= �

⇣@l

@Y

⇥ �⌘+

�@l

@�⇥ �

�,

d

dt

�@l

@↵

�� @l

@↵

= 0,d

dt

⇣@l

@'

⌘� @l

@'

= 0,

� = � !!

s

s

⇥�.


M(�,↵,')

2

64

!!

s

s

↵'

3

75 =� d

dt(M(�,↵,'))

2

4!!

s

s

↵'

3

5+

2

64ad⇤

!

ss

⇣@lc@!

ss

⌘

@T (�,↵,')@↵

@T (�,↵,')@'

3

75

+

2

64

@l

��⇥ �

� @V (�,↵,')@↵

� @V (�,↵,')@'

3

75+

2

4� �

@l

@Y

�⇥ �00

3

5+

2

40⌧↵

⌧'

3

5 .



References I

D. D. Holm, T. Schmah and C. Stoica.Geometric Mechanics and Symmetry.Oxford University Press Inc., New York, 2009.

J. E. Marsden and T. S. Ratiu.Introduction to Mechanics and Symmetry.Springer-Verlag, New York, 1994.

A. D. Lewis and F. Bullo.Geometric Control of Mechanical Systems.Springer-Verlag, New York, 2005.

C, D Eui, and J. E. Marsden.Asymptotic stabilization of the heavy top using controlledLagrangians.In Proc. Conference on Decision and Control(2000): 269-273.

D. D. Holm, J. E. Marsden, and T. S. Ratiu.The Euler-poincare equations and semidirect products withapplications to continuum theories.Advances in Mathematics, volume 137,1998.



References II

D. Schneider.Non-holonomic Euler-Poincare equations and stability in Chaplygin’ssphere.Dynamical Systems, pages 87-130, 2002.

Gajbhiye, S and Banavar, R N.The Euler-Poincare equations for a spherical robot actuated by apendulum.Proceedings of the 4th IFAC Workshop on Lagrangian and Hamiltonianmethods for Non-Linear Control, pages 72-77, 2012.


Documents

Interconnected rigid bodies: Dynamic modelling - …banavar/resources/notes/sneha... · Rigid body dynamics Symmetry breaking potentials Rolling constraints Interconnected rigid bodies: