the euler-poincaré equation for a spherical robot …banavar/resources/talks/sprobot... · ·...

The Euler-Poincaré equation for a spherical

robot actuated by a pendulum

Sneha Gajbhiye1, Ravi N Banavar1

1Systems and Control Engineering,

IIT Bombay, India

August 29, 2012

Outline

Introduction

The Setting

Modeling of spherical robot

Dynamic equation

Equilibrium Configuration

Controllability

Outline

Introduction

The Setting

Dynamic equation

Controllability

Spherical robot

Construction

A spherical shell with a driving mechanism mounted inside to make the

sphere roll.

Figure: Prototype of the spherical robot

Mechanism

pendulum

Figure: Schematic of the spherical robot

Sphere rolling on a plane

Internal driving mechanism consists of a yoke and a pendulum

Movement of the pendulum causes a change in the CG and the sphere

to roll

Yoke movement may be interpreted as a steering input

Broad objectives and methodology

Control objective

To move the sphere from one point and orientation to another speci�ed

point and orientation.

Devise motion planning algorithm for the robot to achieve the desired

orientation and point.

Steps to achieve the objective

Dynamic model of the robot.

Study equilibrium con�gurations.

Study controllability and devise motion planning algorithms .

Broad objectives and methodology

Control objective

To move the sphere from one point and orientation to another speci�ed

point and orientation.

Devise motion planning algorithm for the robot to achieve the desired

orientation and point.

Steps to achieve the objective

Dynamic model of the robot.

Study equilibrium con�gurations.

Study controllability and devise motion planning algorithms .

Outline

Introduction

The Setting

Dynamic equation

Controllability

Lagrangian Mechanics

The set of all possible con�gurations of a mechanical system is a

smooth manifold Q.

The set of con�gurations and velocities is the tangent bundle TQ.

The Lagrangian is a map L : TQ −→ R

A distribution of velocities is a linear subspace D ⊂ TQ (appears in the

context of nonholonomic systems.)

The equations of motion on TQ are given by the principle of least

action applied to a Lagrangian function L.

Symmetry

The Lagrangian function is invariant under a Lie group action

L(g · q) = L(q) ∀q ∈ TQ,∀g ∈ G,

where G is a Lie group.

Lagrangian Mechanics

The set of all possible con�gurations of a mechanical system is a

smooth manifold Q.

The set of con�gurations and velocities is the tangent bundle TQ.

The Lagrangian is a map L : TQ −→ R

A distribution of velocities is a linear subspace D ⊂ TQ (appears in the

context of nonholonomic systems.)

The equations of motion on TQ are given by the principle of least

action applied to a Lagrangian function L.

Symmetry

The Lagrangian function is invariant under a Lie group action

L(g · q) = L(q) ∀q ∈ TQ,∀g ∈ G,

where G is a Lie group.

Lagrangian reduction

By identifying the group symmetry and utilizing the associated

conservation law, the dynamics are expressed on a reduced space.

Start with Q, de�ne a Lie group G action. If the Lagrangian and

distribution are invariant with respect to this group action, express the

reduced Lagrangian on TQ/G 1.

Factor the symmetry on the semidirect product (Euclidean space) to

obtain the Euler-Poincaré equation on reduced space 2.

1Bloch, AM, Krishnaprasad PS, Marsden JE and Murray RM: Archive for Rational

Mechanics and Anal.,136, pp 21-99, 1996.2Cendra H, Holm DD, Marsden JE and Ratiu TS : AMS Trans.(2), vol.186, 1998.

Symmetry Breaking

The full Lie group symmetry is sometimes broken - results in an

isotropy subgroup (eg. with a gravity term).

The Lagrangian function's G-invariance is now expressed with an

advected parameter. (the terminology "advected" �nds its source in

�uid modeling as invariants of a �ow .3)

The equation of motion on a reduced space, given by the principle of

least action on a reduced Lagrangian function l, is called the

Euler-Poincaré equation (EP).

The Euler-Poincaré framework for the Chaplygin's sphere where the center

of mass coincides with the geometric center of the sphere has been

discussed by Schneider4.

3D. D. Holm al: Geometric Mechanics and Symmetry, Oxford Texts, 2009.4Schneider D:Dynamical Systems, pp 87-130, 2002.

Symmetry Breaking

The Euler-Poincaré equation - with potential energy terms

Start with the extended con�guration space Q and the associated

Lagrangian L, which is assumed invariant under G.

System con�guration Q is an immersed submanifold of Q and the

system Lagrangian is invariant under the isotropy subgroup - Gk .

The velocity constraints expressed as a distribution - D ⊂ TQ - give

rise to a reduced constrained-Lagrangian.

Incorporating the advection dynamics, we obtain the Euler-Poincaré

equation.

Outline

Introduction

The Setting

Dynamic equation

Controllability

Coordinate frames

(xc, yc)

Sphere

pendulum

Figure: Coordinate frames for the system

Configuration space

Q = SO(3)× R2 × S1 × S1

Coordinate frames

(xc, yc)

Sphere

pendulum

Figure: Coordinate frames for the system

Configuration space

Q = SO(3)× R2 × S1 × S1

Notations

Rs ∈ SO(3)- orientation of the sphere with respect to the inertial

frame,

Rα - orientation of the yoke with respect to the sphere body frame,

Rϕ - orientation of the pendulum with respect to the yoke frame,

(ωs)s, (ωα)Y , (ωϕ)P - angular velocity of sphere in sphere frame,

angular velocity of yoke in yoke frame and angular velocity of

pendulum in pendulum body frame respectively.

rs -linear velocity of the centre of mass of the sphere.

Lagrangian

2ms‖rs‖2 +

2〈Iωss , ωss〉︸︷︷︸

K.E.ofsphere

+mpgl〈e3, RsRαRϕkp〉︸︷︷︸P.E.ofpendulum

2mp‖rs + [Rsω

ss +RsRα(ωα)Y +RsRαRϕ(ωϕ)P ]×RsRαRϕkp‖2︸︷︷︸

K.E.ofpendulum

Rolling constraint: rs = (ωs)I × re3 =⇒ rs = (ωs)

Symmetry

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

L and D are invariant when RT1 e3 = e3. (Remain unchanged if we

translate the inertial frame anywhere on the XY-plane and rotate it

about e3, the direction of gravity.)

Symmetry group

Ge3 = {(Rs, b) ∈ SO(3) n R3|RTs e3 = e3} = SO(2) n R2.

The advected quantity here is Γ(t) = RTs e3

Lagrangian

2ms‖rs‖2 +

K.E.ofsphere

2mp‖rs + [Rsω

K.E.ofpendulum

Symmetry

Symmetry group

Lagrangian

2ms‖rs‖2 +

K.E.ofsphere

2mp‖rs + [Rsω

K.E.ofpendulum

Symmetry

Symmetry group

Lagrangian

2ms‖rs‖2 +

K.E.ofsphere

2mp‖rs + [Rsω

K.E.ofpendulum

Symmetry

Symmetry group

Mappings

Adjoint and Co-Adjoint operation for SE(3) = SO(3)nR3

Ad(R,x)(ξ, υ) =(RξR−1, Rυ −RξR−1x

)Ad∗(R,x)−1(µ, β) = (RµR−1 + x � (Rβ), Rβ)

where (ξ, υ) ∈ se(3) = so(3) n R3, (µ, β) ∈ se∗(3) = so∗(3) n (R3)∗. Rβ

denote the induced left action of R on β i.e. the left action of SO(3) on

R3 induces a left action of SO(3) on (R3)∗.

Adjoint and Co-adjoint action of se(3) = so(3)nR3

ad(η,ν)(ξ, υ) = ([η, ξ], ηυ − ξν)

where induced action of so(3) on R3 is denoted by ην.

ad∗(η,ν)(µ, β) = (−[η, µ] + β � ν,−ηβ) .

Con�guration space S = SO(3) n R3 × S1 × S1

Original Lagrangian L : T (SO(3)× R2 × S1 × S1) −→ R

Reduced Lagrangian l : t×M × S1 × S1 −→ R t ∈ so(3)× R3

Constrained Lagrangian lc : h×M × S1 × S1 −→ R h ∈ so(3)

M is the orbit space of G/Ge3 acting on e3 in R3.

L(Rs, e3, Rs, X, Rα,Rϕ, Rα, Rϕ)

= l(e,RTs Rs, RTs X, R

Ts e3, Rα, Rϕ, Rα, Rϕ),

= l(ωss , Y ,Γ, Rα, Rϕ, Rα, Rϕ),

= lc(ωss , rω

ssΓ,Γ, Rα, Rϕ, Rα, Rϕ).

Rolling constraint Y = rωssΓ.

ωss = RTs Rs is the (left-invariant) sphere-body angular velocity.

= lc(ωss , rω

Outline

Introduction

The Setting

Dynamic equation

Controllability

The Euler-Poincaré equation

(∂lc∂ωs

)− ad∗ωs

(∂lc∂ωs

)= −

(∂l∂Y� Γ)

+(∂l∂Γ� Γ),

)− ∂l

∂α= 0,

)− ∂l

∂ϕ= 0,

Γ = −ωss × Γ.

The diamond operator

ρv : so(3)→R3 ρ∗v : so∗(3)→R3∗

R3 × R3∗→ so∗(3) : (v, w)→ v � w 4= ρ∗v(w)

A more explicit equation

Carrying out the di�erentials, the dynamic equation is represented as

M(Γ, α, ϕ)

=− d

dt(M(Γ, α, ϕ))

ad∗ωs

(∂lc∂ωs

)∂T (Γ,α,ϕ)

∂T (Γ,α,ϕ)∂ϕ

∂lδΓ× Γ

− ∂V (Γ,α,ϕ)∂α

− ∂V (Γ,α,ϕ)∂ϕ

−(∂l∂Y

)× Γ

Outline

Introduction

The Setting

Dynamic equation

Controllability

Figure: Equilibrium con�guration manifolds: a) downright position of pendulum

b) upright position.

Set (ωss , α, ϕ) ≡ 0, and assuming constant holding torques τα and τϕ,

mpglX × Γe = 0 =⇒ X × Γe = 0,

∂V (Γ, α, ϕ)

∂α= τα;

∂V (Γ, α, ϕ)

∂ϕ= τϕ.

X = RαRϕk

Observations on equilibria

Vector X is collinear with the gravity vector RTs e3 in sphere-body

frame.

Fix α, ϕ: all con�gurations obtained by a rotation around the vertical

axis passing through the point of contact, constitute the equilibrium

manifold.

If Rse is an arbitrary orientation, then any α & ϕ such that X is in the

downright or upright position constitutes an equilibrium.

The control equilibrium of the reduced system is given as

{(Rs, α, ϕ)|Γ×X = 0} ⇒ {(Rs, α, ϕ)|RαRϕ = RTs }

where Γ = RTs e3 and X = RαRϕk.

frame.

manifold.

frame.

manifold.

frame.

manifold.

Outline

Introduction

The Setting

Dynamic equation

Controllability

Vector fields on the reduced space

The control vector �elds on the reduced space T (SO(3)× S1 × S1) are

Yi = M−1(Γ, α, ϕ)

]The potential vector �eld on the reduced space T (SO(3)× S1 × S1) is

(gradV )˜ = M−1(Γ, α, ϕ)

[mpgΓ×X

where i = α,ϕ and yi is a T∗(S1 × S1)-valued function.

Computational procedure

Calculate the symmetric product⟨Yi : Yj

Evaluate the iterated symmetric product of Sym(Y) = {Y ∪ (gradV )˜}.

The system is local con�guration accessible at equilibrium if the rank

of Lie(Sym(Y)) = dim(Q) at q0.

Every bad symmetric product from {Y ∪ (gradV )˜} is the linearcombination of lower degree good symmetric products, then the system

is small time local con�guration controllable.

References

1 D. Schneider, Non-holonomic Euler-Poincaré equations and stability in

Chaplygin's sphere,Dynamical Systems, 2002, pp.87-130

2 A. M. Bloch.Nonholonomic Mechanics and Control . New York:

Springer-Verlag, 2003

3 Bloch, A. M, krishnaprasad P. S, Marsden J. E and Murray R. M. ,

Nonholonomic mechanical systems with symmetry. Archive for

Rational Mechanics and Analysis, 1996, volume 136 pp 21-99.

4 Holm, D. D, Schmah, T and Stoica, C.Geometric Mechanics and

Symmetry, Oxford University Press Inc., New York, 2009.

5 Cendra H, Holm D. D, Marsden J. E and Ratiu T. S Lagrangian

reduction, the Euler-Poincaré equations, and semidirect products,

American Mathematical Society Translation(2), 1998, Volume 186

6 D. D. Holm, Geometric Mechanics: Rotating, Translating and Rolling,

Imperial College Press, 2008

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