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KINEMATICS of a ROLLING BALL
Wayne Lawton
Department of Mathematics National University of Singapore
Lecture based on my student’s MSc Thesis Symmetry and Its Applications in Mechanics
by Lioe Luis Tirtasanjaya(B.Sc., ITB)
KINEMATICS versus MECHANICSKinematics is the geometry of motion
Example: If the height h of an object is a function h = t^2 of time t, then the graph of the function h is a parabola, one of three conic sections. What are the other two ? Why are they called conics ?
Example: (Kepler) The planets move around the sun on elliptical orbits with the sun at one focus, their radius vectors sweep out equal areas in equal times, and their period is proportional to the 3/2 power of the semi-major axis of their orbits.
Mechanics is the physics of motionExample: (Newton) Acceleration = Force / Mass, this gives the height if an object falling near Earth’s surface as a function of t
Example: (Newton) Force = G M m / R^2, this gives the trajectories of the planets
AFFINE SPACEIs a triple )V,(A,
R
is the set of points in the space
A VA: of real numbers
AV is a vector space over the field
is a translation map that satisfies
)()( vupvup
puqVuAqp !,,
where we define ),( upup
and we define uqp q u p
EUCLIDEAN SPACEIs an quadruplet
is an affine space
positive definite :
VVV :bilinear : a linear function of each argument
is a mapping that is
),V,(A, )V,(A,
symmetric : ),(),( vuvu 0),(,0 uuu
Definition , Vvu 0),( vuare orthogonal if
Definition , Eqp have distance
),(),( qpqpqpd
BASES
3,2,1,Mei
i
Theorem 1. If
33RM][
is three dimensional, then a basis of
by a column vector
V
a representation of
VV:M by a matrix
V
Vu 3R][ u
and gives a representation of a linear map
whose columns are the vectors
where
VM
VN
V VMN
V
3R
[M]3R
[N]3R 33
R[M][N][MN]
R
composition of maps matrix representation
}e,e,{e321
is the standard basis for 3R
The following diagram commutes, this means that [MN] = [M][N]
gives
ROTATION LIE GROUP
)},(),(,linear :{ SO(3) vuMvMuVVM
1M][detM][M][ (3)SM ,1T O
Theorem 2. M][
Proof The adjoint of M, defined by
is the matrix representation of a linear mapIf
VV:M with respect to an orthonormal basis then
T][][M M
)M,(),M(*
vuvu
satisfies , the second statement then
is equivalent to M preserving orientation
RIGID TRANSFORMATIONS
Definition
ucpMc )( p)(,Ep
EE q)d(p,(q))(p),d(
Theorem 3 is a rigid transformation iff
VuSOMEc ),3(,
Proof Exercise, note that M is determined
Remark M is determined, c is arbitrary, v is determined by M and c
Definition This is rotation by M about c followed by translation by u
RIGID TRAJECTORIES
Definition
))0()0()(()( )(p cptMtct
Rt, E)(
E t
Theorem 4 )(t is a rigid trajectory iff
IMSOtMEtc )0(),3()(,)(
Proof Follows from Theorem 3
))0()(()(,)0( pttpEp
ROTATION LIE ALGEBRA
}Alinear, VV:A{ so(3) A
Proof The first assertion follows since
Theorem 5.
B]A,[BAABBA)-AB(B]A,[
so(3) is a Lie Algebra under the commutator product
][][)3( T AAsoA For an orthonormal basis [so(3)] are the skew-symmetric matrices
),(),()A,(),)((*
vuABBvuABvuvuAB
and the second assertion follows since
TT][][],[][),( MMvuvu
VECTOR ALGEBRA
Theorem 6. so(3) is isomorphic to under the vector product 3R
Proof
0
0
0
]ˆ[][
12
13
23
3
2
1
Vω,u,v
Definition3Rω uu ̂VV :̂
]ˆ,ˆ[ˆ vuwvuw
Remark
ANGULAR VELOCITY
Theorem 7. If SO(3)R M
)MM()MM(I01 dt
d
dt
d
dt
d
so(3)R1
MM
is differentiable then
Proof
)MM(MMMMMM
11
Definition1
MMˆ
are the
and VR
so(3)R1
MM
and
VR where
and MMˆ 1
angular velocityin space
angular velocityin the bodyand
ADJOINT REPRESENTATIONSso(3)so(3): AdSO(3),M M
Proof Exercise
Theorem 8 The adjoint representations defined above define (1) a homomorphism of the group SO(3) into the group of linear isomorphisms of so(3), and (2) a homomorphism of the Lie algebra of linear maps of so(3) into so(3) with the commutator product. Furthermore, if M and B are differentiable functions of t
1M MBMBAd
so(3)so(3): ad(3),A A so
B]A,[BadA
BAdadBAdBAd Mˆdtd
MMdtd
VELOCITY
) c(0)p(0)(dt
dM(t)))()(( tctp
dt
d
Theorem 9 The velocity of a point trajectory p(t) is given by
) )0()0()(()()(1
cptMtMtM
) )()(()()(1
tctptMtM
) )()(()( tctpt
Proof Differentiate the following formula obtained from Theorem 4
) )0()0(()(c(t)- p(t) cptM
SURFACE MOTION OF A BALLIntroduce orthonormal bases with x, y, z coordinates for V, consider a spherical ball with radius 1 moving so as to be tangent to and above a plane P that is parallel to the x-y subspace in V
Theorem 10. The rigid trajectory of the center of the ball is
]0)()([)]0( [c(t)T
tytxc
T1]100[)]T([)]0()0([
Mcq
and for every
Proof The first assertion follows since the center of the ball must move parallel to P and hence to the x-y subspace of V, the second assertion follows from Theorem 4 since
lies on the ball at time 0 and at time t = T it lies on the ball and P
RT
P)T(]100[)]T()T([T
qcq
NO SLIDING CONDITIONDefinition We say that a ball moves without sliding if the point on the ball in the plane P has zero velocity
Theorem 11. A ball moves without sliding iff
Proof Follows from Theorems 9 and 10 since at time t = T
0000
001
010
0
y
x,
xy dt
dydt
dx
dt
dy
dt
dx
] 100[)]T([Tt
|)]()([)]([
tctqdt
dtcdt
d
NO TURNING CONDITIONDefinition A ball moves no turning if
Theorem 12. A ball rolls iff
][
000
001
010
]ˆ[dt
dc
.0 z
Definition A ball rolls if it moves with no sliding and no turning
Proof Follows from Theorem 10 and the definitions above
EQUATIONS OF A ROLLING BALL
Theorem 13. The rotation equations of a moving ball are
[M]
0
0
0
]M[
xy
xz
yz
and these together with the condition M(0) = I determine M, and approximate trajectory in SO(3) is given by
)]([)])(ˆ[)((exp)]([,)0(111 nnnnn
tMttttMIM
Proof The first assertion follows from the definition1
MMˆ
the second assertion follows since exp maps so(3) onto SO(3)
ROLLING ON A LINETheorem 14. Rolling on a straight line with unit velocity u results in a rotation around a unit vector v, obtained by rotating u counter clockwise by 90 degrees (so v is parallel to the x-y plane), by an amount t in a counterclockwise direction
0
00
00
]v̂[,
0
[v]
yuxu
y
xu
xu
y
u
u
2
v]v̂[)cos1(]v̂[)(sin]v̂[exp)( ttIttR
Proof Consider the action of both sides of the expression above on vectors parallel and orthogonal to v
TURNING BY ROLLINGTheorem 15 If a ball rolls along the straight line from [0 0 0] to [pi/2 0 0], then along the straight line to [pi/2 -d 0], then along the straight line to [0 d 0], the net effect is a rotation by angle d in the counterclockwise direction around v = [0 0 1].
, )2
(, )2
(001
010
100
001
010
100
ca
RR
)2/()()2/()(abc RdRRtR Proof The net rotation
where ]001[][,]010[c][[a]TT b
0
)(, )(100
0cossin
sincos
cossin0
sincos0
001
b
dd
dd
dd
dd tRdR
MATERIAL COORDINATES
Theorem 16 The unit vector-valued function
such that
VSR:u2
dMu1
is the unit vector
defined by
coordinate of the point of contact of ball and the plane P
ucp )0()0(
Definition The material coordinate of a point q on the ball at time t VSv
2 .)v)0()(( qct
Proof If then
dtccptMtcpttp )())0()0()(()())0()(()( so P)()()( tpdtctp
Definition Let d in V be defined so thatT
]100[][ d
is the trajectory of the material
MATERIAL TRAJECTORY
Theorem 17 The material trajectory SR:u2
,uu
satisfies
and u determines M by
Proof
and
},,{ uuuu
uu
0
0
0
[M]]M[
xy
xz
yz
uudMMMdMdt
du111
is an orthornormal basis of V and the last
||,|||||| u
||||||)(||||||1
pMu since )3(SOM and 0),( p hence the third equation follows since
assertion follows from the definition of
HOLONOMY and CURVATURE
Theorem 18 If material trajectory SR:u2
pTuu )()0(
satisfies
then where area
and that the curvature of this connection, a 2-form with values in the Lie algebra so(2) = R, coincides with the area 2-form induced by the Riemannian metric. The detailed proof is developed in Luis’s MSc Thesis.
(area)RM(T)z
)2(/)3()3( 21
SOSOSSO pMuM
Proof This is an extension of Theorem 15. It is based on the fact that the rolling constraints are described by a connection on the principle SO(2) fiber bundle
is the directed area bounded by the curve2
ST])u([0,
VARIATIONAL EQUATION
Theorem 19 If
and
is a rotation trajectory and
is a small trajectory variation
Proof Follows from the fact that
SO(3)R:M SO(3))(TangentR:M
VR: is defined by
δδthen
1MMˆ
MδδMdt
d
the fact that111
MMMMdt
d ---
and the definition1
MMˆ
OPTIMAL TRAJECTORIES
Theorem 20 The shortest trajectory
S2
with specified satisfies
Proof Follows from Theorem 19 using the calculus of variations, details are in the MSc Thesis of Luis
SO(3)M(T)I,M(0)
c ( [0,T] ) is an arc of a circle in the plane P
SO(3)R:M
u( [0,T]) is an arc of a circle in the sphere
Furthermore, M(T) can be computed explicitly from the parameters of either of these arcs
