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Department of Computing
and Information Science,
Queen's University,
Kingston, Ontario K7L 3N6, Canada
Velocity and Acceleration
Assoc. Mere. ASME
Analysis of Contact Between
V. Kumar
Department of Mechanical Engineering
Three-Dimensional Rigid Bodies
and Applied Mechanics,
University of Pennsylvania, During manipulation and locomotion tasks encountered in robotics, it is often neces-
Philadelphia, PA 19104 sary to control the relative motion between two contacting rigid bodies. In this paper
Assoc. Mem. ASME we obtain the equations relating the motion of the contact points on the pair of
contacting bodies to the rigid-body motions of the two bodies. The equations are
developed up to the second order. The velocity and acceleration constraints for
Xiaoping Yun contact, for rolling, and for pure rolling are derived. These equations depend on the
Department of Electrical and local surface properties of each contacting body. Several examples are presented to
Computer Engineering, illustrate the nature of the equations.
Naval Postgraduate School,
Monterey, CA 93943-5121

1 Introduction limited to the motion of the contact points and hence their study
is restricted to a subspace of the configuration space. In Cai and
In robot manipulation tasks, when the robot arm interacts
Ruth (1988) they extend their work to line contacts. Montana
with an object or the environment, it is beneficial to be able to
(1988), like Pars (1968), considers the five-dimensional space
control the contact motion. For example, when exploring the
but his parameterization is somewhat different than Pars' (see
environment by feeling or touching, it is necessary to be able
Section 2). He derives the equations relating the velocity of
to move the robot arm while in contact with the environment
the contact points on the rigid bodies to the relative velocities
and estimate the geometric properties of the environment (Ma-
of the rigid bodies. It is also worth mentioning the work of
son and Salisbury, 1985; Montana, 1988). When grasping an
Kirson and Yang (1978) who developed equations for moving
object it may be desirable (Brock, 1988) to roll or to slide the
and fixed axodes with a relative rolling and sliding motion.
object over one or more fingers in a specified manner. A similar
This paper derives the velocity and acceleration equations
situation also arises in an actively coordinated vehicle traversing
relating the rigid-body motions of the contacting bodies and the
uneven terrain where it is efficient to maintain rolling contact
motions of the contact points on the surfaces of the rigid bodies.
with the ground at all contacts (Kumar and Waldron, 1989).
The papers by Montana (1988) and Cai and Ruth (1987) are
In all these examples, it is necessary to control the motion of
most closely related to this work. We use Montana's definition
an actively coordinated system relative to the contacting object
of the configuration space but our approach is quite different.
or environment in order to achieve a desired motion of the
Although Montana does derive the velocity equations, as seen
contact point on the surface of the object and on the surface of
later in the paper, they do not lend themselves to straightforward
the robot effector.
differentiation. Our approach and results differ from those of
While extensive work has been done on contact between
Cai and Ruth because we do consider the entire five-dimensional
planar rigid bodies (see, for example, Beggs, 1966; Paul, 1979;
configuration space.
Hall, 1966; Rosenauer and Willis, 1953; Whittaker, 1988), the
work on the kinematics of three-dimensional contact is much
more recent and much less is known in this area. Pars (1968) 2 Preliminaries
describes the configuration space associated with the relative
2.1 Notation. In Fig. 1, we consider two rigid objects
motion between two rigid bodies in point contact. He shows
(obj 1 and obj 2) contacting at a point. The contact point is the
that it is five-dimensional. For the special case of a sphere
coincidence of two points, Pl fixed to obj 1, and P2 fixed to obj
rolling over a plane he shows that the configuration space is
2 at time t. cl and c2 are a pair of points, which do not belong
completely accessible. In other words, from any point in the
to either body but move along the surface of obj 1 and obj 2
configuration space it is possible to reach any other specified
respectively so that they are instantaneously at the point of
contact. We choose reference frames on obj 1 and obj 2 at point
Cai and Ruth (1986; 1987) study the relative motion of two
ol and o2, respectively. These reference frames are attached to
contacting bodies in point contact. In Cai and Ruth (1987) they
the objects. We attach coordinate systems at points cl and c2
derive expressions for the motion of the contact point on each
which move with the contact points. Finally, we define object-
contacting surface in terms of the relative motion and the local
fixed coordinate frames at points pl and P2 in such a way that
geometric properties of each surface. However, their focus is
they coincide with cl and c2 frames, respectively, at time t.
Note that the same symbol is used to denote either a point or
a reference frame attached to this point. So far our notation is
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY
OF MECHANICAL ENGINEERS for publication in the A S M E JOURNAL OF APPLIED identical to that in Montana (1988).
MECHANICS. r is used for position vectors, V is used for linear velocities
Discussion on this paper should be addressed to the Technical Editor, Professor and w, for angular velocities, a and a denotes linear and angular
Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, accelerations, respectively, f~ is used to represent the skew sym-
Houston, TX 77204-4792, and will be accepted until four months after final
publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. metric matrix form of w. R denotes rotation matrices.
Manuscript received by the ASME Applied Mechanics Division, Mar. 4, 1994; A leading superscript is used to denote the reference frame
final revision, Mar. 27, 1996. Associate Technical Editor: E. J. Haug, Jr. in which the quantity is observed. A vector q with two trailing

974 / Vol. 63, DECEMBER 1996 Transactions of the ASME

Copyright 1996 by ASME

Downloaded 24 Mar 2009 to Redistribution subject to ASME license or copyright; see
is, S = tJ~'__~&, where &'s are coordinate patches for S. The
set {& }7=~ is called an atlas for S.
In what follows we will need to compute derivatives of f in
order to characterize the local properties of S. We will assume
t h a t f ( ~ , ~2) is at least of class C 3.
DEFINITION2. Natural Basis and Unit Normal: xi = (Of/
0~') ( i = 1, 2) are linearly independent at a given point and
obj 2 are called the natural basis of the smface, xl and x2 minimally
Fig. 1 Two rigid bodies with point contact span the tangent plane at that particular point. A unit normal
n is a unit vector which is perpendicular to the tangent plane
at a given point and is defined by (xl x~)/(llx, x x~ll).
If (&, xz) = 0 where (,) is the symbol for inner product,
subscripts, a and b denotes the difference between q at point a then (f, U) is called an orrthogonal coordinate system. It should
and q at point b. For example, r~o, is the position vector from be noted that x~ is not necessarily a unit vector.
o~ to c~. ,Vqo, represents the difference between the velocity DEFINITION 3. Contact Frame: This is a local reJerence
of point c~ and that of point o~ as observed in reference frame frame at the contact point consisting of the unit vector triad
o~. On the other hand, when there is only one trailing subscript (x#llx~ll), (x#llx211), and n.
then it denotes the reference frame attached to that point. Thus, We choose the coordinate system (f, U) in such a way that
o, ~,L or o, f~, denotes the angular velocity of reference frame n is an outward pointing normal.
c, when it is observed in the reference frame Ol. Note that, DEFINITION4. Metric Tensor: A metric tensor G is a 2 X
unless otherwise specified, we refer to the vectors themselves 2 symmetric, positive definite matrix whos e coefficients go are
and not to their components in a particular coordinate system. defined as
When we do refer to components, we denote these with sub-
scripts x, y, and z and explicitly specify the coordinate system. g~j= (&,xj) i,j= 1,2.
When components are considered, a R b is a rotation matrix which The g~i are coefficients of the first fundamental form. G is
transforms components of a vector in frame b to components diagonal for an orthogonal coordinate system.
in frame a.
DEFIN]TION 5. Christoffel Symbols: There are two types
In the case of position vectors, a leading superscript is not
of Christoffel symbols. Christoffel symbols of the first kind,
used because a position vector does not depend upon the frame denoted by [ ij, k ], are defined as
in which it is Observed. However, any derivative, and therefore
any velocity or acceleration, does depend on the reference frame
in which the differentiation operation is performed. We follow [ij, k]
=fox, )
\ O ~ j , xk i , j , k = 1, 2.
a notation that is similar to that used in Kane and Levinson
(1985). For example, consider the derivative of r~o,. Since the
point oj is fixed to the reference frame o~, Christoffel symbols of the second kind, F~, are defined as

t d
OI])clOl : " ~ - rlo I
F5 = Ox,,&
,=, \ O U i g,k= ~, [ij, l]g,k
i,j,k= 1,2

is the velocity of point c~ in frame o~. Similarly, where glk are the components of the inverse of the metric tensor
t d G.
_ _ (pip o
dt L. qo,) = ,a~,,,~ It is clear from the definition that both kinds of Christoffel
symbols are symmetric in the paired indices ij, that is, [ij, k]
is the acceleration of point c~ in frame o~. = [ji, k] and F~ = Fj~.. Note that all the Christoffel symbols
Thus, the derivative of rqv ~ in frame P2 is not P,vqp,. Instead, vanish when the coordinate system is Cartesian.
It is useful to note that the derivative of the components of
P~d Pld the metric tensor can be expressed in terms of the Christoffel
d---7"r~,,,, = ~ rqv ' + v~w,,, rc,pt symbols.

= Pll)ctPl "q- I'2COpi X rClpl Ogu

0---7 = [jk, i] + [ i k , j ] = g~tF~k + gjf~,g (1)
The development here is identical to that presented by Kane
and Levinson (1985, p. 23). DEFINmON 6. Gauss's Equations: The derivatives of the
basis vectors are given by
2,2 Local Properties of Surfaces in ~-R~. Here we briefly
discuss a few definitions and concepts that will be used later in Oxi
the paper to derive the contact kinematic equations. Detailed c9~----7= Li? + ~ F~x~.
discussion can be found in any standard differential geometry k
text (e.g., Millman and Parker, 1977; Stoker, 1969; Lipschutz, The L0 are related to the coefficients of the second fundamen-
1969). tal form only through the metric tensor. They measure the nor-
DEFINITION 1. Coordinate System: Let S v be an open and mal component of (0&/O~ J) while the F~ measure the tangen-
connected subset of the smface S containing the point p. Then tial components.
the pair ( f U) is called a coordinate system of S v if there exists Higher Order Derivatives: If f is at least of class C 3, we
an open subset U of g] 2 and an invertible map f ' U ~ S e @ ~ can differentiate Lgj and F~. By straighforward differentiation,
such that the partial derivatives ( Of( { )/O{ ~) and ( Of( { )/O~ ~) it can be shown (Lipschutz, 1969, p. 224) that
are linearly independent for all ~ = ( ~ ~, ~ 2) ~ U. The open
connected subset Sp is called a coordinate patch. 02x~ 5",
It may so happen that not all points of S can be represented [ F ~ik,j + FijFm
~ ~
- LoL~.]x~ + [FuL,~ + Lij.k]n
by a single coordinate patch. In such a case, we can construct
a set of coordinate patches which cover all the points of S, that where F~.j = ( OF~IO~ j) and Lij.~ = ( OLu/O~k).

Journal of Applied Mechanics DECEMBER 1996, Vol. 63 ! 975

O o

x x


2 = const


1= constant

~2= constant Fig. 3 Coordinate curves for a plane

~1 = constant

Fig. 2 Coordinate curves for a sphere

2.3.2 Geometric Properties of a Planar Surface. Consider
the plane (Fig. 3) with the coordinate system:

2.3 Examples. f : U C 9l 2 ~ 913 : (~,, ~2) ~ (~,, ~2, 0).

The following results are obtained from the definitions in the

2.3.1 Geometric Properties of a Sphere. For a sphere with
previous subsection.
radius p, let us define a coordinate system (Fig. 2),

f : U ~ ~tt3 : (~l, ~2) Xl ~ X2 = n

(p sin ~ cos ~2, psin~lsin~Z, pcos~l).

The natural basis for this coordinate system and the correspond- gll = 1 gt2 = 0

xl =
[ cos lco
ing unit normal are

p cos ~l. sinl~2

-- p sln
xz = p sin ~ cos (2
g21 = 0 g22 = 1
Since the natural basis vectors and the components of the
metric tensor are constant for a plane, all other higher order
derivatives are zero.

2.4 Contact Coordinates. We now define five contact

["sin ~1 cos ~z] coordinates that characterize the motion of the point of contact.
n= lsin~'sin~Z . First for obj i, we let ~t = ul and ~z = v~ as shown in Fig. 4.
k cos ~ The point of contact is uniquely defined by the four coordinates
Ul, Vl, u2, and v2. The fifth parameter is qJ, the angle of contact
which is the angle between the u~ and u2 curves (Montana,
So the contact frame is defined by [(xl]l[xll[) (x2[[lx2[[) n]. 1988). In Fig. 5, it is the angle between (x~)1 (tangent to the
The components of the metric tensor G are given by v~ = constant curve) and (x~)2 (tangent to the vz = constant
curve). The sign of ~0 is defined in such a way that a rotation
gll = p2 g12 = 0
of (X~)l about the outward pointing unit normal (n)~ to the
surface at point p~ through -~O aligns the axes (xl)~ and (Xl)2.
g21 = 0 g22 = p 2 s i n ( ~ l ) 2 .
We use a trailing subscript i to denote obj i. For example,
the contact frame for obj i is denoted by [(xl/llxl[])i (xz/llxzll)i
G is diagonal because the coordinate system is orthogonal ((x~,
x2) = 0). (n)i].
The Christoffel symbols and the coefficients L U are t

3 Closure Equations
Fit = 0 F121 = 0
In Fig. 1, using the triangle law of vector addition (Paul,
1979), we can write
F]2 = 0 Fl2z = c o t ~ l

F~l = 0 F~i = cot (~

F~2 = - s i n ~ t c o s ~

LH = - p L~z = 0
/ obj 2

L2~ = 0 /-a~ = - p sin (~l)Z.

The derivatives of F~ and Lo that are not zero are

F~2,1 = --CSC 2 ~1 F221,1 : __CSC2 ~1

Fz~2.t = - c o s (2~ 1) La2.~ = - 2 p sin (~ c o s ~1 V 1 = constant/ ~ Q ~

Note that although the Gaussian curvature and the mean curva- U1 = constant
ture of a sphere are constant, the coefficients, go, F~ and L 0 are
not. Fig. 4 Coordinate curves and contact frames on two contacting bodies

976 / Vol. 63, DECEMBER 1996 Transactions of the ASME

P2aclp2 = /qaclpl + P2~pl X rclpl + P2~!o I X (P2wpL X rcllh )

obj 2 / + 2~2Wp~ P'Vc,p, + "2ap,p2. (13)

Note that rc,p~ = 0 in the above equations. Similarly for angular

velocities, we obtain the velocity and acceleration equations:
( Xl )2
P2WcL = I'lWc~ + v2tOpl (14)

I P2~Cl = Pt(.J)Cl + P20,Jpl PtO.)Cl "1- P2{,~.Jpl. (15)

I Now consider the closed loop: P2 -~ cl + c2 -+P2. The transla-
tion closure equations are
r,,~,2 = rc~c2 + r~j,2 (16)
v2Vqv 2 = qVqc 2 + P2w~2 X rc~.2 + ~'=Vca,2 (17)
P2ac=p2 = C2aclc2 -}- P2~.~)c2 rqc 2 + P2Odc2 X (P20)c2 X rqc2)

+ 2P2wc~ qVc,c~ + P2acw2. (18)

Fig. 5 The definitionof angle 0 By definition, the vectors rq~, qV~,c~ and qa,,~ vanish in the
above equations. The loop equations for the angular velocities
P~wq = ~2w~. + P2co,,2 (19)
r,,o, = rca,, + rp,o,. (2)
P2{.~.)Ci = C2~)Cl "~ P2~A,}C2 C2{.4aICl+ P2~JC2. (20)
Note that, in this notation, we implicitly assume that Eq. (2)
can be written in component form in any convenient coordinate We now manipulate the loop equations to obtain what we
system. (At this point we are only concerned with vector equa- call the contact closure equations. Equating the right-hand sides
tions and not their components in a particular coordinate sys- of Eqs. (12) and (17) we get after setting r~lp,, rq~ and Vqc2
tem.) Differentiating each term of Eq. (2) with respect to time to zero
t in the reference frame o~, we get
~'2V~, 2 = P'Vc,l,, + I'~Vl,~p~. (21)
,V,,,o ' = v,V~,p, + % % x G,p, + 'Vine,. (3)
From Eq. (3), because ,Vp,o, = 0 and 0,%,, = 0, Eq. (21) can
Differentiating Eq. (3) in the reference frame o~, we get be written as
iacioi : Piacipi q-- iO3pi X PiVcipi ..{-. i@i rcipi 02Vc202 = OIVcloi -'~- P2VpiP2. (22)
+ %% x (P,Vc,,,, + o,%,, X re.a,,) + ,a,,,o,. (4) Equating the right-hand sides of Eqs. (14) and (19) and then
substituting from Eq. (9), we get
Applying the addition theorem for angular velocities (Kane and
Levinson, 1985, p. 24), we get lW~l + V2OOpt = c20&1 + 2W~2. (23)
o,we, = o, co,,, + ", c%. (5) Next, we equate the right-hand sides of Eqs. ( 13 ) and (18) and
simplify using Eqs. ( 7 ) - (8) to get
Differentiating each term in Eq. (5) in the o~ frame, we get
2a,.~o= = tat,o, + 2P2wpl X lVqo I + P2aplp2. (24)
'd.',., = 'dh,, + P'G., + 'wp, x P,w~, (6)
Finally, equating the right-hand sides of Eqs. (15) and (20)
where %G denotes the time derivative of "CObin frame a. and simplifying using Eqs. ( 9 ) - (10), we write
Since Pi and o~ are fixed in body i,
td3ct + P2O,)pl IOJCl .-~ P2~,t.Jpl
%%=0, o,@,=0, ,G,o, = 0 .
= qd.'c, + ~wc2 X qwq + ~dJ~2. (25)
Substituting these in Eqs. ( 3 ) - (6) we get
In summary, the velocity contact closure equations are (22)
,g~,o, = /"V~a,, (7)
and (23) and the acceleration contact closure equations are (24)
'a ..... = P,ac,v , (8) and (25).

'w~, = "we, (9)

4 Contact Equations: Velocity Analysis
o,~, = P'~c,. (10) Let the components of the relative linear and angular veloci-
ties of the contacting rigid bodies in the frame P2 (also in the
Consider the closed loop: P2 + cl ~ Pl ~ P2. The triangle frame c2) be given in vector form by
law of vector addition applied to position vectors yields
rqv ~ = rqF I + rt,~p2 . ( 11 ) P2gplp2 = gy P2~a,)pl = ~y .
Vz Wz
Differentiating each term of Eq. ( 11 ) in the reference frame P2
two times, we get
In this section we find expressions for the remaining terms in
P2Vclp2 : PlVclpi + P2(.tdpl X rclpl -t- P2Wplp2 (12) Eqs. (22) and (23) in terms of the contact coordinates and their

Journal of Applied M e c h a n i c s DECEMBER 1996, Vol. 63 / 977

derivatives and thus relate changes in contact coordinates to the Therefore,
relative linear and angular velocities defined above.
The position vector of the contact point c~ of obj i in or is
2 ~ , = 2R~, 2Rq = 0 . (31)
r~o~ = r,,o,(~ ~, ~2) i = 1, 2. (26)
Next we consider the transformations between or and cr, and
between c~ and c2. The rotation matrix ~R~ is quite simply
Here "2ftc, is the angular velocity expressed in the contact
frame c2 in matrix form. Similarly, ,f~c~, in the contact
'R~ = (n)r frame cr, can be simplified to (see A p p e n d i x for intermedi-
i i ate steps) :

= [(Xl)i (X2)i (n)r] [ (~7)-1012 02xl]

.1 (27) o,~, = O,R~O,l~ ' = 0,2 27 (x2)T|
(n)~' j
Note that since the coordinate system is orthogonal, all the
elements of the metric tensor, Gr, are positive and the square
root operation is a valid one. From the definition of ~b, ~Rq
can be seen to be

C~Rq =
sin 0]
- s i n ~O - c o s 0 0
1 2
OX2 *j
On .j
0 0 -1 Otxz

X [(~/)-I
We now proceed to obtain expressions for the velocity terms.
Differentiating Eq. (26) in the or frame, we get the velocity of
the contact point cr in that frame. The resulting equation is
2 We introduce Christoffel symbols, [ij, k[ and F}k and the coef-
d o. 1
'V .... = ~. [,rc, o,({ , ~2)] = ~ (xj)i~i i = 1, 2. (28) ficients of the second fundamental form, Lu, and after some
j=l algebraic manipulation we get

=_[ ]77 ([11, 1],ti, + [12, 11113,) 0

0 --([21, l [ d ~ + [22, 111131)
0 0
1 1 1 1
--([11, 1]lt~l + [12, 111131) ([11,211& + [12, 2]Oh) -- - - {(Lll)lUl + (L12)t131 }
(glt)l (gH)l (g=)l (gll)l

+ 1 1 1 1
-- ([11,211t~ + [12, 2111)1) --([21, 1]lal + [22, 111131) -- - - [(Z21)lal -.}.-(L22)1131}
(gll)l (g22)1 (g22)1 (g22)1
1 1
- - [(LiI)i/il + (Li2)i0l } - - {(Lzl)lal + (L22)101} 0
(g")' (g22),

which can be simplified to

Expressing the velocities of the contact points in contact frame
c2 (using Eq. (28)): 0 (-crlFlOl)
o,~, = (c~FlO~) 0
OlV ~2RclCIRoi[(XI)I~I~-(X2)lOI]~IRI~GIU'1 (29)
L u 1 {(~)-'ClO, })f2 0
where cri = { (g=)i/(gH)i } I/2 for i = 1, 2 are the ratios of the
norms of the natural base vectors. For an olrthonormal system,
or1 = 1. F1 and Lr are 1 2 and 2 2 matrices consisting of
the Christoffel symbols of the second kind and the coefficients
where Uj = [ul 1)1] T and U2 = [b/2 1)2] T. of the second fundamental form, respectively, for obj i. F and
The skew symmetric matrix representation of the angular L are defined as follows:
velocity is directly obtained from the rotation matrix (McCar-
thy, 1990). For any rotation matrix eR.,, its derivative in frame F = [F~i F~d
f is given by [L,, L,2]
f~m = RRT. L = [L21 L22J"

978 / Vol. 63, DECEMBER 1996 Transactions of the ASME

Similarly, L = [Lu 2L12 L2~]

0 ( - ~:r:0:) (r~, - 1 Or~,

F11)r. + - -

02f~C2 = (c~:r:0:) 0
2 2
= (rz21 1 2 1 2 0rt2 Or11
( ( ~ ) - ' L ~ 0 : ] ),~~ 0
- + - + -b-U + T
(33) or~2
( r ~ - F ~~) F ~ + - -
Substituting for ~Vc~o~and lVc,ol from Eqs. ( 2 9 ) - ( 3 0 ) in 0( ~
Eq. (22), expressing each vector in the c~ frame, we get
l FllLu OLii
0~ 1 OLIl~"
FI,Ltz + rl2Ll, O(' 0(2~
r I OLI2 /
In Eq. (23), after substituting for ,w~, from Eq. (32), ~co~ 12L12 0~ 2
from Eq. (33), and ~w~, from Eq. (31), we get =

R(~,)-'L, Oi + I wYwx]= -(qrGa)-'I-aOz F221L21 0~1

0La2 0/12
~,L~ + F~:L~, o~' ~l
Simplifying the above equations we obtain the first-order con-
tact kinematics relating the rate of change of contact coordinates F~2L22- 0~'-'5-
to the rigid-body velocities:
We first consider the linear accelerations of the contact points.
In the oi frame,

'a..... =--ftt[,V~,o,] : (x,)/~' + ~-~)

j=l j,k=l
0 2 = ('~2)--1(/-11 '-[-H~)-' - -Wx +/-It V, (35)

Expressing this in the contact frame c2,

(36) [ (Oxl~ )2
=o (37) ,ac,o, = "2Rc]qRo, (Xl)lal + (X2)11)1 + ~OUl/] (/~1
where Hi = (,fG/)-'Li ( ~ / ~ ) - ' and .H, = R~H,R,.
These equations are the velocity contact equations which
were first presented by Montana (1988). However, our notation
+ L~1 ) 1"1'1)1 -~- 1(01) 2
and derivation are slightly different. First, we use standard dif-
ferential geometric notation (gig, r~, and L u) to describe the
surface properties. Second, we resolve all vectors in the frame o [ + r,W,)]
c2 and not in c~ like Montana. ,a.... = L -EIWI J

5 Contact Equations: Acceleration Analysis where Wi = [ (g~) 2 ( u~0~) (0~) z] r represent the nonlinear veloc-
Let the components of the relative linear and angular acceler- ity terms.
ations of the contacting rigid bodies in the frame c2 be Similarly,

P2 aplp2 = ar P2@, = % . 2ac~2= ~ W2 "

az az
We next find an expression for ,~c,, the skew symmetric
In this section we find expressions for the remaining terms in matrix for the angular acceleration of the frame ci relative to
Eqs. (24) and (25) in terms of the contact coordinates and their the frame oi which is attached to obj i, in the contact frame ci.
derivatives in order to relate changes in contact coordinates to
the relative linear and angular accelerations defined above. ol~~c i = o,Re,To.,R~,-
"" (,R~,
o To ,Rc,)(

It is convenient to define other matrix functions of the surface
properties (analogous to F and L). = o,R,.,
To. ""
,Rc~- o ,f~ o.'~c~

We substitute from (32) for ,[]c, and from (27) for ~Rc,,and
r= Lr , after considerable simplification (see Appendix) we obtain

Journal of Applied Mechanics DECEMBER 1996, Vol. 63 / 979

o.. [ o (-a,(r,O, + P,w,))
,fie, = (cri(FiOi + Y'iWi)) 0 ((~ffb-' (L W,o- L,O,)m,]
L ((~/)-l(~im -- riui))lTx2

Note that the 1 3 matrix ~ and the 2 3 matrix ~ representing

third-order surface properties enter into the equations.
Other terms in Eqs. (24) and (25) in the c2 frame are as L R,E,H,.~I~ - E~H2"~ J

I O~E, R S ~ O , x [ R~Ei ( ~ / ~ ) - 1~1

]w,+[ El ( ~ 2 )
I ~2 J W2
P2COplX lVc~ol = ( y)
__CO T
+ [-2cozEiRo~.~ 0

P2COpl X lcocl = o2x r02x lax/l} (43)
-(;~Y) TRIpEI(~.~I)-ILI~-yl J \ - cox/ ~ Ozxl J


2COCl Equation (40) is an acceleration constraint equation which can
be rearranged in the following manner:

where a z = ~1 w' .--I-~L2W2 -Jr-2 Rq;f~lO,. (45)

- - x

6 Examples
Substituting the relevant expressions in Eq. (24) we get 6.1 Two Spheres in Contact. Here we derive the first
and second-order contact kinematic equations for two spheres
~ 2 ( ~ J 2 + F2W2) = R,~i(U 1 "b F,Wl) in contact (obj 1 and obj 2). The choice of coordinates and
notation is according to Section 2.3. Using Eqs. ( 3 4 ) - ( 3 6 ) we
obtain the following first-order contact kinematic equations:
+ 2weEIR,J~U~ + (39)
t~l t92 [COx sin t/, + coy cos ~p]

~W2 = -T, iW1 + 2

()T --COY Rt//~IU 1 + a~.

Pt + P2
(40) + [Vy sin tp - Vx cos q,]
COx (46)
Pl + Pz
And from Eq. (25), we get
1)1 = [p2(0.)x COS ~/ -- 0)y sin 0 )
R~oE1( ' ~ 1 ) - 1 (K1W1 _ LiO, ) -- R 0 ( ~ i ) -'LIUlcoz (Pl + 102) sin uj
+ (Vxsin~0 + Vycos~p)] (47)
+ chF, O, + = ____ __(~/~)-tL20<~b
COx Oly f12 -- Plcoy + Vx (48)
Pl + P2
+ Ei(~G2)-i(Lw2 -- L2G) (41)
- Plcox + Vy
~2 = (49)
-o'i(FiU ' + ~lWl) - RoEi(~i)-ILiOI + olz (P~ + P2) sin u2

~ b = - c o ~ + ~ l [ t a vi
n Pp'+p2 (cox cs ~ - coy sin qt)
= -~b + cr2(F2G + r2W2). (42)
Solving Eqs. (39), (41), and (42), we obtain the five second- + Vx sin ~p + Vy cos 0 ]
order contact kinematic equations relating double derivatives (50)
of the contact coordinates to the rigid-body velocities and accel- Pl + P2 J
At the acceleration level, Eq. (43) can be written in the form

980 / Vol. 63, DECEMBER 1996 Transactions of the ASME

iil I pz sin @ sin u, sin u~ p2 cos O sin ul sin u2 c o s O s i n u , sinu2 sinOsinulsinuz a~

u'a = A
I p2 cos tp sin u2
Pl sin ul
p2 sin ~b sin u2
pt sin ul sin u2
sin ~ sin u2
sin Ul sin u2
where h is a nonlinear function of the rigid-body velocities and the rates of change of contact coordinates given by
cos q, sin u2
sin ul -(()) %
+h (51)

~//~2 + tOz/~l COS if/ -- tOyO 1 COS b/1 -- /~101 COS []/ COS U 1 + 0202 COS U 2 -- tOz01 sin ~ sin u~,
h = tO~0t cos u~ - tO~a~ sin ~ + a101 cos ul sin ~ - tO~0~ cos ~ sin Ul + ~b02 sin u2,
2w,plal sin ~ - 2pla101 cos u~ sin ~b + 2tOePlO1 COS ~b sin ul - p10~ cos ~b cos ul sin ul + p20~ cos u2 sin u2,
2to~ptal cos ~ - 2plat01 cos ~ cos ut - 2p2a202 cos uz - 2to,p1% sin ~ sin ul + p~0~2 cos u~ sin ~ sin ul

and 6.2 A Sphere Contacting a Plane. We consider the ex-

ample of a plane (obj 1) contacting a sphere (obj 2). The
A = (Pl + P2) sin ul sin u2. velocity equations are
From Eq. (44), we get al = p(wy cos 0 + tax sin ~O - Vx cos ~p + Vy sin ~p)
~ ~- --OLz + ~1 COS U 1 + U2 COS ~2
% = p ( - W y sin ~ + cox cos ~b + Vx sin 0 + Vy cos 0 )
- tOy(a1 sin ~ + Ol. sin ul cos ~ ) - /~lOl sin //~1
/~2 = tOy
- u2~2 sin u2 + tO~(01 sin Ul sin q, - a~ cos ,) (52)
02 = --tOx CSC /'/2
and the acceleration constraint from Eq. (40) is
= -(tO~ cot u2 + tOt).
az = - ( p l a ~ 2 + /91(01 sin ul) 2 + p21i22
The acceleration equations are
+ p2(02 sin u2) ~ - 2p~to~al + 2pltO~% sin Ul). (53)
a'l = p~b(a2 sin tp + 02 sin u2 cos ~O)
The development in Cai and Roth (1987) deals with the
special case in which the coordinate curves on the two surfaces - pa2% cos u2 sin ~b - p(02) 2 sin u2 cos u2 cos ~b - 2tOz01
are aligned so that the angle ~b is zero. For this special case,
we get the same results by substituting + p(ax sin ~b + ay cos ~b) - (a~ cos ~b - ay sin ~b)

71" 71" ih = p~b(a2 cos ~b - % sin uz sin 0 )

I[/ = 0 , Ul : ~ , 1)1 = O, U2 = ~ , 1)2=0
- pa202 cos u2 cos ~b + p(O2) 2 sin u2 cos u2 sin ~ + 2Wza~
in our equations above.
+ p(a~ cos O - ay sin ~b) + (a~ sin ~b + ay cos ~b)
P2tOy- V~
/~1 -- - - a2 = 02~b sin u2 + O/y
Pl + P2
P2tO~ + Vy 1)2 = a 2 ~ CSC U 2 -- a202 cot u2 - ax csc u2
Pl + P2 = 1)2 sin u2 -- a2~2 COS u2 -- %.

//-/2 = PltOy + V.
Pt + P2 7 Kinematics of Rolling Contact
Two bodies are said to be in a condition of roiling contact if
- PltOx + Vy
02 = the velocity of the point of contact on one body is equal to the
Pl + P2 velocity of the point of contact on the other body. In other
words, the sliding velocity (Cai and Roth, 1986; Cai and Roth,
-a~ + o~;,p2 - tO,tO,P2 - 2wzvy
Ul = 1987) or the relative velocity between the points of contact is
Pl + P2 zero. This definition is well known and can be found in standard
kinematics texts (see, for example, Paul, 1979). Because this
iJl = ay + Olxp 2 + toytozP2 -- 2tozVx
definition imposes a condition on velocities (and not on higher
Pl + P2 order derivatives), it defines rolling contact up to the first order
a~ + O~ypl + tO~tOzpl (Cai and Roth, 1987). We will refer to this as the first-order
a2= condition f o r rolling.
pl + p2 However, if the relative velocity between the points of con-
tact, in addition to being zero, stays constant through a small
i~2 = ay - axpl + WytOzpl time interval, the two bodies are in a condition of rolling contact
Pl + P2 up to the second order. In other words, the derivative of the
The paper (Cai and Roth, 1987) does not deal with the rate of sliding velocity is zero. We will refer to this as the second-
change of ~b which evolves (at ~b = 0) according to the equations order condition f o r rolling.
In Fig. 6, Pi and/~i are points attached to obj i. p~ is the point
I~= --tOz of contact at time t while/5~ is the contact point at time L t and
are considered to be time instants that are separated by a small
-- tOX(tOyPl + I)X) interval At.
Pl + P2 For the first-order condition f o r rolling we have

Journal of Applied Mechanics DECEMBER 1996, Vol. 63 / 981

It can be shown (Sarkar, 1993) that the third scalar equation
is actually the same equation as (45), provided the first-order
ohj,1 PlyJ , obj 1 condition for rolling is satisfied.
In addition to the rolling conditions defined previously, if we
impose another condition known as the no-spin condition, we
achieve what is called pure rolling (Johnson, 1985, p. 242).
The pivoting component of the angular velocity is zero for pure
rolling (Neimark and Fufaev, 1972, p. 18). In other words, in
the frame c2,

at time [ at time t ~ = 0. (61)

Fig. 6 Roiling motion of two rigid bodies For pure rolling up to the second order, the derivative of the
pivoting velocity or the z component of the derivative of the
relative angular velocity must vanish (Sarkar, 1993).
P2Vp,p2(t ) = O. (54)
a~ = 0 (62)
In the contact frame c2 this is
7.1 Example: Two S p h e r e s in Contact. In the previous
section we considered the example of a sphere contacting an-
Vy =0. (55) other sphere. The conditions for rolling up to the first order are
Vx=O G=0 v~=0
For the second-order condition for rolling we have For rolling up to the second order, from Eq. (60),

-(p~a~ sin qJ + P21)l COS @ ) W z ]

--(pla! COS ~ -- p21)l)~z J (63)
(p~U~ sin ~ + P21)1cos ~b)~x + (pd~l cos ~t -- /921)1 sin $)~y

where di and 1)~are given by Eqs. (46) - (49). Further, for pure
"W,,,pd t ) - ~V,~,~(~) rolling, we have
lim = 0. (56)
<,-~)~o t -
~z = 0 (64)
We rewrite Equation (56) to get
az = 0. (65)
" ~ G , ~ ( t) - P~V~,,p~(~)
lim This yields, upon substitution for ai and 0i from Eqs. ( 4 6 ) -
(t-~)~o t -- (49),
~V,,,p~(~) - ~V,~,,5~(~) ax = 0 (66)
+ lim = 0. (57)
(t-~)~o t --
ay = 0 (67)
The first term of the above equation is nothing but P~ap~p2(t).
Recognizing that P2Vj~,~2 = P2V~,p~, the second term can be writ- (w~ + Wy2)p,p2
a~ (68)
ten as lim (-P2wp, x Ptr&pJt -- t). AS &t ~ 0 Pl + P2

Pl rlsIpl 8 Discussion
lim = -PlVclpl.
(t-~)~o t - - We have derived and presented in this paper the velocity
equations (Eqs. ( 3 4 ) - ( 3 7 ) ) and the acceleration equations
Thus, we get (Eqs. ( 4 3 ) - ( 4 4 ) ) for contact between two three-dimensional
P2aptp2(t ) + P2~pl X (PlVclpl) -- O, (58) bodies. This is the first time that the equations have been pre-
sented in this general form. The work by Cai and Roth (1986;
Because 1987) comes closest to the work presented here. They derived
similar equations that are valid for a Cartesian coordinate system
PtVclm = ~Vqoj, whose origin is coincident with the contact frame. However,
Eq. (58) simplifies to because of this assumption, their coordinate covering of the
contacting surfaces changes as the contact point moves and as
P2aplp2 ( t ) : --PzO.)pl X l Vclol . (59) the relative orientation between the two rigid bodies changes. In
our approach the surface coordinates are independent of where
This is the second-order condition for rolling. In the contact contact occurs and the relative orientation between the two
frame c2, it becomes bodies. Also, our coordinate systems are, in general, curvilinear,
- (- E,R,~/~G)~z" and include the special case of a Cartesian coordinate system.

taxi ay
= ( Wy ) r R , X O
\ -- o) x /
t (60)
Finally, the equations derived by Cai and Roth predict the mo-
tion of the contact point over each surface. However, they do
not consider the evolution of the fifth contact coordinate ~/,.
The special case of pure rolling is particularly interesting
in robotic applications. Rolling contact is generally preferred
where we have substituted the expression for ,Vqo~ from Eq. because it is more efficient and also because rolling motion is
(29) and P~o:p~ = [wx O~ywz] r. easier to control. In order to maintain a condition of rolling the

982 / Vol. 63, DECEMBER 1996 Transactions of the ASME

relative motion between the two contacting bodies must be such Acknowledgment
that the relative acceleration at the contact point is given by This work was in part supported by the grants NSF MSS-
Eq. (60). In general, this acceleration includes tangential (a~ 91-57156, ARPA N0014-88-K-0632 and NATO grant No.
and ay) components as well as a normal component (a~). This 0224/85.
is unlike the planar case in which the tangential acceleration
must be zero for rolling. In three dimensions, only in the special References
Beggs, J. S., 1966, Advanced Mechanism, Macmillan, New York.
case of pure rolling do the tangential components have to be Brock, D. L., 1988, "Enhancing the Dexterity of a Robot Hand Using Con-
zero. The application of the rolling constraint equations to the trolled Slip," Proceedings of the IEEE International Conference on Robotics and
derivation of the equations of motion and the controller for a Automation, Philadelphia, PA, pp. 249-251.
two-arm manipulation system is presented in Sarkar et al. Cai, C., and Roth, B., 1986, "On the planar motion of rigid bodies with point
contact," Mechanism and Machine Theory, Vol. 21, No. 6, pp. 453-466.
(1993) and Sarkar (1993). Cai, C., and Roth, B., 1987, "On the spatial motion of rigid bodies with point
The acceleration level equations depend on first, second, and contact," Proceedings of the IEEE International Conference on Robotics and
third-order properties of the contacting surfaces. In other words, Automation, Raleigh, NC, pp. 686-695.
Cai, C., and Roth, B., 1988, "On the spatial motion of rigid bodies with
we encounter upto the third derivative of the coordinate map, line contact," Proceedings of IEEE International Conference on Robotics and
f, for each object in these equations. Similarly, the velocity Automation. Philadelphia, PA, pp. 1036-1041.
equations depend on first and second order properties. It is Hall, A. S., 1966, Kinematics and Linkage Design, Bait Publishers, West Lafa-
yette, IN.
worth noting that while an object such as a sphere can be Johnson, K. L., 1985, Contact Mechanics. Cambridge University Press, London.
described by a second-order equation in the Cartesian space, it Kane, T. R., and Levinson, D. A., 1985, Dynamics, theory and applications,
does not mean that all third-order partial derivatives are zero. McGraw-Hill, New York.
Kirson, Y., and Yang, A. T., 1978, "Instantaneous lnvariants in Three-Dimen-
In fact, as shown earlier in the example of a sphere, the coeffi- sional Kinematics," ASME JOURNALOF APPLIEDMECHANICS,Vol. 45, pp. 409-
cients, gij, F ~ , a n d L o are not constant and the partial derivatives, 414.
(02x~ / 0~ k0~ J), are, in general, nonzero. Kumar, V., and Waldron, K. J., 1989, "Actively coordinated mobility sys-
Finally a comment about the two examples considered in the tems," ASME Journal of Mechanisms, Transmissions, and Automation in Design,
Vol. 1ll, No. 2 pp. 223-231.
paper. At first sight they may appear trivial and may not serve Lipschutz, M. M., 1969, Differential Geometry, McGraw-Hill, New York.
to provide adequate justification for the theoretical development Mason, M., and Salisbury, J. K., 1985, Robot Hands and the Mechanics of
in this paper. But Eqs. ( 3 4 ) - ( 3 6 ) and Eqs. ( 4 3 ) - ( 4 5 ) are Manipulation, MIT Press, Cambridge, MA.
McCarthy, J. M., 1990, An Introduction to Theoretical Kinematics, MIT Press,
completely general, and they work for any contacting surfaces. Cambridge, MA.
The only information needed are the local differential geometric Millman, R. S., and Parker, G. D., 1977, Elements of Differential Geometry,
surface properties (the coefficients of the first and second funda- Prentice-Hall Englewood Cliffs, NJ.
Montana, D. J., 1988, "The kinematics of contact and grasp," The International
mental forms and their derivatives). The main motivation for Journal of Robotics Research, Vol. 7, No. 3, pp. 17-32.
presenting these "simpler examples" is because for these exam- Neimark, J. I., and Fufaev, N. A., 1972, Dynamics of Nonholonomic Systems,
pies, the equations are well known for planar motions, and for American Mathematical Society, Providence, RI.
Pars, L. A., 1968, A Treatise on Analytical Dynamics, John Wiley and Sons,
spatial motions under the condition of pure rolling. Therefore, New York.
for these examples, there is some opportunity for comparing Paul, B., 1979, Kinematics and Dynamics of Planar Machinery, Prentice-Hall,
the general contact kinematics equations derived here with these Englewoods Cliffs, NJ.
Rosenauer, N., and Willis, A. H., 1953, Kinematics of Mechanisms, Associated
special equations. And in fact, it can be seen from Section 2.3.1 General Publications Pty Ltd., Sydney, Australia.
that, even in these "simple" cases, the derivatives of g~, F~ Sarkar, N., 1993, "Control of Mechanical Systems with Roiling Contacts:
and L 0 do not vanish and that a spherical surface does not really Applications to Robotics," Ph.D. thesis, Department of Mechanical Engineering
and Applied Mechanics, University of Pennsylvania, Philadelphia, PA.
simplify the second-order equations. Sarkar, N., Yun, X., and Kumar, V., 1993, "Dynamic control of 3-D rolling
Current work addresses the application of these equations to in multi-arm manipulation," Proceedings of the IEEE International Conference
better understand the stability of multi-contact grasps and the on Robotics and Automation, Atlanta, GA, pp. 978-983.
Stoker, J. J., 1969, D~ffferential Geometry, John Wiley and Sons, New York.
optimization of fixtures for restraining three-dimensional ma- Whittaker, E. T., 1988, A Treatise on the Analytical Dynamics of Particles &
chine components. Rigid Bodies, Cambridge University Press, Cambridge, UK.

A Expressions for o,~c, and o,~c~
For any rotation matrix IR,,, its derivatives in frame f is given by

i n . , = I~R r

I~,. = KR r _ f~f~

Thus in contact frame ci,

o,~c, = ,.,R~O,R
i ci

o ~0 [ (~/~)-'
o f~,, = ,Rc, ,Rc, = L 012 (X2) T1 [(X|)i (X2) / (n)i]
1 (n)TJ 0,2 o

2 Oxt .j
Ox2 .j
On .j
1 __1fl "

We introduce Christoffel symbols, [ij, k] and F}k and the coefficients of the second fundamental form, Lij, and simplify the
derivative of the metric tensor through Eq. ( 1 ). After some algebraic manipulation we get

Journal of Applied Mechanics DECEMBER 1996, Vol. 63 / 983

[----~----([11, 1]lR, + [12, 111'0,)




1],& + [22, 1]lOt)


--([11, 1],Ri + [12, 11l~,) -- ([11,211& + [12, 21101) --- { (Ll.),Ul + (L12)tS. }
(g.). (gH)i (g22), (gH)l
1 1 1 1
-- ([11, 2],fil -4- [12, 2]~3,) --([21, 1],g, + [22, 1If0,) - - { (./.a.),g, + (L22),0, }
(gl,), (g=)l (g=)l (g22)1
1 1
- - { ( L u ) l & + (L12)11)1 } - - { ( Z 2 1 ) l / g 1 "4- (L22)101 } 0
(g.), (g22),
which can be simplified to
0 (--~iFIOl)
o,~, = (~,rlo,) o {-(~)-iL'o')}='| q (69)

{ (t. G i ) - i L i U i } ) iTx2 0 J
where cr~ = { ( g z E ) ~ / ( g H ) , } ~:2 for i = 1, 2 are the ratios of the norms of the natural base vectors.
In order to obtain an expression for o,h~, we start with Eq. (69).

o- [ 1 ] (x=)~/(n)/T_] (~-~)f ,=1

~ (~7)f /='
~ (~'7)f ] [ ((x/G/)-')01x2 02,0]
t 0,X2

+ ,,~:, + Voe'oCl?" \~:, + ,=,

2 Oxl ; On j

o (-o-,r,G) {-(~,)-'L,O,}ax~ 0 (-~,F,G)

{ -- ( ~ i ) - l L i O i }2X1
- (~riGG) 0 x (GiFiVi) 0
{6/~7)-'L,G }fx~ o [ (,/~7)-*L,o, }f2 0
After simplification, we finally write ,he. in the following compact form:
0 (-oi(FiG + riWi))

,h~, = (o/(Fig + PiWi)) 0 ((,~)-'(LW,o - c,8,))21] .

((~)-i(Lw, - a,g))fx2

984 / Vol. 63, DECEMBER 1996 Transactions of the ASME
