A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

## References

1.
Newton
,
I.
,
1686, Philosophiae Naturalis Principia Mathematica
,
Royal Society Press
,
London
.
2.
Poisson
,
S. D.
,
1827
, “
Note sur i'extension des fils et des plaques élastiques
,”
Ann. Chim. Phys.
,
36
, pp.
384
387
.
3.
Stronge
,
W. J.
,
1990
, “
Rigid Body Collisions With Friction
,”
Proc. R. Soc. London A
,
431
(
1881
), pp.
168
181
.
4.
Stronge
,
W. J.
,
2000
,
Impact Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
5.
Cross
,
R.
,
2010
, “
Impact of a Ball on a Surface With Tangential Compliance
,”
Am. J. Phys.
,
78
(
7
), pp.
716
720
.
6.
Jia
,
Y.-B.
,
2013
, “
Three-Dimensional Impact: Energy-Based Modeling of Tangential Compliance
,”
Int. J. Rob. Res.
,
32
(
1
), pp.
56
83
.
7.
Brach
,
R. M.
,
1989
, “
Rigid Body Collisions
,”
ASME J. Appl. Mech.
,
56
(
1
), pp.
133
137
.
8.
Smith
,
C. E.
,
1991
, “
Predicting Rebounds Using Rigid-Body Dynamics
,”
Trans. ASME
,
58
(
3
), pp.
754
758
.
9.
Glocker
,
G.
, and
Pfeiffer
,
F.
,
1995
, “
Multiple Impacts With Friction in Rigid Multibody Systems
,”
Nonlinear Dyn.
,
7
(
4
), pp.
471
497
.
10.
Stewart
,
D. E.
,
2000
, “
Rigid-Body Dynamics With Friction and Impact
,”
SIAM Rev.
,
42
(
1
), pp.
3
39
.
11.
Chatterjee
,
A.
, and
Ruina
,
A.
,
1998
, “
A New Algebraic Rigid-Body Collision Law Based on Impulse Space Considerations
,”
ASME J. Appl. Mech.
,
65
(
4
), pp.
939
951
.
12.
Routh
,
E. J.
,
1905
,
Dynamics of a System of Rigid Bodies
,
MacMillan
,
London
.
13.
Wang
,
Y.
, and
Mason
,
M. T.
,
1992
, “
Two-Dimensional Rigid-Body Collisions With Friction
,”
ASME J. Appl. Mech.
,
59
(
3
), pp.
635
642
.
14.
Darboux
,
G.
,
1880
, “
Etude géométrique sur les percussions et le choc des corps
,”
Bull. Sci. Math. Astron.
,
4
(
1
), pp.
126
160
.
15.
Keller
,
J. B.
,
1986
, “
Impact With Friction
,”
ASME J. Appl. Mech.
,
53
(
1
), pp.
1
4
.
16.
Zhao
,
Z.
, and
Liu
,
C.
,
2007
, “
The Analysis and Simulation for Three-Dimensional Impact With Friction
,”
Multibody Syst. Dyn.
,
18
(
4
), pp.
511
530
.
17.
Zhang
,
Y.
, and
Sharf
,
I.
,
2007
, “
Rigid Body Impact Modeling Using Integral Formulation
,”
ASME J. Comput. Nonlinear Dyn.
,
2
(
1
), pp.
98
102
.
18.
Bhatt
,
V.
, and
Koechling
,
J.
,
1995
, “
Three-Dimensional Frictional Rigid-Body Impact
,”
ASME J. Appl. Mech.
,
62
(
4
), pp.
893
898
.
19.
Batlle
,
J. A.
,
1996
, “
The Sliding Velocity Flow of Rough Collisions in Multibody Systems
,”
Trans. ASME
,
63
(
3
), pp.
804
809
.
20.
Elkaranshawy
,
H. A.
,
2007
, “
Rough Collision in Three-Dimensional Rigid Multi-Body Systems
,”
Proc. Inst. Mech. Eng., Part K
,
221
(
4
), pp.
541
550
.
21.
Kane
,
T. R.
, and
Levinson
,
D. A.
,
1985
,
Dynamics: Theory and Applications
,
McGraw-Hall
,
New York
.
22.
Pressley
,
A.
,
2001
,
Elementary Differential Geometry
,
Springer-Verlag
,
London
.
23.
Liu
,
C.
,
Zhao
,
Z.
, and
Brogliato
,
B.
,
2008
, “
Frictionless Multiple Impacts in Multibody Systems. I. Theoretical Framework
,”
Proc. R. Soc. London A
,
464
(
2100
), pp.
3193
3211
.
24.
Jia
,
Y.-B.
,
Mason
,
M.
, and
Erdmann
,
M.
,
2013
, “
Multiple Impacts: A State Transition Diagram Approach
,”
Int. J. Rob. Res.
,
32
(
1
), pp.
84
114
.