This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

## References

1.
Marsden
,
J. E.
, and
Ratiu
,
T. S.
,
1999
,
Introduction to Mechanics and Symmetry
,
Springer-Verlag
,
New York
.
2.
Bloch
,
A. M.
,
,
P. S.
,
Marsden
,
J. E.
, and
Murray
,
R. M.
,
1996
, “
Nonholonomic Mechanical Systems With Symmetry
,”
Arch. Ration. Mech. Anal.
,
136
(
1
), pp.
21
99
.
3.
Bloch
,
A. M.
,
2003
,
Nonholonomic Mechanics and Control
,
Springer
, New York.
4.
Koon
,
W. S.
, and
Marsden
,
J. E.
,
1997
, “
The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems
,”
Rep. Math. Phys.
,
40
(
1
), pp.
21
62
.
5.
Chhabra
,
R.
, and
Emami
,
M. R.
,
2016
, “
A Unified Approach to Input-Output Linearization and Concurrent Control of Underactuated Open-Chain Multi-Body Systems With Holonomic and Nonholonomic Constraints
,”
J. Dyn. Control Syst.
,
22
(
1
), pp.
129
168
.
6.
Chhabra
,
R.
,
2016
, “
Dynamical Reduction and Output-Tracking Control of the Lunar Exploration Light Rover (LELR)
,” 2016
IEEE
,
Aerospace Conference
, Mar. 5–12, Big Sky, MT.
7.
Bullo
,
F.
, and
Zefran
,
M.
,
2002
, “
On Mechanical Control Systems With Nonholonomic Constraints and Symmetries
,”
Syst. Control Lett.
,
45
(
2
), pp.
133
143
.
8.
Olfati-Saber
,
R.
,
2001
, “
Nonlinear Control of Underactuated Mechanical Systems With Application to Robotics and Aerospace Vehicles
,”
Ph.D. thesis
, Massachusetts Institute of Technology, Cambridge, MA.
9.
Ferrario
,
C.
, and
Passerini
,
A.
,
2000
, “
Rolling Rigid Bodies and Forces of Constraint: An Application to Affine Nonholonomic Systems
,”
Meccanica
,
35
(
5
), pp.
433
442
.
10.
Sun
,
W.
,
Wu
,
Y. Q.
, and
Sun
,
Z. Y.
,
2014
, “
Tracking Control Design for Nonholonomic Mechanical Systems With Affine Constraints
,”
Int. J. Autom. Comput.
,
11
(
3
), pp.
328
333
.
11.
Fassó
,
F.
, and
Sansonetto
,
N.
,
2015
, “
Conservation of Energy and Momenta in Nonholonomic Systems With Affine Constraints
,”
Regular Chaotic Dyn.
,
20
(
4
), pp.
449
462
.
12.
Noether
,
E.
,
1918
, “
Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen
,” Mathematisch-Physikalische Klasse, pp.
235
257
.
13.
Marsden
,
J. E.
, and
Weinstein
,
A.
,
1974
. “
Reduction of Symplectic Manifolds With Symmetry
,”
Rep. Math. Phys.
,
5
(
1
), pp.
121
130
.
14.
Routh
,
E. J.
,
1882
,
A Treatise on the Dynamics of a System of Rigid Bodies. With Numerous Examples: The Elementary Part
,
Macmillan
, London.
15.
Marsden
,
J. E.
,
1992
,
Lectures on Mechanics
,
Cambridge University Press
, New York.
16.
Planas-Bielsa
,
V.
,
2004
, “
Point Reduction in Almost Symplectic Manifolds
,”
Rep. Math. Phys.
,
54
(
3
), pp.
295
308
.
17.
Marsden
,
J. E.
,
Misiolek
,
G.
, and
Ortega
,
J. P.
,
2007
,
Hamiltonian Reduction by Stages
, 1st ed.,
Springer-Verlag
,
Berlin
.
18.
Chhabra
,
R.
, and
Emami
,
M. R.
,
2015
, “
Symplectic Reduction of Holonomic Open-Chain Multi-Body Systems With Constant Momentum
,”
J. Geom. Phys.
,
89
, pp.
82
110
.
19.
Koon
,
W. S.
, and
Marsden
,
J. E.
,
1998
, “
Poisson Reduction for Nonholonomic Mechanical Systems With Symmetry
,”
Rep. Math. Phys.
,
42
(1–2), pp.
101
134
.
20.
Cendra
,
H.
,
Marsden
,
J. E.
, and
Ratiu
,
T. S.
,
2001
,
Lagrangian Reduction by Stages
, Vol.
722
,
American Mathematical Society
, Providence, RI.
21.
Marsden
,
J. E.
, and
Scheurle
,
J.
,
1993
, “
The Reduced Euler–Lagrange Equations
,”
Fields Inst. Commun.
,
1
, pp.
139
164
.
22.
Marsden
,
J. E.
, and
Scheurle
,
J.
,
1993
, “
Lagrangian Reduction and the Double Spherical Pendulum
,”
Z. Angew. Math. Phys.
,
44
(
1
), pp.
17
43
.
23.
Chaplygin
,
S.
,
2008
, “
On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier Theorem
,”
Regular Chaotic Dyn.
,
13
(
4
), pp.
369
376
[Matematicheskiĭ sbornik, 28(1) (1911)].
24.
Koiller
,
J.
,
1992
, “
Reduction of Some Classical Non-Holonomic Systems With Symmetry
,”
Arch. Ration. Mech. Anal.
,
118
(
2
), pp.
113
148
.
25.
van der Schaft
,
A. J.
, and
Maschke
,
B. M.
,
1994
, “
On the Hamiltonian Formulation of Nonholonomic Mechanical Systems
,”
Rep. Math. Phys.
,
34
(
2
), pp.
225
233
.
26.
Yoshimura
,
H.
, and
Marsden
,
J. E.
,
2006
, “
Dirac Structures in Lagrangian Mechanics Part II: Variational Structures
,”
J. Geom. Phys.
,
57
(
1
), pp.
209
250
.
27.
Cendra
,
H.
,
Marsden
,
J. E.
, and
Ratiu
,
T. S.
,
2001
, “
Geometric Mechanics, Lagrangian Reduction and Nonholonomic Systems
,”
Mathematics Unlimited-2001 and Beyond
,
Springer-Verlag
, Berlin, pp.
221
273
.
28.
Chhabra
,
R.
, and
Emami
,
M. R.
,
2014
, “
Nonholonomic Dynamical Reduction of Open-Chain Multi-Body Systems: A Geometric Approach
,”
Mech. Mach. Theory
,
82
, pp.
231
255
.
29.
Ohsawa
,
T.
,
Fernandez
,
O. E.
,
Bloch
,
A. M.
, and
Zenkov
,
D. V.
,
2011
, “
Nonholonomic Hamilton-Jacobi Theory Via Chaplygin Hamiltonization
,”
J. Geom. Phys.
,
61
(
8
), pp.
1263
1291
.
30.
Hochgerner
,
S.
, and
Garcia-Naranjo
,
L.
,
2009
, “
G-Chaplygin Systems With Internal Symmetries, Truncation, and an (Almost) Symplectic View of Chaplygin's Ball
,”
J. Geom. Mech.
,
1
(1), pp.
35
53
.
31.
Bates
,
L.
, and
Śniatycki
,
J.
,
1993
, “
Nonholonomic Reduction
,”
Rep. Math. Phys.
,
32
(
1
), pp.
99
115
.
32.
Cushman
,
R.
,
Kemppainen
,
D.
,
Śniatycki
,
J.
, and
Bates
,
L.
,
1995
, “
Geometry of Nonholonomic Constraints
,”
Rep. Math. Phys.
,
36
(
2/3
), pp.
275
286
.
33.
Cushman
,
R.
, and
Śniatycki
,
J.
,
2002
, “
,”
Reg. Chaotic Dyn.
,
7
(
1
), pp.
61
72
.
34.
Cushman
,
R.
,
Duistermaat
,
H.
, and
Śniatycki
,
J.
,
2009
,
Geometry of Nonholonomically Constrained Systems
,
World Scientific Publishing Company
, Singapore.
35.
Śniatycki
,
J.
,
1998
, “
Nonholonomic Noether Theorem and Reduction of Symmetries
,”
Rep. Math. Phys.
,
42
(
1/2
), pp.
5
23
.
36.
Śniatycki
,
J.
,
2001
, “
Almost Poisson Spaces and Nonholonomic Singular Reduction
,”
Rep. Math. Phys.
,
48
(
1/2
), pp.
235
248
.
37.
Gay-Balmaz
,
F.
, and
Yoshimura
,
H.
,
2015
, “
Dirac Reduction for Nonholonomic Mechanical Systems and Semidirect Products
,”