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## A and Hussein I, Optimal Control of Underactuated Nonholonomic Mechanical Systems

Venue: | IEEE Transactions on Automatic Control |

Citations: | 13 - 2 self |

### Citations

832 | Introduction to Mechanics and Symmetry,
- Marsden, Ratiu
- 1994
(Show Context)
Citation Context ...nge of coordinates, complexity of notation and the lack of a geometric picture. For more on differential-geometric mechanics and its use in the context of dynamics and control, we refer the reader to =-=[7]-=-, [3], [4]. For the treatment of under-actuated systems using affine connections, we refer the reader to [8]. Aside from [9], [10], previous results usually treat kinematic systems that usually aim at... |

759 |
Morse Theory,
- Milnor
- 1963
(Show Context)
Citation Context ... that the curvature tensor, R, associated with the unconstrained connection ∇, satisfies the identity given by equation (1). Since W is arbitrary, and hence independent of q, then [W,v] = 0 such that =-=[19]-=- ∇W∇vv = ∇v∇Wv +R (W,v)v. Finally, we conclude that ∇W∇̄vv = (∇WP) (∇vv) +P (∇v∇Wv +R (W,v)v) . Using this identity, we obtain δJ = ∫ T 0 gE ( ∇̄Wu,u + ξ ♯ gE ) + µ ( d dt W−∇Wv ) + η ( P (∇v∇Wv +R (W... |

568 |
An Introduction to Differentiable Manifolds and Riemannian Geometry.
- Boothby
- 1986
(Show Context)
Citation Context ...ion we give brief definitions of the various objects from affine connection theory that are essential to this paper. For more complete studies, we refer the reader to the mathematically-oriented text =-=[14]-=- or the more mechanically-oriented text [4]. Let Q be a smooth (C∞) Riemannian manifold with the Riemannian metric defined by gq : TqQ × TqQ → R at some point q ∈ Q, where TQ = ∪qTqQ is the tangent bu... |

226 |
Nonholonomic Mechanics and Control,
- Bloch
- 2003
(Show Context)
Citation Context ...s Professor of Mathematics at the University of Michigan, Ann Arbor, abloch@umich.edu. Nonholonomic mechanical control systems have a long and complex history which is described in, for example, [2], =-=[3]-=- (in particular, Chapter 5) and [4]. Of much interest in the present work are the recent developments that utilize a geometric approach [5], [3] and, in particular, the theory of affine connections [6... |

215 | Nonholonomic mechanical systems with symmetry,
- Bloch, Krishnaprasad, et al.
- 1996
(Show Context)
Citation Context ...l be squandered by creating only more reaction forces (that maintain the constraints) with no net useful motion, or by violating the constraints altogether. For example, for the rolling vertical coin =-=[5]-=-, if excessive torque is applied in the rolling direction, the rolling constraint may be violated. Moreover, the application of any side forces will not contribute to the net motion of the system due ... |

140 |
Geometric Control of Mechanical Systems,
- Bullo, Lewis
- 2004
(Show Context)
Citation Context ...niversity of Michigan, Ann Arbor, abloch@umich.edu. Nonholonomic mechanical control systems have a long and complex history which is described in, for example, [2], [3] (in particular, Chapter 5) and =-=[4]-=-. Of much interest in the present work are the recent developments that utilize a geometric approach [5], [3] and, in particular, the theory of affine connections [6], [4]. These methods offer a coord... |

109 |
do Carmo, Riemannian Geometry,
- P
- 1992
(Show Context)
Citation Context ...Wv) dt, where we made use of the fact that P∗η = η and that η ∈ D∗. Next, we refer the reader to the properties of a curvature tensor R̃ defined in terms of the connection ∇̃ and a metric g̃ found in =-=[20]-=-, Proposition 2.5 on page 91. From these properties, one can show that the curvature satisfies g̃ ( R̃ (W,v)v,X ) = g̃ ( R̃ (X,v)v,W ) , where R̃ is the curvature tensor based on a connection ∇̃ that ... |

73 | Lagrangian reduction by stages,
- Cendra, Marsden, et al.
- 2001
(Show Context)
Citation Context ... and Lagrange’s multiplier method relate to results based on the momentum equation form that appear in [9], [10]. For more on systems with symmetry we refer the reader to, for example, [1], [5], [7], =-=[12]-=-, [13] and references therein. 1These packages are available online for which a reference is provided in [11]. The paper is arranged as follows. In Section II, we briefly describe how nonholonomic mec... |

49 |
Simple mechanical control systems with constraints,”
- Lewis
- 2000
(Show Context)
Citation Context ...3] (in particular, Chapter 5) and [4]. Of much interest in the present work are the recent developments that utilize a geometric approach [5], [3] and, in particular, the theory of affine connections =-=[6]-=-, [4]. These methods offer a coordinate-free differential approach to mechanics and control that avoids many of the issues that arise in classical mechanics such singularity and change of coordinates,... |

45 | Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction
- Koon, Marsden
- 1997
(Show Context)
Citation Context ... and its use in the context of dynamics and control, we refer the reader to [7], [3], [4]. For the treatment of under-actuated systems using affine connections, we refer the reader to [8]. Aside from =-=[9]-=-, [10], previous results usually treat kinematic systems that usually aim at minimizing energy. In this paper the cost function is the square of the norm of the total applied control. We treat second ... |

40 |
Applicable Differential Geometry,
- Crampin, Pirani
- 1988
(Show Context)
Citation Context ...optimal curve q(t). In the above expression, we used the fact that ∇̃Xλ (Z) = ( ∇̃Xλ ) (Z) + λ ( ∇̃XZ ) , for any affine connection ∇̃, vector fields X and Z and any co-vector field λ (see page 78 in =-=[18]-=-). When λ is a Lagrange multiplier, say µ or η, the terms (∇Wµ) (q̇ − v) and (∇Wη) ( ∇̄vv − P (u) ) give us the equations of motion (8) later when we set δJ equal to zero. Hence, as usually done in op... |

24 | Computing reduced equations for robotic systems with constraints and symmetries
- Ostrowski
- 1999
(Show Context)
Citation Context ...hicle, such as robots on wheels and or tracks. The fact that most of these robotic systems apply torques and forces internal to the system, which makes these system move in an undulatory fashion (see =-=[1]-=- and references therein for more on undulatory locomotion), without the application of any external forces, makes the system under-actuated. In fact, control inputs that are applied through the shape ... |

18 | Simple mechanical control systems with constraints and symmetry
- Cortés, Martínez, et al.
(Show Context)
Citation Context ...the Constrained Affine Connection In this section we introduce the affine connection viewpoint of mechanical control systems. The discussion presented here is based on the material found in [6], [4], =-=[16]-=-. Let Q be a C∞ n-dimensional manifold with the tangent and cotangent bundles denoted by TQ and T∗Q, respectively. An under-actuated constrained simple mechanical control system is given by the quadru... |

14 |
Lagrangian mechanics in invariant form,
- Vershik, Faddeev
- 1981
(Show Context)
Citation Context ... affine connection and is given by ∇̄XY = ∇XY + (∇XQ) (Y) = P (∇XY) +∇X (Q(Y)) , (9) for all X,Y ∈ TQ. Note that ∇̄XY ∈ D for all Y ∈ D and X ∈ TQ [4], [6]. The constrained connection also appears in =-=[17]-=-. We now give further properties of the nonholonomic connection ∇̄, in particular, how they operate on functions and one-forms. Lemma II.1. ∇̄Xf = ∇Xf for all f ∈ C∞(Q). Proof. This is obvious since f... |

11 |
A.M.: Optimal control on Riemannian manifolds with potential fields
- Hussein, Bloch
- 2004
(Show Context)
Citation Context ...This should be realized in order to completely (and rigorously) separate variations in configuration, velocity and control variables. For more on this, see for example the discussion in Section II in =-=[21]-=- and the definitions of the operators B and δ therein. It turns out that if we simply ignore this separation step and treat ∇Wu and ∇Wv as terms that involve variations in control and velocity variabl... |

8 |
The Geometry of Cubic Polynomials on Riemannian Manifolds
- Camarinha
- 1996
(Show Context)
Citation Context ...q(vq,vq). A Riemannian connection on Q, denoted∇, is a mapping that assigns to any two smooth vector fields X and Y on Q a new vector field, ∇XY. For the properties of ∇, we refer the reader to [14], =-=[15]-=-, [3]. The operator ∇X, which assigns to every vector field Y the vector field ∇XY, is called the covariant derivative of Y with respect to X. The Lie bracket of the vector fields X and Y will be deno... |

7 |
Reduction, Reconstruction and Optimal Control for Nonholonomic Mechanical Systems with Symmetry
- Koon
- 1997
(Show Context)
Citation Context ...its use in the context of dynamics and control, we refer the reader to [7], [3], [4]. For the treatment of under-actuated systems using affine connections, we refer the reader to [8]. Aside from [9], =-=[10]-=-, previous results usually treat kinematic systems that usually aim at minimizing energy. In this paper the cost function is the square of the norm of the total applied control. We treat second order ... |

3 |
On the history of the development of the nonholonomic dynamics
- Borisov, Mamaev
- 2002
(Show Context)
Citation Context ...och is Professor of Mathematics at the University of Michigan, Ann Arbor, abloch@umich.edu. Nonholonomic mechanical control systems have a long and complex history which is described in, for example, =-=[2]-=-, [3] (in particular, Chapter 5) and [4]. Of much interest in the present work are the recent developments that utilize a geometric approach [5], [3] and, in particular, the theory of affine connectio... |

2 |
Optimal control of under-actuated systems: A coordinate-free approach
- Hussein, Bloch
- 2005
(Show Context)
Citation Context ...metric mechanics and its use in the context of dynamics and control, we refer the reader to [7], [3], [4]. For the treatment of under-actuated systems using affine connections, we refer the reader to =-=[8]-=-. Aside from [9], [10], previous results usually treat kinematic systems that usually aim at minimizing energy. In this paper the cost function is the square of the norm of the total applied control. ... |

1 |
Supplementary chapters for Geometric
- Bullo, Lewis
(Show Context)
Citation Context ... data, one can specialize the result to the specific problem at hand. This process can be automated using symbolic manipulation packages such as Mathematicar and toolboxes such as those introduced in =-=[11]-=-1. While most of the systems appearing in robotics naturally posses symmetries with respect to a group action, which leads to the reduced equations of motion for the system, in this work we provide a ... |

1 |
trajectory tracking on the group of rigid body motions
- “Optimal
- 2005
(Show Context)
Citation Context ...nection by ∇. Since the metric is coordinate independent, the unconstrained Christoffel symbols Γijk are all zero. The curvature on SE(2) is identically zero. (For why curvature is zero on SE(2), see =-=[22]-=-.) For the constrained system, one can check that the Christoffel symbols corresponding to the constrained connection ∇̄ are given by Γ̄113 = −Γ̄ 2 23 = sin(2q3), Γ̄ 1 23 = Γ̄ 2 13 = − cos(2q3), (22) ... |