## Controllability Tests for Mechanical Systems with Symmetries and Constraints (1997)

Venue: | J. Applied Mathematics and Computer Science |

Citations: | 15 - 8 self |

### BibTeX

@ARTICLE{Ostrowski97controllabilitytests,

author = {Jim Ostrowski and Joel Burdick},

title = {Controllability Tests for Mechanical Systems with Symmetries and Constraints},

journal = {J. Applied Mathematics and Computer Science},

year = {1997},

volume = {7},

pages = {101--127}

}

### OpenURL

### Abstract

This paper derives controllability tests for a large class of mechanical systems characterized by nonholonomic constraints and symmetries. Recent research in geometric mechanics has led to a single, simplified framework that describes this class of systems, which includes examples such as wheeled mobile robots; undulatory robotic and biological locomotion systems, such as paramecia, inchworms, and snakes; and the reorientation of satellites and underwater vehicles. This geometric framework has also been applied to more unusual examples, such as the snakeboard robot, the wobblestone, and the reorientation of a falling cat. Using modern results from nonlinear control theory, we develop accessibility and controllability tests based on this reduced geometric structure. We also discuss parallels between these tests and the construction of open-loop control algorithms, with analogies to the generation of locomotive gaits for robotic systems. 1 Introduction M ECHANICAL systems provide a f...

### Citations

287 | Nonholonomic motion planning: Steering using sinusoids
- Murray, Sastry
- 1993
(Show Context)
Citation Context ...oretical bridge between systems with two different types of nonholonomic constraints. On one hand, there are systems with external (often called kinematic) constraints which include wheeled vehicles (=-=Murray and Sastry, 1993-=-), grasping with point-finger contacts, and some models of snakes (Krishnaprasad and Tsakiris, 1994), paramecia (Shapere and Wilczek, 1989), and even legged locomotion (Goodwine and Burdick, 1996; Kel... |

166 | Nonholonomic mechanical systems with symmetry
- Bloch, Krishnaprasad, et al.
- 1996
(Show Context)
Citation Context ...of these problems fall generally into two realms: one in which certain of the conservation laws may remain after the addition of constraints, such as in the rolling penny or the constrained particle (=-=Bloch et al., 1996-=-); and one in which the conservation laws are transformed into what Bloch et al. term a generalized momentum equation, where the momenta are governed by a differential equation. There is strong eviden... |

121 |
Lie Algebras and Lie Groups
- Serre
- 1965
(Show Context)
Citation Context ...der to use these results, we first need to develop a notion of degree of a Lie bracket. This development can be done much more formally using the formalism of free Lie algebras (see (Ostrowski, 1995; =-=Serre, 1992-=-)), but instead we rely on common intuition to provide the necessary understanding of what is meant by the two definitions of degree developed here. Let X = (X 0 ; : : : ; Xm ) be a finite set of vect... |

96 |
Control and stabilization of nonholonomic dynamic systems
- Bloch, Reyhanoglu, et al.
- 1992
(Show Context)
Citation Context ...c constraints. We will highlight some of the more recent theoretical results on controllability for these types of systems here. While certain initial results do exist for dynamic mechanical systems (=-=Bloch et al., 1992-=-), they generally require that the unconstrained dynamics be fully actuated. While this is a stronger result than those derived for kinematic systems, these assumptions still require the continual mot... |

74 | Nonlinear Dynamical Control Systems - Nijmeijer, Schaft - 1990 |

70 | Burdick: Nonholonomic mechanics and locomotion: The snakeboard example
- Ostrowski, Lewis, et al.
- 1994
(Show Context)
Citation Context ...example, we will examine the snakeboard model, which has been an important motivating example behind the theoretical progress for this mixed kinematic and dynamic constraint case (Bloch et al., 1996; =-=Ostrowski et al., 1994-=-; Ostrowski et al., 1995). 2 Background and Problem Formulation The use of Lie groups will be important for the analysis performed in this paper. The principal motivation for using Lie groups arises f... |

61 | The geometric mechanics of undulatory robotic locomotion
- Ostrowski, Burdick
- 1998
(Show Context)
Citation Context ...sent velocities in the unconstrained directions. Doing so is usually an ad-hoc procedure that we have found to be greatly improved by using the formalism and structure of Lie groups (Ostrowski, 1995; =-=Ostrowski and Burdick, 1996-=-b). For systems in which the Lagrangian and the constraints are left-invariant, i.e., for which L(\Phi h q; T q \Phi hsq) = L(q;sq) and ! i (q)sq = ! i (h \Gamma1 q)T q \Phi h \Gamma1sq; forall h 2 G;... |

58 |
The Mechanics and Control of Undulatory Robotic Locomotion
- Ostrowski
- 1995
(Show Context)
Citation Context ...rdinates to represent velocities in the unconstrained directions. Doing so is usually an ad-hoc procedure that we have found to be greatly improved by using the formalism and structure of Lie groups (=-=Ostrowski, 1995-=-; Ostrowski and Burdick, 1996b). For systems in which the Lagrangian and the constraints are left-invariant, i.e., for which L(\Phi h q; T q \Phi hsq) = L(q;sq) and ! i (q)sq = ! i (h \Gamma1 q)T q \P... |

51 |
Geometry of self-propulsion at low Reynolds number
- Shapere, Wilczek
- 1989
(Show Context)
Citation Context ...n called kinematic) constraints which include wheeled vehicles (Murray and Sastry, 1993), grasping with point-finger contacts, and some models of snakes (Krishnaprasad and Tsakiris, 1994), paramecia (=-=Shapere and Wilczek, 1989-=-), and even legged locomotion (Goodwine and Burdick, 1996; Kelly and Murray, 1994). Recent work by Kelly and Murray (Kelly and Murray, 1994) provides kinematically constrained models for a wide range ... |

46 |
Isoholonomic problems and some applications
- Montgomery
- 1990
(Show Context)
Citation Context ...tum conservation laws. Examples of systems with internal nonholonomic constraints include satellites in space (Krishnaprasad, 1990; Nakamura and Mukherjee, 1993) and the problem of the "falling c=-=at" (Montgomery, 1990-=-). Naturally, there exist problems for which both internal and external constraints may exist and interact in a nontrivial manner. Examples of these problems fall generally into two realms: one in whi... |

35 | G-snakes: nonholonomic kinematic chains on Lie groups
- Krishnaprasad, Tsakiris
- 1994
(Show Context)
Citation Context .... Doing so leads to a stronger comprehension of the mechanics of locomotion, but leaves open some very basic questions about the control of such systems. Preliminary research (Kelly and Murray, 1994; =-=Krishnaprasad and Tsakiris, 1994-=-) suggests that the geometric tools used to formulate the mechanics will also provide a basis for unlocking the answers to many of the questions regarding the control theoretic issues involved. An imp... |

28 |
Symmetrical gaits of horses
- Hildebrand
- 1965
(Show Context)
Citation Context ...ect of locomotion that is intricately related to the study of control for these types of systems. A very common observation of locomotion is that it is most often generated by cyclical shape changes (=-=Hildebrand, 1965-=-; Collins and Stewart, 1992). The motion takes on a characteristic form, called a gait. Definition 6.1 A locomotive gait is a specified cyclic pattern of internal shape changes (inputs) which couple t... |

19 |
Aspects of Geometric Mechanics and Control of Mechanical Systems
- LEWIS
(Show Context)
Citation Context ... of a Lie bracket. There are several results on generating a complete set (basis) of iterated Lie brackets, e.g., a Philip Hall basis (Murray and Sastry, 1993; Serre, 1992). We use a result given in (=-=Lewis, 1995-=-): Proposition 5.2 Every element of the free Lie algebra LX can be written as a linear combination of repeated brackets of the form [X k ; [X k\Gamma1 ; [: : : ; [X 2 ; X 1 ] : : : ] ] ]; where X i 2 ... |

15 | 1995] The Mechanics of Undulatory Locomotion: The Mixed Kinematic and Dynamic Case
- Ostrowski, Burdick, et al.
- 1945
(Show Context)
Citation Context ... the snakeboard model, which has been an important motivating example behind the theoretical progress for this mixed kinematic and dynamic constraint case (Bloch et al., 1996; Ostrowski et al., 1994; =-=Ostrowski et al., 1995-=-). 2 Background and Problem Formulation The use of Lie groups will be important for the analysis performed in this paper. The principal motivation for using Lie groups arises from our studies of robot... |

14 |
Geometric phases and optimal reconfiguration for multibody systems
- Krishnaprasad
- 1990
(Show Context)
Citation Context ...etimes called dynamic) constraints on the system, which very often take the form of momentum conservation laws. Examples of systems with internal nonholonomic constraints include satellites in space (=-=Krishnaprasad, 1990; Nakamura-=- and Mukherjee, 1993) and the problem of the "falling cat" (Montgomery, 1990). Naturally, there exist problems for which both internal and external constraints may exist and interact in a no... |

13 | High-order small-time local controllability
- Kawski, Kawski
- 1990
(Show Context)
Citation Context ...r showing small-time local controllability of nonlinear control systems with drift (Sussman, 1987). For a nice review of the issues involved in local controllability tests, the reader is referred to (=-=Kawski, 1990-=-). Also, since the structure of the equations was largely motivated by developments in locomotion, some mention will be given to the relationship between the controllability tests derived here and tra... |

13 |
Exploiting Nonholonomic Redundancy of Free-Flying Space Robots
- Nakamura, Mukherjee
- 1993
(Show Context)
Citation Context ...) constraints on the system, which very often take the form of momentum conservation laws. Examples of systems with internal nonholonomic constraints include satellites in space (Krishnaprasad, 1990; =-=Nakamura and Mukherjee, 1993) and the -=-problem of the "falling cat" (Montgomery, 1990). Naturally, there exist problems for which both internal and external constraints may exist and interact in a nontrivial manner. Examples of t... |

12 |
Symmetry-breaking bifurcation: a possible mechanism for 2:1 frequency-locking in animal locomotion
- Collins, Stewart
- 1992
(Show Context)
Citation Context ...that is intricately related to the study of control for these types of systems. A very common observation of locomotion is that it is most often generated by cyclical shape changes (Hildebrand, 1965; =-=Collins and Stewart, 1992-=-). The motion takes on a characteristic form, called a gait. Definition 6.1 A locomotive gait is a specified cyclic pattern of internal shape changes (inputs) which couple to produce a net motion. One... |

9 | Controllability with unilateral control inputs
- Goodwine, Burdick
- 1996
(Show Context)
Citation Context ...hicles (Murray and Sastry, 1993), grasping with point-finger contacts, and some models of snakes (Krishnaprasad and Tsakiris, 1994), paramecia (Shapere and Wilczek, 1989), and even legged locomotion (=-=Goodwine and Burdick, 1996-=-; Kelly and Murray, 1994). Recent work by Kelly and Murray (Kelly and Murray, 1994) provides kinematically constrained models for a wide range of systems that locomote and controllability results for ... |

5 |
Con controllability of a class of mechanical systems
- LEWIS, MURRAY
- 1995
(Show Context)
Citation Context ...e condition that (oesrsr ) ii j 0 in Proposition 5.4 may appear slightly artificial, it is required in order to satisfy Sussman's criterion for controllability. In fact, research by Lewis and Murray (=-=Lewis and Murray, 1995-=-) suggest that similar conditions may be needed for general mechanical systems. They study accessibility and controllability for unconstrained mechanical systems, and report similar conditions on thes... |

5 |
Computing reduced equations for mechanical systems with constraints and symmetries
- Ostrowski
- 1999
(Show Context)
Citation Context ...sent velocities in the unconstrained directions. Doing so is usually an ad-hoc procedure that we have found to be greatly improved by using the formalism and structure of Lie groups (Ostrowski, 1995; =-=Ostrowski and Burdick, 1996-=-b). For systems in which the Lagrangian and the constraints are left-invariant, i.e., for which L(\Phi h q; T q \Phi hsq) = L(q;sq) and ! i (q)sq = ! i (h \Gamma1 q)T q \Phi h \Gamma1sq; forall h 2 G;... |

3 |
Geometric Phases and Locomotion. Available electronically via http://avalon.caltech.edu/cds/reports/cds94014
- Kelly, Murray
- 1994
(Show Context)
Citation Context ...ng and insightful manner. Doing so leads to a stronger comprehension of the mechanics of locomotion, but leaves open some very basic questions about the control of such systems. Preliminary research (=-=Kelly and Murray, 1994-=-; Krishnaprasad and Tsakiris, 1994) suggests that the geometric tools used to formulate the mechanics will also provide a basis for unlocking the answers to many of the questions regarding the control... |

1 | Geometric Phases and Locomotion - appear, CIT, et al. - 1995 |

1 |
Nonholonomic control and guage theory. Pages 343-- 378 of
- Montgomery
- 1993
(Show Context)
Citation Context ... the geometric relationship between the two wheels, and the paths they must follow. 3.2 Unconstrained Systems with Symmetries In the same manner as for the principal kinematic case above, Montgomery (=-=Montgomery, 1993) sh-=-owed that similar tests can be used to show controllability for an unconstrained dynamical system with Lie group symmetries. His result applies to the case where the spatial momentum �� is zero (a... |

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