## Rolling Bodies with Regular Surface: Controllability Theory and Applications (2000)

Citations: | 29 - 5 self |

### BibTeX

@MISC{Marigo00rollingbodies,

author = {Alessia Marigo and Antonio Bicchi},

title = {Rolling Bodies with Regular Surface: Controllability Theory and Applications},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

Pairs of bodies with regular rigid surfaces rolling onto each other in space form a nonholonomic system of a rather general type, posing several interesting control problems of which not much is known. The nonholonomy of such systems can be exploited in practical devices, which is very useful in robotic applications. In order to achieve all potential benefits, a deeper understanding of these types of systems and more practical algorithms for planning and controlling their motions are necessary. In this paper, we study the controllability aspect of this problem, giving a complete description of the reachable manifold for general pairs of bodies, and a constructive controllability algorithm for planning rolling motions for dexterous robot hands. Index Terms---Nonholonomic systems, nonlinear controllability theory, robotic manipulation. I. INTRODUCTION N ON-HOLONOMIC systems have been attracting much attention in control literature recently, due to both their relevance to practical ap...

### Citations

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Citation Context ...ch a reasonably satisfactory understanding has been reached recently is the class of two--inputs nilpotentizable systems that can be put, by feedback transformation, in the so-- called "chained&q=-=uot; form [8]-=-. A complete characterization of such systems (i.e., necessary and sufficient conditions for the existence of a feedback transformation to chained-- form) has been provided by Murray [9], while an alg... |

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Citation Context ... plate-ball system is a 5--dimensional, 2--input nonholonomic system, which is not differentially flat ([19]), and can be regarded as an instance of the famous 5--variables problem of Cartan. Montana =-=[20]-=- derived a differential--geometric model of the rolling constraint between general bodies, and discussed applications to robotic manipulation; Li and Canny [21] showed that the plate--ball system is c... |

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Citation Context ...nfiguration �� q, the control system (13) has a singularity in �� q. Due to analyticity of the manifold and of the control equations, there will be maximal integral submanifolds for the system=-= motion [24]-=-. However, the dimension of these submanifolds may vary from one (as in the case of two cylinders rolling on each other with aligned axes), to five (e.g., a cone rolling on a cylinder, starting with a... |

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Citation Context ...bly applied. As for the problem of feedback stabilization to a point, time--varying or nonsmooth feedback schemes have been proposed that achieve the goal for chained systems (see for instance Samson =-=[17]-=- and Sordalen and Egeland [18], and references therein). Unfortunately, the system of rolling bodies considered in this paper differs substantially from the class of chained form systems. Consider e.g... |

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Citation Context ...5--variables problem of Cartan. Montana [20] derived a differential--geometric model of the rolling constraint between general bodies, and discussed applications to robotic manipulation; Li and Canny =-=[21]-=- showed that the plate--ball system is controllable, and that the same holds for two rolling spheres, provided that their radius is different. Jurdjevic [22] studied the problem of finding the path th... |

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Citation Context ...cribed by several authors: in his early work Brockett [12] used sinusoidal inputs, that were subsequently investigated in more detail by Murray and Sastry [8]. The methods of Lafferriere and Sussmann =-=[13]-=-, Monaco and Normand--Cyrot [14], and Jacobs [15], using piecewise IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED OCTOBER 29, 1998 3 constant inputs in different arrangements, are particularly well... |

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Citation Context ...s early work Brockett [12] used sinusoidal inputs, that were subsequently investigated in more detail by Murray and Sastry [8]. The methods of Lafferriere and Sussmann [13], Monaco and Normand--Cyrot =-=[14]-=-, and Jacobs [15], using piecewise IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED OCTOBER 29, 1998 3 constant inputs in different arrangements, are particularly well--suited to chained systems, whe... |

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Citation Context .... Examples of systems designed according to such philosophy have been reported by Brockett [1], Nakamura and Mukherjee [2], Ostrowski and Burdick [3], Sordalen and Nakamura [4], Bicchi and Sorrentino =-=[5]-=-. Nonholonomy of rolling is particularly relevant to robotic manipulation, one of the main goals of which is to manipulate an object grasped by a robot end--effector so as to relocate and reorient it ... |

27 |
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Citation Context ...ssary coordinate transform has been presented by Tilbury, Murray, and Sastry [10]. As an example, a car pulling an arbitrary number of trailers has been shown to be a chained--form system by Sordalen =-=[11]-=-. Planning algorithms for chained--form systems in free space have been described by several authors: in his early work Brockett [12] used sinusoidal inputs, that were subsequently investigated in mor... |

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Citation Context ...ystems that are feedback equivalent to chained or nilpotent forms, only iterative planning schemes (such as the generic loops method of Sontag [26], or the continuation method of Sussmann and Chitour =-=[27]-=-) exist for general nonholonomic systems. Because of their generality, such schemes offer limited performance in computational terms, and more efficient algorithms should be seeked that exploit any st... |

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Citation Context ...or not shown in the picture. Examples of systems designed according to such philosophy have been reported by Brockett [1], Nakamura and Mukherjee [2], Ostrowski and Burdick [3], Sordalen and Nakamura =-=[4]-=-, Bicchi and Sorrentino [5]. Nonholonomy of rolling is particularly relevant to robotic manipulation, one of the main goals of which is to manipulate an object grasped by a robot end--effector so as t... |

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Citation Context ...on, while there exist efficient planning algorithms for systems that are feedback equivalent to chained or nilpotent forms, only iterative planning schemes (such as the generic loops method of Sontag =-=[26]-=-, or the continuation method of Sussmann and Chitour [27]) exist for general nonholonomic systems. Because of their generality, such schemes offer limited performance in computational terms, and more ... |

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Citation Context ...d to be maximally nonholonomic. For a system which is maximally nonholonomic in all points of a connected subset V ae M, any two configurationssq 0 ; q 1 in V can be linked through an admissible path =-=[0; 1]-=- !M; s ! q(s) such that q(0) = q 0 ; q(1) = q 1 , and A(q(s))sq(s) = 0; 8s 2 [0; 1]. Manuscript submitted October, 1998. The work has been conducted with partial support of NATO CR grant no. 960750 an... |

22 |
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Citation Context ...ions to robotic manipulation; Li and Canny [21] showed that the plate--ball system is controllable, and that the same holds for two rolling spheres, provided that their radius is different. Jurdjevic =-=[22]-=- studied the problem of finding the path that minimizes the length of the curve traced out by the sphere on the fixed plane. It turns out that optimal paths also minimize the integral of their geodesi... |

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Citation Context ...whole gripper, by an actuator not shown in the picture. Examples of systems designed according to such philosophy have been reported by Brockett [1], Nakamura and Mukherjee [2], Ostrowski and Burdick =-=[3]-=-, Sordalen and Nakamura [4], Bicchi and Sorrentino [5]. Nonholonomy of rolling is particularly relevant to robotic manipulation, one of the main goals of which is to manipulate an object grasped by a ... |

15 | Planning Motions of Rolling Surfaces
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(Show Context)
Citation Context ...ossess an interesting property when one of the bodies has planar surface (as it happens, e.g., in the dexterous grippers of Figures 1 and 2, ). In fact, the following proposition was shown to hold in =-=[28]-=-: Theorem 3: The kinematic equations (13) of a strictly convex body rolling on a planar surface are feedback equivalent to a strictly triangular form. In other words, for (13) with say surface 2 a pla... |

14 | Arbitrary path tracking control of articulated vehicles using nonlinear control theory - Sampei, Tamura, et al. - 1995 |

11 |
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Citation Context ...urve traced out by the sphere on the fixed plane. It turns out that optimal paths also minimize the integral of their geodesic curvature, so that solutions are those of Euler's elastica problem. Levi =-=[23]-=- gave explicit formulas for evaluating the final configuration of the ball after a circular motion of the plate. The paper is organized as follows: in section II we derive a complete mathematical mode... |

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6 |
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Citation Context ...tion. In fact, while rolling between the surfaces of the manipulated object and that of fingers has been previously regarded as a complication to be neglegted, or compensated for, work of Cole et al. =-=[6] and Bicch-=-i et al. [5] tried to exploit rolling for achieving dexterity with simpler mechanical hardware. In particular, several prototype dexterous grippers have been developed at Centro "E. Piaggio"... |

5 |
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Citation Context ...stics of the system (such as e.g. controllability). Simple (continuous, time--invariant) feedback control laws, on the other hand, can not be applied to stabilizing nonholonomic (nonsingular) systems =-=[7]-=-. An important class of nonholonomic systems for which a reasonably satisfactory understanding has been reached recently is the class of two--inputs nilpotentizable systems that can be put, by feedbac... |

4 | On the Recti cation of Vibratory Motion - Brockett - 1989 |

3 |
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Citation Context ...ional mechanics, brought to the attention of the control community by Brockett and Dai [12]. The plate-ball system is a 5--dimensional, 2--input nonholonomic system, which is not differentially flat (=-=[19]-=-), and can be regarded as an instance of the famous 5--variables problem of Cartan. Montana [20] derived a differential--geometric model of the rolling constraint between general bodies, and discussed... |

3 | do Carmo, Di erential Geometry of - P - 1976 |

2 |
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(Show Context)
Citation Context ...ertical translation of the whole gripper, by an actuator not shown in the picture. Examples of systems designed according to such philosophy have been reported by Brockett [1], Nakamura and Mukherjee =-=[2]-=-, Ostrowski and Burdick [3], Sordalen and Nakamura [4], Bicchi and Sorrentino [5]. Nonholonomy of rolling is particularly relevant to robotic manipulation, one of the main goals of which is to manipul... |

2 | Exploiting nonholonomic redundancy of free ying space robots - Nakamura, Mukherjee - 1993 |

2 | Conversion of the kinematics of a car withn trailers into a chained form - Sordalen - 1993 |

2 | A continuation method for nonholonomic path- nding problem - Sussmann - 1993 |

1 |
Motion planning by piecewise constant or polynomial inputs
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(Show Context)
Citation Context ...kett [12] used sinusoidal inputs, that were subsequently investigated in more detail by Murray and Sastry [8]. The methods of Lafferriere and Sussmann [13], Monaco and Normand--Cyrot [14], and Jacobs =-=[15]-=-, using piecewise IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED OCTOBER 29, 1998 3 constant inputs in different arrangements, are particularly well--suited to chained systems, where they achieve e... |

1 |
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(Show Context)
Citation Context ...chieve exact planning (only approximate, iterative planning schemes are obtained in the general case). Further, chained systems are differentially flat, and therefore the techniques of Rouchon et al. =-=[16]-=- can be profitably applied. As for the problem of feedback stabilization to a point, time--varying or nonsmooth feedback schemes have been proposed that achieve the goal for chained systems (see for i... |

1 | Kinematics and control of amulti ngered robot hand with rolling contact - Cole, Hauser, et al. - 1989 |

1 | Exensions of di erentially at elds and liouvillian systems - Chelouah - 1997 |

1 | Orbits of families of vector elds and integrability of distributions - Sussmann - 1973 |