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Autonomous Robots
Erschienen in: Autonomous Robots 4/2012

01.05.2012

A dynamical system approach to realtime obstacle avoidance

verfasst von: Seyed Mohammad Khansari-Zadeh, Aude Billard

Erschienen in: Autonomous Robots | Ausgabe 4/2012

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Abstract

This paper presents a novel approach to real-time obstacle avoidance based on Dynamical Systems (DS) that ensures impenetrability of multiple convex shaped objects. The proposed method can be applied to perform obstacle avoidance in Cartesian and Joint spaces and using both autonomous and non-autonomous DS-based controllers. Obstacle avoidance proceeds by modulating the original dynamics of the controller. The modulation is parameterizable and allows to determine a safety margin and to increase the robot’s reactiveness in the face of uncertainty in the localization of the obstacle. The method is validated in simulation on different types of DS including locally and globally asymptotically stable DS, autonomous and non-autonomous DS, limit cycles, and unstable DS. Further, we verify it in several robot experiments on the 7 degrees of freedom Barrett WAM arm.
Vorheriger Artikel Improving biped walk stability with complementary corrective demonstration

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Fußnoten
1
The development of Eq. (4) was partly inspired by the complex potential function that models the uniform flow around a circular cylinder (Milne-Thomson 1960). In both formulations the modulation of the flow due to the object’s presence decreases quadratically with the distance to the center of the object (see the second term in Eq. (4)). The main difference between the two approaches lies in their functionality. Equation (4) is a d-dimensional vector and its Jacobian is a d×d matrix which can be used to modulate the original flow. In contrast, the complex potential function is a scalar value, and its derivative directly gives the modified flow in the presence of the obstacle.
 
2
In case \(\partial\varGamma (\tilde{\xi}) / \partial\xi_{1}\) vanishes, the vectors are no longer linearly independent and one should choose another index for the derivative which is non-zero.
 
3
Derivation of Eqs. (15)–(16) are inspired from the proof of Theorem 1. For a spherical obstacle, these equations yield to the same result given by Eq. (6).
 
4
From Theorem 2 we know that the normal velocity at the boundary points vanishes. Hence, if f(ξ) is aligned with the normal vector of the tangential hyperplane at a boundary point, we have \(M(\tilde{\xi })f(\xi)=0\).
 
5
One can also define different safety factors along the positive and negative directions of each object’s axis by considering an if-else condition on the sign of each \(\tilde{\xi}_{i}\).
 
6
Equation (25) is in spirit very similar to the weighting coefficients proposed in Waydo and Murray (2003) with the difference that we use Γ k (ξ) to compute weights (rather than the distance between the obstacles).
 
7
For example, we consider the k-th obstacle is locally relevant in the current position of the robot if: \(| \lambda_{i}^{k}(\tilde{\xi}^{k}) - 1 | > \varsigma, \forall i = 1..d\), where ς is a small positive threshold.
 
8
The same principle can be used if the SEDS motions are modeled with a second or higher order DS.
 
9
Note that the motions across θ i , i=3..7 would become uncoupled if the obstacles were placed at θ o,1=[−100;45;0;60;0;−30;0] and θ o,2=[−80;45;0;60;0;−30;0].
 
10
Note that this paper does not claim that the cyclic behavior is always preserved in the presence of the obstacles.
 
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Metadaten
Titel
A dynamical system approach to realtime obstacle avoidance
verfasst von
Seyed Mohammad Khansari-Zadeh
Aude Billard
Publikationsdatum
01.05.2012
Verlag
Springer US
Erschienen in
Autonomous Robots / Ausgabe 4/2012
Print ISSN: 0929-5593
Elektronische ISSN: 1573-7527
DOI
https://doi.org/10.1007/s10514-012-9287-y

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