A dynamical system approach to realtime obstacle avoidance

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.

Appendices

Appendix A: Proof of Theorem 1

Consider a hyper-surface corresponding to boundary points of a hyper-sphere obstacle in ℝ^d with a center ξ ^o and a radius r ^o. Impenetrability of the obstacle’s boundaries is ensured if the normal velocity at boundary points vanishes:

(30)

where n(ξ ^b) is the unit normal vector at a boundary point ξ ^b:

(31)

The eigenvalue decomposition of the square matrix $M^{s}(\tilde{\xi },r^{o})$ is given by:

$$M^s\bigl(\tilde{\xi},r^o\bigr) = V^s\bigl(\tilde{\xi},r^o\bigr) \: D^s\bigl(\tilde{\xi },r^o\bigr) \: V^s\bigl(\tilde{\xi},r^o\bigr)^{(-1)} $$

(32)

where $D^{s}(\tilde{\xi},r^{o})$ is a d×d diagonal matrix composed of the eigenvalues:

$$\begin{cases}\lambda^1 = 1 - \frac{r^2}{\tilde{\xi}^T\tilde{\xi}} \\\lambda^i = 1 + \frac{r^2}{\tilde{\xi}^T\tilde{\xi}} \quad \forall i\in2..d \\\end{cases} $$

(33)

and $V^{s}(\tilde{\xi},r^{o}) = [\upsilon^{1} \ \cdots\ \upsilon^{d}]$ is the matrix of eigenvectors with:

$$\begin{cases}\upsilon^1 = \tilde{\xi} \\\upsilon^i_j =\begin{cases}-\tilde{\xi}_i & j = 1\\\tilde{\xi}_1 & j = i\\0 & j \neq1,i\\\end{cases} \ \forall i \in2..d , \; j \in1..d\end{cases} $$

(34)

Substituting Eqs. (31), (32) and (7) into Eq. (30) yields:

(35)

Since ξ ^b is equal to the first eigenvector of $V^{s}(\tilde{\xi }^{b},r^{o})$, Eq. (35) reduces to:

(36)

where [0] _d−1 is a zero column vector of dimension d−1. For all points on the obstacle boundary, the first eigenvalue is zero, i.e. λ ¹=0, . Thus, we have:

(37)

Appendix B: Proof of Theorem 2

The proof of Theorem 2 follows directly from that of Theorem 1. Observe that:

$$n\bigl(\xi^b\bigr)^T \: \dot{\xi^b} = n\bigl(\xi^b\bigr) E\bigl(\tilde{\xi}^b,r^o\bigr) \: D\bigl(\tilde{\xi}^b,r^o\bigr) \: E\bigl(\tilde{\xi}^b,r^o\bigr)^{(-1)} f(.) $$

(38)

Considering the fact that n(ξ ^b) is equal to the first eigenvector of $E(\tilde{\xi}^{b},r^{o})$, and the first eigenvalue is zero for all points on the obstacle boundary yields:

(39)