Decidability of Robot Manipulation Planning: Three Disks in the Plane

Abstract

This paper considers the problem of planning collision-free motion of three disks in the plane. One of the three disks, the robot, can autonomously translate in the plane, the other two move only when in contact with the robot. This represents the abstract formulation of a manipulation planning problem. Despite the simplicity of the formulation, the decidability of the problem had remained unproven so far. We prove that the problem is decidable, i.e., there exists an exact algorithm that decides whether a solution exists in finite time.

B. Mishra—This work is supported by the EU FP7 ICT-287513 SAPHARI project.

Appendix

In this section we propose a constructive geometric proof of the reduction property for paths in configuration space constrained by contact between the robot and both objects. Preliminary to this proof is the conceptual illustration of the contact manifolds.

1.1 Single-Contact Manifold

Paths corresponding to motion in contact with only one object lie in a 5-dimensional manifold immersed in that foliates with the position of the obstacle that is not in contact. On each leaf the reduction property in [5] can be applied to transform any path in contact into a sequence of transfer and transit paths. In principle, there exist two identical spaces of this kind, one for each object, and they are transversal to each other. We call these spaces $\mathcal{C}_{c_1}$ and $\mathcal{C}_{c_2}$. Figure 5 provides a conceptual illustration of $\mathcal{C}_{c_1}$ and the paths in $\mathcal{C}_{c_1}$ and $\mathcal{C}_{c_1} \cap \mathcal{C}_{c_2}$ represented in $\mathcal{C}_{c_1}$.

Fig. 6

Illustration of the “reduction property” to be proven: is the dashed green path equivalent to the sequence of dotted and blue (continous) path?

1.2 Double-Contact Manifold

Paths of the robot in contact with both objects belong to the 4-dimensional manifold $\mathcal{C}_{c_1,c_2} = \mathcal{C}_{c_1} \cap \mathcal{C}_{c_2}$ at the intersection between $\mathcal{C}_{c_1}$ and $\mathcal{C}_{c_2}$. A path in contact with both objects is represented by the green dashed path in Fig. 6 as a path “across” the foliation of one of the single-contact manifolds.

We start with the following claim: Because of the foliations of $\mathcal{C}_{c_1}$ and $\mathcal{C}_{c_2}$, any path in this manifold should be equivalent to a sequence of transfer paths with two contacts and paths in either $\mathcal{C}_{c_1}$ or $\mathcal{C}_{c_2}$. Figure 6 shows an example of such a decomposition: the green dashed path in contact with both objects can be reduced to the sequence composed by the black dotted path and the blue continuous path. Along the black dotted path both objects are in contact and the contact points do not change along the path. The path terminates where one of the object has reached the desired position. The blue path is a single-contact path lying on a leaf of one of the single contact manifolds. We know that the reduction property applies to paths in contact lying on either of these two manifolds, therefore, we only need to show that the green dashed path is equivalent to the sequence of black and blue paths. Figure 7 illustrates the property through an example: given the initial and the final configurations, respectively ${{\varvec{q}}}_s$ and ${{\varvec{q}}}_g$, any path in the double-contact manifold is admissible. Figure 8 shows how to reduce it to a sequence of transfer and transit paths. A formal proof to this Generalized Reduction Property follows.

Fig. 7

Any path in the double-contact manifold is admissible to go from ${{\varvec{q}}}_s$ to ${{\varvec{q}}}_g$ but not any path is a manipulation path

Fig. 8

Can any path in the double-contact manifold be “reduced” to a sequence of transfer and transit paths as in the figure?

1.3 Generalized Reduction Property

Generalized Reduction Property: Any two configurations belonging to the same connected component of the double-contact manifold can be connected by a manipulation path.

Proof

It is a direct generalization of the reduction property proof in [5]. Let ${{\varvec{q}}}_a$ and ${{\varvec{q}}}_b$ be two configurations in the double-contact manifold connected by a collision-free path in $\mathcal{C}_{c_1,c_2}$. Note that, since the robot is not allowed to move in contact with static obstacles, this path is actually contained in the subset $\tilde{\mathcal{C}}_{c_1,c_2}$ of $\mathcal{C}_{c_1,c_2}$ of all configurations such that the robot is not in contact with any static obstacle. This is an open set in $\mathcal{C}_R$ but might not be in $\mathcal C$.

Denoted the collision-free path as ${{\varvec{p}}}: [0,1] \rightarrow \tilde{\mathcal{C}}_{c_1,c_2}$, with ${{\varvec{p}}}(0)={{\varvec{q}}}_a$ and ${{\varvec{p}}}(1)={{\varvec{q}}}_b$, some preliminary definitions are in order:

• ${{\varvec{p}}}_R$: projection of ${{\varvec{p}}}$ on $\mathcal{C}_R$;

• ${{\varvec{p}}}_{O_1}$: projection of ${{\varvec{p}}}$ on $\mathcal{C}_{O_1}$;

• ${{\varvec{p}}}_{O_2}$: projection of ${{\varvec{p}}}$ on $\mathcal{C}_{O_2}$;

• ${{\varvec{p}}}_{R - {O_1}}$: contact configuration relative to object $O_1$ on ${{\varvec{p}}}$;

• ${{\varvec{p}}}_{R - {O_2}}$: contact configuration relative to object $O_2$ on ${{\varvec{p}}}$.

Assume that the objects can neither be in contact with obstacles nor in contact between themselves (quite unrealistic, to be removed later) and let ${{\varvec{q}}}={{\varvec{p}}}(s)$, $s \in [0,1]$, be a configuration on the path. Due to the non-contact hypothesis, it is always possible to define an open ball $B_{1}$ in the collision-free single-contact configuration space , centered on the contact configuration ${{\varvec{p}}}_{R - {O_1}}(s)$ ³ and without considering $O_2$. Its projection $D_{\epsilon _1}$ in $\mathcal{C}_R$ is homeomorphic to a disk of radius $\epsilon _1~>~0$. The object $O_1$ will not collide with obstacles as long as it is in contact with $R \in D_{\epsilon _1}$. In the same way there exists a ball $B_{2}$ in the collision-free single-contact configuration space , centered on the contact configuration ${{\varvec{p}}}_{R - {O_2}}(s)$. Its projection $D_{\epsilon _2}$ in $\mathcal{C}_R$ is a disk of radius $\epsilon _2~>~0$.

Denote by $\epsilon =\min \{\epsilon _1, \epsilon _2\}$. Due to the continuity of ${{\varvec{p}}}$, there exists an $\eta _R >0$ such that

$$ {\forall \tau \in ]s-\eta _R, s+\eta _R[, \, {{\varvec{p}}}_R(\tau ) \in D_{\epsilon /2}}, $$

an $\eta _1 >0$ such that

$$ \scriptstyle {\forall \tau \in ]s-\eta _1, s+\eta _1[, ||({{\varvec{p}}}_R(\tau ) - {{\varvec{p}}}_{O_1}(\tau )) -({{\varvec{p}}}_R(s) - {{\varvec{p}}}_{O_1}(s))|| < \epsilon /4}, $$

and an $\eta _2 >0$ such that

$$ \scriptstyle {\forall \tau \in ]s-\eta _2, s+\eta _2[, ||({{\varvec{p}}}_R(\tau ) - {{\varvec{p}}}_{O_2}(\tau )) -({{\varvec{p}}}_R(s) - {{\varvec{p}}}_{O_2}(s))|| < \epsilon /4}. $$

Denote by $\eta _3=\min \{\eta _1, \eta _2 \}$, and conclude that

$$ \scriptstyle {\forall \tau \in ]s-\eta _3, s+\eta _3[, ||({{\varvec{p}}}_{O_2}(\tau ) - {{\varvec{p}}}_{O_1}(\tau )) -({{\varvec{p}}}_{O_2}(s)- {{\varvec{p}}}_{O_1}(s))|| < \epsilon /2}. $$

Consider now $\eta =\min \{\eta _R ,\eta _3 \}$ and two configurations along the path: ${{\varvec{q}}}_1={{\varvec{p}}}(\tau _1)$ and ${{\varvec{q}}}_2={{\varvec{p}}}(\tau _2)$, with $\tau _1 < \tau _2$ and both in the interval $]s-\eta ,s+\eta [$.

The path ${{\varvec{p}}}(\tau )=({{\varvec{p}}}_R(\tau ), {{\varvec{p}}}_{O_1}(\tau ), {{\varvec{p}}}_{O_2}(\tau )), \tau \in [\tau _1, \tau _2]$ that transfers of $O_1$ in double-contact can be written as

$$\begin{aligned} {{\varvec{p}}}_R(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau )+( {{\varvec{p}}}_R(\tau _1)-{{\varvec{p}}}_{O_1}(\tau _1))\\ {{\varvec{p}}}_{O_1}(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau )\\ {{\varvec{p}}}_{O_2}(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau ) + ({{\varvec{p}}}_{O_2}(\tau _1)- {{\varvec{p}}}_{O_1}(\tau _1)), \end{aligned}$$

and the transfer path ${{\varvec{p}}}(\tau )=({{\varvec{p}}}_R(\tau ), {{\varvec{p}}}_{O_1}(\tau ), {{\varvec{p}}}_{O_2}(\tau )), \tau \in [\tau _1, \tau _2]$ of $O_2$ in single-contact to its goal position

$$\begin{aligned} {{\varvec{p}}}_R(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau _2)+( {{\varvec{p}}}_{O_2}(\tau )-{{\varvec{p}}}_{O_1}(\tau )) + ( {{\varvec{p}}}_R(\tau _1)-{{\varvec{p}}}_{O_2}(\tau _1))\\ {{\varvec{p}}}_{O_1}(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau _2)\\ {{\varvec{p}}}_{O_2}(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau _2) + ( {{\varvec{p}}}_{O_2}(\tau )-{{\varvec{p}}}_{O_1}(\tau )). \end{aligned}$$

Finally, transit path ${{\varvec{p}}}(\tau )=({{\varvec{p}}}_R(\tau ), {{\varvec{p}}}_{O_1}(\tau ), {{\varvec{p}}}_{O_2}(\tau )), \tau \in [\tau _1, \tau _2]$ of $R$ to its goal:

$$\begin{aligned} {{\varvec{p}}}_R(\tau )= & {} {{\varvec{p}}}_{O_2}(\tau _2)+( {{\varvec{p}}}_R(\tau )-{{\varvec{p}}}_{O_2}(\tau ))\\ {{\varvec{p}}}_{O_1}(\tau )= & {} {{\varvec{p}}}_{O_1}(\tau _2)\\ {{\varvec{p}}}_{O_2}(\tau )= & {} {{\varvec{p}}}_{O_2}(\tau _2). \end{aligned}$$

As a result of the choice of $\eta$ these paths should all be feasible, i.e., collision-free. A symmetric argument can be provided if $O_2$ is transferred first to its goal position.$\diamond$

This proof could be completed by considering the case of objects-obstacles and object-objects contacts, but omitted due to lack of space. The critical point in this case is that double-contact motion could not be allowed because it would not be possible to define an open disk in either of the two one-contact manifolds. It is then necessary to prove that a motion in double contact can be reduced to a sequence of motions in single contact. To achieve this reduction it is sufficient to break both contacts and move back to one of the two single-contact manifolds where the reduction property holds. It is, in fact, possible to show that there always exists a set of “escape” directions allowing the robot to un-grasp both obstacles.