Grasping and Fixturing as Submodular Coverage Problems

Abstract

Grasping and fixturing are concerned with immobilizing objects. Most prior work in this area strives to minimize the number of contacts needed. However, for delicate objects or surfaces such as glass or bone (in medical applications), extra contacts can be used to reduce the forces needed at each contact to resist applied wrenches. We focus on the following class of problems. Given a polyhedral object model, set of candidate contacts, and a limit on the sum of applied forces at the contacts or a limit on any individual applied force, compute a set of k contact points that maximize the radius of the ball in wrench space that can be resisted. We present an algorithm, SatGrasp, that is guaranteed to find near-optimal solutions in linear time. At the core of our approach are (i) an alternate formulation of the residual radius objective, and (ii) the insight that the resulting problem is a submodular coverage problem. This allows us to exploit the submodular saturation algorithm, which has recently been derived for applications in sensor placement. Our approach is applicable in situations with or without friction.

Appendix

Proof of Proposition 1

We use the fact that for two convex sets A and B, A ⊂ B if and only if $h_{A} {\preceq} h_{B}$ (i.e., h _A (x) ≤ h _B (x) for all x). Let B _r be the ball of radius r around the origin. If $\hat{r} = r_{\text{res}} (C)$, then it follows that

$$ B_{{\hat{r}}} \subseteq C $$

(24)

$$ h_{{B_{{\hat{r}}} }} (y) \le h_{C} (y){\text{for}}\,{\text{all}}\,\left\| y \right\| = 1 $$

(25)

$$ \hat{r} = r_{\text{res}} (C) \le \mathop { \hbox{min} }\limits_{\left\| y \right\| = 1} \,h_{C} (y) $$

(26)

On the other hand, let $\tilde{r} = { \hbox{min} }_{\left\| y \right\| = 1} h_{C} (y)$. Then

$$ h_{{B_{{\tilde{r}}} }} (y) \le h_{C} (y){\text{for}}\,{\text{all}}\,\left\| y \right\| = 1 $$

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$$ h_{{B_{{\tilde{r}}}}} \preceq h_{C} \,{\text{since}}\,h\,{\text{is}}\,{\text{homogeneous}} $$

(28)

$$ B_{{\tilde{r}}} \, \subseteq \,C $$

(29)

$$ \tilde{r} = \mathop { \hbox{min} }\limits_{\left\| y \right\| = 1} \,h_{C} (y) \le r_{\text{res}} (C) $$

(30)

Thus $\min_{\left\| y \right\| = 1} h_{C} (y) = r_{\text{res}} \left( C \right)$ ⃞

1.1 Discretization Error in Formula for Residual Radius

We will consider a particular scheme for deterministically sampling the p-dimensional sphere (a subset of ${\mathbb{R}}^{p + 1}$) and bound the error that results when one evaluates the support function h _C (y) at only the sampled points y to approximate its minimum. We take an n × n × · · · × n grid on every p-dimensional facet of the p + 1-dimensional hypercube. This requires (2p + 2)n ^p points.

Let y _n  = argmin h _C (y _i ) be the optimal sampled point, and let y _*  = argmin h _c (y) be the exact optimal point. Let x _n  = argmax $y_{n}^{T} x$.

$$ y^{T} x_{n} = \left\| {x_{n} } \right\|{ \cos }\,\theta_{{x_{n} ,y}} $$

(31)

$$ \frac{d}{d\theta }y^{T} x_{n} = - \left\| {x_{n} } \right\|{ \sin }\,\theta_{{x_{n} ,y}} $$

(32)

$$ {\text{y}}_{n}^{T} x_{n} - y_{*}^{T} x_{n} = - \left\| {x_{n} } \right\|{ \sin }\,\tilde{\theta }\varDelta \theta $$

(33)

where $\theta_{{y^{n} ,x_{n} }} \le \tilde{\theta } \le \theta_{{y_{*} ,x_{n} }}$, and $\varDelta \theta = \theta_{{y_{n} ,x_{n} }} {-}\theta_{{y_{*} ,x_{n} }}$. Next, note that $h(y_{*} ) \ge y_{*}^{T} x_{n}$ so

$$ h(y_{n} ) - h(y_{*} ) \le \left\| {x_{n} } \right\||{ \sin }\,\tilde{\theta }\varDelta \theta | $$

(34)

The largest possible ∆θ occurs at the nearby part of each face, between the vector (1, 0,…, 0) and $\left( {1,\frac{1}{2n}, \ldots ,\frac{1}{2n}} \right)$, where ${ \cos }(\varDelta \theta ) = \sqrt {1 + p/4n^{2} }$. It follows that

$$ \varDelta \theta { \le }\frac{\sqrt p }{2n} $$

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Thus

$$ h(y_{n} ) - h(y_{ * } ) \le \left\| {x_{n} } \right\||{ \sin }\,\tilde{\theta }\varDelta \theta | $$

(36)

$$ \le \left\| {x_{n} } \right\||\sin \,\theta_{{x_{n} ,y_{n} }} |(\frac{\sqrt p }{2n} + \frac{p}{{4n^{2} }}) $$

(37)

1.2 Submodular Saturation Algorithm

SATURATE finds solutions to problems where we are simultaneously trying to optimize a collection of submodular objectives.

$$ \mathop {\hbox{max} }\limits_{{S^{{\prime }} \subset S}} \mathop {\hbox{min} }\limits_{i} F_{i} (S^{{\prime }} ),\quad {\text{subject}}\,{\text{to}}\, | {\text{S}}^{{\prime }} |\le k $$

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If we run saturate and request a αk-element solution, it will give a solution that’s better than the optimal k-element solution, where

$$ \alpha = 1+ { \log }(\mathop { \hbox{max} }\limits_{s \in S} \sum\limits_{i} {F_{i} (s))} $$

(39)

This bound applies when the functions F _i take integer values. Thus to apply this result to a general problem of the form in Eq. (38), we must typically rescale and then round the objective functions. In the present problem, we can rescale and round the values $h_{{FC_{i} }} (y_{j} )$, or, in the frictionless case $y_{j}^{T} w_{i}$. Now the αk-element solution found by SATURATE is no longer guaranteed to be better than the optimal k-element solution, however, the difference is small and due to rounding error.

Here we describe the SATURATE algorithm for self-containedness. The key idea is as follows: if the optimal value is c, then there is no benefit if an objective function exceeds c. Thus we define the truncated objective functions

$$ \widehat{F}_{i,c} (S) = { \hbox{max} }\{ F_{i} (S),c\} $$

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If the original objectives F _i are submodular, then the truncated versions are also submodular. Now our goal is to saturate all of the objective functions $\widehat{F}_{i,c} (S)$, i.e., achieve the value c. The mean of the truncated functions, $\overline{F}_{c}$, is also submodular, and it describes progress towards this goal.

$$ \overline{F}_{c} (S) = \sum\limits_{i} {\widehat{F}_{i,c} (S)} $$

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To optimize $\overline{F}_{c}$, we can use the greedy algorithm, which is guaranteed to give good solutions since it is submodular. (The greedy k-element solution is bested by the optimal k-element solution by at most a factor of 1 − 1/e).

We initially don’t the largest value of c that our greedy algorithm will successfully achieve (i.e., saturate all of the objective functions), so we use a binary search over the range of possible values. For each c, we greedily optimize $\overline{F}_{c}$. If we saturate all of the $\widehat{F}_{i,c}$, then we next try a larger c. If we fail to saturate them all, we next use a lower value of c.

SATURATE runs the greedy algorithm about 10 times with the transformed objective function. Thus the running time is roughly linear in |S| and k. It is actually somewhat faster than O(k) because we can use a “lazy greedy” algorithm that does not test elements that are guaranteed to give less improvement than some element that we’ve already tested.