Robot Grasp Planning: Force Closure and Quality Metrics

Every act of manipulation ends the same way: some part of the robot touches some part of the world, and the physics of that interface decides what happens next. Kinematics can place the fingers and motion planning can route the arm, and perception is what hands this module the known object pose in the robot's frame that every wrench computation below starts from. But once the fingers close, the question is mechanical. Will this hold, or will the object slip? Grasp planning is the discipline that answers that question from geometry: where to put the contacts, how hard to squeeze, and how to know before lifting that the grasp will survive the task.

The answer turns out to be a convex polytope. Map everything the contacts can do into a space of forces and moments, take a convex hull, and ask whether the origin sits inside it. That one geometric fact, force closure, organizes most of classical grasp theory, and the distance from the origin to the hull's boundary became the field's standard quality metric.

The wrench space, live

The lab below is that polytope. A square object, contacts sliding along its perimeter, and the hull of everything they can do plotted beside it with the origin marked. Drag the friction coefficient slider from 0.7 down toward zero and watch the hull shrink until the origin falls outside it: that is the grasp failing. The rest of this module derives what you are looking at.

μ friction coefficient0.70

The cones narrow as μ drops; the wrench hull shrinks with them.

contact positions

contact 10.125

contact 20.875

contact 30.625

Position along the perimeter: 0.00 at the top-right corner, increasing counterclockwise. The exact edge midpoints are 0.125, 0.375, 0.625, and 0.875.

wrench hullcone-edge wrenchzero wrench (inside hull)

contacts 3 force closure yes ε 0.444

3 frictional contacts on the unit square at mu 0.70 open inward cones of half-angle 35.0 degrees; each contact can push along its cone but cannot pull.

Current object contacts and cones

contacts: 3
mu: 0.70
cone half-angle: 35.0°
regime: push-only cones, closed grasp

The grasp wrench hull of 3 contacts currently reports force closure yes with Ferrari-Canny quality epsilon 0.444; that radius is the largest origin-centered wrench ball that still fits inside the hull.

Current wrench-space quality

force closure: yes
epsilon: 0.444
origin: inside the hull
contacts: 3

Left: the object and its friction cones, half-angle arctan μ. Right: the grasp wrench space, the convex hull of the cone-edge wrenches in (fx, fy, τ), with τ scaled per half side of the object. Force closure holds exactly when the origin sits strictly inside the hull; ε is the radius of the largest origin-centered wrench ball that still fits. Things worth trying: remove contact 3 and watch the hull collapse onto the origin, then slide contact 2 to 0.625 so the pair is antipodal; or drag μ down and watch ε shrink.

Contact models

Planar contacts, frictional

minimum for a generic object (Murray, Li, Sastry)

Spatial contacts, frictional

same bound in three dimensions

7-12

Frictionless spatial

Somoff's lower bound, Markenscoff's upper

6.7M

Dex-Net 2.0 training datapoints

synthetic depth/grasp examples with analytic binary labels; not robot trials

A contact model says which forces a fingertip can transmit. The simplest is the frictionless point contact: it pushes along the surface normal and nothing else. Real fingers have friction, and the standard point contact with friction obeys Coulomb's law: the tangential component of the contact force can grow to at most a fixed fraction $\mu$ of the normal component before the contact starts to slide Murray 1994 .

\lVert f^t \rVert \le \mu \, f^n

Geometrically, the admissible forces form a cone around the inward normal, the friction cone, with half-angle $\alpha = \arctan \mu$ Prattichizzo 2016 . In the plane the cone is a wedge bounded by two edge rays; in three dimensions it is a quadratic cone, usually approximated by a polyhedral one so the analysis stays linear. A soft-finger contact adds one more channel: a torsional moment about the contact normal, bounded by a torsional coefficient times the normal load, which models the small contact patch of a real fingertip Murray 1994 .

One choice comes before any of this analysis: what kind of grasp you want at all. Cutkosky's taxonomy separates power grasps, which wrap the object and trade dexterity for stability, from precision grasps, which hold the object with the fingertips and trade stability for the ability to reorient it Cutkosky 1989 . The mechanics below applies to both, but the wrench budget a task demands depends heavily on which style you picked.

The grasp wrench space

A force applied at a contact point does two things to the object: it pushes the center of mass, and it twists. Package both into one vector, the wrench. For a contact at position $r_i$ applying force $f_i$ about the object's reference frame, the wrench is the force stacked over its moment:

w_i = \begin{bmatrix} f_i \\ r_i \times f_i \end{bmatrix}

In the plane this is a three-vector $(f_x, f_y, \tau)$ with $\tau = r_x f_y - r_y f_x$ . Every admissible contact force maps to a point in wrench space, and since a grasp's contacts act simultaneously, the net wrench the grasp can produce is the Minkowski sum of what each contact can produce. Friction cones are convex, so the sum is a convex hull: the grasp wrench space is the convex hull of the primitive wrenches, the wrenches along the cone edges at unit normal force Murray 1994 Bicchi 2000 .

\mathcal{W} = \operatorname{conv}\left\{ w_{ij} \right\}, \qquad w_{ij} = \begin{bmatrix} n_i \pm \mu \, t_i \\ r_i \times (n_i \pm \mu \, t_i) \end{bmatrix}

This construction is the whole trick. Questions about contact forces, which live on separate fingertips and interact through friction inequalities, become questions about one polytope.

Force closure

A grasp has force closure when it can resist any externally applied wrench: push the object in any direction, twist it either way, and some feasible combination of contact forces inside the friction cones cancels the disturbance. In wrench space, resisting any direction means the hull must surround the zero wrench from every side. Force closure holds exactly when the origin lies strictly inside the grasp wrench hull Nguyen 1988 Murray 1994 :

\exists \, \varepsilon > 0 : \quad B_\varepsilon(0) \subset \mathcal{W}

The strictness is load-bearing. An origin resting exactly on the hull's boundary means there is a wrench the grasp cannot quite resist, and since the task gets to choose the disturbance, "almost everywhere" is a failure mode, not a pass. This is why friction does so much work in practice. Without friction, the cone collapses to the normal ray, and the hull has no volume to spare: form closure, closure by geometry alone, demands at least four contacts in the plane and seven in space. That lower bound is older than robotics: Reuleaux stated it in 1875 and Somoff in 1897, as Markenscoff, Ni, and Papadimitriou recount on their way to the matching sufficiency results. Four contacts achieve form closure for any planar object with a piecewise smooth boundary, a circle being the one exception, and in space twelve contacts suffice for any object without rotational symmetry, with seven enough under very general conditions Markenscoff 1990 . Mishra, Schwartz, and Sharir supplied the algorithmic side: tight finger-count bounds for each equilibrium case of a frictionless grip, and linear-time synthesis of such grips on polyhedral objects Mishra 1987 . Friction relaxes all of this: with point contacts with friction, three contacts are necessary and sufficient in the plane and four in space Markenscoff 1990 Murray 1994 .

The cleanest sufficient condition in the whole theory is Nguyen's antipodal grasp theorem: a planar two-fingered grasp with frictional contacts is force closure if and only if the line connecting the two contact points lies inside both friction cones Nguyen 1988 . The proof sketch is worth keeping: either contact can lay off any force in the wedge where the two cones overlap, the partner squeezes harder along the shared line to absorb the excess, and because the line is interior the cone edges supply torques of both signs about the center. This is the theorem a parallel-jaw gripper runs every time it pinches opposite sides of a box, and its strictness is visible in hardware: grasp the box too close to an edge of the cone and a light sideways nudge slides it out.

Grasp quality: the epsilon metric

Force closure is binary, but grasps are not. Two closed grasps can differ enormously in how much disturbance they absorb before a contact saturates. Ferrari and Canny turned the hull geometry into a number: the quality $\varepsilon$ is the radius of the largest ball centered at the origin that still fits inside the grasp wrench space Ferrari 1992 .

\varepsilon = \min_{F_k \in \partial \mathcal{W}} \; \operatorname{dist}\!\left( 0, F_k \right)

Because the hull is a polytope, the largest inscribed ball touches a facet, so $\varepsilon$ is just the distance from the origin to the nearest hull facet. The unit-normal-force convention makes it concrete: $\varepsilon$ is the worst-case disturbance wrench, per unit of normal force at the contacts, that the grasp guarantees it can absorb.

Roa and Suárez review quality measures based on contact locations and hand configuration, and illustrate their behavior with simple examples rather than recommend a universally best measure. For the largest-minimum resisted wrench criterion, they identify two choices that affect its value: the origin used to compute torques, and the metric that scales torques relative to forces. They also distinguish a shared total-force limit from independent per-finger limits; a grasp optimal under one constraint need not be optimal under the other. Most of their analysis is quasi-static. Those choices and the task assumptions need to accompany interpretation of the reported value Roa 2015 .

Reading the hull

Now the lab above reads as mechanics rather than shapes. On the left, the object with each contact's friction cone. On the right, the grasp wrench space projected from $(f_x, f_y, \tau)$ : the hull of the primitive cone-edge wrenches. The origin ring is green exactly when the grasp is force closure, and the readout reports $\varepsilon$ .

The default tripod of top, right, and bottom contacts is force closure at $\mu = 0.7$ . Remove the third contact and the remaining pair, one on the top edge and one on the right, fails Nguyen's test: the line between them sits at 45 degrees to each normal while the cones open only $\arctan 0.7 \approx 35$ degrees, and the hull collapses onto the origin. Now slide contact 2 down onto the bottom edge, opposite contact 1. The pair becomes antipodal, the connecting line is the shared normal, and the hull re-inflates around the origin for any positive friction at all. Finally, drag $\mu$ down and watch $\varepsilon$ fall: the metric is pricing the friction you just removed.

Where this meets the learned stack

Dex-Net 2.0 uses analytic grasp metrics to label synthetic training data. The paper reports 6.7 million training datapoints drawn from 1,500 3D object models; its Figure 3 describes over 6.7 million aligned grasp images, not a count of physical trials or necessarily distinct scene views. A positive training label requires expected epsilon quality above 0.002 and no modeled collision with the object or table. The expected quality accounts for uncertainty in object pose, gripper pose, and friction. The GQ-CNN consumes a grasp-aligned depth crop and the gripper depth relative to the camera, and predicts robustness under this binary success model, rather than regressing the raw epsilon score. The paper’s label threshold should not be treated as a threshold for the lab above Mahler 2017 .

The basic planner samples antipodal parallel-jaw candidates from depth-image gradients, ranks them with the GQ-CNN, and executes the highest-ranked candidate that is reachable and does not collide with the table. Its stated assumptions are isolated rigid objects on a planar worksurface and a single-view depth image; synthetic dataset generation additionally assumes known gripper geometry and camera intrinsics. The physical benchmark used an ABB YuMi with custom silicone fingertips and a Primesense Carmine 1.08 camera. A grasp counted as successful only if it held the object after lifting, transporting, and shaking Mahler 2017 .

For the eight known, 3D-printed adversarial objects, Table III reports 93% success for GQ-L-Adv over 80 trials, ten per object. GQ-L-Adv was initialized from the model trained on the large synthetic dataset and then fine-tuned on synthetic examples of those adversarial objects. A separate, CEM-augmented experiment ran 100 trials on 40 novel household objects, including articulated and deformable objects: it reported 94% success overall and 99% precision, with 68 successes among 69 grasps classified as robust. Here precision is success among grasps whose estimated robustness exceeds 50%, not overall success or synthetic classification accuracy. These results come from separate physical evaluations, not a pooled success rate Mahler 2017 . Missing depth on thin parts and object collisions remained observed failure modes Mahler 2017 .

That pattern generalizes. Modern learned manipulation, including the diffusion policies covered in the manipulation modules, rarely re-derives contact mechanics; it learns from data what the classical theory says a good contact looks like, and it inherits the theory's failure modes when it strays from the data. And the reason grasping resists the sim-to-real recipe that conquered locomotion is precisely the physics of this module: millimeter-accurate contact geometry and friction coefficients are what simulators approximate worst. The classical stack, kinematics through control, ends here, at the interface where the robot finally touches the world.

Grasp Planning