Grasp analysis: the grasp matrix, force closure and form closure

A grasp is a set of contacts between a hand and an object. Lesson 3 described what a single contact can transmit. This lesson puts several contacts together and asks the central question of grasp analysis: can these contacts, pushing within their friction cones, hold the object against any disturbance that might act on it? If the answer is yes, the grasp is said to have force closure. If the contacts trap the object purely by geometry, with no help from friction, the grasp has form closure. Between those ideas lie the tools engineers use every day to compare grasps, reject weak ones and understand why a grasp failed.

The mathematics is linear algebra plus a little convex geometry. You do not need to memorize proofs, but you do need to be able to build the grasp matrix for a simple case, recognize when a set of contact wrenches surrounds the origin, and interpret a grasp quality number. Those skills transfer directly to the grasp planners and learned grasp detectors in lessons 5 and 6, which either compute these quantities or are trained on data labelled by them.

From contact forces to object wrenches

Place a coordinate frame at a reference point on the object, often its centre of mass. A force \( \mathbf{f}_i \) applied at contact point \( \mathbf{p}_i \) produces a wrench on the object: the force itself and the moment \( \mathbf{p}_i \times \mathbf{f}_i \) about the reference point. With several contacts the total wrench is the sum. Stacking every contact force into one long vector \( \mathbf{f}_c \), the relationship becomes a single matrix product.

The matrix \( G \) is the grasp matrix. Each block \( G_i \) maps the force transmitted at contact \( i \), expressed in that contact's local frame (normal plus tangential directions), into a wrench in the object frame. For a point contact with friction in 3-D, each block has three columns; for a soft-finger contact it has four, the extra column representing torsion about the normal. In the plane, a point contact with friction has two columns: the normal and the single tangential direction, and wrenches have three components, \( f_x \), \( f_y \) and the moment \( \tau \).

Two facts about \( G \) matter immediately. First, if \( G \) does not have full row rank (rank 3 in the plane, 6 in space), there is some wrench direction the contacts cannot produce at all, no matter how hard they push. Second, even with full rank, the contact forces are constrained: normal components must be non-negative and every contact force must lie inside its friction cone. Full rank is necessary but not sufficient. That is why grasp analysis always combines the grasp matrix with the friction cones.

Equilibrium and internal forces

For a held object to stay still, the contact wrench must balance any external wrench, such as gravity, an inertial load during motion or a push from the environment: \( G \mathbf{f}_c = -\mathbf{w}_{\text{ext}} \). Because \( G \) usually has more columns than rows, many contact force vectors satisfy that equation. The difference between any two solutions lies in the null space of \( G \). Forces in the null space produce no net wrench; they are internal forces, the squeeze that the fingers apply against each other.

Internal forces are what make friction grasps work. Squeezing two fingers together does nothing to the object's motion, but it raises the normal force at each contact and so enlarges the tangential force each contact can carry before slipping. A gripper controller that holds an object chooses an internal force large enough to keep every contact inside its cone under the expected loads, but not so large that it crushes the object or wastes energy. Lesson 7 turns this into a sizing calculation.

Force closure

A grasp has force closure if, for every external wrench, there exist contact forces inside the friction cones that balance it. An equivalent and more useful statement uses the linearized cones from lesson 3. Each edge of each friction pyramid, mapped through the grasp, gives a primitive contact wrench. Any feasible contact force produces a non-negative combination of these primitive wrenches. The grasp has force closure exactly when non-negative combinations of the primitive wrenches cover the whole wrench space, which happens when the origin lies strictly inside their convex hull.

That geometric test is easy to picture in the plane, where wrenches have three components. Each contact with friction contributes two primitive wrenches, one for each edge of its two-dimensional friction cone. Two contacts give four vectors in three dimensions. Their convex hull is a solid; if the origin sits strictly inside that solid, the grasp resists any disturbance given enough squeeze. If the origin lies on a face or outside, some disturbance direction cannot be resisted.

For two contacts in the plane there is a beautiful shortcut. Two point contacts with friction achieve force closure if and only if the line segment joining the contact points lies inside both friction cones. Contacts that satisfy this condition are called antipodal. The test needs only angles: measure the angle between each contact normal and the line joining the contacts, and check that both angles are smaller than the cone half-angle \( \arctan\mu \). Lesson 5 uses this test to sample grasps quickly on arbitrary shapes.

Form closure

Form closure is stricter. A grasp has form closure if the contacts prevent all motion of the object by geometry alone, treating every contact as frictionless. A peg in a well-fitting hole and a part clamped in a fixture with locating pins are classic examples. Because frictionless contacts can only push along their normals, form closure needs more contacts than force closure: at least four in the plane and seven in space for a rigid body with no special symmetry. Those numbers are lower bounds; a badly arranged set of seven contacts can still leave the object free to move.

Industrial fixtures aim for form closure because it does not depend on friction that may change with oil, wear or temperature. Robot hands rarely achieve it, because surrounding an object with many contacts is slow and needs complex fingers. A useful middle ground is a grasp that has form closure in the directions of the largest expected loads, for example fingers that cup under a part so gravity is resisted geometrically, plus friction for the rest. Shaped fingertips that match a part, sometimes called form-fit fingers, follow this idea and are widely used in machine tending.

Measuring how good a grasp is

Force closure is a yes-or-no property, but two grasps that both have it can differ greatly in robustness. Grasp quality metrics put a number on that difference. The most widely used is the epsilon metric, introduced by Ferrari and Canny in 1992. Normalize the contact forces so the total, or the largest, normal force is one unit. The set of wrenches the grasp can then produce is the convex hull of the primitive wrenches. The epsilon metric is the radius of the largest ball, centred at the origin, that fits inside that hull. A larger radius means the grasp can resist a larger disturbance in its weakest direction for the same squeeze.

The formula reads: in each direction \( \mathbf{u} \), find how far the hull extends, which is the largest projection of any primitive wrench \( \mathbf{w}_j \) onto \( \mathbf{u} \); then take the smallest of those distances over all directions. If the origin is outside the hull, some direction has a negative support value, and \( \varepsilon \) is negative, signalling no force closure. Because moments and forces have different units, implementations divide moments by a characteristic length, often the object's radius, so both parts of the wrench are comparable.

Other metrics capture different concerns. The volume of the wrench hull rewards grasps that are strong on average rather than in the worst case. Metrics based on the smallest singular value of \( G \) measure how close the grasp is to losing rank. Task-oriented metrics weight directions by the disturbances the task will actually apply: a grasp for pouring needs to resist moments about the handle axis more than a grasp for lifting straight up. No single number is best; planners often combine several.

# Planar force closure and an approximate epsilon metric from sampled directions
import numpy as np
def primitive_wrenches(contacts, mu, L):
    """contacts: list of (point, inward unit normal). Returns 2 cone-edge wrenches per contact."""
    ws = []
    a = np.arctan(mu)
    for p, n in contacts:
        for s in (+1, -1):
            c, si = np.cos(s * a), np.sin(s * a)
            f = np.array([c * n[0] - si * n[1], si * n[0] + c * n[1]])   # rotate normal by +/- alpha
            f = f / np.linalg.norm(f)
            tau = p[0] * f[1] - p[1] * f[0]
            ws.append([f[0], f[1], tau / L])                             # scale moment by length L
    return np.array(ws)
def epsilon(ws, n=20000, seed=1):
    rng = np.random.default_rng(seed)
    u = rng.normal(size=(n, 3)); u /= np.linalg.norm(u, axis=1, keepdims=True)
    return (u @ ws.T).max(axis=1).min()
L = 0.04                                      # half-width of the block, m
antipodal = [((-0.04, 0.0), (1.0, 0.0)), ((0.04, 0.0), (-1.0, 0.0))]
offset    = [((-0.04, -0.02), (1.0, 0.0)), ((0.04, 0.02), (-1.0, 0.0))]
for name, g in (("antipodal", antipodal), ("offset", offset)):
    for mu in (0.0, 0.2, 0.5):
        e = epsilon(primitive_wrenches(g, mu, L))
        print(f"{name:9s} mu={mu:.1f}  epsilon={e:+.3f}  {'closure' if e > 1e-3 else 'no closure'}")

antipodal mu=0.0  epsilon=+0.000  no closure
antipodal mu=0.2  epsilon=+0.139  closure
antipodal mu=0.5  epsilon=+0.304  closure
offset    mu=0.0  epsilon=-0.499  no closure
offset    mu=0.2  epsilon=-0.294  no closure
offset    mu=0.5  epsilon=+0.001  no closure

Read the output carefully. With no friction (\( \mu = 0 \)) neither grasp has closure: the antipodal pinch sits exactly on the boundary, because frictionless fingers cannot resist a sideways push. With friction, the antipodal grasp has a clearly positive epsilon that grows with \( \mu \), because wider cones surround the origin more generously. The offset grasp is negative at low friction. At \( \mu = 0.5 \) it lands almost exactly on zero, and that is no accident: the line joining its contacts tilts \( \arctan(0.04/0.08) \approx 26.6^\circ \) from the normals, which equals \( \arctan 0.5 \), the cone half-angle. The sampled metric and the antipodal angle test agree, and both say the offset grasp would need slightly more friction, or better contact placement, to be safe.

Grasp analysis in three dimensions

In space, wrenches have six components and each contact with friction contributes a cone in force space. With a four-edge pyramid per contact, a three-finger grasp gives twelve primitive wrenches in six dimensions. The same convex-hull test and epsilon metric apply, computed with a convex hull or linear programming library rather than by hand. Soft-finger contacts add a torsion column per contact, which is often what turns a two-finger grasp from marginal to robust.

Two simple 3-D facts are worth remembering. Two hard point contacts can never achieve force closure in 3-D, because the object can always rotate about the line joining them. Two soft-finger contacts can, provided the torsional friction is enough for the expected moments about that line. Three hard contacts with friction can achieve force closure on most shapes when their friction cones jointly cover the space, which is why many robot hands have three fingers.

Limits of the analysis

Grasp analysis rests on assumptions that real grasps violate. Contacts are not points: fingers deform and objects have corners. Friction is uncertain and changes with contamination. Objects are not perfectly rigid, and their mass properties may be unknown. Most importantly, the analysis assumes the planned contacts are achieved exactly, but perception and positioning errors shift the actual contacts by millimetres. A grasp that sits just inside its friction cones on paper may fail in practice.

Engineers respond in three ways. They prefer grasps with margin: contacts far inside their cones and large epsilon values. They evaluate a planned grasp under sampled pose errors and friction values, estimating the probability that closure survives, which lesson 5 calls robust grasp planning. And they verify grasps physically, with force-torque or tactile sensing after closing and with a small shake or lift test before fast motion. Analysis tells you which grasps are worth trying; testing tells you which ones work.