Robot Gripper Force: Friction Cone, Twist and Acceleration

The contact friction model follows Howe and Cutkosky (1996), with the ellipsoidal soft-finger form stated explicitly by Fakhari et al. (2019), Eqs. (1)–(5). Each contact obeys

Its zero-torque section is the Coulomb friction cone, with half-angle $\tan^{-1}\mu$. Integrating $\mu p\rho$ over a circular patch gives $r_e=2a/3$ for uniform pressure and $r_e=3\pi a/16$ for Hertz pressure. The radius is an input measured at the relevant load, not a prediction of pad deformation.

Contacts are at $\mathbf r_i=R(\cos\theta_i,\sin\theta_i,0)$ with inward normals. The required contact force is $\mathbf F=m(a_x,a_y,a_z+g)$; the required moment about the ring centre is $\mathbf M=\mathbf e\times\mathbf F+\mathbf C$. All three force and all three moment equations are enforced, including moments generated by the contact lever arms and torsional friction about each inward normal.

The objective is to minimize the largest contact normal force, subject to these six equilibrium equations and the coupled friction constraints. For two fingers, equilibrium fixes the shear split and normal-force difference; a bracketed scalar solve optimally splits contact torsion. For three to six fingers a feasible-start logarithmic-barrier Newton method solves the second-order cone problem. Its final barrier parameter is $10^7$ and its force-ceiling optimality bound is shown in the calculation steps. The allocation sum is reported for this optimum; the objective is the per-finger ceiling, not the sum.

The design load factor multiplies the entire wrench. Slip reserve is $100(1-N_{\min}/N_{\mathrm{available}})$; design reserve uses the factored ceiling. SMC MHL2, printed p. 5 / attached PDF p. 6, provides the zero-offset gravity-only reference.

Along the acceleration sweep, the minimum force ceiling is a convex, positively homogeneous function of the required wrench. Adaptive chord sampling uses these properties to retain zero-load reversals and bound interpolation error without rounding physical corners. The COM moment includes the gyroscopic term in the Newton–Euler equation (Liu and Jain, Eq. 23).

Rigid object; identical isotropic static friction; fixed circular contact patches with uniform or Hertz pressure; contact centres remain fixed in a regular ring in the horizontal x–y plane. Positive z is upward, and the COM offset is measured from the ring centre. Contact torsional moments are signed about the inward normal.

Each contact's normal force can be allocated independently up to the entered ceiling, and the gripper can realize the listed tangential reactions and torque. The moment input is the required contact moment at COM, $\mathbf C=I\boldsymbol\alpha+\boldsymbol\omega\times(I\boldsymbol\omega)-\mathbf M_{\mathrm{applied}}$, expressed in the same frame. COM acceleration includes tangential and centripetal acceleration when the arm rotates. The design factor scales gravity, inertia and additional moments together.

This is instantaneous no-slip feasibility, including an entered peak acceleration. It does not simulate force rise, vibration, impact impulses, controller delay or post-slip travel. A held object can slip during a short transient even when the commanded final grip exceeds this threshold. Use measured peak acceleration and force response.

The ellipsoidal limit surface is an approximation to distributed Coulomb friction; intermediate shear–twist combinations and load-dependent contact area require validation for the actual pads. No contact rolling, peeling, adhesion, suction, crushing or finger strength limit is included. A conventional coupled actuator or an underactuated hand must be checked for its ability to produce the calculated force distribution; a scalar catalogue force alone does not establish that distribution.

Only regular radial arrangements of two to six contacts are covered. The contact-patch radius must not exceed the ring radius. The layout shows the x–y projection; the separate load arrow shows the y–z projection. The cone card is schematic and its curved arrow denotes the extra soft-contact torsion.

Current-load outputs are solved directly. The acceleration curve scales only the entered acceleration vector from zero to twice its value. The chart retains the cancellation point of every wrench component and the least-norm wrench point as solved knots. Load paths along a single wrench direction use positive homogeneity, with separate direct solves on either side of a reversal, so a zero-load corner remains at its exact multiplier.

Other intervals are checked against direct midpoint solves and refined where the convex force curve's chord error exceeds 0.2% of the local force scale. Linear interpolation preserves the retained corners. Excluding contact-solver tolerance, convexity bounds an interval's chord error by twice its midpoint deviation; the largest remaining bound is shown in the calculation steps. The plot includes every solved knot as well as 161 regularly spaced display points.

The initial meshes contain 41, 15, 13, 11 and 9 points for two through six fingers. Midpoint checks and refinement have budgets of 257, 33, 29, 25 and 21 direct solves respectively; mandatory cancellation knots are always retained. A 61-point six-finger sweep previously exceeded 8 s, so refinement is bounded to keep typical runs near 3 s. If the budget is reached before the interpolation target, a condition reports it and the residual bound remains visible in the calculation steps. Enter an acceleration of interest to obtain its directly solved force. The equilibrium residual and force-ceiling optimality bound separately monitor contact-solver accuracy.