# Friction cones

Consider the 2D example depicted in figure to the right. A 2D mass is in contact with a single surface. This contact will remain fixed as long as the contact force $$\bff^c = m \bfg - \bff^{\mathit{ext}}$$ lies within the Coulomb friction cone $$\calC$$. As soon as $$\bff^c$$ exits $$\calC$$, the contact switches to the sliding mode. The property $$\bff^c \in \calC$$ is called the contact-stability condition for this contact mode: as long as it is fulfilled, the contact remains fixed. Deriving contact-stability conditions for multibody systems (such as ZMP support areas) is of key interest as robots are commonly controlled using only a small number of contact modes.

## Coulomb friction

Consider the set of points $$\{C_i\}$$ where the robot contacts its environment. Assuming that the environment surface is smooth enough, one can consider its unit normal $$\bfn_i$$ at $$C_i$$, pointing from the environment to the robot. Let $$\bff^c_i$$ denote the contact force exerted at $$C_i$$ by the environment onto the robot:

• the normal component $$p_i \defeq (\bfn_i \cdot \bff^c_i)$$ is called pressure, and
• the tangential component $$\bff^t_i \defeq \bff^c_i - (\bfn_i \cdot \bff^c_i) \bfn_i$$ is the friction force at $$C_i$$.

A point contact remains in the fixed contact mode while its contact force $$\bff^c_i$$ lies inside the friction cone directed by $$\bfn_i$$:

$$\bff^c_i \cdot \bfn_i \ > \ 0, \quad \textrm{and} \quad \left\| \bff^t_i \right\|_2 \ \leq \ \mu_i (\bff^{c}_i \cdot \bfn_i),$$

where $$\mu_i$$ is the static friction coefficient at contact $$C_i$$. The Euclidean norm $$\| \cdot \|_2$$ in this definition represents friction cones with circular sections, which models the isotropy of friction. Although more realistic, this model presents some computational challenges down the control pipeline. A common practice is to consider its linear approximation:

## Linearized friction cones

A point contact remains in the fixed contact mode while its contact force $$\bff^c_i$$ lies inside the \emph{linearized friction cone} directed by $$\bfn_i$$:

$$\bff^c_i \cdot \bfn_i \ > \ 0, \quad \textrm{and} \quad \frac{\bff^t_i}{\tilde{\mu}_i (\bff^{c}_i \cdot \bfn_i)} \ \in \ \calP_n,$$

where $$\calP_n$$ is the regular $$n$$-sided polygon inscribed in the 2D unit circle.

This approximation can be made as close as desired to the original model by increasing the number of edges $$n$$ of the section polygon $$\calP_n$$. The equation above provides a set of linear inequality constraints. For example, the four-sided friction pyramid obtained for $$n=4$$ can be written:

\begin{eqnarray} \bff^c_i \cdot \bfn_i & > & 0 \\ | \bff^{c}_i \cdot \bft_i | & \leq & \tilde{\mu}_i (\bff^{c}_i \cdot \bfn_i), \\ | \bff^{c}_i \cdot \bfb_i | & \leq & \tilde{\mu}_i (\bff^{c}_i \cdot \bfn_i), \end{eqnarray}

with $$(\bft_i, \bfb_i)$$ any basis of the tangential contact plane such that $$(\bft_i, \bfb_i, \bfn_i)$$ is a direct frame. For $$\tilde{\mu}_i = \mu_i$$, the linearized Coulomb cone is an outer approximation of the circular one, while it is an inner approximation for $$\tilde{\mu}_i = \mu_i / \sqrt{2}$$:

Coulomb friction cone (left) with outer (middle) and inner (right) linear approximations.

## To go further

Individual friction cones can be combined into frictional wrench cones using tools from polyhedral geometry. See for instance pages 69-87 of my PhD thesis, or the discussion and algorithm introduced in this paper.

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