Contact stability
Contact stability is a condition used to check the feasibility of robot motions. It should not be confused with the notion of (Lyapunov) stability, a notion from control theory used to formalize the performance of a controller.
Feasibility of a contact wrench
Everything starts with a mobile robot, i.e., a robot that is able to establish and break temporary contacts with its environment. Think for instance of a humanoid that alternatively plants its feet on the ground to walk. At each contact, the sum of forces exerted by the environment on the robot is represented by a contact wrench. As long as the contact is maintained, this wrench will lie inside its 6D friction cones. The contact breaks are switches to a different contact mode as soon as its wrench hits the boundaries of this friction cone.
Contact stability, modes and switches
As the robot moves, its NewtonEuler equations of motion induce a coupling between the motion's wholebody momentum (linear and angular) and all contact wrenches. If there exists no set of wrenches that simulatenously lie in their friction cones and sum up to the wholebody momentum, the motion is infeasible. When such wrenches exist, the motion may be feasible. This latter condition is known as weak contact stability:
Definition (Pang and Trinkle): a motion of the robot is weakly contactstable if and only if there exists contact wrenches summing up to its wholebody momentum.
Weak contact stability is the underlying criterion used in a variety of humanoid motion planners and pattern generators. Note that, despite its use of the term "stability", this notion is only a feasibility condition. The "weakness" of its definition is relative to the more general definition of contact stability in the context of contact modes:
 A motion is contactstable when all its contacts stay in the same contact mode, i.e., there is no contact switch.
 A motion is weakly contactstable if it may keep all its contacts in the same contact mode, i.e., there exists contact wrenches that support the motion without changing contact mode.
 A motion is strongly contactstable if it must keep all its contacts in the same contact mode, that is, all contact wrenches that support the motion can only be realized without changing contact mode. (This condition is hard to achieve and rarely used in practice.)
The bottom line of these definitions is to avoid contact switches as best as possible. For a concrete example, think of these switches as follows:
In practice, only a few contact modes are exploited by current humanoid controllers. Roughly, only two contact modes: fixed (full) contacts, and no contact. Detecting contact switches on real robot, or providing a controller that would be general enough to deal with multiple contact modes, is still a widely open question, as of 2016.
Checking contact stability
In practice, checking contact stability for a given robot motion amounts to solving a Quadratic Program (QP). Let \((\pdd_G, \dot{\bfL}_G)\) denote the centroidal momentum of the motion, which is the wholebody momentum taken at the center of mass \(G\). Then, feasible contact wrenches exist if and only if a solution to the following QP can be found:
where \(\bfw_{C_i} = (\bff_i, \bftau_{C_i})\) is the vector of coordinates for the \(i^\mathrm{th}\) contact wrench taken at a contact point \(C_i\) and in the world frame. The matrix \(\bfF\) describes a linearized friction cone model in the local frame, and is thus applied after a rotation \(\bfR_i^T\) of wrench coordinates from the world to the local contact frame.
Centroidal wrench cone
The centroidal wrench cone (CWC) is a polyhedral convex cone that characterizes all feasible motions, without solving an optimization problem for each of them. It is the 6D frictional wrench cone computed for the net contact wrench, i.e. the sum of all contact wrenches, using numerical polyhedral projection algorithms.
By construction, a solution can be found to the QP above if and only if the centroidal momentum \((\pdd_G, \dot{\bfL}_G)\) lies in the CWC. When there is a large number of momenta to check, it is computationally more efficient to compute the CWC as a matrix \(\bfU\) first, and then perform each check as \(\bfU [\pdd_G\ \dot{\bfL}_G] \leq \boldsymbol{0}\) rather than solving a large number of quadratic programs.
On an advanced note: in the figure above, the CWC is represented by a red force cone and a green moment cone. This is a drawing convenience: in practice, the CWC is a 6D cone where force and moment are not independent. If you choose a resultant force in the red cone, it will affect the shape of the green one. And conversely, fixing the resultant moment to a given value (e.g. zero, see below) will affect the shape of the red cone.
Contactstability areas and volumes
When additional constraints are imposed on the centroidal motion, the 6D centroidal wrench cone reduces to lowerdimensional areas and volumes that can be used for planning or control:

When the robot is not moving, contact stability is characterized by the static equilibrium polygon: the configuration of the robot is feasible (sustainable) if and only if the center of mass lies in a specific polygon, which can be computed efficiently.

When the robot moves in the Linear Inverted Pendulum Mode (LIPM, i.e. with conserved angular momentum and keeping the COM in a plane), the CWC reduces to a ZMP support area.

When the robot moves with a conserved angular momentum \((\Ld_G = 0)\), the CWC reduces to a 3D cone over COM accelerations that can be used e.g. for multicontact locomotion.
To go further
Stability conditions are used extensively in motion generation. A description of the computation of the CWC can be found in this paper, while the algorithms themselves are implemented in the ContactSet class of the pymanoid library. For a practical application, you can take a look at this multicontact humanoid walking pattern generator.