#### 12.1.1. First-Order Analysis of a Single Contact

This video introduces contact kinematics, the study of the feasible motions of bodies in contact. The analysis is first-order, meaning it considers the contact locations and the contact normal directions, but not the local contact curvature.

Contact kinematics is the study of the motion constraints due to contact between bodies. For example, if these two bodies are in contact, I could ask what motions of the bodies will keep them in contact and what motions will cause breaking contact.

Let's say that q_1 and q_2 are coordinate representations of the rigid-body configurations, q is the combined configuration, and d of q is the distance between the two bodies. We can create a table of possibilities governing whether the two bodies are in contact depending on their trajectories q_1 of t and q_2 of t. If d is greater than zero, the two bodies are not in contact. If d is less than zero, the two bodies are in penetration, and therefore the combined configuration is not allowed. If d is equal to zero, the bodies are currently in contact, but if d-dot is greater than zero, this contact is about to break. If d-dot is less than zero, the bodies are about to penetrate, so the trajectories q_1 of t and q_2 of t are not allowed. If d and d-dot are zero, the bodies are in contact, but if d-double-dot is greater than zero, the contact is about to break. We could continue this analysis for increasing derivatives of d. The bodies only remain in contact if all time derivatives of d are equal to zero.

If we assume the bodies are initially in contact, we can express the time derivative of the distance between two bodies as d-dot equals the vector of partial derivatives of d with respect to q times q-dot. The acceleration of d is d-double-dot, which is the sum of the partial derivatives times q-double-dot and a velocity-product term depending on the matrix of second derivatives of d with respect to q. The vector of partial derivatives carries first-order information about the contact geometry, called the contact normal, which I'll define shortly. The matrix of second derivatives carries second-order information about the contact geometry, namely the curvature at the contact. For simplicity, in this chapter we assume that the second-order and higher-order information on the contact geometry is not available, and we focus on first-order contact geometry. I will highlight cases where the effect of this decision has consequences.

Consider this planar disk contacted by a stationary constraint. This constraint could be a robot finger, a workpiece fixture, or some other part of the robot or the environment. We define the contact tangent line to be the line tangent to the bodies at the contact. We also define the contact normal n to be a unit vector orthogonal to the tangent line. The contact normal could be defined either upward or downward.

Now imagine the disk is in contact with a constraint with a different curvature. The contact normal is the same relative to the disk, and by our first-order analysis, which ignores curvature, the constraints on the disk's motion are identical.

If the movable body is a spatial body contacted by another spatial body, the unit normal is orthogonal to the tangent plane. Again, because we ignore curvature, this pencil provides the same motion constraints on the movable body.

In this chapter we assume contacts between rigid bodies can be modeled as a finite set of point contacts. A planar contact that looks like this is modeled as two contacts, one on each edge adjacent to the vertex, with these contact normals. A line segment contact is modeled as a contact at each end of the line segment. A planar patch contact is modeled as a set of contacts at the vertices of the planar patch. A degnerate contact like this is not allowed, as there is no uniquely defined tangent plane or contact normal.

In the next video we will derive the constraints on the twists of bodies in contact and we will categorize contacts as breaking, sliding, or rolling.