12.2.2. Planar Graphical Methods

This video shows how the positive span of a set of planar wrenches (a wrench cone) can be represented graphically by labeling all points in the plane according to their moment labels: +, -, +/-, or no label. The label + indicates that the wrenches cannot make negative moment about the point, the label – indicates that the wrenches cannot make positive moment, the label +/- indicates that the wrenches can only make zero moment about that point, and no label means the wrenches can make positive or negative moment about that point.

A planar wrench has 3 components: moment about the z-axis out of the plane and linear forces in the x and y directions. If the wrench has a nonzero linear component, then it can be represented as an arrow in the plane, where the tail of the arrow is at (x,y) and the arrow tip is at (x + f_x, y + f_y). The point (x,y) must satisfy the condition that m_z equals x times f_y minus y times f_x. But since this is only one constraint on the two coordinates (x,y), we could place (x,y) at any point on the line of action of the arrow and get an equivalent representation of the wrench F.

To add two wrenches whose lines of action intersect, we can simply slide the arrows along their lines of action until the tails are coincident. Then we use the parallelogram vector sum to get the arrow representation of the new wrench, F_1 plus F_2.

If a wrench F represents a contact force, such as the edge of a friction cone, we often need to represent the set of all nonnegative scalings of that contact force. A convenient graphical method for representing all nonnegative scalings of a wrench is to label all points to the left of the wrench with a plus sign. These are the points about which the scaled wrench cannot create a negative moment. Similarly, all points to the right of the wrench are labeled with a minus sign, since the scaled wrench cannot create a positive moment about these points. Finally, all points on the line of action are labeled with a plus-minus sign. These labels of all the points in the plane are called moment labels. Moment labels allow a convenient graphical representation of planar wrench cones.

This is the representation for the nonnegative scaling of a single wrench. If we add a second wrench, we simply intersect the labels for the individual wrenches. Points in the plane that have no consistent labeling lose their labels. In this case, we have a region labeled plus, a region labeled minus and a point labeled plus-minus. This is a representation of the wrench cone of the positive span of the wrenches F_1 and F_2. This cone could be viewed the wrench cone for a single frictional contact.

These moment labels are properly interpreted as a single convex connected region. As with the rotation center representation, the plus and minus regions are connected at infinity.

If we add a third wrench F_3, then the consistently labeled region is just a small triangle labeled plus. This representation means that the positive span of the three wrenches can create any line of force that passes in a counterclockwise direction about the triangle. For example, this wrench is in the positive span of F_1, F_2, and F_3, since it makes positive moment about all points in the triangle. This wrench, which just passes through a vertex of the triangle, can also be generated. Remember that the plus sign just means that the combination of wrenches cannot make negative moment about the point, and this arrow does not make negative moment about the vertex.

On the other hand, the positive span does not include this wrench, which clearly makes negative moment about the entire region labeled plus. Also, the positive span does not include any wrench that passes through the labeled region.

In short, the moment-labeling representation of the wrench cone due to a friction cone looks like this, and it is easy to graphically combine multiple wrench cones to get a graphical representation of a composite wrench cone.

In the next video we'll use our understanding of contact wrench cones to study force closure.