#### Six Axis Stress Gauge

Consider two adjacent rigid bodies bound together to form a larger rigid body. There are forces and torques on the two bodies but together they are in equilibrium. We want to characterize, and measure, the forces and torques between the two bodies at the binding. Six numbers suffice to describe these forces and torques. Three numbers describe the thrust which is the vector sum of the forces, summed across all the points of the interface. Torque requires three numbers as well and the expression of the torque, as an axial vector, requires selecting a point at which to describe the torque.

I propose a stress gauge configuration in the shape of a regular octahedron. Three of the vertex coordinates are at (1, 0, 0), (0, 1, 0) & (0, 0, 1). These three are rigidly attached to a handle. The other three vertices are at (−1, 0, 0), (0, −1, 0) & (0, 0, −1) and are rigidly anchored to the ground. There are six edges of the octahedron that connect the handle vertices and the ground vertices. Along each of these six edges is a one dimensional structural element that includes a stress gauge measuring force along the axis of the element. These six elements form a non-planar hexagon. The inspiration for this shape comes from the well known six axis Stewart platform supported by a similar set of six hydraulic pistons which can, within limits move the platform with six degrees of freedom.

An interesting exercise is to develop the relations between the six gauges and the forces as conventionally expressed. The relations are linear. I have found derivations based on geometric symmetry to be easier than more analytic means.

Imagine a 12 stick regular octahedron and a one dyne force pushing towards the center on the two points (0,0,1) and (0,0,−1). By symmetry, the four sticks at the top each support 1/4 dyne thrust in the z-direction. The thrust along their axis is sqrt(2)/4. Now consider the vertex (1, 0, 0). Two of the sticks meeting that vertex are under compression with force sqrt(2)/4. By symmetry the other two sticks must be in tension with force sqrt(2)/4 for that point to be in equilibrium. By symmetry we now know all 12 stick forces.

I claim that when a down (−z) force on the handle, centered at the octahedron’s center will produce the same stress on the six gauges as in the more symmetric example above. This is due in part to the fact that the octahedron is not over constrained which means that the length of any of the 12 edges can be changed a little without producing an impossible figure.

The torque relations are simpler!

See this too.

The Stewart Platform is a closely related idea. That uses six links whose length can be controlled to move a rigid element thru 6 degrees of freedom. Here and here are references to the stewart platform. Here is a brief comment about the stewart platform being able to convey 6 force values. Two sites, 1 and 2 on haptic ideas implicitly assume some such idea.