From: Adrian Freed <adrian@cnmat.berkeley.edu>
To: Vangelis L <vl_artcode@yahoo.com>, John MacCallum <john.m@ccallum.com>
Cc: Sha Xin Wei <shaxinwei@gmail.com>, Andy W.Schmeder <andy@cnmat.berkeley.edu>
Now we have the dot product, cross product, velocity and acceleration formulae in "o."
building in too many damning assumptions.
Working on Saturday with Teoma I got confirmation of my concerns about the notion of your "self" coordinate system referenced to a single point of the body, i.e. in mid-hips. She asked me to map hip movement so she could stand in one place for part of our piece. I used shoulder/hip distances for this which might not be what you would expect until you realize that the hips and shoulders have to work in counter motion in order for someone not to fall over.
One way of thinking about this is that Teoma can move the several interesint points of origin dynamically. Another more mature way perhaps is screw theory which models multiple connected bodies.
The quick hack without the full screw theory model is to acknowledge the hierarchy of connected limbs with tapering masses. Angular velocities as we have will therefor be useful here, but I think swept areas are probably better. You can weight the vector lengths with masses to get kinetic energy. Now the exercise is to decide which triangles to use for the swept areas. The formula for swept area is half the norm of the cross product but there is a problem here interpreting this as "effort". We have an asymmetry due to gravity.
It is a lot more work to raise a leg than sweep one. The problem is we have been focussing on kinetic energy without incorporating
changes in potential energy. I suspect you can project things to form a triangle with a point on the ground to reflect an area swept that estimates the potential energy change. Even better is to bite the bullet and move everything into tensors. Then you have enough
traction to do things like continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century, but research in the area continues today.
Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through a constitutive relation.
Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.
and exterior algebra which we need to do vector fields for orientation correlation.