Equations of motion in matrix form and energy consumption

Question

I am working on Lagrangian derived high-dimensional motion equations for a robot in matrix form. The structure of such an equation is known:

$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=0$

In here, $M(q)$, $C(q,\dot{q})$ are matrices, and $G(q)$ are vector.

Are there any properties of $M(q)$, $C(q,\dot{q})$, $G(q)$ related to the power consumption of the robot and is it possible to optimize power consumption by operating with these very properties?

Remark: I mean they are matrices, vectors, etc., i.e. is it possible to somehow use their traces, determinant, norm and other to use for the study of energy consumption?

Answer 1

Can you set up the problem so that the quantities you care about (e.g. power) are more explicitly represented? Reasoning physically, where could the power go?

• Accelerating masses, including rotation
• Pushing against gravity, electromagnetics, other potentials
• Stretching springs and whatever
• Dissipation through friction

You say "optimize power consumption". One could minimize

• peak instantaneous power over some cycle of operation
• total energy (average power) over cycle
• RMS power over cycle
• heat dissipation (tight thermal constraints?)

I would choose one of those to optimize, and express it in terms of your system's M, C and G. I would expect properties of the resulting matrix equation(s) to be easier to reason about than the original M, C and G. I know just enough about multidimensional functional minimization of (potentially) nonlinear matrix equations to know I don't want to go there right now (if ever!)

Good luck!

Answer 2

First, you need to get rid of the damping matrix C as it transforms kinetic energy into heat. Second, you should make the mass matrix as small as possible. (lightweight construction). After that you can think about the best way to distribute the mass that you cant get rid off by taking the desired movements into account.