# Equations of motion in matrix form and energy consumption

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.

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?

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?

• My suggestion is that you start from a two dimensional system (the 2dof system is a system that is easy to physically interpret). Then investigate the methods that work for "energy consumption" (you are using it very vaguely), and then try to extrapolate at higher dimensions.
– NMech
Commented Jul 19, 2021 at 5:51
• @NMech Yes, I agree. I need to start with something simple. And "vague", because I need to use a robot model to optimize energy properties and, as it seems to me, they should be embedded in the matrices of the equation of motion, but the criteria and cost functions are unknown to me. Therefore, it seems to me that it is quite an urgent task to establish the relationship between them.
– ayr
Commented Jul 19, 2021 at 5:56
• Could you provide a description what is the physical interpretation of G?
– NMech
Commented Jul 19, 2021 at 8:39
• IMHO this is very specific to robotics (and the abstract way you are presenting your question is not helpful either), you could try the Robotics SE, where there is a higher probability you will have someone acquainted with your particular problem.
– NMech
Commented Jul 19, 2021 at 10:21
• You can't do much with the very general equation in your OP unless you have some context of what you are modelling. For example, on the face of it $M(q)\ddot q$ says the inertia of the system depends on its position. Really? Do you mean the robot is picking up mass, moving it, and dropping it, or what? Or is inertia really constant and this is just "over-pedantic mathematical notation"? Commented Jul 19, 2021 at 13:29