# 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.

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?

• Posting this as a comment, because I don't think there's enough here to consider it as an "answer" exactly, but my gut reaction is that (1) the Lagrangian deals with the physical system, relating the kinetic and potential energies, so I don't know that there's an appropriate way to use the Lagrangian to optimize for power unless you're meaning you're going to optimize by making changes to the physical structure, and (2) if it is possible, the key would be to focus on the $\ddot{q}$ acceleration terms, because that's where your power will go. Minimize $\left(-M^{-1}C-M^{-1}G\right)$. Jul 19, 2021 at 13:27
• Great question, though - starred it to follow. I'm hoping you get some enlightening answers :) Jul 19, 2021 at 13:48
• @Chuck Of course, I will change the design parameters of the robot, which in turn will affect the moments of inertia, etc. And since the movement of the robot is mathematically described through a matrix differential equation of the second order, I thought that the energy properties of the system somehow find the greatest reflection in traces, determinants, norms, etc. Which can be used as an additional optimization criterion, i.e. cost functions.
– dtn
Jul 19, 2021 at 13:52
• @Chuck I agree that while the structure of the robot is not visible, it can be difficult to answer. As we progress in this, I plan to add new information to the question.
– dtn
Jul 19, 2021 at 13:52
• @r-bryan trank you for your answer, and I work on that - explicit representation of matrix
– dtn
Jul 21, 2021 at 5:57

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!

• Converted previous comments to answer. If it looked like an answer to you, it doesn't matter what I think! Jul 21, 2021 at 17:13

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.

• Could you tell us a little more about the elimination of the dumping matrix and the "distribution" of masses?
– dtn
Jul 25, 2021 at 7:02
• For example, you use an aerodynamic shape if you build a car. The damping does not go away but it gets as small as possible.
– Emil
Jul 25, 2021 at 9:47
• What is "make the mass matrix as small as possible" and "distribute the mass"? Can this be expressed as some kind of criterion relating the distribution to the trace, determinant or norm and their maximum/minimum?
– dtn
Jul 25, 2022 at 16:07