# Transformation of the dynamics of mechanical system under coordinate change

It is well known that the dynamics equation for a mechanical system (e.g. a robotics manipulator) is given be the Euler-Lagrange equation which takes the particular form (in the simplified version),

$$M(q)\ddot{q}+N(q,\dot{q})=u$$

where $M$ is the inertia tensor, $N$ the Coriolis/centripetal vector and $u$ the input (torque). In a given coordinate change of the state e.g.

$$q=h(y)$$

following a similar procedure as in "Robot manipulator control: Theory and Practice" pp 148-149, by Frank Lewis, we get,

$$\dot q=J\dot y$$

where $J=J(y)=\frac{\partial h }{\partial y}$ is teh Jacobian of $h$. Taking second derivatives gives,

$$\ddot q=\dot J\dot y+J\ddot y$$

Plugging into the dynamics we have,

$$M(q)(\dot J \dot y+J\ddot y)+N(q,\dot q)= u \Rightarrow$$

$$M(q)J\ddot y+(N(q,\dot q)+M(q)\dot J \dot y)= u \Rightarrow$$

Apparently this can be shorthanded as,

$$\bar{M}\ddot{y} + \bar{N} =u$$

by collecting terms. The problem is that $M$ and $N$ are still functions of $q$. Given that $M$ is a tensor and $N$ a vector, they transform as such. Thus

$$\hat{M}(y)=J^{T}M(h(y))J$$ and

$$\hat{N}(y)=JN(h(y))$$

Substituting in the transformed equation we get,

$$J^{T}M(h(y))JJ\ddot{y}+(JN(h(y))+J^{T}M(h(y))J\dot{J}\dot{y})= u$$

And the question is, are the last three equations correct? i.e. does the inertial tensor and the Coriolis vector transform this way?

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