Deep learning of multibody minimal coordinates for state and input estimation with Kalman filtering

Abstract

In general, multibody models are described with a set of redundant coordinates and additional constraints. Their dynamics is thus expressed through differential algebraic equations. As an alternative, the minimal coordinate formulation permits to describe a rigid system with the minimal number of variables leading to ordinary differential equations which can be employed in a coupled state/input estimation scheme. However, in some cases the explicit relation between the full-system coordinates and the minimal coordinates may not be available or analytically obtainable, as for closed-loop mechanisms. In this work, a previously presented deep learning framework to find the non-linear mapping and reduce a generic multibody model from redundant to minimal coordinates is employed. The resulting equations are then exploited in an extended Kalman filter where the unknown inputs are considered as augmented states and jointly estimated. The necessary derivatives are given and it is shown that acceleration measurements are sufficient for the estimation. The method is experimentally validated on a slider–crank mechanism.

Appendix A: Input projection matrix

Here, the generic input projection into natural coordinates is obtained. Assuming that the external input (to be estimated) $\boldsymbol{a}$ is a force or a torque in one of the 6 directions:

$$ \boldsymbol{a} = \left [ \textstyle\begin{array}{l} \boldsymbol{a}_{t} \\ \boldsymbol{a}_{r} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{l} a_{t,x} \\ a_{t,y} \\ a_{t,z} \\ a_{r,x} \\ a_{r,y} \\ a_{r,z} \end{array}\displaystyle \right ] $$

(59)

where $\boldsymbol{a}_{t}, \ \boldsymbol{a}_{r} \in \mathbb{R}^{3}$ are, respectively, the translational and rotational components.

If the force/torque is acting on the body $b$, its projection into natural coordinates $\boldsymbol{a}_{n,b} \in \mathbb{R}^{12}$ can be obtained:

$$ \boldsymbol{a}_{n,b} = \left [ \textstyle\begin{array}{c@{\quad }c} \boldsymbol{I}_{3} & \boldsymbol{a}_{t} \\ \boldsymbol{\Theta }_{b}^{\mathrm{T}}& \boldsymbol{a}_{r} \end{array}\displaystyle \right ] = \boldsymbol{S}_{n,b} \ \boldsymbol{a} $$

(60)

where $\boldsymbol{\Theta }_{b} \in \mathbb{R}^{3 \times 9}$ is the NC angular transformation matrix:

$$ \boldsymbol{\Theta }_{b} (\boldsymbol{r}_{b}) = \left ( \frac{\partial \boldsymbol{\theta }_{b}}{\partial \boldsymbol{r}_{b}} \right )^{\mathrm{T}}$$

(61)

with $\boldsymbol{r}_{b} \in \mathbb{R}^{9}$ representing the body rotation matrix components as in Eq. (16) and $\angle \boldsymbol{\theta }_{f} \in \mathbb{R}^{3}$ as the angular displacement. Knowing [26] the NC force due to a torque along a certain unit vector, the NC angular transformation matrix can be expressed as

$$ \boldsymbol{\Theta }_{b} = - \frac{1}{2} \ \left [ \textstyle\begin{array}{l} \left [\boldsymbol{r}_{b,x}\right ]_{\times }\\ \left [\boldsymbol{r}_{b,y}\right ]_{\times }\\ \left [\boldsymbol{r}_{b,z}\right ]_{\times }\end{array}\displaystyle \right ]^{\mathrm{T}}.$$

(62)

Therefore, the obtained NC force can be projected into minimal coordinates as:

$$ \boldsymbol{a}_{m} = \left ( \frac{\partial \psi }{\partial \boldsymbol{q}_{m}} \right )^{\mathrm{T}}\ \boldsymbol{S}_{b \in n}^{\mathrm{T}}\ \boldsymbol{a}_{n, b} $$

(63)

where $\boldsymbol{S}_{b \in n} \in \mathbb{Z}^{12 \times n_{n}}$ is the (generalized Kronecker delta) sparse matrix to select the 12 body coordinates from the $n_{n}$ natural coordinates and $\boldsymbol{a}_{m} \in \mathbb{R}^{n_{m}}$ is the MC projection of the unknown external forces. If, instead of the reference body frame, the input is applied on the generic body-attached frame $f$, a projection as in Eq. (51) has to be performed.

It is finally noted that while for the generic case the derivative of the input projection matrix $\boldsymbol{S}_{n}$ has to be taken into account in Eq. (54), in the case of purely translational forces it is constant, simplifying the expression.