Sensitivity Analysis of Multibody Dynamic Systems Modeled by ODEs and DAEs

Abstract

The optimization of the dynamic response of multibody dynamic systems is a complex and open problem. It relies on using the equations of motion of the system, which amplifies the level of complexity of the problem substantially, compared to other types of optimization. In the context of this kind of optimization, the sensitivity analysis of the dynamic response of the system is a key element. Two main techniques are currently available for the sensitivity analysis of the response of a dynamical system: the direct differentiation method and the adjoint variable method. In this work, different formulations of the equations of motion with dependent coordinates are employed and their sensitivity equations obtained. Direct and adjoint sensitivities are convenient for different types of problems but both methods require accurate derivatives of the equations of motion considered. Both approaches are employed in this work; the direct and adjoint sensitivity equations are obtained for index-3 differential-algebraic equations (DAE), index-1 DAE, and penalty formulations. The adjoint sensitivity for the penalty formulations introduced here is completely novel.

Appendix: Nomenclature and Differentiation Rules

• $\left( \cdots \right) _0$: means evaluation at the initial time $\left( \cdots \right) \left( t_0\right)$.

• $\left( \cdots \right) _F$: means evaluation at the final time $\left( \cdots \right) \left( t_F\right)$.

• $\mathbf{q} \in {\mathbb {R}}^{n}$: vector of coordinates of the system.

• ${\varvec{\rho }} \in {\mathbb {R}}^{p}$: vector of parameters.

• $\delta \left( \right)$: means variation.

• $\left( \right) _\mathbf{q} = \displaystyle \dfrac{\partial \left( \right) }{\partial \mathbf{q}}; \; \left( \right) _{\varvec{\rho }} = \displaystyle \dfrac{\partial \left( \right) }{\partial {\varvec{\rho }}}$

• $\dot{\left( \right) } = \displaystyle \dfrac{\mathrm{d}\left( \right) }{\mathrm{d}t}; \; \ddot{\left( \right) } = \displaystyle \dfrac{\mathrm{d}^2\left( \right) }{\mathrm{d}t^2} \; \left( \right) _{t} = \displaystyle \dfrac{\partial \left( \right) }{\partial t}$

• $\mathbf{M}\left( \mathbf{q},{\varvec{\rho }}\right) \in {\mathbb {R}}^{n \times n}$: generalized mass matrix of the system.

• $\mathbf{Q}\left( t,\mathbf{q},\dot{\mathbf{q}},{\varvec{\rho }} \right) \in {\mathbb {R}}^{n}$: vector of generalized forces of the system.

• ${\varvec{\Phi }}\left( t,\mathbf{q},{\varvec{\rho }} \right) \in {\mathbb {R}}^{m}$: vector of constraints that relate the dependent coordinates.

• $\mathbf{A}_\mathbf{x} = \left[ \begin{array}{c c c c c} \displaystyle \frac{\partial \mathbf{A}}{\partial x_1}&\ldots&\displaystyle \frac{\partial \mathbf{A}}{\partial x_i}&\ldots&\displaystyle \frac{\partial \mathbf{A}}{\partial x_{s}} \end{array}\right] \in {\mathbb {R}}^{q{\times }r{\times }s}$. Third order tensor of derivatives of matrix $\mathbf{A} \in {\mathbb {R}}^{q{\times }r}$ w.r.t. vector $\mathbf{x} \in {\mathbb {R}}^{s}$.

• $\mathbf{A}_\mathbf{x}^\mathrm{T} =\left[ \begin{array}{c c c c c} \displaystyle \frac{\partial \mathbf{A}^\mathrm{T}}{\partial x_1}&\ldots&\displaystyle \frac{\partial \mathbf{A}^\mathrm{T}}{\partial x_i}&\ldots&\displaystyle \frac{\partial \mathbf{A}^\mathrm{T}}{\partial x_{s}} \end{array}\right] \in {\mathbb {R}}^{r{\times }q{\times }s}$.

• $\mathbf{A}_\mathbf{x}\mathbf{b} = \mathbf{A}_\mathbf{x}\otimes \mathbf{b} = \left[ \begin{array}{c c c c c} \displaystyle \frac{\partial \mathbf{A}}{\partial x_1} \mathbf{b}&\ldots&\displaystyle \frac{\partial \mathbf{A}}{\partial x_i} \mathbf{b}&\ldots&\displaystyle \frac{\partial \mathbf{A}}{\partial x_{s}} \mathbf{b} \end{array}\right] \in {\mathbb {R}}^{q{\times }s}$, where $\mathbf{b} \in {\mathbb {R}}^{r}$ is a vector.

• $\mathbf{A}_\mathbf{x}\mathbf{B} = \mathbf{A}_\mathbf{x}\otimes \mathbf{B} = \left[ \begin{array}{c c c c c} \displaystyle \frac{\partial \mathbf{A}}{\partial x_1} \mathbf{B}&\ldots&\displaystyle \frac{\partial \mathbf{A}}{\partial x_i} \mathbf{B}&\ldots&\displaystyle \frac{\partial \mathbf{A}}{\partial x_{s}} \mathbf{B} \end{array}\right] \in {\mathbb {R}}^{q{\times }t{\times }s}$, where $\mathbf{B} \in {\mathbb {R}}^{r{\times }t}$ is a matrix.

• $\mathbf{C} \mathbf{A}_\mathbf{x}\mathbf{B} = \mathbf{C} \otimes \mathbf{A}_\mathbf{x}\mathbf{B} = \left[ \begin{array}{c c c c c} \mathbf{C} \displaystyle \frac{\partial \mathbf{A}}{\partial x_1} \mathbf{B}&\ldots&\mathbf{C} \displaystyle \frac{\partial \mathbf{A}}{\partial x_i} \mathbf{B}&\ldots&\mathbf{C} \displaystyle \frac{\partial \mathbf{A}}{\partial x_{s}} \mathbf{B} \end{array}\right] \in {\mathbb {R}}^{r{\times }t{\times }s}$, where $\mathbf{C} \in {\mathbb {R}}^{r{\times }q}$ is a matrix.