Skip to content

Overview

The equations of motion of a flying vehicle can be organized as a set of simultaneous first-order differential equations, explicitly solved for the derivatives. For \(n\) independent variables, \(x_i\) (such as components of position, velocity, etc.), and \(m\) control inputs, \(u_i\) (such as throttle, control surface deflection, etc.), the general form will be

\[ \left\{ \begin{array}{rl} \dot{x}_1 & {}= f_1\big( x_1\,,\, x_2\,,\, \ldots \,,\,x_n\,,\,u_1\,,\, u_2\,,\, \ldots \,,\,u_m \big) \\ \dot{x}_2 & {}= f_2\big( x_1\,,\, x_2\,,\, \ldots \,,\,x_n\,,\,u_1\,,\, u_2\,,\, \ldots \,,\,u_m \big) \\ \cdots & \quad \cdots \\ \dot{x}_n & {}= f_n\big( x_1\,,\, x_2\,,\, \ldots \,,\,x_n\,,\,u_1\,,\, u_2\,,\, \ldots \,,\,u_m \big) \end{array} \right. \label{eq:EoM:ODE} \]

where the functions \(f_i\) are the nonlinear functions that arise from modeling vehicle's real subsystems. The variables \(x:i\) constitute the smallest set of variables that, together with given inputs \(u_i\), completely describe the behavior of the system (i.e. allow to deterministically compute its evolution in time), and called the set of state variables for the system, and Equations (\(\ref{eq:EoM:ODE}\)) are a state-space description of the system. The functions \(f_i\) are single-valued continuous functions. Equations (\(\ref{eq:EoM:ODE}\)) are often written symbolically as

\[ \dot{\boldsymbol{x}} = \boldsymbol{f}\big( \boldsymbol{x}\,,\,\boldsymbol{u} \big) \label{eq:EoM:ODE:Compact} \]

where the state vector \(\boldsymbol{x}\) is an \(n \times 1\) column array of the \(n\) state variables, the control vector \(\boldsymbol{u}\) is an \(m \times 1\) column array of the control variables, and \(\boldsymbol{f}\) is an array of nonlinear functions.

When \(\boldsymbol{u}\) is held constant, the nonlinear state equations (\(\ref{eq:EoM:ODE}\)), or a subset of them, usually have one or more equilibrium points in the multidimensional state and control space, where a given set of state variable derivatives vanish (usually derivatives having the meaning of translational or angular accelerations).

A major advantage of the state-space formulation is that the nonlinear state equations can be solved numerically. The simplest numerical solution method is Euler integration, described by

\[ \boldsymbol{x}(t_{k+1}) = \boldsymbol{x}(t_k) + \boldsymbol{f}\big( \boldsymbol{x}_k\,,\,\boldsymbol{u}_k \big) \, \Delta t \label{eq:EoM:ODE:Euler:Integration} \]

in which \(\boldsymbol{x}(t_k)\) is the value of the state vector computed at discrete times \(t_k = k \Delta t\), with \(k = 0,1,2, \ldots\), starting from an assigned initial condition \(\boldsymbol{x}(t_0) = \boldsymbol{x}_0\). The integration time step, \(\Delta t\), must be made small enough that, for every \(\Delta t\) interval, \(\boldsymbol{u}\) can be approximated by a constant value \(\boldsymbol{u}(t_k)\), and \(\dot{\boldsymbol{x}} \Delta t\) provides a good approximation to the increment in the state vector. This numerical integration allows the state vector to be stepped forward, in time increments of \(\Delta t\), to obtain a time-history simulation.

TBD