source: Chapter 4 [[alfriendSpacecraftFormationFlying2010]] ## Nonlinear Relative Spacecraft Dynamics ![[Pasted image 20240214152227.png]] The derivation of the Clohessy-Wiltshire Dynamics starts with deriving the inertial equations of the chief (which we will call the target) $ \ddot{\mathbf{r}_0}=-\frac{\mu}{r_0^3} \mathbf{r}_0 $ and the deputy (which we will call thee chaser from hereon). $ \ddot{\mathbf{r}_1}=-\frac{\mu}{r_1^3} \mathbf{r}_1 $ Indeed, the assumption is that the dynamics of each of these is derived from two-body dynamics. Our interest is in studying the relative dynamics of the chaser relative to the target. The position of the deputy relative to the chief is $ \boldsymbol{\rho}= \mathbf{r}_1 - \mathbf{r}_0 $ We can derive the second-order realtive by subtracting the inertial equations of motion of the chaser and target to get $ \ddot{\boldsymbol{\rho}}=-\frac{\mu\left(\mathbf{r}_0+\boldsymbol{\rho}\right)}{\left\|\mathbf{r}_0+\boldsymbol{\rho}\right\|^3}+\frac{\mu}{r_0^3} \mathbf{r}_0 $ In order to express the relative acceleration in frame $\mathscr{L}$, we recall that $ \begin{aligned} \ddot{\boldsymbol{\rho}}= & \frac{d^{2 \mathscr{L}} \boldsymbol{\rho}}{d t^2}+2^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}} \times \frac{d^{\mathscr{L}} \boldsymbol{\rho}}{d t} \\ & +\frac{d^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}}}{d t} \times \boldsymbol{\rho}+{ }^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}} \times\left({ }^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}} \times \boldsymbol{\rho}\right) \end{aligned} \tag{1} $ where ${ }^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}}$ denotes the angular velocity vector of frame $\mathscr{L}$ relative to frame $\mathscr{I}$. As ${ }^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}}$ is normal to the orbital plane, we may write $ { }^{\mathscr{I}} \boldsymbol{\omega}^{\mathscr{L}}=\left[0,0, \dot{\theta}_0\right]^T $ The position vector of the chief can be written as $ \mathbf{r}_0=\left[r_0, 0,0\right]^T $ Also, let $ [\boldsymbol{\rho}]_{\mathscr{L}}=[x, y, z]^T $ Substituting the above into. Equation (1) yields the following scalar equations for relative motion: $ \begin{gathered} \ddot{x}-2 \dot{\theta}_0 \dot{y}-\ddot{\theta}_0 y-\dot{\theta}_0^2 x=-\frac{\mu\left(r_0+x\right)}{\left[\left(r_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}}+\frac{\mu}{r_0^2} \\ \ddot{y}+2 \dot{\theta}_0 \dot{x}+\ddot{\theta}_0 x-\dot{\theta}_0^2 y=-\frac{\mu y}{\left[\left(r_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \\ \ddot{z}=-\frac{\mu z}{\left[\left(r_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \end{gathered} \tag{2} $ Equations (2) together with the target's scalar equations of inertial motion $ \begin{gathered} \ddot{r}_0=r_0 \dot{\theta}_0^2-\frac{\mu}{r_0^2}\\ \ddot{\theta}_0=-\frac{2 \dot{r}_0 \dot{\theta}_0}{r_0} \end{gathered} $ constitute a 10-dimensional system of nonlinear differential equations. For $\theta_0 \neq 0$, these equations admit a single relative equilibrium at $x = y = z = 0$, meaning that the deputy spacecraft will appear stationary in the chief frame if and only if their positions coincide on a given elliptic orbit. ## Simplification 1: Circular Chief/Target Orbit So far, we have derived the general nonlinear equations of relative motion for arbitrary target orbits. A simpler, autonomous, form of the relative motion can be derived using assuming the circular chief/target orbit. In this case, $\dot{\theta}_0=n_0=$ const., $\ddot{\theta}_0=0$ and $r_0=a_0=$ const. Substituting into Eqs. (2) results in $ \begin{gathered} \ddot{x}-2 n_0 \dot{y}-n_0^2 x=-\frac{\mu\left(a_0+x\right)}{\left[\left(a_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}}+\frac{\mu}{a_0^2} \\ \ddot{y}+2 n_0 \dot{x}-n_0^2 y=-\frac{\mu y}{\left[\left(a_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \\ \ddot{z}=-\frac{\mu z}{\left[\left(a_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \end{gathered} \tag{3} $ These equations admit an equilibria continuum $(\text{i.e., }\dot{x}=\ddot{x}=\dot{y}=\ddot{y}=\dot{z}=\ddot{z}=0)$ given by $ \begin{gathered} z=0\\ \left(x+a_0\right)^2+y^2=a_0^2 \end{gathered} $ $a_0$ is the target’s orbit semimajor axis. These equations, in fact, are the equations of a circle centered at $x = −a_0, y = 0$. > [!faq] What is Libration? > From the dynamical systems perspective, we expect that there exist small perturbations near equilibria that will generate periodic orbits about the equilibria. The resulting periodic motion is called libration. > > A similar terminology is used for describing the relative motion between celestial bodies. The dynamic structure of the problem in this case is much more involved, as it includes the mutual gravitation of the bodies ## Simplification 2: Linearization of Equation (3) There are a number of ways to develop and solve the $\mathrm{CW}$ equations. We will discuss some of these methods, as each method contributes insight into the relative motion problem. A straightforward approach for developing the $\mathrm{CW}$ equations is to expand the right-hand side of Eqs. (4.74)-(4.76) into a Taylor series about the origin. Taking only the first-order terms and denoting $n_0=\sqrt{\mu / a_0^3}$ we get $ \begin{gathered} -\frac{\mu\left(a_0+x\right)}{\left[\left(a_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \approx n_0^2\left(2 x-a_0\right) \\ -\frac{\mu y}{\left[\left(a_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \approx-n_0^2 y \\ -\frac{\mu z}{\left[\left(a_0+x\right)^2+y^2+z^2\right]^{\frac{3}{2}}} \approx-n_0^2 z \end{gathered} $ Rearranging and omitting the subscript 0 (so that $n \equiv n_0$ and $a \equiv a_0$ ) yields the CW equations $ \begin{gathered} \ddot{x}-2 n \dot{y}-3 n^2 x=0 \\ \ddot{y}+2 n \dot{x}=0 \\ \ddot{z}+n^2 z=0 \end{gathered} \tag{4} $ Addition of disturbing and/or control accelerations gives the non-homogeneous form of the CW equations $ \begin{gathered} \ddot{x}-2 n \dot{y}-3 n^2 x=d_x+u_x \\ \ddot{y}+2 n \dot{x}=d_y+u_y \\ \ddot{z}+n^2 z=d_z+u_z \end{gathered} $