Low thrust relative orbital transfer

The continuous low-thrust relative transfer can be described in mathematical form by adding components of specific thrust which will act as a control input in the equations of motion model for relative orbital transfer. Although a number of linearized models have been developed since 1960s which gives simplified set of equations, one popular model was developed by W. H. Clohessy and R. S. Wiltshire, and is modified to account for continuous motion and can be written as:

${\displaystyle {\ddot {x}}=3n^{2}x+2n{\dot {y}}+u_{x}}$

${\displaystyle {\ddot {y}}=-2n{\dot {x}}+u_{y}}$

${\displaystyle {\ddot {z}}=-n^{2}z+u_{z}}$

where:

• ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are the relative distance components of the chaser in the target fixed frame of reference
• ${\displaystyle u_{x},u_{y}}$ and ${\displaystyle u_{z}}$ are the specific thrust components in the form of control input along ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$-axis of the target fixed frame of reference
• ${\displaystyle n}$ is the orbital frequency of the target orbit

Since in continuous low-thrust transfers the thrust magnitude is limited, such type of transfers are usually subjected to certain performance index and final state constraints, posing the transfer as an optimal control problem with defined boundary conditions.^[3] For the transfer to have optimal control input expenditure, the problem can be written as:

${\displaystyle J={\frac {1}{2}}\int _{t_{0}}^{t_{f}}({\vec {u}}^{T}\cdot R\cdot {\vec {u}})dt}$

subjected to dynamics of the relative transfer:

${\displaystyle {\dot {\vec {x}}}=A{\vec {x}}+B{\vec {u}}}$

and boundary conditions:

${\displaystyle {\vec {x}}(t_{0})={\vec {x}}_{0}}$

${\displaystyle {\vec {x}}(t_{f})={\vec {x}}_{f}}$

where:

• ${\displaystyle {\vec {x}}}$ is the state-vector defined as ${\displaystyle {\vec {x}}={\begin{bmatrix}x&{\dot {x}}&y&{\dot {y}}&z&{\dot {z}}\end{bmatrix}}^{T}}$
• ${\displaystyle {\vec {u}}}$ is the control input vector defined as ${\displaystyle {\vec {u}}={\begin{bmatrix}u_{x}&u_{y}&u_{z}\end{bmatrix}}^{T}}$
• ${\displaystyle R}$ is the weight matrix
• ${\displaystyle A}$ is the state matrix obtained from the Clohessy-Wiltshire equations, such that, ${\displaystyle A={\begin{bmatrix}0&1&0&0&0&0\\3n^{2}&0&0&2n&0&0\\0&0&0&1&0&0\\0&-2n&0&0&0&0\\0&0&0&0&0&1\\0&0&0&0&-n^{2}&0\\\end{bmatrix}}}$
• ${\displaystyle B}$ is the input matrix, such that, ${\displaystyle B={\begin{bmatrix}0&0&0\\1&0&0\\0&0&0\\0&1&0\\0&0&0\\0&0&1\\\end{bmatrix}}}$
• ${\displaystyle t_{0}}$ is the time of start of transfer
• ${\displaystyle t_{f}}$ is the time of end of transfer
• ${\displaystyle {\vec {x}}_{0}}$ is the initial value of the state vector
• ${\displaystyle {\vec {x}}_{f}}$ is the final value of the state vector

Sometimes, it is also useful to subject the system to control constraints because in case of continuous low-thrust transfer, there are always bounds on the availability of thrust. Hence, if the maximum quantity of thrust available is ${\displaystyle u_{max}}$, then, an additional inequality constraint can be imposed on the optimal control problem posed above as:

${\displaystyle ||{\vec {u}}(t)||\leq u_{max}}$

Additionally, if the relative transfer is occurring such that the chaser and the target spacecraft are very close to each other, the collision-avoidance constraints can also be employed in the optimal control problem in the form of a minimum relative distance, ${\displaystyle r_{min}}$ as:

${\displaystyle ||{\vec {x}}(t)||\geq r_{min}}$

and because of obvious reasons, the final value of state-vector cannot be less than ${\displaystyle r_{min}}$.

• Bi-elliptic transfer
• Delta-v budget
• Geostationary transfer orbit
• Halo orbit
• Lissajous orbit
• List of orbits
• Orbital mechanics