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Polynomial Guidance in Spacecraft: A Simplified Perspective

Updated: Jun 10, 2023


Polynomial guidance is a concept used in various fields, including guidance, navigation, and control (GNC) engineering. This approach is particularly relevant in space and aeronautics technology.


The main idea behind polynomial guidance is using polynomial equations to plot a trajectory for a vehicle (like a spacecraft or missile) from point A to point B. This trajectory is designed to meet certain criteria such as minimizing fuel consumption, time, or distance, or to avoid obstacles. The shape and properties of the trajectory are determined by the choice of polynomial degree and coefficients.


In simple terms, consider a 2nd degree polynomial (a quadratic equation), which describes a parabolic path. If you graph y = ax^2 + bx + c, you get a curve that might describe the arc of a thrown ball, for instance.


Guiding a spacecraft from Earth to Mars, however, requires a more complex path that might involve 4th, 5th, 6th degree polynomials or even higher, depending on the complexity of the path and the number of constraints that need to be considered.


Typically, the polynomial is chosen such that its derivative at the beginning and end points matches the desired velocities at those points. For instance, in a 3rd degree polynomial, there are four coefficients, which can be solved if you have four boundary conditions (e.g., initial and final position and velocity).


Polynomial guidance is implemented using optimization algorithms to select the coefficients that meet your mission criteria. The resulting polynomial is then fed into the vehicle's guidance system, which guides the vehicle along the polynomial path to its destination.


Let's consider a simplified example with a 3rd-degree polynomial, often used in guidance, navigation, and control (GNC) systems. A 3rd-degree polynomial can be written in this form:


f(t) = at³ + bt² + c*t + d.


Here, t represents time, and f(t) is the position of the vehicle at time t. The coefficients a, b, c, and d will determine the shape of the trajectory.


Now, imagine we want to guide a vehicle from an initial position to a final position over a specified time period. For simplicity, we'll assume one-dimensional motion (say, along the x-axis), and we'll ignore factors like gravity and air resistance. We'll define our initial conditions at time t=0 and our final conditions at time t=T.


Given these 4 conditions, we can solve for the 4 unknowns (a, b, c, and d) in our polynomial. The vehicle's guidance system would then use this polynomial to compute the desired position and velocity at each moment in time, and control the vehicle's engines to achieve those values.


In real-world applications, the scenario is much more complex - we'd have to deal with three dimensions plus factors like gravity, vehicle dynamics, and potential changes in the target position. But the basic concept remains the same: define a polynomial that satisfies your initial and final conditions, and then follow that polynomial to guide the vehicle to its target.


Expanding this to spacecraft, the Clohessy-Wiltshire (C-W or Hill's) equations are a set of simplified equations that describe the relative motion between two spacecraft in a circular orbit. These equations are often used in analyzing orbital rendezvous problems. They are essentially an approximation in a local frame where one spacecraft (the target) is at the origin and the other (the chaser) is close by.


The C-W equations, in their simplest form, are:

dx'' = 3n²x + 2ny'

dy'' = -2nx'

dz'' = -n²z


Here, (x, y, z) are the relative positions of the chaser spacecraft with respect to the target, n is the mean motion of the target spacecraft's orbit (essentially its orbital speed), and the primes denote time derivatives.


Now, suppose we want to guide the chaser spacecraft from an initial relative position and velocity to a final relative position and velocity over a specified time period. We can use 3rd degree polynomials for this, similar to the drone example.


We'll have:

f_x(t) = axt³ + bxt² + cxt + dx

f_y(t) = ayt³ + byt² + cyt + dy

f_z(t) = azt³ + bzt² + cz*t + dz


With the initial and final conditions:

Initial:

f_x(0) = x0,

f_x'(0) = vx0 (and similarly for y and z)

Final:

f_x(T) = xT,

f_x'(T) = vxT (and similarly for y and z)


We can solve for the coefficients (a, b, c, d) in the same way as before. Here's where it gets interesting: the C-W equations are second-order differential equations, but our polynomials are only third degree, so their second derivatives are only first degree. This means they can't exactly satisfy the C-W equations at all points in time, only at the initial and final times.


However, for small maneuvers where the chaser is close to the target and the time period is short, this approximation can be satisfactory. The spacecraft's guidance system would use the polynomials to compute the desired position and velocity at each moment in time, and control the spacecraft's thrusters to achieve those values.


Cite this article as:

Y. Kumar, “Polynomial Guidance in Spacecraft: A Simplified Perspective,” Space Navigators, May 13, 2023. [Online]. Available: https://www.spacenavigators.com/post/polynomial-guidance-in-spacecraft-navigation-a-simplified-perspective

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