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corrections from lee

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Cat Flynn 2024-04-26 19:03:38 +01:00
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blogs/2024/4/25/interactive-astrodynamics.md
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@ -19,7 +19,7 @@ So how does Kerbal Space Program do it?
Kepler's laws make a simplification known as the two-body problem. Kepler's laws make a simplification known as the two-body problem.
More complex systems, such as Sun-Earth-Moon or the entire Solar System can be modelled as a composition of several two-body systems, rather than the _n_-body systems they really are. More complex systems, such as Sun-Earth-Moon or the entire Solar System can be modelled as a composition of several two-body systems, rather than the _n_-body systems they really are.
This conveniently sidesteps the requirement to accumulate forces and therefore error over time. This conveniently sidesteps the requirement to use approximate forces and therefore accumulate error over time.
The two body problem describes the motion of one body around another, or more precisely a common barycentre. The two body problem describes the motion of one body around another, or more precisely a common barycentre.
In the case of one massive and one tiny object, the barycentre is approximately in the same position as the centre of the larger object and can therefore be ignored. In the case of one massive and one tiny object, the barycentre is approximately in the same position as the centre of the larger object and can therefore be ignored.
In a two-body system, an orbit is perfectly stable and cyclical. In a two-body system, an orbit is perfectly stable and cyclical.
@ -28,15 +28,15 @@ A body's position can thus be calculated directly as a function of time.
The ability to calculate positions directly as a function of time has some extremely valuable properties. The ability to calculate positions directly as a function of time has some extremely valuable properties.
One of the many challenges faced in deep space exploration is the sheer duration required by such voyages - the Voyager probes were launched nearly half a century ago, and even reaching Mars takes the better part of a year. One of the many challenges faced in deep space exploration is the sheer duration required by such voyages - the Voyager probes were launched nearly half a century ago, and even reaching Mars takes the better part of a year.
To be of any real use, then, interactive astrodynamic simulations need a level of time control. To be of any real use, then, interactive astrodynamic simulations need a level of time control.
Kerbal Space Program has time warp feature, allowing players to navigate missions taking years of simulated time in an evening of real time. Kerbal Space Program has a time warp feature, allowing players to navigate missions taking years of simulated time in an evening of real time.
This would not be possible in with an iterative physics simulation, as the accuracy of the result is directly dependent on the resolution of the time step. This would not be possible with an iterative physics simulation, as the accuracy of the result is directly dependent on the resolution of the time step.
Games have to produce several frames per second to both sell the illusion of motion and stay responsive, meaning each frame must be computed in a small fraction of a second. Games have to produce several frames per second to both sell the illusion of motion and stay responsive, meaning each frame must be computed in a small fraction of a second.
Depending on the platform, target frame rates can vary from as low as 30Hz, for a mobile device preserving its battery, to 120Hz and beyond for competitive or virtual reality applications. Depending on the platform, target frame rates can vary from as low as 30Hz, for a mobile device preserving its battery, to 120Hz and beyond for competitive or virtual reality applications.
This means most games have about 10ms to compute each frame. This means most games have about 10ms to compute each frame.
Processing limitations are of particular interest in games, which are severely constrained in the amount of processing they are able to complete in a single rendered frame. Processing limitations are of particular interest in games, which are severely constrained in the amount of processing they are able to complete in a single rendered frame.
In practice, each sub-systems comprising a game has significantly less than 10ms, since it must use its budget conservatively to allow other sub-systems to use the same resources. In practice, each sub-system comprising a game has significantly less than 10ms, since it must use its budget conservatively to allow other sub-systems to use the same resources.
An astrodynamics physics simulation may be allowed only a single millisecond or less in which to calculate the paths of objects, due to constraints imposed by resourcing requirements of other sub-systems. An astrodynamics physics simulation may be allowed only a single millisecond or less in which to calculate the paths of objects, due to constraints imposed by resourcing requirements of other sub-systems.
Simulating physics to a high degree of accuracy with an iterative model requires high-resolution time steps. Simulating physics to a high degree of accuracy with an iterative model requires high-resolution time steps.
@ -68,7 +68,7 @@ With a lower number of iterations, the calculated value for _E_ is erratic and i
This presents a massive problem in a real-time system like a game. This presents a massive problem in a real-time system like a game.
To be fit for purpose a physics simulation needs to be stable in all reasonable cases the player might encounter, so a Newton-Raphson-based approach would need to be given the resources to run as many iterations as needed to determine a suitably accurate value for _E_. To be fit for purpose a physics simulation needs to be stable in all reasonable cases the player might encounter, so a Newton-Raphson-based approach would need to be given the resources to run as many iterations as needed to determine a suitably accurate value for _E_.
Since the behaviour is of an exponential increase in computation as orbits become increasingly eccentric, the worst-case performance many orders of magnitude worse than the best- or even average-case performance. Since the behaviour is of an exponential increase in computation as orbits become increasingly eccentric, the worst-case performance is many orders of magnitude worse than the best- or even average-case performance.
Even if the worst case could be computed accurately while maintaining the target frame rate, in the majority of cases the simulation would use but a fraction of the resources allocated to it. Even if the worst case could be computed accurately while maintaining the target frame rate, in the majority of cases the simulation would use but a fraction of the resources allocated to it.
This inefficiency comes at the cost of taking resources away from other sub-systems that could do with it, impacting the quality of the whole application. This inefficiency comes at the cost of taking resources away from other sub-systems that could do with it, impacting the quality of the whole application.
@ -122,7 +122,7 @@ Translated to pseudo-C from BASIC source.
--- ---
This approach, while obtuse and archaic, has the property of running in a predictable, constant amount of time, and computes a result to a particular level of precision determined by the number of iterations. This approach, while obtuse and archaic, has the property of running in a predictable, constant amount of time, and computes a result to a particular level of precision determined by the number of iterations.
On average, it performs worst than Newton-Raphson, taking more runtime to arrive at a result of similar accuracy. On average, it performs worse than Newton-Raphson, taking more runtime to arrive at a result of similar accuracy.
However, in the worst-case, it performs just as well in terms of accuracy and runtime as the best-case, which makes it suitable for real-time applications in a way that Newton-Raphson is not. However, in the worst-case, it performs just as well in terms of accuracy and runtime as the best-case, which makes it suitable for real-time applications in a way that Newton-Raphson is not.
Instead of having to pad the physics budget to account for a worst-case scenario, we can make much more efficient use of a smaller budget, despite the algorithm almost always costing more. Instead of having to pad the physics budget to account for a worst-case scenario, we can make much more efficient use of a smaller budget, despite the algorithm almost always costing more.