Magnetorquer Optimization State of the Art

to unfortunate circumstances the satellite exhibits a residual magnetic dipole which is very close to the one the actuators are able to produce (cf. [88]).

An interesting behavior is seen for coils (5) to (8), which were developed for the COMPASS-1 mission. The four different entries in the diagram are in fact one coil with dipole and power calculated for temperatures of −60^∘C, 0^∘C, 50^∘C and 100^∘C [89]. To better distinguish them from the other coils, they are connected by a thin line. Entry (5) is for the coldest temperature, entry (8) for the hottest. Due to the non-linear dependence of dipole and power on the coil temperature, the distances between neighboring coils on that line get closer for increasing temperatures. The dipole-to-power ratio stays constant over the temperature range, which means that coil quality does not degenerate.

Coils (16), (17) and (18) are taken from a dissertation researching optimization of all magnetic actuators [90]. Starting from coil (18), those three coils apparently show a way how to decrease the power consumption by sacrificing magnetic dipole. With a dipole-to-power ratio of 1:2 and a mass of 9.90 g, coil (16) would already be a good candidate for 1 U CubeSat missions. However, the optimization of magnetic actuators presented in [90] is highly doubted.

While the work presents a well-founded mathematical description of coil properties (cf. section 4.2.2), the allowed degree of freedom in the parameter space appears to be overly constrained and no in-depth analysis of the related consequences is provided.

4.2.1 Experiment-Based Optimization

Dildar et al. in [91] and in succession Ali et al. in [92, 93, 94] present the design of reconfigurable embedded air coils for CubeSat applications. Reconfigurable coils allow to study the impact of multiple sets of magnetic and electric properties on small satellite missions. With this approach, optimization would take place in the laboratory or on orbit by experimenting with different setups for those configurable coils.

Experiment-based optimization is inefficient, as appropriate coil candidates need to be identified using best guesses and manufactured before they can be tested in the laboratory. So other methods that depend on the mathematical modeling of magnetic actuators are superior to experimental methods.

4.2.2 Model-Driven Optimization

Model-driven optimization in general is based on mathematical models that describe the system to be optimized. They are then used together with an optimization technique to find optimum candidates. Model-driven optimization of magnetic actuators is applied by Bellini in [90]. His work starts with the definition of mathematical models of embedded air coils, wound air coils, and wound torque rods. From each coil type, a single specimen is manufactured and subjected to test, and the mathematical models are validated against the results. Based on the mathematical models, a design tool is implemented, which promises to find an optimal candidate from all three coil types. The tool uses a three-staged approach to finding the optimal candidate, which also incorporates consideration of manufacturing costs.

The work presented by Bellini in [90] is criticized for multiple aspects. First, his mathematical models use many simplifications instead of implementing a more realistic model. This leads to the introduction of corrective factors to compensate the inaccuracies.

Second, the model for calculating embedded air coil magnetic dipole is applied to all layers in the PCB, which would lead to a coil that does not create any magnetic dipole, due to alternating winding sense of the current through the

coil. Bellini further identifies a residual magnetic force for a single layer, and proposes to use the same layout rotated by 90° in four subsequent layers. This would render coils with a number of layers not divisible by four infeasible.

Third, only a small number of variable input parameters are taken into account for parameter optimization in the presented use cases [90]. This leads to an over-constrained parameter space, and prevents better optimization.

Finally, properties of the coils resulting from the optimization procedure appear to be poorly optimized in comparison to the coils shown in figure 4.1. Dipole-to-power ratio of all resulting coils is below 1:2, and masses are higher than the average in the respective class. Of the optimized embedded air coils, no coil is even shown inside the diagram area.

4.2.3 Sequential Quadratic Programming

Optimization of magnetic actuators using a sequential quadratic programming (SQP) method to find the maximum magnetic dipole for a given power consumption and mass of a wound torque rod is presented by Miller in [79].

Those methods require the formulation of a cost function, and find the minimum of this function from an initial guess of parameters, which are bound by a set of additional equality and inequality constraints. In [79], the cost function expresses magnetic dipole. Four inequality constraints are used to constrain mass, power, total number of turns, and ratio of core length to radius. As pointed out in the original source, the applied optimization method has several limitations which are summarized here:

– SQP may only be used with continuous parameters.

– Depending on the initial guess, only a local optimum may be found.

– For infeasible problems, the SQP method attempts to minimize the maximum constraints, leading to a violation of the constraints.

Some samples of optimized coils are presented in [79]. Well optimized coils from there have good dipole-to-power ratios of above 1:1, but feature in general larger masses as the coils presented in figure 4.1.

Especially the inability to use discrete values for input parameters to the cost function is criticized. For a real world magnetic actuator the enameled wire will only be available in discrete diameters. And for embedded air coils the number of turns per layer and the number of layers are also discrete integer values. Finding optimized solutions for such a setup requires the use of so-called mixed-discrete sequential quadratic programming methods. They are more difficult to apply and have even bigger problems in finding the global optimum.

A solution to the problem of finding the global optimum is not addressed by Miller in [79], and no proof is shown, that the selected coil actually represents the global minimum. One step towards that direction is the discussion led on the complex shape of the design parameter space, where the proposed cost function was found not to be strictly convex.