Design Space Reduction

The drone should reach high angular acceleration around all possible axes perpendicular to the thrust direction. By finding a rotor layout guaranteeing these high angular accelerations, the necessary thrust can then be realised by choosing the appropriate rotor dimensions. This results in an overall good manoeuvrability and allows to focus on energy efficiency at a later evaluation step.

The criteria were formulated as follows:

1. For a given rotor layout find the thrust distribution realising the maximum angular acceler-ation, for each axis perpendicular to the thrusting direction.

2. Then detect the axis with the minimum of these maximum angular acceleration, hence the worst direction regarding angular acceleration for each concept.

3. The best concepts are then the ones that exhibit the highest minimal angular acceleration, hence, those with the best worst-case behaviour.

These criteria prefer layouts that are able to accelerate well around any axis in the x-y-plane (layouts without striking weaknesses). This guarantees a good overall performance, with even better manoevrability around certain axes.

For four axes on a quadcopter, the thrust distributions maximising angular acceleration are shown around these axes. They can be seen in figure 5.5.

(a) Axis +Y (b) Diagonale +X +Y

(c) Axis +X (d) Diagonale +X -Y

Figure 5.5.: Optimal Thrust Distributions for Four Axes of a Quadrotor

5.5. Combine and Firm Up into Concept Variants

Figure 5.6.: Setup to Find the Optimal Rotor Distance The Rotor Distance with Best Thrust per Weight

When taking a look at a single rotor pair, one can find simple formulas for torque around the mounting axis. Figure 5.6 shows the setup for this short analysis. Assuming a constant, concen-trated center inertia I_y,base (e.g. the battery and hitting mechanism), as well as weightless levers with length Lx and rotors with thrust Fz being a point mass with the mass mrotor, the torque around the concentrated inertia T_y results to:

T_y = 2F_zl_x (5.14)

The moment of inertia Iy can be derived as:

I_y =I_y,base+ 2m_rotorL_x² (5.15)

With this, the angular acceleration around the mounting axis is α_y = T_y

I_y = 2F_zL_x

I_y,base+ 2m_rotorL_x² (5.16)

The critical point for the angular acceleration can be found from this equation by solving for the

∂αy

∂Lx = 0. The critical point is at

L_x,opt =

r Iy,base

2m_rotor (5.17)

Investigating the second derivative of α_y shows, that it is a maximum.

For the rotor masses found at the suppliers’ pages and the inertia of a battery pack of 500g, the optimal distances lay beyond the build limits. Closer to the center from this optimal point, the equation for the optimal acceleration behaves linearly in good approximation, so the best mounting point for a rotor is on the border of the build limit.

The Inferiority of Angled Rotors

The effects of rotors slightly tilted against the quadcopter’s z-axis are sketched in figure 5.7. Here a symmetric (both rotors tilted outward by the same angle) geometry was analysed, which does not limit the validity of the conclusions. If the second rotor is tilted inward by the same angle instead of outward, the effects, that will be discussed in this section, occur as with the symmetric tilting, but at switched loadcases. Any other geometry can be seen as a linear combination of both cases. Moreover, the investigated load cases (pure thrust in the copters z-direction and

pure rotation around axes in x-y-plane) are sufficient, any other load case can be modeled as a linear combination of thrust and rotation.

The following effects can be derived:

• In case of thrust in the z direction both rotors generate the same amount of force, F₁ and F₂. Let us also introduce the absolute value of both forces pointing in vertical direction, named F_z. It can be seen by summing up the forces graphically (see figure 5.7(a)), that less thrust is resulting from F₁ and F₂ than from2F_z.

• When it comes to generate torque around the center point, two effects occur. First, tilted rotors suffer from a shorter lever - simply compareL_proj to the original lengthL_x. A shorter lever then means less torque, a clear drawback.

The second effect can be seen from the addition ofF₁and−F₂, resulting in a (small) lateral thrust. If it can be included in the drone control, lateral flight might be beneficial. Still, for pure lateral flight a compensation of the rotation is needed, causing losses and adding rotors (=weight). Altogether, lateral thrust could be generated much more efficiently by an additional rotor dedicated to lateral flight only.

From this discussion it can be seen, that there is no use in pointing rotors out- or inwards.

Variable-tilt rotors (see section 5.4.1) facing in z, but tilted to provide lateral thrust, have not been analysed here. Simulations done later during this thesis have shown, that the added weight from the tilting mechanics, combined with the slow tilting speeds, cancel out the potential of more degrees of freedom.

Yet, for one special case, variable-tilt rotors are the preferred rotor type: rotation around the z-axis can be executed most easily by flipping all rotors either clockwise or counter-clockwise. The vertical thrust decreases only with (1−cos(α)), α measured as the deviation from the vertical, whereas the torque T_z grows with sin(α).

Variable-pitch rotors (see section 5.4.1) can only adjust the amount of generated thrust, but not its direction. Hence, during this examination they can be treated like fixed propellers.

As a conclusion, for best performance, fixed rotors are to be mounted vertically.

Design Space Reduction by Evaluating Random Concepts

With a Matlab Script, large numbers of concepts were generated and evaluated against the simple criteria developed in section 5.5.3. By keeping certain parameters of a concept pre-selected, the influence of the other ones could be investigated.

The following design rules could be retrieved (or, if already found analytically, confirmed):

• Figure 5.8 shows the results of a study (amongst many) where rotor sizes and the distance l_r between a rotor and the coordinate center were set to fixed values. For each combination of size and distance, ten thousand concepts with randomly generated positions on the ring with radiusl_r were evaluated. The plot contains the “maximum angular acceleration in the worst direction” (as defined in section 5.5.3; I will refer to it using MAA) plotted against the distance between the center of inertia and the rotors. Each datapoint (defined by rotor size and radial distance) shows the intermediate MAA of its ten thousand concepts.

Overall, the results were as shown analytically - the angular acceleration has its optimum at or beyond the build limits. Despite the large number of generated concepts, the results showed large disturbances around the trendlines, indicating the following:

5.5. Combine and Firm Up into Concept Variants

(a) Thrust Reduction on Vertical Flight

(b) Lateral Thrust and Shorter Lever on Rota-tion

Figure 5.7.: Impacts of Angled Rotors

– The “maximum angular acceleration in the worst direction” is very sensitive to the position of the rotors.

– It needs more than ten thousand concepts to reliably find the optimum.

– The influence of the distance can be separated from the angular rotor distribution.

• The z coordinate has no influence on the evaluation result, but as the build space is a sphere, any z > 0 reduces the horizontal component p

r²−z² =x²+y² < r of the line between the build space center and the rotor position.

• The focus of another series of concepts rested on optimal rotor positioning. It was already found, that the best solutions lie in the x-y-plane, at the outer rim of the build limit. The remaining question was now how to distribute the rotors on this ring. Using Matlab it was possible to evaluate possible configurations with up to 6 rotors over night. Profiles for the best solutions regarding the MAA can be found in figure 5.9. Most remarkably is layout (c, d), “Grouped Hexacopter”, which can also be thought of as a ”Dual Tricopter“.

This configuration was a new finding to the project team and outperformed the regular hexacopter. Later on during the thesis it was proven by further experiments, that the special distribution of rotors (with a ratio of 1 : 3 for the angles between neighbouring rotors) from the grouped hexacopter also improved performance on any drone with an uneven number of rotor pairs (or speaking precisely: let n being odd, all concepts with 2n= 6,10,14, ...rotors).

• All further experiments, especially bigger subspaces of the design space like octocopters, were carried out in a faster software environment developed in section 5.5.6.