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Three-dimensional relative localization and synchronized movement with wireless ranging

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Abstract

Relative localization is a key capability for autonomous robot swarms, and it is a substantial challenge, especially for small flying robots, as they are extremely restricted in terms of sensors and processing while other robots may be located anywhere around them in three-dimensional space. In this article, we generalize wireless ranging-based relative localization to three dimensions. In particular, we show that robots can localize others in three dimensions by ranging to each other and only exchanging body velocities and yaw rates. We perform a nonlinear observability analysis, investigating the observability of relative locations for different cases. Furthermore, we show both in simulation and with real-world experiments that the proposed method can be used for successfully achieving various swarm behaviours. In order to demonstrate the method’s generality, we demonstrate it both on tiny quadrotors and lightweight flapping wing robots.

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Notes

  1. https://www.bitcraze.io/

  2. https://flapper-drones.com/wp/

  3. https://github.com/tudelft/crazyflie-firmware/tree/cf_swarm3d

  4. https://github.com/tudelft/crazyflie-firmware/tree/flapper_swarm3d

  5. https://github.com/tudelft/crazyflie-suite

  6. https://doi.org/10.4121/17372348

  7. https://www.youtube.com/playlist?list=PL_KSX9GOn2P-dzSwBvYYRZuKOiYvx0pIS

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Authors and Affiliations

  1. Faculty of Aerospace Engineering, Technische Universiteit Delft, Kluyverweg 1, 2629HS, Delft, The Netherlands

    Sven Pfeiffer, Shushuai Li & Guido C. H. E. de Croon

  2. Department of Control and Computer Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Veronica Munaro

  3. Department of Electronics and Telecommunications, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Alessandro Rizzo

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  1. Sven Pfeiffer
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  2. Veronica Munaro
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  3. Shushuai Li
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  4. Alessandro Rizzo
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  5. Guido C. H. E. de Croon
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Appendix A. Derivation of a horizontal velocity model for the flapping wing drone

Appendix A. Derivation of a horizontal velocity model for the flapping wing drone

In this derivation, we consider three main forces acting on the flapping wing drone. Since we are mainly interested in accelerations, we already divide out the mass in the following equations, i.e. \(\mathbf {f} = \frac{\mathbf {F}}{m}\). First, the thrust along the drones central axis is given in the drones body frame as

$$\begin{aligned} \mathbf {f}_{T}^{b} =\left| T \right| \cdot \mathbf {e}_3 \end{aligned}$$
(A1)

Second, we consider a simple drag model with drag forces proportional and opposed to the drone’s body velocity:

$$\begin{aligned} \mathbf {f}^{b}_{d} = \left[ \begin{array}{cc} -b_x v^b_x \\ -b_y v^b_y \\ -b_z v^b_z \end{array} \right] \end{aligned}$$
(A2)

Finally, the force of gravity is easily expressed directly in the horizontal frame

$$\begin{aligned} \mathbf {f}_g^h = -\left| g \right| \cdot \mathbf {e}_3 \end{aligned}$$
(A3)

The sum of acceleration in the horizontal frame experienced by the flapper is then given by A4.

$$\begin{aligned} \mathbf {a}^h = \sum \mathbf {f}^h = \mathbf {R}_{hb}\mathbf {f}_T^b + \mathbf {R}_{hb} \mathbf {f}_d^b + \mathbf {f}_g^h \end{aligned}$$
(A4)

We can now put the drag coefficients into a diagonal matrix \(\mathbf {B} = \text {diag}\left( \mathbf {b}\right)\) and expand the equation. Recall that rotation matrices are orthogonal, and therefore, \(\mathbf {R}_{bh} = \mathbf {R}_{hb}^{-1}=\mathbf {R}_{hb}^T\). To simplify notation, we will at this point drop the superscript h for vectors in the horizontal frame.

$$\begin{aligned} \mathbf {a} = \left| T \right| \cdot \mathbf {R}_{hb}\mathbf {e}_3 -\mathbf {R}_{hb}\mathbf {B}\mathbf {R}_{hb}^{T} \cdot \mathbf {v} -\left| g \right| \cdot \mathbf {e}_3 \end{aligned}$$
(A5)

We now write out the equation for each individual axis in the horizontal frame.

$$\begin{aligned} a_x & = \left| T \right| \text {s}\theta \text {c}\phi - (b_x \text {c}^2\theta + b_y \text {s}^2\theta \text {s}^2\phi + b_z \text {s}^2\theta \text {c}^2\phi )\,v_x\nonumber \\&\,\, - (b_y - b_z)\text {s}\theta \text {s}\phi \text {c}\phi \,v_y + (b_x - b_y \text {s}^2\phi - b_z \text {c}^2\phi )\text {s}\theta \text {c}\theta \,v_z \end{aligned}$$
(A6)
$$\begin{aligned} a_{y} = & - \left| T \right|{\text{s}}\phi - (b_{y} - b_{z} ){\text{s}}\theta {\text{s}}\phi {\text{c}}\phi {\mkern 1mu} v_{x} \\ - & \,\,(b_{y} {\text{c}}^{2} \phi + b_{z} {\text{s}}^{2} \phi ){\mkern 1mu} v_{y} - (b_{y} - b_{z} ){\text{c}}\theta {\text{s}}\phi {\text{c}}\phi {\mkern 1mu} v_{z} \\ \end{aligned}$$
(A7)
$$\begin{aligned} a_z & = \left| T \right| \text {c}\theta \text {c}\phi + (b_x - b_y \text {s}^2\phi - b_z \text {c}^2\phi )\text {s}\theta \text {c}\theta \,v_x - (b_y - b_z)\text {c}\theta \text {s}\phi \text {c}\phi \,v_y \nonumber \\ &\,\, - (b_x\text {s}^2\theta + b_y \text {c}^2\theta \text {s}^2\phi + b_z\text {c}^2\theta \text {c}^2\phi )\,v_z - \left| g \right| \end{aligned}$$
(A8)

Assuming small roll and pitch angles, we can neglect any terms that include the multiplication of two sines (i.e. \(\text {s}^2\theta \approx \text {s}^2\phi \approx \text {s}\theta \text {s}\phi \approx 0\))

$$\begin{aligned} a_x= & {} \left| T \right| \text {s}\theta \text {c}\phi - b_x \text {c}^2\theta \,v_x + (b_x-b_z \text {c}^2\phi )\text {s}\theta \text {c}\theta \,v_z \end{aligned}$$
(A9)
$$\begin{aligned} a_y= & {} -\left| T \right| \text {s}\phi - b_y \text {c}^2\phi \,v_y - (b_y - b_z)\text {c}\theta \text {s}\phi \text {c}\phi \,v_z \end{aligned}$$
(A10)
$$\begin{aligned} a_z= & {} \left| T \right| \text {c}\theta \text {c}\phi + (b_x - b_z \text {c}^2\phi )\text {s}\theta \text {c}\theta \,v_x - (b_y - b_z)\text {c}\theta \text {s}\phi \text {c}\phi \,v_y - b_z\text {c}^2\theta \text {c}^2\phi \,v_z - \left| g \right| \end{aligned}$$
(A11)

Due to the difficulty of estimating the generated thrust of the flapper, we further simplify the model by decoupling horizontal and vertical movement of the flapper, i.e. we consider the vertical velocity to be negligible during horizontal motion. This allows us to find an expression for the thrust in terms of \(\left| g \right|\) and the horizontal velocities.

$$\begin{aligned} a_z= & {} \left| T \right| \text {c}\theta \text {c}\phi + (b_x - b_z \text {c}^2\phi )\text {s}\theta \text {c}\theta \,v_x - (b_y - b_z)\text {c}\theta \text {s}\phi \text {c}\phi \,v_y - \left| g \right| = 0 \end{aligned}$$
(A12)
$$\begin{aligned} \left| T \right|= & {} \frac{\left| g \right| }{\text {c}\theta \text {c}\phi } - (b_x-b_z\text {c}^2\phi )\text {s}\theta \, v_x + (b_y - b_z) \text {s}\phi \, v_y \end{aligned}$$
(A13)

This expression for the thrust can now be inserted into A9 and A10, while setting \(v_z = 0\). In both cases, the thrust is multiplied with a sine, which again allows us to neglect some terms due to our small angle approximation. As a result, only the gravity term remains in the final expressions for the acceleration alongside the deceleration caused by drag.

$$\begin{aligned} a_x= & {} \frac{\text {s}\theta }{\text {c}\theta }\left| g \right| - b_x \text {c}^2\theta v_x \end{aligned}$$
(A14)
$$\begin{aligned} a_y= & {} -\frac{\text {s}\phi }{\text {c}\theta \text {c}\phi }\left| g \right| - b_y \text {c}^2\phi v_y \end{aligned}$$
(A15)

We can now integrate these accelerations using the forward Euler method to estimate our horizontal velocities.

$$\begin{aligned} v_{x,k+1}&= v_{x,k} + \text {d}t \left( \frac{\text {s}\theta _k}{\text {c}\theta _k}\left| g \right| - b_x \text {c}^2\theta _k v_{x,k} \right) \end{aligned}$$
(A16)
$$\begin{aligned} v_{y,k+1}&= v_{y,k} + \text {d}t \left( -\frac{\text {s}\phi _k}{\text {c}\theta _k \text {c}\phi _k}\left| g \right| - b_y \text {c}^2\phi _k v_{y,k} \right) \end{aligned}$$
(A17)

The drag coefficients \(b_x\) and \(b_y\) can be estimated from a data set with ground truth using a least squares approach. Specifically, for an overdetermined system of equations \(\mathbf {y} = \mathbf {X}\mathbf {\beta }\) the least-squares estimate for the parameters \(\mathbf {\beta }\) is given by \(\hat{\mathbf {\beta }} = (\mathbf {X}^T\mathbf {X})^{-1}\mathbf {X}^T\mathbf {y}\). To determine the drag coefficients in our horizontal velocity model, we solve two independent least-squares problems with for \(b_x\)

$$\begin{aligned} \mathbf {\beta }&= b_x \end{aligned}$$
(A18)
$$\begin{aligned} y_{k}&= -\frac{v_{x,k+1}-v_{x,k}}{\text {d}t} + \frac{\text {s}\theta _k}{\text {c}\theta _k}\left| g \right| \end{aligned}$$
(A19)
$$\begin{aligned} X_{k}&= \text {c}^2 \theta _k \,v_{x,k} \end{aligned}$$
(A20)

and for \(b_y\)

$$\begin{aligned} \mathbf {\beta }&= b_y \end{aligned}$$
(A21)
$$\begin{aligned} y_{k}&= -\frac{v_{y,k+1}-v_{y,k}}{\text {d}t} - \frac{\text {s}\phi _k}{\text {c}\theta _k \text {c}\phi _k}\left| g \right| \end{aligned}$$
(A22)
$$\begin{aligned} X_{k}&= \text {c}^2 \phi _k \,v_{y,k} \end{aligned}$$
(A23)

From our data, we find \(b_x\) = 4.2 and \(b_y\) = 1.8 for the Flapper Drones.

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Pfeiffer, S., Munaro, V., Li, S. et al. Three-dimensional relative localization and synchronized movement with wireless ranging. Swarm Intell 17, 147–172 (2023). https://doi.org/10.1007/s11721-022-00221-0

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