#
LSTM-Based Projectile Trajectory Estimation in a GNSS-Denied Environment^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

- •
- to detail an LSTM-based approach to estimate projectile positions, velocities and Euler angles from the embedded IMU, the magnetic field reference, pre-flight parameters and a time vector.
- •
- to present BALCO (BALlistic COde) [21] used to generate the dataset. This simulator provides true-to-life trajectories of several projectiles according to the ammunition parameters.
- •
- to investigate different normalization forms of the LSTM input data in order to evaluate their contribution on the estimation accuracy. For this purpose, several LSTMs are trained with different input data normalizations.
- •
- to study the impact of the local navigation frame rotation on the estimation accuracy. Rotating the local navigation frame during the training step allows having similar variation ranges along the three axes, especially for the lateral position, which is extremely small compared to the two other axes. This method shares the same goals as normalization but without any information loss.
- •
- to examine the influence of inertial sensor models on estimation accuracy. For this purpose, two inertial sensor error models are studied in order to evaluate their influence on LSTM predictions.
- •
- to compare the LSTM accuracy to a Dead-Reckoning, performed on finned mortar trajectory. Estimation methods are evaluated through error criteria based on the Root Mean Square Error and the impact point error.

## 2. Related Work

#### 2.1. Model-Based Projectile Trajectory Estimation

#### 2.2. AI-Based Trajectory Estimation

- Surveillance and target recognition where machine learning algorithms applied to computer vision detect, identify and track objects of interest.For example, the Maven project, presented by the US Department of Defense (DoD), focuses on automatic target identification and localization from images collected by Aerial Vehicles [11].
- Predictive maintenance to establish the optimal time to change a part of a system, as the US Army does on F-16 aircraft [12].
- Military training where AI is used in training simulation software to improve efficiency through various scenarios, such as AIMS (Artificial Intelligence for Military Simulation) [13].
- Analysis and decision support to extract and deal with relevant elements in an information flow, to analyze a field or to predict events. The Defense Advanced Research Projects Agency (DARPA) aims to equip US Army helicopter pilots with augmented reality helmets to support them in operations [14].
- Cybersecurity, as military systems are strongly sensitive to cyberattacks leading to loss and theft of critical information. To this end, the DeepArmor program from SparkCognition uses AI to protect, detect and block networks, computer programs and data from cyber threats [15].

#### 2.2.1. Recurrent Neural Networks (RNNs)

#### 2.2.2. Long Short-Term Memory Cell

- –
- the forget gate filters, through a Sigmoid function $\sigma $, data contained in the concatenation of ${x}_{t}$ and ${h}_{t-1}$. Data are forgotten for values close to 0 and are memorized for values close to 1. The forget gate model is:$$\begin{array}{c}\hfill {f}_{t}=\sigma ({W}_{f}.[{h}_{t-1},{x}_{t}]+{b}_{f})\end{array}$$
- –
- the input gate extracts relevant information from $[{h}_{t-1},{x}_{t}]$ by applying a Sigmoid $\sigma $ and a Tanh function. The input gate is represented by the following:$$\begin{array}{c}\hfill \begin{array}{cccc}{i}_{t}=\sigma ({W}_{i}.[{h}_{t-1},{x}_{t}]+{b}_{f})& & & {\tilde{C}}_{t}=tanh({W}_{c}.[{h}_{t-1},{x}_{t}]+{b}_{c}).\end{array}\end{array}$$The memory cell ${c}_{t}$ is updated from the forget gate${f}_{t}$ and the input gate ${i}_{t}$ and ${\tilde{C}}_{t}$, to memorize pertinent data:$$\begin{array}{c}\hfill {c}_{t}={f}_{t}\times {c}_{t-1}+{i}_{t}\times {\tilde{C}}_{t}\end{array}$$
- –
- the output gate defines the next hidden state ${h}_{t}$ containing information about previous inputs. The hidden state ${h}_{t}$ is updated with the memory cell ${c}_{t}$ normalized by a Tanh function and $[{h}_{t-1},{x}_{t}]$ normalized by a Sigmoid function:$$\begin{array}{c}\hfill {h}_{t}=\sigma ({W}_{h}.[{h}_{t-1},{x}_{t}]+{b}_{h})\times tanh\left({c}_{t}\right)\end{array}$$

## 3. Problem Formulation

#### 3.1. The Projectile Trajectory Dataset BALCO (BALlistic COde)

**n**(black frame in Figure 3) tangent to the Earth and assumed fixed during the projectile flight.

**b**(red frame in Figure 3), which is an ideal hypothetical frame placed exactly at the projectile gravity center, in which the IMU must be placed, providing perfect inertial measurements.

**s**(green frame in Figure 3) rigidly fixed to the projectile and misaligned with the projectile gravity center, considered as the frame where the inertial measurements are performed.

- the
**inertial measurements in the body frame****b****and in the sensor frame****s**, i.e., gyrometer $\omega \in {\mathbb{R}}^{3}$, accelerometer $a\in {\mathbb{R}}^{3}$ and magnetometer $h\in {\mathbb{R}}^{3}$ measurements. Three kinds of inertial measurements are available:- –
- the Perfect IMU measurements performed in the body frame
**b**(red frame in Figure 3), in the ideal case where all the three inertial sensors are perfectly aligned with the projectile gravity center and where no sensor default model is taken into account providing ideal inertial measurements, i.e., without any noise or bias. These measurements are not exploited in this work but are necessary to provide realistic inertial data. - –
- the IMU measurements performed in the sensor frame
**s**(green frame in Figure 3): issued from the Perfect IMU measurements where a sensor error model is added. This error model, specific to each sensor axis, includes a misalignment, a sensitivity factor, a bias and a noise (assumed zero mean white Gaussian noise). Thus, this measurement accurately models an IMU embedded in a finned projectile. - –
- the IMU DYN measurements performed in the sensor frame
**s**(green frame in Figure 3): issued from IMU measurements to which a transfer function is added to each sensor. For each sensor, IMU DYN measurements are modeled by:$$\begin{array}{c}\hfill {y}_{sensor{,}_{IMU\phantom{\rule{4pt}{0ex}}DYN}}=\frac{1}{1+as+b{s}^{2}}{y}_{sensor{,}_{IMU}}\end{array}$$

- the
**magnetic field reference ${h}_{n}\in {\mathbb{R}}^{3}$**in the local navigation frame**n**, assumed constant during the projectile flight. **flight parameters**, which are, in the case of a finned projectile, the fin angle ${\delta}_{f}$ to stabilize projectiles, the initial velocity ${v}_{0}$ at barrel exit and the barrel elevation angle $\alpha $, relatively important to obtain ballistic trajectories with short ranges.- a
**time vector $k{\Delta}_{t}$**where ${\Delta}_{t}$ is the IMU sampling period: ${\Delta}_{t}=1\times {10}^{-3}s$. - the
**reference trajectory**, i.e., the projectile position $p\in {\mathbb{R}}^{3}$, velocity $v\in {\mathbb{R}}^{3}$ and Euler angles $\mathsf{\Psi}\in {\mathbb{R}}^{3}$ in the local navigation frame**n**at the IMU frequency. This trajectory is used to evaluate the LSTMs accuracy and to compute errors.

#### 3.2. Data Characteristics and LSTM Requirements

- -
- $\mathcal{M}\in {\mathbb{R}}^{12}$ the inertial data, including IMU or IMU DYN measurements in the sensor frame
**s**and the reference magnetic field ${h}_{n}\in {\mathbb{R}}^{3}$ in the local navigation frame**n**presented in Section 3.1, - -
- $\mathcal{P}\in {\mathbb{R}}^{3}$ the flight parameters. In the case of a finned projectile, the three flight parameters are the fin angle ${\delta}_{f}$, the initial velocity ${v}_{0}$ and the the barrel elevation angle $\alpha $.
- -
- $\mathcal{T}\in {\mathbb{R}}^{1}$ the time vector, such as $\mathcal{T}=k{\Delta}_{t}$ with k the considered time step and ${\Delta}_{t}$ the IMU sampling period.

- -
- $LST{M}_{ALL}$ trained to estimate 9 output features which are the projectile position $p\in {\mathbb{R}}^{3}$, velocity $v\in {\mathbb{R}}^{3}$ and Euler angles $\mathsf{\Psi}\in {\mathbb{R}}^{3}$ in the navigation frame
**n**. - -
- $LST{M}_{POS}$ trained to estimate 3 output features, which are the projectile position $p\in {\mathbb{R}}^{3}$ expressed in the navigation frame
**n**. - -
- $LST{M}_{VEL}$ trained to estimate 3 output features which are the projectile velocity $v\in {\mathbb{R}}^{3}$ expressed in the navigation frame
**n**. - -
- $LST{M}_{ANG}$ trained to estimate 3 output features which are the projectile Euler angles $\mathsf{\Psi}\in {\mathbb{R}}^{3}$ in the navigation frame
**n**.

## 4. LSTM Input Data Preprocessing

#### 4.1. LSTM Input Data Normalization

- Min/Max normalization $MM(.)$: ${x}_{MM}=\frac{x-{x}_{min}}{{x}_{max}-{x}_{min}}$ with ${x}_{max}$ and ${x}_{min}$ the maximum and minimum of x.
- Standard Deviation normalization $STD(.)$: ${x}_{STD}=\frac{x-{\mu}_{x}}{{\sigma}_{x}}$ with x the quantity to normalize, ${\mu}_{x}=\mu \left(x\right)$ its mean and ${\sigma}_{x}=\sigma \left(x\right)$ its standard deviation. Thus, ${x}_{STD}$ is a quantity with a zero-mean and a standard deviation of one.

#### 4.2. Local Navigation Frame Rotation

**n**by a fixed angle $\gamma $ (local rotated navigation frame ${\mathit{n}}_{\mathbf{\gamma}}$), such as ${x}_{\gamma}={R}_{\gamma}x$ with $x\in {\mathbb{R}}^{3}$ defined in

**n**, ${x}_{\gamma}\in {\mathbb{R}}^{3}$ expressed in ${\mathit{n}}_{\mathbf{\gamma}}$ and ${R}_{\gamma}\in SO\left(3\right)$ the transition matrix from the local navigation frame

**n**to the local rotated navigation frame ${\mathit{n}}_{\mathbf{\gamma}}$ as in the following:

**n**to modify its three components in order to have a similar amplitude order for the three components. The local navigation frame rotation provides similar variation ranges of a quantity along the three axes. This approach is used to ensure that the LSTMs adequately estimate a quantity with small magnitudes along one axis, even though the variations are considerably larger along the other two axes.

**n**.

## 5. Results and Analysis

**s**to the local navigation frame

**n**, $g\in {\mathbb{R}}^{3}$ the constant gravity vector, ${p}_{k}\in {\mathbb{R}}^{3}$ and ${v}_{k}\in {\mathbb{R}}^{3}$, respectively, the projectile position and velocity, and ${[.]}_{\times}$ the skew matrix. This algorithm is generally used for Kalman filters for the prediction step to estimate trajectory, as presented in [2,3,4,8,24,40].

#### 5.1. Impact of the Input Data Normalization and the Local Navigation Frame Rotation

#### 5.1.1. Qualitative Validation: One Finned Projectile Fire Simulation

#### 5.1.2. Quantitative Evaluation: Analysis on the Whole Test Dataset

- Success Rate ${\mathcal{C}}_{1}$: number of simulations where a LSTM RMSE is strictly smaller than the Dead-Reckoning.
- Error Rate ${\mathcal{C}}_{2}$: percentage of LSTM error compared to Dead-Reckoning errors.$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathcal{C}}_{1}=\sum _{k=1}^{{N}_{sim}}RMS{E}_{LSTM}<RMS{E}_{DR},& & {\mathcal{C}}_{2}=\frac{100}{{N}_{sim}}\sum _{k=1}^{{N}_{sim}}\frac{RMS{E}_{LSTM}}{RMS{E}_{LSTM}+RMS{E}_{DR}}\end{array}\end{array}$$

#### 5.2. Impact of Inertial Measurement Type and Local Navigation Frame Rotation on Estimation Accuracy

#### 5.2.1. Impact of the Local Navigation Frame Rotation and IMU Measurement

#### 5.2.2. Impact of the Local Navigation Frame Rotation and IMU DYN Measurement

#### 5.2.3. Evaluation Metric

- Mean Absolute Error:$$\begin{array}{c}\hfill MAE=\frac{1}{N}\sum _{k=1}^{N}\left|x-\widehat{x}\right|\end{array}$$
- SCORE: Number of simulations in the test dataset where the considered method obtains the smallest RMSE.

#### 5.2.4. Errors at Impact Point

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Navigation frames (black—local navigation frame

**n**, red—body frame

**b**, green—sensor frame

**s**) and flight parameters for a finned projectile (fin angle ${\delta}_{f}$, initial velocity ${v}_{0}$, barrel elevation angle $\alpha $).

**Figure 4.**(

**a**) Local navigation frame $\mathit{n}$ and local rotated navigation frame ${\mathit{n}}_{\mathbf{\gamma}}$. (

**b**) Projectile position in the local navigation frame

**n**(blue dashed line), projectile position in the local rotated navigation frame ${\mathit{n}}_{\mathbf{\gamma}}$ (red solid line).

**Figure 5.**Estimated projectile position [m] and Euler angles [rad] obtained by the Dead-Reckoning (green), $LST{M}_{ALL,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ (blue), $LST{M}_{ALL,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ (yellow) and the reference (red).

**Figure 6.**Position analysis: (

**a**) Success Rate ${\mathcal{C}}_{1}$, (

**b**) Error Rate ${\mathcal{C}}_{2}$ (in %) of $LST{M}_{ALL,\phantom{\rule{4pt}{0ex}}{V}_{1}-{V}_{8}}$ and $LST{M}_{POS,\phantom{\rule{4pt}{0ex}}{V}_{1}-{V}_{8}}$.

**Figure 7.**Velocity analysis: (

**a**) Success Rate ${\mathcal{C}}_{1}$, (

**b**) Error Rate ${\mathcal{C}}_{2}$ (in %) of $LST{M}_{ALL,\phantom{\rule{4pt}{0ex}}{V}_{1}-{V}_{8}}$ and $LST{M}_{VEL,\phantom{\rule{4pt}{0ex}}{V}_{1}-{V}_{8}}$.

**Figure 8.**Orientation analysis: (

**a**) Success Rate ${\mathcal{C}}_{1}$, (

**b**) Error Rate ${\mathcal{C}}_{2}$ (in %) of $LST{M}_{ALL,\phantom{\rule{4pt}{0ex}}{V}_{1}-{V}_{8}}$ and $LST{M}_{ANG,\phantom{\rule{4pt}{0ex}}{V}_{1}-{V}_{8}}$.

**Figure 9.**Average position, velocity and orientation error histogram obtained by $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ (blue), $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ (green) and Dead-Reckoning (red).

**Figure 10.**Average position, velocity and orientation error histogram obtained by $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ (blue), $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ (green) and Dead-Reckoning (red).

**Figure 11.**Position, velocity and orientation MAE average ${\mathcal{C}}_{MAE}$ obtained by $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{1}}$, $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{6}}$, $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{1}}$, $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ and the Dead-Reckoning.

**Figure 12.**Position, velocity and orientation score obtained by $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{1}}$, $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{6}}$, $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{1}}$, $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ and the Dead-Reckoning.

**Figure 13.**(

**a**) Errors at impact point obtained by $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ and $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ (blue cross), $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ and $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ (green cross) and the Dead-Reckoning (red dot). (

**b**) Impact point error location.

**Table 1.**Version specifications: Influence of the normalization and the local navigation frame rotation.

Name | Normalization | Rotation |
---|---|---|

${V}_{1}$ | No | No |

${V}_{2}$ | $MM\left(\mathcal{T}\right),MM\left(\mathcal{M}\right),MM\left(\mathcal{P}\right)$ | No |

${V}_{3}$ | $MM(\mathcal{T},\mathcal{M},\mathcal{P})$ | No |

${V}_{4}$ | $STD\left(\mathcal{T}\right),STD\left(\mathcal{M}\right),STD\left(\mathcal{P}\right)$ | No |

${V}_{5}$ | $STD(\mathcal{T},\mathcal{M},\mathcal{P})$ | No |

${V}_{6}$ | No | Yes |

${V}_{7}$ | $MM(\mathcal{T},\mathcal{M},\mathcal{P})$ | Yes |

${V}_{8}$ | $STD(\mathcal{T},\mathcal{M},\mathcal{P})$ | Yes |

**Table 2.**Training characteristics of $LST{M}_{ALL,\phantom{\rule{4pt}{0ex}}POS,\phantom{\rule{4pt}{0ex}}VEL,\phantom{\rule{4pt}{0ex}}ANG,\phantom{\rule{4pt}{0ex}}{V}_{1-8}}$.

Dataset | Training Dataset: | 100 Simulations (Validation: 10 Simulations) |

Test Dataset: | 20 Simulations | |

Input data | Batch size: | 64 (${S}_{eq\phantom{\rule{4pt}{0ex}}len}$: 20 timesteps) |

Input data: | ${I}_{n\phantom{\rule{4pt}{0ex}}Features}=(\mathcal{M},\mathcal{P},\mathcal{T})\in {\mathbb{R}}^{16}$ (with IMU measurements) | |

Cost function: | Mean Squared Error (MSE) | |

Training | Optimization algorithm: | ADAM [41] (Learning rate : $1\times {10}^{-4}$) |

LSTM layer: | 2 (Hidden units: 64–128) |

**Table 3.**Training characteristics of $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{1}}$, $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{1}}$, $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{6}}$, $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{6}}$.

Dataset | Training Dataset: | 4000 Simulations (Validation: 400 Simulations) |

Test Dataset: | 400 Simulations | |

LSTM name | No normalization | $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ (with IMU measurements) |

& No rotation | $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{1}}$ (with IMU DYN measurements) | |

No normalization | $LST{M}_{IMU,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ (with IMU measurements) | |

& Rotation | $LST{M}_{IMU\phantom{\rule{4pt}{0ex}}DYN,\phantom{\rule{4pt}{0ex}}{V}_{6}}$ (with IMU DYN measurements) | |

Input data | Batch size: | 64 (${S}_{eq\phantom{\rule{4pt}{0ex}}len}$: 20 timestamp) |

Input data: | ${I}_{n\phantom{\rule{4pt}{0ex}}Features}=(\mathcal{M},\mathcal{P},\mathcal{T})\in {\mathbb{R}}^{16}$ | |

Cost function: | Mean Squared Error (MSE) | |

Training | Optimization algorithm: | ADAM [41] (Learning rate: 1 × ${10}^{-4}$) |

LSTM layer: | 2 (Hidden units: 64–128) |

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## Share and Cite

**MDPI and ACS Style**

Roux, A.; Changey, S.; Weber, J.; Lauffenburger, J.-P.
LSTM-Based Projectile Trajectory Estimation in a GNSS-Denied Environment. *Sensors* **2023**, *23*, 3025.
https://doi.org/10.3390/s23063025

**AMA Style**

Roux A, Changey S, Weber J, Lauffenburger J-P.
LSTM-Based Projectile Trajectory Estimation in a GNSS-Denied Environment. *Sensors*. 2023; 23(6):3025.
https://doi.org/10.3390/s23063025

**Chicago/Turabian Style**

Roux, Alicia, Sébastien Changey, Jonathan Weber, and Jean-Philippe Lauffenburger.
2023. "LSTM-Based Projectile Trajectory Estimation in a GNSS-Denied Environment" *Sensors* 23, no. 6: 3025.
https://doi.org/10.3390/s23063025