📊 Definitions

🧠 Why the Difference?

• Population SD assumes you have access to every data point in the population. No need to correct for bias.
• Sample SD uses Bessel’s correction (dividing by \(N - 1\)) to avoid underestimating the true variability of the population. This correction compensates for the fact that a sample tends to be less variable than the full population.

🧮 Formulas

• Population SD:

• Sample SD:

Where:

• \(x_i\) = each data point
• \(\mu\) = population mean
• \(\bar{x}\) = sample mean
• \(N\) = number of data points

🎯 When to Use Which?

• Use population SD when you have data for the entire group you’re studying (e.g., all students in a school).
• Use sample SD when you’re working with a subset and want to generalize to the whole population (e.g., survey results from 100 out of 10,000 customers).

Note:

Current version of this post is generated partially using generative AI.

Their story is one of two problems: one solved with elegant simplicity, the other that stumped the world’s greatest minds for centuries.

Finding Latitude

Latitude — how far north or south of the equator you are — was the easier of the two problems. The sky itself provided a natural measuring stick.

Using the North Star

The most elegant solution was Polaris, the North Star. Because Earth’s axis points almost directly at it, Polaris appears nearly stationary in the northern sky while all other stars wheel around it through the night. This unique property made it the perfect reference: the angle of Polaris above your horizon equals your latitude almost exactly.

• At the equator, Polaris sits on the horizon (0°)
• At the North Pole, it is directly overhead (90°)
• Everywhere in between, the angle matches your latitude precisely

Early sailors estimated this angle by holding fingers at arm’s length. Later came purpose-built instruments: the astrolabe, the cross-staff, the backstaff, and eventually the optical sextant — each generation refining the measurement with greater precision.

Using the Sun

During the day, sailors measured the Sun’s angle above the horizon at solar noon — its highest point. Cross-referenced against astronomical tables showing the Sun’s declination for each day of the year, this allowed latitude calculation in both hemispheres, on any day clear enough to see the sun.

The Southern Hemisphere

The Southern Hemisphere had no bright pole star. Sailors used the Southern Cross constellation as a rough reference and relied more heavily on noon sun sights. This difficulty partly explains why European exploration reached southern latitudes later than northern ones.

The Longitude Problem

While latitude could be read from the sky, longitude — your east-west position — was an entirely different kind of problem. It stumped navigators, mathematicians, and monarchs for centuries, and its failure cost thousands of lives in shipwrecks.

The reason is deceptively simple: longitude requires knowing time. Earth rotates 360° in 24 hours — exactly 15° per hour. If you know what time it is at your home port and what local time it is where you are, the difference gives your longitude directly.

Dead Reckoning — and Its Deadly Errors

Without accurate clocks, sailors estimated east-west position by dead reckoning: start from a known position, track your compass heading, estimate your speed, and calculate where you must now be. In calm seas over short voyages, this worked reasonably well. But small errors accumulated relentlessly across days and weeks at sea.

The Lunar Distance Method

Some navigators attempted the lunar distance method: measuring the angle between the Moon and bright stars, then consulting elaborate tables to convert that into Greenwich time, and thus longitude. It worked in theory — but required hours of painstaking calculation, exceptional mathematical skill, and a clear sky at precisely the right moment. It was too demanding for routine use at sea.

The Solution: John Harrison and the Marine Chronometer

Longitude remained unsolved until the 18th century. In 1714, the British government established the Longitude Prize — £20,000 for anyone who could determine longitude to within half a degree over a transatlantic voyage. For decades it went unclaimed.

It was eventually solved not by an astronomer or mathematician, but by a self-taught Yorkshire carpenter and clockmaker named John Harrison.

Harrison’s insight was that the problem was not mathematical — it was mechanical. Longitude did not need a better theory. It needed a better clock.

He overcame several engineering challenges that had defeated every previous attempt:

• A balance wheel replaced the pendulum, which was useless on a rolling ship
• Bimetallic strips compensated for temperature changes that expanded or contracted metal parts
• Jeweled bearings minimized friction, keeping the mechanism accurate for months at sea
• His final design, H4, lost only five seconds over an 81-day sea trial in 1762

With an accurate chronometer aboard, finding longitude became almost trivial: determine local noon from the sun, compare it to the time on the chronometer set to Greenwich, and multiply the difference in hours by 15 to get your longitude in degrees.

A Timeline of Navigation

Summary

Latitude was relatively straightforward: measure the angle of Polaris or the noon sun above the horizon and compare it to astronomical tables. The North Star made this especially elegant for northern sailors.

Longitude remained unsolved for centuries because it requires keeping accurate time at sea — a problem that wasn’t cracked until John Harrison’s marine chronometers proved reliable enough for ocean voyages in the 18th century.

Together, one ancient and elegant, one requiring 18th-century engineering, these two methods allowed sailors to navigate the world’s oceans with confidence — and laid the conceptual groundwork for every positioning system since, including the GPS in your pocket.

1. Background: The Pendulum Analogy

Consider a simple pendulum of length \(L\) oscillating under gravity \(g\). Its small-angle period is:

Now ask: what length \(L\) would produce a period equal to the orbital period of a low-Earth-orbit satellite? A circular orbit at radius \(R\) from Earth’s centre satisfies

Setting \(L = R \approx 6{,}371\text{ km}\) yields \(T \approx 84.4\) min. This is the Schuler period, named after Maximilian Schuler who described the condition in 1923.

The physical picture is striking: a pendulum whose bob hangs at the centre of the Earth would be immune to horizontal accelerations of the carrier, because its effective restoring force is always directed toward Earth’s centre regardless of the platform’s motion. An INS tuned to this period emulates that ideal pendulum.

2. Derivation from First Principles

2.1 Platform Tilt Error

Let \(\alpha\) denote a small tilt angle of the navigation platform away from the local level. A tilted accelerometer measures an apparent horizontal acceleration

which, when integrated twice, produces a growing position error. Left unchecked, this is an unbounded (Schuler-unstable) error mode.

2.2 Closing the Loop

A mechanised INS feeds back computed velocity to drive a levelling torque that corrects the platform tilt. Let \(v\) be the northward velocity error and \(R\) the Earth radius. The angular rate needed to maintain local-level alignment as the vehicle moves over Earth’s curvature is \(\dot\theta = v/R\). The coupled error equations become:

This is simple harmonic motion with angular frequency

and period

The system is Schuler-tuned when the feedback gain is chosen to produce exactly this frequency. Critically, the amplitude of the oscillation does not grow — initial tilt errors oscillate rather than diverge.

2.3 State-Space Form

The complete first-order error state for a single horizontal channel is:

The eigenvalues of this matrix are \(\pm j\omega_S\), confirming purely oscillatory (neutrally stable) behaviour.

3. Effect on Inertial Navigation System Errors

3.1 Gyroscope Drift

A constant gyroscope drift rate \(\varepsilon\) (rad/s) acts as an input disturbance. The resulting horizontal position error is:

The error is bounded and oscillates at the Schuler frequency. It does not grow secularly — a direct consequence of Schuler tuning. The peak position error from a drift \(\varepsilon\) is \(2R\varepsilon/\omega_S\).

3.2 Accelerometer Bias

A constant accelerometer bias \(b\) (m/s²) produces a tilt error that also oscillates at \(\omega_S\). The position error envelope is:

3.3 Initial Condition Errors

All errors are bounded and periodic — none diverge. This is the principal practical benefit of Schuler tuning.

4. Interactive: Schuler Oscillation Simulator

The panel below integrates the two-state Schuler error equations in real time. Adjust the initial conditions and sensor errors to observe how position and tilt errors evolve over one or more Schuler periods.

5. Implications for INS Design

5.1 Why Schuler Tuning Matters

An INS that is not Schuler-tuned will have eigenvalues with a non-zero real part, causing position errors to grow exponentially. The 84.4-minute period is the unique tuning condition that converts this exponential growth into bounded oscillation — a form of neutral stability.

5.2 Aided Navigation

In practice, pure inertial navigation accumulates errors at the Schuler frequency. GNSS-aided systems (GPS/INS) exploit this: the Kalman filter estimator observes the oscillating error signature, which allows it to separate and estimate sensor biases far more effectively than a static test would permit. The Schuler oscillation thus becomes a calibration signal.

5.3 Latitude Dependence

The derivation above assumes a spherical Earth. At latitude \(\varphi\), the horizontal components of Earth’s rotation rate (\(\Omega\cos\varphi\), \(\Omega\sin\varphi\)) perturb the error equations, coupling the north and east channels. The Schuler frequency itself is latitude-invariant (it depends only on \(g\) and \(R\), both weakly latitude-dependent), but the cross-channel coupling introduces additional oscillatory modes — the Foucault oscillation at approximately 24 h and a combined mode near 12 h.

5.4 Summary of Error Modes

6. Conclusion

The Schuler period emerges as a natural consequence of Earth’s gravitational geometry: it is precisely the orbital period of a satellite skimming Earth’s surface. Designing an INS to oscillate at this frequency ensures that position errors remain bounded — a property that no amount of sensor quality alone can replace. For high-accuracy applications (submarine navigation, inertial surveying, precision-guided systems), Schuler-period behaviour is the governing constraint on long-term position error growth, and its interaction with GNSS-aiding forms the basis of modern integrated navigation filter design.

The simulator above integrates the linearised Schuler error equations using a fourth-order Runge–Kutta scheme with 800 steps. Physical constants: \(R = 6{,}371\text{ km},\; g = 9.807\text{ m/s}^2,\; T_S = 84.4\text{ min}\).

Earth Model & Notation

All derivations here treat the Earth as a sphere of mean radius

This introduces errors on the order of 0.3 % relative to a WGS-84 ellipsoid — acceptable for most engineering work at scales below a few hundred kilometres. Where sub-metre accuracy is required, consult ellipsoidal formulae (Vincenty, Karney).

1 — Distance Along a Meridian (Latitude Difference)

Assumption. The two points share the same longitude, or their longitude separation is negligible compared with their latitude separation. Under the spherical model, a meridian is a great circle of radius \(R\), so arc length is simply proportional to the subtended angle.

where \(\Delta\varphi\) is in radians. In terms of decimal degrees \(\Delta\varphi_{\circ}\):

Scale factor. One degree of latitude ≈ 111 139 m (111.1 km), and this value is constant across all latitudes under the spherical assumption — meridians are all great circles of the same radius.

2 — Distance Along a Parallel (Longitude Difference)

Assumption. The two points share the same latitude \(\varphi\), or their latitude separation is negligible. A parallel at latitude \(\varphi\) is not a great circle; its radius shrinks with the cosine of latitude.

The radius of a parallel at latitude \(\varphi\) is \(R\cos\varphi\), so the arc length for a longitude change \(\Delta\lambda\) (radians) is:

In decimal degrees:

Key consequence. One degree of longitude spans ≈ 111.1 km at the equator, but shrinks to zero at the poles. At 45° latitude it is ≈ 78.6 km; at 60° ≈ 55.6 km.

3 — General Two-Point Distance: Haversine Formula

For two arbitrary positions \((\varphi_1, \lambda_1)\) and \((\varphi_2, \lambda_2)\), the Haversine formula computes the great-circle distance on a sphere exactly (within the spherical assumption):

where all angles are in radians. The use of \(\arctan2\) rather than \(\arcsin\) avoids numerical instability for antipodal points.

Why Haversine? The naïve law of cosines form \(d = R\arccos(\sin\varphi_1\sin\varphi_2 + \cos\varphi_1\cos\varphi_2\cos\Delta\lambda)\) suffers from catastrophic cancellation for small separations in floating-point arithmetic. Haversine remains numerically stable at all scales.

Summary

All three converge for small \(\Delta\varphi, \Delta\lambda\) — the first two are simply the component projections of the Haversine result in the limit of small separation.

Basic Introduction

A rotation matrix in \(SO(3)\) is a linear transformation between coordinate frames. Let \(\{e\}\) and \(\{b\}\) be two arbitrary Euclidean coordinate frames, and let \(x^e\) and \(x^b\) denote the same vector expressed in each frame, respectively.

The rotation matrix \(R^e_b \in SO(3)\) describes the orientation of frame \(\{b\}\) relative to frame \(\{e\}\). Pre-multiplying a vector expressed in \(\{b\}\) by this matrix yields its representation in \(\{e\}\):

Since \(R^e_b \in SO(3)\), its inverse equals its transpose:

Therefore, \(R^b_e\) transforms a vector from frame \(\{e\}\) into frame \(\{b\}\).

SO(3) Properties

A matrix \(R\) belongs to \(SO(3)\) if and only if it satisfies two conditions:

The first condition enforces orthonormality of columns (and rows); the second rules out improper rotations (reflections). These constraints imply that each column of \(R\) is a unit vector, and any two distinct columns are mutually orthogonal.

Constructing the Rotation Matrix

The key insight is:

The \(i\)-th column of \(R^e_b\) is the unit vector \(b_i\) expressed in the coordinates of frame \(\{e\}\).

Consider two frames illustrated below, where each \(e_i\) and \(b_i\) is a unit basis vector.

To find each column, project \(b_i\) onto each axis of \(\{e\}\) using the dot product:

Inspecting the figure for the example configuration:

• \(b_1\) is aligned with \(-e_3\), with no projection onto \(e_1\) or \(e_2\):

• \(b_2\) is aligned with \(-e_2\):

• \(b_3\) is aligned with \(-e_1\):

Assembling these column vectors gives the rotation matrix:

One can verify the \(SO(3)\) conditions: the columns are mutually orthogonal unit vectors, and \(\det(R^e_b) = +1\).

Interactive Visualizer

The demo below lets you freely orient the body frame \(\{b\}\) relative to the fixed frame \(\{e\}\) using Euler angle sliders. The rotation matrix is updated live, and the bottom panel shows how it transforms an arbitrary vector.

Verifying the Result

Given a valid \(R^e_b\), one can verify the \(SO(3)\) conditions numerically:

\(R^T R = I \quad \Longleftrightarrow \quad \text{columns are orthonormal},\) \(\det(R) = +1 \quad \Longleftrightarrow \quad \text{proper rotation (no reflection)}.\)

For the example in the figure:

Checking orthogonality, \(R^T R = I\) holds since each column is a standard basis vector (up to sign), which are trivially orthonormal.

🔑 LinkedIn was great!

One thing I found interesting the most is how much easy it is to find matching opportunities in LinkedIn, at least for the field I am in. I tried a few other places including Indeed, but LinkedIn suggestions were the most relevant by miles. Also, that was true for the recruiters that reached out to me through the LinkedIn profile.

🦾 Connections get you far …

Another thing I noticed is that all opportunities I applied through a connection (a previous co-worker or a friend) always resulted in at least a callback for the initial screening. On the other hand, a larger percentage (almost 60%) of the jobs I applied through LinkedIn did not provide me any feedback or a response.

However, I do not want to discount the opportunities I had through direct applying through the LinkedIn. I had four positive mid/final stage interviews, and two of them were opportunities I directly applied through LinkedIn, third was a referral by a friend, and fourth was through a recruiter reached out to me through the LinkedIn profile. The one I ultimately went with was a job I directly applied through LinkedIn.

📊 Visualizations are cool! (especially when they are interactive)

You can see all the data I collected in this sankey plot. This is interactive, and generated with plotly. Feel free to move things around and see if you want to explore data. You can always reset the plot by clicking the “Home” icon on the top right of the plot.

You can check this post to learn about embedding interactive plots.

Horizontal Concatenation

Horizontal concatenation joins matrices side-by-side, appending columns. This corresponds to MATLAB’s [A, B] syntax.

Dimension rule: all input matrices must share the same number of rows. The output column count is the sum of each input’s column count.

Example

Given:

Horizontal concatenation yields:

Simulink block setting: set Concatenation method to Horizontal and Number of inputs to 2.

Vertical Concatenation

Vertical concatenation stacks matrices one above the other, appending rows. This corresponds to MATLAB’s [A; B] syntax.

Dimension rule: all input matrices must share the same number of columns. The output row count is the sum of each input’s row count.

Example

Given:

Vertical concatenation yields:

Simulink block setting: set Concatenation method to Vertical and Number of inputs to 2.

Summary

Common Pitfall

Simulink will throw a dimension mismatch error at simulation start if the constrained dimension is inconsistent across inputs. Check signal dimensions using Simulation → Update Diagram (Ctrl+D) before running to surface these errors early.

First, we need to install plotly. Assuming you already have Python installed in your system, it as simple as:

python3 -m pip  install plotly

Then, we need to generate a plot. For that, I am going to use a default dataset that comes with plotly, and an example code.

import plotly.express as px

df = px.data.gapminder()
fig = px.scatter(
        df.query("year==2007"), 
        x="gdpPercap", y="lifeExp", 
        size="pop", color="continent",
        hover_name="country", log_x=True, size_max=60
    )
fig.show()

fig.write_html('plotly_example.html', full_html=False, include_plotlyjs='cdn')

The last line (write_html) exports the image to an HTML file, and the additional arguments are for optimizing the file size of the HTML file. Without those, the write_html will create a self-contained HTML with the plotly.js source-code, which is useful for offline use, but not that much relevant for a website.

Once the HTML file is generated, copy/move that file to the directory where your website resides. Then, add the following line to your markdown page:

{ % include_relative relative/path/to/plotly_example.html % } 

NOTE: Make sure to delete the space between the curly braces and the percentage signs. I had to keep the space here to avoid markdown parsing the line.

Of course, replace relative/path/to/ with the actual relative path. For example, if both markdown and the HTML files are in the same directory, this will be,

{ % include_relative plotly_example.html % } 

Now, you should be able to see the interactive image on your website, similar to the below plot. You can interact with your mouse, zoom in, check data points, or save the image as a PNG file.

The discrete-time Kalman filter [1] provides the minimum-variance unbiased linear estimator for systems with Gaussian process and measurement noise. Its recursive structure makes it computationally attractive for real-time applications, yet this same recursive nature accumulates floating-point errors that can, over extended operation, destroy the estimator’s theoretical properties. The failure modes range from subtle covariance asymmetry to catastrophic filter divergence, and the mitigation strategies span numerical linear algebra, information-theoretic reformulations, and adaptive covariance management.

The Standard Formulation and Its Arithmetic Vulnerabilities

The conventional time-update and measurement-update equations are, respectively,

The state covariance matrix \(P\) is the central object of concern. By construction it must remain symmetric positive definite (SPD) at every step; its eigenvalues represent the uncertainty ellipsoid of the estimate. Three arithmetic pathways undermine this requirement in finite-precision arithmetic.

Long-Run Consequences

Filter Divergence

Covariance Collapse

In systems with poorly observable states, rounding errors can artificially suppress the diagonal elements of \(P\) corresponding to those states. The filter then assigns near-zero uncertainty to directions it has never actually observed, rejecting future measurements in those directions via near-zero Kalman gains. This is particularly pernicious in GPS-denied navigation where specific error modes are unobservable for extended intervals.

Rank Deficiency

Over hundreds of thousands of recursive steps — common in continuous INS/GNSS integration — \(P\) may lose rank entirely. A rank-deficient covariance matrix has a null space that the filter treats as perfectly known, permanently blocking correction in those directions even when valid measurements arrive.

Mitigation Techniques

1. Joseph Stabilized Form

The numerically robust symmetric form of the covariance update replaces the simple update with

This is algebraically equivalent to the standard form but guarantees symmetry by construction, since it is a sum of products of the form \(A M A^\top\). It doubles the number of multiplications but eliminates the asymmetry accumulation pathway [3]. It does not, however, resolve ill-conditioning.

2. Square-Root Filtering (Potter / Carlson)

Rather than propagating \(P\) directly, square-root filters maintain its Cholesky factor \(S\) such that \(P = S S^\top\). The time and measurement updates are formulated in terms of orthogonal triangularization (QR decomposition or Givens rotations applied to \(S\)), ensuring that \(P\) remains SPD by the fundamental property that \(S S^\top \succeq 0\) whenever \(S\) exists.

The square-root time update uses the QR decomposition:

where \(S_Q\) is the Cholesky factor of \(Q_k\). The effective condition number of \(S\) is the square root of that of \(P\), providing roughly 8 additional decimal digits of headroom in double precision [4]. The Carlson [5] and Bierman [6] UD-factorization variants operate on the unit lower-triangular times diagonal factorization \(P = U D U^\top\), which avoids square roots and is particularly efficient for sequential scalar measurement processing.

3. Information Filter Formulation

The information filter propagates the Fisher information matrix \(\Omega = P^{-1}\) and the information vector \(\xi = P^{-1}\hat{x}\). The measurement update becomes purely additive:

4. Covariance Symmetrization

A pragmatic and computationally cheap intervention is to enforce symmetry explicitly after each update:

This is \(O(n^2)\) and adds negligible overhead. While it does not address ill-conditioning, it prevents the asymmetry pathway from corrupting the eigenstructure and is universally recommended as a baseline hygiene measure [8]. Some implementations combine symmetrization with a periodic eigenvalue check: if the minimum eigenvalue of \(P\) falls below a threshold (e.g., \(10^{-12}\) times the trace), the matrix is projected back to the SPD cone via \(P \leftarrow P + \delta I\).

5. Sequential Measurement Processing

When the measurement noise covariance \(R_k\) is diagonal, each scalar component of \(z_k\) can be processed independently. This reduces the simultaneous update to a sequence of rank-1 updates, each of which requires only a scalar division rather than a full matrix inversion. Sequential processing is thus both numerically safer (smaller, better-conditioned operations) and enables early termination when measurements fail consistency checks [6].

6. Adaptive Covariance Inflation

Filter divergence driven by model mismatch — unmodeled dynamics, incorrect noise statistics — is not a purely numerical problem, but its symptoms are indistinguishable from arithmetic degradation from the filter’s perspective. Adaptive methods [9] estimate the innovation sequence statistics online:

Practical Recommendations

The choice of implementation should be driven by the system’s observability structure and the expected duration of operation:

For short-duration, well-conditioned systems (e.g., tightly coupled GPS/INS over minutes), the Joseph form with explicit symmetrization provides a good balance of simplicity and robustness.

For long-duration or poorly observable systems (e.g., submarine INS, deep-space navigation), square-root or UD-factored implementations are essentially mandatory. The Bierman-Thornton UD filter [6] remains the standard in precision navigation due to its numerical stability, computational efficiency, and amenability to sequential processing.

For distributed or high-measurement-rate architectures (e.g., multi-sensor SLAM), the information filter’s additive fusion structure offers structural advantages that outweigh its more expensive time update.

In all cases, the normalized innovation squared,

where \(m\) is the measurement dimension, provides an online diagnostic. Sustained NIS values outside the \(95\%\) confidence interval \([\chi^2_{m,0.025},\, \chi^2_{m,0.975}]\) indicate either filter divergence or model mismatch and should trigger corrective action.

References

1. R. E. Kalman, "A new approach to linear filtering and prediction problems," Journal of Basic Engineering, vol. 82, no. 1, pp. 35–45, 1960.
2. G. J. Bierman, Factorization Methods for Discrete Sequential Estimation. Academic Press, 1977. (Dover reprint, 2006.)
3. P. D. Joseph, as cited in A. H. Jazwinski, Stochastic Processes and Filtering Theory. Academic Press, 1970.
4. A. Andrews, "A square root formulation of the Kalman covariance equations," AIAA Journal, vol. 6, no. 6, pp. 1165–1166, 1968.
5. G. L. Carlson, "Fast triangular factorization of the square root filter," AIAA Journal, vol. 11, no. 9, pp. 1259–1265, 1973.
6. G. J. Bierman and C. L. Thornton, "Numerical comparison of Kalman filter algorithms: Orbit determination case study," Automatica, vol. 13, no. 1, pp. 23–35, 1977.
7. H. E. Durrant-Whyte and T. Bailey, "Simultaneous localization and mapping: Part I," IEEE Robotics & Automation Magazine, vol. 13, no. 2, pp. 99–108, 2006.
8. R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering, 4th ed. Wiley, 2012.
9. R. Mehra, "Approaches to adaptive filtering," IEEE Transactions on Automatic Control, vol. 17, no. 5, pp. 693–698, 1972.

There are two main modules here:

1. uav_gazebo
2. uav_control

The uav_gazebo includes all the necessary file to create and launch a Gazebo environment with the UAV. The uav_control includes the control plugin that will act as the PID controller, and the relevant custom messages.

The default behavior is as follows:

1. When the Gazebo environment is launched, the UAV will start at the origin.
2. Desired position is defined inside the control_plugin.cpp.
3. The UAV controller will wait for 2 seconds for the system to settle in.
4. Then, the UAV forces and torques will be set using the PID controller such the UAV converges to the desired position.
5. After the convergence, the UAV will stay there.

The gains, desired position, and other physical parameters are hard-coded. Again, the objective here is to familiar with the followings:

1. Reading states of the UAV in the Gazebo environment
2. Defining a basic PID controller
3. Transformation of desired forces and torques to the coordinate frame defined in the Gazebo environment

If are already familiar with the basics and you want to have a more interactive and in-depth ROS + Gazebo UAV control, please check fdcl-gwu/uav_simulator.