Intro to Brazilian Jiu-Jitsu Modeling Language [BJJML]

7 min readApr 30, 2024

If you know the enemy and know yourself, you need not fear the result of a hundred battles. If you know yourself but not the enemy, for every victory gained you will also suffer a defeat. If you know neither the enemy nor yourself, you will succumb in every battle.
-“Art of War” by Sun Tzu

The Mathematical Canvas of Brazilian Jiu Jitsu

Beneath every the grappling match lies a battle of wits and strength, and beneath the intensity lies a hidden language, a strategic canvas that charts the ebb and flow of combat. Enter the Brazilian Jiu-Jitsu Modeling Language (BJJML), a pioneering framework that transforms the dynamic essence of this martial art into a structured, analytical system.

Imagine a world where every grip, sweep, and submission is mapped with precision, allowing practitioners and coaches to navigate the labyrinth of positions and transitions with surgical accuracy. BJJML presents a paradigm shift in the way we approach the complexities of grappling, offering a unified platform to decode the strategic underpinnings of this captivating sport.

But how can such mathematical models capture the fluidity and dynamics of Brazilian Jiu-Jitsu? By delving into the BJJML’s foundational structures in its first iteration, BJJML 1.0, we will unveil the potential to unlock new frontiers in grappling analytics, paving the way for a groundbreaking platform, Graphling — a submission grappling analytics tool that promises to revolutionize the way we think about combat sports strategy.

BJJML 1.0 | The Mathematical Foundations

At the heart of BJJML lies a comprehensive approach to modeling the dynamic and strategic aspects of Brazilian Jiu-Jitsu like a coordinate system. The framework presents a mathematical blueprint that captures the essence of the sport, from the intricacies of position control to the finesse of technique execution.

The BJJML system is built upon a foundation of meticulously defined components, each serving as a building block in the larger strategic tapestry. Position tuples, for instance, provide a structured way to represent the myriad of configurations a grappler can find themselves in, encoding factors such as the type of positions, upper body, and lower body details.

Actions, on the other hand, are defined as the unit of transitions between these position tuples, allowing the model to chart the dynamic flow of movement and the corresponding strategic implications. By mapping these actions and their outcomes, BJJML lays the groundwork for a deeper understanding of the tactical decision-making process.

Dissecting BJJML’s Charting System

At the core of the BJJML framework is the concept of position tuples — a mathematical representation of the various configurations a grappler can assume during a match. Each position is encapsulated in a four-dimensional tuple, comprising the following components:

Family: The broad category of the position, such as Guard, Mount, Back control, Standing, etc.
State: The specific state of the position, whether it’s closed guard, open guard, Russian Tie Clinch, etc.
Upper: The configuration of the upper body in a state, denoting grip detail, position of the torso, and/or limb placement.
Lower: The configuration of the lower body in a state, denoting leg and hip position and engagement.

By breaking down positions into these granular elements, BJJML 1.0 enables a precise and comprehensive charting of the strategic landscape. Imagine a cartographic representation of the grappling universe, where each position is a distinct coordinate on the map, with neighboring possibilities waiting to be explored and mastered.

Actions and Transitions: The Language of Movement

The fluidity of Brazilian Jiu-Jitsu is captured in the BJJML framework through the concept of Actions — the unit of transition between positions. These actions are mathematically defined as functions, mapping one position to another, allowing the model to trace the dynamic flow of the grappling encounter. Technique is then defined as a sequence of actions. The framework is set to evolve with BJJML 2.0, introducing algorithms that refine predictive capabilities and strategy development.

For example, the action of executing a successful guard pass can be represented as a transition from the guard position to the top position. This transition is not just a physical movement, but a strategic shift that alters the balance of power on the mats. By charting these actions and their outcomes, BJJML provides a language to describe the tactical decision-making process, unlocking new avenues for strategic analysis and optimization.

Scoring and Environment: The Game Within the Game

The BJJML framework also encompasses the scoring and environmental factors that influence and update the progression of the match. The scoring function, denoted as F_S, maps the executed actions to their corresponding scoring outcomes, taking into account the current match context and the applicable ruleset.

Moreover, the environment update function, U_R, models how the execution of an action affects the overall match environment, adjusting variables such as points scored for each athlete. This integrated approach to scoring and environment allows BJJML to capture the intricate interplay between tactical decisions and their immediate, as well as long-term, consequences. BJJML 2.0 intends to include strategic analysis via simulation tools based on the deterministic framework. It envisions a scoring system that influences match outcomes, with real-time adjustments and scenario simulations.

Strategic Modeling: From Tactics to Meta-strategy

The true power of the BJJML framework lies in its capability to project analytical framework from various depths and levels. Assuming that all actions and positions are comprehensively mapped out in between within BJJML 1.0, practitioners will be able to construct Paths, which we define as the pair of initial & terminal position tuples and intermediate actions and positions based on their preference and navigation. Further on, users will be able to aggregate Paths into Trees, which consists of multiple Paths, ranging from multiple initial positions and terminal positions with varying sequences. BJJML 2.0 will enable evaluation functions (Φ) on user-created Paths and Trees to extract strategic elements. Tactical Patterns (TP), defined as sequences of position-action pairs from Paths and Trees based on conditions, can be aggregated and synthesized into higher-level strategies (Σ).

These strategic frameworks, in turn, can be subjected to meta-strategic optimization, leveraging factors such as the opponent’s model, the practitioner’s attributes, and the environmental conditions to determine the most advantageous course of action. This hierarchical modeling approach enables a holistic understanding of the grappling landscape, empowering practitioners and coaches to make informed, data-driven decisions. The BJJML framework doesn’t merely capture static positions; it aims to project from tactical sequences to comprehensive strategies. This hierarchical modeling approach leads to the probabilistic models and computer vision analysis of BJJML 3.0, setting the stage for the innovative Graphling platform.

The Future Mapped Out: Graphling’s Role

The BJJML framework lays the groundwork for Graphling, a pioneering submission grappling analytics platform that aims to revolutionize the way combat sports are studied and strategized. Graphling will take the mathematical structure of BJJML and apply it across a wide range of grappling disciplines, creating a universal language for modeling and analyzing the intricacies of various combat sports. Graphling is poised to become a universal language for combat sports analytics. It builds on BJJML’s state modeling and integrates computer vision, preparing to encompass a wide range of grappling disciplines.

By establishing this comprehensive analytical framework, Graphling will empower practitioners, coaches, and researchers to uncover new insights, optimize training regimens, and develop innovative strategies. This platform will serve as a hub for collaboration and innovation, fostering a deeper understanding of the strategic nuances that define the world of submission grappling.

Charting New Territories

The BJJML framework and the Graphling platform represent a transformative shift in the way we approach the complex and captivating world of Brazilian Jiu-Jitsu and submission grappling. By translating the dynamic essence of these martial arts into a mathematical language, we unlock new avenues for strategic analysis, decision-making, and optimization. They herald a future ripe with strategic insights, inviting the combat sports community to join a journey into unexplored territories, redefining what’s possible in martial arts.

Our next step in BJJML is to synthesize a holistic cartography of positions and submissions available in the form of nested nodes and edges. As we venture into this uncharted territory, we invite the combat sports and tech community alike to join us in this groundbreaking journey. Together, we can explore the hidden depths of grappling tactics, uncover novel strategic insights, and redefine the boundaries of what is possible in the pursuit of excellence. The future of submission grappling lies in the power of this mathematical canvas, and with Graphling, we are poised to chart new territories, pushing the limits of what we thought possible.