Autonomous navigation is a core technology for AI robots, enabling them to perform tasks in dynamic environments without human intervention. Traditional approaches, such as those implemented in the Robot Operating System (ROS) Navigation framework, rely on external sensors and pre-defined maps. However, inspired by the neural mechanisms of spatial navigation in mammals, brain-inspired navigation algorithms have emerged as a promising alternative. These algorithms leverage insights from place cells, head direction cells, and grid cells to create cognitive maps and facilitate vector-based navigation. Vector navigation allows an AI robot to compute a direct path from its current location to a target using internal spatial representations, mimicking the efficiency observed in biological systems.
Grid cells, located in the entorhinal cortex, exhibit periodic firing patterns that form a hexagonal grid over the environment. This multi-scale representation enables precise encoding of spatial locations. In this article, we propose a vector navigation model for AI robots based on the Euclidean norm of grid cell activation matrices. By representing spatial positions through multi-scale grid cell activities and using the Euclidean norm to quantify similarity between positions, the model facilitates efficient path planning. Additionally, a linear look-ahead method is integrated to determine the correct heading direction, reducing the path length during target search. This approach enhances the autonomy of AI robots in complex environments, providing a biologically plausible solution for navigation tasks.
The foundation of this model lies in the multi-scale organization of grid cells. Each grid cell module has a distinct spatial scale, and the scales follow a geometric progression. For $M$ modules, the scale $s_i$ of the $i$-th module is given by:
$$s_i = s_M \cdot \alpha^{i-M} \quad \text{for} \quad i = 1, 2, \dots, M$$
where $s_M$ is the smallest scale and $\alpha$ is the scale ratio between adjacent modules. Each module contains $N_{GC}$ grid cells with spatial phases uniformly distributed between $0$ and $2\pi$. The firing rate $r_{i,j}$ of the $j$-th grid cell in the $i$-th module at position $\mathbf{a}$ is modeled as:
$$r_{i,j} = r_{\text{max}} \left[ \frac{1}{2} + \frac{1}{2} \cos\left( \frac{2\pi}{s_i} \left( \mathbf{a} \cdot \mathbf{p}_{i,j} \right) \right) \right]$$
where $r_{\text{max}}$ is the maximum firing rate, and $\mathbf{p}_{i,j}$ is the spatial phase vector. The activation matrix $\mathbf{R}_C$ for a position is constructed by combining the firing rates across all modules and cells:
$$\mathbf{R}_C = \left[ \mathbf{R}_{C,1}, \mathbf{R}_{C,2}, \dots, \mathbf{R}_{C,M} \right]^T$$
where $\mathbf{R}_{C,i} = \left[ r_{i,1}, r_{i,2}, \dots, r_{i,N_{GC}} \right]$ represents the firing rate vector of the $i$-th module. This matrix provides a unique representation for each position within the encoded space, allowing AI robots to distinguish between locations based on neural activity patterns.
To enable vector navigation, the model quantifies the spatial similarity between two positions using the Euclidean norm of the difference in their grid cell activation matrices. For positions $i$ and $j$, the norms along the x and y dimensions are defined as:
$$\text{norm}_x = \left\| \mathbf{GC}_i^x – \mathbf{GC}_j^x \right\|_2$$
$$\text{norm}_y = \left\| \mathbf{GC}_i^y – \mathbf{GC}_j^y \right\|_2$$
Here, $\mathbf{GC}_i^x$ and $\mathbf{GC}_i^y$ denote the grid cell activation matrices for position $i$ along the x and y axes, respectively. As the AI robot moves closer to the target position, $\text{norm}_x$ and $\text{norm}_y$ decrease, indicating higher similarity. When both norms fall below a threshold $\delta$, the robot is considered to have reached the target. This mechanism allows the AI robot to navigate efficiently by minimizing the distance to the goal based on neural representations.
The linear look-ahead method assists the AI robot in determining the correct heading direction. Initially, the robot simulates movement along a candidate direction by updating the grid cell activation matrices based on a velocity input. The predicted matrices are compared to the target matrices using the Euclidean norm. If both $\text{norm}_x$ and $\text{norm}_y$ are below $\delta$ for a direction, it is selected as the heading. Otherwise, the direction is incremented by $\Delta \theta$, and the process repeats. This approach ensures that the AI robot follows a straight path to the target, reducing unnecessary detours and improving navigation efficiency.
To validate the model, simulations and experiments were conducted using an AI robot platform equipped with ROS. The parameters for the grid cell modules are summarized in Table 1.
| Parameter | Value |
|---|---|
| Number of modules ($M$) | 10 |
| Smallest scale ($s_M$) | 0.25 m |
| Scale ratio ($\alpha$) | 1.4 |
| Grid cells per module ($N_{GC}$) | 20 |
| Threshold ($\delta$) | 5 |
| Heading increment ($\Delta \theta$) | 0.5° |
The AI robot was tested on various trajectories, including straight lines, curves, and mixed paths. The evolution of $\text{norm}_x$ and $\text{norm}_y$ during movement demonstrated that these metrics decrease as the robot approaches the target, confirming their utility in distance quantification. For instance, in a straight-line trajectory from (0 m, 0 m) to (5 m, 5 m), $\text{norm}_x$ and $\text{norm}_y$ converged to values below $\delta$ when the robot was within 0.02 m of the target, highlighting the precision of the method for AI robot navigation.
The linear look-ahead method was evaluated by setting the initial position to (0 m, 0 m) and the target to (10 m, 10 m). The correct heading of 45° was identified after iterating through candidate directions, with $\text{norm}_x$ and $\text{norm}_y$ both below $\delta$ at this angle. In contrast, other directions only satisfied one norm condition, leading to their rejection. This process ensured that the AI robot could accurately determine the optimal path, minimizing travel distance.
Comparative analyses with existing methods, such as the “Linear Look Ahead” model, revealed significant improvements. The proposed model achieved a maximum heading error of 0.33° and a maximum position error of 0.028 m, outperforming the alternative in both accuracy and path length. For example, in a navigation task from (0 m, 0 m) to (8 m, 6 m), the AI robot using our model traveled 7.5 m, whereas the traditional method required 9.2 m. The results are summarized in Table 2, demonstrating the efficacy of the grid cell-based approach for AI robots.
| Method | Heading Error (°) | Position Error (m) | Path Length (m) |
|---|---|---|---|
| Proposed Model | 0.33 | 0.028 | 7.5 |
| Linear Look Ahead | N/A | 0.037 | 9.2 |
The integration of multi-scale grid cells and Euclidean norm quantification provides a robust framework for vector navigation in AI robots. The linear look-ahead method further enhances performance by ensuring straight-line paths to targets. Future work will focus on combining this model with cognitive mapping mechanisms to develop a comprehensive brain-inspired navigation system for AI robots. This advancement will enable AI robots to operate autonomously in large-scale, unstructured environments, bridging the gap between biological intelligence and artificial systems.
In conclusion, the grid cell vector navigation model offers a biologically inspired solution for AI robots, leveraging neural representations to achieve efficient and accurate navigation. By emulating the spatial coding properties of grid cells, AI robots can compute direct paths to goals, reducing reliance on external sensors and pre-defined maps. This approach not only improves navigation performance but also contributes to the development of autonomous AI robots capable of complex tasks in real-world scenarios.