In my research on autonomous systems, I have always been fascinated by the concept of bionic robots—machines that emulate biological learning processes to adapt to unknown environments. The core challenge lies in endowing these bionic robots with robust self-learning capabilities without human intervention. Reinforcement learning (RL) offers a promising “perception-action” framework where an agent learns by interacting with its environment, receiving rewards or penalties to optimize its behavior over time. However, in practical applications, especially for complex systems like two-wheeled balancing robots, standard RL algorithms often suffer from large network scales, structural complexity, overfitting, and poor generalization. These issues can lead to prolonged training times and suboptimal performance in dynamic settings. To address this, I propose a novel Memory-Pruning Reinforcement Learning Model (H-RLM) as a learning mechanism for bionic robots, integrating neural network pruning with RL to enhance efficiency and adaptability.
My work focuses on developing an intelligent control system for a two-wheeled bionic robot, which is inherently unstable and requires precise speed tracking for balance. Traditional RL approaches, while effective, often rely on expansive neural networks that are computationally expensive and prone to memorizing noise rather than learning generalizable patterns. Inspired by biological memory consolidation and pruning—where the brain strengthens important neural connections while eliminating redundant ones—I designed H-RLM to dynamically optimize network architecture during learning. This model combines Hessian matrix-based pruning with Markov decision processes, enabling the bionic robot to learn faster, reduce errors, and maintain stability even under disturbances. In this article, I detail the design, implementation, and validation of H-RLM, demonstrating its superiority over conventional RL through extensive simulations.
The foundation of my approach lies in the dynamics of the two-wheeled bionic robot, which I model as a mobile inverted pendulum. This system consists of two independently driven wheels and a pendulum-like body, making it a classic benchmark for balance control. Using Lagrangian mechanics, I derive the equations of motion to capture the nonlinear interactions. The state variables are defined as the left wheel angular velocity $\dot{\theta}_l$, right wheel angular velocity $\dot{\theta}_r$, pendulum angular velocity $\dot{\theta}$, and pendulum angle $\theta$. The control inputs are the motor torques $U_l$ and $U_r$ applied to the wheels. Under small-angle approximations (e.g., $\sin \theta \approx \theta$ and $\cos \theta \approx 1$ for $|\theta| < 5^\circ$), the linearized state-space representation is as follows:
$$ \dot{X} = A X + B U $$
$$ Y = C X + D U $$
where $X = [\dot{\theta}_l, \dot{\theta}_r, \dot{\theta}, \theta]^T$ and $U = [U_l, U_r]^T$. The matrices $A$, $B$, $C$, and $D$ are derived from system parameters. For instance, based on typical robot dimensions and masses, I compute:
$$ A = \begin{bmatrix}
-0.39114 & 0.3602 & 0 & -47.0760 \\
0.3602 & -3.9114 & 0 & -47.0760 \\
2.2414 & 0.2414 & 0 & 13.3323 \\
0 & 0 & 1 & 0
\end{bmatrix}, \quad B = \begin{bmatrix}
2.4145 & -0.2223 \\
-0.2223 & 2.4145 \\
-0.1490 & -0.1490 \\
0 & 0
\end{bmatrix} $$
This model serves as the environment for the bionic robot to interact with during RL training. The goal is to learn a control policy that maintains balance while tracking desired speeds, mimicking how humans learn to ride a bicycle through trial and error.
To enable efficient learning, I developed the H-RLM architecture, which comprises two neural networks: a Critic Neural Network (CNN) and an Actor Neural Network (ANN). The CNN approximates the value function, evaluating how good a state is, while the ANN outputs control actions based on the current state. Both networks are subject to pruning to eliminate redundant weights, akin to synaptic pruning in biological brains. The pruning process uses the Hessian matrix of the loss function to identify and remove less important connections without significantly impacting performance. This not only reduces computational load but also enhances generalization, a critical aspect for bionic robots operating in unpredictable environments.
The key innovation in H-RLM is the integration of Optimal Brain Surgeon (OBS) pruning within the RL framework. I define the loss function for the neural networks as the mean squared error (MSE) between predicted and target outputs. For a weight vector $W$, the change in loss $\Delta E$ due to a perturbation $\Delta W$ can be approximated using a second-order Taylor expansion:
$$ \Delta E \approx \frac{1}{2} \Delta W^T H \Delta W $$
where $H$ is the Hessian matrix of the loss with respect to $W$. To prune a weight $w_i$, I set the constraint $l_i^T \Delta W + w_i = 0$, where $l_i$ is a unit vector. Using Lagrange multipliers, the optimal weight change and the resulting loss increase (saliency $L_i$) are:
$$ \Delta W = – \frac{w_i}{[H^{-1}]_{i,i}} H^{-1} l_i, \quad L_i = \frac{w_i^2}{2 [H^{-1}]_{i,i}} $$
Weights with small saliency relative to the overall loss are pruned iteratively. This process ensures that the network remains compact while preserving essential learning capacity. I implement this online during RL training, allowing the bionic robot to adapt its neural architecture in real-time as it learns from environmental feedback.
The RL algorithm in H-RLM employs a Markov Decision Process (MDP) formulation. The state space $S$ includes all possible configurations of the bionic robot, and the action space $A$ consists of motor control signals. The transition probabilities $P_{s_t s_{t+1}}(a_t)$ describe how the robot’s state evolves given an action. The reward function $R(s_t, a_t)$ provides feedback: for example, a reward of 0 for maintaining the pendulum angle within a safe range (e.g., $|\theta| \leq 0.0523$ rad) and a penalty of -1 otherwise. This sparse reward scheme encourages the bionic robot to discover balance strategies autonomously. The value function $V^\pi(s)$ under a policy $\pi$ is:
$$ V^\pi(s_t) = R(s_t, \pi(s_t)) + \gamma \sum_{s_{t+1} \in S} P_{s_t s_{t+1}}(\pi(s_t)) V^\pi(s_{t+1}) $$
where $\gamma \in [0,1]$ is a discount factor. The optimal policy $\pi^*$ maximizes the expected cumulative reward, solved via the Bellman optimality equation:
$$ V^*(s_t) = \max_{a_t \in A} \left\{ R(s_t, a_t) + \gamma \sum_{s_{t+1} \in S} P_{s_t s_{t+1}}(a_t) V^*(s_{t+1}) \right\} $$
In practice, I use temporal-difference (TD) learning to update the CNN and policy gradient methods for the ANN. The pruning mechanism is applied after each training episode to refine the networks. This combination of pruning and RL enables the bionic robot to learn efficiently, similar to how biological systems consolidate memories by strengthening useful neural pathways.
To validate H-RLM, I conducted simulation experiments comparing it with a standard RL approach (without pruning). The bionic robot was tasked with speed tracking while balancing. I set the sampling time to 0.01 s and the discount factor $\gamma = 0.9$. The CNN had a structure of $N^3[6,8,1]$ (6 inputs: 4 states plus 2 action feedbacks, 8 hidden neurons, 1 output for value), and the ANN had $N^3[4,8,2]$ (4 state inputs, 8 hidden neurons, 2 outputs for motor controls). Both networks were initialized with random weights small than 1. The pruning threshold was set such that weights with saliency $L_i < 0.001 \times \text{MSE}$ were removed. Training involved multiple episodes until the bionic robot achieved stable balance.
The results demonstrate the advantages of H-RLM. The table below summarizes key performance metrics averaged over 50 simulation runs:
| Metric | Standard RL | H-RLM (Proposed) |
|---|---|---|
| Average failures before balance | 50 | 30 |
| Transition steps to steady state | 400 steps (4 s) | 200 steps (2 s) |
| Final MSE of value function | 0.025 | 0.012 |
| Network size after training (weights) | 120 | 65 |
| Generalization error on unseen disturbances | 0.18 | 0.09 |
As shown, H-RLM reduces the number of failures and shortens the transition time by 50%, indicating faster learning. The pruned network has nearly half the weights, yet achieves lower MSE and better generalization. This efficiency is crucial for bionic robots deployed in real-world scenarios where computational resources are limited. The speed tracking performance is illustrated through the pendulum angle and wheel velocities over time. With H-RLM, the bionic robot maintains balance with smaller oscillations and quicker recovery from perturbations.
I further analyzed the learning dynamics by examining the value function and TD error during training. The value function $V(s)$ converges to near-zero values, reflecting optimal balance, while the TD error diminishes rapidly. The following equation captures the TD update for the CNN:
$$ \Delta V(s_t) = \alpha [r_{t+1} + \gamma V(s_{t+1}) – V(s_t)] $$
where $\alpha$ is the learning rate. In H-RLM, these updates are more stable due to pruning, which reduces noise in the weight updates. I also evaluated the impact of different pruning strategies on the bionic robot’s performance. The table below compares OBS pruning (used in H-RLM) with magnitude-based pruning (removing smallest weights) and random pruning:
| Pruning Method | Balance Success Rate (%) | Average Training Time (s) | Network Sparsity (%) |
|---|---|---|---|
| No Pruning (Standard RL) | 85 | 50.2 | 0 |
| Magnitude-Based | 78 | 48.5 | 45 |
| Random Pruning | 65 | 52.1 | 40 |
| OBS (H-RLM) | 95 | 32.7 | 50 |
OBS pruning outperforms others, achieving higher success rates and faster training, validating its suitability for bionic robot learning. The sparsity level indicates that 50% of weights were pruned without compromising performance, akin to biological neural networks that operate efficiently with sparse connectivity.
The robustness of H-RLM was tested under external disturbances, such as sudden pushes or changes in surface friction. The bionic robot equipped with H-RLM adapted quickly, adjusting its control policy within a few steps. This adaptability stems from the continuous pruning and learning cycle, which prioritizes essential connections for balance. I modeled disturbances as additive noise to the state variables, with a noise variance of 0.1. The reward function was modified to include a term for speed tracking error:
$$ R(s_t, a_t) = -(\theta_t^2 + 0.1 (\dot{\theta}_{l,t} – v_{ref})^2 + 0.1 (\dot{\theta}_{r,t} – v_{ref})^2) $$
where $v_{ref}$ is the desired wheel speed. H-RLM minimized this reward effectively, demonstrating precise tracking. The convergence of the policy is assured by the policy gradient theorem for the ANN:
$$ \nabla J(\theta) = \mathbb{E} \left[ \nabla_\theta \log \pi_\theta(a|s) Q^\pi(s,a) \right] $$
where $J(\theta)$ is the expected return, $\pi_\theta$ is the policy parameterized by $\theta$, and $Q^\pi$ is the action-value function approximated by the CNN. Pruning enhances this process by reducing variance in gradient estimates.
In conclusion, my proposed H-RLM model offers a significant advancement for autonomous bionic robots. By integrating neural network pruning with reinforcement learning, it addresses common issues of overfitting and computational inefficiency. The bionic robot learns to balance and track speeds faster, with fewer failures and better generalization, mimicking biological learning principles. Future work will involve implementing H-RLM on physical bionic robot hardware and exploring more complex tasks like navigation in dynamic environments. This research underscores the potential of brain-inspired algorithms to create more adaptive and intelligent machines.
Throughout this study, I have emphasized the importance of efficient learning mechanisms for bionic robots. The H-RLM framework, with its Hessian-based pruning and Markov decision processes, provides a scalable solution for various robotic applications. As bionic robots evolve, such algorithms will be crucial for achieving human-like adaptability in real-world scenarios.