Combining Imitation Learning with Diffusion Processes on Robot Manipulator

Combining Imitation Learning with Diffusion Processes on Robot Manipulator#

Diffusion Processes#

Diffusion processes can be modeled by Stochastic Differential Equations (SDEs) as follows:

$$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t $$

where:

  • $X_t$ is the state of the process at time $t$.

  • $\mu(X_t, t)$ represents the drift term.

  • $\sigma(X_t, t)$ represents the diffusion term.

  • $W_t$ represents a Wiener process (or Brownian motion).

Imitation Learning#

The objective function in imitation learning can be expressed as:

$$ \min_{\pi} \mathbb{E}_{(s,a^)\sim D}[-\log \pi(a^ | s)] $$

where:

  • $D$ is a dataset of state-action pairs $(s, a^*)$.

  • $a^*$ is the action taken by the expert in state $s$.

  • The expertise dataset was created using an Inverse Kinematics Solver (IK-Solver).

Combining Diffusion Processes with Imitation Learning#

$$ \min_{\pi} \mathbb{E}_{(s,a^)\sim D, X_t\sim SDE}[-\log \pi(a^ | s) + \lambda \cdot L(X_t, \pi(s))] $$

where:

  • $L(X_t, \pi(s))$ represents a loss term under the dynamics $X_t$.

  • $\lambda$ is a regularization parameter.

Neural Network Model#

Given a desired end-effector position $x \in \mathbb{R}^3$, the network computes the joint angles $y \in \mathbb{R}^7$ through a series of transformations:

  1. Layer 1: $h_1 = \text{ReLU}(W_1x + b_1)$

  2. Layer 2: $h_2 = \text{ReLU}(W_2h_1 + b_2)$

  3. Layer 3: $h_3 = \text{ReLU}(W_3h_2 + b_3)$

  4. Output Layer: $y = W_4h_3 + b_4$

Where:

  • $W_i$ and $b_i$ are the weights and biases of the $i$-th layer.

  • $\text{ReLU}(z) = \max(0, z)$ is the Rectified Linear Unit activation function.

Loss Function#

Minimizing the difference between the predicted joint angles and the true joint angles. The loss function used is the Mean Squared Error (MSE), given by:

$$ L(\theta) = \frac{1}{N} \sum_{i=1}^{N} | f(x^{(i)}; \theta) - y^{(i)} |^2 $$

where:

  • $N$ is the number of samples in the dataset.

  • $x^{(i)}$ is the $i$-th desired end-effector position.

  • $y^{(i)}$ is the true joint angles for the $i$-th sample.

  • $| \cdot |$ denotes the Euclidean norm.

Optimization#

The training process seeks to find the optimal parameters $\theta^*$ that minimize the loss function $L(\theta)$. This is typically done using gradient-based optimization methods, such as Adam. The update rule for Adam at each iteration $t$ is:

  1. Compute gradients: $g_t = \nabla_{\theta} L(\theta)$

  2. First moment: $m_t = \beta_1 m_{t-1} + (1 - \beta_1) g_t$

  3. Second moment: $v_t = \beta_2 v_{t-1} + (1 - \beta_2) g_t^2$

  4. Bias-corrected first moment: $\hat{m}_t = m_t / (1 - \beta_1^t)$

  5. Bias-corrected second moment: $\hat{v}_t = v_t / (1 - \beta_2^t)$

  6. Update rule:

$$ \theta = \theta - \eta \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} $$

where:

  • $\beta_1$ and $\beta_2$ control the decay rates. Default: $\beta_1 = 0.9$, $\beta_2 = 0.999$

  • $\eta$ is the learning rate. We used $\eta = 0.001$

  • $\epsilon = 10^{-8}$ adds numerical stability

Training Process#

  • Epochs: 10,000

  • Learning Rate ($\eta$): 0.001

  • Total Time Taken: 11.92 seconds

Demonstration#

We used the matplotlib Python library to draw some trajectories to imitate.

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