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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