Hybrid Model Predictive and Iterative Learning Control for Enhanced Leader-Follower Robotic Tracking#

EERC paper

Tested for “S” Trajectory#

Abstract#

This paper introduces a novel hybrid control strategy combining Model Predictive Control (MPC) and Iterative Learning Control (ILC) to improve leader-follower tracking accuracy and robustness in mobile robots. MPC optimizes control inputs over a predictive horizon by minimizing a cost function, while ILC refines these inputs by learning from past errors.

Mathematical Formulation#

System Dynamics#

The state of the follower robot is:

\[x_f = [x_f, y_f, \theta_f]^T\]

Control inputs:

\[u = [v, \omega]^T\]

Discrete-time kinematic model:

\[\begin{split}x_f(k + 1) = x_f(k) + \begin{bmatrix} v(k) \cos(\theta_f(k)) T \\ v(k) \sin(\theta_f(k)) T \\ \omega(k) T \end{bmatrix}\end{split}\]

Where \(T\) is the sampling time.

MPC Cost Function#

The MPC minimizes the following cost function over a finite horizon \(N\):

\[J = \sum_{k=0}^{N-1}\left[\omega_p\|P_f(k)-P_d(k)\|^2 + \omega_\theta(\theta_f(k)-\theta_d(k))^2 + \omega_u\|u(k)\|^2\right]\]

where:

\(P_f(k) = [x_f(k), y_f(k)]^T\): position of follower.
\(P_d(k), \theta_d(k)\): desired position and orientation.
\(\omega_p, \omega_\theta, \omega_u\): weights for position, orientation errors, and control effort.

Desired Position Calculation#

The desired position \(P_d\) is calculated from the leader’s state \(x_l = [x_l, y_l, \theta_l]^T\) and specified distance \(d\):

\[\begin{split}P_d = \begin{bmatrix} x_l - d\cos(\theta_l) \\ y_l - d\sin(\theta_l) \end{bmatrix}\end{split}\]

Iterative Learning Control (ILC) Update#

Error vector:

\[\begin{split}e(k) = \begin{bmatrix} x_f(k)-x_d(k) \\ y_f(k)-y_d(k) \\ \theta_f(k)-\theta_d(k) \end{bmatrix}\end{split}\]

ILC control update:

\[u_{ILC}(k)=\nu_{MPC}(k)+L[e(k)-e(k-1)]\]

where:

\(\nu_{MPC}(k)\): optimal MPC control input.
\(L\): learning matrix.
\(e(k-1)\): error from previous iteration.

Hybrid Integration#

The hybrid method combines MPC and ILC:

Solve MPC optimization: \(\nu_{MPC}=\arg\min J\)
ILC update: \(\nu=\nu_{ILC}\)
Apply \(\nu\) to follower robot
Update follower and leader states

Algorithm#

Algorithm 1: Hybrid MPC-ILC
Require: Initial states x_f, x_l; past error e_past=0, past input u_past=0
Require: T, N, ω_p, ω_θ, ω_u, d, L
While true:
  1. Obtain current leader state x_l(k)
  2. Compute desired P_d(k), θ_d(k)
  3. Solve MPC to get u_MPC
  4. Calculate tracking error e(k)
  5. Update control: u_ILC = u_past + L(e(k)-e_past)
  6. Apply u_ILC to follower
  7. Update u_past = u_MPC, e_past = e(k)
  8. Update robot states
End While

Experimental Results#

Experiments with TurtleBot 3 robots validated the hybrid MPC-ILC method, demonstrating significant improvements in position tracking accuracy, orientation accuracy, and balanced control efforts compared to standalone MPC and ILC.

Leader velocity: \(v=0.3\,\text{m/s}, \omega=0.3\,\text{rad/s}\)
MPC parameters: \(N=20, T=0.2\,s, \omega_p=7, \omega_\theta=1, \omega_u=1\)
ILC learning matrix: \(L = \mathrm{diag}([0.1, 0.1])\)

Conclusion#

The hybrid MPC-ILC approach offers improved accuracy, robustness, and efficiency for leader-follower robotic tracking. Future work includes integrating with advanced control methods and testing in more complex scenarios.

Source Code

Hybrid MPC-ILC Demo

Comparison between MPC and LMPC and Obstacle Avoidance#

We included a safe set process to keep the follower robot at a fixed distance from the leader robot, and then we added obstacle avoidance to the follower robot using LMPC.