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:

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

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 $d$:

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

Iterative Learning Control (ILC) Update#

Error vector:

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

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: $L = \mathrm{diag}([0.1, 0.1])$

Comparison of Cost

Results

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

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.