MPC gait generation

This repository is an implementation in python code of the MPC gait generation described in the "MPC for Humanoid Gait Generation: Stability and Feasibility"

Requirements

matplotlib

​numpy

​scipy

​qpsolvers

Control Framework

QP problem

$$ \min_{\begin{array}{c}\dot{X}_z^k,\dot{Y}_z,X_f,Y_f \end{array}}|\dot{X}_z^k|^2+|\dot{Y}_z^k|^2+\beta(|X_f-\hat{X}_f|^2+|Y_f-\hat{Y}_f|^2) $$

Constrains:Kinematic Constrains,ZMP Position Constrains,Stability Constraint and ZMP Velocity Constraint

Constrains

Kinematic Constrains

$$ \pm \binom{0}{\ell} - \frac{1}{2} \binom{d_{a,x}}{d_{a,y}} \leq R_{j-1}^T \binom{\hat{x}_f^j - \hat{x}_f^{j-1}}{\hat{y}f^j - \hat{y}f^{j-1}} \leq \pm \binom{0}{\ell} + \frac{1}{2} \binom{d{a,x}}{d{a,y}} $$

ZMP Position Constrains

Single Support

$$ -\frac{1}{2} \binom{d_{z,x}}{d_{z,y}} \leq R_j^T \begin{pmatrix} \Delta t \sum_{l=0}^i \dot{x}z^{k+l} - x_f^j \ \Delta t \sum{l=0}^i \dot{y}z^{k+l} - y_f^j \end{pmatrix} + R_j^T \begin{pmatrix} x_z^k \ y_z^k \end{pmatrix} \leq \frac{1}{2} \binom{d{z,x}}{d_{z,y}} $$

Double Support(moving constrains)

$$ x_{mc}(t) = \left( 1 - \alpha^j(t) \right) x_f^j + \alpha^j(t) x_f^{j+1} \\ y_{mc}(t) = \left( 1 - \alpha^j(t) \right) y_f^j + \alpha^j(t) y_f^{j+1} \\ \theta_{mc}(t) = \left( 1 - \alpha^j(t) \right) \theta_f^j + \alpha^j(t) \theta_f^{j+1} \\ \alpha^j(t) = \frac{t - t_s^j}{T_{ds}^j}, \quad t \in [t_s^j, t_s^j + T_{ds}^j] $$

and

$$ -\frac{1}{2} \binom{d_{z,x}}{d_{z,y}} \leq R_{mc}^T \begin{pmatrix} \Delta t \sum_{l=0}^i \dot{x}z^{k+l} - x{mc}^i \ \Delta t \sum_{l=0}^i \dot{y}z^{k+l} - y{mc}^i \end{pmatrix} + R_{mc}^T \begin{pmatrix} x_z^k \ y_z^k \end{pmatrix} \leq \frac{1}{2} \binom{d_{z,x}}{d_{z,y}} $$

Stability Constraint

$$ \sum_{i=0}^{C-1} e^{-i \eta \delta} \dot{x}z^{k+i} = - \sum{i=C}^{\infty} e^{-i \eta \delta} \dot{x}_z^{k+i} + \frac{\eta}{1 - e^{-\eta \delta}} (x_u^k - x_z^k) $$

ZMP Velocity Constraint

$$ |\dot{x}_z| \leq v^{\max}, \quad |\dot{y}_z| \leq v^{\max} $$

Results

simulation 1

simulation 2

simulation 3