3D LIPM Walking Pattern Visualizer

Linear Inverted Pendulum Model

T_c = 0.286 s | 5 steps | 505 samples

Presets

Number of steps3.00

Physical Parameters

CoM Height z_c0.80 m

Support Period T_sup0.80 s

Gravity g9.81 m/s²

Foot Placement Weights

Position weight a10.00

Velocity weight b1.00

Step Data

Key Equations

\ddot{x} = \frac{g}{z_c}\, x \qquad \ddot{y} = \frac{g}{z_c}\, y

Mass travels on a horizontal plane at constant height $z_c$ . Kick force $f = Mgr/z_c$ counteracts gravity.

z_c = 0.80 m

g = 9.81 m/s²

g/z_c = 12.262 s⁻²

T_c \equiv \sqrt{\frac{z_c}{g}}

x(t) = x_0\cosh\!\frac{t}{T_c} + T_c\dot{x}_0\sinh\!\frac{t}{T_c}

\dot{x}(t) = \frac{x_0}{T_c}\sinh\!\frac{t}{T_c} + \dot{x}_0\cosh\!\frac{t}{T_c}

T_c = 0.2856 s

C = cosh(T_sup/T_c) = 8.2644

S = sinh(T_sup/T_c) = 8.2037

E = \tfrac{1}{2}\dot{x}^2 - \frac{g}{2z_c}x^2 = \text{const}

Conserved during single support. $E > 0$ : continuous motion. $E \leq 0$ : reversal.

p_x^{(n)} = p_x^{(n-1)} + s_x^{(n)}

p_y^{(n)} = p_y^{(n-1)} - (-1)^n s_y^{(n)}

*2/*3

\bar{x}^{(n)} = s_x^{(n+1)}/2,\quad \bar{v}_x^{(n)} = \bar{x}\tfrac{C+1}{T_c S}

\ddot{x} = \frac{g}{z_c}(x - p_x^*)

x^d = p_x^{(n)} + \bar{x}^{(n)},\quad \dot{x}^d = \bar{v}_x^{(n)}

N \equiv a(x^d - x_f)^2 + b(\dot{x}^d - \dot{x}_f)^2

Minimized analytically to find $p_x^*$ .