Example that explains critical damping

examples/plot_critical_damping.py

@@ -0,0 +1,32 @@
+"""
+================
+Critical Damping
+================
+
+The transformation system of a DMP converges to the goal and the convergence is
+modeled as a spring-damper system. For an optimal convergence, the constants
+defining the spring-damper system (spring constant k and damping coefficient c)
+have to be set to critical damping for optimal convergence, that is, as quickly
+as possible without overshooting. To illustrate this, we use a standalone
+spring-damper system and explore several values for these parameters.
+"""
+print(__doc__)
+
+
+import numpy as np
+import matplotlib.pyplot as plt
+from movement_primitives.spring_damper import SpringDamper
+
+
+k = 100
+start_y = np.zeros(1)
+goal_y = np.ones(1)
+for c in [10, 20, 40]:
+    attractor = SpringDamper(n_dims=1, k=k, c=c, dt=0.01)
+    attractor.configure(start_y=start_y, goal_y=goal_y)
+    T, Y = attractor.open_loop(run_t=2.0)
+    plt.plot(T, Y[:, 0], label=f"$k={k}, c={c}$")
+plt.scatter(1.0, 1.0, marker="*", s=200, label="Goal")
+plt.legend(loc="best")
+plt.title(r"Condition for critical damping: $c = 2 \sqrt{k}$")
+plt.show()