2.4 Ballistic Walking Model

McMahon states that his ballistic walking model pictured in Figure 2.9 of the course reader “is the least complicated mechanical configuration... (for) thinking about the dynamics of walking. A compound pendulum alone or an inverted pendulum alone is not enough.”

Let’s explore this statement while analyzing a simple inverted pendulum. Below is a picture of an inverted pendulum of mass m and length L.  Its angular position with respect to the ground is given by .

(a) Based on the figure above, derive expressions for the x and y positions of the mass, m, as a function of the length of the limb, L, and specified rotation angle, 

(b) Derive expressions for the vertical and horizontal ground reaction forces as a function of the length of the limb, L, and rotation angle, . (Hint: assume the presence of gravity that is directed in the –y direction. Use this term if you need it)

(c) Given the following:

L = 1 m          m = 1 kg                       

Plot the vertical position of the mass as  varies over a range from 120° to 60° ( to  rads). Briefly discuss how this plot relates to the motion paths of the center of mass in walking.

(d) For the same constants given in part (c), plot the vertical ground reaction force as  varies over a range from 120° to 60° ( to  rads). Briefly discuss how your plot compares to the plot of vertical ground reaction force.

(e) Plot the vertical ground reaction force for  (not zero) and the initial conditions shown below (θ over the same range as before). Briefly discuss how your plot compares to the plot of vertical ground reaction force in part (d).

(f) What are the assumptions made in modeling gait as an inverted pendulum? 

(g) We took a very simple approach to this model in this problem. (i.e., constant velocities and angular accelerations). Can you think of modifications of an inverted pendulum model so that it would more accurately represent aspects of normal locomotion? (If you’re really adventurous, try out some ideas and plot your results)