Biped Model

Many of the essential characteristics of the human walking motion may be captured with a 5-link planar biped walking in the two-dimensional sagittal plane, the vertical plane bisecting the front of the biped. The model contains two links for each leg plus a large, massive torso, which also functions as the base of the tree-structured multibody system. Though the motion is constrained to the 2-dimensional vertical sagittal plane, in our experiments we model the links with a 3-dimensional elliptical shape and a uniform distribution of mass. The physical data corresponding to the model used in our experiments can be found in Table 1.

Table 1: Biped Model Physical Data

Though feet are not included in our biped model, much of their influence may be modeled in ways which do not increase the dimension of the system. From the control perspective, two of the main contributions of the feet, when not expressly considering friction, are the introduction of ankle torques and the liftoff force produced as the heel comes off of the ground. It is possible to include ankle torques in the model by treating these as external forces influencing the tips of each leg at the points of contact. Rather than modeling a liftoff force which lasts the entire duration of the double contact phase, as is normally the case with the foot, we model the liftoff force as an instantaneous impulsive force occurring at the moment of liftoff. This last technique has certain numerical advantages though it cannot completely reproduce the effect of the foot as will be shown in the reports of our numerical experiments.

There are a total of 14 states, 6 control variables, and 1 control parameter in our control problem if an impulse liftoff force is modeled.

   -   torso orientation and position in the vertical plane

  -                 		 torso angular and linear velocity

  ,                  		 angle position and velocity of leg 1 hip

  ,                 		 angle position and velocity of leg 1 knee

  ,                		 angle position and velocity of leg 2 hip

  ,                		 angle position and velocity of leg 2 knee

  ,                      		 applied torque at leg 1 hip and knee

  ,                      		 applied torque at leg 2 hip and knee

  ,                      		 applied torque at leg 1 & 2 ankles