M. Hardt 
K. Kreutz-Delgado 
J.W. Helton 
O. von Stryk
Zentrum Mathematik, Techn. Univ. München, D-80290 München, Germany
Dept. of Elec. Eng., UCSD, 9500 Gilman Dr., La Jolla, CA 92093-0407, U.S.A.
Dept. of Math., UCSD, 9500 Gilman Dr., La Jolla, CA 92093-0112, U.S.A.
We discuss our solution to the problem of generating symmetric, periodic gaits for a 5-link biped robot. We seek to approximate natural motion through the minimization of its injected energy. Our model stands out in that we consider the complete nonlinear dynamical model for the robot moving in the sagittal plane of forward motion. Both phases of walking, the swing and double-support phases, are explicitly modeled, including the contact and collision effects characteristic of each phase. A large number of constraints involving the contact forces, conditions for periodicity, and the range of motion must be considered which ensure the validity of the calculated motion. The solution of this complex problem was made possible through the use of various symbolic, dynamical algorithms relating to multibody systems in combination with powerful numerical optimal control software. Recent improvements in both areas have also further increased the potential to treat even more complex biped models. The symbolic nature of the recursive multibody algorithms, used for evaluating the dynamics and influences of contact and collision, facilitate any changes made to the number of limbs, points of actuation, or to the mass and inertia properties of the system. This flexibility allows one to readily treat many different cases such as the underactuated case when no ankle torques are allowed, and the introduction of impulsive forces for control purposes. The minimum energy walking problem becomes a path planning problem on a 14-dimensional state space with saturation and algebraic constraints on the state variables. The solution will satisfy a Hamilton-Jacobi-Bellman type equation along the optimal path. The optimal control software DIRCOL may solve such multi-phase problems with various forms of constraints, and it handles well the high degree of nonlinearity and dimensionality found in our problem. A newly available version of this software has also provided a substantial decrease in the required computing time for generating solutions. We discuss these and other improvements to our solution approach in this paper.