PRELIMINARIES Introduction Two Test Beds for Theory MODELING, ANALYSIS, AND CONTROL OF ROBOTS WITH PASSIVE POINT FEET Modeling of Planar Bipedal Robots with Point Feet Periodic Orbits and Poincare Return Maps Zero Dynamics of Bipedal Locomotion Systematic Design of Within-Stride Feedback Controllers for Walking Systematic Design of Event-Based Feedback Controllers for Walking Experimental Results for Walking Running with Point Feet WALKING WITH FEET Walking with Feet and Actuated Ankles Directly Controlling the Foot Rotation Indicator Point APPENDICES Appendix A: Getting Started Appendix B: Essential Technical Background Appendix C: Proofs and Technical Details Appendix D: Derivation of the Equations of Motion for Three-Dimensional Mechanisms Appendix E: Single Support Equations of Motion of RABBIT NOMENCLATURE END NOTES REFERENCES INDEX SUPPLEMENTAL INDICES

Motivated by the problem of controlling walking in a biped with series compliant actuation, this paper develops two main theorems relating to the stabilization of periodic orbits in systems with impulse effects. The first main result shows that when a periodic orbit of a system with impulse effects lies within a hybrid invariant manifold, there exist local coordinate transforms under which the Jacobian linearization of the Poincare return map has a block upper triangular structure. One diagonal block is the linearization of the system as restricted to the hybrid invariant manifold, also called the hybrid zero dynamics. The other is the product of two sensitivity matrices related to the transverse dynamics-one pertaining to the impact map and the other pertaining to the closed-loop vector field. When either of these sensitivity matrices is sufficiently close to zero, the stability of the return map is determined solely by the stability of the hybrid zero dynamics. The second main result of the paper details the construction of a hybrid invariant manifold, such as that required by the first main theorem. Forward invariance follows from the methods of Byrnes and Isidori, and impact invariance is achieved by a novel construction of impact-updated control parameters. In addition to providing impact invariance, the construction allows entries of the impact sensitivity matrix of the transverse dynamics to be made arbitrarily small. A simulation example is provided where stable walking is achieved in a 5-link biped with series compliant actuation.