Generating optimal walking and running gaits for biped robot locomotion is an interesting problem.  This group has conducted extensive computer simulations to investigate optimal gaits. Links to some of our publications are listed below.

Macnab Publications  

Dadashzadeh, B., M. Esmaeili, and C. J.B. Macnab. Arbitrary symmetric running gait generation for an underactuated biped model.
     PloS one
     DOI: 10.1371/ journal.pone.0170122, 2017.
 
Hasaneini, S. J.,  John E. A. Bertram and Chris J. B. Macnab. Energy-optimal relative timing of stance-leg push-off and swing-leg retraction in walking.
    Robotica
    Vol. 35, No. 4,  pp. 654-686, 2017.
pape

Dadashzadehab, B.,  M.J. Mahjoobb, M. Nikkhah Bahramib and C.J.B. Macnab. Stable active running of a planar biped robot using Poincare map control
    Advanced Robotics
    Vol. 28, No. 4, pp. 231-244, 2014.
 
Hasaneini, J., C.J.B. Macnab, J. Bertram and H. Leung.  Swing-leg retraction efficiency in bipedal walking
   IEEE/RSJ International Conference on Intelligent Robots and Systems
   (Chicago) pp.  2515 - 2522, 2014.

Dadashzadeh, B.,  M.J. Mahjoob, M. Nikkhah Bahrami, and C.J.B. Macnab. Compliant leg architectures and a linear control strategy for the stable running of planar biped robots
    International Journal of Advanced Robotic Systems
    Vol. 10, DOI: 10.5772/56806, 2013.

Hasaneini, J., C.J.B. Macnab, J.E.A. Bertram, and H. Leung. The dynamic optimization approach to locomotion dynamics: human-like gaits from a minimally-constrained biped model
    Advanced Robotics
    The dynamic optimization approach to locomotion dynamics: human-like gaits from a minimally-constrained biped model Vol. 27, No. 11, pp. 845–859, 2013.

Hasaneini, S.J. C.J.B. Macnab, J.E.A. Bertram,  H. Leung.  Optimal relative timing of stance push-off and swing leg retraction.
    IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
    (Tokyo, Japan) pp. 3616 - 3623, 2013.

Hasaneini, J., C.J.B. Macnab, H. Leung, and J. Bertram. Seven reasons to brake the swing leg just before heel strike.
    Dynamic Walking
    (Pittsburgh) June, 2013.

Guo, Q, C.J.B. Macnab, and J.K. Pieper. Robust control of a rigid articulated hopper.
    International Journal of Robotics and Automation
    Vol. 27, No. 1, pp. 1-14,  2012

Hasaneini, J., C.J.B. Macnab, H. Leung, and J. Bertram. Dynamic optimization clearly shows the determinants of bipedal gaits: reduced gravity predictions and verication.
   Dynamic Walking,   
   (Pensacola, Florida) May 2012.

Guo, Q.,  C.J.B. Macnab, and J.K. Pieper. Generating efficient rigid biped running gaits with calculated take-off velocities.
     Robotica,
    Vol. 29, No. 04, pp. 627-640, 2011

Hasaneini, S.J., C.J.B.  Macnab, J.E.A.  Bertram, JEA, and H. Leung, Gravity effects on energetics and kinematics of walking and running using dynamic optimization.
    Dynamic Walking,
    (Jena, Germany) pp. 101-102, July 2011.


Hasaneini, J., C.J.B. Macnab, H. Leung, and J. Bertram. Global optimum human-like gaits for an articulated one-legged hopper.
    Proc. IEEE/ASME Int. Conf. Advanced Intelligent Mechatronics,
    (Montreal), pp. 391-396, July, 2010.


Guo, Q., C.J.B. Macnab, and J.K. Pieper,  Hopping on even ground and up stairs with a single articulated leg.
    International Journal of Intelligent and Robotic Systems
    Vol. 53 (4), pp. 331-358, 2008

Guo, Q., C.J.B. Macnab, and J.K. Pieper,  Hopping with nearly-passive flight phases.
    Proc. IEEE Int. Confs. on Cybernetics and Intelligent Systems and Robotics, Automation and Mechatronics
   (Chengdu, China), pp. 743-748, 2008

Guo, Q., C.J.B. Macnab, and J.K. Pieper,  Bipedal running with nearly-passive flight phases.
    Proc. IEEE Int. Conf. on Advanced Intelligent Mechatronics
   (Xi'an, China), pp. 564 - 569, 2008.