This is an extended work of IEEE ICRA 2020, “SL1M: Sparse L1-norm Minimization for contact planning on uneven terrain”:


One challenge of legged locomotion on uneven terrains is to deal with both the discrete problem of selecting a contact surface for each footstep and the continuous problem of placing each footstep on the selected surface. Consequently, footstep planning can be addressed with a Mixed Integer Program (MIP), an elegant but computationally-demanding method, which can make it unsuitable for online planning. We reformulate the MIP into a cardinality problem, then approximate it as a computationally efficient l1-norm minimisation, called SL1M. Moreover, we improve the performance and convergence of SL1M by combining it with a sampling-based root trajectory planner to prune irrelevant surface candidates.

Our tests on the humanoid Talos in four representative scenarios show that SL1M always converges faster than MIP. For scenarios when the combinatorial complexity is small (<10 surfaces per step), SL1M converges at least two times faster than MIP with no need for pruning. In more complex cases, SL1M converges up to 100 times faster than MIP with the help of pruning. Moreover, pruning can also improve the MIP computation time. The versatility of the framework is shown with additional tests on the quadruped robot ANYmal.

Experimental Results

The results show the dynamically consistent whole-body motions generated using an open-source whole-body motion generation framework based on our contact plans.

Humanoid, Talos (X3 speed)

bridge stairs

rubbles rns

Quadruped, ANYmal (X2 speed)



Latest version (10 Nov. 2020, arxiv), extension with pruning the irrelevant combinations using trajectory planning:

      title={Solving Footstep Planning as a Feasibility Problem using L1-norm Minimization},
      author={Daeun Song and Pierre Fernbach and Thomas Flayols and Andrea Del Prete and Nicolas Mansard and Steve Tonneau and Young J. Kim},

IEEE ICRA 2020, “SL1M: Sparse L1-norm Minimization for contact planning on uneven terrain”:

  author={S. {Tonneau} and D. {Song} and P. {Fernbach} and N. {Mansard} and M. {Taïx} and A. {Del Prete}},
  booktitle={2020 IEEE International Conference on Robotics and Automation (ICRA)},
  title={SL1M: Sparse L1-norm Minimization for contact planning on uneven terrain},


  • 1 Ewha Womans University, Korea,
  • 2 CNRS, LASS, Toulouse, France,
  • 3 University of Trento, Italy,
  • 4 University of Edinburgh, U.K.