  A hierarchical control architecture is presented for energy-efficient control
of legged robots subject to variety of linear/nonlinear inequality constraints
such as Coulomb friction cones, switching unilateral contacts, actuator
saturation limits, and yet minimizing the power losses in the joint actuators.
The control formulation can incorporate the nonlinear friction cone constraints
into the control without recourse to the common linear approximation of the
constraints or introduction of slack variables. A performance metric is
introduced that allows trading-off the multiple constraints when otherwise
finding an optimal solution is not feasible. Moreover, the projection-based
controller does not require the minimal-order dynamics model and hence allows
switching contacts that is particularly appealing for legged robots. The
fundamental properties of constrained inertia matrix derived are similar to
those of general inertia matrix of the system and subsequently these properties
are greatly exploited for control design purposes. The problem of task space
control with minimum (point-wise) power dissipation subject to all physical
constraints is transcribed into a quadratically constrained quadratic
programming (QCQP) that can be solved by barrier methods.
