Legged robots have many potential applications

in real-world scenarios where the tasks are too dangerous for

humans, and compliance is needed to protect the system against

external disturbances and impacts. In this paper, we propose a

model-based controller for hierarchical tasks of legged systems

subject to external disturbance. The control framework is

based on projected inverse dynamics controller, such that the

control law is decomposed into two orthogonal subspaces,

i.e., the constrained and the unconstrained subspaces. The

unconstrained component controls multiple desired tasks with

impedance responses. The constrained space controller maintains

the contact subject to unknown external disturbances,

without the use of any force/torque sensing at the contact

points. By explicitly modelling the external force, our controller

is robust to external disturbances and errors arising from

incorrect dynamic model information. The main contributions

of this paper include (1) incorporating an impedance controller

to control external disturbances and allow impedance shaping

to adjust the behaviour of the motion under external disturbances,

(2) optimising contact forces within the constrained

subspace that also takes into account the external disturbances

without using force/torque sensors at the contact locations. The

techniques are evaluated on the ANYmal quadruped platform

under a variety of scenarios.