  This paper studies the safety-critical control problem for Euler-Lagrange
(EL) systems subject to multiple ball obstacles and velocity constraints in
accordance with affordable velocity ranges. A key strategy is to exploit the
underlying inner-outer-loop structure for the design of a new cascade
controller for the class of EL systems. In particular, the outer-loop
controller is developed based on quadratic programming (QP) to avoid ball
obstacles and generate velocity reference signals fulfilling the velocity
limitation. Taking full advantage of the conservation-of-energy property, a
nonlinear velocity-tracking controller is designed to form the inner loop. One
major difficulty is caused by the possible non-Lipschitz continuity of the
standard QP algorithm when there are multiple constraints. To solve this
problem, we propose a refined QP algorithm with the feasible set reshaped by an
appropriately chosen positive basis such that the feasibility is retained while
the resulting outer-loop controller is locally Lipschitz. It is proved that the
constraint-satisfaction problem is solvable as long as the ball obstacles
satisfy a mild distance condition. The proposed design is validated by
numerical simulation and an experiment based on a $2$-link planar manipulator.
