  The interplay between crystalline symmetry and band topology gives rise to
unprecedented lower-dimensional boundary states in higher-order topological
insulators (HOTIs). However, the measurement of the topological invariants of
HOTIs remains a significant challenge. Here, we define a {multipole winding
number} (MWN) for chiral-symmetric HOTIs by applying a corner twisted boundary
condition. The MWN, arising from both bulk and boundary states, accurately
captures the bulk-corner correspondence including boundary-obstructed
topological phases. To address the measurement challenge, we leverage the
perturbative nature of the corner twisted boundary condition and develop a
real-space approach for determining the MWN in both two-dimensional and
three-dimensional systems. The real-space formula provides an experimentally
viable strategy for directly probing the topology of chiral-symmetric HOTIs
through dynamical evolution. Our findings not only highlight the twisted
boundary condition as a powerful tool for investigating HOTIs, but also
establish a paradigm for exploring real-space formulas for the topological
invariants of HOTIs.
