  This paper presents the accessibility and small-time local controllability
(STLC) results for $N$-link horizontal planar manipulators with only one
unactuated joint. STLC is important in controls, both for design considerations
and because large and swinging maneuvers may be avoided for close
reconfiguration if a system is STLC. Despite the fact that controllability of
underactuated horizontal planar manipulators has been extensively studied, most
work focused only on three-link and global controllability. This paper thus has
two contributions: 1) using Lie brackets to study the accessibility and STLC
for underactuated two-link manipulators with different actuator configurations,
and illustrating the results from a perspective of system dynamics, 2)
obtaining the accessibility and STLC results for $N$-link manipulators with one
unactuated joint by considering realistic models and different actuator
configurations. It is found that an $N$-link ($N\geq 3$) with the first joint
actuated is STLC for a subset of equilibrium points based on Sussmann's general
theorem for STLC. Moreover, with the dynamics of $N$-link considered in the
controllability analysis, it gives relatively simple forms for the nontrivial
vector fields, which make it easy to determine at which configurations a model
loses full rank condition for accessibility.
