Abstract
Manipulators with free joints are second-order nonholonomic systems whose dynamical constraints are noninte-grable. Such systems are known to be difficult to control and are underactuated systems which have possibility to steer many joints by only one motor. Previously proposed methods to control free-joint manipulators have been based on an assumption of perfectly frictionless free joints.In this paper, 2R and 3R free joint manipulators with only one actuated joints are to be studied with consideration of friction at free joints. Averaging analysis clarifies the frictionless free-joint manipulators are Hamiltonian systems with a conservation of an energy-like quantity. Various models of friction at the free joint are considered for 2R free-joint manipulators and the averaging analysis and simulations reveal that the systems with friction show dissipative behaviors. For 3R free-joint manipulators, the Poincare map of their frictionless behaviors in response to periodic inputs show torus-like invariant manifolds in 4D phase space. From experiments and simulations, 3R free joint manipulators with friction show the behaviors converging to an equilibrium point.Considering these dissipative behaviors with friction, a method using the energy-like quantity to stabilize to the equilibrium point is proposed for 3R free joint manipulators. Several control methods to position to any destination via stabilization to desired invariant manifold are proposed for 2R free-joint manipulators and simulations and experiments show their effectiveness.
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