Free-joint Manipulator Research Articles

ON HOMOCLINIC CHAOS AND A CONTROL CHAOS STRATEGY APPLIED TO A FREE-JOINT NONHOLONOMIC MANIPULATOR

In this work, the occurrence of chaos (homoclinic scene) is verified in a robotic system with two degrees of freedom by using Poincaré–Mel'nikov method. The studied problem was based on experimental results of a two-joint planar manipulator — first joint actuated and the second joint free — that resides in a horizontal plane. This is the simplest model of nonholonomic free-joint manipulators. The purpose of the present study is to verify analytically those results and to suggest a control strategy.

非ホロノミック自由関節マニピュレータの制御

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.

2P2-L10 重力ポテンシャル下における自由関節マニピュレータ(53. ノンホロノミックシステム)

2P2-L10 重力ポテンシャル下における自由関節マニピュレータ(53. ノンホロノミックシステム)

複数の拘束条件が課せられる最適軌道計画問題の一数値解法とその非ホロノミック系への適用

In this paper, we propose a novel gradient-type iterative algorithm for solving the optimal trajectory planning problems with multiple constraints. Since it is too difficult to generate trajectories considering all constraints at the same time, we introduce the concept of the order of priority into the trajectory generation procedure. In the proposed algorithm, a nominal trajectory is iteratively improved considering the order of priority of the constraints. The procedure is executed based on the gradient function which is synthesized in a hierarchical manner so that the convergence rate of the constraint with lower priority does not change the one with higher priority. The convergence properties of the proposed algorithm are examined. A method to improve the convergence properties when we apply the algorithm to the problem of nonholonomic systems is also proposed. The simulation results of planning trajectories for a 3-DOF planar free-joint manipulator show the effectiveness of the proposed algorithm.

非ホロノミック機械系の平均化法による挙動解析と振幅変調制御 平面２Ｒ自由関節マニピュレータの位置制御

Manipulators with free joints fall in a class of second-order nonholonomic systems whose dynamical constraints are nonintegrable. Control of such systems is one of major research topics in robotics and control engineering. In the literature, the authors analyzed nonlinear behaviors of a two joint manipulator with a free joint in response to a periodic input. The manipulator showed chaotic behaviors for larger input-amplitudes. A control method to position both joints via modulation of the input-amplitude was also proposed. Although the strategy was effective, it sufferes from heuristics and difficulty for generalization. In this paper, behaviors of the free-joint manipulator with a periodic input are approximated via the 1st-order and 2nd-order averaging methods. The invariant manifolds of the averaged behavior are analyzed and identified by defining a “pseudo-Hamiltonian” on an “input-amplitude normalized phase plane”. A control method to reach a desired invariant manifold via modulation of the input-amplitude is also proposed and a termination control at the destination is formulated. The effectiveness of the proposed control methods is verified by computer simulations.

Nonlinear behavior and control of a nonholonomic free-joint manipulator

The nonlinear motion of underactuated mechanisms has drawn recent interests of researchers. Underactuated mechanical systems are often subject to so-called nonholonomic constraints which are related to many both theoretical and practical issues. We first analyze the nonlinear behavior of a two-joint planar manipulator with the second joint free, from nonlinear dynamics point of view. We then discuss the simultaneous positioning of both joints. We use a time-periodic input and propose an amplitude modulation of the feedback error. The analysis via the Poincare map shows that the behavior becomes chaotic with large amplitude. The effectiveness of the proposed positioning control is verified by experiments.