Impedance control is a widely adopted approach that ensures the compliant behavior of robot manipulators as they interact with their environment according to specifically designed dynamics. For tasks involving six degrees of freedom (DoF), it is crucial to appropriately manage the position and orientation of the end-effector by controlling dynamic behavior. However, describing orientational displacement and designing the corresponding rotational impedance can be challenging, especially when we use a minimal representation. The well-known minimal representation for orientation, the Euler angle, suffers from representation singularity. As a remedy, the quaternion or dual quaternion can be an alternative, but with non-minimal representations. This lack of minimal representation, which does not suffer from the representation singularity, often leads to handling the impedance design by directly defining the potential energy function in the matrix Lie group. This paper proposes a framework for the six-DoF impedance control design that takes advantage of Lie group theory with minimal representation, known as the exponential coordinate. Since the exponential coordinate can be treated as the Euclidean variable within the injectivity radius, it allows for the formulation of the impedance control more systematically and familiarly. In our framework, a detour strategy is utilized; the impedance is designed in the Lie group$SE(3)$, and the control is designed in the Lie algebra$\mathfrak {se}(3)$, which is isomorphic to the vector space$\mathbb {R}^{6}$. The group structure of$SE(3)$can be maintained using the proposed conversion formula between the Lie group and the Lie algebra, called the differential of the exponential map and its time derivative, with a closed-form expression. Experiments with a 6-DoF robot manipulator verified that the proposed impedance control framework effectively reflects the$SE(3)$group structure and achieves the desired dynamic behavior as the functionality of the impedance control with minimal parameters.

State estimation problem with time delayed measurements is addressed. In dynamic system with noise, after taking measurements, it often requires some time until that is available in a filter. A filter not considering this time delay cannot be used since a current measurement is related with a past state. These delayed measurements problem is solved with augmented state Kalman filter, and uncertainty of the delayed time is also resolved based on the probability distribution of the delay. The proposed method is analyzed by a simple example, and its consistency is verified.

An online pattern generator for bipedal walking control is designed based on the notion of ZMP preview control using the full dynamics model. The method is called the general ZMP preview control. Conventional implementation by Kajita is based on an approximated table-cart model, which neglects the actual system dynamics. This leads to erroneous ZMP tracking for real robots, even if the system tracks the generated COM pattern exactly. Numerical simulations are provided to show the advantage and character of the method

In order for stable control of humanoid robots, ground contact forces should be properly controlled for compensating the dynamic disturbances caused by unactuated body movement. The stability in the sense of the zero-moment point (ZMP), guaranteeing secure contacts during control, is a necessary condition for stable motion control. Therefore, we propose a method to control the ground interaction at the ZMP, or ZMP interaction in short, by modifying the system reference acceleration. We also show that simultaneous control of the ZMP interaction and the body movements is not allowed in general. Simulation result is provided to corroborate the theoretical result.

Second-order contact kinematics for regular contacts, such as surface-surface, curve-curve, curve-surface, and vertex-surface, are formulated in a unified framework, extending Montana's (1988) first-order contact kinematics for surface-surface contact only. Given two rigid bodies in contact with each other, the contact kinematics is the mathematical expression of the relative motion between them and the five dimensional contact coordinate vector. Second-order contact kinematics describes the second-order derivative of the contact coordinate vector in terms of the derivative of the relative twist. The result can be directly applied to geometric integration of contact dynamics.

Numerical integration methods based on the Lie group theoretic geometrical approach are applied to articulated multibody systems with rigid body displacements, belonging to the special Euclidean group SE(3), as a part of generalized coordinates. Three Lie group integrators, the Crouch-Grossman method, commutator-free method, and Munthe-Kaas method, are formulated for the equations of motion of articulated multibody systems. The proposed methods provide singularity-free integration, unlike the Euler-angle method, while approximated solutions always evolve on the underlying manifold structure, unlike the quaternion method. In implementing the methods, the exact closed-form expression of the differential of the exponential map and its inverse on SE(3) are formulated in order to save computations for its approximation up to finite terms. Numerical simulation results validate and compare the methods by checking energy and momentum conservation at every integrated system state.

Based on the linear complementarity formulation of a unilaterally inextensible wire model, the behavior of motion and stability of wire-suspended dynamical systems is addressed in this paper. The concept of tension properness is developed to determine kinematical motion property, particularly the instantaneous degrees-of-freedom. Further, motion determinacy in spite of tension indeterminacy is proven. The notion of tension closure is described to help analyze the stability behavior of wire-suspended dynamical systems. By analyzing the maximal closure-domain of relative tension closure, one can assess the stability behavior of wire-suspended systems qualitatively as well as somewhat quantitatively. Some numerical examples and simulations will clarify and corroborate the theoretical results of this paper.

Numerical integration methods based on Lie group theoretic geometrical approaches are extended to articulated multi-body systems with rigid body displacements belonging to the special Euclidean group SE (3) as a part of generalized coordinate. Two Lie group integrators, Crouch-Grossman method and Munthe-Kaas method, are formulated for the equation of motion for articulated multi-body systems. The proposed methods provide singularity-free integration, unlike the Euler-angle method, while the integration always evolves on the underlying manifold structure, unlike the quarternion method. Numerical simulation result validates the methods by checking energy and momentum conservation at even integrated system state.

This paper addresses one of standard problems in enveloping grasp analysis, the so-called contact force distribution. First, a subspace structure is analyzed using projection matrices into four subspaces of contact force space. Motivated by an example demonstrating that a careless definition of contact force subspaces leads to completely invalid result, we propose a proper set of projection matrices using a special weighted pseudo-inverse. Next, an example of statical ambiguity is provided in obtaining contact force distribution, which implies that contact forces can not be determined in light of statical viewpoint. To resolve such ambiguity, we propose the notion of dynamical balance. By presenting a tool for identification of dynamically balanced motions and forces, we can have a complete characterization of the force distribution for a general enveloping grasp system.

In redundant manipulator research community, the phenomenon called torque instability has not yet been completely characterized using unanimous agreement on its mechanism. This article shows that the behavior is indeed caused by incomplete compensation of null dynamics related with self-motion. This instability can be avoided by proper control with compensation of nonlinear null dynamics, and we propose a method to stabilize conventional redundancy resolution schemes extended to dynamic controls.

Enveloping grasp systems not in static equilibrium result in dynamical motion. In this article, we develop a method based on dynamical-balance to solve motion analysis problem for an active-external grasp, that is to determine contact forces and accelerations consistent with the dynamics and friction law for a given torque-wrench pair. Since the method is analytic in nature, it may give the chance for better efficiency and more direct insight than a relevant linear complementarity formulation, although the solutions are the same. The method is verified using two examples: one is a simple grasp system with one frictional contact, and the other is a relatively general one with all frictionless contacts.

One of recent achievements in the field of nonlinear /spl Hscr//sub /spl infin// optimal control theories for Euler-Lagrange systems is the analytic solution to the Hamilton-Jacobi-Isaacs (HJI) equation associated to the so-called nonlinear /spl Hscr//sub /spl infin// inverse-optimal control. In this paper, we address the problem of nonlinear /spl Hscr//sub /spl infin// optimal control design for an Euler-Lagrange system, rather than the inverse-optimal problem. By introducing a technique of control weight loosening and state weight strengthening, we show that the associated HJI inequality, not the equation, for nonlinear /spl Hscr//sub /spl infin// optimal control can be solved also analytically using the inverse-optimal solution.

This paper proposes a method to accommodate multiple tasks for redundancy utilization, which is based on a specific weighted pseudo-inverse. The proposed method also has task priority imposition property same as those conventional task priority based methods. In order to deal with general situations of task specification, the so-called semi-definitely weighted pseudo-inverse is devised.

The enveloping grasp feasibility inequality is formulated in this article using a coordinate transformation based on the concept of frictional degrees-of-freedom. Various analysis and synthesis problems pertinent to enveloping grasp of whole-limb systems can be handled within the same mathematical framework, even under the situation where statically indeterminate contact forces exists. Simulations are also shown to explain the basic idea.

The problem due to the statically indeterminate contact forces arising in indeterminate frictional enveloping grasps is addressed. First, we show that the statical model for the contact forces is incomplete in the sense that the mathematical frictional forces may have an infeasible component. To resolve this infeasibility, we directly derive the enveloping grasp infeasibility condition on the frictional forces based on the coordinate transformation developed for frictional enveloping grasps. Then the indeterminate grasp can be analyzed in cooperation with the enveloping grasp feasibility inequality, as shown in a numerical example.

The kinematically decoupled joint space decomposition (KD-JSD) is proposed as a systematic method for synthesis of dynamic control of kinematically redundant manipulators. The method leads to a proper control for direct specification and control of the self-motion of kinematically redundant manipulators, as well as the task motion. To cover various possibilities in specifying desired self-motion behaviors the conventional gradient projection method and task priority based method are reformulated within the framework of the KD-JSD. The validity of the proposed control method is shown experimentally adopting a three degrees-of-freedom direct-drive planar redundant manipulator.

The success in nonlinear H/sub /spl infin// control design is applied to the control of Euler-Lagrange systems. It is known that the existence of H/sub /spl infin// optimal control depends on solvability of the so-called Hamilton-Jacobi-Isaacs H/sub /spl infin// partial differential equation. In the article, the associated HJI equation for nonlinear H/sub /spl infin// inverse-optimal control problem for a Euler-Lagrangian system is solved analytically. The resulting control is referred to as the reference error feedback, which takes conventional PID controller form. Consequently, robust motion control can be designed for robot manipulators using L/sub 2/-gain attenuation from exogenous disturbance and parametric error.

Industrial manipulators are under various limitations against high quality motion control, for example friction and dynamic uncertainties, and simple control structure. A robust linear PID motion controller, called the reference error feedback, is proposed, which solves the nonlinear L/sub 2/-gain attenuation control problem. Making use of the fact that the single parameter L/sub 2/-gain /spl gamma/ controls the performance and robustness, we propose a simple and stable method of tuning the controller called the "square law". The analytical results are verified through experiments of six degrees of freedom industrial manipulator.

The kinematically decoupled joint space decomposition is proposed as a general coordinate transformation between the joint space and the task space of kinematically redundant manipulators. The method can describe the kinematic, force, and dynamic redundancy in a unified way using a minimal number of systematic motion variables. Based on the method, a set of equivalent dynamical models of redundant manipulators is constructed and its characteristics are studied focusing on the dynamical decoupling between the task and the null motion dynamics.

Recent successes in nonlinear H/sub /spl infin// control design has been applied to control of robot manipulator systems. It is known that the existence of H/sub /spl infin// optimal control reduces to solvability of the Hamilton-Jacobi-Isaacs partial differential equation, which is very difficult to solve. In this article, a robust control in the sense of L/sub 2/-gain attenuation from an external disturbance and one due to model uncertainties is designed for a class of Euler-Lagrange system based on a class of analytic solution to the associated Hamilton-Jacobi-Isaacs equation.

A robust motion control with special focus on kinematically redundant manipulators is proposed based on the kinematically decoupled (KD) joint space decomposition and the analytic nonlinear /spl Hscr//sub /spl infin// control. Within the /spl Hscr//sub /spl infin// control design, robustness can be achieved by formulating the disturbance input to include model uncertainties and external disturbances as well. The proposed /spl Hscr//sub /spl infin// control is based on the analytic solution of the related Hamilton-Jacobi-Isaccs equation, and does not require any further assumptions except the Euler-Lagrange properties. Two components of motion, task and null motion, of a redundant manipulator are robustly controlled by the proposed KD reference motion compensation (RMC) reference error feedback (REF) control.

The kinematics and dynamics of kinematically redundant manipulators are to be decomposed in consistent way with the joint space decomposition induced by the weighted pseudo-inverse operation. The weighted kinematically decoupled joint space decomposition results to a family of decomposed kinematics and dynamics, parametrized by the symmetric positive-definite weight matrix, in the task motion and weighted null motion space. The inertially-decoupled weighted decomposition is shown to be only possible with the inertia as the weight. Also, the feedback motion controller which is consistent with a specific joint space decomposition is developed. The controller realizes the decoupled and linearized reference acceleration tracking controller in the task motion and weighted null motion space.

Design of a compliant motion controller using a redundant manipulator is addressed in this article. Specifically, a hybrid controller and an impedance controller are extended to general redundant manipulators based on the kinematically and geometrically decoupled joint space decomposition. Geometrically decoupled task space decomposition provides a systematic ways of geometric modeling of general constraints. Both controllers have the advantage that the closed-loop dynamics of the manipulator are decomposed into position-controlled, force-controlled, and null motion-controlled subspaces, and each are linearized, reshaped to desired dynamics, and decoupled from the others. A simple numerical simulations prove the applicability of the proposed algorithms.

Various physical limitations which intrinsically exist in the manipulator control system, for example kinematic limits and torque limit, cause some undesirable effects. Specifically, when one or more actuators are saturated in torque the expected control performance can not be achieved. The effect of torque limit, especially for redundant manipulators, is studied in this article, and a local and analytic method to reconstruct the control input using the redundancy is proposed based on the kinematically decoupled modeling of redundant manipulators. It results in exact compensation of the task motion dynamic error at the cost of the least null motion dynamic error. Numerical simulations help to verify the advantages of the proposed scheme.

Various physical limitations which intrinsically exist in the manipulator inverse kinematic system, for example joint travel and velocity limits, cause some undesirable effects. Specifically, when one or more joints are limited the desired task velocity can not be anticipated. The effect of joint kinematic limits, especially for redundant manipulators, is studied in this article, and an analytic method to reconstruct the inverse kinematic solution using the redundancy is proposed based on the kinematically decoupled modeling of redundant manipulators. It results in no error in the task motion at the cost of the least error in the null motion. Numerical simulations help to verify the advantages of the proposed scheme.

As a redundancy resolution method, the conventional extended Jacobian method (EJM) has two problems: One is the algorithmic singularity, and the second is nonexistence of a sufficient condition. Their impact on the inverse kinematic performances are exemplified. To remedy the related problems, we propose the notion of solution characteristics. It is based on an analytic sufficient condition for the EJM. This serves to confirm the right direction of optimization in the EJM and to characterize the algorithmic singularity problem. The above local characterization of an optimal solution by solution characteristics is globally extended by the invariance of the solution characteristics of the EJM. Specifically the characteristic of a solution with the EJM is invariant before crossing an algorithmic singularity. The ideas in this article also have practical implications since the EJM can now be fully analyzed with the theoretical results. To demonstrate its exactness and usefulness, planar three degrees of freedom (DOF) and spatial 4-DOF regional redundant manipulators are analyzed using the proposed solution characteristics.

Useful specification and stable control of motions for redundant manipulators are two important application aspects. In this article, the dynamic controller which can stabilize the output motion and the null motion is proposed, which has also a unified null motion specification algorithm in it. It is distinctive in that it renders the closed loop system dynamics linear and decoupled in the null space as well as the output space. All the work in this article are based on the kinematically decomposed modeling.

The characteristic of an inverse kinematic solution for redundant manipulators using EJM is exploited with the sufficient condition for EJM. A new analytic perspective other than the well-known algebraic and less-known geometric one on algorithmic singularity of EJM is proposed, on the basis of optimization process. The characteristic of a solution with EJM is shown to be able to change only by crossing the algorithmic singularity thereof. A 3-DOF planar redundant robot is analyzed when EJM is applied with the manipulability as an example.>