Motion planning for mobile manipulators with a non-holonomic constraint using the FSP (full space parameterization) method

The efficient utilization of the motion capabilities of mobile manipulators, i.e., manipulators mounted on mobile platforms, requires the resolution of the kinematically redundant system formed by the addition of the degrees of freedom (DOF) of the platform to those of the manipulator. At the velocity level, the linearized Jacobian equation for such a redundant system represents an underspecified system of algebraic equations, which can be subject to a varying set of contraints such as a non-holonomic constraint on the platform motion, obstacles in the workspace, and various limits on the joint motions. A method, which we named the Full Space Parameterization (FSP), has recently been developed to resolve such underspecified systems with constraints that may vary in time and in number during a single trajectory. In this article, we first review the principles of the FSP and give analytical solutions for constrained motion cases with a general optimization criterion for resolving the redundancy. We then focus on the solutions to (1) the problem introduced by the combined use of prismatic and revolute joints (a common occurrence in practical mobile manipulators), which makes the dimensions of the joint displacement vector components non-homogeneous, and (2) the treatment of a non-holonomic constraint on the platform motion. Sample implementations on several large-payload mobile manipulators with up to 11 DOF are discussed. Comparative trajectories involving combined motions of the platform and manipulator for problems with obstacle and joint limit constraints, and with non-holonomic contraints on the platform motions, are presented to illustrate the use and efficiency of the FSP approach in complex motion planning problems. © 1996 John Wiley & Sons, Inc.     

Spherically actuated platform manipulator

This article presents the inverse and forward pose and rate kinematics solutions for a novel 6‐DOF platform manipulator, actuated by two base‐mounted spherical actuators. The moving platform is connected to the fixed base by two identical spherical‐prismatic‐universal serial chain legs. The S‐joint is active, and the remaining two joints in each chain are passive. An analytical solution is presented for the inverse pose problems, a semi‐analytical solution is presented for the rate problems, and the numerical Newton–Raphson technique is employed to solve the forward pose problem. Unfortunately, the passive joint variables cannot be ignored in the kinematics solutions as they can for the Gough–Stewart platform. Examples are presented and hardware has been built, using two Rosheim Omni‐Wrists on loan from NASA as the spherical actuators. © 2001 John Wiley & Sons, Inc.     

Analytical Solution of the Forward Position Analysis of Parallel Manipulators That Generate 3-RS Structures

In this work the forward position analysis (FPA) of 3-RS structures (R, S, P and C = revolute, spherical, prismatic and cylindrical, respectively) is carried out applying recursively the Sylvester dialytic elimination method. The analytical solution provided in this contribution yields 16 possible poses of the moving platform given the limb lengths of the manipulator and it is applicable to a wide class of parallel manipulators, e.g., the 3-RP*S mechanism, 3-CP*S mechanism, 6–3 Gough–Stewart platforms and 3-RR*S mechanism. A case study is included which consists of solving the FPA of a 3–3 Gough–Stewart platform, also known as an octahedrical mechanism.     

Nonlinear Control of a Robot Manipulator with a Nonholonomic Jerk Constraint

We study the control of a prismatic‐prismatic‐revolute (PPR) robot manipulator subject to a nonholonomic jerk constraint, i.e., a third‐order nonintegrable design constraint. The mathematical model is obtained using the method of Lagrange multipliers. The control inputs are two forces and a torque applied to the prismatic joints and the revolute joint, respectively. The control objective is to control the robot end‐effector movement while keeping the transverse jerk component as zero. The main result of the paper is the construction of a feedback control algorithm that transfers the manipulator from any initial equilibrium configuration to the zero equilibrium configuration in finite time. The effectiveness of the algorithm is illustrated through a simulation example.     

Simulation of Industrial Manipulators Based on the UDU T Decomposition of Inertia Matrix

The UDU ^T – U and D are respectively the upper triangular and diagonal matrices – decomposition of the generalized inertia matrix of an n-link serial manipulator, introduced elsewhere, is used here for the simulation of industrial manipulators which are mainly of serial type. The decomposition is based on the application of the Gaussian elimination rules to the recursive expressions of the elements of the inertia matrix that are obtained using the Decoupled Natural Orthogonal Complement matrices. The decomposition resulted in an efficient order n, i.e., O(n), recursive forward dynamics algorithm that calculates the joint accelerations. These accelerations are then integrated numerically to perform simulation. Using this methodology, a computer algorithm for the simulation of any n degrees of freedom (DOF) industrial manipulator comprising of revolute and/or prismatic joints is developed. As illustrations, simulation results of three manipulators, namely, a three-DOF planar manipulator, and the six-DOF Stanford arm and PUMA robot, are reported in this paper.     

Improving the absolute positioning accuracy of robot manipulators

The absolute positioning accuracy of robot manipulator can be increased substantially by updating the nominal link parameters in the control software. This paper presents a general method to estimate the link parameter errors for any serial link manipulator (i.e., n links, and any combination of revolute and prismatic joints). The parameters are estimated through a linear kinematic model which relates the link bias errors to the end-effector positioning error. Only end-effector measurements are required instead of individual link measurements to implement this method.     

Forward displacement analysis and uncertainty configurations of parallel manipulators with a redundant branch

The effect of adding a redundant branch in terms of reduction of the number of assembly modes and elimination of potential uncertainty configuration types is investigated for a class of parallel manipulators. Considered is a broad class that includes all three-branch manipulators where each branch is comprised of a serial arrangement of three main-arm joints supporting a common payload platform through a passive spherical branch end joint-group. The addition of a redundant branch effectively yields a four-branch manipulator class. Considered in particular is a 3–4 form of the manipulator where two branch ends meet at one point on the mobile platform. Symmetric main-arm joint sensing and actuation (two sensed/acutated main-arm joints per branch) is utilized. Synthetic geometry is used to study the number of assembly configurations of the resulting 3–4 four-branch parallel manipulators. It is presented that the number of assembly modes of three-branch parallel manipulators with passive spherical branch end joints can be reduced by utilizing a redundant branch. It is shown that there exist up to eight and up to four assembly modes when all unsensed joints are revolute and when all unsensed joints are prismatic, respectively. Combinations of unsensed prismatic and revolute joints are also investigated. It is determined that there are up to eight and up to four assembly modes when the unsensed main-arm joint of one of the concurrent branches is prismatic and when the unsensed joints of both concurrent branches are prismatic joints, respectively. Resolving the potential assembly modes require only the consideration of, at highest, second-order single-variable polynomials. In addition, kinematic design considerations allowing reduction of feasible assembly modes are discussed. The investigation of potential uncertainty configuration types is based on examining degeneracies of the screw systems formed by wrenches associated with the forces that the actuated-joints can apply. All linear dependency cases that could potentially cause uncertainties for the class of four-branch manipulators are identified. It is shown that while significantly reducing potential uncertainty configuration types, the addition of a redundant branch number cannot eliminate all potential dependency (uncertainty) cases completely. For the remaining potential uncertainty configuration types, the characteristics of the corresponding unconstrained instantaneous degrees of freedom are discussed. © 1996 John Wiley & Sons, Inc.     

Type synthesis of 5‐DOF parallel manipulators based on screw theory

With the introduction of virtual chains to represent the motion patterns of 5‐DOF motions, a classification of 5‐DOF PMs (parallel manipulators) is proposed at first. A general method for the type synthesis of 5‐DOF PMs is then proposed based on screw theory and using the concept of virtual chains. The type synthesis of US‐equivalent PMs is presented in detail to show the application of the proposed approach. US‐equivalent PMs are the parallel counterparts of the 5‐DOF US serial manipulators. For a US‐equivalent PM, the moving platform can rotate arbitrarily about a point moving along a spherical surface. The type synthesis of legs for US‐equivalent PKCs (parallel kinematic chains), the type synthesis of US‐equivalent PKCs, as well as the selection of actuated joints of US‐equivalent PMs are dealt with in sequence. US‐equivalent PKCs with and without inactive joints are synthesized. Several US‐equivalent PMs as well as other classes of 5‐DOF PMs with identical type of legs are obtained. © 2005 Wiley Periodicals, Inc.     

A Family of Symmetrical Lower‐Mobility Parallel Mechanisms with Spherical and Parallel Subchains

This paper presents a new family of symmetrical lower‐mobility parallel mechanisms (PMs) with spherical and parallel subchains, which consists of two 5‐DOF (degrees of freedom) PMs, one 4‐DOF PM and five 3‐DOF PMs. The basic feature of this family is that each limb consists of five revolute pairs and can be constructed with two subchains, a 2R pointing subchain and a 3R parallel subchain, or a 3R spherical subchain and a 2R parallel subchain. Different geometrical arrangements of the limbs lead to different degrees of freedom. All the PMs of this family can be modularized easily due to the simple structure of the subchains. © 2003 Wiley Periodicals, Inc.     

A new method for the geometric modeling of lower pairs and its application to the kinematic spaces of spatial robots

A new notation is developed to model and analyze spatial robots. This method is formulated based on the homogeneous cylindrical coordinates and Bryant angles transformation matrices and thus termed “C-B notation.” At first, shape matrix for a binary link is derived. Then, the characteristic matrices for the five most commonly used kinematic lower pairs—revolute (R), prismatic (P), cylindrical (C), helical (H), and spherical (S)—are formulated. The “exact” joint positions, that is, the actual location of the physical joint center in space, can be defined using this method. The governing kinematic equations for the analysis of spatial robots are derived as well. Finally, the “kinematic spaces” for the open-chain robot are defined as “the group of space of work (displacement) space, velocity space (the velocity envelope in the displacement differential V_x-V_y-V_z coordinates), and acceleration space (the acceleration envelope in the velocity differential A_x-A_y-A_z coordinates).” Numerical examples for parallel projections of kinematic spaces on both sagittal and frontal planes are illustrated by the industrial Unimate 2000 spherical (SP/RRR) robot, Bendix AA/CNC (RRP/RRR) robot, and 6R Cincinnati Milacron T3 robot.     

Dynamic optimization of N‐joint robotic limb deployments

We describe an approach using a stochastic optimization framework (SOF) for operating complex mobile systems with several degrees of freedom (DOFs), such as robotic limbs with N joints, in environments that can contain obstacles. As part of the SOF, we have employed an efficient simulated annealing algorithm normally used in computationally highly expensive optimization and search problems such as the traveling salesman problem and protein design. This algorithm is particularly suited to run onboard industrial robots, robots in telemedicine, remote spacecraft, planetary landers, and rovers, i.e., robotic platforms with limited computational capabilities. The robotic limb deployment optimization approach presented here offers an alternative to time‐intensive robotic arm deployment path planning algorithms in general and in particular for robotic limb systems in which closed‐form solutions do not exist. Application examples for a (N = 4)‐DOF arm on a planetary exploration rover are presented. © 2009 Wiley Periodicals, Inc.     

Analytical Identification of Limb Structures for Translational Parallel Manipulators

A systematic analytical method, based on the theory of screws, is presented for identification of limb structures of general and over‐constrained 3‐degree‐of‐freedom (DOF) translational parallel manipulators. Given a system of wrenches of constraint, the corresponding reciprocal basis screws are determined. Then, the joint screws of a limb are obtained by a linear combination of these basis screws. Feasible limbs that can be used for construction of translational platforms are enumerated according to the type of constraint and the number of joints making up the limbs. © 2004 Wiley Periodicals, Inc.     

OPTIMAL MECHANISM DESIGN AND DYNAMIC ANALYSIS OF A 3‐LEG 6‐DOF LINEAR MOTOR BASED PARALLEL MANIPULATOR

This paper presents the optimal mechanism design and dynamic analysis of a prototype 3‐leg 6‐DOF (degree‐of‐freedom) parallel manipulator. Inverse kinematics, forward kinematics, inverse dynamics and working space characterizing the platform motion are derived. In the presented architecture, the base platform has three linear slideways individually actuated by a synchronous linear servo motor, and each extensible vertical link connecting the upper and base platforms is actuated by an inductive AC servo motor. The linear motors contribute high‐speed movements to the upper platform. This kind of architecture using hybrid (linear and AC) motors yields high level performance of motions, especially in the working space. The novel result of maximal working angles is the significant contribution of this architecture. The Taguchi Experimental Method is applied to design the optimal mechanism of the platform system, and the result is used as the actual data to build this system.     

Type synthesis,unified kinematic analysis and prototype validation of a family of Exechon inspired parallel mechanisms for 5-axis hybrid kinematic machine tools

To fully disclose potential configurations of parallel mechanisms (PMs) that are performance-comparable to the commercially successful Exechon, a family of one translational two rotational (1T2R) over-constrained Exechon inspired PMs is synthesized through the screw theory. The synthesized PMs are further comparatively analyzed in terms of inverse kinematics, singularity occurrence and reachable workspace based on a unified kinematic model. The kinematic analyses indicate that the inherited overconstraints benefit for eliminating the constraint singularity of all synthesized Exechon inspired PMs and the workspace distribution of an Exechon inspired PM is closely related to its topological arrangement. Based on the kinematic evaluations, a preferred configuration of 2UPR-1RPS PM (‘R’: revolute joint, ‘U’: universal joint, ‘S’: spherical joint, ‘P’: prismatic actuated joint) is selected as a candidate for 1T2R spindle head. A laboratory prototype is fabricated and experimentally tested to verify the feasibility of the type synthesis and the correctness of the kinematic analyses. By integrating the selected PM with a 2-degree of freedom sliding gantry, a full scale prototype of a novel hybrid kinematic machine tool is developed to perform 5-axis machining tasks. The well match between the machined workpiece and its designed shape fully proves the engineering potential of the synthesized PMs that are expected to be employed as the functional module to construct 5-axis serial-parallel hybrid machining devices.     

Inverse Velocity and Singularity Analysis of Low‐DOF Serial Manipulators

This paper presents a method for exact inverse velocity analysis of low‐DOF (degrees of freedom n<6) serial manipulators. For a low‐DOF serial manipulator, the number of independently controllable variables in the Cartesian space is equal to the number of joint variables in the joint space, and the remaining 6?n variables are linearly dependent on these independent variables. This paper employs the theory of reciprocal screws to determine a mapping between the independent velocity components in the Cartesian space and the joint rates in the joint space. It is shown that singular conditions of a low‐DOF manipulator depend on choice of independent variables. A 5‐DOF and a 4‐DOF manipulator are analyzed, and a numerical example in which the end effector of a 4‐DOF manipulator is commanded to follow a straight line is used to demonstrate the methodology. © 2003 Wiley Periodicals, Inc.     

Numerical inverse kinematics for modular reconfigurable robots

The inverse kinematics solutions of a reconfigurable robot system built upon a collection of standardized components is difficult to obtain because of its varying configurations. This article addresses the formulation of a generic numerical inverse kinematics model and automatic generation of the model for arbitrary robot geometry including serial and tree‐typed geometries. Both revolute and prismatic types of joints are considered. The inverse kinematics is obtained through the differential kinematics equations based on the product‐of‐exponential (POE) formulas. The Newton–Raphson iteration method is employed for solution. The automated model generation is accomplished by using the kinematic graph representation of a modular robot assembly configuration and the related accessibility matrix and path matrix. Examples of the inverse kinematics solutions for different types of modular robots are given to demonstrate the applicability and effectiveness of the proposed algorithm. ©1999 John Wiley & Sons, Inc.     

CeDAR: A real-world vision system

We report on the development of a multi-purpose active visual sensor system for real-world application. The Cable-Drive Active-Vision Robot, CeDAR, has been designed for use on a diverse range of platforms to perform a diverse range of tasks. The novel, biologically inspired design has evolved from a systems-based approach. The mechanism is compact and lightweight and is capable of motions that exceed human visual performance and earlier mechanical designs. The control system complements the mechanical design to implement the basic visual behaviours of fixation, smooth pursuit and saccade, with stability during high-speed motions, high precision and repeatability. Real-time algorithms have been developed that process stereo colour images, resulting in a suite of basic visual competencies. We have developed a scheme to fuse the results of the visual algorithms into robust task-oriented behaviours by adopting a statistical framework. CeDAR has been successfully used for experiments in autonomous vehicle guidance, object tracking and visual sensing for mobile robot experiments.Published online: 25 October 2004 Correspondence to: A. Zelinsky     

Control of Discrete‐Time Systems Composed of Linear Blocks in Series with Saturation Components

We consider the problem of controlling cascade systems consisting of two linear dynamic blocks and two saturation elements arranged according to the N‐L‐N‐L series configuration. A cascade controller is considered and its performances are formally analyzed using input–output stability tools. In addition to global boundedness of all signals of the closed‐loop system, the controller is formally shown to enjoy a l₂ ‐tracking performance in presence of arbitrary‐shape inputs (i.e. reference signal, disturbance).     

Translational Parallel Manipulators: New Proposals

Translational parallel manipulators are parallel manipulators wherein the end‐effector performs only spatial translations. This paper presents a new family of translational parallel manipulators. The manipulators of this family are independent constraint manipulators. They have three limbs that are topologically identical and have no rotation singularity. The limbs of these manipulators feature five one‐degree‐of‐freedom kinematic pairs in series. Four joints are revolute pairs and the remaining one, called T‐pair, is a kinematic pair that can be manufactured in different ways. In each limb, three adjacent revolute pairs have parallel axes and the remaining revolute pair has an axis that is not parallel to the axes of the other revolute pairs. The mobility analysis of the manipulators of this new family is addressed by taking into account two different choices for the actuated pairs. One of the results of this analysis is that the geometry of a translational parallel manipulator free from singularities can be defined for a particular choice of the actuated pairs. © 2002 Wiley Periodicals, Inc.