Recent Advances in Robot Kinematics

This paper presents a method for solving the direct kinematics of general 6–5 Stewart-Gough platform mechanisms. The method relies on elimination techniques to reduce a system of equations to a single univariate polynomial. It is shown that, once trivial solutions are eliminated, this polynomial is of degree 40. The resulting computations are simpler than those currently required by the analogous algorithmic method for the general 6–6 configuration.

The kinematic sensitivity of a manipulator is the effect that a ‘small’ change in the required task produces on the posture of the manipulator. Redundant manipulators being capable of performing both a primary and a secondary task admit, correspondingly, two kinematic sensitivities. We refer to them as the primary and the secondary sensitivities, each being associated with a sensitivity matrix. The primary sensitivity matrix turns out to be the partial derivative of the joint rates with respect to the twist of the end-effector, the secondary sensitivity matrix being defined as the partial derivative of the joint rates with respect to the secondary task. It is shown that the primary sensitivity can be minimized by design, while the secondary sensitivity is constant.

This article investigates the design of redundant mechanisms for improved dynamic performance. Our study has resulted in a new methodology which allows the decomposition of the design problem into smaller, more manageable problems. With this method, a redundant mechanism is subdivided into a chain of non-redundant subsystems which can be designed separately. The result is a highly efficient design process.

The goal of this work is to report an experimental study of second-order kinematic control of robot manipulators with robustness to the occurrence of kinematic singularities. A damped least-squares inverse of the Jacobian is used within a closed-loop inverse kinematics algorithm to compute joint accelerations, as well as velocities and positions by integration. These are input to a joint space computed torque control. Results are presented for a six-joint industrial robot with open control architecture.

The present paper treats the problems of modeling and control of manipulators with elastic joints in terms of vector-parametrization of the SO(3) group. Since the computational effectiveness of the vector-parameter approach increases with the increasing number of the revolute degrees of freedom, this fact is successfully used in the problems of elastic joint manipulators, where except the real n links, n fictious links are included as well as additional n revolute degrees of freedom. A dynamic model in “pure” vector-parameter form is developed and the inverse dynamic control is discussed.

In this paper, the motion of a spatial two-arm robot manipulator undergoing grasping and lifting a three dimensonal object is investigated. The manipulator has rigid links. The joints are modeled as flexible, continuous shafts. The object is travelling on a conveyor. A mechanical model is derived for the problem. The effects of grasping, when the end effector of the manipulator approaches, suddenly grasps and lifts are analysed. Simulation results are presented.

This work 1 presents a suitable mathematical formulation of robot and obstacles such that for on-line collision recognition only robot joint positions in the workspace are required. This reduces calculation time essentially, because joint positions in workspace can be computed at every time from the joint variables through robot geometry. It is supposed that the obstacles in the workspace of the manipulator are represented by convex polygons in 2D. The recognition of collisions of the links of the manipulator with obstacles results on-line through a nonsensory method. For every link of the redundant robot and every obstacle a boundary ellipse in 2D is defined in workspace such that there is no collision if the robot joints are outside these ellipses. In addition to this, two collision avoidance methods are presented which allow the use of redundant degrees of freedom such, that a manipulator can avoid obstacles in workspace while tracking the desired end-effector trajectory. Moreover, the new methods are applied directly in workspace. This paper gives also a comparison of these two methods. The effectiveness of the porposed methods is discussed by theoretical considerations and illustrated by simulation of the motion of a planar four-link manipulator between obstacles.

The paper presents a reliable method of bringing any nonredundant serial robot out of a singular configuration the shortest way. The method implies the combination of motion performed in direction of gradient (or anti-gradient) of Jacobian or Gramian with approximation of the twist of the target end-effector motion.

The objective of the project described in this paper is to provide a motion base for a low-cost driving simulator which is a research topic in the development of an automated vehicle/highway system. The motion platform is driven by a parallel mechanism with three degrees of freedom. The kinematics for the position problem of this type of mechanism was quite well developed. The velocity problem, on which there is a focus in the paper, is less stressed by other researchers. A closed form of Jacobian which could be computed efficiently by feedback data from sensors is presented. Static force analysis is carried out before the velocity problem is solved. Prom the result, the singular positions of the structure, which constrain the range of motions, are located and described in the paper. The design of a table-top model, which was built to assist the development of control software, is also briefly described in the paper.

This paper presents a coordinate-invariant differential geometric formulation of manipulability for closed kinematic chains containing active and passive joints. The formulation treats both redundant and nonredun-dant mechanisms, as well as over actuated and exactly actuated ones, in a uniform manner. We illustrate the formulation with several planar closed chain examples.

This paper investigates the global kinematics of generic 3-R regional manipulators through singularity analysis. Generic manipulators are those manipulators whose singularities form smooth manifolds in the jointspace. Starting from a preliminary homotopy based classification scheme recently proposed in the literature, the complete categorisation of all generic regional manipulators is established. It is shown that there are only eight classes of homotopic generic manipulators. Several illustrative examples are provided.

In the paper we study the Cartesian stiffness matrix using methods of differential geometry. We show that the stiffness of a conservative mechanical system is described by a (₂0) tensor and that components of the Cartesian stiffness matrix are given by evaluating this tensor on a pair of basis twists. Our formulation leads to three important results: (a) The stiffness matrix does not depend on the parameterization of the manifold; (b) The stiffness matrix depends on the choice of a connection on the manifold; and (c) The standard definition of the Cartesian stiffness matrix assumes an asymmetric connection and this is the reason that the matrix is, in general, asymmetric.

This paper describes how a stability analysis for compliant mechanisms may be performed using catastrophe theory. The philosophy is demonstrated by an analysis of the stability of the planar two-spring and spatial three-spring systems. Catastrophe locus plots are used to predict where a change in stability will occur. The approach presented yields a stability analysis which does not require an inverse analysis of the mechanism.

It is shown that the acceleration analysis of closed chains provide a more accurate characterization of their singular positions, than that obtained by the velocity analysis only. Further, recent results have shown that the acceleration analysis of closed chains can be formulated very compactly via screw theory. Hence, it is possible to formulate a new methodology for the characterization of singular positions of closed chains. The authors believe that the approach expounded in this paper has some advantages over previously known methods.

This paper discusses some special applications of the theory of rational b-spline motions to the computer aided design of robot trajectories. The main advantage of this approach is the data conformity to state of the art CAD systems which allows straightforward data transfer from CAM to CAD systems. In particular the paper focusses on applications in which the motion of the end effector is implicitly defined by task level considerations. Such applications include for example arc welding, spray painting and the scanning of surfaces with robot equipment.

The problem of optimum design of closed kinematic chains with one or more degrees of freedom and obstacles in the workspace is considered. An optimum design method with a penalty function embedding the collision control algorithm is developed. Several numerical examples which show the effectiveness of the proposed method are presented.

Discrete actuators have a finite number of states, e.g. stepper motors and bistable pneumatic cylinders. Given that the number of distinct configurations attainable by discretely actuated manipulators grows exponentially in the number of actuated degrees of freedom, i.e. Kn for n actuators or actuated modules each with K states, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. Approximating the workspaces of segments of discrete manipulators as density functions on the Euclidean group (which describes workspace positions and orientations) the whole workspace can be approximated as an n-fold convolution of these functions. A generalization of the Fourier transform (and the convolution theorem) is used in this paper to efficiently compute convolutions and to solve the inverse problem of synthesizing discrete manipulators which have prescribed workspace properties.

In this paper a method is presented, for estimating the best location, from dexterity point of view, of a path following task into the workspace of an industrial manipulator. The best location of the path is defined as the location with the higher mean value of the manipulability along the path. The optimisation problem is formulated in a procedural way and solved using Genetic Algorithms (GA). The efficiency of the method is demonstrated through experiments carried out on a planar manipulator and on a PUMA robot in a simulation environment.

In this paper the concept of feasible workspace regions for prescribing workspace in a two-revolute manipulator design has been proposed to determine the limits of arbitrarily given workspace points. A design formulation has been used to deduce analytical procedures for individuating the feasible regions as regions within which feasible solutions for the manipulator design can be obtained. Particularly, the feasible workspace regions have been geometrically characterised and specific topologies have been shown for prescribing hole existence in the workspace in order to demonstrate the feasibility of the procedure with respect to additional design constraints.

The paper treats the workspace of a 4R manipulator. The emphasis is given to the existence domain of the basic case, the classification of the particular cases, the classification of the envelope branches and geometry of the workspace.

This paper deals with the functional dependence of closure equations on unknown pair variables. It is shown that the problem can be studied by considering a small set of metric relations obtained from displacement group theory. Symbolic expressions for such relations are provided and their systematic analysis is performed in order to determine the algebraic and geometric conditions of the functional dependence. Obtained results are also interpreted as relations among invariant geometric properties of displacement groups.

Tongue position, which shapes the vocal airway, is a key link in the chain between the intention to speak and production of an acoustic wave. Scientific study of how the brain controls tongue shape to produce speech has motivated the collection of point-parameterized measurements of the tongue with a computer-controlled x-ray tracking system. The sparse nature of these data, necessitated by exposure limitation, makes interpretation challenging. Curve fitting methods developed from work in robot motion planning can 1) visualize tongue shape, 2) reduce the number of kinematic parameters representing tongue shape, and 3) map tongue position relative to the roof of the mouth in an acoustically relevant manner. These new tools for linguistic data visualization and data reduction are applicable to the scientific description of a large (n=60) database of speakers and the synthesis of speech from an articulatory model. (Work supported by USPHS grant DC-00820.)

The paper presents a new and efficient method for symbolical computation of the Jacobian matrix available for any general redundant or non-redundant n degrees-of-mobility manipulator. Compared with the existing methods the number of required mathematical operations is considerably smaller, making the method remarkably more efficient than the previous methods.

Force-torque analysis of open serial kinematic structures is considered in this work. The close relation between vector and pure quaternion representation of force and torque balance equalities is presented together with the quaternion position and orientation representation of links. These equalities embody the common starting point of the relevant problems concerned with the static analysis. One task is to establish prismatic joint forces and revolute joint torques caused by an external endpoint force, torque, and the gravity of links in the given endpoint position and orientation. Since the inverse kinematics is involved in this problem, redundant structures admit additional restrictions in the form of the cost function depending on joint variables. Another task is to establish the external force and torque, having measured forces in prismatic and torques in revolute joints in the given position of joints. The aim of the paper is to present the common base of the vector force-torque balance equalities and the quaternion representation of positions and orientations of any open multiple link redundant or nonredundant serial kinematic structure.

Dual vectors are a very efficient mathematical tool for the treating of line vectors and screws. Their great advantage is the consequent analogy to the algebra with the usual vectors. The real parts of dual vectors are in total accordance with the usual vectors. As there does not yet exist commonly a software using the dual unit we apply either subroutines or matrices. Matrices are into the bargain very useful in mechanics. Therefore we decide in this paper on the application of matrices and transfer the dual arithmetic and the dual vector algebra to matrix operations. The transition from kinematics to dynamics and vice versa is performed by conversion matrices for instance of inertia or of elastic compliance which convert real parts into dual parts and dual parts into real parts with changed physical dimensions (correlation of screws in ray coordinates with screws in axis coordinates). The dual vectors for the kinematic state of a rigid body with respect to another are represented in a coordinate system fixed in the considered body instantaneously coinciding in the base system. The absolute kinematic dual quantities in the body fixed system have to be multiplied after transformations with the inertia matrix of the body and also regarded the absolute turn of the body. The results are matrix formulas which yield kinetic wrenches as dual vectors verifiable by other methods [11], [13], [14], [17].

The ordinary Euler angles and Briant angles for describing rotations are well known and often used in robotics. However it is sometimes necessary to analyze the product of three general rotations R₁R₂R₃. We point out some dual invariants which are independent of the articular parameters and can be calculated, so all the quantities of the form {X { RY} can be determined for X and Y in the Lie algebra D;. Using these results, the 6R inverse kinematics problem can be solved.

An approach is described whereby, for a given position of the end effector, the target backbone curve for a planar multiple-link manipulator is obtained when a polygonal obstacle is placed in the workspace. The curve approximates an optimum static configuration of the manipulator relative to the minimum joint torques caused by an arbitrary force applied to the end-effector. The proposed approach is numerically effective and can be implemented in real-time procedures.

This paper describes an algorithm for simulating the motion of a nonholonomic robot vehicle and its trailer. The algorithm assumes that both the vehicle and its trailer move along circular arcs when the incremental time step used to update position and heading is small. The purpose of the simulation is to accurately represent the motion of a Navigation Test Vehicle (NTV) so that the ability of a vehicle’s control algorithm to track an explicit path can be evaluated. An auto-tuning algorithm is currently being developed in simulation to autonomously tune the control parameters for a nonholonomic robot vehicle, and this will require an accurate representation of the kinematics of the NTV. An accurate representation of the trailer motion is necessary to develop an autonomous backing-up control algorithm and path tracking control algorithms where the control point is on the trailer. As with the NTV, the positioning subroutine of the simulation program provides the control algorithm with vehicle and trailer pose data at 10 Hz. Locally, the positioning subroutine updates this data at 0.01 second intervals. The simulation currently runs on a Silicon Graphics IRIS Crimson/VGX workstation.

ScrewBall is an extension of the ArcBall implementation of a notional thumb-ball, visible on the screen and activated by desk-top mouse movements, which provides an acceptable user interface for specifying rotational displacements of computer-graphical objects. The ScrewBall extension permits specification of general displacements — i.e. of mixed translations and rotations — by allowing the user to specify a finite displacement screw or, equivalently, the vector part of a unit biquaternion. The paper describes the mathematical basis of ScrewBall, its relationship with ArcBall, and the computer-graphical context in which it would be applied.

This paper presents the first part of a research work devoted to simulation, identification and control of flexible robots based on their Newton-Euler model. First, we recall the generalization of the Newton Euler equations for an open chain of flexible links. These equations constitute an original dynamic model because they represent the dynamic behavior of the chain with a mixed formalism since it is both Lagrangian and Eulerian. This formalism gives the linear intrinsic dynamic model of links. Second, we exhibit the constrained nature of these equations. We use a numerical integration method for a constrained system and an updating procedure well suited to our particular formalism. Finally, simulation results on a two links spatial kinematic are reported.

Kinematic analysis of a braced robot is described. The strategy of bracing is intended for absolute robot end-point pose accuracy and repetability improvement. The principle of mechanical lever is explained with the theoretical point-to-line contact-type of robot bracing. The kinematics of braced robot with true surface-to-surface contact-type are derived: 1) with the local surface geometry of contacting bodies, 2) by an original method of constructing an equivalent four-bar mechanism. Pose uncertainty due to inaccuracy and repetability of free and braced robot is analyzed. An estimation function of accuracy improvement by bracing together with simulation results are presented. Advantages and disadvantages of the proposed method of equivalent four-bar mechanism with regard to the uncertainty reduction are discussed.

Given 3 congruently parametrised axial Darboux motions ζ _ζi = Σ _i \ Σ₀ (i=1,2,3) with pairwise orthogonal and skew axes, which are gained of ζ₀ by rotation of 120, 240 degrees, resp. round a certain axis r₀. Then we are able to show, that each relative motion Σ _i \ Σ _j (i ≠ j; i, j = 1,2,3) has a two-parametric familiy of (real) points situated on hyperboloids of one sheet Φ _i,j , which is moved on spheres centered on an analogue surface Φ _j,i . The intersection of two hyperboloids in a system Σ _i splits into a straight line g _i and a cubic circle c _¡ . It is shown that it is possible to form stiff triangles connecting corresponding points of c₁, c₂, c₃, such that the motions are not disturbed. For the points on the straight lines an analogous result holds: There the stiff triangles degenerate to straight lines, which in Σ₀ determine a regulus on a hyperboloid of rotation at each moment of the motion. Finally as an example two pairs of points on g _i and c _i are linked to get an overconstrained linkage.

The extension of Roberts’ Law concerns inverted slider-cranks with an eccentricity and a general tracing-point attached to the moving plane of the slider. As both curve-cognates degenerate in this case, the infinite turning-joint has first been replaced by a finite far-away joint, but such that the coupler-plane also meets two accuracy-positions of the erstwhile slider.One of the curve-cognates may then be adapted or modified until up to six accuracy-positions are met. They seem to be sufficient to attain an extraordinary good approximation of the entire branch of the original curve. Besides, even the transmission-angle attains a value about twice as large as the one obtained with the auxiliary four-bar representing the first approximation.Cases, where the inverted slider-crank turns into a crank-and-rocker, into a double crank or even into a stretchable four-bar, are shown. The latter corresponds with one of Grashof’s border cases for the inverted slider-crank. Roberts’ Law then ensures that each inverted slider-crank, having a fully revolving crank, possesses three four-bar branch-cognates this way.

In this paper we shall introduce an algorithm to determine self-motions of a special class of platform mechanisms. This class of mechanisms has a geometry that generalizes the geometry of existing flight simulator platforms. The algorithm uses the Study representation of the Lie group of space congruences and the quadratic constaint equation which decribes the condition that a point of the moving system is bound to move on a sphere.

In this paper we present a technique for designing parallel planar manipulators capable of reaching any set of points in a prescribed workspace. The manipulator consists of a platform connected to ground by RPR chains. The set of positions and orientations available to the end-effector of a single RPR chain is mapped into the space of planar quaternions to obtain a quadratic manifold. The coefficients of this constraint manifold are functions of the locations of the base and platform R joints and the distance between them. The structure of these equations is used to define a solution technique that yields the desired chains. Parallel manipulators that can reach the prescribed workspace are assembled from these chains. An example shows the determination of three RPR chains that construct a manipulator that reaches the prescribed workspace.

In the main part of this paper there is described a new spatial mechanism which exhibits the very peculiar property that a variation of the position parameter(s) can change its mobility. Although many linkages of this type might exist, only two are to be found in the relevant literature. Linkages of this type will be labelled here as kinematotropic linkages. In passing a singularity position (in which a certain transitory infinitesimal mobility is attained) these mechanisms permanently change their global mobilities.

A new kinematic design of a parallel spherical wrist with actuator redundancy is presented. The particularity of this type of parallel manipulator is that all the actuator axes are collinear, thus permitting unlimited rotation around any axis inside a given workspace. Detailed kinematic analysis has shown that actuator redundancy not only removes singularities but also increases workspace while improving dexterity. Optimisation of the structure was guided by a global criterion of dexterity. Comparison with other, non-redundant, structures of the same type, carried out with the help of a conditioning measure has shown a significant increase in dexterity in the workspace.

In this paper a special type of symmetric Stewart-Gough platform is analyzed where the six attachment points in each of the two bodies are arranged symmetrically with respect to a centerpoint. The lengths of the six actuators, which connect the two bodies, are given, and the relative position of the two bodies to each other has to be computed (forward kinematics). In a preliminary step a particular symmetric platform with three pairs of equal actuator lengths is studied. It is shown that such a platform can pose in up to 24 different configurations, and that it can become finitely mobile if the geometric dimensions fulfill certain conditions which are provided in the paper. For the more general type of symmetric platform the forward kinematic analysis reveals again that there might exist up to 24 possible postures. A final numerical example even shows that all 24 postures might be real.

In this paper, the singularity loci of general six-degree-of-freedom spatial parallel manipulators are studied and a graphical representation of these loci in the manipulator’s workspace is obtained. The algorithm presented is based on analytical expressions of the determinant of the Jacobian matrix. As mentioned elsewhere, two different types of singularities can occur when parallel manipulators are actuated. Both types are considered here and it is shown that one of the two types leads to a trivial description while the second type is more challenging. Moreover, it is shown that, for a given orientation of the platform, the singularity locus in the Cartesian space is represented by a polynomial of degree four. Examples illustrating these results are given. A graphical representation in the Cartesian space is also obtained.

A three degree-of-freedom manipulator that has a fairly large translational workspace is presented. The mechanism consists of a fixed base, a moving platform, and three extensible limbs. Each limb consists of a prismatic joint and two universal joints connecting the moving platform to the fixed base. Both the direct and inverse kinematics are investigated. The inverse kinematics problem yields two equal and opposite limb lengths for each limb while the direct kinematics problem is reduced to a second-degree polynomial in one unknown. Further, the workspace and singular conditions of the manipulator are discussed.

This paper describes a class of fully parallel manipulators, with planar base and end effector platforms, that have closed-form solutions to their forward position kinematics. “Closed-form” means that only linear, quadratic, and quartic equations have to be solved.

A planar parallel manipulator with holonomic higher pairs is introduced. The end effector is a circular disk which rolls with out slip along the straight lines of the non-grounded rigid links of each of three 2R legs. The contact points between the disk and legs are holonomic higher pairs. The forward kinematic problem of this manipulator is unlike that of planar Stewart-Gough type platforms because the initial assembly configuration must be included in the analysis. A procedure using kinematic mapping to solve the forward kinematics is discussed and a numerical example is given.

The forward analysis of a parallel planar manipulator (PPM) with two degrees of freedom is developed. The two degree of freedom PPM has been applied to an articulated mobile robot and the forward analysis is one of the tools developed to aid in the design and control of the articulated mobile robot. The forward analysis is derived from the loop closure equation for a quadrilateral applied to two quadrilaterals that make up the PPM. One side and two angles are common between the two quadrilaterals. An input-output equation is derived that generates six closures for the mechanism. The forward analysis is applied to the geometry of the PPM utilized in the articulated mobile robot, and determines four real and two imaginary closures.

The paper discusses closed plane mechanism applications to organize spatial parallel manipulators. It presents a classification of these mechanisms and a kinematic scheme is also extracted from this array that looks the most promising for multi-DOF robot creation. The structural features of this manipulator determine its properties that essentially simplify robot control. By using the graphic simulation method, this mechanism motion was examined.

High accuracy in robotic manipulation is an important requirement in order to ensure task specification and constraint. In this case, off line robot geometric parameter identification is very useful. In this paper, the authors present a new 3-D sensor for a 6 DOF parallel robot calibration. The forward and inverse geometric models for both the C5 parallel robot and the LGMPB-sensor are given. Then the parameter observability is studied by looking for the best minimum experimental configurations using some criteria based on the augmented Jacobian. A non-linear identification algorithm is proposed and first results are given.