Fundamentals of manipulator stiffnessmodeling using matrix structural analysis
Matrix, which is the main outcome of the manipulator stiffness analysis. In addition, here the influence of the external loading and buckling analysis cannot be executed in a simple way. Another contribution in this area  deals with MSA-based stiffness analysis of a parallel manipulator composed of several L-structures. The authors applied classical MSA, with the connectivity matrix for lines/columns merging. Although they took into account both the link and joint flexibility, there are still several open questions related to the definition of link/joint stiffness properties and invertibility of local stiffness matrices describing the L-structures. Another useful extension of the MSA for the case of links and joints with non-linear stiffness was proposed in . Here, a passive revolute joint with ball bearings was presented as an element with a rank-deficient force-dependent stiffness matrix, whose parameters were estimated experimentally. The relevant computational procedure included several iterations of conventional MSA linear model. This technique was applied to PARAGRIP handling system and validated by measurements of the end-effector deflection under vertical load. There are also several works the deal with the MSA application to the stiffness analysis of particular parallel and serial manipulators. In  the MSA method was applied to EAST articulated maintenance arm with 11 degrees of freedoms (EAMA robot), which is used for remote inspection of inner components inside the vacuum vessel. In  the MSA technique was employed to obtain a dynamic model of the industrial machining robot ABB IRB 6660 in order to predict vibration instability in machining (chatter). In  the MSA was applied to derive the static stiffness model of 9-dof redundant reconfigurable 3×PPPRS parallel manipulator for meso-Milling Machine Tool (RmMT).
4 Application example
Let us illustrate the efficiency of the developed stiffnessmodeling technique by applying it to the compliance error compensation in the robotic-based machining performed by the industrial robot KUKA KR-270. This robot is equipped with the spring-based gravity compensator located between the first and second links (which leads to the influence on the second actuated joint). In accordance with the considered specifications, the technological process should be performed in the square area of the size 2000 mm2000 mm located at the height 500 mm over the floor level (see Fig, 3 for more details). For comparison purpose, it is assumed that machining force is constant throughout the working area and it is equal to F = (0, 360N, 560N, 0, 0, 0) T , which corresponds to a typical milling process.
The stiffness analysis becomes a critical issue in design optimization of robotic manipulators that currently is targeted at achieving high dynamic performances with relatively small link masses and low energy consumption in actuators. This tendency motivates a revision of existing stiffnessmodeling techniques that must take into account the external and internal loading. To meet this demand, the paper proposes a new systematic method for stiffnessmodeling of robotic manipulators with passive joints in loaded mode. It is based on a multidimensional lumped-parameter model, which presents the links as pseudo-rigid bodies with 6-d.o.f. virtual springs whose parameters are evaluated via the FEA-modeling and describe both the translational/rotational compliances and the coupling between them. The developed technique allows computing the loaded equilibrium configurations, evaluating their stability, and finding the full-scale “load-deflection” path for any given displacement of the end-effector. It is also proposed the linearization procedure for computing the Cartesian stiffness matrix, which is based on the inversion of the dedicated matrix of larger dimension, composed of the stiffness parameters of the virtual springs and the Jacobians/Hessians of the active and passive joints. These results enable designer to evaluate critical forces that may provoke non-linear behavior of the manipulators, such as sudden failure due to elastic instability (geometrical buckling) which has not been previously studied in robotic literature.
Keywords: Stiffnessmodeling, matrix structural analysis, serial robots, parallel robots.
In many modern robotic applications, manipulators are subject to essential external loadings that affect the positioning accuracy and provoke non-negligible compliance errors . For this reason, manipulator stiffness analysis becomes one of the most important issues in the design of robot mechanics and control algorithms. It allows the designer to achieve required balance between the dynamics and accuracy. However, to make the stiffness analysis efficient, it should rely on a simple and computationally reasonable method that is able to deal with flexible links and rigid connections, passive and elastic joints that are common for serial and parallel robots.
The paper presents generalization of the non-linear stiffnessmodeling technique for manipulators under internal and external loadings. The developed technique includes computing of the static equilibrium configuration corresponding to the loadings. It is able to obtain the non-linear force-deflection relation, the Cartesian stiffness matrix for the loaded mode as well as the matrices defining linear mappings from the end- point displacement into the deflections in passive and virtual joints. The obtained results allow us to extend the classical notion of "conservative congruence transformation" for the case of manipulators with auxiliary loading.
Stiffnessmodeling for perfect and non-perfect parallel manipulators under internal and external loadings
Matrix Structural Analysis method (MSA). This method incorporates the main ideas of the FEA but operates with rather
large compliant elements such as beams, arcs, cables, etc. . This obviously leads to the reduction of the computational expenses and, in some cases, allows us to obtain an analytical stiffness matrix for the specific task. Similar to the FEA-modeling the MSA method gives forces/torques and displacements for each node, but here it has a clear physical interpretation (manipulator active or passive joint), which can be useful for some tasks [14,36]. For parallel robots, this method has been developed in works [37,38], where a general technique for stiffnessmodeling of the manipulator with rigid/flexible links and passive joints was proposed. It has been illustrated by stiffness analysis of parallel manipulator of Delta architecture where the links were approximated by regular beams. The latter causes some doubts in the model accuracy compared to the combination of the FEA and VJM techniques that are being developed here. Besides, this result was obtained under the assumption that the external forces/torques are relatively small (i.e. for the unloaded mode), and it is unlikely that such approach can be enhanced to take into account particularities of manipulator behavior in loaded mode. In addition, here there exists a problem of the stiffness matrix computation for the manipulator singular configurations.
Abstract— The paper focuses on the stiffnessmodeling of
parallel manipulators composed of non-perfect serial chains, whose geometrical parameters differ from the nominal ones. In these manipulators, there usually exist essential internal forces/torques that considerably affect the stiffness properties and also change the end-effector location. These internal load- ings are caused by elastic deformations of the manipulator ele- ments during assembling, while the geometrical errors in the chains are compensated for by applying appropriate forces. For this type of manipulators, a non-linear stiffnessmodeling tech- nique is proposed that allows us to take into account inaccuracy in the chains and to aggregate their stiffness models for the case of both small and large deflections. Advantages of the developed technique and its ability to compute and compensate for the compliance errors caused by different factors are illustrated by an example that deals with parallel manipulators of the Orthog- lide family.
*** CNRS, Nantes, France, Le Laboratoire des Sciences du Numérique de Nantes (LS2N) (e-mail: email@example.com)
Abstract: The paper deals with stiffnessmodeling of NAVARO II transmission system, which is a novel variable actuation mechanism based on active and passive pantographs. The desired models are obtained using the enhanced matrix structural analysis (MSA) approach that is able to analyze the under-actuated and over-constrained structures with numerous passive joints. Depending on the pantograph type, the models operate with the matrices of size 252x288 and 264x294 suitable for parametric optimization of the entire manipulator.
5.2 Stiffness of manipulator elements
The desired stiffness model for the entire manipulator (Figure 6) incorporates, as the parameters, the stiffness matrices of all principal links. Each of them was estimated using the FEA-based technique proposed in this paper. The principal components of the mechanism are presented in Figure 4, where the elements (a, b, c) are threaded as flexible ones and the element (d) is assumed to be rigid. allelogram axe d – foot, r – end-effector ) For all flexible links, the compliance matrixes were computed via the FEA-based technique proposed in this paper. Also, for comparison purposes, there were computed similar matrices corresponding to the link approximations, which are presented in Table 7 These results confirm advantages of the proposed technique, that give essential increase of accuracy. Besides, for the manipulator component (b), it was detected extreme difference (13 times) in the values of k 44 evaluated by different methods. The latter is cased by the compliancy of the joint that is taken into account in contrast to previous studies.
Stiffness is a crucially important performance specifica- tion of parallel kinematic machines. In order to get a real industrial machine, the H4 prototype (Fig. 1) must be opti- mized. Because of its long arms and rods, designers must be particularly careful with the machine stiffness, which has direct consequences on manipulation accuracy . Several studies were performed by inventors to determine the geometric model, the usable workspace and the forces into the machine components . The work presented in this paper is about stiffnessmodeling of H4 robot and can be easily extended to lower manipulability parallel
recursive procedure that sequentially modifies the original
matrix in accordance with the geometry of each passive joint. Advantages of the developed technique are illustrated by application examples that deal with stiffnessmodeling of two Stewart-Gough platforms. Future work will focus on the extension of these results for the case of parallel manipulators with non-rigid platform and essential external loading.
required positioning accuracy is rather high. For example, from our experience it is known that the end-effector deflection of heavy industrial robots under loading of 1kN may vary from 1 to 10 mm within the robot workspace, while demanded accuracy for the machining process is typically about 0.1 mm. These compliance errors can be reduced down to admissible level using both on-line and off-line error compensation techniques, which are based on the appropriate stiffness model , either “complete” or “reduced”. The complete stiffness model of an industrial robot is complicated and takes into account all manipulator links and actuators compliances  (Fig. 2a). In practice, a number of manipulator components may be treated as rigid ones (e.g., some links), while the main compliance is concentrated in the actuator transmissions. This allows us to apply so-called reduced models that take into account the joint elasticities only . Such models are quite common for stiffnessmodeling of heavy industrial robots where the links are massive and their deflections under the force of 1kN are much lower than 0.1 mm. The reduced stiffness model is also quite useful at the design stage since it gives valuable approximation of the manipulator stiffness behavior, which is required for optimization. Moreover, in most cases these models could be used to compare stiffness properties of the manipulators of different architectures. For this reason, the presented in this paper comparison study is based on the reduced stiffness model. To make the comparison comprehensible, let us perform further simplification of the considered models. At first, in the frame of the comparison study, it is possible to exclude the robot base from the stiffness analysis. It is obvious that this component influences the stiffness properties of both serial and quasi-serial manipulators in the same way. Further, since the wrist dimensions are essentially smaller than the lengths of the links #2 and #3, we simplify the model by excluding the robot wrist components.
Keywords: Stiffness analysis; joint stiffness identification; Cartesian stiffness matrix; complementary stiff- ness matrix; serial robots; robot machining.
Serial robots are mainly used in industry for tasks that require good repeatability but not necessarily good global pose accuracy (position + orientation as defined in ISO9283) of the robot end-effector (EE). For example, these robots are generally used for pick-and-place, painting and welding operations. Nevertheless, they are now being used for machining operations, such as the trimming, deflashing, degating, sanding and sawing of composite parts, that require high precision and stiffness. Therefore, to perform these operations, the robots must show good kinematic and elastostatic performance. In this context, it appears that conventional machine tools such as the gantry CNC are still more efficient than serial robots. Therefore, it is important to pay attention to robot performance to optimize their usefulness for machining operations. Some research works discuss the
Abstract: The atomic stiffness parameter (ASP) in dense phases:
S = 9νBε -1 where v stands for atomic volume, B for bulk modulus and ε for atomic cohesive energy expresses the competition between metallic attraction and core-core repulsion. ASP is driven by the repulsive interaction between electronic more or less filled internal shells since attractive metallic bonding is common for all metals. The square root of ASP was early introduced as anharmonicity parameter η by Rose et al. ASP controls the occurrence of local defects in crystalline structures. Since crystalline defects are essential for cluster structures, surface rearrangements and grain boundary structures, i.e. nanostructures, ASP also controls bulk malleability and ductility. The ASP set of composite materials controls their tendency to form quasicrystals and amorphous materials. Nanometric friction between materials also depends on respective ASP values. ASP values define several classes of metals, according to their ability to bear non- homogeneity. So called “extra stiff” metals have completely filled internal electronic shells, noble metals are “very stiff” while numerous metals are more or less smooth. This stiffness classification is linked with structural properties of nanostructures.
Data are expressed as mean 6 SD or as percentages unless otherwise specified. Data on b index and BNP were skewed and were thus loga- rithmically transformed. Log BNP and log b index values were used in correlation and regression analyses as appropriate. Relationships be- tween different parameters were assessed by linear correlation anal- ysis and Pearson’s correlation coefficient. To determine the impact of carotid artery stiffness on LV diastolic function, LV filling pressure, BNP plasma level, and symptoms, stepwise linear or logistic multiple regression analyses were performed. Variables with P values < .10 on univariate analysis were incorporated into the multiple regression models, with special care to avoid collinearity among a subset of several variables measuring the same phenomenon. Two-sided P values < .05 were considered significant. Continuous and nominal variables were compared using Student’s t test. Carotid b index was compared between symptomatic and asymptomatic patients using a one-way analysis of variance followed by Tukey’s test. All statistical analyses were performed using Statistica version 6 (StatSoft Inc, Tulsa, OK).
Viscous thread instability (VTI) is a well-characterized effect 16–18 , familiar to anyone who has watched honey
drizzle onto a surface. The phenomenon has been studied in the past in 1D and 2D systems where a fluid is allowed to flow from a height over a moving belt. For a certain range of process parameters, a variety of patterns of movement of the strands can be produced. Patterns are classified into shapes such as “W’s”, meanders, alternating loops, and translating coils 17 (Fig. 1). In the past, these meandering motions have been studied for modeling in
The vibration analysis of cables with a small bending stiffness is a problem encountered in many engineer- ing applications such as the fatigue assessment of stay cables (cable stayed bridges or prestressed concrete beams), the modeling of pipeline laying operation or the determination of bending stresses in drillpipe as- semblies. The elastic theory of rods models the static and dynamic deflections of general elastic beams and cables (Antman 2005, Coleman et al. 1993). How- ever, the smallness of the bending stiffness leads to consider a singularly perturbed equations and the ex- istence of boundary layers near the anchors (Hinch 1991). In these small regions, the curvature and the bending moment change sharply compared with the rest of the cable especially. This phenomenon is mag- nified for clamped beams. Standard numerical tech- niques fail to solve these problems efficiently.
As vehicle safety-oriented control systems become more advanced, their dependence on accurate information on the state of the vehicle and its surroundings increases. For in- stance, the performance of driver-assistance technologies, such as the antilock braking system (ABS), is greatly influenced by the characteristics of the friction force between the tyre and the road. Therefore, by taking into account the external driving conditions of the vehicle, the effectiveness of such active safety systems can be greatly improved . Tyre-road friction, however, cannot be directly measured in real-time; hence its estimation has been an intensive research area in the last years. Numerous different approaches to estimate the tyre-road friction coefficient and its maximum value have been proposed in the literature —see, e.g., , , and references therein. In several of these works it is proposed to estimate the peak tyre-road friction under the premise that the tyre braking stiffness indicates the peak value of the friction-slip curve . The braking stiffness is the slope of the friction with respect to the wheel slip at the zero-friction operating point . In this work we are interested in a generalization of this concept that is known as extended braking stiffness (XBS) and may be defined as the slope of the friction-slip curve at any operating point —see  and . The interest of estimating the XBS is that, in contrast to the unknown optimal value of wheel slip,
4.2. Design optimization setting
In the optimization studies, it is essential to set upper and lower bounds for the design variables such that the solutions are feasible
and meaningful, e.g., the size of the petals should not be too big to overlap with other parts; the relative width of the components should not be too small to cause signiﬁcant manufacturing diﬃculty; the curvature of the curved parts should not be too big to introduce high stress concentrations. For the problems shown in this paper, the upper and lower bounds for the design variables w 1,2,3,4 are set to be [3, 3, 3, 1] and [0 . 8, 0 . 5, 0 . 5, 0 . 25], respectively, to avoid overlaps between the petals and connecting bars. The upper and lower bounds for design variable h 4 are set to be 13.5 and 7, respectively, such that the petals are small enough to ﬁt within a unit cell, and yet of a suf- ﬁcient size to generate a relatively large curvature at the vertices. Referring to Fig. 2 (a), the upper bound of the offsetting variable d 1 is set to be 1, such that a reasonable gap exists between the two arms of the petal, and the lower bound is set to be 0.3 such that a mini- mal distance is maintained between control points C 1 and C 2 . For the petal and connecting bar widths d 2 and d 3 , the upper bound is set as 1 to control the parametrization modeling. To account for a possi- ble minimum thickness constraint in the manufacturing process, the lower bound for d 2 and d 3 is set as 0.2.