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Showing papers in "Robotics and Computer-integrated Manufacturing in 1991"


Journal ArticleDOI
TL;DR: The axiomatic approach discussed in this paper provides decision making tools for software design in addition to systematic means of knowledge and data representation, synthesis and analysis of software, and the construction of the module-junction structure diagram.
Abstract: The ability to utilize fully automated flexible manufacturing systems (FMS) or develop reliable computer-integrated manufacturing (CIM) systems will depend on our ability to develop reliable and reusable software for large complex systems on a timely basis. To date, software design has not gone very far beyond the ad hoc trial-and-error stage. Consequently, the development of software is slow, expensive, unreliable, and unmanageable. The purpose of this paper is to provide a scientific basis for designing software. The approach used here is that of axiomatic design, which is based on two design axioms: the Independence Axiom and the Information Axiom. The axiomatic approach is based on the recognition of the following common elements in design: the existence of independent domains (i.e. the consumer domain, the functional domain, the physical domain, and the process domain); the need to map between various domains during the design process; the decomposition of the characteristic vectors (i.e. functional requirements, design parameters, and process variables) in their respective domains; the zig-zagging required between the domains for decomposition; and the need to satisfy the design axioms during the design process. The axiomatic approach discussed in this paper provides decision making tools for software design in addition to systematic means of knowledge and data representation, synthesis and analysis of software, and the construction of the module-junction structure diagram.

114 citations


Journal ArticleDOI
TL;DR: A calibration procedure is presented which allows the automatic determination of all kinematic parameters, gear parameters, and static elasticity effects and it is possible to increase pose accuracy up to the robot's system limit as determined by its repeatability.
Abstract: For many applications, robots are needed which have an absolute Cartesian pose accuracy of the same order of magnitude as their repeatability. Calibration procedures must be applied to reach this aim. In this paper, a calibration procedure is presented which allows the automatic determination of all kinematic parameters, gear parameters, and static elasticity effects. With the aid of these results it is possible to increase pose accuracy up to the robot's system limit as determined by its repeatability. Moreover, information can be acquired which suggests improvements in robot design, thereby also improving repeatability.

54 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated how various process variables such as speed of the tool center point and payload effect robot repeatability in a multistation assembly operation on a flat table top.
Abstract: This study investigated how various process variables such as speed of the tool center point and payload effect robot repeatability. The study was a simulated multistation assembly operation done on a flat table top. Various combinations of the process variables were used for the study. The results of the investigation showed that these process variables affect robot performance in varying degrees, with higher speeds and weights having the most significant effect. The repeatability of the robot deteriorated as the speed increased beyond 50% of the status speed (508 mm/s) and the weight increased beyond 0.91 kg. The findings do not support what might be a general practice of selecting robots with higher speed or payload over those with lower ones, everything else being equal. This is important to the production engineer concerned with the deployment of robotic systems.

52 citations


Journal ArticleDOI
TL;DR: In this article, an accurate model is developed for the contact interface at each locating and clamping region on the workpiece's surface, and an end milling operation is simulated to analyze the effects of various factors on workpiece accuracy and demonstrate the advantage of the simulation approach.
Abstract: This paper presents a methodology for dynamic modeling and simulation of a fixture-workpiece system. A simulation approach is required since standards typically do not exist for dynamic situations like machining operations. In addition, an accurate model is developed for the contact interface at each locating and clamping region on the workpiece's surface. An end milling operation is simulated to analyze the effects of various factors on workpiece accuracy and demonstrate the advantage of the simulation approach. The clamping forces required to keep the workpiece in contact with its locators are obtained, and the influences of locator placement, clamp placement, clamping forces, and clamping sequence on linear and angular errors are reported. Elastic effects of the locator-workpiece and clamp-workpiece contacts, yielding nonlinear dynamic equations of motion, are included in the model. Since system dynamics are considered, results are obtained as a function of time. The study compares well with previous experimental work by other investigators, and the method shows promise as a fixture design tool.

47 citations


Journal ArticleDOI
TL;DR: In this paper, a non-standard parametrization of rotation motions is proposed for manipulator modeling and control through a Lie group structure, and the advantage of the method is the computational efficiency at the kinematic level, not from efficient presentation but from fundamental topological considerations of the configurational manifold.
Abstract: This paper discusses manipulator modelling and control through a nonstandard parametrization of rotation motions. The advantage of the method is the computational efficiency at the kinematic level, not from efficient presentation but from fundamental topological considerations of the configurational manifold, described by a Lie group structure.

22 citations


Journal ArticleDOI
TL;DR: In this paper, a methodology based on Taguchi methods was implemented to determine/optimize robot process capability (RPC) for path following, which consists of the characterization of the robot path data, the experimental design, the data analysis procedure, and the verification of the results.
Abstract: A methodology based on Taguchi methods was implemented to determined/optimize robot process capability (RPC) for path following. The methodology consists of the characterization of the robot path data, the experimental design, the data analysis procedure, and the verification of the results. Four figures of merit (FOMs) for accuracy and four FOMs for repeatability were used to characterize a robot's controlled path capability. The experiment considered seven controllable factors (i.e. load, speed, shape, size, direction, orientation, and height) and used an L 27 (3 13 ) orthogonal array. From the results of the data analysis, the optimal factor-level combinations for the best path capability and minimum process variation were determined. The experimental runs were verified. The recommended methodology has the advantages of cost effectiveness, efficiency, data compression, and flexibility.

19 citations


Journal ArticleDOI
TL;DR: In this paper, an active end effector is developed to improve precision in robotic deburring by performing position adjustments normal to the part edge for constant cutting force, and an extended discrete PID controller based on an ARMAX plant model is designed and simulated off-line prior to realtime implementation.
Abstract: An active end effector is developed to improve precision in robotic deburring. The design objectives are obtained from a dynamic analysis of the combined system of robot arm, end effector, and deburring process dynamics. The unit allows robot-independent positioning, over a range of ±15 mm in two orthogonal directions with an accuracy of better than 0.01 mm and a bandwidth of 20 Hz. A combination of d.c. servo motors with linear ball screws achieves both high precision and a large mechanical advantage. A high quality chamfer is obtained by performing position adjustments normal to the part edge for constant cutting force. An extended discrete PID controller based on an ARMAX plant model is designed and simulated off-line prior to real-time implementation. In real-time force control tests, the active end effector system is shown to improve the precision of a PUMA-560 robot by an order of magnitude over the open-loop case.

14 citations


PatentDOI
Mayumi Ohya1
TL;DR: A music interpreter apparatus automatically interprets a notational music composition to provide its performance realization information which is specific and complete.
Abstract: A music interpreter apparatus automatically interprets a notational music composition (written or printed music), which is abstract and incomplete to some extent, to provide its performance realization information which is specific and complete. The apparatus includes a score memory, an interpreter and a performance memory. The score memory stores a coded music notational symbol string representing a written music composition. The interpreter reads and interprets the coded music notational string based on an artificial intelligence simulating knowledge and experience of a human music interpreter or performer to provide a performance symbol string specifying the performance realization and containing played pitch, note-on time, duration and loudness parameters of specified values with respect to each note in the music composition. The resultant performance symbol string is stored into the performance memory.

13 citations


Journal ArticleDOI
TL;DR: The behavior of collaborative systems can be interpreted in terms of games of strategy; this approach permits a better understanding of the conditions for globally optimal behavior, as well as strategies for their attainment.
Abstract: The effectiveness of a system relates to its ability to fulfill its functional requirements. When a system consists of a set of collaborating agents, the overall performance may depend more on their ability to work together than on optimizing individual behavior. In fact, the goal of an agent is often at odds with the interest of the collective. The limitations of centralized control for complex systems suggest the need for decentralized coordination. This can be achieved by having each agent take explicit account of the cost of engaging in an activity as well as a measure of reward for effective behavior. The consideration of payoffs and penalties leads to an economic perspective of multiagent systems. In consequence, it is possible to draw on previous work in diverse fields, ranging from game theory to welfare economics and social policy. The promise and limitations of the explicit valuation approach are examined. To illustrate, the behavior of collaborative systems can be interpreted in terms of games of strategy; this approach permits a better understanding of the conditions for globally optimal behavior, as well as strategies for their attainment. The notions of agents and explicit valuation of action are versatile concepts. One indication of the versatility lies in the fact that these concepts cover as a special case the idea of genetic algorithms as a mechanism for learning systems.

12 citations


Journal ArticleDOI
TL;DR: In this article, the problem of a heavy robot performing highly accelerated motion is discussed, and a suitable redundancy is added to improve the acceleration capabilities of the original six degree of freedom configuration.
Abstract: The problem of a heavy robot performing highly accelerated motion is discussed. Since the original six degree of freedom configuration could not solve the task, a suitable redundancy was added to improve the acceleration capabilities. This redundant configuration equipped with the DPS (distributed positioning system) software was able to perform the motion required. The theoretical background and simulation results are presented.

10 citations


Journal ArticleDOI
TL;DR: Algebraic and stochastic models of innovative decision making are presented and suggestions are made for extending the work along both theoretical and empirical lines.
Abstract: Real-world tasks are often complex, uncertain, and constrained. The complexity arises from the need to consider numerous factors that are of varying degrees of relevance to the problem at hand. The uncertainty springs from imperfect information concerning the state of the world, the repertoire of feasible alternatives, and the consequences of each action. The constraints are attributable to time, money, and computational resources as well as individual tastes and societal values. Despite the rich nature of practical tasks, previous work in decision making—whether in engineering, statistics, management, or economics—has focused solely on partial aspects of the problem. This state of affairs is reflected in the nomenclature, which involves categories such as “constrained optimization” or “decisions under uncertainty”. If real-world tasks are to be addressed in a coherent fashion, it is imperative to develop a systematic framework providing an integrated view. The framework may then serve as the foundation for a general theory of decision making that can capture the full richness of realistic problems. This paper explores how these goals might be achieved. Algebraic and stochastic models of innovative decision making are presented. This is followed by an examination of idea generation in product design. Finally, suggestions are made for extending the work along both theoretical and empirical lines.

Journal ArticleDOI
TL;DR: This paper has two objectives: to present the difficulties in modeling and simulating such distributed, concurrent systems, and to describe a new implementation paradigm towards these objectives using concurrent logic programming.
Abstract: Multi-robot systems can be viewed as a distributed, asynchronous and concurrent mechanism. The components of such a system communicate with each other and with the external world and perform discrete tasks. Such tasks exhibit nondeterminism, concurrency, mutual exclusiveness, and similar properties. Due to the system complexity, it is desired to model it for purposes of verification, analysis, and simulation. This paper has two objectives: to present the difficulties in modeling and simulating such distributed, concurrent systems, and to describe a new implementation paradigm towards these objectives using concurrent logic programming. This approach is described and examples given using FCP (Flat Concurrent Prolog) for a robotic manufacturing cell.

Journal ArticleDOI
TL;DR: In this article, an analysis of the 3D coordinate tolerance problem is presented in conjunction with the requirements of the tolerance transfer principle and the limitations imposed by the accuracy-related manufacturing cost and process and machine tool capabilities.
Abstract: Design and/or functional dimensions and tolerances do not always coincide with the coordinate machining axes. For manufacturing reasons, these dimensions and tolerances must often be transferred to the orthogonal datum system of the particular application. Whereas the derivation of the new dimensions or coordinates does not present any major problems this is not the case with new coordinate tolerances. The tolerance transfer principle alone is not sufficient for their systematic evaluation. Additional constraints towards an optimized problem approach must be introduced. In this paper, an analysis of the 3D coordinate tolerance problem is presented in conjunction with the requirements of the tolerance transfer principle and the limitations imposed by the accuracy-related manufacturing cost and process and machine tool capabilities. The application of the procedure developed for the evaluation of coordinate tolerances is further demonstrated in a case study.

Journal ArticleDOI
TL;DR: This paper adds the innovation of variable-node spacing, with the node density in various parts of the network reflecting the need for precise position control in each local area of the workspace, to ensure precise motion control is possible without an undue computational burden.
Abstract: This paper describes a research effort in the area of collision avoidance path planning for robotic manipulators. A robotic arm with known geometry is to perform a spatial manipulation in the presence of known obstacles in its workspace. The task is to generate a series of waypoints for its path to pass through which will guarantee a safe, collision-free trajectory from its predefined starting point to its predefined goal position. This is an important topic in the area of automated manufacturing. Automated factories are becoming increasingly important for the goals of greater manufacturing efficiency, better equipment utilization, and lower overall manufacturing costs. Robotic devices, including manipulator arms and assembly devices, are finding more uses in these factories. The approach is to discretize the robot's workspace into a transition network. The optimal path through this network, in terms of angular displacement of the manipulator's joints, is generated by dynamic programming. While this approach has been used previously, this paper adds the innovation of variable-node spacing, with the node density in various parts of the network reflecting the need for precise position control in each local area of the workspace. In this way, precise motion control is possible without an undue computational burden.

Journal ArticleDOI
TL;DR: An advanced robot control system based on the three parallel processor boards dedicated to controlling industrial robots with up to six degrees of freedom, based on a disk-oriented real-time operating system.
Abstract: This paper describes an advanced robot control system based on the three parallel processor boards. The controller has been designed as a general-purpose robot control system dedicated to controlling industrial robots with up to six degrees of freedom. The controller is based on a disk-oriented real-time operating system. The entire set of robot parameters can be monitored and changed by the user on-line. Source files, compilers, linkers and various utilities are provided as well. The main software layers include robot program interpreter, manipulator control facility, robot kinematics, dynamic feed-forward compensation, and digital servos. Basis, hand and joint coordinates are supported. Point-to-point and continuous path motions are provided. Digital and analog IO modules are included for synchronization with an environment. An acquisition system for monitoring and graphical presentation of robot coordinates, velocities and IO signals is provided. The paper ends with some experimental results for a six-degree of freedom robot.

Journal ArticleDOI
TL;DR: An algorithm for positioning and orientation of the hand for a redundant or non-redundant manipulator along a continuous path in space that minimizes the distance between the actual position of the tip of the end-effector and the desired path.
Abstract: This paper presents an algorithm for positioning and orientation of the hand for a redundant or non-redundant manipulator along a continuous path in space. This algorithm minimizes the distance between the actual position of the tip of the end-effector and the desired path. The algorithm does not use the Jacobian matrix for the inverse kinematics of the robot. It takes full advantage of the resolution of the joint drives, avoids singularity problems, and can be used for both redundant manipulators. The algorithm can be used in any situation where continuus motion of the end-effector is required in an open loop mode.

Journal ArticleDOI
TL;DR: In this article, the analysis of a prosthesis of the lower limb of a human lower limb is presented. But the analysis is restricted to a single prosthesis and does not cover the entire lower limb.
Abstract: Biomechanics and mechatronics are important fields that are emerging rapidly. This paper combines elements of both these fields in explaining the analysis of a prosthesis of the lower limb.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of extremely nonuniform motion for large robots in arc welding and proposed a parallel degree of freedom distribution of motion to parallel degrees of freedom.
Abstract: This paper considers some problems concerning robotic welding of large structures. If spot welding is applied, then the task consists of fast motions between welding points, and at each point the end effector stops and maintains position for a short time. Similar examples can be found in arc welding applications. Since large structures and, accordingly, large robots are considered, a significant problem appears when extremely nonuniform motion is required. One way to solve this problem is to introduce parallel degrees of freedom. Two methods for this distribution of motion to parallel degrees of freedom are discussed. The theory resulted in the design of a large portal robot with high dynamic capabilities.

Journal ArticleDOI
TL;DR: The task of concurrent design is to ensure that the product performs as specified, with fidelity and consistency as discussed by the authors, and ensuring product performance is the objective of quality assurance. Software packages can be developed to assist in this effort, ranging from the initial phase of requirements definition to production control and maintenance.
Abstract: Engineering design involves the satisfaction of customer needs, These needs, whether explicitly stated or only implicitly defined, must be translated into a set of specifications for a product. The subsequent stages of design, fabrication, and maintenance are dedicated to the fulfillment of these specifications. The task of concurrent design is to ensure that the product performs as specified, with fidelity and consistency. Ensuring product performance is the objective of quality assurance. Software packages can be developed to assist in this effort, ranging from the initial phase of requirements definition to production control and maintenance.

Journal ArticleDOI
TL;DR: The program, named SyK, is written in Pascal and can derive symbolic equations for robot position, orientation, angular and translational velocity, angularand translational acceleration, Jacobian and Hess matrices in backward and forward recursive form.
Abstract: A new program for automatic symbolic derivation of robot kinematic equations was developed. The program, named SyK, is written in Pascal and can derive symbolic equations for robot position, orientation, angular and translational velocity, angular and translational acceleration, Jacobian and Hess matrices in backward and forward recursive form. It eliminates superfluous multiplications and additions in treating a specific robot manipulator. The main advantage of the program is its simple portability and notable efficiency. Symbolic equations for a desired quantity can be obtained in a few milliseconds (Micro VAX). The main disadvantage is that the kinematic equations obtained have a fixed recursive form. SyK is commercially available.

Journal ArticleDOI
Y. Yao1
TL;DR: In this paper, it is shown that the rate is limited by the robot end effector's lateral transient response to the contact forces, while the response depends mainly on the link inertia characteristics and joint compliances.
Abstract: This paper is concerned with achieving a higher rate at which a robot operation involving part mating can be carried out while free of excessive contact forces. It is shown that the rate is primarily limited by the robot end effector's lateral transient response to the contact forces, while the response depends mainly on the link inertia characteristics and joint compliances. Both factors were modelled and simulated for robots of SCARA configuration for the chamfer-crossing stage. The transient response predicted by the model led to determination of a varying-speed insertion motion for the chamfer-crossing stage to avoid large initial impacts between peg and hole and to limit the contact force during the entire stage. The discrepancy angle between the contact force and the resultant lateral motion was also considered. Good agreement was obtained between the transient response predicted in simulation and that observed in experiment.

Journal ArticleDOI
TL;DR: In this paper, the simulation of the dynamic behavior of an underwater robot is described, and the motion of a robot arm under water is analyzed and the dynamics of the robot discussed.
Abstract: The simulation of the dynamic behavior of an underwater robot is described in this paper. The motion of a robot arm under water is analyzed and the dynamics of the robot discussed. The interactions between the robot and its environment and between the robot and its base are examined. The results of the simulation will yield basic knowledge for developing the positioning control units and sizing of the drives.

Journal ArticleDOI
TL;DR: This paper will discuss how the expert system approach is used to integrate the quality judgement process into the production process, and bring the factory one step further toward total CIM.
Abstract: True computer-integrated manufacturing requires that many links between factory processes become more automated, more computerized, and more intelligent. Many links between processes, however, are based on human judgement, experience, and intuition. These are naturally difficult tasks to automate, but the advent of artificial intelligence, and specifically expert system programming, has made such links possible. This paper presents a research effort toward one of these links, that of quality control judgements of ongoing processes. The use of expert system programming, combined with in-process metrology and system integration, allows the factory to be more fully automated and computer-integrated, resulting in higher precess precision and lower production errors. This paper will discuss how the expert system approach is used to integrate the quality judgement process into the production process, and bring the factory one step further toward total CIM.

Journal ArticleDOI
TL;DR: In this article, the effects of chopper current control on the torque and acceleration characteristics of a hybrid step motor are discussed, and the control angle in the current control mode, yielding greater torques and improved acceleration of hybrid step motors than for the optimal control angle without phase current control.
Abstract: Optimal closed-loop control of a hybrid step motor control with chopper drive is described. Expressions for the average torque and optimal control angle giving maximal average torque in the stationary state without phase current chopper control are presented. The effects of chopper current control on the torque and acceleration characteristics of a hybrid step motor are discussed. Expression have been developed for the control angle in the chopper current control mode, yielding greater torques and improved acceleration of hybrid step motors than for the optimal control angle without phase current control. The speed response of a hybrid step motor operated in a closed-loop system has been studied by computer simulation for different supply voltages and torque loads, using various control angle algorithms.

Journal ArticleDOI
TL;DR: In this article, a production control policy for an automated printed circuit board (PCB) assembly system is developed which satisfies demand requirements for a realistic environment, i.e., an environment which allows machine failures and repairs as well as demand changes to occur.
Abstract: Production control policies for an automated printed circuit board (PCB) assembly system are developed which fulfill a number of characteristics. First, they satisfy demand requirements for a realistic environment, i.e. an environment which allows machine failures and repairs as well as demand changes to occur. Second, they constitute solutions of intermediate generality, i.e. they are not specific to the system under study. Third, they are easily implementable in real time for a computerized system. Policies seek to satisfy both total production volume and balance among part types and to minimize work-in-process. The system's main controlling variables are the part entry decisions; that is, the time and type of part or parts to enter the system. Entry decisions are based on cumulative performance measures and current system capacity constraints like machine failures, bottleneck cell status, and maximum allowed work-in-process. Dispatching in front of production cells is done with the use of simple priority rules. Policies are tested on a simulation model of the PCB assembly implemented on an AI environment, the KEE-SIMKIT package on an Explorer Lisp machine.

Journal ArticleDOI
TL;DR: In this paper, the authors examine some critical issues and unifying concepts in concurrent design, grounded in an application to printed circuit board design, and propose a solution to solve them.
Abstract: A key objective of concurrent design is to bring to bear all the relevant knowledge of an organization into the design of a product. A second objective is to reduce the product development schedule, thereby maintaining competitive capability in a dynamic industrial environment. All aspects of the parallel development activities must be in accord with the final objective of product performance and customer satisfaction. This paper examines some critical issues and unifying concepts in concurrent design. The discussion is grounded in an application to printed circuit board design.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear model-reference adaptive contrll system based on Liapunov stability considerations for robotic manipulators is presented for trajectory tracking in contour cutting or precise wedding.
Abstract: This paper presents a nonlinear model-reference adaptive contrll system based on Liapunov stability considerations for robotic manipulators. Asymptotic stability of the computed torque method is proven using a model-reference formulation. Robustness of the control algorithm against model uncertainties are studied. For trajectory tracking in contour cutting or precise wedding, the accuracy is very important. Tracking accuracy for contour paths is studied as a function of commanded trajectory step resolution and the natural frequency of the reference model. The higher the natural frequency of the second order reference model, the faster the response; but the simulations show that there is a trade-off between the magnitude of natural frequency and the desired trajectory tracking accuracy. In addition, as the natural frequency of the reference model increases, the robotic system needs finer trajectory step resolution in order to guarantee the same levels of accuracy as achieved with low natural-frequency reference models.

Journal ArticleDOI
TL;DR: In this paper, an optimal level of precision at which maximizes the productivity of a continuous path manufacturing system is defined at the programming level, and the problem is formulated as a stochastic optimization problem.
Abstract: As in many manufacturing processes, there is a trade off between the production rate and the quality of the product in a robotic continuous path manufacturing system. The precision of the process may be increased at the programming level by defining the path by more points, employing a more sophisticated path generation method, or utilizing a feedback control system. However, increasing the number of points will reduce the maximum operation speed of the robotic operation; therefore the production rate will suffer as the operation cycle time increases. On the other hand, speeding up the operation may cause low precision which may eventually lead to unacceptable product quality. As a result, productivity in terms of the number of acceptable parts per unit time may not be improved. In this paper, the precision at which the robot should operate is quantified at the programming level. Furthermore, it is determined that an optimum level of precision exists for which the productivity of the continuous path manufacturing system is maximized. Due to inherent random errors present in the control and operational phases, the problem is formulated as a stochastic system and is solved through a stochastic optimization method.

Journal ArticleDOI
TL;DR: In this article, the load influence on a d.c. drive motor servo system for all six degrees of freedom of industrial robots with an articulated configuration was analyzed, and the analytical equations of total inertia and gravitation moments of an industrial robot service system for external coordinate function were obtained.
Abstract: This paper analyses the load influence on a d.c. drive motor servo system for all six degrees of freedom of industrial robots with an articulated configuration. Through the analyses, the analytical equations of total inertia and gravitation moments of an industrial robot servo system for external coordinate function were obtained. A mathematical model of an electro-mechanical industrial robot system was also developed.

Journal ArticleDOI
TL;DR: In this paper, a two-digit claw grasps an object and a set of elementary contact points, each characterized by its coordinates and pressure force applied, constitutes a representative image of each contact spot.
Abstract: When a two-digit claw grasps an object, a contact spot on each jaw is involved in the grasping. A set of elementary contact points, each characterized by its coordinates and pressure force applied, constitutes a representative image of each contact spot. If a six-axis sensor in each jaw measures the forces/torques acting between the object and the claw, the risk of slipping can be calculated from the representative images. The risk of release by translation of the contact spot along the grasping surfaces is easy to calculate, but the risk of slipping by rotation round an instantaneous virtual pivot axis is more difficult. It requires the location of the pivot point of each contact spot, and then the calculation of the limit resistant torque relative to this point. This paper deals with the general aspect of the problem, and particularly with the means of locating the pivot point.