Showing papers in "Multibody System Dynamics in 2018"

A unified formulation for mechanical joints with and without clearances/bushings and/or stops in the framework of multibody systems

Abstract: Virtually all machines and mechanisms use mechanical joints that are not perfect from the kinematic point of view and for which tolerances, in the fitting of their components, are specified. Together with such controlled clearances, mechanical joints may require the use of bushing elements, such as those used in vehicle suspensions. Furthermore, in many situations the joints exhibit limits (stops) in their translational or rotational motion that have to be taken into account when modeling them. The dynamic response of the mechanical systems that use such realistic mechanical joints is largely dependent on their characteristic dimensions and material properties of the compliant elements, implying that correct models of these systems must include realistic models of the bushing/clearance joints and of the joint stops. Several works addressed the modeling of imperfect joints to account for the existence of clearances and bushings, generally independently of the formulation of the perfect kinematic joints. This work proposes a formulation in which both perfect and clearance/bushing joints share the same kinematic information making their modeling data similar and enabling their easy permutation in the context of multibody systems modeling. The proposed methodology is suitable for the most common mechanical joints and easily extended to many other joint types benefiting the exploration of a wide number of modeling applications, including the representation of cut-joints required for some formulations in multibody dynamics. The formulation presented in this work is applied to several demonstrative examples of spatial mechanisms to show the need to consider the type of imperfect joints and/or joints with stops modeling in practical applications.

Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments

Abstract: Coordinate singularities and gimbal lock are two phenomena that present themselves in models for the dynamics of mechanical systems. The former phenomenon pertains to the coordinates used to parameterize the configuration manifold of the system, while the latter phenomenon has a distinctive physical manifestation. In the present paper, we use tools from differential geometry to show how gimbal lock is intimately associated with an orthogonality condition on the applied forces and moments which act on the system. This condition is equivalent to a generalized applied force being normal to the configuration manifold of the system. Numerous examples, including the classic bead on a rotating hoop example and a gimbaled rigid body, are used to illuminate the orthogonality condition. These examples help to offer a new explanation for the elimination of gimbal lock by the addition of gimbals and demonstrate how integrable constraints alter the configuration manifold and may consequently eliminate coordinate singularities.

Screw and Lie group theory in multibody dynamics

Abstract: After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user-friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies and the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive $O ( n ) $ algorithms, for which the so-called “spatial operator algebra” is one example, and allows for use of readily available geometric data. In this paper, three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, that is, joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit–Hartenberg parameters, redundant. Four different definitions of twists are recalled, and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.

On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning

Abstract: This paper describes the fractional modeling and control of an industrial selective compliant assembly robot arm (SCARA); the fractional model was obtained by using the Euler–Lagrange and Hamilton formalisms. Each joint of the robot manipulator was driven by an induction motor. In this work, the fractional model of each induction motor was formulated, and the matching of the induction motors with the SCARA robot is shown. For comparison purposes, the SCARA robot control was formulated by conventional PI and PD and by fractional PIς and PDδ controllers. So each induction motor was controlled by using PI and fractional PIς controllers, and for trajectory tracking control, PD and fractional PDδ controllers were designed. For tuning the PI, PIς, PD, and PDδ controllers, the PSO algorithm was used; the same restrictions were used for the PI and PD classical controllers, and ITAE index was used as a cost function to be minimized. For computing the fractional derivatives and to obtain the numerical solution of the system, the Riemann–Liouville and Grunwald–Letnikov approaches were used. The numerical simulations have shown the effectiveness of the use of fractional PIς and PDδ controllers.

Multibody systems with 3D revolute joints with clearances: an industrial case study with an experimental validation

Abstract: This article is devoted to the analysis of the influence of the joint clearances in a mechanism of a circuit breaker, which is a 42 degree-of-freedom mechanism made of seven links, seven revolute joints, and four unilateral contacts with friction. Spatial (3D) revolute joints are modeled with both radial and axial clearances taking into account contact with flanges. Unilateral contact, Coulomb’s friction and Newton impact laws are modeled within the framework of nonsmooth mechanics without resorting to some regularizations or compliance/damping at contact. The nonsmooth contact dynamics method based on an event-capturing time-stepping scheme with a second order cone complementarity solver is used to perform the numerical integration. Furthermore, the stabilization of the constraints at the position level is made thanks to the stabilized combined projected Moreau–Jean scheme. The nonsmooth modeling approach together with an event–capturing time-stepping scheme allows us to simulate, in an efficient and robust way, the contact and impacts phenomena that occur in joints with clearances. In particular, comparing with the event-detecting time-stepping schemes, the event-capturing scheme enables us to perform the time-integration with a large number of events (impacts, sliding/sticking transitions, changes in the direction of sliding) and possibly with finite-time accumulations with a reasonable time-step length. Comparing with compliant contact models, we avoid stiff problems related with high stiffnesses at contact which generate some issues in contact stabilization and spurious oscillations during persistent contact periods. In the studied mechanisms of the circuit breakers, the numerical method deals with more than 70 contact points without any problems. Furthermore, the number of contact parameters is small—one coefficient of restitution and one coefficient of friction. Though they are sometimes difficult to measure accurately, the sensitivity of the simulation result with respect to contact parameters is low in the mechanism of the circuit breaker. It is demonstrated that this method, thanks to its robustness and efficiency, allows us to perform a sensitivity analysis using a Monte Carlo method. The numerical results are also validated by careful comparisons with experimental data, showing a very good correlation.

Dynamic modeling and vibration characteristics analysis of flexible-link and flexible-joint space manipulator

Abstract: With the development of space technology, lighter and larger space manipulators will be born, of which flexible characteristics are more obvious. The manipulator vibration caused by the flexibility not only reduces the efficiency of the manipulator but also affects the accuracy of the operation. The flexibility of space manipulator mainly comes from structural flexibility of links and transmission flexibility of harmonic gear reducer in joints. The vibrations generated by these two kinds of flexibility are coupled and transformed mutually, making the dynamics characteristics of space manipulator system complicated. Therefore it is difficult to assess respective effects of these flexibilities on vibrations of the manipulator tip. And the characteristics of integrated vibration of manipulator tip with different link and joint stiffnesses are not very clear. In this paper, the dynamic equations of multi-link multi-DOF flexible manipulator are established. Then, vibration responses of the tip under different elastic modulus, damping and joint stiffness were studied, and vibration characteristics of the tip with both link and joint were also analyzed. Moreover, the effects of motion planning on the vibration of the tip were analyzed. Finally, the vibration characteristics of the manipulator with flexible joints and links are verified by a two-degree-of-freedom manipulator experimental system. Dynamics analysis results presented some useful rules for the path planning and control to suppress the vibration of the flexible space manipulator.

A contact force model considering constant external forces for impact analysis in multibody dynamics

Abstract: A continuous contact force model, which considers the influence of constant external forces, is presented for the dynamic analysis of a multibody system. In this model, the Hertz contact law is applied to represent the nonlinear nature of contact, and a damping force is derived for evaluating the energy loss during impact. Together with the restitution coefficient, the external force influence factor defined in this paper is required for calculating the hysteresis damping factor associated with damping force. Moreover, the expression of hysteresis damping factor is deduced based on the energy-based method, which is adopted frequently in literature, and then it is improved by a weighted combination method with an exponential function due to the fact that the energy-based method has great errors when the restitution coefficient is low. Meanwhile, the exponential function is obtained by fitting the parametric surface of hysteresis damping factor gained from a numerical approach. Finally, four contact force models, including the new model, are utilized to compare the dynamic response of a special bouncing ball. The results illustrate that the described model is more suitable for impact analysis in multibody dynamics. In addition, the external forces and the energy loss are the main reasons for the multibody system to enter a steady contact state from repeated impact state.

Online prediction model for wheel wear considering track flexibility

Abstract: The objective of this study is to develop a new online model for wheel wear that takes into account the track flexibility. The proposed model consists of two parts that interact with each other, namely, (a) a locomotive/track coupled dynamics model considering the track flexibility, which is validated by field measurement results, and (b) a model for the wear estimation. The wheel wear prediction model can be employed in online solutions rather than in post-processing. The effect of including the track flexibility on the wear estimation is investigated by comparing the results with those obtained for a rigid track. Moreover, the effect of the wheel profile updating strategy on the wheel wear is also examined. The simulation results indicate that the track flexibility cannot be neglected for the wheel wear prediction. The wear predicted with the rigid track model is generally larger than that predicted with the flexible track model. The strategy of maintaining unchanged wheel profiles during the dynamic simulation coincides with the online updating strategy in terms of the predicted wear.

Dynamic modeling of tree-type robotic systems by combining 3 × 3 $3\times3$ rotation and 4 × 4 $4\times4$ transformation matrices

Abstract: The objective of this article is to present a systematic method of deriving the mathematical formulation for the oblique impact of a tree-type robotic manipulator. The dynamic response of this system (confined within a curved-wall environment) is expressed by two diverse models. A set of differential equations is employed to obtain the dynamic behavior of the system when it has no contact with any object in its environment (flying phase), and a set of algebraic equations is used to describe the collision of the system with the curved walls (impact phase). The Gibbs–Appell formulation in recursive form and the Newton’s impact law are utilized to derive the governing equations of this robotic system for the flying and impact phases, respectively. The main innovation of this article is the development of an automatic approach based on the combination of $3\times3$ rotation and $4\times4$ transformation matrices. In fact, this is the first time the merits of $3\times3$ rotation matrices (i.e., improving the computational efficiency of the developed algorithm) have been merged with the capabilities of $4\times4$ transformation matrices (i.e., deriving more compact motion equations by combining rotations with translations). Finally, a case study involving a tree-type robotic system with 12 degrees of freedom has been simulated to show the efficiency of the proposed dynamic modeling.

Walking of biped with passive exoskeleton: evaluation of energy consumption

Abstract: The paper aims to theoretically show the feasibility and efficiency of a passive exoskeleton for a human walking and carrying a load. The human is modeled using a planar bipedal anthropomorphic mechanism. This mechanism consists of a trunk and two identical legs; each leg consists of a thigh, shin, and foot (massless). The exoskeleton is considered also as an anthropomorphic mechanism. The shape and the degrees of freedom of the exoskeleton are identical to the biped (to human)—the topology of the exoskeleton is the same as of the biped (human). Each model of the human and exoskeleton has seven links and six joints. The hip-joint connects the trunk and two thighs of the two legs. If the biped is equipped with an exoskeleton, then the links of this exoskeleton are attached to the corresponding links of the biped and the corresponding hip, knee, and ankle joints coincide. We compare the walking gaits of a biped alone (without exoskeleton) and of a biped equipped with exoskeleton; for both cases the same load is transported. The problem is studied in the framework of a ballistic walking model. During ballistic walking of the biped with exoskeleton, the knee of the support leg is locked, but the knee of the swing leg is unlocked. The locking and unlocking can be realized in the knees of the exoskeleton by any mechanical brake devices without energy consumption. There are no actuators in the exoskeleton. Therefore, we call it a passive exoskeleton. The walking of the biped consists of alternating single- and double-support phases. In our study, the double-support phase is assumed instantaneous. At the instant of this phase, the knee of the previous swing leg is locked and the knee of the previous support leg is unlocked. Numerical results show that during the load transport the human with the exoskeleton spends less energy than human alone. For transportation of a load with mass 40 kg, the economy of the energy is approximately 28%, if the length of the step and its duration are equal to 0.5 m and 0.5 s, respectively.

Development of a multibody system for crashworthiness certification of aircraft seat

Abstract: This work proposes a multibody approach in the simulation of 16-g aircraft seats, referred to the front-row of seats located behind bulkheads compliance with the Head Injury Criteria (HIC) requirement. The multibody model of the seat structure has been developed and analysed by using a home-made algorithm implemented in Matlab® code, as a 2D system of rigid bodies interconnected by springs and joints. The research has been oriented to assess the capability of simulating a 16g frontal impact of a sled equipped with the seat of a regional aircraft on which an anthropomorphic dummy is arranged. This sled test, for which experimental data were available, has been used as test case; inertial and structural properties of the system have also been experimentally and numerically evaluated in order to make the numerical model compliant with the real one. One of the primary goals of the paper is to provide an intuitive, easily extendable numerical tool to support designers in multibody simulation and to define a tool able to obtain global sled-test results in very short time, especially if compared to the computational time of a detailed finite element simulation. This tool will allow running sensitivity analysis and first level optimisation of key design parameters, integrating itself in the design cycle, not in place of, but as a support to the main simulation tools.

A 3D ellipsoidal volumetric foot–ground contact model for forward dynamics

Abstract: Foot–ground contact models are an important part of forward dynamic biomechanic models, particularly those used to model gait, and have many challenges associated with them. Contact models can dramatically increase the complexity of the multibody system equations, especially if the contact surface is relatively large or conforming. Since foot–ground contact has a large potential contact area, creating a computationally efficient model is challenging. This is particularly problematic in predictive simulations, which may determine optimal performance by running a model simulation thousands of times. An ideal contact model must find a balance between accuracy for large, conforming surfaces, and computational efficiency. Volumetric contact modelling is explored as a computationally efficient model for foot–ground contact. Previous foot models have used volumetric contact before, but were limited to 2D motion and approximated the surfaces as spheres or 2D shapes. The model presented here improves on current work by using ellipsoid contact geometry and considering 3D motion and geometry. A gait experiment was used to parametrise and validate the model. The model ran over 100 times faster than real-time (in an inverse simulation at 128 fps) and matched experimental normal force and centre of pressure location with less than 7% root-mean-square error. In most gait studies, only the net reaction forces, centre of pressure, and body motions are recorded and used to identify parameters. In this study, contact pressure was also recorded and used as a part of the identification, which was found to increase parameter optimisation time from 10 to 164 s (due to the additional time needed to calculate the pressure distribution) but helped the results converge to a more realistic model. The model matched experimental pressures with 33–45% root-mean-square error, though some of this was due to measurement errors. The same parametrisation was done with friction included in the foot model. It was determined that the velocity-based friction model that was used was inappropriate for use in an inverse-dynamics simulation. Attempting to optimise the model to match experimental friction resulted in a poor match to the experimental friction forces, inaccurate values for the coefficient of friction, and a poorer match to the experimental normal force.

Dynamics analysis and fuzzy anti-swing control design of overhead crane system based on Riccati discrete time transfer matrix method

Abstract: This paper describes an efficient method called Riccati discrete time transfer matrix method of multibody system (MS-RDTTMM) for studying the dynamic modeling and anti-swing control design of a two-dimensional overhead crane system, which consists of a trolley, rope, load, and control subsystem. Regarding the rope as a series of rigid segments connected by hinges, a multibody model of the overhead crane system can be developed easily by using MS-RDTTMM. Then three separate fuzzy logic controllers are designed for positioning and anti-swing control. For improving the performance of the predesigned fuzzy control system, the genetic algorithm based on MS-RDTTMM is presented offline to tune the initial control parameters. Using the recursive transfer formula to describe the system dynamics, instead of the global dynamics equation in ordinary dynamics methods, the matrices involved in this method are always very small, and the computational cost of the dynamic analysis and control system optimization can be greatly reduced. The numerical verification is carried out to show the computational efficiency, numerical stability, and control performance of the proposed method.

Combined semi-recursive formulation and lumped fluid method for monolithic simulation of multibody and hydraulic dynamics

Abstract: The use of multibody simulation tools allows complex machinery to be described in detail while still providing a solution for the system in real time. As mechanical components are often accompanied by other dynamical systems, such as hydraulics, description of each subsystem is required to fully describe the dynamics of complex machinery. A potential candidate for solving the multiphysics problem at hand is known as the unified or monolithic approach. This strongly coupled approach yields a single set of equations to be integrated and, compared to co-simulation and co-integration approaches, a relatively simple integration procedure. In this paper, a monolithic formulation for a combined simulation of multibody and hydraulic dynamics using an efficient semi-recursive formulation and the lumped fluid method is introduced. The results indicate that the proposed method shows potential for efficient simulation of combined multibody and hydraulic problems. The robustness of the multibody method is maintained when combined with the hydraulic dynamics description and higher efficiency is observed than with an equivalent global approach.

Feedback control of multibody systems with joint clearance and dynamic backlash: a tutorial

Abstract: The problem of feedback control of mechanisms with joint clearance is analysed. Various control strategies are reviewed: impactless trajectories with persistent contact, control through collisions, the stabilization of equilibrium points, and trajectory tracking control. This article sets a general control framework, brings some preliminary answers and leaves some problems open, which are mentioned throughout the article and in the conclusions.

Study of the dynamic vehicle-track interaction in a railway turnout

Abstract: This paper analyzes the vertical dynamic response of a railway track subjected to traffic loads in a turnout, especially around the switch blades and the crossing nose. In order to study the influence of vehicle and track parameters on the vehicle-track dynamics, a numerical feedback interaction between a multi-body model of the vehicle and a 3D finite elements model of the track is carried out. For this purpose, two different models are developed. In the first one, the track is modeled by means of a FE model in the time domain through ANSYS software; while in the second one, the vehicle is simulated as a multi-body model by means of VAMPIRE PRO software. The influence of different parameters (e.g., speed, sprung masses, track stiffness and vehicle eigenfrequencies) on the generation of dynamic loads in a turnout, especially in the switch blades and the crossing nose, is studied. Finally, the vibrations induced by the passing of the vehicle are calculated for different scenarios.

Real-time trajectory control of an overhead crane using servo-constraints

Abstract: In this paper, the system dynamics of an overhead crane are inverted by servo-constraints. The inversion provides a feedforward control for trajectory tracking of the system output. The overhead crane is inherently underactuated and modeled as a two-dimensional mechanical system with nonlinear system dynamics. Actuators are modeled as first-order systems to simplify implementation and account for velocity-controlled drives. The control based on servo-constraints is shown to be an effective method of trajectory control for overhead cranes. It will be demonstrated that the formulation is solvable in real-time using linear implicit Euler integration. The feedforward solution is made robust by an augmentation with LQR as well as a sliding mode controller. Experiments are conducted on a laboratory crane of 13 m motion range.

Flexible multibody dynamics using coordinate reduction improved by dynamic correction

Abstract: This paper is concerned with the efficient dynamic analysis of flexible multibody systems using a robust coordinate reduction technique. Unlike conventional static correction, the formulation is derived by dynamic correction that considers the inertia effect. In this formulation, the constraint and fixed-interface normal modes, which are representative modes in the typical coordinate reduction, are corrected by considering the truncated modal effect with the residual flexibility. Therefore, the proposed method can offer a more precise reduced system without increasing the dimension, which consequently leads to a more accurate and efficient flexible multibody simulation. We implement here the proposed method under augmented formulations of the floating reference frame approach, and test its performance with numerical examples.

An EMG-marker tracking optimisation method for estimating muscle forces

Abstract: Existing algorithms for estimating muscle forces mainly use least-activation criteria, which do not necessarily lead to physiologically consistent results. Our objective was to assess an innovative forward dynamics-based optimisation, assisted by both electromyography (EMG) and marker tracking, for estimating the upper-limb muscle forces. A reference movement was generated, and EMG was simulated to reproduce the desired joint kinematics. Random noise was added to both simulated EMG and marker trajectories in order to create 30 trials. Then, muscle forces were estimated using (1) the innovative EMG-marker tracking forward optimisation, (2) a marker tracking forward optimisation with a least-excitation criterion, and (3) static optimisation with a least-activation criterion. Approaches (1) and (2) were solved using a direct multiple shooting algorithm. Finally, reference and estimated joint angles and muscle forces for the three optimisations were statistically compared using root-mean-square errors (RMSEs), biases, and statistical parametric mapping. The joint angles RMSEs were qualitatively similar across the three optimisations: (1) $1.63 \pm 0.51$ °; (2) $2.02 \pm 0.64$ °; (3) $0.79 \pm 0.38$ °. However, the muscle forces RMSE for the EMG-marker tracking optimisation ( $20.39 \pm 13.24$ N) was about seven times smaller than those resulting from the marker tracking ( $124.22 \pm 118.22$ N) and static ( $148.15 \pm 94.01$ N) optimisations. The originality of this novel approach is close tracking of both simulated EMG and marker trajectories in the same objective function, using forward dynamics. Therefore, the presented EMG-marker tracking optimisation led to accurate muscle forces estimations.

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

Abstract: In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.

Implicit co-simulation method for constraint coupling with improved stability behavior

Abstract: This paper deals with a novel co-simulation approach for coupling mechanical subsystems in time domain. The submodels are assumed to be coupled by algebraic constraint equations. In contrast to well-known coupling techniques from the literature, the here presented index-1 approach uses a special technique for approximating the coupling variables so that the constraint equations together with the hidden constraints on velocity and acceleration level can be enforced simultaneously at the communication time points. The method discussed here uses second- and third-order approximation polynomials. Because of the high approximation order, the numerical errors are very small, and a good convergence behavior is achieved. A stability analysis is carried out, and it is shown that—despite the fact that higher-order approximation polynomials are applied—also a good numerical stability behavior is observed. Different numerical examples are presented, which illustrate the practical application of the approach.

Flexible multibody modeling of reeving systems including transverse vibrations

Abstract: This paper presents a discretization procedure for the flexible multibody modeling of reeving systems. Reeving systems are assumed to include a set of rigid bodies connected by wire ropes using a set of sheaves and reels. The method is capable to model the deformation of the varying-length wire-rope spans. Wire ropes are assumed to deform axially, transversally and in torsion. This paper shows the capability of the presented method to model transverse vibrations. The discretization procedure uses a combination of absolute position coordinates, relative-transverse deformation coordinates and longitudinal material coordinates. Each wire-rope span is modeled using a single two-noded element under an arbitrary Lagrangian–Eulerian approach. The discretization method is validated using analytical and numerical reference solutions found in the literature that describe the dynamics of varying-length strings. In addition, the dynamics of a three-dimensional tower crane is simulated.

The discrete adjoint method for parameter identification in multibody system dynamics.

Abstract: The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.

Dynamic balance preservation and prevention of sliding for humanoid robots in the presence of multiple spatial contacts

Abstract: The main indicator of dynamic balance is the $\mathit{ZMP}$ . Its original notion assumes that both feet of the robot are in contact with the flat horizontal surface (all contacts are in the same plane) and that the friction is high enough so that sliding does not occur. With increasing capabilities of humanoid robots and the higher complexity of the motion that needs to be performed, these assumptions might not hold. Having in mind that the system is dynamically balanced if there is no rotation about the edges of the feet and if the feet do not slide, we propose a novel approach for testing the dynamic balance of bipedal robots, by using linear contact wrench conditions compiled in a single matrix (Dynamic Balance Matrix). The proposed approach has wide applicability since it can be used to check the stability of different kinds of contacts (including point, line, and surface) with arbitrary perimeter shapes. Motion feasibility conditions are derived on the basis of the conditions which the wrench of each contact has to satisfy. The approach was tested by simulation in two scenarios: biped climbing up and walking sideways on the inclined flat surface which is too steep for a regular walk without additional support. The whole-body motion was synthesized and performed using a generalized task prioritization framework.

Modelling of flexible bodies with minimal coordinates by means of the corotational formulation

Abstract: This paper is concerned with the extension of the minimal coordinates approach to flexible bodies. When using minimal coordinates, the number of configuration parameters corresponds exactly to the number of degrees of freedom and they can be chosen arbitrarily as far as there is a one-to-one relationship between the configuration of the system and the configuration parameters. In the rigid case, the equations of motion are obtained from the description of the translational and rotational motion of a frame attached to each body in terms of the chosen configuration parameters, and from the forces acting on each body. The extension to the simulation of flexible bodies naturally leads to a description of the motion of a flexible body from the one of its nodes. However, the relationship between the latter and the full internal motion of the body is not unique and is the subject of various developments. It was then proposed for the sake of generality to systematically treat flexible bodies as superelements, implemented according to the corotational approach, with a floating corotational frame. This allows to model any flexible body from its mass and stiffness matrices obtained from any available finite element code. Moreover, it doesn’t impose any restriction on the kinematics of the nodes which can then be expressed indifferently from absolute or relative coordinates as usually encountered with minimal coordinates. After a description of the adopted framework, the paper develops the equations of motion. Some test examples are presented, where the proposed approach will be compared to the ones obtained with the classical body reference frame approach and results from the literature. In some cases, the influence of the chosen corotational frame is analysed. The examples confirm that the corotational formulation should be restricted to flexible bodies involving only small deformations and rotational velocity. It is also shown that modelling can be adapted to improve the quality of the results.

Dynamic contact model of shell for multibody system applications

Abstract: In a multibody system consisting of shell structures, the contact may appear in any area of shells. It is difficult to simulate the contact of shells with large deformation because of the geometric nonlinearity of deformation and the boundary nonlinearity of contact. This study presents a rotation-free shell formulation and an extended contact discretization in multibody systems using a corotational frame. This model is different from previous formulations in the definition of the local frame and the processing of local large curvature. In order to deal with the shell contact, a unified contact discretization scheme including edge-to-edge contact for facet triangle shell elements is proposed to solve the large penetration problem. A series of numerical examples of multibody dynamics have validated the approach of the nonlinear shell model and contact treatments. Moreover, a practical application of deployment of solar cells shows the capability of the proposed formulation in solving large-scale problems of flexible multibody system with large deformation and contact.

Skeletal-level control-based forward dynamic analysis of acquired healthy and assisted gait motion

Abstract: Gait analysis is commonly addressed through inverse dynamics. However, forward dynamics can be advantageous when descending to muscular level, as it allows activation and contraction equations to be integrated with motion thus providing better dynamic consistency, or when studying assisted gait, as it enables the estimation of the interaction forces between subject and devices even if the motion capture process doesn’t provide enough resolution to distinguish the motions of limb and device. Control-based methods seem to be the most natural choice to carry out the forward-dynamics analysis of an acquired gait, but several options exist in their application. The paper explores such options for healthy and assisted gait, and concludes that the computed torque control of all the subject’s degrees of freedom is the alternative that provides the most accurate results. Moreover, the study of its more problematic underactuated variant accompanied by contact models showed to be connected to neighbor challenging topics as gait prediction or walking simulation of humanoids.

Analytic solution for planar indeterminate impact problems using an energy constraint

Abstract: This work proposes an analytic method for resolving planar multi-point indeterminate impact problems for rigid-body systems. An event-based approach is used to detect impact events, and constraints consistent with the rigid-body assumption are used to resolve the indeterminacy associated with multi-point impact analysis. The work-energy relation is utilized to determine post-impact velocities based on an energetic coefficient of restitution to model energy dissipation, thereby yielding an energetically consistent set of post-impact velocities based on Stronge’s energetic coefficient of restitution for the treatment of rigid impacts. The effect of stick–slip transition is analyzed based on Coulomb friction. This paper also discusses the transition from impact to contact. This analysis is essential for considering the rocking block problem that is used as an example herein. The predictions of the model for the rocking block problem are compared to experimental results published in the literature. An example of a planar ball undergoing two-point impact is also presented.

Design and validation of a multi-body model of a front suspension bicycle and a passive rider for braking dynamics investigations

Abstract: The global spread of Electric Bicycles (EBs) is increasing more and more. In addition to supporting the ease of cycling, the electric energy available can also be used for innovative braking control systems. The project BikeSafe picks up on this idea and aims at developing an active Braking Dynamics Assistance system (BDA) for EBs equipped with hydraulic brakes. A simulation model taking into account all substantial braking dynamics influences is necessary for the model-based design of the BDA. This paper presents a Multi-Body Model (MBM) of a front suspension bicycle and a passive rider. This MBM has been experimentally validated for in-plane braking dynamics using road tests. It has real-time ability and can therefore be used as a virtual representation of the plant for the model-based design process of the BDA.