Inertia Shape Research Articles

Partially Decoupled Impedance Motion Force Control Using Prioritized Inertia Shaping

Partially Decoupled Impedance Motion Force Control Using Prioritized Inertia Shaping

Beam modeling in a floating frame of reference for torsion dynamics of helicopter rotor blades

AbstractIn the ongoing development of DLR’s Versatile Aeromechanics Simulation Tool, an elastic-beam model is integrated into the multibody system based on the floating frame-of-reference formulation. Although the application of this formulation for one-dimensional beam models has already been addressed in the literature, the challenge remains to properly model the torsion dynamics of rotor blades – especially under high centrifugal loads. To this aim, this work suggests the consideration of rotational shape functions in the inertia shape integrals and in the application of gravitational, inertial, and external loads. This modified approach is validated based on the structural analysis of a rotor blade with complex geometrical properties.

A unified model with inertia shaping for highly dynamic jumps of legged robots

A unified model with inertia shaping for highly dynamic jumps of legged robots

Improved Landscape Expansion Index and Its Application to Urban Growth in Urumqi

Automatic determination of quantitative parameters describing the pattern of urban expansion is extremely important for urban planning, urban management and civic resource configuration. Though the widely adopted LEI (landscape expansion index) has exhibited the potential to capture the evolution of urban landscape patterns using multi-temporal remote sensing data, erroneous determination still exists, especially for patches with special shapes due to the limited consideration of spatial relationships among patches. In this paper, we improve the identification of urban landscape expansion patterns with an enhancement of the topological relationship. We propose MCI (Mean patch Compactness Index) and AWCI (Area-Weighted Compactness Index) in terms of the moment of inertia shape index. The effectiveness of the improved approach in identifying types of expansion patches is theoretically demonstrated with a series of designed experiments. Further, we apply the proposed approaches to the analysis of urban expansion features and dynamics of urban compactness over Urumqi at various 5-year stages using available SUBAD-China data from 1990–2015. The results achieved by the theoretical experiments and case application show our approach effectively suppressed the effects induced by shapes of patches and buffer or envelope box parameters for the accurate identification of patch type. Moreover, the modified MCI and AWCI exhibited an improved potential in capturing the landscape scale compactness of urban dynamics. The investigated 25-year urban expansion of Urumqi is dominated by edge-expansion patches and supplemented by outlying growth, with opposite trends of increasing and decreasing, with a gradual decrease in landscape fragmentation. Our examination using the proposed MCI and AWCI indicates Urumqi was growing more compact in latter 15-year period compared with the first 10 years studied, with the primary urban patches tending to be compacted earlier than the entire urban setting. The historical transformation trajectories based on remote sensing data show a significant construction land gain—from 1.06% in 1990 to 6.96% in 2015—due to 289.16 km2 of recently established construction, accompanied by fast expansion northward, less dynamic expansion southward, and earlier extension in the westward direction than eastward. This work provides a possible means to improve the identification of patch expansion type and further understand the compactness of urban dynamics.

Momentum-Aware Trajectory Optimization and Control for Agile Quadrupedal Locomotion

In this letter, we present a versatile hierarchical offline planning algorithm, along with an online control pipeline for agile quadrupedal locomotion. Our offline planner alternates between optimizing centroidal dynamics for a reduced-order model and whole-body trajectory optimization, with the aim of achieving dynamics consensus. Our novel momentum-inertia-aware centroidal optimization, which uses an equimomental ellipsoid parameterization, is able to generate highly acrobatic motions via “inertia shaping”. Our whole-body optimization approach significantly improves upon the quality of standard DDP-based approaches by iteratively exploiting feedback from the centroidal level. For online control, we have developed a novel convex model predictive control scheme through a linear transformation of the full centroidal dynamics. Our controller can efficiently optimize for both contact forces and joint accelerations in single optimization, enabling more straightforward tracking for momentum-rich motions compared to existing quadrupedal MPC controllers. We demonstrate the capability and generality of our trajectory planner on four different dynamic maneuvers. We then present one hardware experiment on the MIT Mini Cheetah platform to demonstrate the performance of the entire planning and control pipeline on a twisting jump maneuver.

Practical consequences of inertia shaping for interaction and tracking in robot control

In trajectory tracking and interaction control of robots, two fundamentally different concepts define the boundaries within which most nonlinear model-based approaches can be located. On the one hand controllers such as the PD+ preserve the natural inertia and avoid feedback of external forces and torques. On the other hand controllers based on feedback linearization, as used in most inverse dynamics approaches, enforce linear closed-loop dynamics by means of external force/torque feedback. Here, these two basic concepts of keeping and shaping of the natural inertia are investigated and compared including aspects such as interaction behavior, tracking performance, tuning parameters, influence of modeling errors, and effective feedback gains. Exemplary case studies on a standard torque-controlled robot are performed. The understanding of these features and differences is of major importance for the proper selection and deployment of interaction and tracking controllers in practice.

The nodal-based floating frame of reference formulation with modal reduction

In a recent paper of the authors, a novel nodal-based floating frame of reference formulation (FFRF) for solid finite elements has been proposed. The nodal-based approach bypasses the unhandy inertia shape integrals ab initio, i.e. they neither arise in the derivation nor in the final equations of motion, leading to a surprisingly simple derivation and computer implementation without a lumped mass approximation, which is conventionally employed within commercial multibody codes. However, the nodal-based FFRF has so far been presented without modal reduction, which is usually required for efficient simulations. Hence, the aim of this follow-up paper is to bring the nodal-based FFRF into a suitable form, which allows the incorporation of modal reduction techniques to reduce the overall system size down to the number of modes included in the reduction basis, which further reduces the computational complexity significantly. Moreover, this exhibits a way to calculate the so-called FFRF invariants, which are constant “ingredients” required to set up the FFRF mass matrix and quadratic velocity vector, without integrals and without a lumped mass approximation.

A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

The Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation.

Impedance control on rack steering vehicle for inertia shaping on cornering track

Overdriven factor in a vehicle motion is one of the issues that need to be tackled for safety and energy efficiencies, especially in the cornering track. This issue is crucial especially for rack steering vehicle with non-holonomic configuration, whereby inertia will contribute to high collisions to the peer walls or offroad incidents. Therefore this study has taken the initiative to propose a dynamic control technique that considers the interaction between the vehicle and the terrain using impedance control. This control technique allows coping with the issue by indirectly shaping inertia forces. The proposed impedance control is derived by handling the vehicle dynamic developed and shaping the vehicle steering angle. For the study purposes, a rack steering four wheels vehicle (RT4WV) is used as a platform, and its dynamic model was derived for analysis. The implementation of torque feedback based impedance control for inertia shaping is emphasized on both vertical (x axis) and horizontal (y axis) of the vehicle body, during which inertia could happen. The kinodynamic input for the system control input is the difference between steering angular changes in which representing front wheel angular changes. This proposed dynamic control strategy is verified by simulating the derived impedance control on RT4WV system model with road terrain and aerodynamic frictions as disturbances. The result shows that the proposed impedance control able to reduce the inertia forces via shaping the steering angular input to the vehicle although there have both road terrain and aerodynamic frictions, especially in cornering tracks.

Further investigation on the coupling between the reference and elastic displacements in flexible body dynamics

A new semi-analytical multibody system (MBS) model derived using general nonlinear flexible body dynamics formulation is developed in this investigation. The semi-analytical model can serve as a benchmark example for testing the implementation of the finite element (FE) floating frame of reference (FFR) formulation in commercial MBS software. In particular, this new model sheds light on the important concept of the reference conditions which is fundamental in the computer implementation of the FE/FFR formulation. In order to develop the new FE/FFR semi-analytical model, the reference motion of the flexible body is expressed in terms of specified motion trajectories, allowing for developing a set of second-order ordinary differential equations (ODE's) with time-varying coefficients. The FE/FFR inertia shape integrals are evaluated and used to define the generalized forces associated with the deformation coordinates. The simplifications that result from the use of the mean-axis reference conditions are highlighted and the significance of these simplifications on the solution accuracy when using more appropriate sets of reference conditions is discussed. It is shown that if a MBS computer algorithm restricts the choice of the reference conditions to one set, the use of other reference conditions can lead to a homogeneous ODE system with time-varying coefficients which has zero solution in the case of zero initial conditions and zero external forces regardless of the flexible body reference rotation.

Inertial Properties in Haptic Devices: Non-Linear Inertia Shaping vs. Force Feedforward

Abstract The inertia of haptic devices limits the user’s manipulation and dynamically couples the Cartesian motion, which influences the transparency and fidelity in haptic feedback. By employing force-torque sensing, we investigate two approaches to reduce the apparent inertial effect of haptic devices and to overcome dynamic coupling. First, in order to shape the apparent inertia felt by the user, non-linear inertia shaping (NIS) is presented and introduced to the field of haptics. NIS is based on non-linear dynamic decoupling (NLD). Second, as a standard approach, force feedforward control (FF) is presented that uniformly scales down the apparent inertia. We demonstrate that FF is a special case of NIS, under the assumption that gravitational, centripetal and Coriolis terms are neglected. Simulations and experiments were conducted on DLR’s bi-manual haptic device HUG. It is shown that NIS is suited to compensate for the coupling effects, while FF can reduce the apparent inertia more effectively.

Inertia forces and shape integrals in the floating frame of reference formulation

Modeling and analysis of complex dynamical systems can be effectively performed using multibody system (MBS) simulation software. Many modern MBS packages are able to efficiently and reliably handle rigid and flexible bodies, often offering a wide choice of different formulations. Despite many advances in modeling of flexible systems, the most widely used formulation remains the well-established floating frame of reference formulation (FFRF). Although FFRF usually allows inclusion of only small elastic deformations, this assumption is adequate for many industrial applications. In addition, FFRF is computationally efficient if implemented with appropriate model order reduction techniques and effective handling of system inertia terms by utilization of so-called inertia shape integrals. However, derivation of the system of equations of motion for FFRF bodies is a complex and often error-prone task. The main goal of this paper is to provide a reliable, detailed, universal and clear set of inertia terms within the FFRF. The paper concentrates on detailed derivation of the inertia forces with focus on accurate determination and exploitation of the inertia shape integrals. Two standard methods are employed, namely the Lagrangian and Virtual Work approaches. Additionally, the introduced derivations are executed without selection of specific rotational parameters. Direct application of Euler parameters and Euler angles is presented. It is found that the derived expressions are well suited for direct implementation and simplify derivation of force components.

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.

Modeling Flexible Bodies in Multibody Systems in Joint-Coordinates Formulation Using Spatial Algebra

This paper presents an efficient approach of using spatial algebra operator to formulate the kinematic and dynamic equations for developing capabilities to model flexible bodies within general purpose multibody dynamics solver. The proposed approach utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are initially formulated using the Cartesian body coordinates (CBC) and the joint coordinates (JC) to form an augmented set of differential algebraic equations. The system connectivity matrix is derived from system topological relations and is used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations. The modal transformation matrix is used to describe the finite element kinematics in terms of a small set of generalized modal coordinates. Although the resulting stiffness matrix is constant, the mass matrix depends on the generalized elastic modal coordinates and needs to be updated at each time step. To reduce the computational efforts, a set of precomputed inertia shape invariants (ISI) can be identified and used to update the flexible body mass matrix. In this proposed joint-coordinates formulation, the transformation operations required for the flexible body inertia matrix are different from those in case of CBC formulation. The necessary ISI and the algorithm to reconstruct the modal mass matrix will be presented in this paper.

Direction-changing fall control of humanoid robots: theory and experiments

Humanoid robots are expected to share human environments in the future and it is important to ensure the safety of their operation. A serious threat to safety is the fall of such robots, which can seriously damage the robot itself as well as objects in its surrounding. Although fall is a rare event in the life of a humanoid robot, the robot must be equipped with a robust fall strategy since the consequences of fall can be catastrophic. In this paper we present a strategy to change the default fall direction of a robot, during the fall. By changing the fall direction the robot may avoid falling on a delicate object or on a person. Our approach is based on the key observation that the toppling motion of a robot necessarily occurs at an edge of its support area. To modify the fall direction the robot needs to change the position and orientation of this edge vis-a-vis the prohibited directions. We achieve this through intelligent stepping as soon as the fall is predicted. We compute the optimal stepping location which results in the safest fall. Additional improvement to the fall controller is achieved through inertia shaping, which is a principled approach aimed at manipulating the robot's centroidal inertia, thereby indirectly controlling its fall direction. We describe the theory behind this approach and demonstrate our results through simulation and experiments of the Aldebaran NAO H25 robot. To our knowledge, this is the first implementation of a controller that attempts to change the fall direction of a humanoid robot.

Efficient calculation of the inertia terms in floating frame of reference formulations for flexible multibody dynamics

One of the characteristics of floating frame of reference (FFR) formulations for flexible multibody dynamics is the fact that the inertia terms are highly non-linear. At every time-step, both the mass matrix and the velocity-dependent forces vector must be updated, and this can become the most CPU intensive task. This work studies the efficiency of two different methods for performing this operation, when applied to both a formulation in absolute coordinates and another in relative coordinates. The first method calculates the inertia terms by projecting the finite element (FE) mass matrix into the generalized coordinates, by means of a variable projection matrix. The second one calculates the inertia shape integrals at a preprocessing stage and uses them for obtaining the inertia terms in a more efficient way, at the cost of a more involved implementation. Both methods have been tested when used in combination with either the FFR absolute or relative formulation, by simulating a vehicle with 12 flexible elements. The results show that the performance can be considerably increased by means of the preprocessing method, especially in the case of large FE models, whereas, for small models, the projection method can be more convenient due to its simplicity.

On the Passivity-Based Impedance Control of Flexible Joint Robots

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, a novel type of impedance controllers for flexible joint robots is proposed. As a target impedance, a desired stiffness and damping are considered without inertia shaping. For this problem, two controllers of different complexity are proposed. Both have a cascaded structure with an inner torque feedback loop and an outer impedance controller. For the torque feedback, a physical interpretation as a scaling of the motor inertia is given, which allows to incorporate the torque feedback into a passivity-based analysis. The outer impedance control law is then designed differently for the two controllers. In the first approach, the stiffness and damping terms and the gravity compensation term are designed separately. This outer control loop uses only the motor position and velocity, but no noncollocated feedback of the joint torques or link side positions. In combination with the physical interpretation of torque feedback, this allows us to give a proof of the asymptotic stability of the closed-loop system based on the passivity properties of the system. The second control law is a refinement of this approach, in which the gravity compensation and the stiffness implementation are designed in a combined way. Thereby, a desired static stiffness relationship is obtained exactly. Additionally, some extensions of the controller to viscoelastic joints and to Cartesian impedance control are given. Finally, some experiments with the German Aerospace Center (DLR) lightweight robots verify the developed controllers and show the efficiency of the proposed control approach. </para>

How Dynamic Consumer Response, Competitor Response, Company Support, and Company Inertia Shape Long-Term Marketing Effectiveness

Long-term marketing effectiveness is a high-priority research topic for managers, and emerges from the complex interplay among dynamic reactions of several market players. This paper introduces restricted policy simulations to distinguish four dynamic forces: consumer response, competitor response, company inertia, and company support. A rich marketing dataset allows the analysis of price, display, feature, advertising, and product-line extensions. The first finding is that consumer response differs significantly from the net effectiveness of product-line extensions, price, feature, and advertising. In particular, net sales effects are up to five times stronger and longer-lasting than consumer response. Second, this difference is not due to competitor response, but to company action. For tactical actions (price and feature), it takes the form of inertia, as promotions last for several weeks. For strategic actions (advertising and product-line extensions), support by other marketing instruments greatly enhances dynamic consumer response. This company action negates the postpromotion dip in consumer response, and enhances the long-term sales benefits of product-line extensions, feature, and advertising. Therefore, managers are urged to evaluate company decision rules for inertia and support when assessing long-term marketing effectiveness.

Dynamic and Vibration Analysis of a Vehicle Rear Axle System

A detailed finite element model for the rear axle system of a sport utility vehicle is developed in this investigation. The axle system is treated as a multibody system that consists of nine bodies that include the input shaft, two output shafts, the carrier and tube system, four control arms and a track bar. The rotating input and output shafts are mounted on the carrier and tube system using six bearings. The four control arms and the track bar are connected to the carrier system and the frame of the vehicle using rubber bushings. In the model developed in this investigation, three dimensional beam elements are used to develop the finite element model for the input and output axle shafts, the control arms, and the track bar. A non-conventional finite element formulation is used to develop the equations of motion of the rotating input and output shafts in order to account for the effect of their angular velocities. These equations are expressed in terms of inertia shape integrals that depend on the assumed displacement field. The inertia shape integrals are first evaluated for each finite element. The inertia shape integrals of the rotating shafts are obtained by assembling the inertia shape integrals of its finite elements using a standard finite element assembly procedure. A conventional finite element formulation is used for the control arms and the track bar. The model developed in this investigation includes the effect of the bearing stiffness, the effect of the stiffness of the helical springs of the suspension system, and the effect of the stiffness of the tires. Using the Lagrangian dynamics and the finite element method, the equations of motion of the axle system are developed and expressed in terms of the nodal coordinates of the shafts, the control arms and the track bar as well as the degrees of freedom of the carrier. This finite dimensional model is used to determine the mode shapes and the natural frequencies of the axle system. The discrepancies between several of the natural frequencies predicted using the dynamic model developed in this investigation and natural frequencies determined experimentally are found to be less than 2%. A parametric study is performed in order to investigate the effect of the axle system parameters on the natural frequencies and mode shapes. Using the modal transformation, a set of differential equations of motion of the axle system is developed and used to examine the system dynamics under given loading conditions. The solutions of the resulting equations of motion are obtained using numerical methods.

Equivalence of the floating frame of reference approach and finite element formulations

The floating frame of reference formulation which is widely used in flexible multibody simulations leads to a highly non-linear inertia matrix. This matrix which exhibits a strong inertia coupling between the reference motion and the elastic deformation can be expressed in terms of a unique set of inertia shape integrals that depend on the assumed displacement field. In this paper, we demonstrate that the floating frame of reference formulation leads to the same dynamic relationship obtained using large deformation finite element formulations. This result clearly demonstrates that the floating frame of reference formulation can also be used in the large deformation analysis of constrained flexible bodies provided that no infinitesimal or finite rotations are used as nodal coordinates; instead, displacements and slopes are used to describe the element configuration. The relationship between the local and the global slopes is defined and used to establish a coordinate transformation which is used to demonstrate the equivalence of the dynamic relationships and the inertia forces obtained using two different finite element formulations. Using this equivalence relationship, a simple and systematic procedure for evaluating all the inertia shape integrals that appear in the floating frame of reference formulation from the constant mass matrix that appears in linear structural dynamics is developed. In order to develop this procedure, the concepts of the local and global shape functions are introduced. The local shape function, which is used in the floating frame of reference formulation, does not have any rigid body modes and is defined using an appropriate set of reference conditions. The global shape function, on the other hand, has a complete set of rigid body modes that can describe an arbitrary rigid body displacement. The analysis presented in this paper demonstrates that the floating frame of reference formulation does not lead to a separation between the rigid body motion and the elastic deformation, and in such a formulation, the reference motion cannot be considered as the rigid body motion of the deformable body.