Interactive simulation of solid rigid bodies

The article describes the implementation of an interactive system for simulating rigid bodies with contact and friction. The system can simulate moderately complex mechanical systems at interactive rates (20-30 Hz on low end Silicon Graphics workstations). New objects and user specified constraints can be added into the simulation environment on the fly. The system uses analytical methods to compute contact forces, as opposed to the penalty methods common in other interactive systems. Currently, the system's weakest feature is that it can fail to detect high speed collisions. Objects that move at high speeds (relative to the step size of the simulation) are subject to a form of aliasing and may tunnel through other objects without causing a collision. Other simplifications of the system involve approximating collision times and locations by interpolation methods, and periodic error correction adjustments of geometric tolerances. It is difficult to try to quantify the error incurred by a given approximation or tradeoff. Some of the design choices will likely curtail the system's use for highly predictive applications. However, they do not seriously affect simulating the basic dynamics of a mechanism like a feeder. In general, the system performs with sufficient accuracy and realism to be considered a viable interactive simulation environment     

Dynamic simulation of articulated rigid bodies with contact and collision

We propose a novel approach for dynamically simulating articulated rigid bodies undergoing frequent and unpredictable contact and collision. In order to leverage existing algorithms for nonconvex bodies, multiple collisions, large contact groups, stacking, etc., we use maximal rather than generalized coordinates and take an impulse-based approach that allows us to treat articulation, contact, and collision in a unified manner. Traditional constraint handling methods are subject to drift, and we propose a novel prestabilization method that does not require tunable potentially stiff parameters as does Baumgarte stabilization. This differs from poststabilization in that we compute allowable trajectories before moving the rigid bodies to their new positions, instead of correcting them after the fact when it can be difficult to incorporate the effects of contact and collision. A poststabilization technique is used for momentum and angular momentum. Our approach works with any black box method for specifying valid joint constraints and no special considerations are required for arbitrary closed loops or branching. Moreover, our implementation is linear both in the number of bodies and in the number of auxiliary contact and collision constraints, unlike many other methods that are linear in the number of bodies, but not in the number of auxiliary constraints.     

Bounded attitude control of rigid bodies: Real-time experimentation to a quadrotor mini-helicopter

A quaternion-based feedback is developed for the attitude stabilization of rigid bodies. The control design takes into account a priori input bounds and is based on nested saturation approach. It results in a very simple controller suitable for an embedded use with low computational resources available. The proposed method is generic not restricted to symmetric rigid bodies and does not require the knowledge of the inertia matrix of the body. The control law can be tuned to force closed-loop trajectories to enter in some a priori fixed neighborhood of the origin in a finite time and remain thereafter. The global stability is guaranteed in the case where angular velocity sensors have limited measurement range. The control law is experimentally applied to the attitude stabilization of a quadrotor mini-helicopter.     

Almost global stabilization of fully-actuated rigid bodies

This paper addresses the problem of stabilizing a fully-actuated rigid body. The problem is formulated by considering the natural configuration space for rigid bodies, the Special Euclidean group SE(3). The proposed solution consists of a landmark-based controller for force and torque actuation that guarantees almost global asymptotic stability of the desired equilibrium point. As such the equilibrium point is asymptotically stable and only a nowhere dense set of measure zero lies outside its region of attraction. The controller uses velocity measurements and the position coordinates of a collection of landmarks fixed in the environment. As an additional feature, the control law is designed so as to verify prescribed bounds on the actuation.     

Drift-free attitude estimation for accelerated rigid bodies

In this paper we study the attitude estimation problem for an accelerated rigid body using gyros and accelerometers. The application in mind is that of a walking robot and particular attention is paid to the large and abrupt changes in accelerations that can be expected in such an environment. We propose a state estimation algorithm that fuses data from rate gyros and accelerometers to give long-term drift free attitude estimates. The algorithm does not use any local parameterization of the rigid body kinematics and can thus be used for a rigid body performing any kind of rotations. The algorithm is a combination of two non-standard, but in a sense linear, Kalman filters between which a trigger based switching takes place. The kinematics representation used makes it possible to construct a linear algorithm that can be shown to give convergent estimates for this nonlinear problem. The state estimator is evaluated in simulations demonstrating how the estimates are long-term stable even in the presence of gyro drift.     

Real-time snowing simulation

A snowing scene has a unique fascination for people due to its incomparable beauty. However, little work has been presented on the real-time generation of a dynamic snowing scene, partially due to the difficulty that the simulation of a dynamic snowing process involves the complex modeling of the wind field and the interaction between wind and snow. In this paper, by fully considering the physical characteristics of wind and snow, we construct a three-dimensional wind field based on the discrete form of the Boltzmann equation. According to the interaction laws between wind and snow, we simulate the falling of snow, deposition and erosion in 3D space. Experimental results show that realistic wind-driven snow scenes under different speeds of wind with different amounts of snowfall can be rendered in real-time.     

Real-time knot-tying simulation

The real-time simulation of rope, and knot tying in particular, raises difficult issues in contact detection and management. Some practical knots can only be achieved by complicated crossings of the rope, yielding multiple simultaneous contacts, especially when the rope is pulled tight. This paper describes a graphical simulator that allows a user to grasp and smoothly manipulate a virtual rope and to tie arbitrary knots, including knots around other objects, in real time. A first component of the simulator computes the global configuration of the rope based on user interactions. Another component of the simulator precisely detects self-collisions in the rope as well as collisions with other objects. Finally, a third component manages collisions to prevent penetration, while making the rope slide with some friction along itself and other objects, so that knots can be pulled tight in a realistic manner. An additional module uses recent results from knot theory to identify, also in real time, which topological knots have been tied. This work was motivated by surgical suturing, but simulation in other domains, such as sailing and rock climbing, could also benefit from it.     

Forward dynamics based realistic animation of rigid bodies

This paper presents a new methodology for model and control of the motion of an (articulated) rigid body for the purposes of animation. The technique uses a parameter optimization method for forward dynamic simulation to obtain a good set of values for the control variables of the system. We model articulated rigid bodies using a moderate number of control nodes, and we linearly interpolate control values between adjacent pairs of these nodes. The interpolated control values are used to determine the forces/torques for the body actuators. We can control total motion duration time, and the control is more flexible than in any other dynamics based animation techniques. We employ a parameter optimization, (or nonlinear programming) method to find a good set of values for the control nodes. We extend this method by using a musculotendon skeletal model for the human body instead of the more commonly used robot model to provide more accurate human motion simulations. Skeletal and musculotendon dynamics enable us to do the human body animation more accurately than ever because the muscle force depends on the geometry of a human as well as on differential kinematic parameters. We show various levels of motion control for forward dynamics animation: ranging from piecewise linear forces/torques control for joints to muscle activation signal control for muscles to generate highly nonlinear forces/torques. This spectrum of control levels provides various nonlinear resulting motions to animators to allow them to achieve effective motion control and physically realistic motion simultaneously. Because our algorithms are heavily dependent on parameter optimization, and since the optimization technique may have difficulty finding a global optimum, we provide a modified optimization method along with various techniques to reduce the search space size. Our parameter optimization based forward dynamic animation and musculotendon dynamics based animation present the first use of such techniques in animation research to date.     

A simulation testbed based on lightweight processes

The Si lightweight-process based system for simulating process interactions is an enhancement to the C programming languge in the form of library primitives with sets of predefined data structures. The Si system encapsulates an existing lightweight-process library to provide a discrete-event simulation environment supporting the process view. It was developed as a research testbed for investigating methods which support simulations efficiently. Easy extensions and modifications to the Si system are a major design objective, accomplished through modularity and layering. This paper describes the system, our experiences with its implementation, and its applicability to simulation modeling. We report on performance measurements of different implementations of the simulation scheduler, and of different algorithms for simulating service disciplines.     

Optimized parametrization of systems of incidences between rigid bodies

Graphs of pairwise incidences between collections of rigid bodies occur in many practical applications and give rise to large algebraic systems for which all solutions have to be found. Such pairwise incidences have explicit, simple and rational parametrizations that, in principle, allow us to partially resolve these systems and arrive at a reduced, parametrized system in terms of the rational parameters. However, the choice of incidences and the partial order of incidence resolution strongly determine the algebraic complexity of the reduced, parametrized system—measured primarily in the number of variables and secondarily in the degree of the equations.     

A 2-dimensional multi rigid bodies skiing model

This paper develops a 2-dimensional model to simulate skiing movements based on multibody system. The model includes one rigid ski and five rigid skier’s segments (arms, trunk, thighs, shanks, and boots). The ski is modeled as a discrete-point rigid body. A spring-damper element is used to describe the interaction of every point between the ski and snow. The snow ground can be planar or uneven terrains. The air resistance of skier and ski are considered. Finally, several simple skiing movements were simulated. On the one hand, these simulations included some general skiing characteristics such as downhill, rotation, flying, and landing. On the other hand, the snow surfaces could be planar slope, sinusoidal slope, and rampant slope. The results show that the model can simulate simple types of skiing, and can provide kinematic and kinetic data such as the skier’s displacement, velocity, acceleration, joint moment and force, and ground reaction force. The model may be useful for building 3-dimensional skiing model, and for further studies of skiing movements.     

Rigid multibody system dynamics with uncertain rigid bodies

This paper is devoted to the construction of a probabilistic model of uncertain rigid bodies for multibody system dynamics. We first construct a stochastic model of an uncertain rigid body by replacing the mass, the center of mass, and the tensor of inertia by random variables. The prior probability distributions of the stochastic model are constructed using the maximum entropy principle under the constraints defined by the available information. The generators of independent realizations corresponding to the prior probability distribution of these random quantities are further developed. Then several uncertain rigid bodies can be linked to each other in order to calculate the random response of a multibody dynamical system. An application is proposed to illustrate the theoretical development.     

The control of angular momentum for asymmetric rigid bodies

A linear feedback law is obtained in closed form for the regulation of the angular momentum of an arbitrary rotating rigid body, without linearizing hypotheses of symmetry, yet optimal for a suitable quadratic energy cost. The evolution of the amplitude of the controlled angular momentum is thereby obtained in closed form, whence asymptotic closed-loop stability and exact open-loop terminal rate nulling are derived. Earlier steady-state results are recovered as particular cases.     

Kinematic control of free rigid bodies using dual quaternions

This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method.     

Real-time simulation of time-of-flight sensors

Today’s time-of-flight (TOF) sensors measure full-range distance information by estimating the elapsed time between emission and receiving of active light in real-time. Such sensors are inexpensive, compact, and they have a high performance, which especially fits real-time applications, e.g. in the fields of automotive, robotics, 3D imaging, and visualization. The simulation of such sensors is an essential building block for hardware design and application development. Therefore, the simulation data must capture the major sensor characteristics.This paper introduces a simulation approach, which is motivated by physics, for the Photonic Mixing Device (PMD) sensor which is a specific type of time-of-flight sensor. Dynamic motion blurring and resolution artifacts such as flying pixels as well as the typical deviation error are prominent effects of real world systems. Flying pixels arise when an area of inhomogeneous depth is covered by a single PMD-pixel whereas the deviation error is based on the anharmonic properties of the optical signal. The modeling of these artifacts is essential for an authentic simulation approach. We present a detailed comparison between a real PMD-device and the simulation data regarding the sensor characteristics.The proposed algorithms are implemented in a hardware accelerated solution which makes use of the programmability of modern Graphics Processing Units (GPUs). This way, an interactive simulation feedback is provided for applications and further data processing. The simulation takes place in real-time and thus all required control mechanisms are accessible in real-time, too.     

Synthesis of subminimum-time bounded control of system of rigid bodies

An algorithm of control of a system of rigid bodies in a form of feedback in special coordinates is discussed. The control is bounded, and it is also subminimum-time in the sense that it coincides with the optimal in linear approximation and in the case of a system with one degree of freedom. It is shown that the controller endows the unperturbed motion of the system with the property of asymptotic stability in large and is also robust. The results of numerical experiments, demonstrating the closeness of the trajectories to the time-optimal ones, are presented.     

The dynamics of articulated rigid bodies for purposes of animation

Curves and surfaces satisfying continuity and smoothness conditions are used in computer graphics to fit spatial data points. In a similar fashion, smooth motions of objects should be available to animators in such a way that the dynamics are correct to the degree required for realism. The motion, like a curve or surface shape, should be controllable by easy manipulations of a set of control parameters or by real-time interaction between the animator and a scene generated by dynamic simulation. In this paper, the objects considered have the form of rigid links joined at hinges to form a tree. This is a reasonable first approximation to human and animal bodies. The equations of motion are formulated with respect to hinge-centered coordinates, and are solved by an efficient technique in time which grows linearly with the number of links.     

Distributed finite-time attitude containment control for multiple rigid bodies

Distributed finite-time attitude containment control for multiple rigid bodies is addressed in this paper. When there exist multiple stationary leaders, we propose a model-independent control law to guarantee that the attitudes of the followers converge to the stationary convex hull formed by those of the leaders in finite time by using both the one-hop and two-hop neighbors’ information. We also discuss the special case of a single stationary leader and propose a control law using only the one-hop neighbors’ information to guarantee cooperative attitude regulation in finite time. When there exist multiple dynamic leaders, a distributed sliding-mode estimator and a non-singular sliding surface were given to guarantee that the attitudes and angular velocities of the followers converge, respectively, to the dynamic convex hull formed by those of the leaders in finite time. We also explicitly show the finite settling time.     

Issues in computing contact forces for non-penetrating rigid bodies

In rigid-body simulation it is necessary to compute the forces that arise between contacting bodies to prevent interpenetration. This paper studies the problem of rigid-body simulation when the bodies being simulated are restricted to contact at only finitely many points. Some theoretical and practical issues in computing contact forces for systems with large numbers of contact points are considered. Both systems of rigid bodies with and without Coulomb friction are studied. Complexity results are derived for certain classes of configurations and numerical methods for computing contact forces are discussed.A preliminary version of Sections 2.2, 2.3, 7, and 8.2, and the proof in Section 6 has appeared in conference form [3], [4].     

A theory for learning based on rigid bodies dynamics

A new learning theory derived from the study of the dynamics of an abstract system of masses, moving in a multidimensional space under an external force field, is presented. The set of equations describing system's dynamics may be directly interpreted as a learning algorithm for neural layers. Relevant properties of the proposed learning theory are discussed within the paper, along with results of computer simulations performed in order to assess its effectiveness in applied fields.