The body and spatial representations of rigid body motion correspond, respectively, to the convective and spatial representations of continuum dynamics. With a view to developing a unified computational approach for both types of problems, the discrete Clebsch approach of Cotter and Holm for continuum mechanics is applied to derive (i) body and spatial representations of discrete time models of various rigid body motions and (ii) the discrete momentum maps associated with symmetry reduction for these motions. For these problems, this paper shows that the discrete Clebsch approach yields a known class of explicit variational integrators, called discrete Moser-Veselov (DMV) integrators. The spatial representation of DMV integrators are Poisson with respect to a Lie-Poisson bracket for the semi-direct product Lie algebra. Numerical results are presented which confirm the conservative properties and accuracy of the numerical solutions.
Similar approaches derive discrete equations of rigid body motion for optimal control problems. Bloch, Crouch, Marsden and Ratiu @cite_15 , for example, derive the symmetrised rigid body equations by introducing optimality constraints in the action principle. We distinguish our approach from theirs in two ways. Firstly, although they consider the rigid body motion as an optimal control problem with an associated constrained action principle, they do not identify the constraints as Clebsch variables and derive the momentum maps. Secondly, they present left and right trivialisations of @math where as we present body and spatial representations of a left SO(3) action invariant Lagrangian only. The authors make this point when distinguishing their approach from that of Holm and Kupershmidt @cite_27 . We use the expression for the (left) momentum map to prove that the flow on the cotangent bundle preserves spatial angular momentum and derive the equations of motion.
@cite_27 @cite_15
2081703719 2109132421
This paper analyses continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser–Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in this paper may be found in (Bloch AM, Crouch P, Marsden J E and Ratiu T S 1998 Proc. IEEE Conf. on Decision and Control 37 2249–54). The paper describes the variational formulation of smooth particle hydrodynamics for both fluids and solids applications. The resulting equations treat the continuum as a Hamiltonian system of particles where the constitutive equation of the continuum is represented via an internal energy term. For solids this internal energy is derived from the deformation gradient of the mapping in terms of a hyperelastic strain energy function. In the case of fluids, the internal energy term is a function of the density. Once the internal energy terms are established the equations of motion are developed as equations of Lagrange, where the Lagrangian coordinates are the current positions of the particles. Since the energy terms are independent of rigid body rotations and translations, this formulation ensures the preservation of physical constants of the motion such as linear and angular momentum. A new image-based control strategy for visual servoing of a class of under-actuated rigid body systems is presented. The proposed control design applies to "eye-in-hand" systems where the camera is fixed to a rigid body with actuated dynamics. The control design is motivated by a theoretical analysis of the dynamic equations of motion of a rigid body and exploits passivity-like properties of these dynamics to derive a Lyapunov control algorithm using robust backstepping techniques. The proposed control is novel in considering the full dynamic system incorporating all degrees of freedom (albeit for a restricted class of dynamics) and in not requiring measurement of the relative depths of the observed image points. A motivating application is the stabilization of a scale model autonomous helicopter over a marked landing pad. We propose a new view-invariant measure for action recognition. For this purpose, we introduce the idea that the motion of an articulated body can be decomposed into rigid motions of planes defined by triplets of body points. Using the fact that the homography induced by the motion of a triplet of body points in two identical pose transitions reduces to the special case of a homology, we use the equality of two of its eigenvalues as a measure of the similarity of the pose transitions between two subjects, observed by different perspective cameras and from different viewpoints. Experimental results show that our method can accurately identify human pose transitions and actions even when they include dynamic timeline maps, and are obtained from totally different viewpoints with different unknown camera parameters. Optimal control approaches in combination with trajectory optimization have recently proven to be a promising control strategy for legged robots. Computationally efficient and robust algorithms were derived using simplified models of the contact interaction between robot and environment such as the linear inverted pendulum model (LIPM). However, as humanoid robots enter more complex environments, less restrictive models become increasingly important. As we leave the regime of linear models, we need to build dedicated solvers that can compute interaction forces together with consistent kinematic plans for the whole-body. In this paper, we address the problem of planning robot motion and interaction forces for legged robots given predefined contact surfaces. The motion generation process is decomposed into two alternating parts computing force and motion plans in coherence. We focus on the properties of the momentum computation leading to sparse optimal control formulations to be exploited by a dedicated solver. In our experiments, we demonstrate that our motion generation algorithm computes consistent contact forces and joint trajectories for our humanoid robot. We also demonstrate the favorable time complexity due to our formulation and composition of the momentum equations. INTRODUCTION: Brief History. Multifingered Hands and Dextrous Manipulation. Outline of the Book. Bibliography. RIGID BODY MOTION: Rigid Body Transformations. Rotational Motion in R3. Rigid Motion in R3. Velocity of a Rigid Body. Wrenches and Reciprocal Screws. MANIPULATOR KINEMATICS: Introduction. Forward Kinematics. Inverse Kinematics. The Manipulator Jacobian. Redundant and Parallel Manipulators. ROBOT DYNAMICS AND CONTROL: Introduction. Lagrange's Equations. Dynamics of Open-Chain Manipulators. Lyapunov Stability Theory. Position Control and Trajectory Tracking. Control of Constrained Manipulators. MULTIFINGERED HAND KINEMATICS: Introduction to Grasping. Grasp Statics. Force-Closure. Grasp Planning. Grasp Constraints. Rolling Contact Kinematics. HAND DYNAMICS AND CONTROL: Lagrange's Equations with Constraints. Robot Hand Dynamics. Redundant and Nonmanipulable Robot Systems. Kinematics and Statics of Tendon Actuation. Control of Robot Hands. NONHOLONOMIC BEHAVIOR IN ROBOTIC SYSTEMS: Introduction. Controllability and Frobenius' Theorem. Examples of Nonholonomic Systems. Structure of Nonholonomic Systems. NONHOLONOMIC MOTION PLANNING: Introduction. Steering Model Control Systems Using Sinusoids. General Methods for Steering. Dynamic Finger Repositioning. FUTURE PROSPECTS: Robots in Hazardous Environments. Medical Applications for Multifingered Hands. Robots on a Small Scale: Microrobotics. APPENDICES: Lie Groups and Robot Kinematics. A Mathematica Package for Screw Calculus. Bibliography. Index Each chapter also includes a Summary, Bibliography, and Exercises A spatial motion analysis model for high-mobility tracked vehicles was constructed for evaluation of ride performance, steerability, and stability on rough terrain. Ordinary high-mobility tracked vehicles are equipped with independent torsion bar type suspension system, which consists of road arms and road wheels. The road arm rotates about the axis of torsion bar, and rigidity of the torsion bar and cohesion of damper absorb sudden force change exerted by interaction with the ground. The motion of the road arms should be considered for the evaluation of off-road vehicle performance in numerical analysis model. In order to obtain equations of motion for the tracked vehicles, the equations of motion for the vehicle body and for the assembly of a road wheel and a road arm were constructed separately at first. Two sets of equations were reduced with the constraint equations, which the road arms are mechanically connected to the vehicle body. The equations of motion for the vehicle have been expressed with minimal set of variables of the same number as the degrees of freedom for the vehicle motion. We also included the effect of track tension in the equations without constructing equations of motion for the tracks. Numerical simulation based on the vehicle model and experiment of a scale model passing over a trapezoidal speed bump were performed in order to examine the numerical model. It was found that the numerical results reasonably predict the vehicle motion. A numerical method is given for the solution of a system of ordinary differential equations and algebraic, unilateral constraints. The equations govern the motion of a mechanical system of rigid bodies, where contacts between the bodies are created and disappear in the time interval of interest. The ordinary differential equations are discretized by linear multistep methods. In order to satisfy the constraints, a quadratic programming problem is solved at each time step. The fact that the variation of the objective function is small from step to step is utilized to save computing time. A discrete friction model, based on Coulomb’s law of friction and suitable for efficient computation, is proposed for planar problems where dry friction cannot be neglected. The normal forces and the friction forces are the optimal solution to a quadratic programming problem. The methods are tested on four model problems. A data structure and possible generalizations are discussed. We present an energy-based approach to visual odometry from RGB-D images of a Microsoft Kinect camera. To this end we propose an energy function which aims at finding the best rigid body motion to map one RGB-D image into another one, assuming a static scene filmed by a moving camera. We then propose a linearization of the energy function which leads to a 6×6 normal equation for the twist coordinates representing the rigid body motion. To allow for larger motions, we solve this equation in a coarse-to-fine scheme. Extensive quantitative analysis on recently proposed benchmark datasets shows that the proposed solution is faster than a state-of-the-art implementation of the iterative closest point (ICP) algorithm by two orders of magnitude. While ICP is more robust to large camera motion, the proposed method gives better results in the regime of small displacements which are often the case in camera tracking applications. This paper studies segmentation of multiple rigid-body motions in a 3-D dynamic scene under perspective camera projection. We consider dynamic scenes that contain both 3-D rigid-body structures and 2-D planar structures. Based on the well-known epipolar and homography constraints between two views, we propose a hybrid perspective constraint (HPC) to unify the representation of rigid-body and planar motions. Given a mixture of K hybrid perspective constraints, we propose an algebraic process to partition image correspondences to the individual 3-D motions, called Robust Algebraic Segmentation (RAS). Particularly, we prove that the joint distribution of image correspondences is uniquely determined by a set of (2K)-th degree polynomials, a global signature for the union of K motions of possibly mixed type. The first and second derivatives of these polynomials provide a means to recover the association of the individual image samples to their respective motions. Finally, using robust statistics, we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers. We conduct extensive simulations and real experiments to validate the performance of the new algorithm. The results demonstrate that RAS achieves notably higher accuracy than most existing robust motion-segmentation methods, including random sample consensus (RANSAC) and its variations. The implementation of the algorithm is also two to three times faster than the existing methods. The implementation of the algorithm and the benchmark scripts are available at http: perception.csl.illinois.edu ras . We introduce a new method to generate agile and natural human landing motions in real-time via physical simulation without using any mocap or pre-scripted sequences. We develop a general controller that allows the character to fall from a wide range of heights and initial speeds, continuously roll on the ground, and get back on its feet, without inducing large stress on joints at any moment. The character's motion is generated through a forward simulator and a control algorithm that consists of an airborne phase and a landing phase. During the airborne phase, the character optimizes its moment of inertia to meet the ideal relation between the landing velocity and the angle of attack, under the laws of conservation of momentum. The landing phase can be divided into three stages: impact, rolling, and getting-up. To reduce joint stress at landing, the character leverages contact forces to control linear momentum and angular momentum, resulting in a rolling motion which distributes impact over multiple body parts. We demonstrate that our control algorithm can be applied to a variety of initial conditions with different falling heights, orientations, and linear and angular velocities. Simulated results show that our algorithm can effectively create realistic action sequences comparable to real world footage of experienced freerunners. We propose a new template-based approach for view invariant recognition of body poses, based on geometric constraints derived from the motion of body point triplets. In addition to spatial information our templates encode temporal information of body pose transitions. Unlike existing methods that study a body pose as a whole, we decompose it into a number of body point triplets, and compare their motions to our templates. Using the fact that the homography induced by the motion of a triplet of body points in two identical body pose transitions reduces to the special case of a homology, we exploit the equality of two of its eigenvalues to impose constraints on the similarity of the pose transitions between two subjects, observed by different perspective cameras and from different viewpoints. Extensive experimental results show that our method can accurately identify human poses from video sequences when they are observed from totally different viewpoints with different camera parameters. A humanoid robot can be viewed as a constrained dynamic system with constraints imposed by manipulation tasks, locomotion tasks, and the environment. This paper focuses on dealing with constraints in the upper-body of humanoid robots for manipulation tasks that involve coordinated motion of two arms. Inspired by research on human bimanual actions in the biomechanics area, we have developed the Extended-Cooperative-Task-Space (ECTS) representation that efficiently describes various coordinated motion tasks performed by a humanoid robot. Furthermore, we present a general whole-body control framework as an optimal controller based on Gauss's principle of least constraint. We show that all the constraints imposed on a humanoid system can be handled in a unified manner. The proposed framework is verified by numerical simulations on a Hubo II+ humanoid robot model. Abstract : To plan dynamic, whole-body motions for robots one conventionally faces the choice between a complex, fullbody dynamic model containing every link and actuator of the robot, or a highly simplified model of the robot as a point mass. In this paper we explore a powerful middle ground between these extremes. We present an approach to generate whole-body motions using a simple dynamics model, which enforces that the linear and angular momentum of the robot be consistent with the external wrenches on the robot, and a full-body kinematics model that enforces rich geometric constraints, such as end-effector positioning or collision avoidance. We obtain a trajectory for the robot and profiles of contact wrenches by solving a nonlinear optimization problem (NLP). We further demonstrate that we can plan without pre-specifying the contact sequence by exploiting the complementarity conditions between contact forces and contact distance. We demonstrate that this algorithm is capable of generating highly-dynamic motion plans with examples of a humanoid robot negotiating obstacle course elements and gait optimization for a quadrupedal robot. We propose a framework for the full two-way coupling of rigid and deformable bodies, which is achieved with both a unified time integration scheme as well as individual two-way coupled algorithms at each point of that scheme. As our algorithm is two-way coupled in every fashion, we do not require ad hoc methods for dealing with stability issues or interleaving parts of the simulation. We maintain the ability to treat the key desirable aspects of rigid bodies (e.g. contact, collision, stacking, and friction) and deformable bodies (e.g. arbitrary constitutive models, thin shells, and self-collisions). In addition, our simulation framework supports more advanced features such as proportional derivative controlled articulation between rigid bodies. This not only allows for the robust simulation of a number of new phenomena, but also directly lends itself to the design of deformable creatures with proportional derivative controlled articulated rigid skeletons that interact in a life-like way with their environment. We propose a momentum-conserving two-way coupling method of SPH fluids and arbitrary rigid objects based on hydrodynamic forces. Our approach samples the surface of rigid bodies with boundary particles that interact with the fluid, preventing deficiency issues and both spatial and temporal discontinuities. The problem of inhomogeneous boundary sampling is addressed by considering the relative contribution of a boundary particle to a physical quantity. This facilitates not only the initialization process but also allows the simulation of multiple dynamic objects. Thin structures consisting of only one layer or one line of boundary particles, and also non-manifold geometries can be handled without any additional treatment. We have integrated our approach into WCSPH and PCISPH, and demonstrate its stability and flexibility with several scenarios including multiphase flow. In this paper a new time-stepping method for simulating systems of rigid bodies is given. Unlike methods which take an instantaneous point of view, our method is based on impulse-momentum equations, and so does not need to explicitly resolve impulsive forces. On the other hand, our method is distinct from previous impulsive methods in that it does not require explicit collision checking and it can handle simultaneous impacts. Numerical results are given for one planar and one three dimensional example, which demonstrate the practicality of the method, and its convergence as the step size becomes small. A number of conceptually simple but behavior-rich "inverted pendulum" humanoid models have greatly enhanced the understanding and analytical insight of humanoid dynamics. However, these models do not incorporate the robot's angular momentum properties, a critical component of its dynamics. We introduce the reaction mass pendulum (RMP) model, a 3D generalization of the better-known reaction wheel pendulum. The RMP model augments the existing models by compactly capturing the robot's centroidal momenta through its composite rigid body (CRB) inertia. This model provides additional analytical insights into legged robot dynamics, especially for motions involving dominant rotation, and leads to a simpler class of control laws. In this paper we show how a humanoid robot of general geometry and dynamics can be mapped into its equivalent RMP model. A movement is subsequently mapped to the time evolution of the RMP. We also show how an "inertia shaping" control law can be designed based on the RMP. The ability to predict how objects behave during manipulation is an important problem. Models informed by mechanics are powerful, but are hard to tune. An alternative is to learn a model of the object's motion from data, to learn to predict. We study this for push manipulation. The paper starts by formulating a quasi-static prediction problem. We then pose the problem of learning to predict in two different frameworks: (i) regression and (ii) density estimation. Our architecture is modular: many simple, object specific, and context specific predictors are learned. We show empirically that such predictors outperform a rigid body dynamics engine tuned on the same data. We then extend the density estimation approach using a product of experts. This allows transfer of learned motion models to objects of novel shape, and to novel actions. With the right representation and learning method, these transferred models can match the prediction performance of a rigid body dynamics engine for novel objects or actions. This paper explores a method, first proposed by Wei and Chai [1], for estimating 3D human pose from several frames of uncalibrated 2D point correspondences containing projected body joint locations. In their work Wei and Chai boldly claimed that, through the introduction of rigid constraints to the torso and hip, camera scales, bone lengths and absolute depths could be estimated from a finite number of frames (i.e. ≥ 5). In this paper we show this claim to be false, demonstrating in principle one can never estimate these parameters in a finite number of frames. Further, we demonstrate their approach is only valid for rigid sub-structures of the body (e.g. torso). Based on this analysis we propose a novel approach using deterministic structure from motion based on assumptions of rigidity in the body's torso. Our approach provides notably more accurate estimates and is substantially faster than Wei and Chai's approach, and unlike the original, can be solved as a deterministic least-squares problem.