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
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In this paper we consider the motion planning problem of humanoid with articulated-spine for object manipulation in double support phase. Complexity of motion planning increases due to higher degrees-of-freedom (DOF), redundancy arising due to articulated spine, inherent underactuation and stability constraint. Additionally, loop-closure constraints arise during double-support phase, and self-collision constraints exist independent of the task to be performed. These problems make motion planning of humanoids a challenging task. In this work, we address the above issues by proposing a sampling based approach for planning the motion of the humanoid. Our approach to the problem is based on RRT* which can generate asymptotically optimal paths. The proposed approach deals with the stability constraints by rejective sampling approach which divides random configurations generated into valid and invalid configurations. Additionally, the loop-closure constraints are tackled by separating the humanoid into two open-kinematic sub-chains, and then generating random configurations in one sub-chain whereas the remaining sub-chain uses inverse kinematics for closure. Efficacy of the proposed approach is demonstrated for whole-body motion planning of a 25-DOF humanoid in generating asymptotically optimal end-effector paths. 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). We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the first provably intractable problem in robotics. We address stable synchronization of a network of rotating and translating rigid bodies in three-dimensional space. Motivated by applications that require coordinated spinning spacecraft or diving underwater vehicles, we prove control laws that stably couple and coordinate the dynamics of multiple rigid bodies. We design decentralized, energy shaping control laws for each individual rigid body that depend on the relative orientation and relative position of its neighbors. Energy methods are used to prove stability of the coordinated multi-body dynamical system. To prove exponential stability, we break symmetry and consider a controlled dissipation term that requires each individual to measure its own velocity. The control laws are illustrated in simulation for a network of spinning rigid bodies. In this paper, we present a convex optimization problem to generate Center of Mass (CoM) and momentum trajectories of a walking robot, such that the motion robustly satisfies the friction cone constraints on uneven terrain. We adopt the Contact Wrench Cone (CWC) criterion to measure a robot's dynamical stability, which generalizes the venerable Zero Moment Point (ZMP) criterion. Unlike the ZMP criterion, which is ideal for walking on flat ground with unbounded tangential friction forces, the CWC criterion incorporates non-coplanar contacts with friction cone constraints. We measure the robustness of the motion using the margin in the Contact Wrench Cone at each time instance, which quantifies the capability of the robot to instantaneously resist external force torque disturbance, without causing the foot to tip over or slide. For pre-specified footstep location and time, we formulate a convex optimization problem to search for robot linear and angular momenta that satisfy the CWC criterion. We aim to maximize the CWC margin to improve the robustness of the motion, and minimize the centroidal angular momentum (angular momentum about CoM) to make the motion natural. Instead of directly minimizing the non-convex centroidal angular momentum, we resort to minimizing a convex upper bound. We show that our CWC planner can generate motion similar to the result of the ZMP planner on flat ground with sufficient friction. Moreover, on an uneven terrain course with friction cone constraints, our CWC planner can still find feasible motion, while the outcome of the ZMP planner violates the friction limit. Constrained motion planning and Manipulation planning, for generic non-linear constraints, highly rely on the ability of solving non-linear equations. The Newton-Raphson method, often used in this context, is discontinuous with respect to its input and point wise path projection can lead to discontinuous constrained paths. The discontinuities come from the pseudo-inverse involved at each iteration of the Newton-Raphson algorithm. An interval of continuity for one iteration is derived from an upper bound of the norm of the Hessian of the constraints. Such a bound is easy to compute for constraints involving joint positions and orientations. We introduce two heuristical path projection algorithms and empirically show that they generate continuous constrained paths. The algorithms are integrated into a motion planning algorithm to empirically assert the continuity of the solution. This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP-hard even if B has no velocity modulus bound, where, in both cases, B has 6 degrees of freedom. To prove these results, we use a unique method of simulation of a Turing machine that uses time to encode configurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom). We also investigate a natural class of dynamic problems that we call asteroid avoidance problems : B, the object we wish to move, is a convex polyhedron that is free to move by translation with bounded velocity modulus, and the polyhedral obstacles have known translational trajectories but cannot rotate. This problem has many applications to robot, automobile, and aircraft collision avoidance. Our main positive results are polynomial time algorithms for the 2-D asteroid avoidance problem, where B is a moving polygon and we assume a constant number of obstacles, as well as single exponential time or polynomial space algorithms for the 3-D asteroid avoidance problem, where B is a convex polyhedron and there are arbitrarily many obstacles. Our techniques for solving these asteroid avoidance problems use “normal path” arguments, which are an intereting generalization of techniques previously used to solve static shortest path problems. We also give some additional positive results for various other dynamic movers problems, and in particular give polynomial time algorithms for the case in which B has no velocity bounds and the movements of obstacles are algebraic in space-time. 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 The seemingly simple everyday actions of moving limb and body to accomplish a motor task or interact with the environment are incredibly complex. To reach for a target we first need to sense the target's position with respect to an external coordinate system; we then need to plan a limb trajectory which is executed by issuing an appropriate series of neural commands to the muscles. These, in turn, exert appropriate forces and torques on the joints leading to the desired movement of the arm. Here we review some of the earlier work as well as more recent studies on the control of human movement, focusing on behavioral and modeling studies dealing with task space and joint-space movement planning. At the task level, we describe studies investigating trajectory planning and inverse kinematics problems during point-to-point reaching movements as well as two-dimensional (2D) and three-dimensional (3D) drawing movements. We discuss models dealing with the two-thirds power law, particularly differential geometrical approaches dealing with the relation between path geometry and movement velocity. We also discuss optimization principles such as the minimum-jerk model and the isochrony principle for point-to-point and curved movements. We next deal with joint-space movement planning and generation, discussing the inverse kinematics problem and common solutions to the problems of kinematic redundancy. We address the question of which reference frames are used by the nervous system and review studies examining the employment of kinematic constraints such as Donders' and Listing's laws. We also discuss optimization approaches based on Riemannian geometry. One principle of motor coordination during human locomotion emerging from this body of work is the intersegmental law of coordination. However, the nature of the coordinate systems underlying motion planning remains of interest as they are related to the principles governing the control of human arm movements. We consider the problem of segmenting multiple rigid-body motions from point correspondences in multiple affine views. We cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two, three or four. Our algorithm involves projecting all point trajectories onto a 5-dimensional subspace using the SVD, the PowerFactorization method, or RANSAC, and fitting multiple linear subspaces representing different rigid-body motions to the points in ?5 using GPCA. Unlike previous work, our approach does not restrict the motion subspaces to be four-dimensional and independent. Instead, it deals gracefully with all the spectrum of possible affine motions: from two-dimensional and partially dependent to four-dimensional and fully independent. Our algorithm can handle the case of missing data, meaning that point tracks do not have to be visible in all images, by using the PowerFactorization method to project the data. In addition, our method can handle outlying trajectories by using RANSAC to perform the projection. We compare our approach to other methods on a database of 167 motion sequences with full motions, independent motions, degenerate motions, partially dependent motions, missing data, outliers, etc. On motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions. 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. We consider the design of optimal localized feedback gains for one-dimensional formations in which vehicles only use information from their immediate neighbors. The control objective is to enhance coherence of the formation by making it behave like a rigid lattice. For the single-integrator model with symmetric gains, we establish convexity, implying that the globally optimal controller can be computed efficiently. We also identify a class of convex problems for double-integrators by restricting the controller to symmetric position and uniform diagonal velocity gains. To obtain the optimal non-symmetric gains for both the single- and the double-integrator models, we solve a parameterized family of optimal control problems ranging from an easily solvable problem to the problem of interest as the underlying parameter increases. When this parameter is kept small, we employ perturbation analysis to decouple the matrix equations that result from the optimality conditions, thereby rendering the unique optimal feedback gain. This solution is used to initialize a homotopy-based Newton's method to find the optimal localized gain. To investigate the performance of localized controllers, we examine how the coherence of large-scale stochastically forced formations scales with the number of vehicles. We establish several explicit scaling relationships and show that the best performance is achieved by a localized controller that is both non-symmetric and spatially-varying. Rigid body dynamics is a well-established framework in robotics. It can be used to expose the analytic form of kinematic and dynamic functions of the robot model. So far, two major algorithms, namely the recursive Newton-Euler algorithm (RNEA) and the articulated body algorithm (ABA), have been proposed to compute the inverse dynamics and the forward dynamics in a few microseconds. Evaluating their derivatives is an important challenge for various robotic applications (optimal control, estimation, co-design or reinforcement learning). However it remains time consuming, whether using finite differences or automatic differentiation. In this paper, we propose new algorithms to efficiently compute them thanks to closed-form formulations. Using the chain rule and adequate algebraic differentiation of spatial algebra, we firstly differentiate explicitly RNEA. Then, using properties about the derivative of function composition, we show that the same algorithm can also be used to compute the derivatives of ABA with a marginal additional cost. For this purpose, we introduce a new algorithm to compute the inverse of the joint-space inertia matrix, without explicitly computing the matrix itself. All the algorithms are implemented in our open-source C++ framework called Pinocchio. Benchmarks show computational costs varying between 3 microseconds (for a 7-dof arm) up to 17 microseconds (for a 36-dof humanoid), outperforming the alternative approaches of the state of the art. This paper proposes a new method for rigid body pose estimation based on spectrahedral representations of the tautological orbitopes of SE(2) and SE(3). The approach can use dense point cloud data from stereo vision or an RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data as input. The method is a convex relaxation of the classical pose estimation problem, and is based on explicit linear matrix inequality (LMI) representations for the convex hulls of SE(2) and SE(3). Given these representations, the relaxed pose estimation problem can be framed as a robust least squares problem with the optimization variable constrained to these convex sets. Although this formulation is a relaxation of the original problem, numerical experiments indicates that it is indeed exact - i.e. its solution is a member of SE(2) or SE(3) - in many interesting settings. We additionally show that this method is guaranteed to be exact for a large class of pose estimation problems. In this letter, we propose a two-stage strategy for optimal control problems of robotic mechanical systems that proves to be more robust, and yet more efficient, than straightforward solution strategies. Specifically, we focus on a simplified humanoid model, represented as a two-dimensional articulated serial chain of rigid bodies, in the tasks of getting up (sitting down) from (to) the supine and prone postures. Interactions with the environment are integral parts of these motions, and a priori unscheduled contact sequences are discovered by the solver itself, opportunistically making or breaking contacts with the ground through feet, knees, hips, elbows, and hands. The present investigation analyzes the effects on the computational performance of: 1) the explicit introduction of contact forces among the optimization variables, 2) the substitution of undesired contact forces with geometric constraints that prevent interpenetrations, and 3) the splitting of the planning problem into two consecutive phases of increasing complexity. To the best of our knowledge, these tests represent the only quantitative analysis of the performances achievable with different solution strategies for optimization-based, whole-body dynamic motion planning in the presence of contacts. This paper presents a generic and efficient approach to generate dynamically consistent motions for under-actuated systems like humanoid or quadruped robots. The main contribution is a walking pattern generator, able to compute a stable trajectory of the center of mass of the robot along with the angular momentum, for any given configuration of contacts (e.g. on uneven, sloppy or slippery terrain, or with closed-gripper). Unlike existing methods, our solver is fast enough to be applied as a model-predictive controller. We then integrate this pattern generator in a complete framework: an acyclic contact planner is first used to automatically compute the contact sequence from a 3D model of the environment and a desired final posture; a stable walking pattern is then computed by the proposed solver; a dynamically-stable whole-body trajectory is finally obtained using a second-order hierarchical inverse kinematics. The implementation of the whole pipeline is fast enough to plan a step while the previous one is executed. The interest of the method is demonstrated by real experiments on the HRP-2 robot, by performing long-step walking and climbing a staircase with handrail support. This paper develops probabilistic methods for visual tracking of a three-dimensional geometric hand model from monocular image sequences. We consider a redundant representation in which each model component is described by its position and orientation in the world coordinate frame. A prior model is then defined which enforces the kinematic constraints implied by the model's joints. We show that this prior has a local structure, and is in fact a pairwise Markov random field. Furthermore, our redundant representation allows color and edge-based likelihood measures, such as the Chamfer distance, to be similarly decomposed in cases where there is no self-occlusion. Given this graphical model of hand kinematics, we may track the hand's motion using the recently proposed nonparametric belief propagation (NBP) algorithm. Like particle filters, NBP approximates the posterior distribution over hand configurations as a collection of samples. However, NBP uses the graphical structure to greatly reduce the dimensionality of these distributions, providing improved robustness. Several methods are used to improve NBP's computational efficiency, including a novel KD-tree based method for fast Chamfer distance evaluation. We provide simulations showing that NBP may be used to refine inaccurate model initializations, as well as track hand motion through extended image sequences. Few computational models have addressed the spatiotemporal features of unconstrained three-dimensional (3D) arm motion. Empirical observations made on hand paths, speed profiles, and arm postures during point-to-point movements led to the assumption that hand path and arm posture are independent of movement speed, suggesting that the geometric and temporal properties of movements are decoupled. In this study, we present a computational model of 3D movements for an arm with four degrees of freedom based on the assumption that optimization principles are separately applied at the geometric and temporal levels of control. Geometric properties (path and posture) are defined in terms of geodesic paths with respect to the kinetic energy metric in the Riemannian configuration space. Accordingly, a geodesic path can be generated with less muscular effort than on any other, nongeodesic path, because the sum of all configuration-speed-dependent torques vanishes. The temporal properties of the movement (speed) are determined in task space by minimizing the squared jerk along the selected end-effector path. The integration of both planning levels into a single spatiotemporal representation simplifies the control of arm dynamics along geodesic paths and results in movements with near minimal torque change and minimal peak value of kinetic energy. Thus, the application of Riemannian geometry allows for a reconciliation of computational models previously proposed for the description of arm movements. We suggest that geodesics are an emergent property of the motor system through the exploration of dynamical space. Our data validated the predictions for joint trajectories, hand paths, final postures, speed profiles, and driving torques. Traditional motion planning approaches for multilegged locomotion divide the problem into several stages, such as contact search and trajectory generation. However, reasoning about contacts and motions simultaneously is crucial for the generation of complex whole-body behaviors. Currently, coupling theses problems has required either the assumption of a fixed gait sequence and flat terrain condition, or nonconvex optimization with intractable computation time. In this letter, we propose a mixed-integer convex formulation to plan simultaneously contact locations, gait transitions, and motion, in a computationally efficient fashion. In contrast to previous works, our approach is not limited to flat terrain nor to a prespecified gait sequence. Instead, we incorporate the friction cone stability margin, approximate the robot's torque limits, and plan the gait using mixed-integer convex constraints. We experimentally validated our approach on the HyQ robot by traversing different challenging terrains, where nonconvexity and flat terrain assumptions might lead to suboptimal or unstable plans. Our method increases the motion robustness while keeping a low computation time. The use of Coulomb's friction law with the principles of classical rigid-body dynamics introduces mathematical inconsistencies. Specifically, the forward dynamics problem can have no solutions or multiple solutions. In these situations, compliant contact models, while increasing the dimensionality of the state vector, can resolve these problems. The simplicity and efficiency of rigid-body models, however, provide strong motivation for their use during those portions of a simulation when the rigid-body solution is unique and stable. In this paper, we use singular perturbation analysis in conjunction with linear complementarity theory to establish conditions under which the solution predicted by the rigid-body dynamic model is stable. We employ a general model of contact compliance to derive stability criteria for planar mechanical systems. In particular, we show that for cases with one sliding contact, there is always at most one stable solution. Our approach is not directly applicable to transitions between rolling and sliding where the Coulomb friction law is discontinuous. To overcome this difficulty, we introduce a smooth nonlinear friction law, which approximates Coulomb friction. Such a friction model can also increase the efficiency of both rigid-body and compliant contact simulation. Numerical simulations for the different models and comparison with experimental results are also presented.