Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

We describe a new physics engine tailored to model-based control. Multi-joint dynamics are represented in generalized coordinates and computed via recursive algorithms. Contact responses are computed via efficient new algorithms we have developed, based on the modern velocity-stepping approach which avoids the difficulties with spring-dampers. Models are specified using either a high-level C++ API or an intuitive XML file format. A built-in compiler transforms the user model into an optimized data structure used for runtime computation. The engine can compute both forward and inverse dynamics. The latter are well-defined even in the presence of contacts and equality constraints. The model can include tendon wrapping as well as actuator activation states (e.g. pneumatic cylinders or muscles). To facilitate optimal control applications and in particular sampling and finite differencing, the dynamics can be evaluated for different states and controls in parallel. Around 400,000 dynamics evaluations per second are possible on a 12-core machine, for a 3D homanoid with 18 dofs and 6 active contacts. We have already used the engine in a number of control applications. It will soon be made publicly available.

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MuJoCo: A physics engine for model-based control

A survey of models, analysis tools and compensation methods for the control of machines with friction

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

We use reinforcement learning (RL) to learn dexterous in-hand manipulation policies that can perform vision-based object reorientation on a physical Shadow Dexterous Hand. The training is performed...

Learning dexterous in-hand manipulation:

In this paper a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks. Unlike other methods which take an instantaneous point of view, this method does not need to identify explicitly impulsive forces. Instead, the treatment is similar to that of J. J. Moreau and Monteiro-Marques, except that the numerical formulation used here ensures that there is no inter-penetration of rigid bodies, unlike their velocity-based formulation. Numerical results are given for the method presented here for a spinning rod impacting a table in two dimensions, and a system of four balls colliding on a table in a fully three-dimensional way. These numerical results also show the practicality of the method, and convergence of the method as the step size becomes small.

An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction

This paper is a summary of a comprehensive study of the problem of predicting the possible acceleration(s) of a set of rigid, three-dimensional bodies in contact in the presence of Coulomb friction. We begin with a brief introduction to this problem and a survey of related work and previous approaches. This is followed by the introduction of two novel complementarity formulations for the contact problem under two friction laws: Coulomb''s Law and an analogous law in which Coulomb''s quadratic friction cone is approximated by a pyramid. Under a full co umn rank assumption on the system Jacobian matrix, we establish the existence and uniqueness of a solution to our new models in the case where the friction coefficients are nonnegative and sufficiently small. For the model based on the friction pyramid law, we also show that the classical Lemke almost-complementary pivot algorithm and our new feasible interior point method are guaranteed to compute a solution. Extensive computational results are presented to demonstrate the effectiveness and characteristics of these solution methods for solving the model with friction pyramids. Finally, results are summarized and conclusions are drawn.

On Dynamic Multi-Rigid-Body Contact Problems with Coulomb Friction

Three fundamental problems in the study of grasping and dextrous manipulation with multifingered robotic hands are as follows, a) Given a robotic hand and a grasp characterized by a set of contact points and the associated contact models, determine if the grasp has force closure, b) Given a grasp along with robotic hand kinematic structure and joint effort limit constraints, determine if the fingers are able to apply a specified resultant wrench on the object, c) Compute "optimal" contact forces if the answer to problem b) is affirmative. In this paper, based on an early result by Buss et al., which transforms the nonlinear friction cone constraints into positive definiteness constraints imposed on certainty symmetric matrices, we further cast the friction cone constraints into linear matrix inequalities (LMI) and formulate all three of the problems stated above as a set of convex optimization problems involving LMI. The latter problems have been extensively studied in optimization and control communities. Currently highly efficient algorithms with polynomial time complexity have been developed and made available. We perform numerical studies to show the simplicity and efficiency of the LMI formulation to the three grasp analysis problems.

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Grasp analysis as linear matrix inequality problems

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.

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An implicit time-stepping scheme for rigid body dynamics with Coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulomb's law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established.

Jeff Trinkle

Papers

An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction

On Dynamic Multi-Rigid-Body Contact Problems with Coulomb Friction

Grasp analysis as linear matrix inequality problems

An implicit time-stepping scheme for rigid body dynamics with Coulomb friction

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction