An effective paradigm for simulating the dynamics of robots that locomote and manipulate is multi-rigid body simulation with rigid contact. This paradigm provides reasonable tradeoffs between accuracy, running time, and simplicity of parameter selection and identification. The Stewart-Trinkle Anitescu-Potra time stepping approach is the basis of many existing implementations. It successfully treats inconsistent (Painleve-type) contact configurations, efficiently handles many contact events occurring in short time intervals, and provably converges to the solution of the continuous time differential algebraic equations (DAEs) as the integration step size tends to zero. However, there is currently no means to determine when the solution has largely converged, i.e., when smaller integration steps would result in only small increases in accuracy. The present work describes an approach that computes the event times (when the set of active equations in a DAE changes) of all contact impact events for a multi-body simulation, toward using integration techniques with error control to compute a solution with desired accuracy. We also describe a first-order, variable integration approach that ensures that rigid bodies with convex polytopic geometries never interpenetrate. This approach permits taking large steps when possible and takes small steps when contact is complex.
Mirtich pioneered (CA), which uses the distance between bodies and bodies' velocities and accelerations, to integrate rigid bodies and multi-rigid body systems without missing contact events (a guarantee root finding approaches cannot make @cite_4 ). Assuming the typical first order integration process for multibody dynamics with contact, CA (depicted in Figure ) can use the formula below, where variables @math represent linear velocities, @math represent angular velocities, @math is the minimum bounding sphere for the body, and @math :
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This paper presents a method for generating dynamically feasible, keyframe-interpolating motions for robots undergoing contact, such as in legged locomotion and manipulation. The first stage generates a twice-differentiable interpolating path that obeys kinematic contact constraints up to a user-specified tolerance. The second stage optimizes speeds along the path to minimize time while satisfying dynamic constraints. The method supports velocity, acceleration, and torque constraints, and polyhedral contact friction constraints at an arbitrary number of contact points. The method is numerically stable, and empirical running time is weakly linear in the number of degrees of freedom and polynomial in the time-domain grid resolution. Experiments demonstrate that full-body motions for robots with 100 degrees of freedom and dozens of contact points are calculated in seconds. In this letter, we present a whole-body nonlinear model predictive control approach for rigid body systems subject to contacts. We use a full-dynamic system model which also includes explicit contact dynamics. Therefore, contact locations, sequences, and timings are not prespecified but optimized by the solver. Yet, using numerical and software engineering allows for running the nonlinear Optimal Control solver at rates up to 190 Hz on a quadruped for a time horizon of half a second. This outperforms the state-of-the-art by at least one order of magnitude. Hardware experiments in the form of periodic and nonperiodic tasks are applied to two quadrupeds with different actuation systems. The obtained results underline the performance, transferability, and robustness of the approach. Our work builds largely on Nagasaka's stabilizer in multi-contact motion [1]. Using a sequence of contact stances from an offline multi-contact planner, we use first a Model Predictive Controller to generate a dynamic trajectory of the center of mass, then a whole-body closed-loop model-based controller to track it at best. Relatively to Nagasaka's work, we allow frame changes of the preferred force, provide a heuristic to compute the timing of the transition from purely geometrical features and investigate the synchronization problem between the reduced-model preview control and the whole-body controller. Using our framework, we generate a wide range of 3D motions, while accounting for predictable external forces, which includes transporting objects. Simulation scenarios are presented and obtained results are analyzed and discussed. A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates.