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96 CHAPTER 7. SIMULATOR Simulation groups are advanced in contexts of chained transactions, where each transaction advances a corresponding simulation group to a desired time. If we have n simulation groups denoted G1, . . . , Gn then there are n chained transactions, where the i-th transaction advances Gi, the first transaction is triggered by the simulator and the j-th transaction (j ≥ 2) by the (j − 1)-th transaction. The chaining of transactions works as follows. The simulator triggers the first transaction to advance G1 (while advancing G1 the other simulation groups are not updated). If G1 reaches the desired time, then the second transaction is triggered to advance G2 and the first transaction waits for the second transaction to complete. If the second transaction commits then the first transaction must commit too and if the second transaction aborts then the first transaction must abort. The same rule applies to the i-th and the (i + 1)-th transactions (the (i + 1)-th transaction is triggered by the i-th transaction when Gi reaches the desired time, the i-th transaction commits or aborts depending on the completion status of the (i + 1)-th transaction). As a result, either all transactions 1, . . . , n commit if all Gi reach the desired time or transactions 1 ≤ j ≤ i abort if Gi fails to advance. The transactional processing is required in order to ensure consistency and do proper individual rollbacks when certain ODE solver (due to Gi) can not advance to the desired time. It remains to show why transaction i can ever fail (besides the obvious reasons, such as numerical problems when penetration can not be avoided, for example). The reason is the need for simulation group merging. Although force object targets can not change while advancing the simulation, new geometry hull overlaps (and contacts) might emerge which requires the simulator to simulate the corresponding bodies within a common group in order to produce consistent results. To illustrate the problem, let us assume that we have two bodies A and B, so that the first body belongs to G1 and the second body to G2 and the corresponding geometry hulls do not overlap at time t. Now let us advance the simulation state. G1 is advanced first and while advancing G1, it can happen that the geometry hull of A will overlap the hull of B. The trouble is that when the overlap is detected, at time tcur = t, the geometry hull of A is valid with respect to time tcur but the geometry hull of B is valid with respect to t, because G2 was not updated yet. Even if we could merge the corresponding simulation groups straight in the middle of the transaction so that we could handle the overlap (which we can not), potential contacts would be inconsistent because they would be computed with respect to geometry states at different times — hence the inconsistency. The (chained) transaction is thus aborted, the simulation state restored to the state at time t, bodies merged to a common simulation group and the transaction rerun. Advancing Simulation <strong>Group</strong> Simulation groups maintain lists of rigid bodies and force objects assigned to them so that they could build the ODE that determines the evolution of the group in time and compute the equation right-hand side (evaluate ODE state derivative) upon the request of the ODE solver that is specific to the simulation group. A particular simulation group is then advanced in terms of its ODE. We will now outline the steps taken by evaluate derivative, • Collision detection is performed Higher-level collision detection runs and the list of overlapping geometry hulls is updated. New geometry hull overlaps among bodies from different simulation groups indicate that the transaction should abort and bodies involved in the overlaps merged to common groups (the evaluation of the ODE state derivative ends with a request to abort the transaction).
7.3. COMPONENTS 97 Lower-level collision detection involving bodies in the simulation group is performed. Penetrations indicate that the current state is invalid and the ODE solver should rollback and locate the latest valid state (the evaluation of the ODE state derivative ends with an indication that the current state is invalid). Otherwise, contacts among bodies in the simulation group are computed and temporary contact joints, represented by the instances of classContactJoint, created to handle contact. • External forces and torques are computed Force and torque accumulators of the bodies in the group are cleared, the list of force objects assigned to the simulation group and the list of global forces traversed to accumulate the force and torque contributions. If we are in the middle of an ODE step (the derivative is evaluated for an intermediate state), impulses are ignored. • Constraints are processed Dynamics constraints represented by the instances of Joint class assigned to this simulation group (including the implicit joints due to contacts) are traversed to determine what constraints are to be maintained on the velocity and acceleration levels and constraint impulses and forces are computed and applied. Non-penetration is enforced by stopping bodies at critical contacts. Contact joints are finally removed. • The equation right-hand side is returned Force and torque accumulators and the rigid body states are used to fill the right-hand sides of (5.18) due to individual bodies in the group. If impulses were applied at this invocation of the function, the ODE solver is notified that the ODE state has changed. 7.3.5 Constraint Solver Constraints on the acceleration and velocity levels are represented by a special kind of binary forces called joints (instances of Joint class). Joints provide functions to describe the constraints in the form of (6.17) and (6.26) and their corresponding soft variants. Constraint solver traverses the list of joints and invokes these functions to retrieve the velocity level or acceleration level formulations of the constraints, described in terms of the instances of Joint::BasicInformation (providing the information on the constraint DOFs, the dimensionality mi of the constraint) and Joint::ExtendedInformation (providing the values of the Jacobian blocks Ji,Ai and Ji,Bi , constraint right-hand side vector ci or ki, Lagrange multiplier limits λ lo i and λ hi i and softness constants si). The formulations are in turn used to develop conditions on the multipliers λ, the multipliers are computed and constraint force (impulse) exerted on the bodies in the group. Velocity level constraints are solved first, yielding a change of the ODE state. Acceleration level constraints are then computed with respect to the new ODE state. We will now describe this process more thoroughly. There are some differences from the outline we have just presented. First, bodies in the simulation group are further partitioned to so called islands defined as the groups of bodies (subsets of the bodies in the simulation group) that are currently connected by a joint, [22]. If we have two islands A and B, then the constraint force due to constraints in the island A can not
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Univerzita Karlova v Praze Matemati
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