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112 CHAPTER 9. RESULTS desks and non-penetration constraints due to the contacts between the cube and the desks can not be satisfied. • TestFriction Static, dynamic and impulsive friction approximation at contacts is evaluated by this test. The scene consists of a table, desk and a stack of boxes. The boxes are stacked on the desk that lies on the table and there is no friction between the desk and the table. A ball strikes the desk and the impulsive friction between the ball and the desk makes the desk (as well as the stacked boxes) move. The whole desk can be dragged by just pushing onto the top-most box. 9.1.2 Joint Tests Simulator library implements various geometric joints that can be used to attach bodies together and provides a way to control angles or velocities about joint axes, impose joint angle limits, control displacements along joint axes, etc. This functionality is presented by TestHinge- Joint, TestUniversalJoint, TestBallAndSocketJoint, TestSliderJoint, TestMotorJoint and TestLimitedSpringJoint tests that evaluate the corresponding joint classes. It is assumed here that the user provides appropriate external influences by dragging the attached bodies to validate the functionality of the joints and their motors. 9.1.3 Performance and Robustness Tests To test the performance of the constraint solver and its fitness to handle highly constrained articulated structures whose segments can collide, the following two tests exercising certain types of articulated structures were implemented. • TestChain A chain consisting of 32 segments connected by various joint types, where each joint removes at least 3 DOFs, corresponding to a linear articulated structure is presented by this test. • TestArticulatedStructure This test implements and evaluates a non-linear highly articulated structure consisting of 127 segments whose all DOFs are constrained and presents the use of soft constraints to attract articulated structures to initial states — soft constraints are used to constrain joint angles to zeroes. Unless the structure’s segments contact, the constraint forces are computed in the linear time. TestJointCycles test verifies that articulated structures with loops (rigid body systems with cycles in the body-joint graph and other auxiliary constraints) are handled correctly by the linear time constraint solver and that the LCP solver can solve problems whose matrices A are not positive-definite. The test also demonstrates the handling of redundant constraints, because more constraints are imposed than is actually needed to achieve the effect. 9.1.4 Integration Precision Tests These tests simulate the same test scene by integrating the equations of motion using a set of distinct ODE solvers so that the simulation effects (integration precision of the individual solvers) could be compared. An articulated structure consisting of a rotating carousel with four chains
9.1. TESTS 113 Figure 9.1: The illustration of the average constraint error at the carousel links. Different ODE solvers were used to integrate the equations of motion, the values of C i p due to PointToPoint constraints at the links measured each 0.1 seconds (= the time axis unit) and the average values plotted. attached to its construction is chosen to demonstrate the effects, constraint errors at the chain links measure the integration precision (the precision can be compared visually quite easily — the chain links either hold or the chain is broken). To stress the precision differences attained by the solvers, all position-level constraints Cp = 0 are implemented on the acceleration level only3 , an integration error creeps into both ˙ Cp and Cp and it is nice to see how well the particular ODE solver performs to keep Cp and ˙ Cp at zero vectors. The tests are implemented by TestIntegrationPrecisionEuler, TestIntegrationPrecisionMidpoint and TestIntegrationPrecisionRungeKutta that use the corresponding ODE solvers (Euler, Midpoint, Runge-Kutta) to integrate the equations of motion. As can be seen by running the tests or looking at figure (9.1), Euler solver fails miserably. However when the constraints are implemented directly on the velocity level (contrary to the other tests), the method performs quite well — yet it is no wonder, because velocity-level constraints affect the states of constrained bodies directly, independently on the used integration method. In that case the only external force integrated by the Euler solver is the force due to gravity and so there is nothing to mess up. This is demonstrated by TestIntegrationPrecisionEulerVelocityLevel. Since the maximum time step sizes are currently determined by the upper velocity bounds of the bodies in the corresponding simulation groups (see page 79), the step sizes are often too limited, higher-order integration methods can not manifest their potential (the ability to take relatively large time steps with relatively small integration error) and the extra computational cost required to take a step is not balanced by the step size. That said, Euler solver is currently 3 By default, these constraints would be maintained on the velocity level at regular states (to stabilize the velocity level errors) and on the acceleration level otherwise.
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