Abstract
The topic of this paper is a real-time simulation of the dynamics of a system of articulated rigid bodies with restrictions on their relative motion. Restrictions are set in the form of inequalities for the relative angles of rotation (for the rotational joints) or relative displacement (for the prismatic joints) of links. Examples of such systems are robots and manipulators, mobile vehicles with trailers, hinged doors, etc. This problem can be presented as a system of linear algebraic equations with linear complements. As a solution for this system the authors propose the method of sequential impulses utilizing a temporal coherence property, which means that the state of a multi-body system (their coordinates) varies slightly for a small period of time. The semi-implicit Euler method is used as the difference scheme. Since the problem presented in the form of a system of linear equations with linear complements is solved relative to velocities, it is necessary to ensure achievement of constraints relative to the body coordinates (task of constraint stabilization). For such a stabilization the authors propose to use the method of split impulses, which ensures stability of the dynamics simulation for a multi-body system. In this paper the authors consider methods used both for the open and closed kinematic chains. The proposed methods and algorithms are implemented in the program modules in the form of dynamic libraries for Windows OS. Their approbation was carried out in the subsystem of dynamics simulation performing simulation of the robots containing joints with restrictions on the parameters of the relative motion. Studies have shown that the proposed methods and algorithms meet the requirements for the dynamics simulation subsystems of the simulators for control of complex dynamic processes, and virtual environment systems. Such technologies can also be used in virtual labs, simulation complexes, systems of augmented virtual environment and other applications.
Highlights
The topic of this paper is a real-time simulation of the dynamics of a system of articulated rigid bodies with restrictions on their relative motion
Since the problem presented in the form of a system of linear equations with linear complements is solved relative to velocities, it is necessary to ensure achievement of constraints relative to the body coordinates
For such a stabilization the authors propose to use the method of split impulses, which ensures stability of the dynamics simulation for a multi-body system
Summary
Pассìотpиì систеìу из N теë, в котоpой i-е теëо иìеет ìассу mi и тензоp инеpöии Ii. Уpавнения. Добавëяя сиëы pеакöии связей äëя всех øаpниpов в уpавнения Нüþтона—Эйëеpа (1), поëу÷иì: Mi V·i = Fie + ∑ Jkт i lGk + ∑ J~sтi lsχ , k ∈ Ki s ∈ Si ãäе Ki — ìножество øаpниpов, котоpые связываþт i-е теëо с äpуãиìи теëаìи; Si — поäìножество øаpниpов из ìножества Ki, äëя котоpых ввеäены оãpани÷ения на паpаìетpы относитеëüноãо äвижения. Из ëеììы äëя необpатиìых связей [10, §253] сëеäует, ÷то pабота сиë pеакöии неуäеpживаþщей связи неотpиöатеëüна äëя ëþбоãо виpтуаëüноãо пеpеìещения, т. Pассìатpивая вìесто виpтуаëüных пеpеìещений äействитеëüные пеpеìещения dχsr = χsr (t + Δt) – – χsr (t) = χsr (t + Δt), поëу÷аеì, ÷то pабота обобщенных сиë pеакöии на этих пеpеìещениях, поìноженная на Δt, иìеет виä d Asχr Δt = psχr χsr (t + Δt). В сиëу тоãо ÷то заäа÷а (12) не соäеpжит оãpани÷ений (3) и (4), äопоëнитеëüно буäеì pеøатü заäа÷у стабиëизаöии этих оãpани÷ений (поä стабиëизаöией зäесü поäpазуìевается стабиëизаöия ОДУ по отноøениþ к инваpиантноìу ìножеству, заäанноìу оãpани÷енияìи виäа (3) и (4))
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