State Space Representation Of The Equations Of Motion

Assume that the state of the system is known and defined by the collection of generalized coordinates (Q, Q, Q) corresponding to effective joint positions, velocities, and accelerations, respectively. Furthermore, assume that the mass distribution of each rigid element of the multibody is completely characterized, given a localized center of mass and a known inertia tensor. It then follows that for a constrained molecular dynamics model, the EOM under Newton's formulation are represented in state space as

where F denotes the vector of generalized forces (e.g., torques) applied at the joint, N are any other conservative forces (e.g., potential) acting on each generalized coordinate, M denotes the articulated body inertia matrix, and C denotes the non-linear velocity-dependent terms of force (e.g., Coriolis, centrifugal, and gyroscopic forces). Solving Eq. (1) leads to a direct calculation of joint actuator forces required for the system to follow a specific spatial trajectory. On the other hand, the dynamics of motion are obtained by solving Eq. (1) for effective joint accelerations,

This entry explores the dependence of both Eqs. (1) and (2) on the articulated body inertia operator M and particular forms of this operator that lead to reduced complexities in both the inverse and forward dynamics solutions expressed above.

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