Force method application in calculating spatial motion of a manipulator with the massive solid body taking into account the links and rotary drives elasticity

The paper considers spatial motion dynamics of a three-link manipulator consisting of two elastic rods bending in the vertical and lateral planes and spinning with a massive absolutely rigid body (gripper with a load) attached to them at the end. The first link is connected to a fixed base. The links are connected to each other by the hinged units with the given (controlled) relative rotation angles. Calculation model took into account the mechanism compliance at these angles. Mathematical model of the system non-stationary oscillations under the arbitrary kinematic influence was developed. Rods’ deformations were assumed to be insignificant (linear), their inertia was not taken into account. The system dynamics equations were obtained using the force method based on the Castigliano principle, and inertial forces of the arbitrary rigid body with a gripper were replaced by unknown reactions in the fastening unit. Using the force method, an algorithm to determine elastic displacements and rotation angles in the fixed coordinate system at the body attachment point was constructed, and the time-dependent compliance matrix was found. The body inertial forces and moments were first determined in the motion coordinate system associated with the body, and then they were registered in the initial fixed coordinate system using the body motion linearized equations at its low angular velocities. As a result, the problem was reduced to six differential equations of the rigid body oscillations in the fixed coordinate system with the attached to it elastic rod system of variable structure. As a numerical example, the problem of controlled plane motion of a symmetrical manipulator with two elastic rod links and transferred rigid body was solved taking into account elastic compliance in the articulated joints. Comparison was made with solution to the displacement problem estimating the rods inertia influence and compliance in the connections.

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