Reference member dynamic equation

A brief review of previous work related to the dynamic simulation of simple closed-chain mechanisms is given in the second section of this chapter. A model for simple closed-chain mechanisms is described in the third section, and the basic problem statement is also discussed. The dynamic equations of motion for the individual chains and the common reference member are summarized in the fourth section. 

In [31], Oh and Orin extend the basic method of Orin and McGhee [33] to include simple closed-chain mechanisms with m chains of N links each. The dynamic equations of motion for each chain are combined with the net face and moment equations for the reference membo and the kinematic constraint equations at the chain tips to form a large system of linear algebraic equations. The system unknowns are the joint accelerations for all the chains, the constraint fcwces applied to the reference memba, and the spatial acceleration of the reference member, lb find the Joint accelerations, this system must be solved as a whole via standard elimination techniques. Although this approach is sbmghtforward, its computational complexity of 0(m N ) is high. 

As in the single closed chain problem, we will begin our analysis with the dynamic equations of motion for the entire simple closed-chain system. First, we will consider the dynamic equations for each supporting chain, and then we will formulate an appropriate dynamic equation for the reference member al ie. 

The dynamic behavira of the reference member may be described using a spatial force balance equation for that body. The sum of the spatial forces exerted by each chain on the reference member and any other external spatial forces (including gravity) are equal to the resultant force on the reference member. Using spatial notation, we may write the face balance equation as follows ... 

The vector ao refers to the motion of the coordinate oigin of frame 0. The spatial inertia matrix, lo, is also defined at this point, and it is known and constant Because wq is givra fw the present state, the velocity-dependent term, bo, may be computed directly. If we combine Equations 6.6 and 6.7, we finally obtain the following dynamic equation for the refnence member. 

The dynamic equation for the reference member given in Equation 6.9 may be rewritten as follows ... 

The triplication of Equation 6.42 to ev actuated chain in the simple closed-chain mechanism results in a complete solution to the Direct Dynamics problem for this robotic system. The next state positions and velocities may be computed by integrating the appropriate quantities for each chain and the reference member. As discussed in Chapter 5, small amounts of negative position and rate feedback may be employed to countra t the drift which is a result of the integration process. 

Equations for the dynamic reactions of typical structural members are available from the same sources which provide the transformation factors. Refer to Tables 6.1,... 

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