Introduction

In Chapter 5 we studied how the postures of some mechanisms and multibody systems can be found analytically using hand calculations to find closed-form solutions. Typically, this requires forming the necessary transformation matrices, and ensuring that all dependent position variables are made consistent with the constraints expressed by the loop-closure equations. In Chapter 5 we solved several example problems, in both 2-D and 3-D, to illustrate the process, but we also found that the calculations quickly became burdensome, even for problems with only a few unknown joint variables. In principle the methods look powerful, but in practice they quickly reach a limit on practicality.

Does this mean that the methods are not adequate? Not exactly; rather, it means that we are in need of a better means of calculating. Perhaps these tedious computations should be automated for solution by numeric methods using a computer.

Let us reflect on the nature of the problem of posture analysis of a multibody system. In general, the number of bodies (ℓ) is usually reasonably small, typically limited by cost and the desire for simplicity and reliability to tens of moving parts or less. The number of joints (n) is of the same order. The number of closed loops (NL) is usually much smaller. The number of joint variables (φ) is of the same order as the number of joints. However, the number of independent variables (ψ) is almost always very small. After all, the whole point of our multibody system is to control the movements of the parts to only those required for proper function of the system. Thus, the mobility (f) is often only one, and is very rarely as many as ten.