5-DOF Robots with Closed-Form Solution
Certain Five-degree-of-freedom robots with simple geometries may yield inverse kinematic equations that can be solved directly and without need for a numeric technique such as Newton-Raphson. In Chapter 5, some particular 4-DOF robot structures were found for which the reduced system of equations (5.13)-(5.16) was overspecified i.e. the matrix H of the linear system was singular. The analysis of these special geometries showed that one or two of the four equations of the reduced system became constraint equations on pose elements, particularly, elements tz, Pz, t.p, and p.p.
In the case of 5-DOF robots, the quantities uz, qz, u.q, and q.q (u playing the role of t and q that of p) are either linear functions of S1 and C1 as shown in Eqs. (6.1l)-(6.14) or linear functions of S5 and C5 as shown in Eqs. (6.34)-(6.37). Either way, the constraint equations described in the ten cases of chapter 5 can be used to solve for the correct value of el or e5 directly without need for an iterative technique. This result means that if a 5-DOF robot manipulator has a 4-DOF section with one of the special geometries discussed in Chapter 5, then the arm can be solved in closed form. We now proceed to prove this point.
The 5-DOF inverse kinematics problem of Eq. (6.1) can be reduced to the 4-DOF one of Eq. (6.2). In this case, the
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