24.1: A Robot Arm

In the earlier units we have frequently taken inspiration from applications related to robots navigation, kinematics, and dynamics. In these earlier analyses we considered systems consisting of relatively few "lumped" components and hence the computational tasks were rather modest. However, it is also often important to address not just lumped components but also the detailed deformations and stresses within say a robot arm: excessive deformation can compromise performance in precision applications; and excessive stresses can lead to failure in large manufacturing tasks.

The standard approach for the analysis of deformations and stresses is the finite element (FE) method. In the FE approach, the spatial domain is first broken into many (many) small regions denoted elements: this decomposition is often denoted a triangulation (or more generally a grid or mesh), though elements may be triangles, quadrilaterals, tetrahedra, or hexahedra; the vertices of these elements define nodes (and we may introduce additional nodes at say edge or face midpoints). The displacement field within each such element is then expressed in terms of a low order polynomial representation which interpolates the displacements at the corresponding nodes. Finally, the partial differential equations of linear elasticity are invoked, in variational form, to yield equilibrium equations at (roughly speaking) each node in terms of the displacements at the neighboring nodes. Very crudely, the coefficients in these equations represent effective spring constants which reflect the relative nodal geometric configuration and the material properties. We may express this system of \(n\) equations - one equation for each node - in \(n\) unknowns - one displacement (or "degree of freedom") for each node - as \(K u=f\), in which \(K\) is an \(n \times n\) matrix, \(u\) is an \(n \times 1\) vector of the unknown displacements, and \(f\) is an \(n \times 1\) vector of imposed forces or "loads." 1

We show in Figure \(24.1\) the finite element solution for a robot arm subject only to the "self-load" induced by gravity. The blue arm is the unloaded (undeformed) arm, whereas the multi-color arm is the loaded (deformed) arm; note in the latter we greatly amplify the actual displacements for purposes of visualization. The underlying triangulation - in particular surface triangles associated

\({ }^{1}\) In fact, for this vector-valued displacement field, there are three equations and three degrees of freedom for each (geometric) node. For simplicity we speak of (generalized) nodes equated to degrees of freedom.

Figure 24.1: Deflection of robot arm.

with volumetric tetrahedral elements - is also shown. In this FE discretization there are \(n=60,030\) degrees of freedom (for technical reasons we do not count the nodes at the robot shoulder). The issue is thus how to effectively solve the linear system of equations \(K u=f\) given the very large number of degrees of freedom. In fact, many finite element discretizations result not in \(10^{5}\) unknowns but rather \(10^{6}\) or even \(10^{7}\) unknowns. The computational task is thus formidable, in particular since typically one analysis will not suffice - rather, many analyses will be required for purposes of design and optimization.