The problem of where to place reference frames in order to describe the motion of flexible robot links modelled as beams has two major solutions. One is designated
as the inertial (also designated as fixed) reference frame approach and the other is designated as the body (moving or shadow) reference frame approach. The distinction between these two approaches is well established in the work of Simo and Vu-Quoc (1986). The body reference frame approach is especially motivated by the assumption of small displacements. On the other hand, the inertial reference frame approach requires the use of beam theories capable of accounting for large rotations of the beam, that is, the strain measures must be invariant under superposed rigid body motion.
The main advantages of the body reference frame approach is that it yields a formulation of conceptual closeness to the flexible robotics control problem and it allows the use of shape functions of the structure for spatial discretisation of the partial differential equations (PDEs) of motion. The inertial reference frame approach, on the other hand, requires the use of the finite element method yielding a finite dimensional model of much larger dimension. Simo and Vu-Quoc (1986) argued that, from the computational standpoint, in the general case, the use of shape functions does not pose a significant advantage due to the non-linear and highly coupled inertia terms emanating from the body reference frame approach. In the inertial reference frame approach, the non-linearity of the problem is shifted to the stiffness part of the equations of motion leading to a simpler structure of the same equations. This aspect has not been investigated here on account of the introductory statement of the paragraph; in robotic manipulators, joint actuator torque is applied along a body reference frame.
A pitfall in the body reference frame approach appears when small displacements are assumed. Under this assumption, the resulting equations of motion predict an incorrect behaviour in relation to the effects of centrifugal force on the deformation of the link. As was eloquently shown in Simo and Vu-Quoc (1987), appropriate account of the influence of centrifugal force on the bending stiffness of a flexible beam requires the use of a geometrically non-linear (at least second-order) beam theory. This does not necessarily imply assuming large strains. Even with infinitesimal strains, the geometry of the beam may allow for large displacements that necessarily must be treated with a non-linear beam theory. Small displacements imply small strains but the converse does not necessarily apply. This leads to capturing the kinematic effect known as beam foreshortening (Kane et al., 1987; Smith and Baruh, 1991).
Under these settings the kinematic description of a flexible link is achieved here through the use of a body reference frame. The inertial reference frame is designated as {OI, XIYIZI}, and the body reference frame is designated as {On, XnYnZn} (see Figure 3.1). The latter is rigidly attached to the first point of neutral axis of the beam, and describes the rigid body motion of the beam. It gives the orientation of the beam in space, in its reference, un-deformed configuration. In an inertial reference frame formulation this reference frame does not exist.
3.2.1 Deformation assumptions
The adoption of the body reference frame allows the separation of rigid-body motion from elastic deformation. The former is described by the body reference frame and the latter is described through the introduction of a third reference frame. The choice
50 Flexible robot manipulators
rn
xp
up e3,Zk t
Zn e1,X
k
e2,Yk
Xp
rp YI
ZI OI XI
E3,Zn On
Ok Yp E2,Yn
E1,Xn
Xn
On n
no
Yn
Figure 3.1 Kinematic description of a flexible link
of this last reference frame is based on the following deformation assumptions for the flexible beam:
(i) Plane beam cross-sections before deformation, remain plane after deforma-tion (warping is not considered).
(ii) Bending and shear deformation are considered (Timoshenko beam theory).
(iii) The beam neutral fibre does not suffer extension.
(iv) The beam neutral axis in the un-deformed configuration is a straight line.
(v) The beam cross-section is of constant specific mass, and is symmetrical relative to its principal axis.
(vi) The shear strains and bending strains are considered to be small, that is, they are accounted for up to the first order, O(1).
Assumptions (i) and (ii) fall in the theory of Timoshenko beams (Bayo, 1989;
Timoshenko et al., 1974). This constitutes a general case of applications, where the slenderness of robot links are relaxed beyond the Euler–Bernoulli beam theory assumptions to a slenderness ratio of length/width>10.
Assumptions (iii), (iv) and (v) are motivated by the physical reality of flexible manipulator arms. The stiffness of the manipulator links in the direction of its axis is significantly larger than in the transverse directions, therefore, the beam axis is con-sidered inextensible. Assumptions (iv) and (v) are common design criteria. Although these can be exploited to improve the design, this is not attempted here.
The motivation for assumption (vi) is twofold. First, if local deformation is large, most probably the material is not adequate to build a robotic manipulator! Second, and more determinant, common materials used in applications retain their linearity for extremely small strain. It is worth stressing that what is enforced in this assumption is small strain, and not small displacement.
The last reference frame is now placed, taking advantage of the assumption that the beam cross-section is rigid during deformation. This reference frame is designated as cross section reference frame{Ok, XkYkZk}. Its origin is placed at the point where the beam neutral axis intersects the beam cross-section (Figure 3.1), and its axes are directed along the axes of symmetry of the cross section. The assumption of rigid cross sections has also implications on the reference frame{On, XnYnZn}. This reference frame is now not just rigidly attached to the first point of the neutral axis, but rigidly attached to the first cross section of the beam. For this cross section, reference frames {On, XnYnZn} and {Ok, XkYkZk} coincide with one another. Similarly, when referring to the end cross section of the beam, the cross-section reference frame becomes {Oˆn, XˆnYˆnZˆn}. Notice that these reference frames are of the co-rotational type and not of the convected basis type (Simo, 1985). In the latter, the axis normal to the cross section is removed, and placed tangent to the neutral axis, rendering the reference frame non-orthogonal.
In summary, a flexible beam is seen as a group of rigid cross sections distributed along an inextensible line, kept together by elastic forces and moments due to small local shear deformation and bending curvature, respectively. Attached to each cross section there is a reference frame{Ok, XkYkZk} whose position and orientation relative to reference frame{On, XnYnZn} describes the beam deformation. The position and orientation of the reference frame{On, XnYnZn} relative to the inertial reference frame, {OI, XIYIZI}, describes the rigid-body motion.
3.2.2 Kinematics of a flexible link
The kinematic description presented in this section is similar to that of (Géradin and Cordona, 2001) with the exception of introducing the moving reference frame. The orthogonal matrix expressing the orientation of{Ok, XkYkZk} relative to {On, XnYnZn} is designated as Rek, and is given by
Rek =
e1 e2 e3
(3.1)
where e1, e2 and e3 are the unit vectors along the axes of the cross-section refer-ence frame, written in the body referrefer-ence frame. The orthogonal matrix expressing the orientation of{On, XnYnZn} relative to {OI, XIYIZI} is designated as Rn, and is given by
Rn=
E1 E2 E3
(3.2)
where E1, E2 and E3are unit vectors along the axes of the body reference frame, expressed in the inertial reference frame.
The position vector of a material point p of the beam belonging to a certain cross section, relative to the origin of the cross-section reference frame and with components along the same reference frame is defined as Yp. Owing to the assumption of rigid cross section, the components of this vector remain constant during deformation.
However, relative to the moving reference frame, the position coordinates of point p change because of deformation. The reference position of point p is designated as Xpand the displaced position of point p is defined as xp. The components of Xpare
52 Flexible robot manipulators
the material coordinates of the beam. The displacement vector carrying point p from position Xpto position xpis defined as up. The components of the displacement vector, just like the components of Xpand xpare along the reference frame{On, XnYnZn}.
In component form Yp=
0 X2 X3T