3
ANALYTICAL FRAMEWORK
This chapter is devoted to the description of the mathematical framework de-veloped to optimize the design phase. As an opening, a brief introduction will be made to the characteristics of the material used for the manufacture.
3.1 Introduction to Elastomers
The silicone rubber belongs to the family of the elastomeric materials. ASTM D-156611 de nes an elastomer as a 'macromolecular material that returns rapidly to approximately the initial dimensions and shape after substantial deformation by a weak stress and release of the stress'. Such elongations typically exceed 100%. It is a polymer with viscoelasticity, low Young's modulus and high failure strain compared with other materials. The earliest-used elastomer was natural rubber, obtained from the sap of the rubber tree, which contains around 95% of a polymer whose repeating unit is isoprene. Rubber is one of the materials which mankind has bene tted rather immensely and in innumerable ways. However, rubber in its original form as derived from latex nds limited use without further processing it to suit a speci c need. Actually, natural rubber does not have stable properties as it can become plastic even at moderately higher temperatures or become brittle in cold conditions. It was left to Charles Goodyear in the year 1839 to acciden-tally discover the "vulcanization" process by adding sulfur to natural rubber and heating that resulted in a harder substance (Figure ??). Synthetic rubbers have come to be known as elastomers in the modern polymer era, with names such as Neoprene, Isoprene, Styrene-Butadiene, Butyl, Nitril, Acrylic, Butadiene, Ure-thane, etc. Elastomers consist of long chain-like molecules, linked together to form
Figure 3.1: Example of a vulcanized elastomer chain, showing disul de cross-link. a three dimensional network. Typically, an average of about 1 in 100 molecules are cross-linked: when this number rises to about 1 in 30, the material becomes more rigid and brittle. Most elastomers are thermoset materials, and cannot be remoulded, an exception being the class of materials known as 'thermoplastic elas-tomers'. In general the elastomers are a class of polymers having the following chemical properties:
• they are amorphous in nature and are comprised of long molecular chains; • the molecular chains are highly twisted, coiled, and randomly oriented in an
undeformed state;
• these chains become partially straightened and untwisted under tensile load-ing and upon removal of teh load;
• material bounce back to original shape and size;
• straightening of rubber is achieved by forming cross-links and this is achieved by using vulcanization process.
The industrial and commercial use of rubber products are immense due to the many outstanding properties of rubber, namely, its elasticity, its mouldability into any desired shape or size, the adaptability for changing it to suit the desired en-gineering properties, e.g., strength or toughness, through chemical, pressure and thermal treatments. Vulcanization process changes the structure of the polymer chains of natural rubber permanently, resulting in changed physical and mechan-ical properties. Synthetic rubbers can be manufactured to the needs of the
ap-plications they are put to and the properties vary accordingly. However, certain characteristics of rubbers make them unique, e.g.
• Ability to undergo large elongations, usually up to 700%; • High Poisson ratio- resulting in large lateral deformation;
• Usually, in tension, the material softens then stiens again. On the other hand, in compression, the response becomes immediately quite sti.
Figure 3.2: Elastomer mechanical characteristics
These special deformational characteristics of rubber are responsible for its wide spread usage in industry e.g., tires, sealants, dampers, etc., as it can take shocks, deform locally, stretch and bend, or deform like a liquid under pressure and con- nement.
3.2 Mechanical Model
It has been developed a mathematical framework to model the mechanical behavior of SCG nipper, which can be assimilated to a cantilever beam with em-bedded tendons, that undergo large de ections. To develop the model, it was necessary to closely study the eects of tendon actuation and evaluate its eect on the internal tensional state of the beam. Then, by means of the material con-stitutive laws, the beam kinematics was derived. The model is valid for trunks for which the ratio between the section major dimension and the beam length is much lower than 1(it is called aspect ratio). In this case the nipper has an aspect ratio less than 10, the lowest limit to assimilate a segment to a beam. This limits the usability of the model from a quantitatively point of view but it can be used
as an useful tools for optimizing the design phase.
3.2.1 Assumptions
In the classical theory, the DOFs of one section are six: the curvatures along the two axes lying on the section plane, the torsion on the third perpendicular axis, two shear strains and the longitudinal strain. It has been assumed the shear strains to be negligible; therefore in this case each section of the beam has four DOF. In other words, the Euler − Bernoulli hypothesis has been adopted, so the perpendicular axis of the section is parallel to the tangent vector of the beam backbone. It is assumed that the beam is made of hyper-elastic material and the constitutive equations are linear. It is important to clearly highlight in advance that in order to unburden the algorithmic complexity, the gravity eect has not been taken into count. This aspect could be implemented into a more complete dynamical model. By the way, the gravity eect is proportional to the body sizes and does not play a relevant role in the scale of the nipper described in this work. More important, this model can approximately describe the kinematics of the system, i.e. the deformed state knowing the cable tension and the geometrical sizes, but since the case of study is a soft beam, its control in position diers between loaded and unloaded state, i.e. any action externally explicated, which diers from the actuation force, produces a deformation that can not be negligible. To accurately de ne the pose of points of interest on the robot, it is necessary to characterize forces and moments applied to the body by both its own actuators and the external environment. Thus if the kinematics description is only approximative, the inverse kinematics can not be performed with the simpli cations done, and deserves further examination.
3.2.2 Kinematics
The nipper actuation is performed by pulling together the cable. It produces a bending with increasing curvature from the base to the tip, since the transverse section is decreasing towards the tip, thus giving less resistance to in exion. Unlike kinematics for traditional rigid-link robots where the position of any point on the robot can be fully de ned (in closed form) by link lengths and joint angles, the inherent compliance of continuum robots requires consideration of elasticity. It results in modeling a continuous robot as a curve in space shaped by shear, extension, and bending. An orientation assigned to each point along this curve gives the plane in which the cross-section of the rod lies. This formulation is
termed Cosserat rod model. A Cosserat beam is regarded as an in nite collection of in nitesimal rigid bodies. An orthogonal reference frame centered on the center of mass is attached to each cross section. The geometric transformation between two consecutive local frames is determined by the strain values of the section. Therefore, we can reconstruct the shape of the beam by referring it to a xed frame knowing the strains quantities along the beam and the position-orientation of the base local reference (Figure 3.3). The generic section of the beam is represented
Figure 3.3: Beam backbone.
by a tern of directors. Two of them ~n and ~b lie on the cross-section plane and the third, perpendicular to the plane, comes out in the direction of the tip of the beam ~t. For all the local reference frames ~t×~n = ~b holds. κ and ξ represent respectively the curvature with respect to the axes ~b and ~n and τ represents the torsion of the section with respect to the axis ~t (Figure 3.4).
The following expressions show the kinematic equations of this Cosserat beam: d ds(~t) = κ(1 + q)~n − ξ(1 + q)~b d ds(~n) = −κ(1 + q)~t + τ (1 + q)~b d ds(~b) = ξ(1 + q)~t − τ (1 + q)~n d ds(¯u) = (1 + q)~t (3.1)
Figure 3.4: Global and local reference frames.
3.2.3 Cable parametrization
Figure 3.5: Generic cross section of the beam.
By pulling the cable the segment bends on the side of the cable in a way that depends on the cable tension and on the arrangement of the tendon inside the robot body. The cable path is parameterized with its distance from the mid-line. The bending motion increases with the anchorage distance to the midline. De ning d(s) the distance of the cable from the mid-line (Figure 3.5), it results for the left
and right part of the cable: dl(s) = ( al− bl L )s − bl dr(s) = ( ar− br L )s − br (3.2) where b and a are respectively the distance of the tendon from the mid-line at base level and anchorage level and L is the length of the segment (Figure 3.6). The distance d(s) is de ned by its components on ~n and ~b. The kinematics of the robot arm and function d(s) determine the kinematics of the cables as follows:
¯
uc= ¯u + dn(s)~n + db(s)~b (3.3)
Figure 3.6: Tendon parametrization inside the nipper body.
3.2.4 Internal loading by tendon
There are two key assumptions we must mention with respect to the tendon. The rst is that, internally, the tendon can only resist axial tension all the way through the termination point. Externally, the tendon can only resist locally transverse loads along its length (i.e., no friction, only contact forces). These assumptions are the equivalent of saying that the tension in the tendon is constant along its length or that it is a pure tension element. Let us assume a generic beam with a tendon embedded stuck at the base of a global frame; then the cables allow a point load Lc located where the cables are fastened, equal to the cable tension and
tangent to the local t vector and a distributed centripetal load Ld along the cable
con guration, due to the contact between the cable and the silicone body. The friction is neglected, so the contact force is orthogonal to the cable (Figure 3.7). For a detailed mechanical analysis of this tendon load see the fundamental work of [Camarillo et al., 2008]. The distributed force is constant along the length of the tendon-beam interface. The magnitude of Lc and Ld is equal to T . The
Figure 3.7: Concentrated(Lc) and distributed(Ld) load due to the tendon tension
on longitudinal beam section containing cable.
previously results refer to the situation of a single-tendon xed at the tip. In this case, the special path of the cable immersed inside the nipper body produces a distributed fastening, hence a distributed load to the tip of the segment, where the path becomes parallel to the section and two distributed centripetal loads along the symmetrical cable con guration. The magnitude of the load to the tip is still equal to the cable tension and tangent to the local ~t vector of the section, while the magnitude of the two distributed centripetal loads is equal to T/2.
3.2.5 Statics
The motion of the beam is described by equilibrium and constitutive equations. The following expressions show the equilibrium equations (under the steady-state condition): d ds(F ) +n = 0 d ds(M ) + d ds(¯u) × F + m = 0 (3.4) where F is the vector of the internal contact force and M is the vector of the internal torque force. n and m are respectively the external applied force and torque for unit of reference length. Everything is expressed in the local reference
frame. For the details about the formulation written above please refer to: [Simo, 1985; Simo and Vu-Quoc, 1986, 1988]. The equilibrium balance at the tip section led to:
F (L) = −T ~tL
Mb = T an
Mn = 0
(3.5)
where Mb and Mn are the rotational equilibrium around the axis ~b and ~n. From
the assumption of linear constitutive equations: GI 0 0 0 EJn 0 0 0 EJb τ (s) ξ(s) κ(s) = M (3.6) EAq(s) = N (s) (3.7) N is the ~t component of F, E is the Young modulus obtained from a tensile test executed with standard dog bone silicone specimens, G is the shear modulus, I, Jn
and Jb are respectively the moment of inertia of the section with respect to axes t,
n and b. It is assumed that both the cross section and the constitutive equations do not change with the deformation. The rotational DOF has been considered to belong to C1 class function; thus equation (3.6) can be derived obtaining the
following expression: GI 0 0 0 EJn 0 0 0 EJb d dsτ d dsξ d dsκ + G ˙I 0 0 0 E ˙Jn 0 0 0 E ˙Jb × τ (s) ξ(s) κ(s) = ˙M (s) (3.8) where the dot means the derivative with respect to s. It results more useful to solve equation (3.8) instead of equation (3.6). The nger has a rectangular cross section; therefore the section moments of inertia are:
Jn(s) = 1 12H(s) B 3 Jb(s) = 1 12B H(s) 3 I(s) = Jn(s) + Jb(s) (3.9)
where B is the constant base of the rectangle and H(s) is the variable height of the rectangle. The area A is equal to BH(s) . The shear modulus G can be calculated
with E/2(1 + ν) where ν is the Poisson ratio.
3.2.6 Results
Finally, handling the constitutive equations (3.6) and (3.7) the motion of the manipulator has been obtained. For the details about the manipulator equilibrium equations and the dierential equations which describe the DOFs please refer to [Renda and Laschi, 2012]. The found equations allow to obtain the deformations value (in a planar steady-state condition) of the nger by knowing the geometric parameters and the tension of the cable embedded inside the body. The graphics obtained from the numerical simulation in MathWorks MATLAB environment can be seen in Figure 3.8 and Figure 3.9. Torsion module and the curvature on ~n are null because bending happens around ~b (Figure 3.8a, 3.8b). Curvature on ~b and compression increase (in module) toward the tip of the segment, because the cross section decreases (Figure 3.8c, 3.8d).
(a) (b)
(c) (d)
Figure 3.8: Graphic results of the numerical simulation:(a) torsion module, (b) curvature on n, (c) curvature on b and (d) compression module.
(a)
(b)
Figure 3.9: Graphic results of the numerical simulation: backbone deformed con- guration on (a) XY-plane and (b) 3-D.
