IEEE Robotics & Automation Magazine - September 2023 - 115

system's complexity, is an example of a multi-DoF joint, parallel
link, and closed-loop component, which the SoRoSim
toolbox can model. We assign two divisions for these soft
links, enable all rotational DoF, and assign constant strain
bases. Using appropriate reference strain values, we set the
precurvatures of the soft buffer links. An angle-controlled
rigid disk attached at the end of the buffer assists further orientation
of the grippers by rotating about its local x-axis as
in Figure 8(b)(5).
Finally, we have a soft gripper with two fingers that are
cable actuated. The thin sections of the finger act as flexible
links, while the thick sections act as rigid links. We assign a
strain basis with constant bending about the local y-axis to
the thin sections. This design of the gripper, where the cable
is positioned outside the thin soft section, is widely used in the
soft robotic community [13]. This particular mode of actuation,
where the actuators go in and out of the system, is an
important class of prototype that SoRoSim is able to simulate.
We compute the static equilibrium configuration of the system
for the actuation inputs given in Figure 8(b). The system is
subject to gravity and a follower point force. The direction
of the gravity and the point force (F) are included in Figure 8(a).
The value of the point force is 2 N. The corresponding equilibrium
configurations are detailed in Figure 8(b)(2)-(5). Figure
8(b)(6) demonstrates the actuation of the grippers when a cable
tension of 2 N is applied. We show a rod as a reference grip
target. There are 39 DoF and seven actuation inputs for the
system. On average, the static equilibrium simulations [shown
in Figure 7(b)(1)-(6)] take 1.2 s.
This example analyzed a system with passive and actuated
rigid joints controlled by joint coordinate values. The toolbox
can also define joints controlled by joint force and moment
values. The user can also assign stiffness values for rigid
joints. For instance, a prismatic joint with a positive stiffness
value is equivalent to a linear spring.
CASE 2: CONTACT DYNAMICS
Analysis of mechanical systems involving contact-impact
events is demanding for many real-world applications. Contact
dynamics involve determining potential contact points
between colliding bodies, evaluating contact-impact forces,
and establishing the transition between different contact scenarios.
The implementation of contact mechanics is one of
the most challenging and complex problems in modeling and
computation [23].
Contact between two points involves the application of equal
and opposite normal and tangential forces on each other. The
current SoRoSim toolbox does not have built-in capability for
contact dynamics. However, the user can apply various kinds
of external loads by enabling a property of the SorosimLinkage
class, namely, " Custom External Force Present, " and then
editing a MATLAB function, " CustomExtForce.m, " to model
the external force. Quantities such as Jacobian J, screw velocity
,h and transformation matrices g are passed as inputs into
the " CustomExtForce " function. The user can apply a custom
external force as a function of these quantities.
To demonstrate contact dynamics using SoRoSim, we analyze
a three-fingered gripper system attempting to grip and
manipulate a spherical rigid body [Figure 8(d)]. The system
consists of two rigid links with prismatic joints and three symmetrically
arranged fingers similar to the previous case. We
choose the dimensions of links arbitrarily. The inputs to the
dynamic simulations are the kinematic inputs of the prismatic
joints (vertical and horizontal translation) and the cable tension,
which actuates the fingers.
We make the following approximations to simplify the estimation
of contact points and the evaluation of contact forces.
1) The three thick rigid sections of each finger are approximated
as spheres with diameters ()r2 i equal to the section
heights and centered at the center of mass of the links.
2) We neglect the tangential (frictional) component of the
contact forces. The normal force due to nine different contact
points (three on each finger) is sufficient to provide a
force closure grasp.
0.5
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Kinetic Energy
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FIGURE 7. The free fall dynamics. (a) The energy change for the
undamped (solid) and damped (dashed) beam. Superimposed
images of (b) the damped case and (c) the undamped case.
Dashed lines denote the tip trajectory.
SEPTEMBER 2023 IEEE ROBOTICS & AUTOMATION MAGAZINE
115
t = 0.75 s
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IEEE Robotics & Automation Magazine - September 2023

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