Design

Design Criteria

With this project, our end goal was to develop a reasonable model of a rigid object grasped by a flexible, non-rigid vacuum gripper in order to enable better planning and actuation in real-world applications where vacuum grippers are currently used.

In order for this project to meet its end goal, we require that we must be able to perform on benchmark tasks. These benchmark tasks will demonstrate that our model for the rigid object - vacuum gripper system is correct and could potentially be used to augment real-world applications.

The first desired functionality is for the robot to position its end effector in the correct orientation in order for the rigid object to achieve the desired orientation without the use of camera sensing. This planning task will demonstrate that the model correctly describes the relationships between the robot’s end-effector and the rigid object being grasped by the flexible, non-rigid vacuum gripper.

The second desired functionality is for the robot to use the dynamics model to determine the mass of unknown objects. Essentially, if the system is able to measure the angular displacement induced in the vacuum gripper by objects of unknown mass and use these measurements along with the dynamics model of the vacuum gripper to determine the mass of the grasped object with some degree of accuracy, this would represent that the model correctly reflects reality.

Our Approach

The key question our project is trying to answer is “how to model a vacuum gripper”. To answer this question, we experimented with different mathematical models for the vacuum gripper.

Model Approach #1: Underactuated Joint

Our first idea was to simply model the bent angle between object and vacuum gripper as an unactuated joint. We believed it could be treated as a freely moving joint with the length being the length of our gripper attachment; however, there were a number of drawbacks to this approach.

With this model, we realized we could not accurately determine how objects of different size, masses, etc would move. Additionally, this model didn’t align well with the non-rigid deformation that we saw on the flexible gripper cup when grasping a large object. Empirically, we noticed the object’s angle would oscillate before it settled down, which was a key indicator that a spring might be a better model.

Gripper could be modeled as first link with gripper angle q1 and object as the second link with angle q2 (note that angle q2 represents the angle that the vacuum cup bends at and is thus an un-actuated joint) [image source: EECS106A]

Model Approach #2: Torsional Spring

After noticing the push and pull nature of the vacuum gripper suction cup, we thought it reminded us of a spring! We decided to try to fit a spring model to the vacuum gripper. In this case a torsional spring would be the proper model due to the rotational behavior of the object at the gripper’s suction cup.

Although a torsional spring is a more complex model than an underactuated joint, we believed that it would better model the non-rigid behavior. In real-engineering applications, using an accurate model will let us better predict object pose for a given gripper pose. Additionally a more accurate model will let us generalize the results to a broader number of objects and gripper poses with higher accuracy.

Energy of torsional spring is determined by displacement angle [image source]

Torque on torsional spring is determined by displacement angle [image source]

Lagrangian Dynamics

To properly model a torsional spring, we need to determine the spring constant K. This constant is best determined through an experimental process where we will observe angles between the gripper and object for a variety of objects (discussed further in the implementation section).

In order to understand the behavior of the system, we were interested in computing the Lagrangian dynamics for a simple system containing the gripper, the gripper suction cup (modeled as a torsional spring), and an object of mass M.

Ultimately we are trying to find the spring constant. Using this Lagrangian dynamics model, it’s difficult to reason about what the spring constant might be considering that the Lagrangian is a continuous time model and any measurements we take of gripper-object angle will be discrete time. Additionally, we aren’t particularly interested in the transient angle of the object over time and are more interested in how the object behaves in a steady state.

Free Body Diagram Model (Steady State Analysis)

This motivated us to work on a simpler free body diagram model of the gripper that still used the torsional spring model, but now in steady state. In this model, we consider balanced forces that will give us equations to relate gripper angle to the object angle based on the spring constant K.

The key relationships from the model are boxed in the diagram below.

This equation is the basis for our modeling and will be used for determining the string constant K given a series of experimental measurements of object and robot angles. Notably this model doesn’t consider time and provides an understanding of the system at steady state. Real world control applications with this vacuum gripper might require understanding transient behavior of the object angle. For this, we can still use this static model to determine the string constant K and then use this K value in the Lagrangian model we used above.

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