Workspace Analysis

Workspace analysis is essential in robotics to understand the capabilities of robotic manipulators or grippers in manipulation tasks. Previously, the Monte Carlo method was employed to simulate all potential positions of the fingertip on the robotic gripper, assuming a 100% success rate in the gripping process. However, this approach did not fully capture the complexity of real-world manipulation as it overlooked the contribution of the entire upper side of the fingers.

To address this limitation, our revised methodology now incorporates a more comprehensive analysis of the finger by including the entire upper side of the finger in our workspace analysis. This involves iteratively combining all the points to ensure complete exploration of the workspace. Furthermore, it is crucial to determine whether the combinations of these points result in a secure grip, taking into account the resultant force and momentum. These factors are particularly important in visualizing the graspability map, as they significantly influence the effectiveness of the grip.

Assumption

All the objects the gripper have interacted with is sphere
Each finger has one and only one contact point with the object

Accessibility

The robotic hand's accessibility is visualized through a point cloud map, where different colors represent different fingers. This map illustrates the hand's reachability, identifying areas within the hand's operational space that can be effectively reached and manipulated by each finger. Regions outside this point cloud map are considered unreachable, providing a clear demarcation of the hand's operational limits.

The Upper image represents a generated point cloud with uniform spacing between each point. In contrast, the lower image, generated using the Monte Carlo method, illustrates potential fingertip positions and includes significant noise, resulting in non-uniform intervals. This disparity suggests that while the method on the right may conserve computational resources, it is prone to producing numerous omissions in the workspace analysis.

Dexterity

Upon establishing the coordinates of these three contact points, we can generate the direction of the force from each point, oriented perpendicularly to the finger's inner surface. Then we can calculate the circle's center and radius using three points. Lastly, by iteratively increasing the force exerted at each contact point in its force direction, we can obtain the minimum resultant forces and moments for each point combination. An object is considered graspable if both the resultant force and moment remain beneath a predetermined threshold affected by the frictional forces and the potential deformation of the fingers. Information such as its centre of circle, radius, resultant force and momentum would be recorded.

Object-size-relative dexterity

The radius of the object would be used to construct a histogram. This histogram contrasts the gripper’s dexterity for specific dimensions (radius) among its total grasping range, showcasing the gripper’s effectiveness with various object sizes.

The histogram clearly indicates that the performance of two and three-finger grippers is comparable when handling small objects (radius ranging from 20 to 40 mm). However, the three-finger gripper demonstrates superior efficacy in grasping larger objects. Conversely, the two-finger gripper exhibits a marginally better performance than the three-finger variant when picking up smaller objects (radius ranging from 40 to 60 mm).

Spatial Dexterity

For spatial dexterity visualization, Kernel Density Estimation (KDE) with a Gaussian Kernel is used to create smooth, continuous density estimates. We apply KDE to the workspace visualization of the proposed gripper, which demonstrates where the gripper is most capable of handling objects of various sizes. It is first by filtering out the specific radius of the objects, then use the KDE on the objects' coordination. Eventually the object with such radius's 3D dexterity map would be generated.

Formula for the KDE:

f(x) represents the estimated density at point x
n is the total number of data points.
h is the bandwidth that controls the width of the kernel.
K is the kernel function, which for a Gaussian kernel is expressed as:

Here u is the standardized distance(x−xi)/h, meaning the relative position from the center xi, scaled by the bandwidth h.

Combining these, we get the complete formula for the Gaussian kernel density estimate:

Below is a example of the three finger gripper dexterity workspace for objects with radius of 90- 95 mm

Three finger dexterity worksapce in Y and X plane projection

Two finger dexterity worksapce in Y and X plane projection

Graspability

Object-size-relative graspability

The two histograms show how the Resultant Momentum and force changes with respect to the radius of the objects. The trend of the momentum keeps increasing, but since the unit is (N × mm) such such momentum can hardly rotate the objects.

While for the resultant force, the area from 40-70 is its lowest value despite the noise in the previous one. Which means object within that range can have the most effect grips.

Spatial Graspability

Upon filtering out coordinate with the lowest momentum and resultant force, the distribution of the objects is depicted in the subsequent diagram.

Objects of Minimum Resultant Force Distribution

Objects of Minimum Resultant Momentum Distribution

The coordinates of both optimal grip essentially overlap, confined within the range of 0 to 100 on the X-axis, -50 to 50 on the Y-axis, and 50 to 100 on the Z-axis. The similar distribution of points in the two diagrams indicates that the gripper's operations within this spatial region can achieve both minimal resultant force and minimal resultant torque, which is an ideal outcome. This suggests that the gripper's design and control strategies are optimized for this specific spatial region, enabling efficient and stable grasping operations.

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