1. Mobility calculation of subsystems

The robotic system consists of three coordinated subsystems that enable horizontal movement, vertical lifting, and object grasping. Each motion is driven by a dedicated motorized mechanism designed to provide smooth and precise control within the workspace. The combined operation of these three motions allows the robot to reach, lift, and place objects accurately while maintaining stable and gentle handling.

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Fig. 1a: Gantry’s Horizontal Translation

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Fig. 1b: Arm’s Vertical Extension

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Fig. 1c: Claw’s Grasping Mechanism

Gantry’s Horizontal Translation Motion: As shown in Fig. 1a, the horizontal motion of the system is driven by a motorized belt–rail mechanism mounted at the top of the gantry. When the motor rotates, it moves a slider along the rail, carrying the entire lifting and grabbing assembly horizontally across the working area. The Gruebler-Kutzbach Equation for this gantry is shown below:

Arm’s Vertical Extension Motion: As shown in Fig. 1b, the arm’s vertical motion is actuated by a scissor link mechanism connected to a slider–crank actuated with a servo motor. As the crank moves, the deployment angle of the scissor linkages extends or retracts accordingly, moving the arm up and down. The Gruebler-Kutzbach Equation for this arm is shown below:

Claw’s Grasping Mechanism Motion: As shown in Fig. 1c, the claw’s grasping action is controlled by a servo motor mounted on a prismatic joint that drives the central slider of the claw. The motion of this slider transmits through linkages to open or close the claw fingers, enabling the system to grip or release objects. The Gruebler-Kutzbach Equation for this claw is shown below:

2. Claw’s grasping mechanism analysis

2.1. Introduction

The claw grasping mechanism is designed to gently grasp and release objects as part of a vertically integrated material-handling system. Unlike a simple single-hinge gripper, this mechanism employs a symmetric linkage to synchronize the motion of both fingers while enabling controlled grasping behavior using a single input.

The main objectives of this analysis are to:

1. Develop a kinematic model of the claw mechanism.

2. Use the vertical constant-velocity motion of the slider at point C as the input.

3. Determine the resulting angular motions of links AB and BC, as well as key-point velocities and acceleration.

4. Use the kinematic results (via power/velocity–force relationships) to evaluate the required actuation force and identify operating regions to avoid toggle (singularity) conditions.

2.2. Mechanism description

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Fig. 2: Claw’s position description

The claw consists of the following functional components:

• Input Slider Link (L4: Point C): The slider driven by a rack-and-pinion mechanism 20mm/s.

• Coupler Link (L3: BC): whose rotation angle is θ3.

• Crank Link (L2: AB): whose rotation angle is θ2.

• Ground Link (L1)

As the slider at point C moves up and down, links BC and AB rotate accordingly, and this motion is then transmitted to the symmetric finger linkage, producing coordinated opening and closing of the claw.

2.3. Position analysis

The linkage geometry is adopted where point A slides on a vertical guide (so Vcx = 0), and uses the slider’s constant vertical velocity  Vcy = 20mm/s as the input, and ​ θ2 are θ3 are determined from the known link lengths and geometric constraints as functions of time.

This GIF illustrates the motion of the claw mechanism. Point B moves along a circular arc about the fixed pivot A, while point C translates strictly along a vertical guide, driving the symmetric rotation of the finger links and producing mirrored opening and closing of the claws. The fingertip paths are elongated and curved due to the combined effects of slider translation and link rotation. A critical condition occurs when links AB and BC become collinear, forming a toggle configuration. At this geometric lock, shown by the pink star marker at point B, an infinitesimal rotation of the input link results in negligible motion of the slider A. In this state, the mechanical advantage becomes extremely high, increasing the potential gripping force but preventing smooth motion and making it difficult for the mechanism to pass through this position without external assistance. Therefore, practical operation should avoid approaching the toggle region to ensure stable and reversible grasping behavior.

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Fig. 3: Claw’s motion profile

2.4. Velocity analysis

With constant input velocity, the angular velocity relations are:

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Fig. 4a: Point’s speed

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Fig. 4b: Link’s angular velocity

2.5. Acceleration analysis

The angular acceleration equations for L2 and L3 are shown below:

The acceleration profile is shown below:

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Fig. 5: Link’s angular acceleration

2.6. Mechanical advantage

The mechanical advantage describes the amplification of motion or force between the input and output:

Where VB is the velocity of point B on the input link and  is the velocity of the slider C. A higher mechanical advantage increases gripping force but reduces the available stroke range, while a lower mechanical advantage increases finger motion range at the cost of output force.

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Fig. 6: Mechanical advantage v.s. time

2.7. Force analysis

The output back-driven slider-crank force will just be mass multiplied by acceleration (F = ma). It could be calculated easily, and the force profile will look similar to Fig. 6.

3. Arm’s vertical extension mechanism analysis

3.1. Introduction

The arm mechanism consists of a slider crank driver and a scissor array for vertical motion with a connection point for the claw at the end. As the angle of the crank increases, the claws connection point is lowered until it reaches the desired z axis position. As one of the main goals for this project is to minimize acceleration on the object being moved, the acceleration at this connection point is very important to manage and minimize wherever possible.

3.2. Mechanism description

As the crank of the slider increases in angle, the slider shifts to the left and the scissor array expands in the z axis. While expanded, small degree changes in the crank cause large position changes at the claw attachment point compared to when the mechanism is almost fully closed. For this system the minimum angle to fully close the array is 0 degrees while the maximum angle to fully expand the array is 90 degrees.

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Fig. 7: Arm’s position description

3.3. Kinematic analysis

The position, velocity, and acceleration were all determined at each point in the arm mechanism. The connection of the mechanism to the crank was modeled to be 46.48mm above the origin to allow for a gap between the designed position for the servo and the end of the linkage system. The kinematic analysis was done in two stages, first the slider crank mechanism was analyzed to determine its motion properties. Next the values of the slider’s motion were used to determine the rest of the scissor array’s motion with the most important value being the acceleration at the claw attachment point. The GIFs below show the position, velocity, and acceleration of each point in the mechanism. Each frame of the GIFs below show what the velocity and acceleration would be using a constant angular velocity of 1 rad/sec and a constant angular acceleration of 1 rad/sec^2.

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Fig. 8a: Velocity cw

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Fig. 8c: Acceleration cw

 

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Fig. 8b: Velocity ccw

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Fig. 8d: Acceleration ccw

 

3.4. Velocity and acceleration analysis

The most important part of the entire kinematic analysis for this mechanism is to determine the acceleration at the claw attachment point. To determine this the kinematic analysis was run iteratively to find the time step that optimized the acceleration and velocity at the claw attachment point. The goal was to minimize acceleration but there has to be a minimum desired velocity or the system will act incredibly slowly. The acceleration had a threshold of 100±5mm/s² that it could not go past while the velocity had a threshold around 50±5mm/s that the system wanted to maintain as much as possible. The iterative process found a timestep for each angle change that would optimize these values and then did so in reverse to find the wind down sequence for the arm movement. This was repeated for the desired z distance required for the regions where the balls will be deposited. This process generated the following graphs using the following parameters.

Angle range for initial position - 0-90 degrees

Angle range for close deposit zone - 0-40 degrees

Angle range for far deposit zone - 0-18 degrees

Length of crank - 41.14 mm

Distance of the ground joints - 46.48 mm

Length of connecting rod - 100 mm

Length of X bars - 135 mm

Length of Link connecting claw to X bar - 67.5 mm

Acceleration threshold - 100±5 mm/sec²

Velocity threshold - 50±5 mm/sec

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Fig.9a: Theta vs Time Initial

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Fig.9d: Velocity vs Theta Initial

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Fig.9g: Acceleration vs Theta Initial

 

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Fig.9b: Theta vs Time Close

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Fig.9e: Velocity vs Theta Close

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Fig.9h: Acceleration vs Theta Close   

 

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Fig.9c: Theta vs Time Far

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Fig.9f: Velocity vs Theta Far position

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Fig.9i: Acceleration vs Theta Far

 

The theta vs time graphs show the angle against the calculated timestep to try and minimize acceleration which is then used to get the timestep array for controlling the arm mechanism driving servo. The velocity and acceleration vs theta show the magnitude of the motion properties at the claw attachment point to ensure that the thresholds are followed and acceleration is minimized as much as possible while maintaining a certain velocity.