1. Kinematic Analysis

1.1. 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.

Open

Fig. 1a: Gantry’s Horizontal Translation

Open

Fig. 1b: Arm’s Vertical Extension

Open

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:

1.2. Claw’s grasping mechanism analysis

1.2.1. Introduction

In this mechanism, link BC can be modeled as the input with a constant angular velocity as 1 rad/s to conveniently determine the resulting motion, velocity, and acceleration of the slider. This assumption allows the kinematic relationships to be clearly expressed and analyzed. However, in practical implementation of the claw mechanism, the situation is reversed: the slider undergoes driven linear motion, typically with acceleration and deceleration phases, while the output link rotates more uniformly to achieve controlled grasping. This configuration ensures smoother finger closure, improved grip stability, and reduced impact forces when contacting the object. Therefore, by first assuming a constant rotational input at link BC, the motion characteristics of the slider can be back-calculated. These results can then be inverted and applied to determine the required linear input profile for the slider in the actual actuation scenario.

The main purpose of this analysis is to:

1. Model the kinematic motion of the claw mechanism.

2. Use the rotation of one driving link (BC) as the input motion, assumed to rotate at constant angular velocity.

3. Determine the resulting linear motion of the central slider (point A).

4. Compute the velocity and acceleration of the slider.

5. Use these results to evaluate the force required to grasp an object.

1.2.2. Mechanism description

The claw consists of the following functional components:

• Input Link (BC): Input constant angular velocity 1 rad/s (L2) in this problem.

• Coupler Link: Transfers motion from link BC to the central slider (L3).

• Central Slider (A): Moves linearly along a prismatic guide and determines the finger opening/closing distance (L4).

• Ground Link: L1.

When the input link BC rotates, it drives the coupler, which pushes or pulls the slider A. The slider then simultaneously rotates both gripper fingers inward or outward.

This ensures:

• Equal displacement of both fingers.

• Smooth and predictable grasping behavior.

Open

Fig. 2: Position description

1.2.3. Position analysis

The linkage geometry is adopted where point A slides on a vertical guide and the input link BC rotates with angle theta_2. Using your relations, the coupler angle theta_3  is:

Open

This GIF below illustrates the motion of the claw mechanism. Point B moves along a circular arc about the fixed pivot C, while point A 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.

Open

Fig. 3: Linkage movement animation

1.2.4. Velocity analysis

With constant input angular velocity, the velocity relations are:

Open

Open

Open

Fig. 4: Velocity analysis

1.2.5. Acceleration analysis

Using the expressions with input angular acceleration:

Open

The slider point A acceleration is:

Open

Open

Fig. 5a: Input angular acceleration

Open

Fig. 5b: Slider acceleration

 

1.2.6. Mechanical advantage

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

Open

Where V_B is the velocity of point B on the input link and V_A is the velocity of the slider A. 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.

Open

Fig. 6: Mechanical advantage

1.2.7. Force analysis

The output slider-crank force will just be mass multiplied by acceleration (F = ma) since it’s a pure prismatic translation. It could be calculated when we get the slider’s physical mass, and the force profile will look similar to Fig. 5b.

1.3. Arm’s Vertical extension mechanism analysis

1.3.1. Introduction

This mechanism is broken up into 2 distinct parts. The first is the slider crank mechanism in the 1st quadrant, and the second is the x bar linkage system underneath it. The main purpose of this mechanism is to adjust the claw position in the z axis up and down to the correct height to pick up and deposit its object.

The main purpose of this analysis is to:

1. Model the kinematic motion of the arm extension mechanism

2. Determine the acceleration felt at the claw connection point

3. Determine the angle change vs time required to minimize acceleration

1.3.2. Mechanism Description

The input that will be controlled is the angle for the crank link in the slider crank, denoted as L1 in Fig. 7. As the angle for L1 increases, the x bar extends and the claw lowers, while the angle for L1 decreasing retracts the x bar which raises the claw. An important requirement for this mechanism is to minimize the acceleration felt by the object at the end of the claw. This correlates with the acceleration at the claw attachment point on the x bar and so a main goal is to minimize acceleration felt at that attachment point.

Open

Fig. 7: Arm extension mechanism

1.3.3. Kinematic Analysis

The linkage system is grounded at the origin of Fig. 7 and the positions, velocities, and accelerations are calculated in 2 steps. The first step is determining the slider’s position at each angle step using the main slider crank mechanism, then using this to determine the position at the end of the x bar. This 2 step process is repeated to determine the velocity and acceleration at these points. 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.

Open

Fig. 8a: Velocity with CCW rotation

Open

Fig. 8c: Acceleration with CCW rotation

 

Open

Fig. 8b: Velocity with CW rotation

Open

Fig. 8d: Acceleration with CW rotation

 

1.3.4. Acceleration Analysis

Due to this mechanism’s main goal being to minimize the acceleration felt at the end point, an iterative process was required to determine how long it must take for each angle step to take place. This was done by calculating the acceleration felt at the claw attachment point for each angle change and iteratively determining the necessary timestep in order to meet the acceleration requirements. If the only goal is minimizing acceleration the velocity the claw will move at will be almost zero. To counteract this a secondary requirement for maintaining a certain velocity for as long as possible was created. This is so there are no sudden shifts in acceleration while allowing for the arm mechanism to move at a reasonable speed.Both an acceleration threshold that the system must remain below and a velocity threshold the system must remain within while not exceeding the acceleration threshold was implemented to accomplish this. The next graphs were determined using this method and the following parameters.

Angle range for L1 - 60-120 degrees

Length of crank - 30 mm

Length of connecting rod - 70 mm

Length of X bars - 90 mm

Length of Link connecting claw to X bar - 45 mm

Acceleration threshold - 20±2 mm^2/sec

Velocity threshold - 10±1 mm/sec

Open

Fig. 9a: Theta vs Time

Open

Fig. 9c: Acceleration vs Time

Open

Fig. 9e: Acceleration vs Theta

 

Open

Fig. 9b: Velocity vs Time

Open

Fig. 9d: Velocity vs Theta

 

The most important graphs here are the theta vs time graph and the acceleration vs theta graph. The these vs time graph shows the timestep required for each change in angle which can then be used directly in the mechanism to ensure the proper acceleration. The acceleration vs theta graph confirms that the magnitude of the acceleration vector doesn’t go above the threshold and the acceleration requirements are met.

2. Physical Prototype

2.1. Claw mechanism prototype

Open

Video 1: This claw mechanism could grasp a D25mm ball successfully

2.2. Gantry-arm-claw system demonstration

Open

Video 2: A combined demonstration of the gantry-arm-claw system, which achieves the translation-extension-grasping goal

3. Iteration Documentation

3.1. Initial design iteration

3.1.1. The 1 -DOF claw

Open

Fig. 10: This 1-DOF claw cannot fulfill the multistep translation-extension-grasping goal

3.1.2. Gantry-arm-claw combined system

Open

Fig. 11: This combined system can fulfill the multistep goal

3.1.3. Different gripper mechanism designs

Open

Fig. 12a: Current design

Open

Fig. 12b: Alternative gripper design

 

Current design in Fig. 12a uses slider-crank mechanism, it has pros and cons as follows:

Pros:

●       Self-centered motion ensures both jaws move symmetrically, reducing misalignment when grasping objects

●       Compact and lightweight

●       Predictable kinematics; single input variable defines full motion path

Cons:

●       Limited grip force as mechanical advantage decreases as jaws fully close

●       Frictional losses in small revolute joints that may reduce precision

●       Limited payload size relative to gear-based or cam-driven alternatives

The alternative design in Fig. 12b replaces the linkage actuation in Fig. 12a with two meshing gears that directly rotate the gripper arms. A central pinion driven by a motor produces synchronized mirrored motion of both jaws. This design allows for high torque transfer and precise angular positioning, beneficial for heavier payloads or tighter controls.

Pros:

●       Increased torque capacity enables handling of heavier or stiffer objects.

●       High positional accuracy due to direct gear engagement

●       Easier to integrate with rotary servos or stepper motors for fine control

Cons:

●       Manufacturing tolerance sensitivity – 3D printed gears may have backlash or even meshing

●       Increased noise and wear under continuous cycling

3.1.4. Horizontal translation system design

Open

Fig. 13: Horizontal translation concept design

Horizontal translation is achieved via a motor-driven pinion engaging a linear rack. As the pinion rotates, it converts rotary motion into linear displacement of the platform, moving it along the x-axis. A linear guide rail ensures stability and eliminates out-of-plane movement.

Pros:

●       High positional precision and repeatability for controlled linear motion

●       Rigid structural support – the guided track prevents lateral or vertical deflection

●       Scalable – easy to extend travel length by increasing rack length

Cons:

●       Manufacturing tolerance sensitivity – 3D printed gears may have backlash or even meshing

●       Gear wear overtime can degrade positioning accuracy

●       Requires precise alignment between pinion and rack to avoid binding

●       No inherent holding torque – power loss can cause unintended direct unless locked

3.1.5. Different arm extension system designs

Open

Fig. 14a: Slider-Crank

 

Open

Fig. 14b: Pulley system

 

Open

Fig 14c: Electric linear solenoid

 

Concept 1: Slider-Crank

The mechanism in Fig. 14a converts rotary motion into vertical linear displacement in the y-direction through a slider-crank linkage. A rotating crank, L3, drives a connecting rod that moves a slider vertically, providing a defined lift stroke.This mechanism would be mounted on a prismatic base in order to allow for the x-axis translation.

Pros:

●       Reliable and well-understood motion profile with predictable output path

●       Smooth sinusoidal motion, ideal for repetitive lifting cycles

●       Mechanically efficient – minimal energy loss through friction compared to cables in a pulley system.

Cons:

●       Fixed stroke length determined by crank radius; limited adjustability

●       Nonlinear vertical profile – slower at extremes, faster mid-stroke

●       Difficult to scale for longer vertical travel without redesigning linkage

Concept 2: Pulley system

In Fig. 14b, a motorized winch drives a cable wound around pulleys to raise and lower the vertical arm assembly. Bearings guide the rope to maintain tension and alignment, offering flexible travel length and smooth lifting.

Pros:

●       Large range of motion achievable with minimal redesign

●       Fewer rigid components making it lightweight

●       Easily reversible motion by changing winch motor direction

Cons:

●       Low positional precision due to cable stretch and slack

●       Requires counterweights or tensioners for stability

●       Safety concern if cable fails under load

Concept 3:  Electric Linear-Control Solenoid

In Fig. 14c, a proportional-controlled solenoid provides direct linear motion, pushing and pulling the connected links and thus creating a scissor type extension/retraction along the y-axis. This type of solenoid has a responsive actuation based on the current input, allowing for variable actuation.

Pros:

●       Fast response with minimal delay

●       Compact and self-contained – no external linkages or pulleys

●       Easy electronic control with PWM signals

●       Low maintenance due to the absence of moving joints

Cons:

●       Limited stroke length – adjusting it would require a new solenoid or shaft

●       Force drop with distance – poor performance for long or heavy lifts

●       Requires continuous power to maintain actuation (no self-locking)

●       Low efficiency compared to pulley or slider-crank for sustained loads

3.2. Prototype iteration

3.2.1. Iteration 1 - 2D claw

Open

Fig. 15: Use 2D mechanism to verify our initial design

3.2.2. Iteration 2 - 3D claw

Open

Fig. 16: This 3D claw holds balls securely

3.2.3. Iteration 3 - Incorporated with other subsystems

Open

Fig. 17: Whole system

4. Draft BOM