9.2 - Prototype & Kinematic Analysis

1. Kinematic Analysis & System Geometry

Description of Ideal Motion and Phase Transition:

The mechanism uses an asymmetric friction model to achieve locomotion, switching between two phases:

• Push Phase (Crank swings forward): The foot anchors into the global ground with a high friction. The linkage acts as a crank-slider pushing against a fixed wall, causing the global Body frame to be pushed to the left.

• Pull Phase (Crank swings backward): The foot lifts slightly off the ground greatly reducing the friction. The global Body frame then becomes fixed, and the foot retracts back toward the body.

System Geometry and Linkage Dimensions:

Based on our updated kinematic schematic, our design evolved into a symmetric slider-linkage mechanism to effectively manage the 1-DOF transition. The physical prototype is built using the following parameters:

• L1 (Ground/Base): = 75 mm (Center to edge)

• L2 (Lower Link): = 55 mm

• L3 (Upper Link): = 55 mm

• L4 (Slider/Clamp): = 20 mm

• Total Base Footprint: 150 mm at maximum extension (flat state).

Mobility Calculation (Gruebler's Equation):

Links (n = 5):

1. Ground (the track/body)

2. Crank (Link 2)

3. Coupler (Link 3)

4. Foot Link (Link 4)

5. Slider (the foot contact sliding on the ground)

Full Joints (J₁ = 5):

1. Revolute: Ground to Crank

2. Revolute: Crank to Coupler

3. Revolute: Coupler to Foot Link (Joint C)

4. Revolute: Foot Link to Slider

5. Prismatic: Slider to Ground

Gruebler’s Calculation:

• M = 3(5 - 1) - 2(5) - 0

• M = 12 - 10

• M = 2 Degrees of Freedom

However, due to friction and the dynamic distribution of forces within the system, one degree of freedom is consistently restricted during operation. This allows us to model the mechanism as swapping between multiple 1-DOF systems across a four-states:

• State 1 (Pull Extended): Low friction allows the foot to slide freely. Internal tension forces hold the knee joint in close to a 180 degree extension, causing the coupler and foot link to act as a single effective link. The system operates as a standard 1-DOF Slider-Crank.

• State 2 (Buckling): High static friction suddenly anchors the foot to the ground, eliminating the prismatic sliding DOF. The driving force redirects to bend the knee, morphing the system into a grounded 1-DOF Four-Bar Linkage.

• State 3 (Push Bent): The foot remains anchored by friction while the bending knee acts on the links to push the mechanism forward. The coupler and foot link behave as a newly bent effective rigid length. The system functions as an inverted 1-DOF Slider-Crank.

• State 4 (Unbuckling): The foot remains temporarily fixed as the crank reverses. The internal pushing forces straighten the knee joint, returning the mechanism to a 1-DOF Four-Bar Linkage until the leg is fully extended, at which point the friction breaks and the cycle repeats.

Attached below are videos which show:

• the 1-DOF behavior when the side near the motor is unable to slide.

• An animation showing the mechanism’s joint positions and translation over time

• Graphs generated by the animation that show the velocity and acceleration analysis.

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2. Physical Prototype Construction

Material & Assembly Overview:

The current physical prototype is a hybrid construction utilizing both additive manufacturing and laser cutting to balance structural rigidity with rapid prototyping capabilities.

• Chassis & Ground Track: The main base frame (L1) was laser-cut from wood, providing a flat, rigid track for the sliding mechanism. The central cutout dictates the linear path and acts as a physical limit for the slider.

• Linkages: The dynamic links (L2, L3) and the stationary mounting brackets were 3D-printed in PLA/PETG. The geometry includes integrated hubs to house metal shafts, acting as robust hinge joints.

• Actuation & Tension: In this initial iteration, two rubber bands are utilized beneath the sliding mechanism to provide tension. This tests the system's ability to return to a flat state and simulates the continuous pulling force required for actuation.

3. Iteration Documentation

Iteration 1: Conceptual Linear Translation

Our initial sketches focused heavily on a linear translation approach to achieve the required Delta x. Early ideas involved strings and pulleys closing the distance between the legs.

• Observation: A string-based system lacked the rigid push-force required to lock the robot into a 3D state and would not translate well into continuous walking motion.

Iteration 2: Rigid Kinematic CAD & Printed Pins

To solve the rigidity issue, we transitioned to a fully 3D-printed rigid multi-bar linkage.

• Observation: Relying purely on basic 3D-printed plastic pins resulted in high friction and significant "slop" (joint play). The torque required to overcome the initial "flat" mechanical disadvantage was too high.

Iteration 3: Hybrid Slider-Linkage Prototype (Current)

We redesigned the system to use a central laser-cut wood track with 3D-printed slider blocks and links. Most importantly, we replaced the printed pins with exact metal hardware (M3 screws and 8mm metal shafts).

• Observation: The inclusion of precision metal hardware drastically reduced internal friction and joint play. The rubber-band tensioned mock-up successfully and smoothly transitions from 2D to 3D, proving the mechanical logic is sound.

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Current model ~ Solidworks representation of model

4. Bill of Materials (BOM)

Current Prototype Hardware (Phase 1 Build):

To eliminate slop and ensure smooth deployment, the current physical model utilizes the following precise hardware:

Note on Pricing: Estimated prices reflect unit costs derived from the nearest standard size and package quantity available via McMaster-Carr. Individual component costs may differ when purchased in small quantities.

Additionally, not all materials were purchased for this prototype and mainly made use of the materials already accessible in the TIW, ( * ) are the materials that were actually purchased in the making of the prototype.