4.4 Kinematic Analysis
Our analysis was broken into two separate parts: the arm rocker and the slider finger assemblies. Code can be found in the subpage, Kinematic Analysis Code - Final Demo.
Arm Analysis
Ideal Motion Profile
The ideal motion profile for the arm mechanism replicates the pre-game ritual performed before rock-paper-scissors matches: a rhythmic pumping motion before each player gestures their chosen state. To achieve this initial motion, the arm employs a planar crank-rocker mechanism, which generates a rounded, cyclic motion. A ternary link serves as the coupler which gives the impression of a wrist joint at the end of the forearm. Adding this offset at the end of the coupler forearm enables a distinct rocking action independent of the crank input. All these design choices culminate to give the illusion of natural wrist movement.
Mobility Calculations
Crank-Rocker Link Lengths:
Crank length = 2.16 in
Coupler length = 9.00 in
Rocker length = 7.95 in
Ground link = 7.125 in
Grashof Condition:
S + L ≤ P + Q
2.16 + 9.00 ≤ 7.95 + 7.125
11.16 ≤ 15.075
L2 (smallest link) can rotate through a complete revolution.
Gruebler Equation:
l = 4 (ground, crank, coupler, rocker)
j₁ = 4 (ground-crank, crank-coupler, coupler-rocker, rocker-ground)
j₂ = 0 (none)
M = 3(l − 1) − 2j₁ − j₂
M = 3(4 − 1) − 2(4) − 0 = 1
Position Analysis
The plots below illustrate the angular displacement of the ternary coupler (θ₃) and the rocker (θ₄) as a function of crank rotation (θ₂) over a full 360° cycle. The coupler exhibits sinusoidal behavior, reflecting the continuous rotational input from the crank. In contrast, the rocker demonstrates an oscillatory motion, but isn’t completely sinusoidal.
The plot below illustrates the position profile of two key points that make up the wrist: the joint between links 3 and 4 (purple dashed curve) and the ternary offset point (green solid curve) at the coupler’s end. The first joint follows a tight elliptical path as expected from a Crank-Rocker mechanism. The ternary offset point, which maintains a rigid offset distance (p) of 6.64 inches, traces a larger, translated trajectory. When visualized together, the different trajectories help depict the vertical rocking behavior at the wrist.
Velocity Analysis
The velocity profiles below demonstrate slight differences in the x and y component speeds between the wrist point and the joint connecting links 3 and 4. These variations illustrate the distinct rocking motion introduced by the offset of the coupler. Specifically, the separation in velocity curves show that the wrist moves with an additional degree of articulation.
Force Analysis
The force analysis was done using the torque specification value of the Greartisan 12V, 100RPM DC Motor. This torque provided is likely the stall torque of the motor at no speed however with no motor curve provided online, this was an approximate bounding case.
With a spec of 3.6 kg/cm torque, the arm provides a minimum of approximately 0.15N of force to the joint between the link 3 and link 4 of the arm. This 0.15N is approximately equal to 0.03 pounds, which is sufficient for our now smaller palm assembly.
Finger Analysis
Ideal Motion Profile
The ideal finger mechanism should enable each finger to rotate up to 90 degrees, repeatable formation of the three distinct gestures: rock, paper, and scissors. To achieve the rotation aspect, each finger utilizes a coupled slider-crank system that transforms rotation within the hand's plane into the bending motion needed to actuate the fingers. The hand crank rotates in a circular path, which pushes each slider in a straight line. This linear motion is coupled into a back-driven slider-crank that redirects the force to rotate each finger freely. In order to achieve each distinct state, a servo motor mounted on the hand drives the central gear which has an attached crank. As this gear rotates, it also turns the surrounding gears (each with its own crank) coordinating motion of multiple fingers. A gear ratio of 3:3:2 ensures that each finger moves in sync with the desired range of motion to depict different states.
Position Analysis
The figure below demonstrates the periodic motion of all three fingers as they cycle through the rock, paper, and scissors states. The 3:3:2 gear ratio used to drive the mechanism influences the timing and amplitude of each finger’s motion. This enables staggered finger actuation which form clear, repeatable gestures. The colored vertical lines mark the crank positions corresponding to each configuration. The intersection of each plot with these lines convey the distinct combinations of finger angles which result in each hand sign.
Velocity Analysis
The angular velocity plots below show the smooth, periodic motion for all three fingers.
Acceleration Analysis
The angular acceleration plots below show dynamic, periodic profiles for all three fingers.