Kinematic Analysis

Animation- Original Design With Capstan Drive

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Fusion 360 Motion Study animation for slider joint and motor shaft A (hip joint) rotation. The capstan drum relationship is defined in a separate motion link

Equations of Motion Used for Animated Analysis with Capstan Drive

• While we used these equations initially to implement into our MATLAB animation, we realized relatively far along in the process that the reason our animation was not running/working as intended was because we forgot to normalize the distance between joints in the XZ plane. Unfortunately, we did not have time to fix this before the presentation, and ended up deciding to implement a new linkage without a capstan drive for simplicity.

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Capstan Drive Assembly- Mobility analyzed using Gruebler’s equation M = 3(N-1) - 2J1 - J2:

• N = 5

  • 1- Ground/Base Link (top horizontal link)

  • 2- Capstan (B-Drive) Effective Link (rocker, on capstan body connecting links 2 and 4)

  • 3- A-Drive Hip Link (bottom horizontal link)

  • 4- Long Leg (output link)

  • 5- Slider Blocks (motors moving vertically)

• J1 = 6

  • Connecting joints, also accounting for capstan joint and slider block

• M = 3(5-1) - 2(6) = 0 DOF

• When evaluated with Gruebler’s equation, the mechanism yields 0 DOF, but there is obviously motion in the system. This indicates the presence of a redundant constraint.

  • θ3 = f(θ1), θ4 = f(θ1)

  • q3 = f(q1), qs1 = f(q1)

• Because all generalized joint variables (q) depend on a single independent coordinate, one independent direction of motion exists, verifying our system’s single degree of freedom

Animation- New Design with Watt’s Six-Bar Linkage (No Capstan)

• The kinematic analysis and MATLAB animation for our capstan-driven linkage proved very difficult, so we shifted our design to a Watt’s Six-Bar Linkage that produces a similar foot-motion profile. Rather than animating the full kinematic six-bar mechanism, we replaced actuator links 2 and 3 with simplified visual indicators that illustrate motor direction.

• If we end up using a third actuator (“actuator 1”), it will serve to laterally rotate the hip/coxa link; however, it is not illustrated in the animation because it rotates the short link out of the screen in the y-direction.

• Actuator 2 is located at the hip, and represents the rotation of the femur servo, which directly controls the femur joint and induces rotation of the upper leg/femur relative to the rest of the body. This actuator controls θ2.

• Actuator 3 is also located at the hip, and indirectly rotates the tibia through the six-bar linkage. This allows the lower leg to move without having to add another motor at the knee, and controls θ3.

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Equations of Motion Used for Animated Analysis with Watt’s Six-Bar Linkage

• Determining foot coordinates (X4, Y4, Z4)

  • x4 =L1cos(θ1) + L2*cos(θ2)sin(θ1) + L3cos(θ2)cos(θ3)sin(θ1) - L3sin(θ1)sin(θ2)sin(θ3)

  • y4 = L1sin(θ1) - L2 cos(θ1)cos(θ2) - L3cos(θ1)cos(θ2)cos(θ3) + L3cos(θ1)sin(θ2)sin(θ3)

  • z4 = L2sin(θ2) + L3cos(θ2)sin(θ3) + L3cos(θ3)sin(θ2)

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Inverse kinematic equations used for θ1, θ2, and θ3 in a simplified linkage considering only the coxa link (L1 = 20mm), the femur link (L2 = 100mm), and the tibia link (L3 = 117mm). The rest of the six-bar links serve as an internal transmission mechanism, and hence are not included in kinematic analysis

• Note: Velocity and acceleration profiles were obtained through finite difference (and second finite difference) of position. For the interior points, we used central difference formulas with forward/backward differences applied at the endpoints.

Watt’s Six Bar Assembly- Mobility is 2 or 3 DOF depending on whether or not we choose to implement a third servo for the coxa joint. If we do implement it, the motion can be completely described by the the three actuators rotating the coxa, femur, and tibia joints.

Iteration Documentation

Iteration 1: Determining Range of Motion

• Before we knew the exact dimensions of the servo motors we will be using, we designed our first iteration with arbitrary geometry dimensions. This assembly does not include the capstan relationship, and only includes one symbolic motor which rotates the hip joint. The goal of this iteration was just to visualize the motion and understand what motion links are needed to achieve the desired motion.

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Cardboard Motion and Joint Analysis

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Minimum slider position- maximum tension in capstan drive

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Maximum slider position- maximum jumping height with no tension in capstan drive

Iteration 2: Testing Motion With Servo Dimensions

• Once we documented the dimensions of the two servo motors we are using, we designed them in Fusion and tailored the geometry of the rest of the mechanism accordingly. For this iteration, we designed the vector loop in the four bar linkage as a parallelogram, where the angle of the long leg is controlled directly by the input angle of the effective link on the capstan drive’s large drum.

• Once we implemented various joints within the mechanism, we came to realize that the large drum of the capstan drive was not actually rotating about its center, altering the accuracy of the motion link and intended contact between the large and small drum. Because we were decently far along in the process of assembling joints, editing the original sketch of our capstan drive’s large drum caused many issues, leading us to create a new assembly.

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Assembly 2 at zero configuration, showing parallelogram geometry and large drum rotation error.

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Addition of properly dimensioned servo motors, and corresponding attachment links

Iteration 3: Assessing Capstan Drive Relationships

• For our third assembly, we aimed to analyze the intended relationship between the capstan drive input, and its corresponding outputs. The large drum of the capstan drive (and the effective link in the same body) must pivot about its radius/center of rotation to maintain contact with the small drum. While the large drum/effective link rotation is driven by the capstan drive (motor B ) and not the top motor (motor A), motor shaft A muss pass through the large drum to effectively drive the hip joint rotation. Using a ball bearing positioned at the large drum’s center of rotation, motor shaft A is able to rotate the hip joint as the large drum/effective link pivots about its center. To further optimize our intended motion, we also got rid of the parallelogram linkage by shortening the link between the hip joint and long leg by 15mm.

• While this assembly got us started on modeling the capstan drive relationship through assembly joints, motion limits, and motion links, the effective link on the capstan drive’s large drum seemed to be too short, and constrained the motion.

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Motion link for the capstan drive small drum rotation- later discovered that this link was ineffective and we needed to add ball bearings to drive the motion effectively with assembly joints

Iteration 4: Dimensioning Capstan Drive for Rope and Fixing Assembly Joints

• To test the motion of the capstan drive’s large and small drum, we created a coil around the small drum fit for 4mm rope, and another coil around the large drum with proportional diameter and pitch measurements. While the pitch on each drum lined up, we missed the critical aspect of both drums needing to be the same width, so this did not end up actually working with rope. Additionally, we added the preexisting 6mm ball bearing file used for our slider crank in place of the arbitrarily dimensioned joint pegs. This allowed for a much easier joint assembly process, producing smoother motion with less technical errors.

• After we created motion links between (1) the slider and hip joint rotation and (2) the capstan large drum and small drum rotation, we were able to execute a motion study

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Capstan drive small drum and large drum rope guide, dimensioned for diameter and pitch proportionality

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Fourth iteration at zero configuration

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Fourth iteration at minimum slider position and maximum capstan drive tension

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Fourth iteration at maximum slider position and no capstan drive tension

Iteration 5: Physical Prototype- No Capstan!

• The capstan drive was a great idea in theory, but we were having too much trouble with motion constraints and how to model them. To make things more simple, we developed a Watts six-bar linkage that executes a similar output, but has a much more widely studied kinematic analysis.

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Bill of Materials