5.3 - Kinematic Analysis
Kinematic Analysis
Ideal Motion and Force Profile
Our robot is able to climb a vertically suspended rope by feeding the rope through L1 and L4. Both these links have three pins that the rope creates an S-shape around. The rope feeds through L1, goes down through L4, and is tensioned at the bottom. As the crank is turned CCW, L4 rotates CCW and pulls the rope down with it due to friction, which allows the rope to feed through the loop in L1 and move L1 higher along the rope. As theta_2 passes 180 degrees, L4 rises back up (rotates CW), and the rope slides through it due to the tension at the bottom. Within one revolution of L2, the robot advances up the rope by about 1 inch.
Our ideal motion for the mechanism moves end of link 4 as vertically as possible, oscillating between a high and low position. The force profile will allow the robot to move up the rope while limiting shaking to allow a stable climb.
Prototype Kinematic Analysis:
Position Profile of Point P:
This a plot of the y-position vs. the x-position for Point P on Link 4 for all positions of Link 2 0-360 degrees. Point P will make one back-and-forth motion of this curve every time L2 rotates fully.
Point P (located at the end of L4) position profile over one full revolution:
Mobility Calculation
Gruebler’s Equation:
Number of Links: 4
Number of 1 DOF Joints: 4
Number of 2 DOF Joints: 0
M = 3 (4 - 1) - 2 (4) - 0 = 1 DOF
Grashof Condition:
L1 = 128.0624 mm, L2 = 70 mm, L3 = 110 mm, L4 = 119mm
S + L < P + Q
70 + 128.0624 = 198.0624
110 + 110 = 220
198.0624 < 220
Based on the Grashof Equation, our preliminary design is a Type I linkage, so link 2 will be able to rotate fully through 360 degrees.
Kinematic Analysis
Point A position profile over one full revolution:
Point B position profile over one full revolution:
Point P (located at the end of L4) position profile over one full revolution:
Theta_3 vs Theta_2 over one full revolution:
Theta_4 vs Theta_2 over one full revolution:
Velocity at Point A:
Velocity at point B:
Velocity at point P:
Acceleration of Point A vs Input Angle:
Acceleration of Point B vs Input Angle:
Acceleration of Point P vs Input Angle:
Force Analysis
The motor we are considering to purchase has a torque of 22 N*mm. This leads to a force input of 0.314 N at the end of the crank. Since the angular velocity of link 4 switches from positive to negative throughout one revolution of the crank, the Torque Ratio, Mechanical Advantage, and theoretical output force go to infinity when omega_4 is zero.
Output Force vs Input Angle:
Output Torque vs Input Angle:
Angular Velocity Ratio vs Input Angle:
Torque Ratio vs Input Angle:
Mechanical Advantage vs Input Angle:
Animation
Final Design Kinematic Analysis:
Here is the animation for our final design:
As seen from the animation, it is very similar to the prototype with a few changes implemented. The shape of L1 has changed (the black lines), and the lengths of all links have been reduced by the same factor. A slider crank mechanism can be seen in yellow, which is controlled by a cam that is attached to the same motor that drives L2 with a belt-pulley system. The slider clamps the rope so that the rope can slide through L4 without the robot falling down the rope. The slider then un-clamps the rope to allow the robot to ascend the rope while it is clamped at L4, due to the one way feed mechanism implemented in L4.
Due to all link lengths being scaled by the same factor, all graphs for position/velocity/acceleration/force analysis have the same shape, so I will not include the duplicate graphs.
Slider position vs input angle (transitions between diamaters in the cam were approximated linearly, although realistically these transition areas should have a slight curve to them.
Our new motor produces a torque of 1.08 N*m (estimated because we are using a brushless motor), which would effectively scale our output force/torque graphs by a factor of 49.