19.3 Kinematic Analysis
Background
The ideal motion for our final jumping rabbit prototype involves a mechanism that efficiently stores energy in a torsion spring during the loading phase and then releases it in a controlled manner during the launch phase, propelling the toy forward. In addition to ensuring the toy generates sufficient forward motion, we aim to guarantee stability upon landing.
The jumping mechanism project consists of a four-bar mechanism and a cam-follower mechanism. In this analysis, we focus solely on the four-bar system, assuming a constant input rotational velocity. The cam-follower interaction, body weight, and friction are not considered in this section, as the goal is to isolate the internal motion of a single leg. Therefore, there is no ground to prevent the leg from extending downward.
Preliminary Performance Evaluation
Before conducting a detailed kinematic analysis, we performed a rough estimation to verify that the system is theoretically capable of achieving airborne motion. The results suggest it is feasible, though not entirely safe or reliable.
Preparation for Kinematic Analysis
Step 1: Kinematic Diagram of a Single Leg
This diagram defines the positive direction of input torque and output forces. The diagram omits the detailed curvature at the foot for simplicity, while keeping the analysis calculation to match the real design.
Step 2: Define System
We converted the four-bar linkage system from the global frame to the local frame and found out that it is an open-circuit configuration.
Step 3: Define the Motion Range of the System
We defined the input angle range in order to define the motion range of the system. The diagram below demonstrates the relationship between the torsional spring angle (theta_spring) and the input angle (thetea_2).
The range of theta_spring is from [60, 110] degrees.
Therefore, the range of theta_2 is [70,120] degrees.
Step 4: Identify Critical Geometry for Output Calculation
The geometry at the foot is critical for determining the velocity, acceleration, and force components along the normal and transverse directions.
Analysis
Mobility
Use the Greubler’s Equation: M = 3*(L-1)-2*(J1)-J2
The mobility of the four-bar linkage system is M = 3*(4-1) - 2*4 -0 = 0 DoF
The result of 0 DoF indicates this system is a structure.
Position Analysis
Velocity Analysis
Acceleration Analysis
Force Analysis
The torsional spring serves as the input force source; therefore, the output force at the foot can be calculated using mechanical advantage.
Step 1: Find the Input torque of the torsional spring
Torque induced by the torsional force can be described by Hooke’s Law below
T = K × Δθ
where:
Δθ is the angle of twist (in radians)
T is the torque (N*mm)
K is the spring constant (N*mm/rad)
K was calculated using the formula below,
where,
E is the Young’s modulus of the spring material
d is the wire diameter of the spring
D is the mean coil diameter
N is the number of active coils
Step 2: Find Mechanical Advantage
Based on the four-bar system, we can therefore calculate the output force translated from the input torque of the spring to the output force at the foot
The mechanical advantage equation is described below
MA = (l_vec(4)/l_vec(2))*(r_input/r_output) * (sind(th_4_vec - th_3_vec) ./ sind(th_2_vec - th_3_vec));
With the calculated mechanical advantage, we calculated the output force at the toe-off point.
Conceptual Analysis Ideas
Analysis of the Overall System
1] From an Energy Perspective
Assuming 100% efficiency of each energy transfer, we expect the potential energy of the robot at jump apex to equal its change in kinetic energy (launched from rest), which equals to energy stored in our actuators (elastic energy). Thus, the maximum height the robot can jump is directly related to:
System’s overall mass
Spring constant
The number of springs is loaded and released (actuator output)
These factors determine the theoretical max height and are independent of our linkage choice and design. Thus, before we proceed deeper into design, we need to make sure our model can at least satisfy these requirements with respect to safety factors that account for real efficiency losses.
2] From a Momentum Perspective
Apart from frictions, the main efficiencies of the system we can control are the four-bar link ratios.
We know from kinematics of rigid bodies (if we assume our system jumps purely vertically and is close to rigid) that in order to maximize a launching object’s maximum height, we need to maximize its initial velocity first when leaving the ground. We also know velocity at the apex is zero, and total impulse equals mass times initial velocity. Thus, in order to maximize our apex height, we need to maximize the total impulse, which is the foot’s output force integrated over time. This output force profile is something we can analyze based on our four-bar designs, and we can effectively tune our four-bar ratio to find the configuration that has the largest total impulse. We can also compare the total impulse from four bar configurations analogous to different animals’ leg anatomy. We predict that the nature-involved leg mechanism has high efficiencies in terms of (if applicable) four-bar length ratios.
Discarded Analysis of Cam-Follower System
We initially designed a cam-follower system, but due to its complexity, we chose not to implement it in this project. The system had the potential to define key parameters such as jump height, duration, and allow for detailed kinematic analysis. However, we have only verified our cam design on SolidWorks and its actual performance remains unproven. Additionally, the first iteration of the cam was too big for the four-bar linkage system.
