Designing a walking mechanism for lunar applications with an unconventional drive input requires extra analysis when compared to conventional walking mechanisms .Similar to conventional walking mechanisms, a nearly perfectly flat position profile along the bottom is ideal however there are some extra considerations. In order for the mechanism to be useful it must be able to navigate on uneven rocky terrain with large obstacles. This requires the rest of the foot profile to have a high step allowing for high ground clearances. Figure 1 below shows an example of an acceptable output position profile. Notice how the bottom of the profile is flat and the return path above has a large clearance height. Furthermore, the use of a nitinol actuator for the mechanisms driving input prevents the use of conventional circular crank inputs common to walking mechanisms. The desired input shape from a nitinol actuator is an elliptical shape. This shape is necessary because it offers opposing working faces that allow the nitinol spring and return spring to pull a follower pin along. Figure 2 below shows a representation of the input shape provided by a nitinol actuator.

Figure 1.
Desired output profile representation

Figure 2.
Representation of nitinol input profile

Initial Design and Analysis

The initial design process began with a base Klann walking mechanism simulated in MotionGEN. Since the exact shape of the nitinol actuator input had not been determined yet, a simplified input ellipsoid was used as a “best guess” for the input shape to allow for initial design iterations. Figure 3 below shows the geometry of the first iteration design. 

Figure 3.
First iteration Klann walking mechanism with ellipsoid generator input

Mobility analysis of this mechanism shows there are 7 linkages, 8 joints, 1 half joint, and 1 grounding. Using equation 1 yields that this system has 1 degree of freedom

DOF = 3 L - 2 J - 1 HJ - 3 G        (1)

At this point in the design process most of the analysis is qualitative and based on visual inspection of the output profiles while making slight adjustments to the geometry. The three figures below show the progression through the design process. The leftmost design has an output position profile with a high step like desired. However it should also be noted that the output motion is not very flat along the bottom. While the bottom section is generally oriented horizontally, there is a significant curve to it that will need to be removed. The middle figure shows that in trying to further flatten the bottom of the output profile, we over adjusted and created an elongated shape with the high step biased towards the front. The rightmost figure shows the final iteration of design from MotionGEN. As can be seen from the figure, the output profile is similar to that of the first iteration but is much more symmetric and has a very flat bottom section.

Figures 4,5,6.
Left to Right - First iteration, Intermediate iteration, final MotionGEN iteration
(Click image to play gif showing mechanism motion)

Custom Simulation and Nitinol Input Integration

After producing a satisfactory design using MotionGEN, the geometry was copied to a custom python simulator and integrated with the finalized nitinol actuator input profile to further refine the output kinematics. Figure 7, below, shows the exact shape of the nitinol actuator input profile. The cusp points at the upper left and lower right corners are toggle points where the nitinol actuator reverses its motion. More information about the function of the nitinol actuator can be found in [Manufacturing and Prototype Iterations]

Figure 7.
Nitinol actuator input profile

As seen in figure 8, the first simulation of the mechanism with the nitinol actuator input changed the output significantly from the simplified ellipsoid input. This change, while expected, means further simulations needed to be performed to regain a flat profile.

Figure 8.
Initial nitinol input integration with undesirable result

For the simulation in figure 8, the nitinol input was rotated to match the rotation of the generic input ellipse used in the initial analysis in MotionGEN. To move forward we decided to simulate every possible orientation of the nitinol input profile. Figure 9 shows a gif with the profile of every joint including the foot output for every rotation angle theta of the nitinol input.

Figure 9.
Nitinol input rotation gif for theta = 0°-360° with 5° increments

From visual inspection of figure 9, the foot tip output profile has a long flat section for angles of theta from 45° to 70°. This reduced range was used to run another simulation with a finer angle increment. This is shown by the gif in figure 10.

Figure 10.
Nitinol input rotation gif for theta = 45°-70° with 0.5° increments

This narrowed search interval was then used to perform a linear regression on the flat section to compare R2 values and find the optimal angle. This method yielded an input rotation of 56.5° with R2=0.992. Even though the output was not horizontal, the mechanism can just be rotated until the flat is horizontal. Analysis of the data shows the entire mechanism should be rotated by 29.8 degrees. Figure 11 below shows the foot output profile for the optimal input rotation angle as well as the linear regression line for the ground contact section.

Figure 11.
Optimal output profile with regression line

Next Section: Manufacturing and Prototype Iterations