Analysis of Walking Leg Mechanisms - Team 1

Overview

Our objective is to create a walking robot that uses two actuators to control the two degrees of freedom: walking forward/backward and steering left/right. Of course, the simplest solution for achieving two degrees of freedom using two actuators would be to power the left and right legs of the robot separately. This method is essentially how a tank drives, changing the speeds of the left and right sides relative to each other to be able to drive and make turns. Our goal is not just to achieve our objective but to achieve it with a more advanced approach. Instead of altering the speed of the left and right gait, we want to alter the stride length of the gaits. Decreasing the stride length of one side and increasing the stride length of the other will cause the robot to turn towards the side with the smaller side length.

In this analysis, we chose to look into three established leg mechanisms that achieve a very flat bottom in the position profile that they produce. We figured out how we could manipulate them to alter their stride length. The three mechanisms are called the Jansen linkage, the Klann linkage, and a more recently established and unnamed eight-bar linkage, which we will just be referring to as the eight-bar linkage. Below are images of each linkage design and the resulting gait profile that each creates.

The only aspects that we can change about these mechanisms (without making them different mechanisms) are their link lengths and grounded joint positions. We chose to analyze how changing the grounded joint positions affects the gait profiles as that seemed to be easier to design from a mechanical standpoint than changing the lengths of links that are moving. 

Our goal for this analysis is to find which of these three mechanisms offers the largest ratio of long to short stride length while keeping the flattest bottom of the gait profile and changing the height of the stride the least through the transition from the short to long stride. What do we mean by long and short stride, flat bottom, and change in height of the stride?

Looking at the images above we can define what we mean by stride height, length, and bottom flatness. These images show a gait profile of the small stride in dark blue, the large stride in green, and the intermediate stride in blue. The first image shows that as the gait alters, it substantially changes the stride length and maintains a flat bottom, but it changes in stride height. If this scenario was applied to our robot, when it would turn the long stride length side would be short in height and the short stride length side would be tall which ultimately tilts our robot. The second image shows that as the gait alters, it maintains a flat bottom and keeps a consistent stride height but does not substantially change the stride length. If this scenario was applied to our robot, it would be able to walk smoothly but would not be able to turn well or at all. The third image shows that as the gait alters, it does substantially change the stride lengths but does not maintain a flat bottom and also changes in stride height. This scenario would both cause the robot to tilt and bob up and down when turning and walking. The fourth image shows that as the gait alters, it meets all of the objectives that we want to meet: substantial change in stride length, consistent stride height, and maintains a flat bottom. This scenario will ensure that the robot does not tilt or bob while walking and turning. Tilting and bobbing are things we need to avoid because if the robots body is moving in any way other than in the direction of travel, that extra work must be covered by the motors. If we can reduce extra motion then we can choose a less expensive and lighter motor to drive our robot.

To understand how changing the grounded joint positions of each linkage mechanism affects the gait we programmed models of these systems to allow us to input certain grounded joint positions and output the resulting gait profile.

Jansen Linkage

One of the mechanisms we looked into for possible solutions was the Jansen mechanism. This mechanism has only one grounded link and one input link which made it a possible candidate for changing gait through grounded link movement.  Figure # below shows a Jansen mechanism.

Our link lengths and initial angles are based on a research paper. This is the gait we obtained as a starting position for grounded link movement:

The stride length is around 45mm(~1.77 in) and stride height around 25mm(~1 in) for a leg which is nearly 120mm (~4.75in) in length. We then explored changing the gait of the mechanism by varying the x and y distance between the input link and the grounded link.  The two images below show the effects that the position of the grounded link has on gait.

Therefore we came to the conclusion that although this a simple and elegant solution, it doesn’t give us the flexibility of the 8-bar mechanism which has multiple ground points which can be moved to change the gait of the mechanism. Thus giving us a more desired solution.

Klann Linkage

The Klann linkage is composed of six links, one of which is a ternary link, and has seven joints, three of which are grounded links. As one of the grounded joints will be responsible for driving the mechanism, O2, we will only be looking at how the gait profile changes as the other two grounded joints, O4 and O7, change the gait profile. The highlighted portions show that the Klann linkage is essentially two four-bars connected to each other and that is the basis of how the gait profile was calculated. The gait profile in this case is the shape that the point E draws as the link L2 is driven through one revolution. With the original link lengths and grounded positions that have been pre-determined through prior research (which we will be referring to as the optimum gait) the gait profile looks like this:

After finding the input angle range responsible for the bottom of the gait, we can take the standard deviation of the Y value of that part of the profile to calculate “flatness”. The flatness score that the optimum gait achieved is 0.09. Below you can see that the half of the cycle responsible for the bottom of the gait is highlighted in orange.

We plotted the gaits for every combination of O4 and O7 positions where they each varied by some distance less than and greater than their x and y values. We then filtered all the gait profiles to show us only the gait profiles that have a flatness below 0.3 which seemed to be a relatively good flatness by eye. This will show us the range for all relatively flat gait profiles that can be achieved by manipulating the positions of O4 and O7. Here are those results:

We can see that the Klann will not be a good solution for changing the stride length substantially and keeping a consistent stride height.

Eight-Bar Linkage

This eight-bar linkage system consists of eight links and seven joints, three of which are grounded, and one of the grounded joints, O2, drives the mechanism. We will again be looking into how changing the grounded joints, O4 and O7, changes the gait profile. There are now three four-bars that were solved to solve for the path of the foot, point E. The optimum gait looks like this:

We plotted the gaits for every combination of O4 and O7 positions where they each varied by the same amount as we did in the Klann linkage. We then filtered all the gait profiles to show us only the gait profiles with relatively good flatness. Here are the results:

These results show a lot of promise. There is a wide range of gait profiles that still achieve a good amount of flatness. Now we can see that the Eight-Bar linkage has good potential for finding a solution to manipulate the grounded joints in a mechanical fashion to achieve a change in stride length while still maintaining flatness and consistent stride height.

We knew that it would be mechanically easier to change the position of one grounded joint only so instead of looking at the combinations of differing O4 and O7 positions, let's change the position of one grounded joint while the other remains in its original position. To find a solution that is most optimal we need a nice way to show the performance characteristics of the gait as a function of the grounded joint position. It would be nice to see a heatmap where the x and y axis is the position of the grounded joint and the colors represent flatness, stride length, and stride height. We made a code to do just that. First, let’s see how the position O4 changes the performance characteristics.

In the graph for flatness, we see that in the center where its original position is the most flat, and as you move away from the original position flatness gets worse. However, the dark spot is somewhat elliptical and if we were to move along the diagonal we could maintain relatively good flatness. In the graph for stride length, we see that we can achieve a maximum stride length of around 8 units and a minimum of around 5 units. To do so we would need to move O4 along the diagonal seen below to achieve relatively large differences in stride length. In the graph for stride height, we see that if we wanted to maintain a consistent stride height then we would have to move along the same shade of color, which in this case would be in a diagonal direction that is contradictory to the direction of travel needed in the other graphs.

O4 has proven to be an invalid solution, but let's see how the position of O7 affects these performance characteristics.

In the graph for flatness, we see a sort of dark oblong arc section that runs through the center or original position. If the O7 moves through this section, we could maintain good flatness. In the graph for stride length, we see that the highest stride length that can be achieved is around 8.5 units and the shortest is around 4.5, which is already a better ratio than O4’s performance. The long and short stride lengths are in the top left and bottom left corners, respectively. In the graph for stride height, we see that there is a sort of aura effect centered around the left bottom corner. If we move O7 along an arc along similarly shaded regions, we could maintain consistent stride height. Looking at all three of these graphs together, we see that there is an arced path that O7 can move along while meeting all three of our performance criteria!

With this arc as the ideal path, we could see that the position of O7 could be moved by another link of radius and center that matches this arc. With just three points along this arc, we can determine the length and joint position needed for this link. Now let's solve for the gait profiles along this arc, and see if the results are actually good.

We can see that we do get a big difference in the length of the long and short stride and that we maintain flatness and consistent height! Unfortunately, we did not find this result until after we were in the process of building the prototype. The prototype allows us to manipulate the x or y position of O7 but not both at the same time. We did this because we saw potential in these adjustments, especially with the change in the y position of O7, which just so happens to be very close to the arc path we found.

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