Naturally, after building a single joint, it only makes sense to design a single leg. From CARA 1.0, I knew that a coaxial 5-bar linkage design would be the most ideal design since it allows for less loading on each link compared to a standard quadruped leg. I also love the fact that most quads don’t use this design, making CARA super unique. Also, a coaxial 5-bar linkage pairs super well with a capstan drive. In fact, I believe it’s the most compact way to create a 3-DOF leg with a capstan drive. Since we already knew what the leg would look like, the design process became an optimization problem. How can we improve cost, weight, and assembly?

The first thing we decided to improve was the number of mirrored parts in the leg. CARA 1.0 used a ton of mirrored parts, which made assembly confusing at times. This time, we decided to make each leg on the bot identical. Ultimately, this bit us in the back, but more on that later. We also decided to reduce the number of screws used in the leg. Why use 4 screws to fasten something that just needs 2? I’ve started asking myself more questions like this since screws add cost, weight, and assembly time to builds. In line with screws, we also got rid of redundant bracing on some parts. Overconstraining a part doesn’t immediately seem like an issue, but it definitely can become one since things in the real world are never perfect. One good example is 4 legged table. You only need 3 legs to fully constrain a table. That 4th leg overconstrains the table. So, if one leg happens to be a little shorter or taller than the others, the whole table becomes unbalanced. To help further reduce weight, we also used much thinner bearings. There are a lot of radially constrained parts in the leg, so lighter bearings go a long way in making the robot itself significantly lighter. The total weight of the single leg was 1.47 kg (3.24 lbs).

One thing I wanted to investigate with CARA 2.0’s leg design was the upper-to-lower link ratio of the 5-bar linkage. For CARA 1.0, I used a 1:2 ratio just on a whim without doing any analysis. This time, I had one of my teammates investigate the ideal ratio using a MATLAB simulation. What we found was that a 1:1 ratio gives you the highest ROM. Unfortunately, a 1:1 ratio looks kinda goofy, so we ended up using a 2:3 ratio to get a bit more ROM while preserving aesthetics. Lastly, the biggest performance change we made with the leg design was switching from a TPU foot to a squash ball foot. TPU feet sucked with CARA 1.0 as it’s simply a flexible plastic with no compliance or traction. Squash balls, on the other hand, excel in compliance and traction. Squash balls are also super cheap and were used in the MIT Mini Cheetah as well as Stanley, both high-performing quads. There are actually 4 main types of squash balls: blue dot (beginner), red dot (progressing), single yellow dot (competition), and double yellow dot (pro). Double yellow dot is the firmest and least bouncy, so we chose those.

Because the squash ball foot could not be placed at the center of the 5-bar linkage pivot (like the TPU foot), new inverse kinematics (IK) equations had to be derived. You can view those derivations as well as the forward kinematics equations (FK) here. A question I often get asked is “how do you learn to derive IK equations?” While it may look difficult, it’s not. It’s just very tedious with lots of room to make mistakes. I’ve never used more than 8th-grade geometry to derive IK or FK equations. Any complex robot structure can be broken up into a series of triangles. I’ve only ever used James Bruton’s video on IK to learn how to derive kinematics equations (How Robots Use Maths to Move). Once you know the working principle behind it, you can basically derive IK and FK equations for any robot, given that you’re proficient in geometry.