M. Plecnik, D. W. Haldane, J. K. Yim, and R. S. Fearing, 2017. “Design Exploration and Kinematic Tuning of a Power Modulating Jumping Monopod,” Journal of Mechanisms and Robotics, 9(1): 011009. link
D. W. Haldane, M. Plecnik, J. K. Yim, and R. S. Fearing, 2016. “Robotic Vertical Jumping Agility via Series-Elastic Power Modulation,” Science Robotics, 1(1): eaag2048. link
Salto is a jumping monopedal robot. It is inspired from a small nocturnal primate, Galago senegalensis. The galago has the remarkable ability to produce jumps at greater powers than what its muscles can output. Salto can do this too. It exhibits special mechanics termed series-elastic power modulation. What this means is that at the beginning of a jump, Salto’s motor stores transient energy in its elastic element (the Galago’s muscles store transient energy in its muscle-tendon complex) which flows immediately thereafter into the jump motion.
For series-elastic power modulation to happen, Salto must have a specific mechanical advantage (MA) profile, that is how Salto’s leg linkage multiplies forces during its jump motion. At the beginning of a jump, when Salto is nearly crouched, MA multiplies the series-elastic input torque into a very small foot force pushing on the ground. With a small ground force, the robot doesn’t move much, but buys the motor a few extra milliseconds (50 or so) to pump energy into the system. It stores energy into Salto’s spring, a cylindrical piece of latex. However, MA does not stay low for long. As Salto’s leg begins to extend, MA rapidly increases, now multiplying the increased series-elastic input torque to greater values at the foot. Toward the end of the jump, MA approaches infinity. The time evolution of energy conversion during a jump motion is shown here:
The mechanics described above are the key to series-elastic power modulation, but are not sufficient to produce a useful jump. There are some more practical considerations, such as ensuring the leg fits into a certain size envelope or that the robot does not spin wildly in the air. When designing Salto, we listed 8 required behaviors to obtain its resulting jumps:
- The foot travels on a straight vertical line
- The foot stroke is long (15 cm of travel)
- Linkage pivots are above the foot
- Linkage lengths are compact
- Input link rotation is large to reduce gearing
- MA is low in the crouched configuration
- MA during extension defines a constant ground force to limit peak loading
- Angular momentum of moving links is balanced
Simultaneously satisfying all 8 requirements is impossible without the use of computational mechanical design. Salto was designed by a two stage procedure: design exploration and kinematic tuning.
Design exploration is the more important phase. This uses polynomial homotopy continuation to find all viable portions of the design space. Essentially what is produced is an ad hoc atlas of design candidates. The mechanical designer may shop through this atlas to determine which one best satisfies the required behaviors.
Click the designs to see them animate.
Following design exploration, a gradient descent based optimizer is run to kinematically tune design candidates of interest. In this way, required behaviors are prioritized and more accurately achieved. Eight design iterations are shown in the video below:
The final design from the kinematic tuning procedure is given below:
This is Salto’s design. It is an eight-bar linkage. It can jump vertically 1 m with nearly zero angular momentum, and a ground reaction force that is spread constant over its stroke. Simultaneously it produces the desired series-elastic power modulating mechanics. These mechanics allow Salto to elevate 1 m every 0.58 sec (vertical jumping agility of 1.7 m/s). The galago is able to elevate 1.7 m every 0.78 sec (vertical jumping agility of 2.2 m/s). Biology still has the edge on our robot, but maybe not for long.