Salto, A Jumping Monopod

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:

  1. The foot travels on a straight vertical line
  2. The foot stroke is long (15 cm of travel)
  3. Linkage pivots are above the foot
  4. Linkage lengths are compact
  5. Input link rotation is large to reduce gearing
  6. MA is low in the crouched configuration
  7. MA during extension defines a constant ground force to limit peak loading
  8. 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

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.

Wing Mechanism

A wing mechanism was designed to reproduce a complex accelerative flapping gait from a single constant RPM motor. The flapping gate was deciphered from data obtained by Tobalske and Dial, 1996, of black-billed magpies. The synthesis process begins by specifying a 4R spatial serial chain that resembles a magpie’s anatomy. Moving the spatial chain through the desired flapping gait defines a function of joint angle over time at each of the four joints. A six-bar Stephenson II function generator was designed for each joint angle function and the whole system was coupled together such that it can be run by a motor spinning at constant RPM.

Finally, compliant joints were added between the wrist and wing tip to mimic this portion of a bird’s anatomy. This joints utilize hard stops in order to limit their compliance to one direction creating aerodynamic check values such that control surfaces remain rigid during downstroke and deflect during backstroke.

The resulting motion has a long, stretched out downstroke followed by a quick, compressed backstroke.

B. W. Tobalske and K. P. Dial, “Flight Kinematics of Black-billed Magpies and Pigeons Over A Wide Range of Speeds,” The Journal of Experimental Biology, 199(2): 263-280, 1996.

Leg Mechanisms

This page shows an application of the design of six-bar path generating linkages for creating a mechanism that makes a walking motion.  Each six-bar is either a Stephenson I, II, or III type, and they were synthesized in order to guide a point path in a straight trajectory across the ground, then lift off the ground and move forward for another cycle. The example demonstrates how finding near complete solution sets to large degree design equations results in a large number of design choices (in the hundreds).  Linkage animations of some of the algorithm results appears here.

Leg walker design by Mark Plecnik

An embodiment of a leg mechanism for use in a small ambulating robot concept appears below. This embodiment accomplishes the walking motion of six legs in an alternating tripod gait without the use of gears or pulleys. Each leg module is made with compliant joints and motion between leg modules is transferred by compliant pantograph linkages. The robot would be actuated by one motor per side. The compliant linkages would be lasercut, then each side would be attached onto the robot, providing a lightweight, easily manufactured, kinematic system that accomplishes a complex motion with a minimal number of actuators.

A physical prototype of the three legged module from above appears below. The leg module was lasercut from polypropylene in 20 min.

A secondary robot concept appears next. An entire side of the robot (8 leg modules) is made from a single piece. In this case, rotary motion to each leg module is transmitted by pulleys.

This net robot envisions a concept of fabricating a six legged alternating tripod robot from rigid links:

Related to this topic, Festo has recently designed a bionic ant that uses a compliant leg mechanism.