To achieve high performance, highly detailed and realistic models of the mechanism, its limits and inefficiencies are required. For example, the motor is subject to speed, torque and thermal limits, and there are energy losses in the motor, transmission, and springs, as well as when the robot’s foot hits the ground.
Robot Skippy’s mechanism was modeled with 56 parameters (the detailed listing of the parameters has been omitted due to its large size). The schematic diagram is presented in Fig. 1.
The robot has three joints: the ankle, the foot and the hip. The ankle is a passive joint (it is not directly controlled) and has a spring attached to it. A second spring is attached in series with the ring screw nut, which is also connected to the main motor of the robot. This motor effectively controls the hip via a 4-bar linkage; both springs are custom made from glass fiber. Finally, Skippy has a crossbar (controlled by a second motor), which will be used to achieve 3D balance and steering, which are not relevant to this study.
We decided that some parameters would be fixed while others were the result of separate optimization studies (such as the 4-bar linkage). For the study presented in this article, we optimized seven parameters: six of these define the profiles of the two springs while the seventh defines the position of the center of mass of the upper link, which has a significant impact on the robot’s performance.
Finally, we developed a simulation environment in which the mechanism’s performance for each of the desired performance objectives was tested. A behavior was defined with 26 parameters of which 13 were independent variables representing the voltage profile fed to the robot’s motor. Fig. 2 shows an example of a voltage profile. The simulation was performed with MATLAB and Simulink.

Fig. 3 - Overall organization of the design optimization process

Fig. 3 illustrates the overall organization of the process has been arranged into modeFRONTIER, platform for process automation, optimization and data mining, which, as mentioned previously, is organized in two layers. The upper layer finds a Pareto front of optimal designs that meet all of the objectives. The MOGAII algorithm was used because it allows multiple objectives and because it is not limited to local exploration, such as gradientbased algorithms. The latter was important because we were managing a high dimensional non-linear search space. The lower layer consists of 12 individual optimization experiments to find a given mechanism’s best performance for each performance objective, and a calculation to determine that mechanism’s balancing ability, which we wanted to maximize. To summarize, these performance objectives were to:

• perform jumps increasing in height (with no angular momentum or horizontal velocity) up to 3m from a resting phase
• return to rest via a series of jumps of decreasing height
• initiate, stop or continue performing travelling hops (with non-zero horizontal velocity and zero angular momentum)
• perform a 2m triple backward somersault after a 2m vertical jump
• maximize the physical balancing ability.

MOGA-II was also selected for this stage, with some of the performance objectives being formulated as multi-objective optimization problems.
Finally, the algorithms were subject to 11 constraints to ensure the realism and feasibility of the simulations. To name just a few, these were to: avoid penetrating the ground, avoid selfcollision, limit the applied current and more. An overview of the algorithms, which were used in their default configurations, and their parameters is presented in the following table (Table 1).

Table 1 - Algorithms and parameters used

The Incremental Space Filler technique and a user-defined initial seed were used to explore the search space. The initial seed is a mechanism which can nearly perform the desired objectives and their corresponding behaviors (it was identified via a separate optimization study using the same framework).

The optimization scheme evaluated 260 designs, of which 46 (~ 18%) passed all the physical performance tests. A design was considered to be a successful candidate when at least one valid behavior per objective was discovered, allowing for a small interval of error.
We discovered mechanisms that were very skillful in one or in some of the objectives, but that performed poorly in others. The mechanism that could jump the highest under-performed in the traveling hops, while the mechanism that excelled in somersaults had difficulties in performing low hops. A new, more extensive experiment, to be conducted in future, will seek a deeper understanding of the physical traits that lead to this natural inclination for specific behaviors.
In this study, our main objective was to identify a single machine with the highest overall merit: a machine that displays high athleticism in all of the performance objectives, is a capable balancer, and expends the least amount of energy while performing these behaviors (this was an extra selection criterion that was not considered for the optimization). By examining the Pareto front solutions, we identified a design that satisfied all of the above criteria and this is currently being manufactured to test the effectiveness of our method.
Fig. 4 presents an sample behavior of our selected design and shows the stacked energy flows during one of the most demanding behaviors (the stance phase of a 2m triple backward somersault). The potential and kinetic energies can be seen as well as energy losses due to friction in the springs, the motor and the ring screw. In Fig. 2, we presented the voltage and current profile of the same stance phase of a 2m triple backward somersault. Notice that the motor is in saturation (at 31 Volts) for a large part of the stance phase, proving that the mechanism is indeed being driven to its limits.