This paper explores the benefits of using multiple gaits in a single robot. Inspired by nature, where humans and animals use different gaits to increase their energetic economy, we analyzed how increasing speed affects the choice of gait, and how the choice of gait influences optimal speed. To this end, we used optimal control as a tool to identify motions that minimize the cost of transport of two detailed models: a planar biped and a planar quadruped. Both of these models are actuated with high compliance series elastic actuators that enable a rich set of natural dynamics. These models have damping in their springs, feet with mass, and realistic limitations on actuator torques and velocities. They therefore serve as an intermediary between past simpler models and hardware. We discovered optimal motions with an established multiple shooting implementation that relies on pre-defined contact sequences, and with a direct collocation implementation in which the footfall pattern was an outcome of the optimization. Both algorithms confirmed findings from biology. For both models, changing gaits as speed varies leads to greatly increased energetic economy. For bipeds, the optimal gaits were walking at low speeds, grounded running at intermediate speeds, and running at high speeds. For quadrupeds, the optimal gaits were four-beat walking at low speeds and trotting at intermediate speeds. At high speeds, galloping and trotting were the best gaits, with nearly equal performance. We found that the transition between gaits was primarily driven by damping losses and negative actuator work, with collisions playing a relatively small role.

In this paper we use optimal control on a geared electric DC motor to compare the energetic efficiency of a simulation of conceptual monoped hoppers with either parallel elastic actuation (PEA) or series elastic actuation (SEA). The energy is measured using three cost functions: positive actuator work, electrical losses, and positive electrical work. For PEA, the presence of the motor inertia in the collision losses leads to increased collision losses at large transmission ratios, which lead to energetically costly compensatory strategies where the SEA is at its most efficient. At small transmission ratios, the motor force increases for both cases, leading to increased thermal losses. In agreement with those theoretical predictions, our work shows that for positive actuator work and positive electrical work the optimal parameter choice for SEA is significantly more energetically efficient than the optimal choice for PEA. For electrical losses, a suitable choice of the transmission ratio can lead to negligible cost values for both actuator concepts.

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a) (b) (c) (d) Fig. 1: Our five-fingered dexterous hand solves a scrambled Rubik's Cube by operating its layers and changing its pose.Our method starts with a random state (a), plans the optimal move sequence (b,c), and reaches the desired state (d).Abstract-We present a learning-based approach to solving a Rubik's cube with a multi-fingered dexterous hand. Despite the promising performance of dexterous in-hand manipulation, solving complex tasks which involve multiple steps and diverse internal object structure has remained an important, yet challenging task. In this paper, we tackle this challenge with a hierarchical deep reinforcement learning method, which separates planning and manipulation. A model-based cube solver finds an optimal move sequence for restoring the cube and a model-free cube operator controls all five fingers to execute each move step by step. To train our models, we build a high-fidelity simulator which manipulates a Rubik's Cube, an object containing high-dimensional state space, with a 24-DoF robot hand. Extensive experiments on 1400 randomly scrambled Rubik's cubes demonstrate the effectiveness of our method, achieving an average success rate of 90.3%.

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