One of the primary challenges in quadrupedal locomotion pertains to the robot’s ability to adapt its gait to the surrounding environment and the desired task. This capability allows quadrupedal robots to select suitable foothold locations and adjust their gait for optimal performance. We address the problem of gait adaptation using trajectory optimization (TO), which takes into account the simplified switched system’s dynamics and optimizes the different phases of motion in which we split the robot’s movement. The robot dynamic model is a single rigid body (SRB) with a rigid contact model and foot positions. We apply contact and friction cone constraints to ensure a physically feasible motion of the real robot. We tackle the optimization using the direct multiple shooting (DMS) method. Leveraging kinematic inversion to map the base and feet positions into joint positions, velocities, and accelerations, we design a controller that combines iterative learning control (ILC) and proportional derivative (PD) feedback control. The iterative controller compensates for the sim-to-real gap, allowing the real robot to learn the task during the execution of the latter. We evaluate the performance of the proposed approach on two different quadrupedal robots and on different terrains.

Soft robots are deformable, compliant, and underactuated systems. During any task, due to their enormous capability of body deformation, the relative degree may not be well-defined. Since the applicability of the large majority of the state-of-the-art control techniques depends on this property, they frequently encounter singularities. This fact can jeopardize the system’s safety, prevent the correct task execution, or reduce performance. In this work, we investigate the relative degree of dependence for a class of compliant underactuated arms. Our method leverages the well-known strong inertial coupling hypothesis that, if holds, guarantees a constant relative degree of two. We generalize it by introducing coupling conditions where the relative degree is assured to be piecewise constant and greater than two. Relying on the design parameters, we analyze the dynamic evolution of the coupling conditions, which are then used to synthesize a classic input-output feedback controller. We also prove the stability of the closed-loop system. Finally, we validate the efficacy of the approach in simulation and on real hardware using a two and three degrees of freedom underactuated compliant arms with varying stiffness profiles, tasks, and disturbances.

Soft robots can execute tasks with safer interactions. However, control techniques that can effectively exploit the systems' capabilities are still missing. Differential dynamic programming (DDP) has emerged as a promising tool for achieving highly dynamic tasks. But most of the literature deals with applying the DDP to articulated soft robots by using numerical differentiation, in addition to using pure feed-forward control to perform explosive tasks. Further, underactuated compliant robots are known to be difficult to control and the use of DDP-based algorithms to control them is not yet addressed. We propose an efficient DDP-based algorithm for trajectory optimization of articulated soft robots that can optimize the state trajectory, input torques, and stiffness profile. We provide an efficient method to compute the forward dynamics and the analytical derivatives of series elastic actuators (SEA)/variable stiffness actuators (VSA) and underactuated compliant robots. We present a state-feedback controller that uses locally optimal feedback policies obtained from the DDP. We show through simulations and experiments that the use of feedback is crucial in improving the performance and stabilization properties of various tasks. We also show that the proposed method can be used to plan and control underactuated compliant robots with varying degrees of underactuation effectively.

Operations involving safe interactions in unstructured environments require robots with adapting behaviors. Compliant manipulators are a promising technology to achieve this goal. Despite that, some classical control problems such as following a trajectory are still open. A typical solution is to compensate the system dynamics with feedback loops. However, this solution increases the effective robot stiffness and jeopardizes the safety property provided by the compliant design. On the other hand, purely feedforward approaches can achieve good tracking performance while preserving the robot intrinsic compliance. However, a feedforward control framework for robots with passive elastic joints is still missing. This article presents an iterative learning control algorithm for purely feedforward trajectory tracking for compliant underactuated arms. Each arm is composed of active elastic joints and a generic number of passive ones connected through rigid links. We prove the convergence of the iterative method, also in the presence of uncertainties and bounded disturbances. Different output functions are analyzed providing conditions, based on the system inertial properties that ensure the algorithm applicability. Additionally, an automatic selection of the learning gain is proposed. Finally, we extensively validate the theoretical results with simulations and experiments.

Trajectory tracking of flexible link robots is a classical control problem. Historically, the link elasticity was considered as something to be removed. Hence, the control performance was guaranteed by adopting high-gain feedback loops and, possibly, a dynamic compensation with the result to stiffen up the dynamic behavior of the robot. Nowadays, robots are pushed more and more towards a safe physical interaction with a less and less structured environment. Hence, the design and control of the robots moved to an on-purpose introduction of highly compliant elements in the robot bodies, the so-called soft robotics, and towards control approaches that aim to provide the tracking performance without a substantial change in the robot dynamic behavior. Following this approach, we present an iterative learning control that relies mainly on a feedforward component, hence preserves the robot dynamics, for trajectory tracking of a one-link flexible arm. We provide a condition, based on the system dynamics and similar to the Strong Inertially Coupled property, that ensures the applicability of the proposed control method. Finally, we report simulation and experimental tests to validate the theoretical results.