Achieving controlled jumping behaviour for a quadruped robot is a challenging task, especially when introducing passive compliance in mechanical design. This study addresses this challenge via imitation-based deep reinforcement learning with a progressive training process. To start, we learn the jumping skill by mimicking a coarse jumping example generated by model-based trajectory optimization. Subsequently, we generalize the learned policy to broader situations, including various distances in both forward and lateral directions, and then pursue robust jumping in unknown ground unevenness. In addition, without tuning the reward much, we learn the jumping policy for a quadruped with parallel elasticity. Results show that using the proposed method, i) the robot learns versatile jumps by learning only from a single demonstration, ii) the robot with parallel compliance reduces the landing error by 11.1%, saves energy cost by 15.2% and reduces the peak torque by 15.8%, compared to the rigid robot without parallel elasticity, iii) the robot can perform jumps of variable distances with robustness against ground unevenness (maximal ±4cm height perturbations) using only proprioceptive perception.

Controlled execution of dynamic motions in quadrupedal robots, especially those with articulated soft bodies, presents a unique set of challenges that traditional methods struggle to address efficiently. In this study, we tackle these issues by relying on a simple yet effective two-stage learning framework to generate dynamic motions for quadrupedal robots. First, a gradient-free evolution strategy is employed to discover simply represented control policies, eliminating the need for a predefined reference motion. Then, we refine these policies using deep reinforcement learning. Our approach enables the acquisition of complex motions like pronking and back-flipping, effectively from scratch. Additionally, our method simplifies the traditionally labour-intensive task of reward shaping, boosting the efficiency of the learning process. Importantly, our framework proves particularly effective for articulated soft quadrupeds, whose inherent compliance and adaptability make them ideal for dynamic tasks but also introduce unique control challenges.

Introducing parallel elasticity in the hardware design endows quadrupedal robots with the ability to perform explosive and efficient motions. However, for this kind of articulated soft quadruped, realizing dynamic jumping with robustness against system uncertainties remains a challenging problem. To achieve this, we propose an impact-aware jumping planning and control approach. Specifically, an offline kino-dynamic-type trajectory optimizer is first formulated to achieve compliant 3-D jumping motions, using a novel actuated spring-loaded inverted pendulum (SLIP) model. Then, an optimization-based online landing strategy, including preimpact leg motion modulation and postimpact landing recovery, is designed. The actuated SLIP model, with the capability of explicitly characterizing parallel elasticity, captures the jumping and landing dynamics, making the problem of motion generation/regulation more tractable. Finally, a hybrid torque control consisting of a feedback tracking loop and a feedforward compensation loop is employed for motion control. Experiments demonstrate the ability to accomplish robust 3-D jumping motions with stable landing and recovery. Besides, our approach can be applied to quadrupedal robots with or without additional parallel compliance.

Stable walking in real-world environments is a challenging task for humanoid robots, especially when considering the dynamic disturbances, e.g., caused by external perturbations that may be encountered during locomotion. The varying nature of disturbance necessitates high adaptability. In this paper, we propose an enhanced Nonlinear Model Predictive Control (NMPC) approach for robust and adaptable walking – we term it versatile locomotion, by limiting both the Center of Pressure (CoP) and Divergent Component of Motion (DCM) movements. Due to utilization of the Nonlinear Inverted Pendulum plus Flywheel model, the robot is endowed with the capabilities of CoP manipulation (if equipped with finitesized feet), step location adjustment, upper body rotation, and vertical height variation. Considering the feasibility constraints, especially the usage of relaxed CoP constraints, the NMPC scheme is established as a Quadratically Constrained Quadratic Programming problem, which is solved efficiently by Sequential Quadratic Programming with enhanced solvability. Simulation experiments demonstrate the effectiveness of our method to recruit optimal hybrid strategies in order to realize versatile locomotion, for the robot with finite-sized or point feet.

Despite plenty of motion planning strategies have been proposed for bipedal locomotion, enhancing the walking robustness in real-world environments is still an open question. This paper focuses on robust body and leg trajectories synthesis through integrating constrained optimization with imitation learning. Specifically, we first propose a Quadratically Constrained Quadratic Programming (QCQP) algorithm to make use of the ankle strategy and stepping strategy. Based on the Linear Inverted Pendulum (LIP) model, body motion can be determined by the modulated Center of Pressure (CoP) position and step parameters (including step location and step duration). After that, we exploit an imitation learning approach Kernelized Movement Primitives (KMP) to plan robot leg motions, which allows for adapting the learned motion patterns to new situations (e.g., passing through various desired points) in a straightforward manner. Several LIP simulations and whole-body dynamic simulations demonstrate that higher walking robustness can be achieved using our framework.

When experiencing disturbances during locomotion, human beings use several strategies to maintain balance, e.g. changing posture, modulating step frequency and location. However, when it comes to the gait generation for humanoid robots, modifying step time or body posture in real time introduces nonlinearities in the walking dynamics, thus increases the complexity of the planning. In this paper, we propose a two-layer hierarchical optimization framework to address this issue and provide the humanoids with the abilities of step time and step location adjustment, Center of Mass (CoM) height variation and angular momentum adaptation. In the first layer, times and locations of consecutive two steps are modulated online based on the current CoM state using the Linear Inverted Pendulum Model. By introducing new optimization variables to substitute the hyperbolic functions of step time, the derivatives of the objective function and feasibility constraints are analytically derived, thus reduces the computational cost. Then, taking the generated horizontal CoM trajectory, step times and step locations as inputs, CoM height and angular momentum changes are optimized by the second layer nonlinear model predictive control. This whole procedure will be repeated until the termination condition is met. The improved recovery capability under external disturbances is validated in simulation studies.

The modulation of step location and duration plays an important role in realizing robust bipedal walking. This paper formulates it as a nonlinear programming problem (NLP) and proposes a novel optimization approach to adjust step location and duration in real time. Based on state feedback, the Linear Inverted Pendulum dynamics is exploited to determine the optimal step parameters. Different from previous works, this work presents three main characteristics: i) the hyperbolic functions of step duration rather than the step duration itself are chosen to be optimization variables; ii) the approach can be switched from baseline two-steps-prediction optimization to one-step-prediction optimization through merely adding several equality constraints in problem formulation; iii) the approach can deal with relative step location tracking (velocity tracking) or absolute step location tracking (position tracking) via changing the reference step parameters. As a result, the first characteristic enables the NLP to be solved in a computational-efficient manner and the latter two endow the approach with versatility under different control modes. The effectiveness has been demonstrated by simulation experiments.

This paper makes a paradigm shift in the assumed notion of common subexpressions for complexity reduction of multiple constant multiplications implementation. Our proposed unified adder/subtractor (UAS)-based common subexpression elimination (CSE) algorithm is inspired by the recent advancement in complex arithmetic component mapping for datapath synthesis of digital systems. A dedicated UAS operator is designed at gate level to achieve arithmetic reduction for concurrent computation of the sum and difference of two input signals. To maximize computation reuse, dual subexpression is defined to enable a UAS to be shared by the otherwise incompatible odd and even common subexpressions. The three different types of common subexpression are uniquely encoded by a quadruple in the proposed data structure. Constant coefficients are represented by signed digits in Cartesian coordinate system from which nonoverlapping pairs of nonzero digits are parsed for dual, even, and odd subexpressions to maximize the reuse of all three types of arithmetic resources. The effectiveness of our proposed UAS-based CSE in overcoming the complexity reduction bottleneck are demonstrated by comparing the synthesis results obtained from six benchmark finite impulse response filters, an electroencephalogram filter bank, fast Fourier transform, and discrete cosine transform multipliers designed by ten algorithms. The results show a noteworthy 27.2% reduction in area-time complexity of our method over the baseline canonical signed digit implementation. Our solutions are also more power efficient, with average power saving of 12.0% over those designed by other algorithms in comparison.

In this paper, a new design algorithm is proposed to synthesize low complexity twiddle factor multipliers in radix-2 FFT. A prudently defined cost function has been proposed as a measure of hardware complexity when generating trigonometric expressions for the complex multiplier coefficients. The selected trigonometric expressions of the coefficients lead to cost efficient implementation of twiddle factor multipliers by maximizing the sharing of both trigonometric expressions and weight-two common subexpressions in the coefficients. The effectiveness of the proposed design algorithm are demonstrated using two design examples where the proposed solution saves up to 31% of arithmetic operator and multiplexer costs over other existing methods.