Cooperative Adaptive Control for Cloud-Based Robotics

This paper studies collaboration through the cloud in the context of cooperative adaptive control for robot manipulators. We first consider the case of multiple robots manipulating a common object through synchronous centralized update laws to identify unknown inertial parameters. Through this development, we introduce a notion of Collective Sufficient Richness, wherein parameter convergence can be enabled through teamwork in the group. The introduction of this property and the analysis of stable adaptive controllers that benefit from it constitute the main new contributions of this work. Building on this original example, we then consider decentralized update laws, time-varying network topologies, and the influence of communication delays on this process. Perhaps surprisingly, these nonidealized networked conditions inherit the same benefits of convergence being determined through collective effects for the group. Simple simulations of a planar manipulator identifying an unknown load are provided to illustrate the central idea and benefits of Collective Sufficient Richness.Comment: ICRA 201

Numerical Methods to Compute the Coriolis Matrix and Christoffel Symbols for Rigid-Body Systems

The growth of model-based control strategies for robotics platforms has led to the need for additional rigid-body-dynamics algorithms to support their operation. Toward addressing this need, this article summarizes efficient numerical methods to compute the Coriolis matrix and underlying Christoffel Symbols (of the first kind) for tree-structure rigid-body systems. The resulting algorithms can be executed purely numerically, without requiring any partial derivatives that would be required in symbolic techniques that do not scale. Properties of the presented algorithms share recursive structure in common with classical methods such as the Composite-Rigid-Body Algorithm. The algorithms presented are of the lowest possible order: $O(Nd)$ for the Coriolis Matrix and $O(Nd^2)$ for the Christoffel symbols, where $N$ is the number of bodies and $d$ is the depth of the kinematic tree. A method of order $O(Nd)$ is also provided to compute the time derivative of the mass matrix. A numerical implementation of these algorithms in C/C++ is benchmarked showing computation times on the order of 10-20 $\mu$s for the computation of the Coriolis matrix and $40-120$ $\mu$s for the computation of the Christoffel symbols for systems with $20$ degrees of freedom. These results demonstrate feasibility for the adoption of these numerical methods within control loops that need to operate at $1$kHz rates or higher, as is commonly required for model-based control applications

Tensor-Free Second-Order Differential Dynamic Programming

This paper presents a method to reduce the computational complexity of including second-order dynamics sensitivity information into the Differential Dynamic Programming (DDP) trajectory optimization algorithm. A tensor-free approach to DDP is developed where all the necessary derivatives are computed with the same complexity as in the iterative Linear Quadratic Regulator~(iLQR). Compared to linearized models used in iLQR, DDP more accurately represents the dynamics locally, but it is not often used since the second-order derivatives of the dynamics are tensorial and expensive to compute. This work shows how to avoid the need for computing the derivative tensor by instead leveraging reverse-mode accumulation of derivative information to compute a key vector-tensor product directly. We benchmark this approach for trajectory optimization with multi-link manipulators and show that the benefits of DDP can often be included without sacrificing evaluation time, and can be done in fewer iterations than iLQR

Online Planning for Autonomous Running Jumps Over Obstacles in High-Speed Quadrupeds

This paper presents a new framework for the generation of high-speed running jumps to clear terrain obstacles in quadrupedal robots. Our methods enable the quadruped to autonomously jump over obstacles up to 40 cm in height within a single control framework. Specifically, we propose new control system components, layered on top of a low-level running controller, which actively modify the approach and select stance force profiles as required to clear a sensed obstacle. The approach controller enables the quadruped to end in a preferable state relative to the obstacle just before the jump. This multi-step gait planning is formulated as a multiple-horizon model predictive control problem and solved at each step through quadratic programming. Ground reaction force profiles to execute the running jump are selected through constrained nonlinear optimization on a simplified model of the robot that possesses polynomial dynamics. Exploiting the simplified structure of these dynamics, the presented method greatly accelerates the computation of otherwise costly function and constraint evaluations that are required during optimization. With these considerations, the new algorithms allow for online planning that is critical for reliable response to unexpected situations. Experimental results, for a stand-alone quadruped with on-board power and computation, show the viability of this approach, and represent important steps towards broader dynamic maneuverability in experimental machines.United States. Defense Advanced Research Projects Agency. Maximum Mobility and Manipulation (M3) ProgramKorean Agency for Defense Development (Contract UD1400731D

Multi-Shooting Differential Dynamic Programming for Hybrid Systems using Analytical Derivatives

Differential Dynamic Programming (DDP) is a popular technique used to generate motion for dynamic-legged robots in the recent past. However, in most cases, only the first-order partial derivatives of the underlying dynamics are used, resulting in the iLQR approach. Neglecting the second-order terms often slows down the convergence rate compared to full DDP. Multi-Shooting is another popular technique to improve robustness, especially if the dynamics are highly non-linear. In this work, we consider Multi-Shooting DDP for trajectory optimization of a bounding gait for a simplified quadruped model. As the main contribution, we develop Second-Order analytical partial derivatives of the rigid-body contact dynamics, extending our previous results for fixed/floating base models with multi-DoF joints. Finally, we show the benefits of a novel Quasi-Newton method for approximating second-order derivatives of the dynamics, leading to order-of-magnitude speedups in the convergence compared to the full DDP method.Comment: https://www.youtube.com/watch?v=C0h6mEpcnA

A Unified Perspective on Multiple Shooting In Differential Dynamic Programming

Differential Dynamic Programming (DDP) is an efficient computational tool for solving nonlinear optimal control problems. It was originally designed as a single shooting method and thus is sensitive to the initial guess supplied. This work considers the extension of DDP to multiple shooting (MS), improving its robustness to initial guesses. A novel derivation is proposed that accounts for the defect between shooting segments during the DDP backward pass, while still maintaining quadratic convergence locally. The derivation enables unifying multiple previous MS algorithms, and opens the door to many smaller algorithmic improvements. A penalty method is introduced to strategically control the step size, further improving the convergence performance. An adaptive merit function and a more reliable acceptance condition are employed for globalization. The effects of these improvements are benchmarked for trajectory optimization with a quadrotor, an acrobot, and a manipulator. MS-DDP is also demonstrated for use in Model Predictive Control (MPC) for dynamic jumping with a quadruped robot, showing its benefits over a single shooting approach