Trajectory Optimization Through Contacts and Automatic Gait Discovery for Quadrupeds

In this work we present a trajectory Optimization framework for whole-body motion planning through contacts. We demonstrate how the proposed approach can be applied to automatically discover different gaits and dynamic motions on a quadruped robot. In contrast to most previous methods, we do not pre-specify contact switches, timings, points or gait patterns, but they are a direct outcome of the optimization. Furthermore, we optimize over the entire dynamics of the robot, which enables the optimizer to fully leverage the capabilities of the robot. To illustrate the spectrum of achievable motions, here we show eight different tasks, which would require very different control structures when solved with state-of-the-art methods. Using our trajectory Optimization approach, we are solving each task with a simple, high level cost function and without any changes in the control structure. Furthermore, we fully integrated our approach with the robot's control and estimation framework such that optimization can be run online. By demonstrating a rough manipulation task with multiple dynamic contact switches, we exemplarily show how optimized trajectories and control inputs can be directly applied to hardware.Comment: Video: https://youtu.be/sILuqJBsyK

Automatic Gait Pattern Selection for Legged Robots

An important issue when synthesizing legged locomotion plans is the combinatorial complexity that arises from gait pattern selection. Though it can be defined manually, the gait pattern plays an important role in the feasibility and optimality of a motion with respect to a task. Replacing human intuition with an automatic and efficient approach for gait pattern selection would allow for more autonomous robots, responsive to task and environment changes. To this end, we propose the idea of building a map from task to gait pattern selection for given environment and performance objective. Indeed, we show that for a 2D half-cheetah model and a quadruped robot, a direct mapping between a given task and an optimal gait pattern can be established. We use supervised learning to capture the structure of this map in a form of gait regions. Furthermore, we propose to construct a warm-starting trajectory for each gait region. We empirically show that these warm-starting trajectories improve the convergence speed of our trajectory optimization problem up to 60 times when compared with random initial guesses. Finally, we conduct experimental trials on the ANYmal robot to validate our method.</p

A Family of Iterative Gauss-Newton Shooting Methods for Nonlinear Optimal Control

This paper introduces a family of iterative algorithms for unconstrained nonlinear optimal control. We generalize the well-known iLQR algorithm to different multiple-shooting variants, combining advantages like straight-forward initialization and a closed-loop forward integration. All algorithms have similar computational complexity, i.e. linear complexity in the time horizon, and can be derived in the same computational framework. We compare the full-step variants of our algorithms and present several simulation examples, including a high-dimensional underactuated robot subject to contact switches. Simulation results show that our multiple-shooting algorithms can achieve faster convergence, better local contraction rates and much shorter runtimes than classical iLQR, which makes them a superior choice for nonlinear model predictive control applications.Comment: 8 page

Contact-Implicit Trajectory Optimization Based on a Variable Smooth Contact Model and Successive Convexification

In this paper, we propose a contact-implicit trajectory optimization (CITO) method based on a variable smooth contact model (VSCM) and successive convexification (SCvx). The VSCM facilitates the convergence of gradient-based optimization without compromising physical fidelity. On the other hand, the proposed SCvx-based approach combines the advantages of direct and shooting methods for CITO. For evaluations, we consider non-prehensile manipulation tasks. The proposed method is compared to a version based on iterative linear quadratic regulator (iLQR) on a planar example. The results demonstrate that both methods can find physically-consistent motions that complete the tasks without a meaningful initial guess owing to the VSCM. The proposed SCvx-based method outperforms the iLQR-based method in terms of convergence, computation time, and the quality of motions found. Finally, the proposed SCvx-based method is tested on a standard robot platform and shown to perform efficiently for a real-world application.Comment: Accepted for publication in ICRA 201

Frequency-Aware Model Predictive Control

Transferring solutions found by trajectory optimization to robotic hardware remains a challenging task. When the optimization fully exploits the provided model to perform dynamic tasks, the presence of unmodeled dynamics renders the motion infeasible on the real system. Model errors can be a result of model simplifications, but also naturally arise when deploying the robot in unstructured and nondeterministic environments. Predominantly, compliant contacts and actuator dynamics lead to bandwidth limitations. While classical control methods provide tools to synthesize controllers that are robust to a class of model errors, such a notion is missing in modern trajectory optimization, which is solved in the time domain. We propose frequency-shaped cost functions to achieve robust solutions in the context of optimal control for legged robots. Through simulation and hardware experiments we show that motion plans can be made compatible with bandwidth limits set by actuators and contact dynamics. The smoothness of the model predictive solutions can be continuously tuned without compromising the feasibility of the problem. Experiments with the quadrupedal robot ANYmal, which is driven by highly-compliant series elastic actuators, showed significantly improved tracking performance of the planned motion, torque, and force trajectories and enabled the machine to walk robustly on terrain with unmodeled compliance

Whole-Body MPC and Online Gait Sequence Generation for Wheeled-Legged Robots

Our paper proposes a model predictive controller as a single-task formulation that simultaneously optimizes wheel and torso motions. This online joint velocity and ground reaction force optimization integrates a kinodynamic model of a wheeled quadrupedal robot. It defines the single rigid body dynamics along with the robot's kinematics while treating the wheels as moving ground contacts. With this approach, we can accurately capture the robot's rolling constraint and dynamics, enabling automatic discovery of hybrid maneuvers without needless motion heuristics. The formulation's generality through the simultaneous optimization over the robot's whole-body variables allows for a single set of parameters and makes online gait sequence adaptation possible. Aperiodic gait sequences are automatically found through kinematic leg utilities without the need for predefined contact and lift-off timings, reducing the cost of transport by up to 85%. Our experiments demonstrate dynamic motions on a quadrupedal robot with non-steerable wheels in challenging indoor and outdoor environments. The paper's findings contribute to evaluating a decomposed, i.e., sequential optimization of wheel and torso motion, and single-task motion planner with a novel quantity, the prediction error, which describes how well a receding horizon planner can predict the robot's future state. To this end, we report an improvement of up to 71% using our proposed single-task approach, making fast locomotion feasible and revealing wheeled-legged robots' full potential.Comment: 8 pages, 6 figures, 1 table, 52 references, 9 equation

A Comparative Analysis of Contact Models in Trajectory Optimization for Manipulation

In this paper, we analyze the effects of contact models on contact-implicit trajectory optimization for manipulation. We consider three different approaches: (1) a contact model that is based on complementarity constraints, (2) a smooth contact model, and our proposed method (3) a variable smooth contact model. We compare these models in simulation in terms of physical accuracy, quality of motions, and computation time. In each case, the optimization process is initialized by setting all torque variables to zero, namely, without a meaningful initial guess. For simulations, we consider a pushing task with varying complexity for a 7 degrees-of-freedom robot arm. Our results demonstrate that the optimization based on the proposed variable smooth contact model provides a good trade-off between the physical fidelity and quality of motions at the cost of increased computation time.Comment: 6 pages, 7 figures, 4 tables, IROS 2018 camera-ready versio

Automatic Differentiation of Rigid Body Dynamics for Optimal Control and Estimation

Many algorithms for control, optimization and estimation in robotics depend on derivatives of the underlying system dynamics, e.g. to compute linearizations, sensitivities or gradient directions. However, we show that when dealing with Rigid Body Dynamics, these derivatives are difficult to derive analytically and to implement efficiently. To overcome this issue, we extend the modelling tool `RobCoGen' to be compatible with Automatic Differentiation. Additionally, we propose how to automatically obtain the derivatives and generate highly efficient source code. We highlight the flexibility and performance of the approach in two application examples. First, we show a Trajectory Optimization example for the quadrupedal robot HyQ, which employs auto-differentiation on the dynamics including a contact model. Second, we present a hardware experiment in which a 6 DoF robotic arm avoids a randomly moving obstacle in a go-to task by fast, dynamic replanning