arXiv subjects

Jessy W. Grizzle

Publications and source records attributed to Jessy W. Grizzle.

26 records · Page 2Linked to original sources

arXiv2019

Rapid Trajectory Optimization Using C-FROST with Illustration on a Cassie-Series Dynamic Walking Biped

One of the big attractions of low-dimensional models for gait design has been the ability to compute solutions rapidly, whereas one of their drawbacks has been the difficulty in mapping the solutions back to the target robot. This paper presents a set of tools for rapidly determining solutions for ``humanoids'' without removing or lumping degrees of freedom. The main tools are (1) C-FROST, an open-source C++ interface for FROST, a direct collocation optimization tool; and (2) multi-threading. The results will be illustrated on a 20-DoF floating-base model for a Cassie-series bipedal robot through numerical calculations and physical experiments.

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arXiv2018

Hybrid Contact Preintegration for Visual-Inertial-Contact State Estimation Using Factor Graphs

The factor graph framework is a convenient modeling technique for robotic state estimation where states are represented as nodes, and measurements are modeled as factors. When designing a sensor fusion framework for legged robots, one often has access to visual, inertial, joint encoder, and contact sensors. While visual-inertial odometry has been studied extensively in this framework, the addition of a preintegrated contact factor for legged robots has been only recently proposed. This allowed for integration of encoder and contact measurements into existing factor graphs, however, new nodes had to be added to the graph every time contact was made or broken. In this work, to cope with the problem of switching contact frames, we propose a hybrid contact preintegration theory that allows contact information to be integrated through an arbitrary number of contact switches. The proposed hybrid modeling approach reduces the number of required variables in the nonlinear optimization problem by only requiring new states to be added alongside camera or selected keyframes. This method is evaluated using real experimental data collected from a Cassie-series robot where the trajectory of the robot produced by a motion capture system is used as a proxy for ground truth. The evaluation shows that inclusion of the proposed preintegrated hybrid contact factor alongside visual-inertial navigation systems improves estimation accuracy as well as robustness to vision failure, while its generalization makes it more accessible for legged platforms.

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arXiv2018

Contact-Aided Invariant Extended Kalman Filtering for Legged Robot State Estimation

This paper derives a contact-aided inertial navigation observer for a 3D bipedal robot using the theory of invariant observer design. Aided inertial navigation is fundamentally a nonlinear observer design problem; thus, current solutions are based on approximations of the system dynamics, such as an Extended Kalman Filter (EKF), which uses a system's Jacobian linearization along the current best estimate of its trajectory. On the basis of the theory of invariant observer design by Barrau and Bonnabel, and in particular, the Invariant EKF (InEKF), we show that the error dynamics of the point contact-inertial system follows a log-linear autonomous differential equation; hence, the observable state variables can be rendered convergent with a domain of attraction that is independent of the system's trajectory. Due to the log-linear form of the error dynamics, it is not necessary to perform a nonlinear observability analysis to show that when using an Inertial Measurement Unit (IMU) and contact sensors, the absolute position of the robot and a rotation about the gravity vector (yaw) are unobservable. We further augment the state of the developed InEKF with IMU biases, as the online estimation of these parameters has a crucial impact on system performance. We evaluate the convergence of the proposed system with the commonly used quaternion-based EKF observer using a Monte-Carlo simulation. In addition, our experimental evaluation using a Cassie-series bipedal robot shows that the contact-aided InEKF provides better performance in comparison with the quaternion-based EKF as a result of exploiting symmetries present in the system dynamics.

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arXiv2018

Legged Robot State-Estimation Through Combined Forward Kinematic and Preintegrated Contact Factors

State-of-the-art robotic perception systems have achieved sufficiently good performance using Inertial Measurement Units (IMUs), cameras, and nonlinear optimization techniques, that they are now being deployed as technologies. However, many of these methods rely significantly on vision and often fail when visual tracking is lost due to lighting or scarcity of features. This paper presents a state-estimation technique for legged robots that takes into account the robot's kinematic model as well as its contact with the environment. We introduce forward kinematic factors and preintegrated contact factors into a factor graph framework that can be incrementally solved in real-time. The forward kinematic factor relates the robot's base pose to a contact frame through noisy encoder measurements. The preintegrated contact factor provides odometry measurements of this contact frame while accounting for possible foot slippage. Together, the two developed factors constrain the graph optimization problem allowing the robot's trajectory to be estimated. The paper evaluates the method using simulated and real sensory IMU and kinematic data from experiments with a Cassie-series robot designed by Agility Robotics. These preliminary experiments show that using the proposed method in addition to IMU decreases drift and improves localization accuracy, suggesting that its use can enable successful recovery from a loss of visual tracking.

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arXiv2017

Correctness Guarantees for the Composition of Lane Keeping and Adaptive Cruise Control

This paper develops a control approach with correctness guarantees for the simultaneous operation of lane keeping and adaptive cruise control. The safety specifications for these driver assistance modules are expressed in terms of set invariance. Control barrier functions are used to design a family of control solutions that guarantee the forward invariance of a set, which implies satisfaction of the safety specifications. The control barrier functions are synthesized through a combination of sum-of-squares program and physics-based modeling and optimization. A real-time quadratic program is posed to combine the control barrier functions with the performance-based controllers, which can be either expressed as control Lyapunov function conditions or as black-box legacy controllers. In both cases, the resulting feedback control guarantees the safety of the composed driver assistance modules in a formally correct manner. Importantly, the quadratic program admits a closed-form solution that can be easily implemented. The effectiveness of the control approach is demonstrated by simulations in the industry-standard vehicle simulator Carsim.

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arXiv2016

Robustness of Control Barrier Functions for Safety Critical Control

Barrier functions (also called certificates) have been an important tool for the verification of hybrid systems, and have also played important roles in optimization and multi-objective control. The extension of a barrier function to a controlled system results in a control barrier function. This can be thought of as being analogous to how Sontag extended Lyapunov functions to control Lyapunov functions in order to enable controller synthesis for stabilization tasks. A control barrier function enables controller synthesis for safety requirements specified by forward invariance of a set using a Lyapunov-like condition. This paper develops several important extensions to the notion of a control barrier function. The first involves robustness under perturbations to the vector field defining the system. Input-to-State stability conditions are given that provide for forward invariance, when disturbances are present, of a "relaxation" of set rendered invariant without disturbances. A control barrier function can be combined with a control Lyapunov function in a quadratic program to achieve a control objective subject to safety guarantees. The second result of the paper gives conditions for the control law obtained by solving the quadratic program to be Lipschitz continuous and therefore to gives rise to well-defined solutions of the resulting closed-loop system.

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arXiv2016

Control Barrier Function Based Quadratic Programs for Safety Critical Systems

Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions -- expressed as control barrier functions -- to be unified with performance objectives -- expressed as control Lyapunov functions -- in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.

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arXiv2010

Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot

This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3D (spatial) bipedal robot consisting of a torso, two legs, and passive (unactuated) point feet. The contact between the robot and the walking surface is assumed to inhibit yaw rotation. The studied robot has 8 DOF in the single support phase and 6 actuators. The interest of studying robots with point feet is that the robot's natural dynamics must be explicitly taken into account to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit. This method allows the computations to be carried out on a 2-DOF subsystem of the 8-DOF robot model. The stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincaré map of the hybrid zero dynamics. Three strategies are explored. The first strategy consists of imposing a stability condition during the search of a periodic gait by optimization. The second strategy uses an event-based controller. In the third approach, the effect of output selection is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.

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