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In this letter we present Hybrid iterative Linear Quadratic Estimation (HiLQE), an optimization based offline state estimation algorithm for hybrid dynamical systems. We utilize the saltation matrix, a first order approximation of the variational update through an event driven hybrid transition, to calculate gradient information through hybrid events in the backward pass of an iterative linear quadratic optimization over state estimates. This enables accurate computation of the value function approximation at each timestep. Additionally, the forward pass in the iterative algorithm is augmented with hybrid dynamics in the rollout. A reference extension method is used to account for varying impact times when comparing states for the feedback gain in noise calculation. The proposed method is demonstrated on an ASLIP hopper system with position measurements. In comparison to the Salted Kalman Filter (SKF), the algorithm presented here achieves a maximum of 63.55% reduction in estimation error magnitude over all state dimensions near impact events. 

Model Predictive Control (MPC) is a popular strategy for controlling robots but is difficult for systems with contact due to the complex nature of hybrid dynamics. To implement MPC for systems with contact, dynamic models are often simplified or contact sequences fixed in time in order to plan trajectories efficiently. In this work, we propose the Hybrid iterative Linear Quadratic Regulator (HiLQR), which extends iLQR to a class of piecewisesmooth hybrid dynamical systems with state jumps. This is accomplished by 1) allowing for changing hybrid modes in the forward pass, 2) using the saltation matrix to update the gradient information in the backwards pass, and 3) using a reference extension to account for mode mismatch. We demonstrate these changes on a variety of hybrid systems and compare the different strategies for computing the gradients. We further show how HiLQR can work in a MPC fashion (HiLQR MPC) by 1) modifying how the cost function is computed when contact modes do not align, 2) utilizing parallelizations when simulating rigid body dynamics, and 3) using efficient analytical derivative computations of the rigid body dynamics. The result is a system that can modify the contact sequence of the reference behavior and plan whole body motions cohesively – which is crucial when dealing with large perturbations. HiLQR MPC is tested on two systems: first, the hybrid cost modification is validated on a simple actuated bouncing ball hybrid system. Then HiLQR MPC is compared against methods that utilize centroidal dynamic assumptions on a quadruped robot (Unitree A1). HiLQR MPC outperforms the centroidal methods in both simulation and hardware tests. 

Changes in Arctic sea ice since the middle of the last century are explored in this study. Both observations and climate model simulations show an overall sea ice expansion during 1953–1970 but a general sea ice decline afterward. Anthropogenic aerosols, nature forcing and atmospheric ozone changes are found to contribute to the sea ice expansion in the early period. Their effects are strong generally in late boreal summer. On the other hand, greenhouse gas warming has a dominant effect on diminishing Arctic sea ice cover during 1971–2005, especially in September. Internal climate variability also plays a role in the Arctic sea ice change during 1953–1970. However, it cannot solely explain the Arctic sea ice decline since the 1970s. 

Hybrid dynamical systems, i.e., systems that have both continuous and discrete states, are ubiquitous in engineering but are difficult to work with due to their discontinuous transitions. For example, a robot leg is able to exert very little control effort, while it is in the air compared to when it is on the ground. When the leg hits the ground, the penetrating velocity instantaneously collapses to zero. These instantaneous changes in dynamics and discontinuities (or jumps) in state make standard smooth tools for planning, estimation, control, and learning difficult for hybrid systems. One of the key tools for accounting for these jumps is called the saltation matrix. The saltation matrix is the sensitivity update when a hybrid jump occurs and has been used in a variety of fields, including robotics, power circuits, and computational neuroscience. This article presents an intuitive derivation of the saltation matrix and discusses what it captures, where it has been used in the past, how it is used for linear and quadratic forms, how it is computed for rigid body systems with unilateral constraints, and some of the structural properties of the saltation matrix in these cases. 

In this paper, we present a method for updating robotic state belief through contact with uncertain surfaces and apply this update to a Kalman filter for more accurate state estimation. Examining how guard surface uncertainty affects the time spent in each mode, we derive a novel guard saltation matrix- which maps perturbations prior to hybrid events to perturbations after - accounting for additional variation in the resulting state. Additionally, we propose the use of parameterized reset functions - capturing how unknown parameters change how states are mapped from one mode to the next - the Jacobian of which accounts for additional uncertainty in the resulting state. The accuracy of these mappings is shown by simulating sampled distributions through uncertain transition events and comparing the resulting covariances. Finally, we integrate these additional terms into the “uncertainty aware Salted Kalman Filter”, uaSKF, and show a peak reduction in average estimation error by 24–60% on a variety of test conditions and systems. 

Many controllers for legged robotic systems leverage open- or closed-loop control at discrete hybrid events to enhance stability. These controllers appear in several well studied phenomena such as the Raibert stepping controller, paddle juggling, and swing leg retraction. This work introduces hybrid event shaping (HES): a generalized method for analyzing and designing stable hybrid event controllers. HES utilizes the saltation matrix, which gives a closed-form equation for the effect that hybrid events have on stability. We also introduce shape parameters, which are higher order terms that can be tuned completely independently of the system dynamics to promote stability. Optimization methods are used to produce values of these parameters that optimize a stability measure. Hybrid event shaping captures previously developed control methods while also producing new optimally stable trajectories without the need for continuous-domain feedback. 

Trajectory optimization is a popular strategy for planning trajectories for robotic systems. However, many robotic tasks require changing contact conditions, which is difficult due to the hybrid nature of the dynamics. The optimal sequence and timing of these modes are typically not known ahead of time. In this work, we extend the Iterative Linear Quadratic Regulator (iLQR) method to a class of piecewise-smooth hybrid dynamical systems with state jumps by allowing for changing hybrid modes in the forward pass, using the saltation matrix to update the gradient information in the backwards pass, and using a reference extension to account for mode mismatch. We demonstrate these changes on a variety of hybrid systems and compare the different strategies for computing the gradients.