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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. 

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. 

Entanglements like vines and branches in natural settings or cords and pipes in human spaces prevent mobile robots from accessing many environments. Legged robots should be effective in these settings, and more so than wheeled or tracked platforms, but naive controllers quickly become entangled and stuck. In this paper we present a method for proprioception aimed specifically at the task of sensing entanglements of a robot’s legs as well as a reaction strategy to disentangle legs during their swing phase as they advance to their next foothold. We demonstrate our proprioception and reaction strategy enables traversal of entanglements of many stiffnesses and geometries succeeding in 14 out of 16 trials in laboratory tests, as well as a natural outdoor environment. 

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.