In this paper 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 extend Hybrid iterative Linear Quadratic Regulator to 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.

I Figure 1: An example 2 mode hybrid system where the domains are shown in black circles D, the dynamics are shown with gray arrows F, the guard for the current domain is shown in red dashed g, and the reset from the current mode to the next mode is shown in blue R. The saltation matrix relies on differentiating the guards B. Saltation matrix derivation and resets so they must be differentiable. Excluding Zeno In this section, the derivation of the saltation matrix (2) is conditions ensures we avoid computing infinite saltation matrices presented, following the geometric derivation from [10] with in finite time, which would clearly be unsound for the addition of reset maps. There are many alternate ways analysis. Transversality ensures that neighboring trajectories to derive (2): a derivation using the chain rule is included in impact the same guard unless the impact point lies on any Appendix A and a derivation using a double limit can be found other guard surface, in which case the Bouligand derivative in [96]. is the appropriate analysis tool [52, 114-117]. Transversality Suppose the nominal trajectory of interest is x(t) as shown also ensures the denominator in (2) does not approach zero. in Figure 1. The trajectory starts in mode I and goes through a In some cases, the saltation matrix for a hybrid transition hybrid transition to mode J at time t. The saltation matrix is a can become an identity transformation.

The ability to generate robust walking gaits on bipedal robots is key to their successful realization on hardware. To this end, this work extends the method of Hybrid Zero Dynamics (HZD) -- which traditionally only accounts for locomotive stability via periodicity constraints under perfect impact events -- through the inclusion of the saltation matrix with a view toward synthesizing robust walking gaits. By jointly minimizing the norm of the extended saltation matrix and the torque of the robot directly in the gait generation process, we demonstrate that the synthesized gaits are more robust than gaits generated with either term alone; these results are shown in simulation and on hardware for the AMBER-3M planar biped and the Atalante lower-body exoskeleton (both with and without a human subject). The end result is experimental validation that combining saltation matrices with HZD methods produces more robust bipedal walking in practice.

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 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 the 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.