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Increasing Reliability of Legged Robots in the Presence of Uncertainty

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posted on 2023-02-28, 22:08 authored by Nathan KongNathan Kong

Legged robots have the potential to traverse a wide variety of environments, but are currently too unreliable to use in mission critical settings. A major factor that hinders the reliability of legged robots is the hybrid dynamics that arises when their legs make varying contact with the environment. The discontinuities introduced by hybrid dynamics interfere with traditional tracking, planning, and state estimation strategies. This thesis presents several novel ideas and tools in overcoming these difficulties: creating robust trajectories through optimally convergent planning, a tutorial on the saltation matrix (the update to the sensitivity equation for hybrid transitions), Kalman filtering on hybrid systems, iterative Linear Quadratic Regulator for hybrid systems, and a model predictive controller which can continuously update the current plan given new information. 

Optimally convergent planning creates trajectories that are robust to state uncertainty in undersensed and underactuated systems. Convergent planning utilizes ideas from contraction analysis and minimizes divergence to find trajectories that naturally shrink state uncertainty. This optimization framework is validated for an undersensed hill navigation problem as well as an underactuated rotary cart pole incline. 

The saltation matrix tutorial provides the necessary information to get started implementing smooth tools for hybrid systems with event-triggered transitions. The tutorial contains a survey of where the saltation matrix is commonly used, as well as expanded proofs for deriving the saltation matrix. We also show an example saltation matrix calculation for a simple rigid body system and show that vital information is lost when not using the saltation matrix, and we calculate the saltation matrix for a generalized rigid body system with unilateral constraints.

The Kalman filter is a widely used optimal state estimation algorithm. We extended the Kalman filter to hybrid systems by creating the “Salted Kalman Filter” which allows hybrid transitions to occur during the a priori and a posteriori updates and by using linearizations about a hybrid transition in order to propagate uncertainty belief. The Salted Kalman Filter is compared against a version of the algorithm but with a naive method of linearization about hybrid transitions. The Salted Kalman Filter is also validated by comparing it against a hybrid particle filter benchmark. 

Iterative Linear Quadratic Regulator (iLQR) is a trajectory optimization algorithm that uses locally linear models of the dynamics and uses a quadratic cost function. We extended iterative Linear Quadratic Regulator to hybrid systems with the following additions: allowing for hybrid transitions on the forward pass, handling mode mismatches with extensions on reference trajectories, and using the linearization about a hybrid transition on the backward pass. We validate the hybrid iterative Linear Quadratic Regulator on a variety of hybrid systems and show that the algorithm can optimally choose contact timing and placements. 

We utilize hybrid iLQR as a Model Predictive Controller (MPC) in order to replan in realtime. By replanning in real-time, the robot will be more robust to unplanned, large disturbances because the controller can generate a new plan instead of rigidly tracking a reference trajectory. We validated the MPC on a quadruped robot both in simulation and on the real robot by applying disturbances to the robot, and we also compared the MPC against an MPC that uses simplified robot dynamics. 

Overall, this thesis makes legged robots more reliable by creating a robust global plan and by reactively replanning in real time to deal with local disturbances such as contact timing errors or unplanned slips. Robustness to the global plan can be achieved through planning convergent trajectories. To alleviate the complexity of hybrid transitions, we heavily utilize linearizations to enable reactive replanning and fast state estimation through hybrid iLQR MPC and the Salted Kalman Filter. 




Degree Type



Mechanical Engineering

Degree Name

  • Doctor of Philosophy (PhD)


Aaron Johnson

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