Stochastic Predictive Control for Legged Robots

Abstract

Humans show distinguished capabilities in decision-making. Whether planning for daily commuting routes, deciding on the next move in a chess game, or shooting a rocket in space, we face the same conundrum of having to make hundreds of present decisions in the face of uncertain events in the future. By incorporating such uncertainties in the decision-making process, one can act with a sense of rationality given the information available about the world. Despite the long history of academic effort formulating this problem in the fields of operational research, numerical optimization, and control theory, planning under uncertainty persists to be a challenging problem in robotics and remains an active area of research. One of the main challenges of planning under uncertainty stems from the computational intractability of the underlying mathematical programs given the limited computational resources against the realm of infinite uncertain future outcomes. Moreover, most of the interesting planning problems need to be solved in real-time, which adds to the complexity of the problem. For instance, If I am (an amateur chess player) to play a chess game against Garry Kasparov where he has to wait for me 30 mins at a time during my turns to simulate as many scenarios as possible in the future to maximize my winning chances, not only it would be the most boring chess match ever played, but my limited computed scenarios might as well not be sufficient enough to win Kasparov. Imagine if we could optimize New York City’s subway network for hours at a time to compute the optimal cycle of train routes and their frequencies, given the uncertainty of the number of daily commuters over all the network stations. This would be great except for the fact that New Yorkers will stop using the subway as they won’t accept such optimization delay time before announcing the next cycle of routes for their commutes. Even if they would, it would be an outdated forecast already. Another important aspect in planning under uncertainty constitutes satisfying a set of constraints along with a desired objective. For example, a robot navigating around a museum needs to approach randomly walking visitors to assist them, while ensuring a collision-free trajectory based on a robustness measure in the presence of randomly walking visitors around the museum. The choice of a robustness measure for satisfying constraints is directly intertwined with the optimality of achieving a task. This is not an easy task, since an overestimation of future uncertainties can lead to a conservative behavior that deteriorates the task objective of a robot, while a very relaxed one can hinder safety by violating the planning constraints. Consider the case of a self-driving car on a highway that keeps skipping all exits on its way to the airport to guarantee a worst-case separation distance measure to avoid colliding with other cars. Although this measure achieves perfect robustness, it fails to achieve main the objective of getting to the airport in time by deteriorating the objective performance. On the other hand imagine the complement of this scenario, where a self-driving car arrives at the airport on time after making a couple of collisions with other cars to achieve its objective. Designing robust trajectory optimizers for legged robots is a very challenging task. This is because of the task complexity of locomotion, and the unstable nature of the non-smooth under-actuated dynamics, which needs to be stabilized by making contact with the environment that respects the physics constraints. Due to this complexity, it became a common practice in the legged locomotion community to design sub-optimal robustness measures for constraint satisfaction either heuristically or using worst-case estimation of uncertainties to bypass the curse of computation using fast re-planning. Not only does this deteriorate the performance of the control policy as mentioned above, but we also argue that high-frequency re-planning is not helpful in the case of legged robots. This is due to the hybrid nature of the fast-switching dynamics of walking robots. Consider a legged robot climbing stairs while losing a step due to an external push or wrong estimates from its sensors. In this scenario, initiating a contact force in the air when it’s supposed to be swinging its feet forward, can only make things worse due to the non-causality of the control policy. This is similar to a scenario of a dog wagging its tail, which does not get better even when the dog does it faster. We claim that anticipating such uncertainties during real-time planning can help achieve more robust locomotion behaviors for legged robots. This thesis attempts to overcome the aforementioned challenges by designing a tractable real-time robust feedback control policy for legged robots subject to additive and parametric stochastic uncertainties on the model dynamics. We use tools inspired by stochastic model predictive control and chance-constraint pro- gramming to design uncertainty-aware constraints in a systematic non-conservative fashion for trajectory optimization of legged robots as a robustness measure. The final policy is designed as a trade-off between robustness and performance.