Differential Dynamic Programming With Nonlinear Safety Constraints Under System Uncertainties

Abstract

Safe operation of systems such as robots requires them to plan and execute trajectories subject to safety constraints. When those systems are subject to uncertainties in their dynamics, it is challenging to ensure that the constraints are not violated. In this letter, we propose Safe-CDDP, a safe trajectory optimization and control approach for systems under additive uncertainties and nonlinear safety constraints based on constrained differential dynamic programming (DDP). The safety of the robot during its motion is formulated as chance constraints with user-chosen probabilities of constraint satisfaction. The chance constraints are transformed into deterministic ones in DDP formulation by constraint tightening. To avoid overconservatism during constraint tightening, linear control gains of the feedback policy derived from the constrained DDP are used in the approximation of closed-loop uncertainty propagation in prediction. The proposed algorithm is empirically evaluated on three different robot dynamics with up to 12 degrees of freedom in simulation. The computational feasibility and applicability of the approach are demonstrated with a physical hardware implementation.

Trajectory Optimization on Matrix Lie Groups with Differential Dynamic Programming and Nonlinear Constraints

“…Note that the second order term of ψ ∧ (ξ ∧ p − ξ ∧ ) is also discarded to obtain the linear dynamics of the configuration error. ψ in (11) is the perturbed configuration represented in Lie algebra. The perturbed velocity and control input are also defined as δξ = ξ p − ξ, and δu = u p − u…”

“…However, these iterations are less expensive to compute and often lead to a faster overall convergence rate. Therefore, in this work, we eliminated the second order dynamics as approximating the perturbed state dynamics as described in (11).…”

“…As a future direction, it would be interesting to investigate the closed loop uncertainty propagation through the prediction horizon, as has been done for Euclidean models [11], in order to design a robust Model Predictive Control (MPC) framework for matrix Lie groups. This would allow the proposed method to be used in more challenging and dynamic environments.…”

“…Differential dynamic programming (DDP) is a numerical method for solving optimal control problems that has gained widespread use in various engineering fields. Originally proposed by Mayne and Jacobson [10], DDP has been applied to a wide range of complex, high-dimensional systems [11]. One of the key advantages of DDP is its scalability, which allows it to handle large and complex systems with many degrees of freedom.…”