Discrete-Time Differential Dynamic Programming on Lie Groups: Derivation, Convergence Analysis, and Numerical Results

Boutselis, George I.; Theodorou, Evangelos A.

doi:10.1109/tac.2020.3034206

Cited by 9 publications

(17 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A wide range of the geometric control literature considers the design of the error or cost function on manifolds [11], [20], [21]. These methods are general, but they introduce difficulties while deriving the gradient or Hessian matrix of a cost function on the manifold.…”

Section: Cost Function Designmentioning

confidence: 99%

“…We compare the proposed framework with the open source discrete time Lie group DDP algorithm in [21]. This baseline algorithm designs the cost function on manifold thus the gradient and Hessian matrix need to be updated at each iteration.…”

Section: B Ilqr On So(3)mentioning

confidence: 99%

“…The work of [20] proposed the Lie group projection operator Newton method for continuous-time control on Lie groups. A discrete-time Differential Dynamic Programming (DDP) algorithm is proposed on Lie groups in [21]. The main procedure of [20], [21] is to derive the local perturbed system and then solve a local optimal control problem via dynamic programming.…”

Section: Introductionmentioning

confidence: 99%

“…A discrete-time Differential Dynamic Programming (DDP) algorithm is proposed on Lie groups in [21]. The main procedure of [20], [21] is to derive the local perturbed system and then solve a local optimal control problem via dynamic programming. Superlinear convergence is possible for both cases.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Lie Algebraic Cost Function Design for Control on Lie Groups

Teng¹,

Clark²,

Bloch³

et al. 2022

Preprint

View full text Add to dashboard Cite

This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to design the control objective on a Lie group and then derive the gradient for controller design are non-trivial and can result in slow convergence in tracking control. We show that with a proper left-invariant metric, setting the gradient of the cost function as the tracking error in the Lie algebra leads to a quadratic Lyapunov function that enables globally exponential convergence. In the PD control case, we show that our controller can maintain an exponential convergence rate even when the initial error is approaching π in SO(3). We also show the merit of this proposed framework in trajectory optimization. The proposed cost function enables the iterative Linear Quadratic Regulator (iLQR) to converge much faster than the Differential Dynamic Programming (DDP) with a well-adopted cost function when the initial trajectory is poorly initialized on SO(3).

show abstract

Section: Cost Function Designmentioning

confidence: 99%

Section: B Ilqr On So(3)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Lie Algebraic Cost Function Design for Control on Lie Groups

Teng¹,

Clark²,

Bloch³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally, a brief discussion about the application of DDP for implicitly defined systems from a Lie theoretic viewpoint is given in [25]. Here, we present a more complete and deep treatment, with extensive comparisons.…”

Section: Prior Work On Differential Dynamic Programming a Differentia...mentioning

confidence: 99%

Trajectory optimization for contact-rich motions using implicit differential dynamic programming

Chatzinikolaidis¹,

2021

Preprint

View full text Add to dashboard Cite

This paper presents a novel approach using sensitivity analysis for generalizing Differential Dynamic Programming (DDP) to systems characterized by implicit dynamics, such as those modelled via inverse dynamics and variational or implicit integrators. It leads to a more general formulation of DDP, enabling for example the use of the faster recursive Newton-Euler inverse dynamics. We leverage the implicit formulation for precise and exact contact modelling in DDP, where we focus on two contributions: (1) Contact dynamics in acceleration level that enables high-order integration schemes; (2) Formulation using an invertible contact model in the forward pass and a closed-form solution in the backward pass to improve the numerical resolution of contacts. The performance of the proposed framework is validated (1) by comparing implicit versus explicit DDP for the swing-up of a double pendulum, and (2) by planning motions for two tasks using a single leg model making multi-body contacts with the environment: standing up from ground, where a priori contact enumeration is challenging, and maintaining balance under an external perturbation.

show abstract

Discrete-Time Differential Dynamic Programming on SO(3) With Pose Constraints

Liu

2022

IEEE Access

View full text Add to dashboard Cite

The motion of many robotics systems, such as the rotation of unmanned aerial vehicles, can be modeled by SO(3). However, the difficulties in the parameterization of the SO(3) makes it hard to implement the state-of-the-art collision avoidance algorithm. In this paper, we present a new method for control on SO(3) via the combination of the nonlinear state constraints and geometric control technique. We first define the nonlinear constraints for trajectory optimization on SO(3). Then we solve the trajectory optimization problem by constrained Differential Dynamic Programming (DDP) in a Riemannian geometry framework. The first and second-order expansion of the cost function and dynamics are derived in a coordinate-free manner. The safety condition represented by nonlinear constraints is incorporated into the DDP via an active set method, where we find the active set in the backward path to obtain the optimal control policy. We validated our methods on motion planning of rigid model on SO(3) manifold where pose constraints are imposed. Our methods outperform the baseline methods in terms of convergence speed and numerical robustness to disturbance. The numerical robustness is essential when the system is initialized far from the local optimum.

show abstract

Discrete-Time Differential Dynamic Programming on Lie Groups: Derivation, Convergence Analysis, and Numerical Results

Cited by 9 publications

References 33 publications

Lie Algebraic Cost Function Design for Control on Lie Groups

Lie Algebraic Cost Function Design for Control on Lie Groups

Trajectory optimization for contact-rich motions using implicit differential dynamic programming

Discrete-Time Differential Dynamic Programming on SO(3) With Pose Constraints

Contact Info

Product

Resources

About