Model-based Optimal Control for Rigid-Soft Underactuated Systems

Technical Summary: Model-based Optimal Control for Rigid-Soft Underactuated Systems

Problem Statement
Continuum Soft Robots (CSRs) possess theoretically infinite Degrees of Freedom (DoFs), offering dexterity and adaptability superior to rigid robots. However, they are inherently underactuated, driven by a finite number of inputs, and subject to intrinsic box constraints due to the unidirectional and distributed nature of soft actuation. While control for CSRs has largely been restricted to quasi-static behaviors, dynamic tasks—such as swing-up maneuvers to unstable equilibria—require accurate exploitation of continuum dynamics. Existing model-based optimal control approaches face significant hurdles: high-fidelity models are computationally expensive, and numerical differentiation for high-dimensional systems is often inaccurate or prohibitive. Furthermore, standard explicit integration schemes struggle with the stiff differential equations governing soft dynamics, and simplified models (e.g., pseudo-rigid or low-order templates) fail to capture complex deformations necessary for dynamic control.

Methodology
This work leverages the Geometrically Variable Strain (GVS) formulation, derived from Cosserat Rod Theory (CRT), to model hybrid rigid–soft systems. The GVS approach parameterizes the strain field using truncated Legendre polynomial expansions, reducing Partial Differential Equations (PDEs) to a finite set of Ordinary Differential Equations (ODEs) while capturing all strain modes (bending, twisting, stretching, shearing). Crucially, recent advances provide analytical derivatives for GVS dynamics, enabling efficient gradient-based optimization.

The paper evaluates three model-based optimal control strategies on high-DoF continuum models without quasi-static assumptions:

1. Direct Collocation (DC): A direct transcription method using trapezoidal collocation to solve the optimal control problem (OCP) as a large-scale Nonlinear Programming (NLP) problem.
2. Box-Implicit Differential Dynamic Programming (Box-IDDP): A novel algorithm developed to address the specific challenges of CSRs. It combines:
   • Implicit Time Integration: To ensure stability for stiff dynamics without requiring prohibitively small time steps.
   • Box Constraints: To enforce actuator limits (input saturation).
   • DDP Framework: Utilizing Bellman's principle of optimality with backward and forward passes. The backward pass computes local optimal policies via constrained quadratic programming, while the forward pass rolls out the trajectory using implicit integration.
3. Nonlinear Model Predictive Control (NMPC): Implemented online using the Box-IDDP solver to handle state feedback and disturbances over a receding horizon.

To enhance computational efficiency and convergence, the authors introduce a resolution-based warm-start strategy. This approach exploits the hierarchical structure of the GVS model: the OCP is first solved on a simplified rigid equivalent, then progressively transferred to a soft model with constant curvature, and finally to the full high-order model with increasing strain modes (curvature, then linear strain).

Key Contributions

• Analytical Differentiation for High-Order Control: The work demonstrates that analytical derivatives of the GVS model render the optimization of high-dimensional continuum dynamics computationally tractable, avoiding the need for pseudo-rigid approximations.
• Box-IDDP Algorithm: The development of Box-IDDP, which integrates implicit time integration with box-constrained control inputs, specifically tailored for the stiff, constrained dynamics of underactuated soft systems.
• Warm-Start Strategy: A novel initialization method that accelerates solver convergence by leveraging the GVS strain parameterization, starting from rigid models and gradually increasing fidelity.
• Comprehensive Benchmarking: The evaluation of DC, Box-IDDP, and NMPC on three increasingly complex benchmarks: the Soft Cart-Pole (1D planar), Soft Pendubot (2-link planar), and Soft Furuta Pendulum (3D hybrid with out-of-plane deformations).

Results
The methods were validated in simulation on the three benchmark systems, all modeled with second-order Legendre polynomial expansions (yielding high DoFs).

• Soft Cart-Pole: All controllers successfully achieved swing-up. The controllers exhibited non-minimum phase behavior (moving the cart opposite to the target) to accumulate energy due to input saturation. DC and Box-IDDP utilized rapid oscillatory inputs to counteract gravity-induced bending, while NMPC achieved smoother convergence with a larger cart displacement.
• Soft Pendubot: The controllers managed the coupling between two soft links. As the system approached the upright equilibrium, optimal policies applied negative torque to counteract gravity-induced bending of the first link caused by the load of the second link.
• Soft Furuta Pendulum: This 3D system highlighted complex interactions. The Box-IDDP and NMPC solutions discovered a strategy involving strong out-of-plane bending and torsion to "wind" the link into a helicoidal shape, storing elastic energy that was released to drive the swing-up. In contrast, DC relied more heavily on in-plane bending.
• Performance Comparison:
  • Convergence: The warm-start strategy significantly improved cost convergence speed for Box-IDDP compared to pseudo-random initialization.
  • Efficiency: Box-IDDP converged faster per iteration than DC (requiring 2.9x less computational time per iteration in the tested setup). However, DC achieved a lower final cost (suboptimal for Box-IDDP), likely due to its ability to navigate the NLP landscape more globally, whereas Box-IDDP is more susceptible to local minima without a good initial guess.
  • Suitability: Box-IDDP is identified as superior for real-time applications (like NMPC) due to speed, while DC remains preferred for offline planning requiring complex constraint handling.

Significance and Claims
The paper claims to provide a systematic framework for the dynamic control of underactuated rigid–soft robots, overcoming the limitations of quasi-static and low-order approximations. By combining the GVS modeling framework with analytical derivatives and the proposed Box-IDDP algorithm, the authors demonstrate that high-fidelity continuum dynamics can be optimized efficiently. The work highlights that accurate exploitation of continuum dynamics is essential for dynamic tasks and that the proposed warm-start strategy is critical for making these high-dimensional optimizations feasible. The authors conclude that their approach enables the control of complex behaviors, including non-minimum phase dynamics and out-of-plane deformations, which were previously difficult to address with existing methods. Future work is noted to focus on experimental validation and assessing robustness to model uncertainties.