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Soft Robot Optimal Control Via Reduced Order Finite Element Models
Sander Tonkens, Joseph Lorenzetti, Marco Pavone
TL;DR
High-dimensional FEM models capture soft robots’ nonlinear dynamics but make real-time control difficult. The paper uses model order reduction to build a nonlinear reduced FEM model for constrained optimal control, solved with sequential convex programming. In simulation, the approach tracks well and generally satisfies constraints, unlike a linearized ROM controller, and is reported as real-time capable.
Problem
High-dimensional FEM models capture nonlinear soft-robot dynamics, but their computational demands challenge real-time control.
Source: §I · p. 1, B. Constrained Optimal Control Problem · p. 3Method
The paper combines model order reduction into a nonlinear piecewise affine FEM model with output-feedback receding-horizon control solved by sequential convex programming.
Source: §I · pp. 1–2Results
The proposed approach and Koopman controller track well and generally satisfy constraints, while the linearized ROM scheme tracks poorly and severely violates constraints.
Source: B. Results · p. 6Takeaways & Limitations
Simulation demonstrates constrained output-feedback optimal control with the reduced model for tasks including setpoint and trajectory tracking.
Source: §VI · p. 6
Abstract
from arXiv · showhide
Finite element methods have been successfully used to develop physics-based models of soft robots that capture the nonlinear dynamic behavior induced by continuous deformation. These high-fidelity models are therefore ideal for designing controllers for complex dynamic tasks such as trajectory optimization and trajectory tracking. However, finite element models are also typically very high-dimensional, which makes real-time control challenging. In this work we propose an approach for finite element model-based control of soft robots that leverages model order reduction techniques to significantly increase computational efficiency. In particular, a constrained optimal control problem is formulated based on a nonlinear reduced order finite element model and is solved via sequential convex programming. This approach is demonstrated through simulation of a cable-driven soft robot for a constrained trajectory tracking task, where a 9768-dimensional finite element model is used for controller design.
I. INTRODUCTION
Soft-robot control is difficult because continuous deformation creates nonlinear, high-dimensional dynamics, while simplified models can be geometry-specific or kinematics-focused. This work uses a reduced nonlinear FEM model and sequential convex programming for constrained optimal control, demonstrating the approach in simulation.
I. INTRODUCTION
- Soft robots’ continuous deformation produces nonlinear, effectively infinite-dimensional dynamics, while design diversity makes generalizable modeling difficult.These challenges have limited practical use of soft robots.Source: §I · p. 1
Related Work
- Hand-engineered piecewise-constant-curvature, beam, and Cosserat models often target specific geometries and model only kinematics, complicating dynamic control.Data-driven models have also been used for kinematic and dynamic control, but lack physics-based principles and a systematic development procedure; reliance on experimental data limits their use in design.Source: §I · p. 1
- FEM offers systematic, physics-based models for varied soft robots, but its high dimensionality makes real-time dynamic control challenging.Linearized reduced-order FEM controllers and inverse-kinematics-based nonlinear reduction do not adequately address dynamic tasks such as trajectory tracking.Source: §I · pp. 1–2
Statement of Contributions
- The proposed controller combines a low-dimensional piecewise affine approximation of a nonlinear FEM model with output-feedback receding-horizon control solved by sequential convex programming.The method targets dynamic tasks including setpoint and trajectory tracking and is demonstrated in simulation with a 9,768-dimensional FEM model.Source: §I · p. 2
- The paper presents a SOFA-based implementation and evaluates constrained trajectory tracking on a cable-actuated Diamond robot in simulation.The model has dimension 9,768; the paper also positions the approach for multiple dynamic control tasks.Source: §I · p. 2
II. PROBLEM FORMULATION
The paper formulates a soft-robot FEM model from node-level motion and forces, then defines measurement and performance outputs for control. This formulation distinguishes the measured state from the variables used to evaluate the control task.
A. Soft Robot Finite Element Model
- FEM discretizes partial differential equations over mesh nodes, producing ordinary differential equations for the state at each node.For soft robots, each node’s state comprises position and velocity, yielding six degrees of freedom per node and total dimension nf = 6N.Source: A. Soft Robot Finite Element Model · p. 2
- The FEM dynamics use Newton’s second law, with node positions, velocities, and control input defining the model variables.The model also includes a constant mass matrix, external forces, nonlinear internal forces, and a configuration-dependent input matrix; internal forces may use a linear-elastic Hooke’s-law model.Source: A. Soft Robot Finite Element Model · p. 2
- A linear measurement model supports output-feedback control, with the combined FEM state formed from node velocities and positions.The measurement is denoted y(t), while the state is xf = [vf T, qf T]T.Source: A. Soft Robot Finite Element Model · p. 2
- Performance variables z can represent task-relevant quantities such as end-effector position and velocity, distinct from the full FEM state.Their linear output model allows the optimal-control cost and constraints to use relevant performance variables rather than the full state.Source: A. Soft Robot Finite Element Model · p. 2
B. Constrained Optimal Control Problem
The constrained optimal-control problem penalizes deviations from target performance and input trajectories while enforcing convex bounds and FEM dynamics. The paper identifies nonlinear, high-dimensional optimization as the obstacle to real-time receding-horizon control and proposes model reduction to address it.
B. Constrained Optimal Control Problem
- The OCP constrains performance variables and control inputs to convex polytopes over a finite horizon.Its cost penalizes deviations from desired performance and input trajectories, including reference trajectories for tracking and equilibrium targets for regulation.Source: B. Constrained Optimal Control Problem · p. 2
- The optimization is subject to the FEM dynamics, which relate acceleration to external forces, nonlinear internal forces, and control inputs.The governing relation is M f ˙vf = P f − F f(qf, vf) + Hf(qf)u.Source: B. Constrained Optimal Control Problem · p. 2
- Receding-horizon control requires repeated real-time OCP solves, but nonlinear constraints and high-dimensional FEM states make this computationally challenging.Even a convex QP has stated complexity O(T(N + m)3); full state dimensions commonly reach thousands to tens of thousands.Source: B. Constrained Optimal Control Problem · pp. 2–3
- The proposed response is to derive a high-fidelity, low-dimensional FEM approximation using model order reduction for real-time soft-robot control.The approximation addresses the computational challenge rather than replacing the physics-based FEM model.Source: B. Constrained Optimal Control Problem · p. 3
III. REDUCED ORDER MODELING FOR SOFT ROBOTS
The paper reduces nonlinear soft-robot FEM dynamics by projecting the model onto a lower-dimensional subspace and efficiently approximating its nonlinear terms. It uses POD to construct the projection basis and reconstructs full states from reduced states.
III. REDUCED ORDER MODELING FOR SOFT ROBOTS
- Nonlinear model reduction comprises projection onto a reduced-order subspace and an efficient approximation of the model’s high-dimensional nonlinear terms.The paper considers projection-based methods, including balanced truncation and proper orthogonal decomposition (POD).Source: §III · p. 3, A. Projection-based Model Order… · p. 3
A. Projection-based Model Order Reduction
- POD defines a Galerkin projection using an orthogonal basis U, with reduced position and velocity coordinates in a subspace of dimension r.The projection applies to the high-dimensional FEM position and velocity vectors; typically r is much smaller than 3N.Source: A. Projection-based Model Order… · p. 3
- The reduced model reconstructs original states from reduced coordinates and projects the FEM dynamics onto the reduced subspace.Reference states are chosen at a static robot equilibrium, with zero reference velocity.Source: A. Projection-based Model Order… · p. 3
- The combined reduced state has dimension n = 2r, and reduced measurement and performance models are obtained by projecting the corresponding FEM models.The resulting ROM dimension can be orders of magnitude smaller than the original FEM model dimension nf.Source: A. Projection-based Model Order… · p. 3
B. Nonlinear Model Reduction
The reduced model’s nonlinear terms remain costly to evaluate because they depend on the high-dimensional state. Piecewise-affine approximations replace those terms with local linearizations, yielding a model suited to optimization-based control.
B. Nonlinear Model Reduction
- Nonlinear reduced-order terms still require high-dimensional evaluation, so piecewise-affine approximations are used to reduce that computational burden.The cited internal-force evaluation scales as O(nf).Source: B. Nonlinear Model Reduction · p. 3
- N linearization points combine local approximations into a global piecewise-affine ROM, using first-order Taylor expansions for internal forces and a zeroth-order approximation for the input matrix.The approximation is defined in the vicinity of each linearization point.Source: B. Nonlinear Model Reduction · p. 3
- The weighting matrix W selects the nearest linearization point, and for this robot W = blkdiag(0, I) because internal forces depend heavily on configuration.Source: B. Nonlinear Model Reduction · pp. 3–4
- Reduced-order matrices make the ROM easy to store and simulate, while allowing its Jacobians to be extracted for optimization-based control.Source: B. Nonlinear Model Reduction · p. 4
C. Building Piecewise Affine ROM
The piecewise-affine ROM is built through an automated offline process using high-dimensional FEM simulations. The process computes a reduced basis and adds linearization points as needed to represent the robot’s motion.
C. Building Piecewise Affine ROM
- The automated ROM-building procedure computes the reduced basis and selects linearization points from FEM simulations, entirely offline.Source: C. Building Piecewise Affine ROM · p. 4
- POD uses FEM snapshots of configuration, velocity, and acceleration to form an SVD basis, retaining the r leading modes for the reduced subspace.The rank is chosen as small as possible while preserving approximation accuracy; truncated-singular-value energy is a common heuristic.Source: C. Building Piecewise Affine ROM · p. 4
- The ROM is built incrementally by adding a linearization point when its predicted reduced-order state diverges too far from the FEM simulation.Source: C. Building Piecewise Affine ROM · p. 4
- To cover possible robot motions, the FEM simulation uses an open-loop sequence sampled from admissible actuations with Latin hypercube sampling.Source: C. Building Piecewise Affine ROM · p. 4
IV. REDUCED ORDER OPTIMAL CONTROL
The reduced-order optimal-control scheme combines an OCP, a reduced-order state estimator, and a feedback control law. The OCP optimizes a trajectory, the estimator uses measurements to estimate the current state, and feedback tracks the optimized trajectory.
IV. REDUCED ORDER OPTIMAL CONTROL
- The control scheme has three components: a reduced-order OCP, a reduced-order state estimator, and a reduced-order feedback control law.Source: §IV · p. 4
- The OCP optimizes a reduced-order trajectory, while the estimator incorporates measurements to estimate the robot’s current reduced-order state.Source: §IV · p. 4
- The feedback law uses the state estimate to drive the robot to track the optimized trajectory.Source: §IV · p. 4
A. Reduced Order Optimal Control Problem
A finite-horizon reduced-order approximation of the original OCP determines a constrained trajectory from the current state. The problem is solved repeatedly in receding-horizon fashion using sequential convex programming.
A. Reduced Order Optimal Control Problem
- The finite-horizon reduced-order OCP optimizes a trajectory subject to the reduced dynamics, initial-state, input, and performance-variable constraints.Source: A. Reduced Order Optimal Control Problem · p. 4
- The problem is solved in receding-horizon fashion, with the initial state reset to the previous solution’s optimal state at each update.Source: A. Reduced Order Optimal Control Problem · p. 4
- Sequential convex programming solves the nonconvex OCP through successive quadratic-program approximations until convergence.Source: A. Reduced Order Optimal Control Problem · p. 5
B. Reduced Order Controller and State Estimator
The controller combines a reduced-order optimal trajectory with state estimation and feedback to make the robot track that trajectory. Gain scheduling supplies the feedback gains, while the solve schedule and trajectory interpolation support efficient operation.
B. Reduced Order Controller and State Estimator
- The reduced-order OCP optimizes a trajectory, the estimator incorporates measurements to estimate the current reduced-order state, and feedback drives the robot to track that trajectory.The estimator uses an extended Kalman filter gain based on the reduced-order dynamics.Source: B. Reduced Order Controller and State… · p. 5
- Feedback gains are selected by gain scheduling from discrete-time LQR gains computed at linearization points.The gains can be computed a priori using the reduced-order dynamics matrices and cost matrices.Source: B. Reduced Order Controller and State… · p. 5
- The reduced-order OCP is initialized from the previous solution and solved every Tr < T time steps, allowing the next solve to start as soon as the previous one finishes.With sampling time h, this provides hTr seconds to solve the next OCP and can help warm-start sequential convex programming.Source: B. Reduced Order Controller and State… · p. 5
- Interpolating the optimized trajectory lets the estimator and control law operate at a higher frequency than the OCP discretization.The passage notes that the OCP trajectory is discretized with sampling time h, while the estimator and control law can run at a higher frequency.Source: B. Reduced Order Controller and State… · p. 5
V. SIMULATION RESULTS
In simulation, the proposed controller handles constrained trajectory tracking for the Diamond soft robot using a reduced-order model of a 9768-dimensional FEM. It performs comparably to the Koopman controller in tracking and constraint satisfaction, while ROMPC performs poorly; the paper also reports real-time capability.
A. Controllers
- The comparisons use a linearized-FEM ROMPC controller and a Koopman model built from the same FEM simulation data.The Koopman model uses delay d = 1, monomials of maximum degree 2, and a lifted state dimension of 66.Source: §V · p. 5, A. Controllers · p. 5
V. SIMULATION RESULTS
- The simulation uses a 9768-dimensional FEM of the cable-actuated Diamond robot to track the end-effector position subject to position constraints.The performance variable is end-effector position; the simulation includes additive Gaussian measurement noise.Source: §V · p. 5
A. Controllers
- The proposed controller uses a piecewise-affine ROM with dimension 42 and 642 linearization points, with the OCP discretized at 0.05 seconds and feedback and estimation at 0.01 seconds.The ROM is built using acceleration snapshots and a 21-dimensional reduced-order subspace.Source: A. Controllers · p. 5
V. SIMULATION RESULTS
- The proposed controller performs comparably to Koopman in tracking and constraint satisfaction, while ROMPC performs poorly; Fig. 2 illustrates constrained trajectory tracking.The desired trajectory is partially infeasible because of the constraints.Source: Figure 3 · p. 6, Figure 2 · p. 6
A. Controllers
- Table I compares controllers by mean square error and QP solve time, and is reported to show state-of-the-art performance and real-time capability.The reported solve time for the proposed method sums QP solve times across the sequential convex programming algorithm.Source: Table I · p. 6
B. Results
In simulation, the proposed approach and the Koopman controller tracked well and generally satisfied constraints, while linearized ROMPC tracked poorly and severely violated them. The proposed scheme was reported as real-time capable and offered advantages over Koopman control; hardware validation and further theoretical analysis remain future work.
B. Results
- The proposed approach and Koopman controller tracked well and generally satisfied constraints, unlike linearized-ROM ROMPC, which tracked poorly and severely violated them.The comparison is shown for the Diamond FEM model in Figures 2 and 3.Source: B. Results · p. 6
- The proposed FEM model-based control scheme is real-time capable, with QP solve time reported as the main computational measure.The proposed sequential convex programming approach may require multiple QPs, unlike ROMPC and Koopman, which solve one QP at a time.Source: B. Results · p. 6
- The proposed approach offers similar performance to Koopman control in this simulation, while its FEM model captures the robot’s entire state and supports different control problems.Unlike Koopman control, it is not limited to outputs that are a subset of measured variables, and its optimized trajectory can be subsampled for higher operating frequencies.Source: B. Results · p. 6
VI. CONCLUSION
- The paper proposes constrained optimal control based on high-fidelity nonlinear FEM models, using a reduced-order piecewise affine approximation for output-feedback setpoint and trajectory tracking.The approach is evaluated in simulation against data-driven and linear FEM-based constrained control methods.Source: §VI · p. 6
- Hardware experiments, broader robot types and contact scenarios, and theoretical analysis of performance, stability, and constraint-satisfaction guarantees remain future work.The authors also identify advanced nonlinear model reduction and data-driven augmentation as future directions.Source: §VI · p. 6