Origami structures exhibit complex dynamical behaviors during reconfiguration due to coupled panel deformation, hinge rotations, and folding-induced changes in stiffness and compliance. This work addresses origami reconfiguration control by developing an integrated control and model identification framework. First, an extended first-order generalized pseudo-Bayesian (E-GPB1) filter is developed to jointly estimate continuous states and identify discrete dynamic modes. Second, a probability-weighted state-dependent Riccati equation (SDRE) controller is designed to regulate the system under input constraints. The combined E-GPB1/SDRE framework enables robust control despite switching dynamics and physical parameter variations. In addition, simulation on a Kresling origami pattern demonstrates accurate mode identification, reliable trajectory regulation, and stable convergence to an otherwise unstable configuration. These results establish the proposed approach as an effective strategy for estimation and control of reconfigurable origami-inspired structures.

The reconfiguration of origami during the folding and unfolding process is governed through a sequence of panel deformations and hinge orientations. To develop an effective model for representing the reconfiguration process, this paper introduces planar straight-line graphs and a novel consensus protocol for reaching the target origami configuration. The convergence and stability properties of the proposed consensus protocol are subsequently analyzed. Furthermore, to account for aggregate material and structural effects in the proposed consensus-based reconfiguration model, effective parameters embedded in the consensus protocol are identified from trajectory data using a fitting algorithm. Lastly, the effectiveness of the proposed modeling approach is shown using simulations of the two-panel structure and the Kresling origami pattern reconfiguration process.

Learning-based Koopman models provide a structured framework for modeling nonlinear controlled systems. However, the finite-time prediction reliability of learned Koopman models under noisy measurements has not been sufficiently investigated. This paper studies multi-step prediction error propagation for neural bilinear Koopman models of continuous-time control-affine systems under noisy state and input measurements. We first derive certified properties of the neural lifting function by exploiting its structure, and then analyze the local model mismatch by decomposing it into lifting inconsistency and discretization mismatch. Based on these results, we establish an explicit finite-horizon prediction error bound that captures the joint effects of noisy measurements and local mismatch, and further reveals a trade-off on finite-time prediction error with respect to the sampling period. Numerical experiments are presented to support the theoretical findings.

We investigate the controllability of an origami system composed of Miura-ori cells. Extensive research has been conducted on the folding architecture, kinematic behavior, and actuation techniques of origami structures. However, understanding their transient dynamics and constructing control models remains a formidable task, primarily due to their innate flexibility and compliance. In light of this challenge, we discretize the origami system into a network composed of interconnected particle masses alongside bar and hinge elements. This yields a state-space representation of the system’s dynamics, facilitating the analysis of the system’s controllability properties. Informed by this computational framework, we explore the controllability Gramian-based method to find the most efficient crease lines for the deployment of single and tessellated Miura-ori cells using servo-motor actuators. We demonstrate that the deployment efficiency guided by this theoretical method shows good agreement with the empirical results derived from the control effort in deploying the origami prototypes. This investigation paves the way toward the efficient design and operation of complex actuation systems for origami-based deployable structures.

The Koopman operator theory provides a global linearization framework for general nonlinear dynamics, offering significant advantages for system analysis and control. However, practical applications typically involve approximating the infinite-dimensional Koopman operator in a lifted space spanned by a finite set of observable functions. The accuracy of this approximation is the key to effective Koopman operator-based analysis and control methods, generally improving as the dimension of the observables increases. Nonetheless, this increase in dimensionality significantly escalates both storage requirements and computational complexity, particularly for high-dimensional systems, thereby limiting the applicability of these methods in real-world problems. In this paper, we address this problem by reformulating the Koopman operator in tensor format to break the curse of dimensionality associated with its approximation through tensor decomposition techniques. This effective reduction in complexity enables the selection of high-dimensional observable functions and the handling of large-scale datasets, which leads to a precise linear prediction model utilizing the tensor-based Koopman operator. Furthermore, we propose an optimal control framework with the tensor-based Koopman operator, which adeptly addresses the nonlinear dynamics and constraints by linear reformulation in the lifted space and significantly reduces the computational complexity through separated representation of the tensor structure.

We examine an effective dynamic reconfiguration model for triangulated origami structures using planar straight-line graphs. In this setup, the origami panels are first represented as edges and vertices in an undirected triangular graph. A triangulated consensus protocol for the corresponding origami formation control problem is then developed, where state of the nodes reach agreement on target configuration while ensuring that the reconfiguration process is realizable by each panelthe setup is then extended to the entire triangulated origami structure. The proposed approach provides a general graph-theoretic framework for expressing the geometric evolution of diverse origami patterns during the folding/unfolding process.

This paper explores the use of separated representation to improve the scalability of the Koopman operator for both continuous- and discrete-time dynamical systems. We propose an approximation method for the Koopman operator using tensor-product bases. The approach comprises two main steps. In the first step, we utilize the Galerkin method to derive a tensor-based approximate Koopman operator, which subsequently aids in the development of a decomposition algorithm for achieving a separable operator. The second step finds the separated representation via an efficient algorithm to decompose the operator by exploiting its low-rank tensor structure. Numerical experiments demonstrate that the proposed method provides high accuracy with significantly reduced computational effort. The results highlight the potential of separated representation for handling high-dimensional systems and advancing the applicability of Koopman operator-based techniques.

Mixed-integer optimal control problems (MIOCPs) frequently arise in the domain of optimal control problems (OCPs) when decisions including integer variables are involved. However, existing state-of-the-art approaches for solving MIOCPs are often plagued by drawbacks such as high computational costs, low precision, and compromised optimality. In this study, we propose a novel multiphase scheme coupled with an iterative second-order cone programming (SOCP) algorithm to efficiently and effectively address these challenges in MIOCPs. In the first phase, we relax the discrete decision constraints and account for the terminal state constraints and certain path constraints by introducing them as penalty terms in the objective function. After formulating the problem as a quadratically constrained quadratic programming (QCQP) problem, we propose the iterative SOCP algorithm to solve general QCQPs. In the second phase, we reintroduce the discrete decision constraints to generate the final solution. We substantiate the efficacy of our proposed multiphase scheme and iterative SOCP algorithm through successful application to two practical MIOCPs in planetary exploration missions.

The paper investigates close range rendezvous, nearby orbital transfer, and collision avoidance maneuvers for a chaser spacecraft moving relative to a target in a 9:2 L2 Near- Rectilinear Halo Orbit (NRHO). A feature learning optimal control method (L-OCM) is proposed to solve optimal control problems corresponding to the maneuvers to minimize fuel consumption. First, the optimal control problems for each operation are formulated using nonlinear relative dynamics for the circular restricted three-body problem (CR3BP), where the controller is a two-finite-burn maneuver. Then, a nonlinear programming (NLP) solver is used to solve the problems offline for a range of initial conditions. The NLP solutions are also compared to solutions from existing methods, i.e., differential corrections. The set of solutions is used to generate datasets with initial conditions as inputs and the identified critical features as outputs. These features are extracted from the optimal solution and are used to reconstruct the entire optimal solution. A deep neural network is trained off-line to map the complex, nonlinear relationship between the inputs and outputs, and then implemented to find on-line solutions to any initial condition. The L-OCM method provides fuel-optimal, real-time solutions that can be implemented by a spacecraft performing operations in cislunar space.

Tensor structures are fundamental in addressing the intricate challenges posed by high-dimensional data across a spectrum of scientific and computational domains. Within this context, low-rank tensor approximation plays a pivotal role in enhancing data processing efficiency. This paper develops a novel adaptive low-rank tensor approximation method by introducing mixed-integer representations to identify an appropriate low-rank approximation for high-dimensional tensors. The approach takes into consideration both tensor rank determination and approximation accuracy, leveraging binary variables to represent tensor ranks that will be optimized as unknowns with the tensor arrays. By integrating the alternating least squares technique with the truncation method, the integrated algorithm effectively achieves a proper low-rank tensor with high approximation accuracy. To substantiate its efficacy and efficiency in solving tensor approximation problems, the paper provides extensive simulation results and analysis.

Sampling-based model predictive control (MPC) has found significant success in optimal control problems with non-smooth system dynamics and cost function. Many machine learning-based works proposed to improve MPC by a) learning or fine-tuning the dynamics/ cost function, or b) learning to optimize for the update of the MPC controllers. For the latter, imitation learning-based optimizers are trained to update the MPC controller by mimicking the expert demonstrations, which, however, are expensive or even unavailable. More significantly, many sequential decision-making problems are in non-stationary environments, requiring that an optimizer should be adaptable and generalizable to update the MPC controller for solving different tasks. To address those issues, we propose to learn an optimizer based on meta-reinforcement learning (RL) to update the controllers. This optimizer does not need expert demonstration and can enable fast adaptation (e.g., few-shots) when it is deployed in unseen control tasks. Experimental results validate the effectiveness of the learned optimizer regarding fast adaptation.

Origami is known as a traditional art of paper folding. It has attracted extensive attention due to its self-folding mechanism, shape-morphing capability, and deployable structures. This article develops network-based methods for designing and controlling a three-dimensional (3D) triangulated origami tessellation to approximate multiple surfaces. The desired surfaces are represented by sets of discrete nodes and the origami tessellation to be designed is composed of triangles. Then, the tessellation design problem is formulated as an optimization problem of minimizing the distance between the origami triangle vertices and the discrete nodes subject to developability and rigid-foldability constraints. Solving the resulting optimization problem leads to an origami tessellation with folding states associated with each target surface. To achieve transformation between different shapes, we first leverage graph rigidity theory to define every 3D origami shape uniquely up to translations and rotations. Next, in order to minimize the control efforts, the shape transformation control problem is formulated as an optimal control problem subject to the derived rigidity conditions, whose feasibility is guaranteed by transformability of the origami and controllability of dynamic vertices. Finally, simulation examples for surface approximation are provided to verify the effectiveness of the network-based design and control methods.

The abort mission refers to the mission where the landing vehicle needs to terminate the landing mission when an anomaly happens and be safely guided to the desired orbit. This paper focuses on solving the time-optimal abort guidance (TOAG) problem in real-time via the feature-based learning method. First, according to the optimal control theory, the features are identified to represent the optimal solutions of TOAG using a few parameters. After that, a sufficiently large data set of time-optimal abort trajectories is generated offline by solving the TOAG problems with different initial conditions. Then, the features are extracted for all generated cases. To find the implicit relationships between the initial conditions and identified features, neural networks are constructed to map the relationships based on the generated data set. Finally, experimental flight tests are conducted to demonstrate the onboard computation capability and effectiveness of the proposed method.

This article investigates the six-degree-of-freedom (6-DoF) entry trajectory optimization problem in a Human-Mars entry, powered descent, and landing mission. During the entry phase, aerodynamic forces are employed to decelerate the vehicle. Instead of being treated as a point mass, both translational and rotational motions of the entry vehicle are considered. Specifically, the 6-DoF rigid body motion of the entry vehicle is modeled using the unit dual quaternion representations to avoid highly nonlinear terms in the flight dynamics expression originally based on the flight-path coordinates. Then, the entry trajectory optimization problem is to minimize the terminal speed subject to dynamical, operational, and mission constraints modeled by the new representation scheme. By applying the discretization technique and polynomial approximation, the entry trajectory optimization problem is reformulated as a nonconvex quadratically constrained quadratic programming problem, which is solved via a hybrid alternating direction method of multipliers (ADMM). The accuracy of the dual-quaternion-based model and the computational efficiency of the hybrid ADMM are validated via numerical simulations.

We propose an integral action nonlinear model predictive controller (NMPC) for trajectory tracking of an articulated vehicle with an uncertain hitching offset. The controller is intended for complex parking maneuvers including forward and backward movement with tight specifications on the lateral positional tracking error of the trailer. In order to assess performance with uncertain hitching offsets, disturbances, and sensor noise, we conduct extensive hardware-in-the-loop simulations using a dSPACE Scalexio unit. With high-grade sensing, we demonstrate that the closed-loop control system achieves a lateral tracking error of < 3 [cm] in expectation, and an absolute terminal error of < 15 [cm] with high probability p > 0.97. The proposed integral action is shown to be essential in achieving this performance, and the efficacy of the proposed NMPC is evaluated by comparison to alternative MPCs.

The abort mission refers to the mission where the landing vehicle needs to terminate the landing mission when an anomaly happens and be safely guided to the desired orbit. This paper focuses on solving the time-optimal abort guidance (TOAG) problem in real-time via the feature-based learning method. First, according to the optimal control theory, the features are identified to represent the optimal solutions of TOAG using a few parameters. After that, a sufficiently large dataset of time-optimal abort trajectories is generated offline by solving the TOAG problems with different initial conditions. Then the features are extracted for all generated cases. To find the implicit relationships between the initial conditions and identified features, neural networks are constructed to map the relationships based on the generated dataset. Simulation examples are provided to verify effectiveness and efficiency of the proposed method.

This paper investigates the three-dimensional (3D) multi-point landing guidance (MLG) problem with hazard avoidance by developing a mixed-input learning-based method to achieve precise and fuel-efficient planetary landing in future Mars missions. Specifically, we aim to find a safe, fuel-efficient landing point and generate a fuel-optimal trajectory simultaneously in real-time. First, by introducing binary variables, the MLG problem is formulated as an optimal control problem with quadratic constraints. Then, by formulating the Hamiltonian function, the necessary conditions of optimality for the MLG problem are obtained, where the critical parameters are identified to represent the complete optimal solution. After that, to find the implicit relationship between the problem inputs and these critical parameters, a hybrid deep neural network is constructed. To be specific, on the one hand, the contour maps of the landing area, which are image inputs, are adopted to reflect the features of the pre-defined landing area. On the other hand, the velocity and position vectors, which belong to numeric inputs, are adopted to reflect the features of the initial state of the powered descent phase. Finally, with the constructed hybrid deep neural network well trained, the mixed-input learning-based optimal control solution can be computed onboard. To verify the effectiveness and accuracy of the proposed method, the simulation results of 3D MLG problems are presented and analyzed.

This paper proposes a multi-stage optimization framework based on iterative second-order cone programming (SOCP) to solve the three-dimensional (3D) multi-point landing guidance (MLG) problem with hazard avoidance. The approach is used to generate the offline optimal trajectories for database construction in Part II of this paper. The 3D MLG problem with hazard avoidance is to choose a safe landing point while finding an optimal path to the selected landing point with minimum fuel consumption. First, by introducing binary variables associated with quadratic constraints, the launch vehicle is enforced to land at one of the pre-specified landing sites. Next, to solve the formulated problem, a multi-stage optimization framework, combined with the relaxation technique, is introduced. To be specific, in the first stage, all the binary constraints are relaxed as upper and lower bounded continuous variables, and the reformulated problem in the first stage is solved via an iterative second-order cone programming algorithm. The solution from the first stage is used as an initial guess for the second stage, where the binary constraints of the original problem are reconsidered to obtain the final solution. Finally, the effectiveness of the proposed method is verified via numerical simulations.

This paper develops a novel approach to design, actuate, and manufacture a space debris collector based on the conical Kresling origami pattern. The deployable nature of origami structures and the radial closability of the conical Kresling pattern are leveraged to form an enclosure volume for collecting space debris at different sizes. We first introduce the geometric, volume, and energy models of the conical Kresling pattern. Based on these models, the debris collector design problem is formulated as a parameter optimization problem to minimize the actuation energy for the folding process while satisfying the minimum volume constraint and geometric/functional constraints. To automatically capture debris in space, an actuation system is designed, which is compatible with space environments. Moreover, the multi-material three-dimensional printing technology is applied to build the designed debris collector, which makes it feasible for manufacturing the product in orbit. The proposed design, actuation, and manufacturing approaches are verified with experimental tests using a designed prototype.