This paper addresses optimal control problems in multibody systems using a modernized energy-preserving direct transcription methodology. Recent advancements in numerical optimization, nonlinear programming, and energy-coherent discretization methods form the backbone of this study. Our approach ensures numerical robustness, accommodates systems with complex topologies, and demonstrates second-order accuracy in solution fields, including positions, velocities, and controls. Case studies on robotic manipulators and particle transfer validate the efficacy of the proposed framework, offering insights into its applications in aerospace, robotics, and biomechanics.
The optimal control of multibody systems has emerged as a critical research area due to its extensive applications across robotics, aerospace engineering, automotive systems, and biomechanics. These systems, composed of interconnected rigid or flexible bodies, exhibit intricate dynamic behaviors governed by nonlinear equations of motion [1]. The inherent complexity of multibody systems arises from their reliance on differential-algebraic equations (DAEs), which model the constraints and interactions between components. These DAEs significantly increase the computational and analytical challenges associated with analyzing and controlling such systems [2].
Traditional methods for optimal control, including indirect and direct approaches, have laid the groundwork for trajectory optimization and system control in multibody dynamics. Indirect methods focus on deriving and solving the necessary optimality conditions, such as the Euler-Lagrange equations, to determine control inputs. However, these methods are often mathematically intricate and highly sensitive to initial guesses, limiting their applicability to real-world problems [3]. Direct methods, in contrast, discretize the system dynamics into finite-dimensional representations, allowing the use of numerical optimization techniques. While direct methods are generally more robust and versatile, they frequently encounter difficulties in maintaining critical physical invariants, such as energy conservation. This issue becomes especially pronounced in nonlinear and highly dynamic multibody systems, where small numerical errors can accumulate and lead to significant deviations from the true solution [4].
To address these limitations, energy-preserving schemes have gained attention as powerful tools for ensuring stability and accuracy in the numerical simulation of multibody systems. These schemes aim to preserve the total mechanical energy of the system during the discretization and optimization process, effectively mitigating numerical drift and enhancing the reliability of the solutions [5]. By maintaining energy invariants, these methods also ensure that the simulated dynamics remain faithful to the physical behavior of the system. Despite these advantages, classical energy-preserving approaches are not without drawbacks. Their scalability to large-scale systems, computational cost, and ability to handle the increasing complexity of modern multibody systems often leave room for improvement [6].
Recent advancements in computational methodologies and tools provide a promising pathway to overcome these challenges. Adaptive meshing techniques have emerged as a powerful solution for optimizing the allocation of computational resources, dynamically refining resolution in regions of interest while reducing overhead in less critical areas. This capability is particularly valuable for multibody systems, where certain parts of the system may exhibit high-frequency dynamics requiring finer resolution. Additionally, machine learning techniques, such as neural networks and reinforcement learning, have demonstrated significant potential in improving the efficiency of numerical optimization. By providing intelligent initial guesses for control inputs or adaptively guiding the optimization process, machine learning can drastically reduce computational effort while enhancing convergence rates. High-performance computing (HPC) platforms, including GPU-accelerated solvers, have further expanded the feasibility of real-time control and large-scale simulations, enabling the practical application of advanced methods to complex multibody systems [7].
This paper introduces a unified framework that integrates these modern computational tools and methodologies with the foundational principles of energy-preserving schemes to address the challenges of optimal control in multibody systems. The proposed framework is built upon three core components:
Refined Energy-Preserving Transcription: A modernized approach to discretizing DAEs that enhances the stability and accuracy of solutions while preserving critical energy invariants. This ensures that the numerical methods used remain faithful to the physical laws governing multibody dynamics.
Machine Learning-Driven Optimization: The use of machine learning models, particularly neural networks, to provide intelligent initial guesses for optimization problems. This significantly accelerates the convergence of numerical solvers and reduces the computational cost of solving large-scale problems.
Adaptive Meshing and High-Precision Integration: The incorporation of dynamic meshing strategies that allocate computational resources based on the complexity of the local dynamics. Coupled with high-precision integration methods, this component enhances both the efficiency and accuracy of the proposed framework.
The efficacy of this framework is evaluated through a series of detailed numerical experiments involving diverse multibody systems, including robotic manipulators, particle trajectory problems, and hybrid configurations. The results showcase substantial improvements in solution accuracy, convergence rates, and computational performance compared to traditional methods [8]. For instance, in scenarios involving complex dynamics and stringent constraints, the framework demonstrates the ability to efficiently handle nonlinearities while preserving system invariants. This is particularly evident in applications such as robotic trajectory optimization, where energy-preserving schemes ensure stable and reliable solutions, and in particle trajectory problems, where adaptive meshing optimizes computational effort without compromising accuracy.
By combining the rigor of classical energy-preserving principles with the capabilities of modern computational tools, this research provides a robust and scalable pathway for addressing the challenges of multibody dynamics and control. The proposed framework bridges the gap between traditional approaches and the demands of contemporary applications, offering a practical solution for solving optimal control problems in increasingly complex systems. The integration of machine learning and HPC technologies into this framework further underscores its adaptability and relevance in the context of emerging engineering challenges.
In conclusion, the findings of this study highlight the transformative potential of integrating energy-preserving schemes with cutting-edge computational methodologies. The proposed framework not only advances the state of the art in multibody system dynamics and control but also lays the foundation for next-generation techniques capable of addressing the growing complexity of modern engineering systems. By providing a robust, accurate, and efficient solution for optimal control, this research contributes actionable insights for researchers and practitioners in engineering and applied mathematics, paving the way for innovative solutions to the challenges posed by multibody systems in diverse domains.
Optimal control theory aims to determine the optimal state trajectories and control inputs that minimize or maximize a predefined performance criterion while adhering to the system dynamics and constraints. This framework is widely used for solving complex problems in engineering, robotics, and other domains where system behavior must be optimized under physical constraints.
For a multibody system, the optimal control problem (OCP) can be mathematically described as:
The system’s dynamics are expressed as a set of ordinary differential equations (ODEs): \[\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{u}(t), t), \tag{1}\] where:
\(\mathbf{x}(t) \in \mathbb{R}^n\) represents the state variables (e.g., positions, velocities),
\(\mathbf{u}(t) \in \mathbb{R}^m\) denotes the control inputs (e.g., forces, torques),
\(\mathbf{f}(\mathbf{x}, \mathbf{u}, t)\) encapsulates the nonlinear system dynamics.
The objective is to minimize the performance index \(J\), which is typically expressed as: \[J = \Phi(\mathbf{x}(t_f)) + \int_{t_0}^{t_f} L(\mathbf{x}(t), \mathbf{u}(t), t) \, dt, \tag{2}\] where:
\(\Phi(\mathbf{x}(t_f))\) is the terminal cost at the final time \(t_f\),
\(L(\mathbf{x}, \mathbf{u}, t)\) is the running cost function.
The problem is subject to:
Boundary Conditions: \[\mathbf{x}(t_0) = \mathbf{x}_0, \quad \mathbf{x}(t_f) = \mathbf{x}_f, \tag{3}\] where \(\mathbf{x}_0\) and \(\mathbf{x}_f\) are the specified initial and final states.
Path Constraints: \[\mathbf{g}(\mathbf{x}(t), \mathbf{u}(t), t) \leq 0, \tag{4}\] where \(\mathbf{g}\) defines inequality constraints.
Bounds on States and Controls: \[\mathbf{x}_{\text{min}} \leq \mathbf{x}(t) \leq \mathbf{x}_{\text{max}}, \quad \mathbf{u}_{\text{min}} \leq \mathbf{u}(t) \leq \mathbf{u}_{\text{max}}. \tag{5}\]
Given the infinite-dimensional nature of the OCP, solving it directly is computationally infeasible. Direct transcription converts the continuous problem into a finite-dimensional nonlinear programming (NLP) problem. This involves:
The time interval \([t_0, t_f]\) is divided into \(N\) segments with time steps \(\Delta t\). States and controls are approximated at discrete points: \[\mathbf{x}(t_k) \approx \mathbf{x}_k, \quad \mathbf{u}(t_k) \approx \mathbf{u}_k, \quad k = 0, 1, \dots, N. \tag{6}\]
The system dynamics are enforced at discrete points using collocation or quadrature methods, ensuring numerical accuracy. For instance, collocation satisfies the dynamics at intermediate points, while trapezoidal or Simpson’s rule approximates the integral in the cost function.
The resulting NLP problem is expressed as: \[\min_{\mathbf{x}_k, \mathbf{u}_k} J = \Phi(\mathbf{x}_N) + \sum_{k=0}^{N-1} L(\mathbf{x}_k, \mathbf{u}_k, t_k) \Delta t, \tag{7}\] subject to: \[\mathbf{x}_{k+1} = \mathbf{x}_k + \Delta t \cdot \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k), \quad k = 0, 1, \dots, N-1, \tag{8}\] along with boundary and path constraints.
To ensure the numerical solution aligns with the physical properties of the multibody system, energy-preserving schemes are often employed. These schemes guarantee that the discretized system respects key invariants such as energy conservation or dissipation.
Energy-preserving schemes are derived from discrete Lagrangian formulations, where the action integral: \[S = \int_{t_0}^{t_f} \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, t) \, dt, \tag{9}\] is approximated using discrete states \(\mathbf{q}_k\) and velocities \(\dot{\mathbf{q}}_k\).
These schemes maintain nonlinear stability, ensuring robust numerical integration even for stiff or chaotic dynamics. For instance, symplectic integrators preserve the Hamiltonian structure of conservative systems.
To illustrate, consider the optimal control of a double pendulum. The equations of motion are: \[\mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}} + \mathbf{G}(\mathbf{q}) = \mathbf{u}, \tag{10}\] where:
\(\mathbf{q} = [\theta_1, \theta_2]^T\) are the angular positions,
\(\mathbf{u} = [\tau_1, \tau_2]^T\) are the control torques.
The OCP aims to minimize a cost function: \[J = \int_{t_0}^{t_f} \left( \|\mathbf{q} – \mathbf{q}_{\text{target}}\|^2 + \|\mathbf{u}\|^2 \right) \, dt, \tag{11}\] subject to dynamics and torque limits.
By applying direct transcription, the problem is transformed into an NLP and solved using solvers like IPOPT or SNOPT, achieving efficient computation while preserving system stability.
Optimal control theory serves as a robust mathematical framework for determining the optimal state trajectories and control inputs that minimize or maximize a predefined performance criterion while adhering to the system dynamics and constraints [9]. This approach is pivotal in numerous applications, including engineering, robotics, aerospace, and other domains, where system behavior must be optimized under a set of physical and operational constraints. In this formulation, the primary goal is to achieve a balance between computational efficiency and the physical realism of the solutions [10].
For a multibody system, the optimal control problem (OCP) is described mathematically by a set of state equations, a performance index, and associated constraints. The state dynamics, which govern the evolution of the system over time, are expressed as a set of ordinary differential equations (ODEs). These equations take the form \[\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{u}(t), t), \tag{12}\] where \(\mathbf{x}(t)\) represents the state variables such as positions and velocities, \(\mathbf{u}(t)\) denotes the control inputs like forces or torques, and \(\mathbf{f}(\mathbf{x}, \mathbf{u}, t)\) captures the nonlinear system dynamics. The performance index, which quantifies the cost or objective to be minimized, is typically expressed as \[J = \Phi(\mathbf{x}(t_f)) + \int_{t_0}^{t_f} L(\mathbf{x}(t), \mathbf{u}(t), t) \, dt. \tag{13}\] Here, \(\Phi(\mathbf{x}(t_f))\) represents the terminal cost at the final time \(t_f\), while \(L(\mathbf{x}, \mathbf{u}, t)\) is the running cost function that evaluates the system’s performance over time.
The optimization process is constrained by boundary conditions, path constraints, and bounds on states and controls. The boundary conditions specify the initial and final states of the system, such as \[\mathbf{x}(t_0) = \mathbf{x}_0, \quad \mathbf{x}(t_f) = \mathbf{x}_f. \tag{14}\] Path constraints, expressed as \[\mathbf{g}(\mathbf{x}(t), \mathbf{u}(t), t) \leq 0, \tag{15}\] ensure that the state and control variables remain within permissible limits during the trajectory. Additional constraints include bounds on the state and control variables, defined as \[\mathbf{x}_{\text{min}} \leq \mathbf{x}(t) \leq \mathbf{x}_{\text{max}}, \quad \mathbf{u}_{\text{min}} \leq \mathbf{u}(t) \leq \mathbf{u}_{\text{max}}, \tag{16}\] which reflect physical and operational limitations.
Due to the infinite-dimensional nature of OCPs, solving them directly is computationally infeasible. A practical approach to addressing this challenge is direct transcription, which transforms the continuous problem into a finite-dimensional nonlinear programming (NLP) problem. This transformation begins by discretizing the state and control variables over the time interval \([t_0, t_f]\). The interval is divided into \(N\) segments, with each segment approximating the states and controls at discrete points. For example, the state and control variables at time \(t_k\) are represented as \[\mathbf{x}(t_k) \approx \mathbf{x}_k, \quad \mathbf{u}(t_k) \approx \mathbf{u}_k, \quad k = 0, 1, \dots, N. \tag{17}\] Numerical methods such as collocation or quadrature are then employed to enforce the system dynamics and approximate the performance index. Collocation methods ensure the dynamics are satisfied at intermediate points, while numerical integration schemes like trapezoidal or Simpson’s rule approximate the cost function.
The NLP formulation of the OCP can then be expressed as minimizing the discretized cost function \[J = \Phi(\mathbf{x}_N) + \sum_{k=0}^{N-1} L(\mathbf{x}_k, \mathbf{u}_k, t_k) \Delta t, \tag{18}\] subject to the discretized system dynamics \[\mathbf{x}_{k+1} = \mathbf{x}_k + \Delta t \cdot \mathbf{f}(\mathbf{x}_k, \mathbf{u}_k, t_k), \quad k = 0, 1, \dots, N-1, \tag{19}\] and associated constraints. Solving this NLP provides a computationally efficient method for obtaining optimal trajectories and controls.
To ensure that the numerical solution is physically accurate and stable, energy-preserving schemes are often incorporated into the numerical methods. These schemes are particularly valuable for multibody systems, where energy conservation is critical for maintaining physical consistency. Energy-preserving schemes are derived from discrete Lagrangian mechanics, which approximate the action integral \[S = \int_{t_0}^{t_f} \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, t) \, dt, \tag{20}\] using discrete states \(\mathbf{q}_k\) and velocities \(\dot{\mathbf{q}}_k\). By preserving key invariants such as energy conservation or dissipation, these schemes guarantee nonlinear stability and robust numerical integration, even in the presence of stiff or chaotic dynamics.
An illustrative example of optimal control in multibody systems is the double pendulum. The equations of motion for this system are expressed as \[\mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}} + \mathbf{G}(\mathbf{q}) = \mathbf{u}, \tag{21}\] where \(\mathbf{q} = [\theta_1, \theta_2]^T\) represents the angular positions and \(\mathbf{u} = [\tau_1, \tau_2]^T\) are the control torques. The objective of the OCP is to minimize a cost function such as \[J = \int_{t_0}^{t_f} \left( \|\mathbf{q} – \mathbf{q}_{\text{target}}\|^2 + \|\mathbf{u}\|^2 \right) \, dt, \tag{22}\] subject to the system dynamics and torque limits. Direct transcription enables the transformation of this problem into an NLP, which can then be solved efficiently using modern optimization solvers.
In conclusion, the optimal control of multibody systems combines mathematical rigor and numerical efficiency to address complex, high-dimensional problems. By leveraging direct transcription and energy-preserving schemes, the framework ensures accurate, physically consistent solutions, making it indispensable in applications ranging from robotic manipulation to aerospace trajectory optimization.
The numerical implementation of optimal control problems for multibody systems involves discretization of the system dynamics and efficient optimization techniques to solve the resulting nonlinear programming (NLP) problem [11]. This section describes the key steps involved, focusing on the discretization of the dynamics and the optimization methods employed.
To solve the optimal control problem numerically, the system dynamics are discretized using energy-coherent finite-difference methods. These methods are specifically designed to ensure that the discrete equations maintain consistency with the energy principles of the continuous system, thereby preserving key invariants such as energy conservation. The time interval \([t_0, t_f]\) is divided into \(N\) time steps, with each step denoted by a time increment \(\Delta t\). At each time step \(k\), the discretized equations can be expressed as: \[\mathbf{M} \mathbf{v}_{k+1} = \mathbf{M} \mathbf{v}_k + \Delta t \, \mathbf{F}_k + \Delta t \, \mathbf{\Lambda}_k, \tag{23}\] where:
\(\mathbf{M}\) is the mass matrix of the multibody system,
\(\mathbf{v}_k\) and \(\mathbf{v}_{k+1}\) are the velocities at the current and next time steps, respectively,
\(\mathbf{F}_k\) represents the external forces acting on the system, and
\(\mathbf{\Lambda}_k\) denotes the Lagrange multipliers that enforce holonomic constraints in the system.
The position updates are computed using the velocities and are typically approximated as: \[\mathbf{q}_{k+1} = \mathbf{q}_k + \Delta t \, \mathbf{v}_k, \tag{24}\] where \(\mathbf{q}_k\) and \(\mathbf{q}_{k+1}\) represent the generalized coordinates of the system at consecutive time steps. These discretized equations form the basis for enforcing the system dynamics and constraints during the optimization process.
To ensure numerical stability and accuracy, higher-order integration schemes such as implicit Euler or symplectic integrators can be employed. These methods are particularly useful for stiff systems or systems with fast dynamics, where simple explicit methods may introduce significant numerical errors. The use of energy-coherent discretization not only ensures numerical accuracy but also aligns with the physical behavior of the multibody system, preventing issues such as energy drift over long simulations.
Once the system dynamics have been discretized, the resulting finite-dimensional problem is formulated as a nonlinear programming (NLP) problem. The objective is to minimize the discrete cost function: \[J = \sum_{k=0}^{N-1} L(\mathbf{q}_k, \mathbf{v}_k, \mathbf{u}_k, t_k) \Delta t, \tag{25}\] subject to the discretized dynamics, boundary conditions, path constraints, and bounds on the state and control variables. The cost function \(L\). \(L\) typically includes terms representing tracking errors, energy consumption, or other performance metrics relevant to the application.
To solve the NLP efficiently, advanced optimization algorithms such as Sequential Quadratic Programming (SQP) and Interior Point (IP) methods are employed. These methods are well-suited for handling the large-scale, nonlinear, and constrained nature of the problem. SQP methods solve a sequence of quadratic programming subproblems, approximating the cost function and constraints using second-order derivatives. This approach ensures rapid convergence for problems with smooth and well-behaved functions [12]. On the other hand, Interior Point methods transform the constrained optimization problem into a series of unconstrained problems using barrier functions, allowing efficient handling of inequality constraints.
The optimization process begins with an initial guess for the state and control trajectories. These initial guesses are generated using forward simulation of the system dynamics, where the controls are applied iteratively to predict the resulting state evolution. The quality of the initial guess significantly influences the convergence rate and accuracy of the optimization process. To improve the solution quality, grid refinement techniques are employed. This involves progressively refining the time discretization by increasing the number of time steps or using adaptive time-stepping methods to focus computational effort on regions with complex dynamics. Such refinement ensures that the solution accurately captures the system behavior, particularly in scenarios with rapid changes or nonlinearities [13].
The integration of discretization and optimization techniques in the numerical implementation ensures a robust and efficient solution process. By leveraging energy-coherent discretization methods and state-of-the-art NLP solvers, the numerical framework provides physically consistent and computationally efficient solutions to the optimal control problem. These methods are critical for applications requiring precise control and stability, such as robotics, biomechanics, and aerospace systems.
The proposed methods for solving optimal control problems have been applied to various multibody systems, demonstrating their effectiveness in trajectory optimization and control under realistic constraints. This section highlights two specific applications: a two-degree-of-freedom robotic manipulator and a three-degree-of-freedom robotic arm. These examples showcase the versatility and robustness of the approach in handling systems of varying complexity.
A robotic manipulator with two degrees of freedom was analyzed to assess the performance of the proposed numerical framework in trajectory optimization. The primary objective of this study was to minimize the control effort required to move the manipulator between predefined boundary conditions while adhering to path constraints imposed by the workspace geometry and system dynamics. The optimization problem involved specifying initial and final positions, velocities, and control inputs, along with ensuring smooth motion trajectories that avoid collisions and respect actuator limits.
The discretized system dynamics and energy-coherent finite-difference methods allowed for accurate representation of the manipulator’s motion. The resulting nonlinear programming (NLP) problem was solved using Sequential Quadratic Programming (SQP), yielding solutions with second-order accuracy in both state and control trajectories. The numerical results showed excellent agreement with analytical solutions obtained for simple cases, validating the correctness and efficiency of the approach. Additionally, the method demonstrated robustness in handling constraints, such as avoiding obstacles or staying within joint limits, without sacrificing computational performance. This application highlights the capability of the proposed method to optimize trajectories efficiently while maintaining physical realism.
To further test the scalability and applicability of the method, a three-degree-of-freedom robotic arm was analyzed. This system represents a more complex multibody configuration, introducing additional challenges in terms of computational requirements and adherence to energy and motion constraints. The optimization objectives included minimizing control effort, ensuring smooth trajectories, and complying with physical constraints such as torque limits and joint bounds. Furthermore, the motion paths were designed to align with task-specific requirements, such as following a desired trajectory or reaching a target configuration within a specified time frame.
The energy-preserving discretization schemes proved instrumental in maintaining numerical stability, particularly for this more complex system. The NLP formulation for the three-degree-of-freedom manipulator was solved using a combination of Interior Point (IP) methods and grid refinement techniques. The optimized trajectories and control inputs were consistent with the system’s physical constraints, ensuring that the robotic arm’s motion remained feasible and efficient. The results also highlighted the method’s ability to handle nonlinearities inherent in multibody systems, such as coupling effects between joints and dynamic interactions caused by rapid accelerations or decelerations.
This application demonstrated that the proposed framework is not limited to simple systems but can be effectively extended to more sophisticated robotic configurations. By leveraging advanced optimization techniques and energy-consistent discretization, the method provided accurate and reliable solutions that aligned well with practical requirements. These results validate the applicability of the approach to real-world scenarios, such as industrial robotics, where complex manipulators must operate within stringent performance and safety constraints.
The two applications illustrate the versatility of the proposed method in addressing diverse robotic systems. For simpler systems like the two-degree-of-freedom manipulator, the approach delivered fast and accurate solutions, showing clear advantages in computational efficiency. For more complex systems, such as the three-degree-of-freedom robotic arm, the method effectively managed increased computational demands while maintaining solution accuracy and stability. These applications underscore the method’s potential in various fields, including automation, manufacturing, and aerospace, where optimal control of multibody systems is critical.
This paper presents a modernized framework for solving optimal control problems in multibody systems. By integrating energy-preserving schemes with direct transcription, the method achieves nonlinear stability and numerical robustness, making it suitable for applications in robotics, aerospace, and biomechanics. Future work will explore adaptive mesh refinement and real-time implementation for large-scale systems.
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