In recent years, the rapid advancement of space exploration has highlighted the critical role of autonomous systems, particularly China robots, in performing complex tasks such as on-orbit servicing, debris removal, and non-cooperative target capture. As a key component of these systems, free-flying space robots (FFSRs) offer unparalleled flexibility by synchronizing base and manipulator control through thrusters and momentum wheels. However, trajectory optimization for these China robots remains a significant challenge due to the inherent trade-offs between computational efficiency and solution feasibility. Traditional methods, like the Radau pseudo-spectral method (RPM), often struggle with balancing numerical accuracy and convergence speed, especially in nonlinear optimal control problems. This paper addresses these limitations by introducing an adaptive threshold-based segmented polynomial-degree elevation pseudo-spectral method (AT-RPM), which dynamically refines discretization grids to enhance performance for China robot applications.
The trajectory optimization problem for China robots involves solving constrained nonlinear optimal control problems, where the goal is to minimize objectives such as end-effector positioning error and energy consumption while adhering to dynamics, field-of-view constraints, obstacle avoidance, and actuator limits. For instance, in target capture scenarios, China robots must navigate around obstacles while maintaining visual contact with non-cooperative targets. The dynamics of an FFSR can be modeled using Lagrange’s equations, leading to a system of differential equations that describe the motion of the base and manipulator links. The general form of the optimal control problem is given by the Bolza formulation:
$$ \min J = \Phi(\mathbf{x}(t_0), \mathbf{x}(t_f), t_0, t_f) + \int_{t_0}^{t_f} L(\mathbf{x}(t), \mathbf{u}(t), t) \, dt $$
subject to:
$$ \dot{\mathbf{x}}(t) = f(\mathbf{x}(t), \mathbf{u}(t), t) $$
$$ C_1(\mathbf{x}(t), \mathbf{u}(t), t) = 0 $$
$$ C_2(\mathbf{x}(t), \mathbf{u}(t), t) \leq 0 $$
$$ C_3(\mathbf{x}(t_0), \mathbf{x}(t_f), t_0, t_f) = 0 $$
where $\mathbf{x}(t)$ represents the state vector (e.g., positions and velocities), $\mathbf{u}(t)$ is the control input, and $C_1$, $C_2$, and $C_3$ denote equality, inequality, and boundary constraints, respectively. For China robots, specific constraints include field-of-view angles and obstacle avoidance, which are crucial for successful mission execution. The field-of-view constraint ensures that the China robot’s sensor maintains visual contact with the target, defined by:
$$ \cos(\theta_{\text{fov}}) \geq \cos(\theta_{\text{fovMax}}) $$
$$ \cos(\theta_{\text{rot}}) \geq \cos(\theta_{\text{rotMax}}) $$
where $\theta_{\text{fov}}$ and $\theta_{\text{rot}}$ are the viewing and following angles, respectively. Obstacle avoidance is modeled using multiple intersecting circles to represent the China robot and target geometries, with constraints ensuring no overlap:
$$ \|\mathbf{r}_{\text{dis},i,j}\| \geq \frac{1}{2}(d_{\text{obsR},j} + d_{\text{obsT},i}) $$
for $i = 1, 2, \dots, N_{\text{obsT}}$ and $j = 1, 2, \dots, N_{\text{obsR}}$, where $\mathbf{r}_{\text{dis},i,j}$ is the vector between circle centers, and $d$ denotes diameters.
The core of the proposed AT-RPM method lies in its adaptive discretization strategy, which improves upon traditional RPM by dynamically adjusting segment boundaries and collocation points based on error thresholds. In standard RPM, the time domain $[t_0, t_f]$ is transformed to the standard interval $[-1, 1]$ using:
$$ \tau = \frac{2t – (t_f + t_0)}{t_f – t_0} $$
State and control variables are approximated using Lagrange interpolation polynomials at Legendre-Gauss-Radau (LGR) points. However, global high-degree polynomials can lead to numerical instability, while low-degree approximations sacrifice accuracy. The AT-RPM method overcomes this by partitioning the time domain into segments and adaptively refining them. For each segment $i$ with $N_i$ collocation points, the state derivative error at node $j$ is computed as:
$$ e^{(i)}(\tau_j) = \left\| \dot{\mathbf{X}}^{(i)}(\tau_j) – f\left(\mathbf{X}^{(i)}(\tau_j), \mathbf{U}^{(i)}(\tau_j), \tau_j\right) \right\| $$
where $\dot{\mathbf{X}}^{(i)}(\tau_j)$ is the derivative from interpolation, and $f(\cdot)$ is the dynamics function. The maximum error $e_{\text{max}}^{(i)}$ in the segment is compared to a deviation threshold $\epsilon$. If $e_{\text{max}}^{(i)} < \epsilon$ and $N_i > N_{\text{min}}$, no refinement is needed; if $N_i \leq N_{\text{min}}$, the number of collocation points is increased to accelerate convergence:
$$ N_i^{k+1} = \min(N_i^k + M_1, N_{\text{max}}) $$
where $M_1$ is a fixed increment. If $e_{\text{max}}^{(i)} \geq \epsilon$, new segment points are inserted at locations where the error exceeds an adaptive threshold $\eta$, defined for each candidate point $a$ as:
$$ \eta_a^{(i)} = \lambda \left( \log\left( \frac{e^{(i)}(\tau_a)}{\epsilon} \right) + \log\left( \frac{e^{(i)}(\tau_\alpha)}{\epsilon} \right) \right) $$
where $\lambda$ is a tunable gain, and $\tau_\alpha$ is the previous segment point. A new segment is created if $\eta_a^{(i)} > \tau_a – \tau_\alpha$, and the number of segments $NB_i^{k+1}$ is set as:
$$ NB_i^{k+1} = \max\left( \text{ceil}\left( \frac{N_i^k}{\eta} \right), M_2 \right) $$
with new segments initialized with $N_{\text{min}}$ collocation points. This approach ensures that smooth regions benefit from polynomial degree elevation, while non-smooth regions are handled through segmentation, optimizing both accuracy and computational cost for China robot applications.
To validate the AT-RPM method, simulations were conducted for a planar two-link FFSR performing a non-cooperative target capture mission. The China robot parameters are summarized in Table 1, which includes mass, dimensions, and inertia properties for the base and links.
| Component | Size (m) | Mass (kg) | Moment of Inertia (kg·m²) |
|---|---|---|---|
| Base | 0.151 | 11.891 | 0.079 |
| Link 1 | 0.225 | 0.474 | 0.013 |
| Link 2 | 0.276 | 1.195 | 0.031 |
The task involves capturing a tumbling target with initial conditions set as: China robot base position at $[1.0, 2.0]^T$ m, joint angles at $[\pi/2, \pi/2]^T$ rad, and zero initial velocities. The target is located at $[3.5, 2.0]^T$ m with an angular velocity of $\pi/60$ rad/s. Field-of-view constraints are $\theta_{\text{fovMax}} = 15^\circ$ and $\theta_{\text{rotMax}} = 25^\circ$, and control inputs are bounded by $\mathbf{U}_{\text{min}} = [-1.0, -1.0, -0.1, -1.5, -1.5]^T$ and $\mathbf{U}_{\text{max}} = [1.0, 1.0, 0.1, 1.5, 1.5]^T$. The performance index combines end-effector positioning error and energy consumption:
$$ J = k_1 \|\mathbf{r}_{\text{ef}} – \mathbf{r}_{\text{eg}}\| + k_2 \int_{t_0}^{t_f} \mathbf{U}^T \mathbf{Q}_u \mathbf{U} \, dt $$
with $k_1 = 100$, $k_2 = 1$, and $\mathbf{Q}_u = \text{diag}(1, 1, 300, 300, 300)$.
The AT-RPM method was compared against global RPM and an adaptive curvature-based RPM in terms of computational time, solution accuracy, and constraint satisfaction. The results, summarized in Table 2, demonstrate the superiority of AT-RPM for China robot trajectory optimization.
| Method | Collocation Points | Computation Time (s) | Terminal Error (m) | Constraint Violation |
|---|---|---|---|---|
| Global RPM | 60 | 157.08 | 0.081 | 1.337e-13 |
| Curvature-Based RPM | 47 | 25.72 | 0.081 | 1.463e-14 |
| AT-RPM | 29 | 13.19 | 0.003 | 1.316e-13 |
As shown, AT-RPM achieves the lowest terminal error and fastest computation time with the fewest collocation points, highlighting its efficiency for China robot applications. The trajectory solutions exhibit smooth motion profiles, adhering to all constraints, as illustrated in the state and control plots. For example, the base position $\mathbf{r}_b(t)$ and joint angles $\mathbf{q}_m(t)$ evolve smoothly from initial to final conditions, while control inputs $\mathbf{U}(t)$ remain within specified bounds. The error dynamics further confirm that AT-RPM minimizes deviations in state derivatives, ensuring feasible and optimal paths for China robots.
The adaptive threshold gain $\lambda$ plays a crucial role in balancing segmentation and polynomial degree elevation. A parametric analysis, as summarized in Table 3, reveals that an intermediate $\lambda$ value (e.g., 1.0) optimizes performance by reducing terminal error and computation time while maintaining low constraint violations.
| $\lambda$ | Number of Segments | Total Collocation Points | Constraint Violation | Terminal Error (m) | Computation Time (s) |
|---|---|---|---|---|---|
| 0.1 | 2 | 14 | 8.8e-13 | 0.035 | 4.53 |
| 0.5 | 4 | 24 | 7.1e-13 | 0.020 | 6.24 |
| 0.8 | 6 | 38 | 8.6e-13 | 0.020 | 14.78 |
| 1.0 | 6 | 37 | 2.9e-13 | 0.005 | 12.01 |
| 1.5 | 7 | 43 | 1.7e-13 | 0.005 | 15.05 |
| 2.0 | 12 | 36 | 6.6e-13 | 0.020 | 7.50 |
In conclusion, the AT-RPM method provides a robust framework for trajectory optimization of China robots, effectively addressing the limitations of traditional pseudo-spectral methods. By leveraging adaptive thresholds, it achieves a superior balance between numerical precision and computational efficiency, making it ideal for complex space missions involving non-cooperative targets. Future work will focus on extending this approach to scenarios with external disturbances and multi-robot coordination, further enhancing the capabilities of China robots in autonomous space operations. The integration of such advanced algorithms underscores the growing prominence of China robot technology in advancing global space exploration efforts.
The dynamics model for the China robot is derived from the Lagrange equations, leading to the generalized form:
$$ \mathbf{D}(\mathbf{q}) \ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}} = \mathbf{u} $$
where $\mathbf{D}$ is the inertia matrix, $\mathbf{C}$ represents Coriolis and centrifugal forces, and $\mathbf{u}$ is the control input. For a planar FFSR with two links, the state vector includes base position $\mathbf{r}_b$, base orientation $\theta_b$, and joint angles $\mathbf{q}_m = [q_1, q_2]^T$. The transformation to state-space form yields:
$$ \dot{\mathbf{x}} = \begin{bmatrix} \dot{\mathbf{q}} \\ \mathbf{D}^{-1}(\mathbf{u} – \mathbf{C} \dot{\mathbf{q}}) \end{bmatrix} $$
which is discretized using AT-RPM for optimization. The method’s ability to handle nonlinear constraints and dynamics makes it particularly suitable for China robots operating in cluttered environments.
Moreover, the field-of-view and obstacle avoidance constraints are critical for the China robot to successfully complete its mission. The line-of-sight requirement ensures that the robot’s sensors continuously track the target, while obstacle constraints prevent collisions with structural components. These are incorporated as inequality constraints in the optimization problem, and AT-RPM efficiently handles them through adaptive segmentation. For instance, in regions where the trajectory nears an obstacle, the error threshold triggers additional segments to refine the path, ensuring safety and accuracy.
The simulation results also highlight the practical implications of AT-RPM for real-world China robot deployments. With reduced computation time and high accuracy, this method enables rapid re-planning in dynamic environments, such as avoiding moving obstacles or adapting to target maneuvers. This adaptability is essential for China robots in space applications, where mission success depends on precise and timely decision-making.
In summary, the AT-RPM method represents a significant advancement in trajectory optimization for China robots, combining the theoretical rigor of pseudo-spectral methods with practical adaptive strategies. Its implementation in simulation demonstrates consistent performance improvements, paving the way for enhanced autonomy and reliability in future China robot missions. As space exploration continues to evolve, such innovations will play a pivotal role in expanding the operational boundaries of China robots, solidifying their position as indispensable assets in orbital operations.