Trajectory Optimization Strategy

• Trajectory Optimization Strategy is a set of mathematical and algorithmic methods that compute optimal state and control trajectories under dynamic and operational constraints.
• Key approaches include Differential Dynamic Programming for real-time feedback control and Gauss Pseudospectral Optimal Control for efficient convergence in long-horizon problems.
• These strategies balance control effort, trajectory smoothness, and constraint adherence, making them essential in robotics, aerospace, and engineered systems.

A trajectory optimization strategy refers to a mathematical and algorithmic methodology for computing system state and control trajectories that optimally achieve a task subject to constraints. This concept is fundamental in robotics, aerospace, and complex engineered systems, where the required system performance is specified not only by the final state but also by metrics such as completion time, control effort, and adherence to physical and operational constraints. The principal goal is to determine, for given dynamics, constraints, and cost functionals, the control and state evolution that minimize (or maximize) the prescribed performance index.

1. Core Approaches in Trajectory Optimization

Two foundational approaches underpin trajectory optimization: BeLLMan’s Dynamic Programming—manifested as Differential Dynamic Programming (DDP)—and Pontryagin’s Maximum Principle—implemented in direct methods such as Gauss Pseudospectral Optimal Control (GPOC) (Gandhi, 2015).

• Differential Dynamic Programming (DDP):
  • Iterative “shooting” method that leverages BeLLMan’s principle of optimality.
  • Employs a forward-backward pass: a forward pass rolls out the current nominal trajectory, and the backward pass computes a local quadratic expansion of the cost-to-go about the nominal, enabling updates to feedforward and feedback gains.
  • Control updates take the form

  $$\delta\mathbf{u}(t_k) = \mathbf{l}(t_k) + \mathbf{L}(t_k)\delta\mathbf{x}(t_k),$$

  where $\mathbf{l}$ and $\mathbf{L}$ are derived from the quadratic expansion and Riccati-like recursion. - Particularly effective for short-horizon problems and scenarios requiring feedback to counter perturbations.

• Pontryagin’s Maximum Principle / Gauss Pseudospectral Optimal Control (GPOC):

  • Transcribes an infinite-dimensional, continuous-time optimal control problem into a finite-dimensional nonlinear program (NLP).
  • Approximates state, control, and costate variables using global polynomial interpolants (Lagrange polynomials) at Legendre-Gauss collocation points.
  • Enforces system dynamics and transversality conditions at collocation nodes via algebraic equations.
  • Favors rapid convergence for long-horizon and constraint-dense problems, but typically yields purely feedforward solutions.

2. Mathematical Formulation and Algorithmic Techniques

Trajectory optimization problems are formalized as minimizing an integral cost subject to system dynamics and path/terminal constraints:

$$\min_u J = \Phi(\mathbf{x}(t_f)) + \int_{t_0}^{t_f} L(\mathbf{x}, \mathbf{u}, t)dt$$

subject to

$$\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, \mathbf{u}, t), \quad \mathbf{x}(t_0) = \mathbf{x}_0, \quad \mathbf{x}(t_f) \in \mathcal{X}_f, \quad (\mathbf{x}, \mathbf{u}) \in \mathcal{C}.$$

• DDP Implementation:
  • Linearizes dynamics and quadratizes cost at each iteration about the nominal trajectory.
  • Solves for optimal control increments by minimizing the second-order expansion over control input increments, yielding both open-loop and feedback terms.
  • Backward pass: efficiently performed via discrete Riccati recursion, updating the value function’s derivatives at each time step.
  • Forward pass: updates the nominal trajectory using the improved control inputs.
• GPOC Implementation:
  • Maps the time domain to standardized $[-1, 1]$ for numerical stability:

  $$\tau = \frac{2t}{t_f-t_0} - \frac{t_f + t_0}{t_f-t_0}.$$ - Interpolates states using Lagrange polynomials at $K$ collocation nodes:

  $$x(\tau) \approx \sum_{i=1}^{K} \mathcal{L}_i(\tau) x(\tau_i),$$

  where $\mathcal{L}_i$ are Lagrange basis functions. - Enforces dynamics at these nodes using a differentiation matrix $D_{ki}$. - Approximates integral costs using Gauss quadrature:

  $$\int_{-1}^{1} f(\tau)d\tau \approx \sum_{i=1}^K w_i f(\tau_i), \quad w_i = \frac{2}{(1-\tau_i^2)[\dot{P}_K(\tau_i)]^2}.$$

3. Comparative Performance and Trade-Offs

The choice between DDP and pseudospectral approaches hinges on the problem characteristics (Gandhi, 2015):

• DDP:

  • Superior for problems with short time horizons and for systems sensitive to disturbances due to local feedback gains.
  • Often achieves lower control effort for the same task when feedback is critical.
  • Simulation examples indicate that, for tasks such as the cart pole and underactuated systems, DDP’s feedback structure is advantageous for stabilization and handling state uncertainties.
• GPOC:
  • Excels in long-horizon or highly constrained problems due to global polynomial approximations enabling rapid convergence.
  • Produces purely feedforward trajectories, meeting terminal and path constraints stringently.
  • In simulation, provides lower runtime (e.g., 0.9 s vs. 7.5 s for DDP in cart pole), albeit sometimes at the expense of greater control usage, particularly when strict boundary conditions are imposed.
• Empirically, DDP-generated trajectories are smoother and consume less control for disturbance-requiring tasks, whereas GPOC achieves precise endpoint specification with efficient global optimization for tasks characterized by long planning horizons or complex constraints.

4. Numerical and System-Level Validation

Validation of these methods is performed on a range of dynamical systems, including:

• Cart Pole: DDP achieves stabilization with less control usage but longer runtime; GPOC provides faster convergence but higher control activity.
• Double Cart Pole: DDP’s feedback mechanism is particularly beneficial due to increased underactuation.
• Quadrotor: DDP leverages the full time horizon to approach the target with minimized control penalties; GPOC converges faster but at a higher control cost.

The trade-offs are scenario-dependent, and method selection should be guided by the system’s sensitivity to disturbances, required feedback capability, horizon length, and constraint landscape.

5. Limitations and Extensions

Both methodologies have inherent limitations:

• DDP: While efficient for short horizons and amenable to local feedback, it relies on local linearizations and second-order approximations, thus its convergence is sensitive to initial guess quality and it may fail in highly nonlinear, long-horizon, or constraint-laden scenarios.
• GPOC: While robust for direct transcription, its performance is sensitive to the discretization grid and may lack responsive feedback, making it vulnerable in the presence of unexpected disturbances or model inaccuracies.

Future research directions include:

• Implementing efficient DDP solvers in compiled languages (e.g., C++) for increased convergence speed.
• Extending pseudospectral optimal control to stochastic domains via polynomial chaos expansions and other uncertainty propagation techniques.
• Hybrid integration with receding horizon or model predictive control (MPC) schemes, leveraging both DDP feedback and GPOC efficiency for real-time applications.
• Adapting both methods to more complex dynamical systems featuring contact, hybrid dynamics, and discontinuities (possibly via linear complementarity system formulations).

6. Concluding Synthesis

Trajectory optimization strategies—embodied by DDP and Gauss Pseudospectral methods—provide complementary toolsets for solving optimal control problems in robotics and engineering systems (Gandhi, 2015). DDP’s forward–backward pass yields both feedforward and feedback control policies, supporting near-optimal performance and robustness to disturbances, particularly within short to moderate time horizons. Gauss pseudospectral methods transcribe the full optimal control problem into a finite-dimensional nonlinear program, achieving high precision and rapid convergence for long-horizon, highly constrained scenarios.

The decision between these strategies should be informed by the system’s need for feedback, planning horizon, and the relationship between control effort and terminal accuracy. Ongoing advancements focus on bridging these approaches for real-time, stochastic, and hybrid systems to broaden the application of trajectory optimization in complex, uncertain, and dynamic environments.