Trajectory Generation

Unlike a simple geometric path, a trajectory explicitly defines the robot's state as a function of time. This includes specifying position, velocity, acceleration, and sometimes jerk (the rate of change of acceleration) at every point along the path. This is essential for dynamic feasibility, ensuring the robot's motors can physically achieve the commanded motions without exceeding torque or velocity limits.

• Key Output: A function (q(t)), where (q) is the robot's configuration (e.g., joint angles) and (t) is time.
• Purpose: Enables precise timing for tasks like synchronization on an assembly line or catching a moving object.

A valid trajectory must respect the kinematic and dynamic constraints of the robot and its environment. This involves complex calculations to ensure the planned motion is physically executable.

• Kinematic Constraints: Limits on joint positions, velocities, and accelerations.
• Dynamic Constraints: Limits on joint torques and forces, derived from the robot's mass, inertia, and actuator capabilities.
• Environmental Constraints: Includes collision avoidance with static and dynamic obstacles, adherence to keep-out zones, and surface contact constraints for manipulation.

Failure to satisfy these constraints results in trajectories that cannot be executed or cause unsafe collisions.

Trajectory generation is typically formulated as an optimization problem. The goal is to find the 'best' trajectory according to a defined cost function, balancing competing objectives.

Common optimization criteria include:

• Minimize Time: Execute the task as quickly as possible.
• Minimize Energy/Jerk: Reduce power consumption or mechanical stress for smoother, more wear-resistant motion (e.g., minimize ∫ jerk² dt).
• Maximize Clearance: Stay as far from obstacles as possible for safety.

Algorithms like Trajectory Optimization and Model Predictive Control (MPC) solve these problems online or offline.

Trajectory generation operates at the lowest level of the Task and Motion Planning (TAMP) hierarchy. It receives a high-level plan (a sequence of symbolic actions) and a geometric goal from a motion planner and converts it into a dynamically executable command stream.

Typical Flow:

1. Task Planner: Decides what to do (e.g., "Pick up block A").
2. Motion/Path Planner: Decides where to go geometrically, outputting a collision-free path.
3. Trajectory Generator: Decides how to move along that path over time, outputting motor commands.

This separation allows efficient reasoning at different levels of abstraction.

The generated trajectory serves as the reference signal for the robot's low-level feedback controller (e.g., a PID or computed-torque controller). The controller's job is to minimize the error between the robot's actual state (measured by sensors) and the desired state specified by the trajectory.

• Feedforward Terms: Optimal trajectories often include predicted control inputs (feedforward) to improve tracking performance.
• Real-time Adjustment: In architectures like Model Predictive Control (MPC), trajectory generation and control are fused into a single, frequently solved optimization loop that reacts to disturbances.

Two primary algorithmic families exist for trajectory generation, often used in sequence.

• *Sampling-Based Planners (e.g., RRT, PRM)**: Efficiently find a feasible geometric path in complex spaces. They typically do not produce time-parameterized trajectories.
• Optimization-Based Smoothers: Post-process the geometric path from a sampler to create a smooth, time-optimal, and dynamically feasible trajectory. CHOMP (Covariant Hamiltonian Optimization for Motion Planning) and STOMP (Stochastic Trajectory Optimization for Motion Planning) are classic examples.

Modern approaches like KOMO (K-Order Motion Optimization) unify path finding and trajectory optimization into a single nonlinear program.

The broader algorithmic process of finding any valid sequence of configurations for a robot to move from a start to a goal state. Trajectory generation is a specific type of motion planning that adds the critical dimension of time, producing a time-parameterized path with velocities and accelerations. While path planning finds a geometric route, motion planning encompasses dynamics, constraints, and sometimes task logic.

• Key Distinction: A path is a curve in space; a trajectory is that curve plus a speed profile.
• Example: Motion planning for a drone involves finding a 3D route (path planning) and then determining how fast to fly each segment to conserve battery (trajectory generation).

The process of refining a candidate trajectory to minimize a cost function (e.g., energy, time, jerk) while satisfying dynamic constraints and avoiding collisions. It is often the final, polishing stage of trajectory generation.

• Common Costs: Minimize total time, mechanical effort, or sudden changes in acceleration (jerk) for smooth motion.
• Methods: Includes non-linear programming and model predictive control (MPC), which solves a rolling-horizon optimization online.
• Application: A robotic arm polishing a car door must optimize its path to maintain constant contact force and speed, which is a trajectory optimization problem.

A short, parameterized, and reusable trajectory segment representing a fundamental robot movement. Trajectory generation for complex tasks often involves sequencing and blending these primitives from a skill library.

• Characteristics: Each primitive is a pre-computed or pre-optimized trajectory for actions like reach, grasp, turn, or move-forward.
• Advantage: Using primitives ensures dynamical feasibility and reduces online computation. Planning becomes a search over sequences of these robust building blocks.
• Example: A warehouse robot's library may contain primitives for 'drive to bin,' 'extend lift,' and 'retract with item.'

The computational process that determines the joint angles required to position a robot's end-effector at a desired location and orientation in space. It is a critical upstream step for trajectory generation in articulated robots.

• Relationship to Trajectory Generation: A trajectory for an end-effector (e.g., a welding torch) is defined in Cartesian space. IK solvers are called at each time step to convert this Cartesian path into a feasible joint-space trajectory.
• Challenges: IK problems can have multiple, infinite, or no solutions. Trajectory generation must account for this to ensure smooth, continuous joint motion.
• Tool: The Jacobian matrix is used in differential IK to map end-effector velocities to joint velocities.

An advanced online control strategy that performs constrained trajectory optimization over a receding horizon. It tightly couples trajectory generation with real-time execution and feedback.

• Mechanism: At each control cycle, MPC solves a finite-time optimization problem from the current state, executes only the first control step, then re-solves at the next cycle with new sensor data.
• Benefit: Provides inherent robustness to disturbances and model inaccuracies, as it constantly re-plans.
• Use Case: Autonomous vehicles use MPC to generate smooth, collision-free steering and acceleration trajectories while reacting to other moving cars.

A fundamental mathematical representation where every possible state of a robot is mapped to a single point. Trajectory generation algorithms primarily operate in this abstract space.

• Definition: For a robotic arm with N joints, its C-space is an N-dimensional space where each axis represents a joint angle. A point in C-space is one specific robot pose.
• Obstacle Mapping: Physical obstacles become forbidden regions (C-obstacles) in C-space. A valid trajectory is a curve through the free C-space that avoids these regions.
• Importance: Planning in C-space transforms the complex problem of moving a multi-link geometry into the simpler problem of moving a point through a mapped space.