Trajectory Generation

The core of trajectory generation is converting a geometric path into a trajectory by defining how position changes with time. This involves assigning velocity and acceleration profiles at every point. Common profiles include:

• Trapezoidal velocity: A profile with constant acceleration, constant velocity, and constant deceleration phases.
• S-curve (Jerk-limited): A smoother profile that limits the rate of change of acceleration (jerk), reducing vibrations and mechanical stress on the robot. Time-parameterization ensures the motion is physically feasible given the robot's actuator limits.

A valid trajectory must respect the robot's dynamic constraints. This includes:

• Joint torque/force limits: The commanded accelerations must not require more torque than the motors can provide.
• Velocity limits: Maximum rotational or linear speeds of the joints and end-effector.
• Kinematic limits: Joint position boundaries to avoid self-collision or hitting mechanical stops. Trajectory optimization algorithms explicitly incorporate these constraints to generate motions the hardware can execute without damage or failure.

High-quality trajectories are smooth, meaning they have continuous derivatives. Discontinuities in velocity (infinite acceleration) or acceleration (infinite jerk) cause shocks, vibrations, and tracking errors.

• Cⁿ Continuity: A trajectory is Cⁿ continuous if its nth derivative is continuous. For robotics, C² continuity (continuous acceleration) is often a minimum for smooth motion.
• Polynomial Trajectories: Commonly used (e.g., quintic polynomials) because they can be easily constrained for smoothness at start and end points. Smoothness minimizes wear on mechanical components and improves control accuracy.

The generated trajectory must be collision-free. This is typically achieved by integrating with a motion planner.

• Configuration Space (C-space): The planner finds a path in the space of all joint angles. The trajectory generator then time-parameterizes this path.
• Online Re-planning: In dynamic environments, trajectory generation must be reactive, using real-time sensor data to modify the planned trajectory for collision avoidance. Techniques like velocity obstacles or model predictive control (MPC) are used for online, safe trajectory adjustment.

Trajectories are often optimized for specific performance criteria:

• Minimum Time: Get to the goal as fast as possible within constraints.
• Minimum Jerk: Prioritize smoothness to reduce vibration (common in tasks like welding or painting).
• Minimum Energy: Reduce total energy consumption, important for mobile or battery-operated robots.
• Task-Specific Goals: Such as maintaining a tool's orientation or applying a specific force profile. These objectives are formalized into a cost function that an optimization solver minimizes.

The generated trajectory serves as the reference signal for the robot's low-level feedback controller (e.g., a PID or computed-torque controller).

• Feedforward Terms: An accurate dynamic model can compute the expected torques for the trajectory, providing a feedforward signal that greatly improves tracking performance.
• Receding Horizon: In Model Predictive Control (MPC), trajectory generation and control are fused. The controller continuously solves for a short-horizon optimal trajectory and executes the first step, then re-plans. This tight integration ensures the robot accurately follows the planned motion in the presence of disturbances.

Path planning is the algorithmic determination of a collision-free geometric route for a robot's end-effector or body from a start configuration to a goal configuration within an environment containing obstacles. It focuses on the spatial sequence of points, not the timing.

• Geometric vs. Temporal: A path is a pure sequence of states; a trajectory adds timing, velocity, and acceleration.
• Algorithms: Common methods include Probabilistic Roadmaps (PRM) and Rapidly-exploring Random Trees (RRT).
• Relationship to Trajectory Generation: Path planning provides the geometric input; trajectory generation converts this into a time-parameterized motion plan suitable for a controller.

Motion planning is the broader computational problem of finding a sequence of valid configurations and control inputs that move a robot from a start to a goal while satisfying constraints like collision avoidance, joint limits, and dynamic feasibility. It often encompasses both path planning and trajectory generation.

• Holistic Process: Integrates geometric planning, kinematic/dynamic constraints, and sometimes task-level logic.
• Constraints: Must account for robot kinematics, dynamics, obstacles, and task-specific requirements.
• Output: The result is a full motion specification, which is typically a trajectory ready for execution by a low-level controller.

Trajectory optimization is the mathematical process of refining a candidate trajectory to minimize a cost function (e.g., energy, time, jerk) while satisfying a set of constraints (e.g., dynamics, torque limits, obstacle avoidance). It transforms a feasible path into an optimal motion plan.

• Optimal Control: Framed as solving an optimal control problem over a finite time horizon.
• Cost Functions: Common objectives include minimizing execution time, mechanical effort, or jerk (rate of change of acceleration) for smooth motion.
• Methods: Techniques range from direct collocation and shooting methods to modern gradient-based optimization using automatic differentiation.

Model Predictive Control (MPC) is an advanced, real-time control strategy that uses an internal dynamic model to predict the robot's future states over a finite horizon. It solves a trajectory optimization problem at each control step and applies only the first control input, then re-plans, making it highly reactive to disturbances.

• Receding Horizon: Continuously re-optimizes based on new sensor feedback.
• For Manipulation: MPC for manipulation explicitly handles contact forces, friction cones, and task constraints, making it ideal for dexterous or contact-rich tasks.
• Key Advantage: Provides a unified framework for control and short-horizon trajectory generation that respects system dynamics and constraints.

Inverse kinematics (IK) is the computational process of determining the joint parameters (angles or displacements) of a robotic manipulator required to achieve a desired position and orientation of its end-effector in Cartesian space. It is a fundamental step in converting a geometric path into joint-space commands.

• Core Function: Solves for joint angles q given a desired end-effector pose T.
• Challenge: Often has multiple or no solutions; requires numerical or analytical methods.
• Trajectory Context: A Cartesian-space trajectory must be converted into a joint-space trajectory via IK at each time step to be executable by the robot's actuators.

Task and Motion Planning (TAMP) is an integrated approach that combines high-level symbolic task planning (deciding what actions to perform) with low-level geometric motion planning (deciding how to move) to solve complex, long-horizon manipulation problems.

• Hierarchical Integration: The task planner sequences symbolic actions (e.g., pick(A), place(A, table)), and the motion planner finds feasible trajectories for each.
• Interaction: The feasibility of a task plan depends on the existence of collision-free trajectories, creating a tight feedback loop.
• Role of Trajectory Generation: Provides the executable motion plans that ground the abstract task plan in physical reality.