What Is Trajectory Planning?

Trajectory planning is the process of computing a time-parameterized motion profile that moves a robot, vehicle, or other dynamical system from an initial state to a goal state while satisfying constraints on velocity, acceleration, and obstacle avoidance. It extends pure geometric path planning by assigning a speed law to the path, specifying where the system will be, when it will be there, and how fast it will be moving at each moment. The resulting trajectory serves as the reference that a tracking controller follows during execution.

The field bridges robotics, mechanical engineering, and control theory. A trajectory planner must account for the kinematic structure of the system, the physical limits of its actuators, the geometry of the environment, and, in many applications, the presence of other moving agents. Trajectory planning is distinct from trajectory optimization in emphasis: planning focuses on finding a feasible trajectory quickly, often in real time, while optimization focuses on finding the best trajectory according to a defined cost criterion, sometimes at greater computational expense.

Path and Trajectory Representation

A trajectory is typically represented as a set of waypoints connected by interpolating functions, or as a continuous mathematical curve parameterized by time. Polynomial splines, particularly cubic and quintic polynomials in joint space, provide trajectories with specified position and velocity at the endpoints and smooth intermediate motion that avoids discontinuities in acceleration. In Cartesian space, B-spline and Bezier curve representations are common because they offer intuitive geometric control over the path shape and guarantee that the trajectory lies within a convex hull of the control points. The ScienceDirect topic overview of trajectory planning in engineering surveys the mathematical representations used across robotic manipulation, CNC machining, and autonomous vehicles, noting that the choice of representation affects both planning speed and the smoothness of the resulting motion.

Planning Algorithms

Trajectory planning algorithms span sampling-based, optimization-based, and learning-based families. Sampling-based methods such as the rapidly-exploring random tree (RRT) and its probabilistically complete variants explore the configuration space by randomly sampling states and extending a tree toward them, eventually connecting the initial and goal configurations. These methods handle high-dimensional spaces and complex obstacle geometries effectively but do not directly optimize trajectory quality. Optimization-based planners formulate the planning problem as a constrained optimization and solve it using gradient-based or convex programming methods, producing trajectories that explicitly minimize cost criteria such as time, energy, or jerk. The CMU Real-Time Motion Planner paper on autonomous vehicles presents a real-time trajectory planner for autonomous vehicles that combines lattice-based state space search with trajectory optimization to generate dynamically feasible paths at planning rates compatible with vehicle speeds.

Kinematic and Dynamic Constraints

Every trajectory planner must respect the physical limits of the system it commands. Kinematic constraints arise from joint range limits, workspace boundaries, and, for wheeled vehicles, non-holonomic constraints that prevent lateral sliding. Dynamic constraints arise from actuator force and torque limits, which bound achievable acceleration and deceleration profiles. Time-optimal trajectory planning under these constraints is a classical problem: given a geometric path, the maximum-speed trajectory is found by computing the maximum allowable speed at each point subject to actuator saturation, then integrating forward and backward along the path to produce a time-parameterized profile. For robot manipulators, this problem involves both joint-space and Cartesian-space constraint satisfaction simultaneously. A PMC article on geometrical optimal navigation and path planning reviews how geometric and algorithmic approaches bridge trajectory planning theory with practical implementation across ground robots, aerial vehicles, and marine systems.

Applications

Trajectory planning has applications in a wide range of fields, including:

• Industrial robot arm programming for assembly and welding tasks
• Autonomous vehicle lane-change and intersection navigation
• Unmanned aerial vehicle mission path generation
• CNC machine tool cutting path optimization
• Surgical robot motion planning for minimally invasive procedures
• Human-robot collaboration with safe motion constraints