Trajectory planning is the process of turning a desired motion into a time-parameterized sequence of positions, velocities and accelerations that a robot can actually execute. In robotic manipulation, it takes a geometric path for an arm, for example from a collision-free motion planner or a list of waypoints, and decides how fast to move along it at every instant while respecting joint limits, torque limits and task requirements.
The distinction between a path and a trajectory is central. A path describes where the robot goes, as a curve in joint or Cartesian space without timing. A trajectory adds time, so it defines when each point is reached. Two robots can follow the same path with very different trajectories, one smooth and fast, the other jerky and slow.
How trajectory planning works
A typical manipulation pipeline runs in stages. A motion planner finds a collision-free geometric path, often as a set of joint-space waypoints. Inverse kinematics converts Cartesian targets into joint values where needed. The trajectory planner then performs time parameterization, assigning timestamps and smooth velocity and acceleration profiles to the path. Finally, the controller tracks the trajectory at a fixed rate, commonly hundreds of hertz to a few kilohertz on industrial arms.
The planner works under kinematic and dynamic constraints. Kinematic constraints include joint position, velocity, acceleration and sometimes jerk limits. Dynamic constraints include motor torque and payload effects. The planner seeks the fastest or smoothest motion that stays within all of them.
| Constraint | Typical source | Effect if ignored |
|---|---|---|
| Joint position limits | Mechanical end stops | Fault or collision with stops |
| Velocity limits | Motor and gearbox ratings | Controller saturation |
| Acceleration limits | Torque and payload | Tracking error, overshoot |
| Jerk limits | Vibration and wear concerns | Oscillation, spilled payloads |
| Cartesian speed limits | Safety standards for collaboration | Unsafe contact forces |
Common trajectory profiles and methods
Trajectory planners use a small set of well-understood mathematical profiles. Trapezoidal velocity profiles accelerate at a constant rate, cruise and decelerate; they are simple and fast but have discontinuous acceleration. S-curve profiles limit jerk for smoother motion. Polynomial and spline interpolation, such as cubic or quintic polynomials and B-splines, pass smoothly through multiple waypoints.
| Profile | Smoothness | Computation | Typical use |
|---|---|---|---|
| Trapezoidal velocity | Continuous velocity only | Very low | Point-to-point moves |
| S-curve, jerk-limited | Continuous acceleration | Low | Fast moves with fragile payloads |
| Cubic spline | Continuous acceleration | Low | Smooth multi-waypoint paths |
| Quintic polynomial | Continuous jerk at ends | Low | Precise start and stop states |
| Time-optimal parameterization | Fastest within limits | Medium | Cycle-time-critical cells |
| Trajectory optimization | Optimizes a cost function | High | Complex, constrained motions |
Robot frameworks such as MoveIt provide time parameterization algorithms that add timing to planned paths, including time-optimal methods that push each joint to its limits along the given path. Industrial controllers expose similar behavior through motion types such as joint moves and linear moves.
Joint space and Cartesian space trajectories
A trajectory can be planned in joint space, interpolating each joint angle directly, or in Cartesian space, interpolating the pose of the tool. Joint-space trajectories are efficient and avoid singularity problems, but the tool follows a curved line. Cartesian trajectories keep the tool on a straight line or defined arc, which is required for welding, gluing or inserting parts, but they can pass near kinematic singularities where joint speeds become very high.
| Aspect | Joint space | Cartesian space |
|---|---|---|
| Tool path shape | Curved, not specified | Straight line or defined curve |
| Singularity risk | Low | Significant near singular poses |
| Computation | Simple interpolation | Inverse kinematics at each step |
| Typical use | Free-space transfers | Process paths and insertions |
Trajectory planning compared with motion planning
In manipulation, motion planning and trajectory planning are often mentioned together and sometimes used as synonyms, but they solve different problems. Motion planning answers which collision-free path connects the start and goal configurations, typically with sampling-based planners. Trajectory planning answers how to traverse that path in time. Collision checking belongs mainly to motion planning, while dynamics and timing belong to trajectory planning. Some optimization-based methods solve both together.
- Inverse kinematics supplies joint targets that the trajectory connects.
- Grasp planning defines the approach and retreat poses at the start or end of a trajectory.
- Pick and place cycles are dominated by trajectory timing in free space.
- Degrees of freedom determine the dimension of the joint-space trajectory.
Pitfalls in trajectory planning
The most common problem is ignoring jerk, which causes vibration, overshoot and wear even when velocity and acceleration are within limits. Another is planning with nominal limits while carrying a heavy payload, which can saturate motors and produce large tracking errors. Cartesian moves that pass near singularities may trigger joint speed faults that stop the robot mid-cycle.
Finally, a trajectory is only valid if it is checked again after timing is added: smoothing a path between waypoints can make the arm cut corners into obstacles that the original path avoided. Re-running collision checking on the final trajectory prevents this.
Collaborative applications add a further layer. When people share the workspace, speed and separation monitoring or power and force limiting may scale the trajectory down in real time, so the planned cycle time is no longer guaranteed. Designers should therefore plan with realistic reduced speeds for collaborative zones and verify the final timing on the actual cell rather than in simulation alone, since friction, payload and controller filtering all change the executed motion.
In everyday language
Planning a route on a map tells you which roads to take. It does not say how fast to drive on each road. Trajectory planning adds that timing for a robot arm. It decides how quickly to speed up, cruise and slow down at every moment.
Good timing keeps the motion fast but smooth. A robot that jerks wastes energy, shakes its load and wears out its joints. A well-planned trajectory moves like a skilled person carrying a full cup of coffee.
Seen in the real world
A packaging line uses a six-axis robot arm to move filled glass jars from a conveyor into cartons. With a trapezoidal velocity profile, the arm reached its cycle time target, but jars occasionally tipped or spilled at the start and end of each move.
Engineers switched to a jerk-limited S-curve profile, lowered the acceleration limit for the loaded leg of the cycle and kept higher limits for the empty return. Free-space transfers were planned in joint space, while the final vertical insertion into the carton used a short Cartesian straight-line move. The overall cycle time stayed almost the same, and spillage stopped, because acceleration now changed gradually instead of in sudden steps.